Result for (* 7.62939453125e-6 (+ (+ (+ (* 2.0 u2) (* (- u3 u1) (* amt 65536))) (* (- (+ (- (* 2.0 u1) (* 5.0 u2)) (* 4.0 u3)) u4) (* (pow amt 2) 65536))) (* (+ (- (- (* 3.0 u2) u1) (* 3.0 u3)) u4) (* (pow amt 3) 65536))))

Specification

\[0.5 \leq amt \land amt \leq 1\]

\[\begin{array}{l} \\ 7.62939453125 \cdot 10^{-6} \cdot \left(\left(\left(2 \cdot u2 + \left(u3 - u1\right) \cdot \left(amt \cdot 65536\right)\right) + \left(\left(\left(2 \cdot u1 - 5 \cdot u2\right) + 4 \cdot u3\right) - u4\right) \cdot \left({amt}^{2} \cdot 65536\right)\right) + \left(\left(\left(3 \cdot u2 - u1\right) - 3 \cdot u3\right) + u4\right) \cdot \left({amt}^{3} \cdot 65536\right)\right) \end{array} \]

(FPCore (u2 u3 u1 amt u4)
 :precision binary32
 (*
  7.62939453125e-6
  (+
   (+
    (+ (* 2.0 u2) (* (- u3 u1) (* amt 65536.0)))
    (*
     (- (+ (- (* 2.0 u1) (* 5.0 u2)) (* 4.0 u3)) u4)
     (* (pow amt 2.0) 65536.0)))
   (* (+ (- (- (* 3.0 u2) u1) (* 3.0 u3)) u4) (* (pow amt 3.0) 65536.0)))))

float code(float u2, float u3, float u1, float amt, float u4) {
	return 7.62939453125e-6f * ((((2.0f * u2) + ((u3 - u1) * (amt * 65536.0f))) + (((((2.0f * u1) - (5.0f * u2)) + (4.0f * u3)) - u4) * (powf(amt, 2.0f) * 65536.0f))) + (((((3.0f * u2) - u1) - (3.0f * u3)) + u4) * (powf(amt, 3.0f) * 65536.0f)));
}

real(4) function code(u2, u3, u1, amt, u4)
    real(4), intent (in) :: u2
    real(4), intent (in) :: u3
    real(4), intent (in) :: u1
    real(4), intent (in) :: amt
    real(4), intent (in) :: u4
    code = 7.62939453125e-6 * ((((2.0e0 * u2) + ((u3 - u1) * (amt * 65536.0e0))) + (((((2.0e0 * u1) - (5.0e0 * u2)) + (4.0e0 * u3)) - u4) * ((amt ** 2.0e0) * 65536.0e0))) + (((((3.0e0 * u2) - u1) - (3.0e0 * u3)) + u4) * ((amt ** 3.0e0) * 65536.0e0)))
end function

function code(u2, u3, u1, amt, u4)
	return Float32(Float32(7.62939453125e-6) * Float32(Float32(Float32(Float32(Float32(2.0) * u2) + Float32(Float32(u3 - u1) * Float32(amt * Float32(65536.0)))) + Float32(Float32(Float32(Float32(Float32(Float32(2.0) * u1) - Float32(Float32(5.0) * u2)) + Float32(Float32(4.0) * u3)) - u4) * Float32((amt ^ Float32(2.0)) * Float32(65536.0)))) + Float32(Float32(Float32(Float32(Float32(Float32(3.0) * u2) - u1) - Float32(Float32(3.0) * u3)) + u4) * Float32((amt ^ Float32(3.0)) * Float32(65536.0)))))
end

function tmp = code(u2, u3, u1, amt, u4)
	tmp = single(7.62939453125e-6) * ((((single(2.0) * u2) + ((u3 - u1) * (amt * single(65536.0)))) + (((((single(2.0) * u1) - (single(5.0) * u2)) + (single(4.0) * u3)) - u4) * ((amt ^ single(2.0)) * single(65536.0)))) + (((((single(3.0) * u2) - u1) - (single(3.0) * u3)) + u4) * ((amt ^ single(3.0)) * single(65536.0))));
end

\begin{array}{l}

\\
7.62939453125 \cdot 10^{-6} \cdot \left(\left(\left(2 \cdot u2 + \left(u3 - u1\right) \cdot \left(amt \cdot 65536\right)\right) + \left(\left(\left(2 \cdot u1 - 5 \cdot u2\right) + 4 \cdot u3\right) - u4\right) \cdot \left({amt}^{2} \cdot 65536\right)\right) + \left(\left(\left(3 \cdot u2 - u1\right) - 3 \cdot u3\right) + u4\right) \cdot \left({amt}^{3} \cdot 65536\right)\right)
\end{array}

Sampling outcomes in binary32 precision:

Initial Program: 71.7% accurate, 1.0× speedup?

(FPCore (u2 u3 u1 amt u4)
 :precision binary32
 (*
  7.62939453125e-6
  (+
   (+
    (+ (* 2.0 u2) (* (- u3 u1) (* amt 65536.0)))
    (*
     (- (+ (- (* 2.0 u1) (* 5.0 u2)) (* 4.0 u3)) u4)
     (* (pow amt 2.0) 65536.0)))
   (* (+ (- (- (* 3.0 u2) u1) (* 3.0 u3)) u4) (* (pow amt 3.0) 65536.0)))))

float code(float u2, float u3, float u1, float amt, float u4) {
	return 7.62939453125e-6f * ((((2.0f * u2) + ((u3 - u1) * (amt * 65536.0f))) + (((((2.0f * u1) - (5.0f * u2)) + (4.0f * u3)) - u4) * (powf(amt, 2.0f) * 65536.0f))) + (((((3.0f * u2) - u1) - (3.0f * u3)) + u4) * (powf(amt, 3.0f) * 65536.0f)));
}

real(4) function code(u2, u3, u1, amt, u4)
    real(4), intent (in) :: u2
    real(4), intent (in) :: u3
    real(4), intent (in) :: u1
    real(4), intent (in) :: amt
    real(4), intent (in) :: u4
    code = 7.62939453125e-6 * ((((2.0e0 * u2) + ((u3 - u1) * (amt * 65536.0e0))) + (((((2.0e0 * u1) - (5.0e0 * u2)) + (4.0e0 * u3)) - u4) * ((amt ** 2.0e0) * 65536.0e0))) + (((((3.0e0 * u2) - u1) - (3.0e0 * u3)) + u4) * ((amt ** 3.0e0) * 65536.0e0)))
end function

function code(u2, u3, u1, amt, u4)
	return Float32(Float32(7.62939453125e-6) * Float32(Float32(Float32(Float32(Float32(2.0) * u2) + Float32(Float32(u3 - u1) * Float32(amt * Float32(65536.0)))) + Float32(Float32(Float32(Float32(Float32(Float32(2.0) * u1) - Float32(Float32(5.0) * u2)) + Float32(Float32(4.0) * u3)) - u4) * Float32((amt ^ Float32(2.0)) * Float32(65536.0)))) + Float32(Float32(Float32(Float32(Float32(Float32(3.0) * u2) - u1) - Float32(Float32(3.0) * u3)) + u4) * Float32((amt ^ Float32(3.0)) * Float32(65536.0)))))
end

function tmp = code(u2, u3, u1, amt, u4)
	tmp = single(7.62939453125e-6) * ((((single(2.0) * u2) + ((u3 - u1) * (amt * single(65536.0)))) + (((((single(2.0) * u1) - (single(5.0) * u2)) + (single(4.0) * u3)) - u4) * ((amt ^ single(2.0)) * single(65536.0)))) + (((((single(3.0) * u2) - u1) - (single(3.0) * u3)) + u4) * ((amt ^ single(3.0)) * single(65536.0))));
end

\begin{array}{l}

\\
7.62939453125 \cdot 10^{-6} \cdot \left(\left(\left(2 \cdot u2 + \left(u3 - u1\right) \cdot \left(amt \cdot 65536\right)\right) + \left(\left(\left(2 \cdot u1 - 5 \cdot u2\right) + 4 \cdot u3\right) - u4\right) \cdot \left({amt}^{2} \cdot 65536\right)\right) + \left(\left(\left(3 \cdot u2 - u1\right) - 3 \cdot u3\right) + u4\right) \cdot \left({amt}^{3} \cdot 65536\right)\right)
\end{array}

Alternative 1: 95.5% accurate, 3.9× speedup?

\[\begin{array}{l} \\ \mathsf{fma}\left(amt, 0.5 \cdot \mathsf{fma}\left(amt, \mathsf{fma}\left(2, u1, \mathsf{fma}\left(amt, \mathsf{fma}\left(-3, u3, u4\right) - u1, u3 \cdot 4\right) - u4\right), u3 - u1\right), u2 \cdot \mathsf{fma}\left(0.5 \cdot \left(amt \cdot amt\right), \mathsf{fma}\left(3, amt, -5\right), 1.52587890625 \cdot 10^{-5}\right)\right) \end{array} \]

(FPCore (u2 u3 u1 amt u4)
 :precision binary32
 (fma
  amt
  (*
   0.5
   (fma
    amt
    (fma 2.0 u1 (- (fma amt (- (fma -3.0 u3 u4) u1) (* u3 4.0)) u4))
    (- u3 u1)))
  (* u2 (fma (* 0.5 (* amt amt)) (fma 3.0 amt -5.0) 1.52587890625e-5))))

float code(float u2, float u3, float u1, float amt, float u4) {
	return fmaf(amt, (0.5f * fmaf(amt, fmaf(2.0f, u1, (fmaf(amt, (fmaf(-3.0f, u3, u4) - u1), (u3 * 4.0f)) - u4)), (u3 - u1))), (u2 * fmaf((0.5f * (amt * amt)), fmaf(3.0f, amt, -5.0f), 1.52587890625e-5f)));
}

function code(u2, u3, u1, amt, u4)
	return fma(amt, Float32(Float32(0.5) * fma(amt, fma(Float32(2.0), u1, Float32(fma(amt, Float32(fma(Float32(-3.0), u3, u4) - u1), Float32(u3 * Float32(4.0))) - u4)), Float32(u3 - u1))), Float32(u2 * fma(Float32(Float32(0.5) * Float32(amt * amt)), fma(Float32(3.0), amt, Float32(-5.0)), Float32(1.52587890625e-5))))
end

\begin{array}{l}

\\
\mathsf{fma}\left(amt, 0.5 \cdot \mathsf{fma}\left(amt, \mathsf{fma}\left(2, u1, \mathsf{fma}\left(amt, \mathsf{fma}\left(-3, u3, u4\right) - u1, u3 \cdot 4\right) - u4\right), u3 - u1\right), u2 \cdot \mathsf{fma}\left(0.5 \cdot \left(amt \cdot amt\right), \mathsf{fma}\left(3, amt, -5\right), 1.52587890625 \cdot 10^{-5}\right)\right)
\end{array}

Derivation

Initial program 75.5%
\[7.62939453125 \cdot 10^{-6} \cdot \left(\left(\left(2 \cdot u2 + \left(u3 - u1\right) \cdot \left(amt \cdot 65536\right)\right) + \left(\left(\left(2 \cdot u1 - 5 \cdot u2\right) + 4 \cdot u3\right) - u4\right) \cdot \left({amt}^{2} \cdot 65536\right)\right) + \left(\left(\left(3 \cdot u2 - u1\right) - 3 \cdot u3\right) + u4\right) \cdot \left({amt}^{3} \cdot 65536\right)\right) \]
Add Preprocessing
Taylor expanded in amt around 0
\[\leadsto \color{blue}{\frac{1}{65536} \cdot u2 + amt \cdot \left(\frac{1}{2} \cdot \left(u3 - u1\right) + amt \cdot \left(\frac{1}{2} \cdot \left(amt \cdot \left(\left(u4 + 3 \cdot u2\right) - \left(u1 + 3 \cdot u3\right)\right)\right) + \frac{1}{2} \cdot \left(\left(2 \cdot u1 + 4 \cdot u3\right) - \left(u4 + 5 \cdot u2\right)\right)\right)\right)} \]
Step-by-step derivation
1. +-commutativeN/A
  \[\leadsto \color{blue}{amt \cdot \left(\frac{1}{2} \cdot \left(u3 - u1\right) + amt \cdot \left(\frac{1}{2} \cdot \left(amt \cdot \left(\left(u4 + 3 \cdot u2\right) - \left(u1 + 3 \cdot u3\right)\right)\right) + \frac{1}{2} \cdot \left(\left(2 \cdot u1 + 4 \cdot u3\right) - \left(u4 + 5 \cdot u2\right)\right)\right)\right) + \frac{1}{65536} \cdot u2} \]
2. lower-fma.f32N/A
  \[\leadsto \color{blue}{\mathsf{fma}\left(amt, \frac{1}{2} \cdot \left(u3 - u1\right) + amt \cdot \left(\frac{1}{2} \cdot \left(amt \cdot \left(\left(u4 + 3 \cdot u2\right) - \left(u1 + 3 \cdot u3\right)\right)\right) + \frac{1}{2} \cdot \left(\left(2 \cdot u1 + 4 \cdot u3\right) - \left(u4 + 5 \cdot u2\right)\right)\right), \frac{1}{65536} \cdot u2\right)} \]
Applied rewrites94.9%
\[\leadsto \color{blue}{\mathsf{fma}\left(amt, \mathsf{fma}\left(amt, 0.5 \cdot \mathsf{fma}\left(amt, \mathsf{fma}\left(3, u2 - u3, u4\right) - u1, \mathsf{fma}\left(2, u1, \mathsf{fma}\left(u3, 4, u2 \cdot -5\right)\right) - u4\right), \left(u3 - u1\right) \cdot 0.5\right), u2 \cdot 1.52587890625 \cdot 10^{-5}\right)} \]
Taylor expanded in u2 around 0
\[\leadsto amt \cdot \left(\frac{1}{2} \cdot \left(amt \cdot \left(\left(2 \cdot u1 + \left(4 \cdot u3 + amt \cdot \left(\left(u4 + -3 \cdot u3\right) - u1\right)\right)\right) - u4\right)\right) + \frac{1}{2} \cdot \left(u3 - u1\right)\right) + \color{blue}{u2 \cdot \left(\frac{1}{65536} + \frac{1}{2} \cdot \left({amt}^{2} \cdot \left(3 \cdot amt - 5\right)\right)\right)} \]
Step-by-step derivation
Applied rewrites96.2%
\[\leadsto \mathsf{fma}\left(amt, \color{blue}{0.5 \cdot \mathsf{fma}\left(amt, \mathsf{fma}\left(2, u1, \mathsf{fma}\left(amt, \mathsf{fma}\left(-3, u3, u4\right) - u1, 4 \cdot u3\right) - u4\right), u3 - u1\right)}, u2 \cdot \mathsf{fma}\left(0.5 \cdot \left(amt \cdot amt\right), \mathsf{fma}\left(3, amt, -5\right), 1.52587890625 \cdot 10^{-5}\right)\right) \]
Final simplification96.2%
\[\leadsto \mathsf{fma}\left(amt, 0.5 \cdot \mathsf{fma}\left(amt, \mathsf{fma}\left(2, u1, \mathsf{fma}\left(amt, \mathsf{fma}\left(-3, u3, u4\right) - u1, u3 \cdot 4\right) - u4\right), u3 - u1\right), u2 \cdot \mathsf{fma}\left(0.5 \cdot \left(amt \cdot amt\right), \mathsf{fma}\left(3, amt, -5\right), 1.52587890625 \cdot 10^{-5}\right)\right) \]
Add Preprocessing

Alternative 2: 95.2% accurate, 4.0× speedup?

\[\begin{array}{l} \\ \mathsf{fma}\left(amt, \mathsf{fma}\left(amt, 0.5 \cdot \mathsf{fma}\left(u4, amt + -1, \mathsf{fma}\left(-5, u2, \mathsf{fma}\left(2, u1, \mathsf{fma}\left(amt, \mathsf{fma}\left(3, u2 - u3, -u1\right), u3 \cdot 4\right)\right)\right)\right), 0.5 \cdot \left(u3 - u1\right)\right), u2 \cdot 1.52587890625 \cdot 10^{-5}\right) \end{array} \]

(FPCore (u2 u3 u1 amt u4)
 :precision binary32
 (fma
  amt
  (fma
   amt
   (*
    0.5
    (fma
     u4
     (+ amt -1.0)
     (fma
      -5.0
      u2
      (fma 2.0 u1 (fma amt (fma 3.0 (- u2 u3) (- u1)) (* u3 4.0))))))
   (* 0.5 (- u3 u1)))
  (* u2 1.52587890625e-5)))

float code(float u2, float u3, float u1, float amt, float u4) {
	return fmaf(amt, fmaf(amt, (0.5f * fmaf(u4, (amt + -1.0f), fmaf(-5.0f, u2, fmaf(2.0f, u1, fmaf(amt, fmaf(3.0f, (u2 - u3), -u1), (u3 * 4.0f)))))), (0.5f * (u3 - u1))), (u2 * 1.52587890625e-5f));
}

function code(u2, u3, u1, amt, u4)
	return fma(amt, fma(amt, Float32(Float32(0.5) * fma(u4, Float32(amt + Float32(-1.0)), fma(Float32(-5.0), u2, fma(Float32(2.0), u1, fma(amt, fma(Float32(3.0), Float32(u2 - u3), Float32(-u1)), Float32(u3 * Float32(4.0))))))), Float32(Float32(0.5) * Float32(u3 - u1))), Float32(u2 * Float32(1.52587890625e-5)))
end

\begin{array}{l}

\\
\mathsf{fma}\left(amt, \mathsf{fma}\left(amt, 0.5 \cdot \mathsf{fma}\left(u4, amt + -1, \mathsf{fma}\left(-5, u2, \mathsf{fma}\left(2, u1, \mathsf{fma}\left(amt, \mathsf{fma}\left(3, u2 - u3, -u1\right), u3 \cdot 4\right)\right)\right)\right), 0.5 \cdot \left(u3 - u1\right)\right), u2 \cdot 1.52587890625 \cdot 10^{-5}\right)
\end{array}

Derivation

Initial program 75.5%
\[7.62939453125 \cdot 10^{-6} \cdot \left(\left(\left(2 \cdot u2 + \left(u3 - u1\right) \cdot \left(amt \cdot 65536\right)\right) + \left(\left(\left(2 \cdot u1 - 5 \cdot u2\right) + 4 \cdot u3\right) - u4\right) \cdot \left({amt}^{2} \cdot 65536\right)\right) + \left(\left(\left(3 \cdot u2 - u1\right) - 3 \cdot u3\right) + u4\right) \cdot \left({amt}^{3} \cdot 65536\right)\right) \]
Add Preprocessing
Taylor expanded in amt around 0
\[\leadsto \color{blue}{\frac{1}{65536} \cdot u2 + amt \cdot \left(\frac{1}{2} \cdot \left(u3 - u1\right) + amt \cdot \left(\frac{1}{2} \cdot \left(amt \cdot \left(\left(u4 + 3 \cdot u2\right) - \left(u1 + 3 \cdot u3\right)\right)\right) + \frac{1}{2} \cdot \left(\left(2 \cdot u1 + 4 \cdot u3\right) - \left(u4 + 5 \cdot u2\right)\right)\right)\right)} \]
Step-by-step derivation
1. +-commutativeN/A
  \[\leadsto \color{blue}{amt \cdot \left(\frac{1}{2} \cdot \left(u3 - u1\right) + amt \cdot \left(\frac{1}{2} \cdot \left(amt \cdot \left(\left(u4 + 3 \cdot u2\right) - \left(u1 + 3 \cdot u3\right)\right)\right) + \frac{1}{2} \cdot \left(\left(2 \cdot u1 + 4 \cdot u3\right) - \left(u4 + 5 \cdot u2\right)\right)\right)\right) + \frac{1}{65536} \cdot u2} \]
2. lower-fma.f32N/A
  \[\leadsto \color{blue}{\mathsf{fma}\left(amt, \frac{1}{2} \cdot \left(u3 - u1\right) + amt \cdot \left(\frac{1}{2} \cdot \left(amt \cdot \left(\left(u4 + 3 \cdot u2\right) - \left(u1 + 3 \cdot u3\right)\right)\right) + \frac{1}{2} \cdot \left(\left(2 \cdot u1 + 4 \cdot u3\right) - \left(u4 + 5 \cdot u2\right)\right)\right), \frac{1}{65536} \cdot u2\right)} \]
Applied rewrites94.9%
\[\leadsto \color{blue}{\mathsf{fma}\left(amt, \mathsf{fma}\left(amt, 0.5 \cdot \mathsf{fma}\left(amt, \mathsf{fma}\left(3, u2 - u3, u4\right) - u1, \mathsf{fma}\left(2, u1, \mathsf{fma}\left(u3, 4, u2 \cdot -5\right)\right) - u4\right), \left(u3 - u1\right) \cdot 0.5\right), u2 \cdot 1.52587890625 \cdot 10^{-5}\right)} \]
Taylor expanded in u4 around 0
\[\leadsto \mathsf{fma}\left(amt, \mathsf{fma}\left(amt, \frac{1}{2} \cdot \left(u4 \cdot \left(amt - 1\right)\right) + \color{blue}{\frac{1}{2} \cdot \left(-5 \cdot u2 + \left(2 \cdot u1 + \left(4 \cdot u3 + amt \cdot \left(3 \cdot \left(u2 - u3\right) - u1\right)\right)\right)\right)}, \left(u3 - u1\right) \cdot \frac{1}{2}\right), u2 \cdot \frac{1}{65536}\right) \]
Step-by-step derivation
Applied rewrites95.9%
\[\leadsto \mathsf{fma}\left(amt, \mathsf{fma}\left(amt, 0.5 \cdot \color{blue}{\mathsf{fma}\left(u4, amt + -1, \mathsf{fma}\left(-5, u2, \mathsf{fma}\left(2, u1, \mathsf{fma}\left(amt, \mathsf{fma}\left(3, u2 - u3, -u1\right), 4 \cdot u3\right)\right)\right)\right)}, \left(u3 - u1\right) \cdot 0.5\right), u2 \cdot 1.52587890625 \cdot 10^{-5}\right) \]
Final simplification95.9%
\[\leadsto \mathsf{fma}\left(amt, \mathsf{fma}\left(amt, 0.5 \cdot \mathsf{fma}\left(u4, amt + -1, \mathsf{fma}\left(-5, u2, \mathsf{fma}\left(2, u1, \mathsf{fma}\left(amt, \mathsf{fma}\left(3, u2 - u3, -u1\right), u3 \cdot 4\right)\right)\right)\right), 0.5 \cdot \left(u3 - u1\right)\right), u2 \cdot 1.52587890625 \cdot 10^{-5}\right) \]
Add Preprocessing

Alternative 3: 95.0% accurate, 4.1× speedup?

\[\begin{array}{l} \\ \mathsf{fma}\left(amt, \mathsf{fma}\left(amt, 0.5 \cdot \mathsf{fma}\left(-5, u2, \mathsf{fma}\left(2, u1, \mathsf{fma}\left(amt, \mathsf{fma}\left(3, u2, u4\right) - u1, u3 \cdot \mathsf{fma}\left(-3, amt, 4\right)\right)\right) - u4\right), 0.5 \cdot \left(u3 - u1\right)\right), u2 \cdot 1.52587890625 \cdot 10^{-5}\right) \end{array} \]

(FPCore (u2 u3 u1 amt u4)
 :precision binary32
 (fma
  amt
  (fma
   amt
   (*
    0.5
    (fma
     -5.0
     u2
     (-
      (fma 2.0 u1 (fma amt (- (fma 3.0 u2 u4) u1) (* u3 (fma -3.0 amt 4.0))))
      u4)))
   (* 0.5 (- u3 u1)))
  (* u2 1.52587890625e-5)))

float code(float u2, float u3, float u1, float amt, float u4) {
	return fmaf(amt, fmaf(amt, (0.5f * fmaf(-5.0f, u2, (fmaf(2.0f, u1, fmaf(amt, (fmaf(3.0f, u2, u4) - u1), (u3 * fmaf(-3.0f, amt, 4.0f)))) - u4))), (0.5f * (u3 - u1))), (u2 * 1.52587890625e-5f));
}

function code(u2, u3, u1, amt, u4)
	return fma(amt, fma(amt, Float32(Float32(0.5) * fma(Float32(-5.0), u2, Float32(fma(Float32(2.0), u1, fma(amt, Float32(fma(Float32(3.0), u2, u4) - u1), Float32(u3 * fma(Float32(-3.0), amt, Float32(4.0))))) - u4))), Float32(Float32(0.5) * Float32(u3 - u1))), Float32(u2 * Float32(1.52587890625e-5)))
end

\begin{array}{l}

\\
\mathsf{fma}\left(amt, \mathsf{fma}\left(amt, 0.5 \cdot \mathsf{fma}\left(-5, u2, \mathsf{fma}\left(2, u1, \mathsf{fma}\left(amt, \mathsf{fma}\left(3, u2, u4\right) - u1, u3 \cdot \mathsf{fma}\left(-3, amt, 4\right)\right)\right) - u4\right), 0.5 \cdot \left(u3 - u1\right)\right), u2 \cdot 1.52587890625 \cdot 10^{-5}\right)
\end{array}

Derivation

Initial program 75.5%
\[7.62939453125 \cdot 10^{-6} \cdot \left(\left(\left(2 \cdot u2 + \left(u3 - u1\right) \cdot \left(amt \cdot 65536\right)\right) + \left(\left(\left(2 \cdot u1 - 5 \cdot u2\right) + 4 \cdot u3\right) - u4\right) \cdot \left({amt}^{2} \cdot 65536\right)\right) + \left(\left(\left(3 \cdot u2 - u1\right) - 3 \cdot u3\right) + u4\right) \cdot \left({amt}^{3} \cdot 65536\right)\right) \]
Add Preprocessing
Taylor expanded in amt around 0
\[\leadsto \color{blue}{\frac{1}{65536} \cdot u2 + amt \cdot \left(\frac{1}{2} \cdot \left(u3 - u1\right) + amt \cdot \left(\frac{1}{2} \cdot \left(amt \cdot \left(\left(u4 + 3 \cdot u2\right) - \left(u1 + 3 \cdot u3\right)\right)\right) + \frac{1}{2} \cdot \left(\left(2 \cdot u1 + 4 \cdot u3\right) - \left(u4 + 5 \cdot u2\right)\right)\right)\right)} \]
Step-by-step derivation
1. +-commutativeN/A
  \[\leadsto \color{blue}{amt \cdot \left(\frac{1}{2} \cdot \left(u3 - u1\right) + amt \cdot \left(\frac{1}{2} \cdot \left(amt \cdot \left(\left(u4 + 3 \cdot u2\right) - \left(u1 + 3 \cdot u3\right)\right)\right) + \frac{1}{2} \cdot \left(\left(2 \cdot u1 + 4 \cdot u3\right) - \left(u4 + 5 \cdot u2\right)\right)\right)\right) + \frac{1}{65536} \cdot u2} \]
2. lower-fma.f32N/A
  \[\leadsto \color{blue}{\mathsf{fma}\left(amt, \frac{1}{2} \cdot \left(u3 - u1\right) + amt \cdot \left(\frac{1}{2} \cdot \left(amt \cdot \left(\left(u4 + 3 \cdot u2\right) - \left(u1 + 3 \cdot u3\right)\right)\right) + \frac{1}{2} \cdot \left(\left(2 \cdot u1 + 4 \cdot u3\right) - \left(u4 + 5 \cdot u2\right)\right)\right), \frac{1}{65536} \cdot u2\right)} \]
Applied rewrites94.9%
\[\leadsto \color{blue}{\mathsf{fma}\left(amt, \mathsf{fma}\left(amt, 0.5 \cdot \mathsf{fma}\left(amt, \mathsf{fma}\left(3, u2 - u3, u4\right) - u1, \mathsf{fma}\left(2, u1, \mathsf{fma}\left(u3, 4, u2 \cdot -5\right)\right) - u4\right), \left(u3 - u1\right) \cdot 0.5\right), u2 \cdot 1.52587890625 \cdot 10^{-5}\right)} \]
Taylor expanded in u3 around 0
\[\leadsto \mathsf{fma}\left(amt, \mathsf{fma}\left(amt, \frac{1}{2} \cdot \left(\left(-5 \cdot u2 + \left(2 \cdot u1 + \left(amt \cdot \left(\left(u4 + 3 \cdot u2\right) - u1\right) + u3 \cdot \left(4 + -3 \cdot amt\right)\right)\right)\right) - \color{blue}{u4}\right), \left(u3 - u1\right) \cdot \frac{1}{2}\right), u2 \cdot \frac{1}{65536}\right) \]
Step-by-step derivation
Applied rewrites95.7%
\[\leadsto \mathsf{fma}\left(amt, \mathsf{fma}\left(amt, 0.5 \cdot \mathsf{fma}\left(-5, \color{blue}{u2}, \mathsf{fma}\left(2, u1, \mathsf{fma}\left(amt, \mathsf{fma}\left(3, u2, u4\right) - u1, u3 \cdot \mathsf{fma}\left(-3, amt, 4\right)\right)\right) - u4\right), \left(u3 - u1\right) \cdot 0.5\right), u2 \cdot 1.52587890625 \cdot 10^{-5}\right) \]
Final simplification95.7%
\[\leadsto \mathsf{fma}\left(amt, \mathsf{fma}\left(amt, 0.5 \cdot \mathsf{fma}\left(-5, u2, \mathsf{fma}\left(2, u1, \mathsf{fma}\left(amt, \mathsf{fma}\left(3, u2, u4\right) - u1, u3 \cdot \mathsf{fma}\left(-3, amt, 4\right)\right)\right) - u4\right), 0.5 \cdot \left(u3 - u1\right)\right), u2 \cdot 1.52587890625 \cdot 10^{-5}\right) \]
Add Preprocessing

Alternative 4: 94.9% accurate, 4.1× speedup?

\[\begin{array}{l} \\ \mathsf{fma}\left(amt, \mathsf{fma}\left(amt, 0.5 \cdot \mathsf{fma}\left(u2, \mathsf{fma}\left(3, amt, -5\right), \mathsf{fma}\left(2, u1, \mathsf{fma}\left(amt, \mathsf{fma}\left(-3, u3, u4\right) - u1, u3 \cdot 4\right) - u4\right)\right), 0.5 \cdot \left(u3 - u1\right)\right), u2 \cdot 1.52587890625 \cdot 10^{-5}\right) \end{array} \]

(FPCore (u2 u3 u1 amt u4)
 :precision binary32
 (fma
  amt
  (fma
   amt
   (*
    0.5
    (fma
     u2
     (fma 3.0 amt -5.0)
     (fma 2.0 u1 (- (fma amt (- (fma -3.0 u3 u4) u1) (* u3 4.0)) u4))))
   (* 0.5 (- u3 u1)))
  (* u2 1.52587890625e-5)))

float code(float u2, float u3, float u1, float amt, float u4) {
	return fmaf(amt, fmaf(amt, (0.5f * fmaf(u2, fmaf(3.0f, amt, -5.0f), fmaf(2.0f, u1, (fmaf(amt, (fmaf(-3.0f, u3, u4) - u1), (u3 * 4.0f)) - u4)))), (0.5f * (u3 - u1))), (u2 * 1.52587890625e-5f));
}

function code(u2, u3, u1, amt, u4)
	return fma(amt, fma(amt, Float32(Float32(0.5) * fma(u2, fma(Float32(3.0), amt, Float32(-5.0)), fma(Float32(2.0), u1, Float32(fma(amt, Float32(fma(Float32(-3.0), u3, u4) - u1), Float32(u3 * Float32(4.0))) - u4)))), Float32(Float32(0.5) * Float32(u3 - u1))), Float32(u2 * Float32(1.52587890625e-5)))
end

\begin{array}{l}

\\
\mathsf{fma}\left(amt, \mathsf{fma}\left(amt, 0.5 \cdot \mathsf{fma}\left(u2, \mathsf{fma}\left(3, amt, -5\right), \mathsf{fma}\left(2, u1, \mathsf{fma}\left(amt, \mathsf{fma}\left(-3, u3, u4\right) - u1, u3 \cdot 4\right) - u4\right)\right), 0.5 \cdot \left(u3 - u1\right)\right), u2 \cdot 1.52587890625 \cdot 10^{-5}\right)
\end{array}

Derivation

Initial program 75.5%
\[7.62939453125 \cdot 10^{-6} \cdot \left(\left(\left(2 \cdot u2 + \left(u3 - u1\right) \cdot \left(amt \cdot 65536\right)\right) + \left(\left(\left(2 \cdot u1 - 5 \cdot u2\right) + 4 \cdot u3\right) - u4\right) \cdot \left({amt}^{2} \cdot 65536\right)\right) + \left(\left(\left(3 \cdot u2 - u1\right) - 3 \cdot u3\right) + u4\right) \cdot \left({amt}^{3} \cdot 65536\right)\right) \]
Add Preprocessing
Taylor expanded in amt around 0
\[\leadsto \color{blue}{\frac{1}{65536} \cdot u2 + amt \cdot \left(\frac{1}{2} \cdot \left(u3 - u1\right) + amt \cdot \left(\frac{1}{2} \cdot \left(amt \cdot \left(\left(u4 + 3 \cdot u2\right) - \left(u1 + 3 \cdot u3\right)\right)\right) + \frac{1}{2} \cdot \left(\left(2 \cdot u1 + 4 \cdot u3\right) - \left(u4 + 5 \cdot u2\right)\right)\right)\right)} \]
Step-by-step derivation
1. +-commutativeN/A
  \[\leadsto \color{blue}{amt \cdot \left(\frac{1}{2} \cdot \left(u3 - u1\right) + amt \cdot \left(\frac{1}{2} \cdot \left(amt \cdot \left(\left(u4 + 3 \cdot u2\right) - \left(u1 + 3 \cdot u3\right)\right)\right) + \frac{1}{2} \cdot \left(\left(2 \cdot u1 + 4 \cdot u3\right) - \left(u4 + 5 \cdot u2\right)\right)\right)\right) + \frac{1}{65536} \cdot u2} \]
2. lower-fma.f32N/A
  \[\leadsto \color{blue}{\mathsf{fma}\left(amt, \frac{1}{2} \cdot \left(u3 - u1\right) + amt \cdot \left(\frac{1}{2} \cdot \left(amt \cdot \left(\left(u4 + 3 \cdot u2\right) - \left(u1 + 3 \cdot u3\right)\right)\right) + \frac{1}{2} \cdot \left(\left(2 \cdot u1 + 4 \cdot u3\right) - \left(u4 + 5 \cdot u2\right)\right)\right), \frac{1}{65536} \cdot u2\right)} \]
Applied rewrites94.9%
\[\leadsto \color{blue}{\mathsf{fma}\left(amt, \mathsf{fma}\left(amt, 0.5 \cdot \mathsf{fma}\left(amt, \mathsf{fma}\left(3, u2 - u3, u4\right) - u1, \mathsf{fma}\left(2, u1, \mathsf{fma}\left(u3, 4, u2 \cdot -5\right)\right) - u4\right), \left(u3 - u1\right) \cdot 0.5\right), u2 \cdot 1.52587890625 \cdot 10^{-5}\right)} \]
Taylor expanded in u2 around 0
\[\leadsto \mathsf{fma}\left(amt, \mathsf{fma}\left(amt, \frac{1}{2} \cdot \left(u2 \cdot \left(3 \cdot amt - 5\right)\right) + \color{blue}{\frac{1}{2} \cdot \left(\left(2 \cdot u1 + \left(4 \cdot u3 + amt \cdot \left(\left(u4 + -3 \cdot u3\right) - u1\right)\right)\right) - u4\right)}, \left(u3 - u1\right) \cdot \frac{1}{2}\right), u2 \cdot \frac{1}{65536}\right) \]
Step-by-step derivation
Applied rewrites95.5%
\[\leadsto \mathsf{fma}\left(amt, \mathsf{fma}\left(amt, 0.5 \cdot \color{blue}{\mathsf{fma}\left(u2, \mathsf{fma}\left(3, amt, -5\right), \mathsf{fma}\left(2, u1, \mathsf{fma}\left(amt, \mathsf{fma}\left(-3, u3, u4\right) - u1, 4 \cdot u3\right) - u4\right)\right)}, \left(u3 - u1\right) \cdot 0.5\right), u2 \cdot 1.52587890625 \cdot 10^{-5}\right) \]
Final simplification95.5%
\[\leadsto \mathsf{fma}\left(amt, \mathsf{fma}\left(amt, 0.5 \cdot \mathsf{fma}\left(u2, \mathsf{fma}\left(3, amt, -5\right), \mathsf{fma}\left(2, u1, \mathsf{fma}\left(amt, \mathsf{fma}\left(-3, u3, u4\right) - u1, u3 \cdot 4\right) - u4\right)\right), 0.5 \cdot \left(u3 - u1\right)\right), u2 \cdot 1.52587890625 \cdot 10^{-5}\right) \]
Add Preprocessing

Alternative 5: 85.1% accurate, 4.3× speedup?

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(amt, 0.5 \cdot \left(u3 + amt \cdot \mathsf{fma}\left(-5, u2, \mathsf{fma}\left(amt, \mathsf{fma}\left(3, u2 - u3, u4\right), u3 \cdot 4\right) - u4\right)\right), u2 \cdot 1.52587890625 \cdot 10^{-5}\right)\\ \mathbf{if}\;u3 \leq -2:\\ \;\;\;\;t_0\\ \mathbf{elif}\;u3 \leq 40000:\\ \;\;\;\;\mathsf{fma}\left(amt, \mathsf{fma}\left(amt \cdot 0.5, \mathsf{fma}\left(-5, u2, \mathsf{fma}\left(amt, \mathsf{fma}\left(3, u2, u4\right) - u1, 2 \cdot u1\right) - u4\right), u1 \cdot -0.5\right), u2 \cdot 1.52587890625 \cdot 10^{-5}\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

(FPCore (u2 u3 u1 amt u4)
 :precision binary32
 (let* ((t_0
         (fma
          amt
          (*
           0.5
           (+
            u3
            (*
             amt
             (fma
              -5.0
              u2
              (- (fma amt (fma 3.0 (- u2 u3) u4) (* u3 4.0)) u4)))))
          (* u2 1.52587890625e-5))))
   (if (<= u3 -2.0)
     t_0
     (if (<= u3 40000.0)
       (fma
        amt
        (fma
         (* amt 0.5)
         (fma -5.0 u2 (- (fma amt (- (fma 3.0 u2 u4) u1) (* 2.0 u1)) u4))
         (* u1 -0.5))
        (* u2 1.52587890625e-5))
       t_0))))

float code(float u2, float u3, float u1, float amt, float u4) {
	float t_0 = fmaf(amt, (0.5f * (u3 + (amt * fmaf(-5.0f, u2, (fmaf(amt, fmaf(3.0f, (u2 - u3), u4), (u3 * 4.0f)) - u4))))), (u2 * 1.52587890625e-5f));
	float tmp;
	if (u3 <= -2.0f) {
		tmp = t_0;
	} else if (u3 <= 40000.0f) {
		tmp = fmaf(amt, fmaf((amt * 0.5f), fmaf(-5.0f, u2, (fmaf(amt, (fmaf(3.0f, u2, u4) - u1), (2.0f * u1)) - u4)), (u1 * -0.5f)), (u2 * 1.52587890625e-5f));
	} else {
		tmp = t_0;
	}
	return tmp;
}

function code(u2, u3, u1, amt, u4)
	t_0 = fma(amt, Float32(Float32(0.5) * Float32(u3 + Float32(amt * fma(Float32(-5.0), u2, Float32(fma(amt, fma(Float32(3.0), Float32(u2 - u3), u4), Float32(u3 * Float32(4.0))) - u4))))), Float32(u2 * Float32(1.52587890625e-5)))
	tmp = Float32(0.0)
	if (u3 <= Float32(-2.0))
		tmp = t_0;
	elseif (u3 <= Float32(40000.0))
		tmp = fma(amt, fma(Float32(amt * Float32(0.5)), fma(Float32(-5.0), u2, Float32(fma(amt, Float32(fma(Float32(3.0), u2, u4) - u1), Float32(Float32(2.0) * u1)) - u4)), Float32(u1 * Float32(-0.5))), Float32(u2 * Float32(1.52587890625e-5)));
	else
		tmp = t_0;
	end
	return tmp
end

\begin{array}{l}

\\
\begin{array}{l}
t_0 := \mathsf{fma}\left(amt, 0.5 \cdot \left(u3 + amt \cdot \mathsf{fma}\left(-5, u2, \mathsf{fma}\left(amt, \mathsf{fma}\left(3, u2 - u3, u4\right), u3 \cdot 4\right) - u4\right)\right), u2 \cdot 1.52587890625 \cdot 10^{-5}\right)\\
\mathbf{if}\;u3 \leq -2:\\
\;\;\;\;t_0\\

\mathbf{elif}\;u3 \leq 40000:\\
\;\;\;\;\mathsf{fma}\left(amt, \mathsf{fma}\left(amt \cdot 0.5, \mathsf{fma}\left(-5, u2, \mathsf{fma}\left(amt, \mathsf{fma}\left(3, u2, u4\right) - u1, 2 \cdot u1\right) - u4\right), u1 \cdot -0.5\right), u2 \cdot 1.52587890625 \cdot 10^{-5}\right)\\

\mathbf{else}:\\
\;\;\;\;t_0\\


\end{array}
\end{array}

Derivation

Split input into 2 regimes
if u3 < -2 or 4e4 < u3
1. Initial program 71.8%
  \[7.62939453125 \cdot 10^{-6} \cdot \left(\left(\left(2 \cdot u2 + \left(u3 - u1\right) \cdot \left(amt \cdot 65536\right)\right) + \left(\left(\left(2 \cdot u1 - 5 \cdot u2\right) + 4 \cdot u3\right) - u4\right) \cdot \left({amt}^{2} \cdot 65536\right)\right) + \left(\left(\left(3 \cdot u2 - u1\right) - 3 \cdot u3\right) + u4\right) \cdot \left({amt}^{3} \cdot 65536\right)\right) \]
2. Add Preprocessing
3. Taylor expanded in amt around 0
  \[\leadsto \color{blue}{\frac{1}{65536} \cdot u2 + amt \cdot \left(\frac{1}{2} \cdot \left(u3 - u1\right) + amt \cdot \left(\frac{1}{2} \cdot \left(amt \cdot \left(\left(u4 + 3 \cdot u2\right) - \left(u1 + 3 \cdot u3\right)\right)\right) + \frac{1}{2} \cdot \left(\left(2 \cdot u1 + 4 \cdot u3\right) - \left(u4 + 5 \cdot u2\right)\right)\right)\right)} \]
4. Step-by-step derivation
  1. +-commutativeN/A
    \[\leadsto \color{blue}{amt \cdot \left(\frac{1}{2} \cdot \left(u3 - u1\right) + amt \cdot \left(\frac{1}{2} \cdot \left(amt \cdot \left(\left(u4 + 3 \cdot u2\right) - \left(u1 + 3 \cdot u3\right)\right)\right) + \frac{1}{2} \cdot \left(\left(2 \cdot u1 + 4 \cdot u3\right) - \left(u4 + 5 \cdot u2\right)\right)\right)\right) + \frac{1}{65536} \cdot u2} \]
  2. lower-fma.f32N/A
    \[\leadsto \color{blue}{\mathsf{fma}\left(amt, \frac{1}{2} \cdot \left(u3 - u1\right) + amt \cdot \left(\frac{1}{2} \cdot \left(amt \cdot \left(\left(u4 + 3 \cdot u2\right) - \left(u1 + 3 \cdot u3\right)\right)\right) + \frac{1}{2} \cdot \left(\left(2 \cdot u1 + 4 \cdot u3\right) - \left(u4 + 5 \cdot u2\right)\right)\right), \frac{1}{65536} \cdot u2\right)} \]
5. Applied rewrites95.6%
  \[\leadsto \color{blue}{\mathsf{fma}\left(amt, \mathsf{fma}\left(amt, 0.5 \cdot \mathsf{fma}\left(amt, \mathsf{fma}\left(3, u2 - u3, u4\right) - u1, \mathsf{fma}\left(2, u1, \mathsf{fma}\left(u3, 4, u2 \cdot -5\right)\right) - u4\right), \left(u3 - u1\right) \cdot 0.5\right), u2 \cdot 1.52587890625 \cdot 10^{-5}\right)} \]
6. Taylor expanded in u1 around 0
  \[\leadsto \mathsf{fma}\left(amt, \frac{1}{2} \cdot u3 + \color{blue}{\frac{1}{2} \cdot \left(amt \cdot \left(\left(-5 \cdot u2 + \left(4 \cdot u3 + amt \cdot \left(u4 + 3 \cdot \left(u2 - u3\right)\right)\right)\right) - u4\right)\right)}, u2 \cdot \frac{1}{65536}\right) \]
7. Step-by-step derivation
8. Applied rewrites86.4%
  \[\leadsto \mathsf{fma}\left(amt, 0.5 \cdot \color{blue}{\left(u3 + amt \cdot \mathsf{fma}\left(-5, u2, \mathsf{fma}\left(amt, \mathsf{fma}\left(3, u2 - u3, u4\right), 4 \cdot u3\right) - u4\right)\right)}, u2 \cdot 1.52587890625 \cdot 10^{-5}\right) \]
if -2 < u3 < 4e4
1. Initial program 78.3%
  \[7.62939453125 \cdot 10^{-6} \cdot \left(\left(\left(2 \cdot u2 + \left(u3 - u1\right) \cdot \left(amt \cdot 65536\right)\right) + \left(\left(\left(2 \cdot u1 - 5 \cdot u2\right) + 4 \cdot u3\right) - u4\right) \cdot \left({amt}^{2} \cdot 65536\right)\right) + \left(\left(\left(3 \cdot u2 - u1\right) - 3 \cdot u3\right) + u4\right) \cdot \left({amt}^{3} \cdot 65536\right)\right) \]
2. Add Preprocessing
3. Taylor expanded in amt around 0
  \[\leadsto \color{blue}{\frac{1}{65536} \cdot u2 + amt \cdot \left(\frac{1}{2} \cdot \left(u3 - u1\right) + amt \cdot \left(\frac{1}{2} \cdot \left(amt \cdot \left(\left(u4 + 3 \cdot u2\right) - \left(u1 + 3 \cdot u3\right)\right)\right) + \frac{1}{2} \cdot \left(\left(2 \cdot u1 + 4 \cdot u3\right) - \left(u4 + 5 \cdot u2\right)\right)\right)\right)} \]
4. Step-by-step derivation
  1. +-commutativeN/A
    \[\leadsto \color{blue}{amt \cdot \left(\frac{1}{2} \cdot \left(u3 - u1\right) + amt \cdot \left(\frac{1}{2} \cdot \left(amt \cdot \left(\left(u4 + 3 \cdot u2\right) - \left(u1 + 3 \cdot u3\right)\right)\right) + \frac{1}{2} \cdot \left(\left(2 \cdot u1 + 4 \cdot u3\right) - \left(u4 + 5 \cdot u2\right)\right)\right)\right) + \frac{1}{65536} \cdot u2} \]
  2. lower-fma.f32N/A
    \[\leadsto \color{blue}{\mathsf{fma}\left(amt, \frac{1}{2} \cdot \left(u3 - u1\right) + amt \cdot \left(\frac{1}{2} \cdot \left(amt \cdot \left(\left(u4 + 3 \cdot u2\right) - \left(u1 + 3 \cdot u3\right)\right)\right) + \frac{1}{2} \cdot \left(\left(2 \cdot u1 + 4 \cdot u3\right) - \left(u4 + 5 \cdot u2\right)\right)\right), \frac{1}{65536} \cdot u2\right)} \]
5. Applied rewrites94.4%
  \[\leadsto \color{blue}{\mathsf{fma}\left(amt, \mathsf{fma}\left(amt, 0.5 \cdot \mathsf{fma}\left(amt, \mathsf{fma}\left(3, u2 - u3, u4\right) - u1, \mathsf{fma}\left(2, u1, \mathsf{fma}\left(u3, 4, u2 \cdot -5\right)\right) - u4\right), \left(u3 - u1\right) \cdot 0.5\right), u2 \cdot 1.52587890625 \cdot 10^{-5}\right)} \]
6. Taylor expanded in u3 around 0
  \[\leadsto \mathsf{fma}\left(amt, \frac{-1}{2} \cdot u1 + \color{blue}{\frac{1}{2} \cdot \left(amt \cdot \left(\left(-5 \cdot u2 + \left(2 \cdot u1 + amt \cdot \left(\left(u4 + 3 \cdot u2\right) - u1\right)\right)\right) - u4\right)\right)}, u2 \cdot \frac{1}{65536}\right) \]
7. Step-by-step derivation
8. Applied rewrites91.7%
  \[\leadsto \mathsf{fma}\left(amt, \mathsf{fma}\left(0.5 \cdot amt, \color{blue}{\mathsf{fma}\left(-5, u2, \mathsf{fma}\left(amt, \mathsf{fma}\left(3, u2, u4\right) - u1, 2 \cdot u1\right) - u4\right)}, -0.5 \cdot u1\right), u2 \cdot 1.52587890625 \cdot 10^{-5}\right) \]
Recombined 2 regimes into one program.
Final simplification89.5%
\[\leadsto \begin{array}{l} \mathbf{if}\;u3 \leq -2:\\ \;\;\;\;\mathsf{fma}\left(amt, 0.5 \cdot \left(u3 + amt \cdot \mathsf{fma}\left(-5, u2, \mathsf{fma}\left(amt, \mathsf{fma}\left(3, u2 - u3, u4\right), u3 \cdot 4\right) - u4\right)\right), u2 \cdot 1.52587890625 \cdot 10^{-5}\right)\\ \mathbf{elif}\;u3 \leq 40000:\\ \;\;\;\;\mathsf{fma}\left(amt, \mathsf{fma}\left(amt \cdot 0.5, \mathsf{fma}\left(-5, u2, \mathsf{fma}\left(amt, \mathsf{fma}\left(3, u2, u4\right) - u1, 2 \cdot u1\right) - u4\right), u1 \cdot -0.5\right), u2 \cdot 1.52587890625 \cdot 10^{-5}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(amt, 0.5 \cdot \left(u3 + amt \cdot \mathsf{fma}\left(-5, u2, \mathsf{fma}\left(amt, \mathsf{fma}\left(3, u2 - u3, u4\right), u3 \cdot 4\right) - u4\right)\right), u2 \cdot 1.52587890625 \cdot 10^{-5}\right)\\ \end{array} \]
Add Preprocessing

Alternative 6: 94.3% accurate, 4.3× speedup?

\[\begin{array}{l} \\ \mathsf{fma}\left(amt, \mathsf{fma}\left(amt, 0.5 \cdot \mathsf{fma}\left(u1, 2 - amt, \mathsf{fma}\left(-5, u2, \mathsf{fma}\left(amt, \mathsf{fma}\left(3, u2 - u3, u4\right), u3 \cdot 4\right) - u4\right)\right), 0.5 \cdot \left(u3 - u1\right)\right), u2 \cdot 1.52587890625 \cdot 10^{-5}\right) \end{array} \]

(FPCore (u2 u3 u1 amt u4)
 :precision binary32
 (fma
  amt
  (fma
   amt
   (*
    0.5
    (fma
     u1
     (- 2.0 amt)
     (fma -5.0 u2 (- (fma amt (fma 3.0 (- u2 u3) u4) (* u3 4.0)) u4))))
   (* 0.5 (- u3 u1)))
  (* u2 1.52587890625e-5)))

float code(float u2, float u3, float u1, float amt, float u4) {
	return fmaf(amt, fmaf(amt, (0.5f * fmaf(u1, (2.0f - amt), fmaf(-5.0f, u2, (fmaf(amt, fmaf(3.0f, (u2 - u3), u4), (u3 * 4.0f)) - u4)))), (0.5f * (u3 - u1))), (u2 * 1.52587890625e-5f));
}

function code(u2, u3, u1, amt, u4)
	return fma(amt, fma(amt, Float32(Float32(0.5) * fma(u1, Float32(Float32(2.0) - amt), fma(Float32(-5.0), u2, Float32(fma(amt, fma(Float32(3.0), Float32(u2 - u3), u4), Float32(u3 * Float32(4.0))) - u4)))), Float32(Float32(0.5) * Float32(u3 - u1))), Float32(u2 * Float32(1.52587890625e-5)))
end

\begin{array}{l}

\\
\mathsf{fma}\left(amt, \mathsf{fma}\left(amt, 0.5 \cdot \mathsf{fma}\left(u1, 2 - amt, \mathsf{fma}\left(-5, u2, \mathsf{fma}\left(amt, \mathsf{fma}\left(3, u2 - u3, u4\right), u3 \cdot 4\right) - u4\right)\right), 0.5 \cdot \left(u3 - u1\right)\right), u2 \cdot 1.52587890625 \cdot 10^{-5}\right)
\end{array}

Derivation

Initial program 75.5%
\[7.62939453125 \cdot 10^{-6} \cdot \left(\left(\left(2 \cdot u2 + \left(u3 - u1\right) \cdot \left(amt \cdot 65536\right)\right) + \left(\left(\left(2 \cdot u1 - 5 \cdot u2\right) + 4 \cdot u3\right) - u4\right) \cdot \left({amt}^{2} \cdot 65536\right)\right) + \left(\left(\left(3 \cdot u2 - u1\right) - 3 \cdot u3\right) + u4\right) \cdot \left({amt}^{3} \cdot 65536\right)\right) \]
Add Preprocessing
Taylor expanded in amt around 0
\[\leadsto \color{blue}{\frac{1}{65536} \cdot u2 + amt \cdot \left(\frac{1}{2} \cdot \left(u3 - u1\right) + amt \cdot \left(\frac{1}{2} \cdot \left(amt \cdot \left(\left(u4 + 3 \cdot u2\right) - \left(u1 + 3 \cdot u3\right)\right)\right) + \frac{1}{2} \cdot \left(\left(2 \cdot u1 + 4 \cdot u3\right) - \left(u4 + 5 \cdot u2\right)\right)\right)\right)} \]
Step-by-step derivation
1. +-commutativeN/A
  \[\leadsto \color{blue}{amt \cdot \left(\frac{1}{2} \cdot \left(u3 - u1\right) + amt \cdot \left(\frac{1}{2} \cdot \left(amt \cdot \left(\left(u4 + 3 \cdot u2\right) - \left(u1 + 3 \cdot u3\right)\right)\right) + \frac{1}{2} \cdot \left(\left(2 \cdot u1 + 4 \cdot u3\right) - \left(u4 + 5 \cdot u2\right)\right)\right)\right) + \frac{1}{65536} \cdot u2} \]
2. lower-fma.f32N/A
  \[\leadsto \color{blue}{\mathsf{fma}\left(amt, \frac{1}{2} \cdot \left(u3 - u1\right) + amt \cdot \left(\frac{1}{2} \cdot \left(amt \cdot \left(\left(u4 + 3 \cdot u2\right) - \left(u1 + 3 \cdot u3\right)\right)\right) + \frac{1}{2} \cdot \left(\left(2 \cdot u1 + 4 \cdot u3\right) - \left(u4 + 5 \cdot u2\right)\right)\right), \frac{1}{65536} \cdot u2\right)} \]
Applied rewrites94.9%
\[\leadsto \color{blue}{\mathsf{fma}\left(amt, \mathsf{fma}\left(amt, 0.5 \cdot \mathsf{fma}\left(amt, \mathsf{fma}\left(3, u2 - u3, u4\right) - u1, \mathsf{fma}\left(2, u1, \mathsf{fma}\left(u3, 4, u2 \cdot -5\right)\right) - u4\right), \left(u3 - u1\right) \cdot 0.5\right), u2 \cdot 1.52587890625 \cdot 10^{-5}\right)} \]
Taylor expanded in u1 around 0
\[\leadsto \mathsf{fma}\left(amt, \mathsf{fma}\left(amt, \frac{1}{2} \cdot \left(u1 \cdot \left(2 + -1 \cdot amt\right)\right) + \color{blue}{\frac{1}{2} \cdot \left(\left(-5 \cdot u2 + \left(4 \cdot u3 + amt \cdot \left(u4 + 3 \cdot \left(u2 - u3\right)\right)\right)\right) - u4\right)}, \left(u3 - u1\right) \cdot \frac{1}{2}\right), u2 \cdot \frac{1}{65536}\right) \]
Step-by-step derivation
Applied rewrites95.0%
\[\leadsto \mathsf{fma}\left(amt, \mathsf{fma}\left(amt, 0.5 \cdot \color{blue}{\mathsf{fma}\left(u1, 2 - amt, \mathsf{fma}\left(-5, u2, \mathsf{fma}\left(amt, \mathsf{fma}\left(3, u2 - u3, u4\right), 4 \cdot u3\right) - u4\right)\right)}, \left(u3 - u1\right) \cdot 0.5\right), u2 \cdot 1.52587890625 \cdot 10^{-5}\right) \]
Final simplification95.0%
\[\leadsto \mathsf{fma}\left(amt, \mathsf{fma}\left(amt, 0.5 \cdot \mathsf{fma}\left(u1, 2 - amt, \mathsf{fma}\left(-5, u2, \mathsf{fma}\left(amt, \mathsf{fma}\left(3, u2 - u3, u4\right), u3 \cdot 4\right) - u4\right)\right), 0.5 \cdot \left(u3 - u1\right)\right), u2 \cdot 1.52587890625 \cdot 10^{-5}\right) \]
Add Preprocessing

Alternative 7: 85.1% accurate, 4.5× speedup?

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(amt, 0.5 \cdot \left(u3 + amt \cdot \mathsf{fma}\left(-5, u2, \mathsf{fma}\left(amt, \mathsf{fma}\left(3, u2 - u3, u4\right), u3 \cdot 4\right) - u4\right)\right), u2 \cdot 1.52587890625 \cdot 10^{-5}\right)\\ \mathbf{if}\;u2 \leq -5000:\\ \;\;\;\;t_0\\ \mathbf{elif}\;u2 \leq 199999996861349900:\\ \;\;\;\;amt \cdot \mathsf{fma}\left(amt, 0.5 \cdot \mathsf{fma}\left(amt, u4 - \mathsf{fma}\left(3, u3, u1\right), \mathsf{fma}\left(2, u1, \mathsf{fma}\left(4, u3, -u4\right)\right)\right), 0.5 \cdot \left(u3 - u1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

(FPCore (u2 u3 u1 amt u4)
 :precision binary32
 (let* ((t_0
         (fma
          amt
          (*
           0.5
           (+
            u3
            (*
             amt
             (fma
              -5.0
              u2
              (- (fma amt (fma 3.0 (- u2 u3) u4) (* u3 4.0)) u4)))))
          (* u2 1.52587890625e-5))))
   (if (<= u2 -5000.0)
     t_0
     (if (<= u2 199999996861349900.0)
       (*
        amt
        (fma
         amt
         (*
          0.5
          (fma amt (- u4 (fma 3.0 u3 u1)) (fma 2.0 u1 (fma 4.0 u3 (- u4)))))
         (* 0.5 (- u3 u1))))
       t_0))))

float code(float u2, float u3, float u1, float amt, float u4) {
	float t_0 = fmaf(amt, (0.5f * (u3 + (amt * fmaf(-5.0f, u2, (fmaf(amt, fmaf(3.0f, (u2 - u3), u4), (u3 * 4.0f)) - u4))))), (u2 * 1.52587890625e-5f));
	float tmp;
	if (u2 <= -5000.0f) {
		tmp = t_0;
	} else if (u2 <= 199999996861349900.0f) {
		tmp = amt * fmaf(amt, (0.5f * fmaf(amt, (u4 - fmaf(3.0f, u3, u1)), fmaf(2.0f, u1, fmaf(4.0f, u3, -u4)))), (0.5f * (u3 - u1)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

function code(u2, u3, u1, amt, u4)
	t_0 = fma(amt, Float32(Float32(0.5) * Float32(u3 + Float32(amt * fma(Float32(-5.0), u2, Float32(fma(amt, fma(Float32(3.0), Float32(u2 - u3), u4), Float32(u3 * Float32(4.0))) - u4))))), Float32(u2 * Float32(1.52587890625e-5)))
	tmp = Float32(0.0)
	if (u2 <= Float32(-5000.0))
		tmp = t_0;
	elseif (u2 <= Float32(199999996861349900.0))
		tmp = Float32(amt * fma(amt, Float32(Float32(0.5) * fma(amt, Float32(u4 - fma(Float32(3.0), u3, u1)), fma(Float32(2.0), u1, fma(Float32(4.0), u3, Float32(-u4))))), Float32(Float32(0.5) * Float32(u3 - u1))));
	else
		tmp = t_0;
	end
	return tmp
end

\begin{array}{l}

\\
\begin{array}{l}
t_0 := \mathsf{fma}\left(amt, 0.5 \cdot \left(u3 + amt \cdot \mathsf{fma}\left(-5, u2, \mathsf{fma}\left(amt, \mathsf{fma}\left(3, u2 - u3, u4\right), u3 \cdot 4\right) - u4\right)\right), u2 \cdot 1.52587890625 \cdot 10^{-5}\right)\\
\mathbf{if}\;u2 \leq -5000:\\
\;\;\;\;t_0\\

\mathbf{elif}\;u2 \leq 199999996861349900:\\
\;\;\;\;amt \cdot \mathsf{fma}\left(amt, 0.5 \cdot \mathsf{fma}\left(amt, u4 - \mathsf{fma}\left(3, u3, u1\right), \mathsf{fma}\left(2, u1, \mathsf{fma}\left(4, u3, -u4\right)\right)\right), 0.5 \cdot \left(u3 - u1\right)\right)\\

\mathbf{else}:\\
\;\;\;\;t_0\\


\end{array}
\end{array}

Derivation

Split input into 2 regimes
if u2 < -5e3 or 1.99999997e17 < u2
1. Initial program 65.1%
  \[7.62939453125 \cdot 10^{-6} \cdot \left(\left(\left(2 \cdot u2 + \left(u3 - u1\right) \cdot \left(amt \cdot 65536\right)\right) + \left(\left(\left(2 \cdot u1 - 5 \cdot u2\right) + 4 \cdot u3\right) - u4\right) \cdot \left({amt}^{2} \cdot 65536\right)\right) + \left(\left(\left(3 \cdot u2 - u1\right) - 3 \cdot u3\right) + u4\right) \cdot \left({amt}^{3} \cdot 65536\right)\right) \]
2. Add Preprocessing
3. Taylor expanded in amt around 0
  \[\leadsto \color{blue}{\frac{1}{65536} \cdot u2 + amt \cdot \left(\frac{1}{2} \cdot \left(u3 - u1\right) + amt \cdot \left(\frac{1}{2} \cdot \left(amt \cdot \left(\left(u4 + 3 \cdot u2\right) - \left(u1 + 3 \cdot u3\right)\right)\right) + \frac{1}{2} \cdot \left(\left(2 \cdot u1 + 4 \cdot u3\right) - \left(u4 + 5 \cdot u2\right)\right)\right)\right)} \]
4. Step-by-step derivation
  1. +-commutativeN/A
    \[\leadsto \color{blue}{amt \cdot \left(\frac{1}{2} \cdot \left(u3 - u1\right) + amt \cdot \left(\frac{1}{2} \cdot \left(amt \cdot \left(\left(u4 + 3 \cdot u2\right) - \left(u1 + 3 \cdot u3\right)\right)\right) + \frac{1}{2} \cdot \left(\left(2 \cdot u1 + 4 \cdot u3\right) - \left(u4 + 5 \cdot u2\right)\right)\right)\right) + \frac{1}{65536} \cdot u2} \]
  2. lower-fma.f32N/A
    \[\leadsto \color{blue}{\mathsf{fma}\left(amt, \frac{1}{2} \cdot \left(u3 - u1\right) + amt \cdot \left(\frac{1}{2} \cdot \left(amt \cdot \left(\left(u4 + 3 \cdot u2\right) - \left(u1 + 3 \cdot u3\right)\right)\right) + \frac{1}{2} \cdot \left(\left(2 \cdot u1 + 4 \cdot u3\right) - \left(u4 + 5 \cdot u2\right)\right)\right), \frac{1}{65536} \cdot u2\right)} \]
5. Applied rewrites92.7%
  \[\leadsto \color{blue}{\mathsf{fma}\left(amt, \mathsf{fma}\left(amt, 0.5 \cdot \mathsf{fma}\left(amt, \mathsf{fma}\left(3, u2 - u3, u4\right) - u1, \mathsf{fma}\left(2, u1, \mathsf{fma}\left(u3, 4, u2 \cdot -5\right)\right) - u4\right), \left(u3 - u1\right) \cdot 0.5\right), u2 \cdot 1.52587890625 \cdot 10^{-5}\right)} \]
6. Taylor expanded in u1 around 0
  \[\leadsto \mathsf{fma}\left(amt, \frac{1}{2} \cdot u3 + \color{blue}{\frac{1}{2} \cdot \left(amt \cdot \left(\left(-5 \cdot u2 + \left(4 \cdot u3 + amt \cdot \left(u4 + 3 \cdot \left(u2 - u3\right)\right)\right)\right) - u4\right)\right)}, u2 \cdot \frac{1}{65536}\right) \]
7. Step-by-step derivation
8. Applied rewrites83.8%
  \[\leadsto \mathsf{fma}\left(amt, 0.5 \cdot \color{blue}{\left(u3 + amt \cdot \mathsf{fma}\left(-5, u2, \mathsf{fma}\left(amt, \mathsf{fma}\left(3, u2 - u3, u4\right), 4 \cdot u3\right) - u4\right)\right)}, u2 \cdot 1.52587890625 \cdot 10^{-5}\right) \]
if -5e3 < u2 < 1.99999997e17
1. Initial program 81.3%
  \[7.62939453125 \cdot 10^{-6} \cdot \left(\left(\left(2 \cdot u2 + \left(u3 - u1\right) \cdot \left(amt \cdot 65536\right)\right) + \left(\left(\left(2 \cdot u1 - 5 \cdot u2\right) + 4 \cdot u3\right) - u4\right) \cdot \left({amt}^{2} \cdot 65536\right)\right) + \left(\left(\left(3 \cdot u2 - u1\right) - 3 \cdot u3\right) + u4\right) \cdot \left({amt}^{3} \cdot 65536\right)\right) \]
2. Add Preprocessing
3. Taylor expanded in u2 around 0
  \[\leadsto \color{blue}{\frac{1}{131072} \cdot \left(65536 \cdot \left(amt \cdot \left(u3 - u1\right)\right) + \left(65536 \cdot \left({amt}^{2} \cdot \left(\left(2 \cdot u1 + 4 \cdot u3\right) - u4\right)\right) + 65536 \cdot \left({amt}^{3} \cdot \left(u4 - \left(u1 + 3 \cdot u3\right)\right)\right)\right)\right)} \]
4. Step-by-step derivation
  1. +-commutativeN/A
    \[\leadsto \frac{1}{131072} \cdot \color{blue}{\left(\left(65536 \cdot \left({amt}^{2} \cdot \left(\left(2 \cdot u1 + 4 \cdot u3\right) - u4\right)\right) + 65536 \cdot \left({amt}^{3} \cdot \left(u4 - \left(u1 + 3 \cdot u3\right)\right)\right)\right) + 65536 \cdot \left(amt \cdot \left(u3 - u1\right)\right)\right)} \]
  2. distribute-lft-inN/A
    \[\leadsto \color{blue}{\frac{1}{131072} \cdot \left(65536 \cdot \left({amt}^{2} \cdot \left(\left(2 \cdot u1 + 4 \cdot u3\right) - u4\right)\right) + 65536 \cdot \left({amt}^{3} \cdot \left(u4 - \left(u1 + 3 \cdot u3\right)\right)\right)\right) + \frac{1}{131072} \cdot \left(65536 \cdot \left(amt \cdot \left(u3 - u1\right)\right)\right)} \]
  3. distribute-lft-outN/A
    \[\leadsto \frac{1}{131072} \cdot \color{blue}{\left(65536 \cdot \left({amt}^{2} \cdot \left(\left(2 \cdot u1 + 4 \cdot u3\right) - u4\right) + {amt}^{3} \cdot \left(u4 - \left(u1 + 3 \cdot u3\right)\right)\right)\right)} + \frac{1}{131072} \cdot \left(65536 \cdot \left(amt \cdot \left(u3 - u1\right)\right)\right) \]
  4. associate-*r*N/A
    \[\leadsto \color{blue}{\left(\frac{1}{131072} \cdot 65536\right) \cdot \left({amt}^{2} \cdot \left(\left(2 \cdot u1 + 4 \cdot u3\right) - u4\right) + {amt}^{3} \cdot \left(u4 - \left(u1 + 3 \cdot u3\right)\right)\right)} + \frac{1}{131072} \cdot \left(65536 \cdot \left(amt \cdot \left(u3 - u1\right)\right)\right) \]
  5. metadata-evalN/A
    \[\leadsto \color{blue}{\frac{1}{2}} \cdot \left({amt}^{2} \cdot \left(\left(2 \cdot u1 + 4 \cdot u3\right) - u4\right) + {amt}^{3} \cdot \left(u4 - \left(u1 + 3 \cdot u3\right)\right)\right) + \frac{1}{131072} \cdot \left(65536 \cdot \left(amt \cdot \left(u3 - u1\right)\right)\right) \]
  6. associate-*r*N/A
    \[\leadsto \frac{1}{2} \cdot \left({amt}^{2} \cdot \left(\left(2 \cdot u1 + 4 \cdot u3\right) - u4\right) + {amt}^{3} \cdot \left(u4 - \left(u1 + 3 \cdot u3\right)\right)\right) + \color{blue}{\left(\frac{1}{131072} \cdot 65536\right) \cdot \left(amt \cdot \left(u3 - u1\right)\right)} \]
  7. metadata-evalN/A
    \[\leadsto \frac{1}{2} \cdot \left({amt}^{2} \cdot \left(\left(2 \cdot u1 + 4 \cdot u3\right) - u4\right) + {amt}^{3} \cdot \left(u4 - \left(u1 + 3 \cdot u3\right)\right)\right) + \color{blue}{\frac{1}{2}} \cdot \left(amt \cdot \left(u3 - u1\right)\right) \]
  8. *-commutativeN/A
    \[\leadsto \frac{1}{2} \cdot \left({amt}^{2} \cdot \left(\left(2 \cdot u1 + 4 \cdot u3\right) - u4\right) + {amt}^{3} \cdot \left(u4 - \left(u1 + 3 \cdot u3\right)\right)\right) + \color{blue}{\left(amt \cdot \left(u3 - u1\right)\right) \cdot \frac{1}{2}} \]
  9. associate-*r*N/A
    \[\leadsto \frac{1}{2} \cdot \left({amt}^{2} \cdot \left(\left(2 \cdot u1 + 4 \cdot u3\right) - u4\right) + {amt}^{3} \cdot \left(u4 - \left(u1 + 3 \cdot u3\right)\right)\right) + \color{blue}{amt \cdot \left(\left(u3 - u1\right) \cdot \frac{1}{2}\right)} \]
  10. *-commutativeN/A
    \[\leadsto \frac{1}{2} \cdot \left({amt}^{2} \cdot \left(\left(2 \cdot u1 + 4 \cdot u3\right) - u4\right) + {amt}^{3} \cdot \left(u4 - \left(u1 + 3 \cdot u3\right)\right)\right) + amt \cdot \color{blue}{\left(\frac{1}{2} \cdot \left(u3 - u1\right)\right)} \]
5. Applied rewrites87.3%
  \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(amt \cdot \left(amt \cdot amt\right), u4 - \mathsf{fma}\left(3, u3, u1\right), amt \cdot \left(amt \cdot \mathsf{fma}\left(2, u1, u3 \cdot 4 - u4\right)\right)\right), \left(amt \cdot \left(u3 - u1\right)\right) \cdot 0.5\right)} \]
6. Taylor expanded in amt around 0
  \[\leadsto amt \cdot \color{blue}{\left(\frac{1}{2} \cdot \left(u3 - u1\right) + amt \cdot \left(\frac{1}{2} \cdot \left(amt \cdot \left(u4 - \left(u1 + 3 \cdot u3\right)\right)\right) + \frac{1}{2} \cdot \left(\left(2 \cdot u1 + 4 \cdot u3\right) - u4\right)\right)\right)} \]
7. Step-by-step derivation
8. Applied rewrites89.2%
  \[\leadsto amt \cdot \color{blue}{\mathsf{fma}\left(amt, 0.5 \cdot \mathsf{fma}\left(amt, u4 - \mathsf{fma}\left(3, u3, u1\right), \mathsf{fma}\left(2, u1, \mathsf{fma}\left(4, u3, -u4\right)\right)\right), 0.5 \cdot \left(u3 - u1\right)\right)} \]
Recombined 2 regimes into one program.
Final simplification87.3%
\[\leadsto \begin{array}{l} \mathbf{if}\;u2 \leq -5000:\\ \;\;\;\;\mathsf{fma}\left(amt, 0.5 \cdot \left(u3 + amt \cdot \mathsf{fma}\left(-5, u2, \mathsf{fma}\left(amt, \mathsf{fma}\left(3, u2 - u3, u4\right), u3 \cdot 4\right) - u4\right)\right), u2 \cdot 1.52587890625 \cdot 10^{-5}\right)\\ \mathbf{elif}\;u2 \leq 199999996861349900:\\ \;\;\;\;amt \cdot \mathsf{fma}\left(amt, 0.5 \cdot \mathsf{fma}\left(amt, u4 - \mathsf{fma}\left(3, u3, u1\right), \mathsf{fma}\left(2, u1, \mathsf{fma}\left(4, u3, -u4\right)\right)\right), 0.5 \cdot \left(u3 - u1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(amt, 0.5 \cdot \left(u3 + amt \cdot \mathsf{fma}\left(-5, u2, \mathsf{fma}\left(amt, \mathsf{fma}\left(3, u2 - u3, u4\right), u3 \cdot 4\right) - u4\right)\right), u2 \cdot 1.52587890625 \cdot 10^{-5}\right)\\ \end{array} \]
Add Preprocessing

Alternative 8: 79.8% accurate, 4.5× speedup?

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \left(u3 - u1\right)\\ \mathbf{if}\;u2 \leq -2.0000000276968557 \cdot 10^{+24}:\\ \;\;\;\;u2 \cdot \mathsf{fma}\left(0.5 \cdot \left(amt \cdot amt\right), \mathsf{fma}\left(3, amt, -5\right), 1.52587890625 \cdot 10^{-5}\right)\\ \mathbf{elif}\;u2 \leq 18000000404716257000:\\ \;\;\;\;amt \cdot \mathsf{fma}\left(amt, 0.5 \cdot \mathsf{fma}\left(amt, u4 - \mathsf{fma}\left(3, u3, u1\right), \mathsf{fma}\left(2, u1, \mathsf{fma}\left(4, u3, -u4\right)\right)\right), t_0\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(amt, \mathsf{fma}\left(amt, 0.5 \cdot \left(u2 \cdot \mathsf{fma}\left(3, amt, -5\right)\right), t_0\right), u2 \cdot 1.52587890625 \cdot 10^{-5}\right)\\ \end{array} \end{array} \]

(FPCore (u2 u3 u1 amt u4)
 :precision binary32
 (let* ((t_0 (* 0.5 (- u3 u1))))
   (if (<= u2 -2.0000000276968557e+24)
     (* u2 (fma (* 0.5 (* amt amt)) (fma 3.0 amt -5.0) 1.52587890625e-5))
     (if (<= u2 18000000404716257000.0)
       (*
        amt
        (fma
         amt
         (*
          0.5
          (fma amt (- u4 (fma 3.0 u3 u1)) (fma 2.0 u1 (fma 4.0 u3 (- u4)))))
         t_0))
       (fma
        amt
        (fma amt (* 0.5 (* u2 (fma 3.0 amt -5.0))) t_0)
        (* u2 1.52587890625e-5))))))

float code(float u2, float u3, float u1, float amt, float u4) {
	float t_0 = 0.5f * (u3 - u1);
	float tmp;
	if (u2 <= -2.0000000276968557e+24f) {
		tmp = u2 * fmaf((0.5f * (amt * amt)), fmaf(3.0f, amt, -5.0f), 1.52587890625e-5f);
	} else if (u2 <= 18000000404716257000.0f) {
		tmp = amt * fmaf(amt, (0.5f * fmaf(amt, (u4 - fmaf(3.0f, u3, u1)), fmaf(2.0f, u1, fmaf(4.0f, u3, -u4)))), t_0);
	} else {
		tmp = fmaf(amt, fmaf(amt, (0.5f * (u2 * fmaf(3.0f, amt, -5.0f))), t_0), (u2 * 1.52587890625e-5f));
	}
	return tmp;
}

function code(u2, u3, u1, amt, u4)
	t_0 = Float32(Float32(0.5) * Float32(u3 - u1))
	tmp = Float32(0.0)
	if (u2 <= Float32(-2.0000000276968557e+24))
		tmp = Float32(u2 * fma(Float32(Float32(0.5) * Float32(amt * amt)), fma(Float32(3.0), amt, Float32(-5.0)), Float32(1.52587890625e-5)));
	elseif (u2 <= Float32(18000000404716257000.0))
		tmp = Float32(amt * fma(amt, Float32(Float32(0.5) * fma(amt, Float32(u4 - fma(Float32(3.0), u3, u1)), fma(Float32(2.0), u1, fma(Float32(4.0), u3, Float32(-u4))))), t_0));
	else
		tmp = fma(amt, fma(amt, Float32(Float32(0.5) * Float32(u2 * fma(Float32(3.0), amt, Float32(-5.0)))), t_0), Float32(u2 * Float32(1.52587890625e-5)));
	end
	return tmp
end

\begin{array}{l}

\\
\begin{array}{l}
t_0 := 0.5 \cdot \left(u3 - u1\right)\\
\mathbf{if}\;u2 \leq -2.0000000276968557 \cdot 10^{+24}:\\
\;\;\;\;u2 \cdot \mathsf{fma}\left(0.5 \cdot \left(amt \cdot amt\right), \mathsf{fma}\left(3, amt, -5\right), 1.52587890625 \cdot 10^{-5}\right)\\

\mathbf{elif}\;u2 \leq 18000000404716257000:\\
\;\;\;\;amt \cdot \mathsf{fma}\left(amt, 0.5 \cdot \mathsf{fma}\left(amt, u4 - \mathsf{fma}\left(3, u3, u1\right), \mathsf{fma}\left(2, u1, \mathsf{fma}\left(4, u3, -u4\right)\right)\right), t_0\right)\\

\mathbf{else}:\\
\;\;\;\;\mathsf{fma}\left(amt, \mathsf{fma}\left(amt, 0.5 \cdot \left(u2 \cdot \mathsf{fma}\left(3, amt, -5\right)\right), t_0\right), u2 \cdot 1.52587890625 \cdot 10^{-5}\right)\\


\end{array}
\end{array}

Derivation

Split input into 3 regimes
if u2 < -2.00000003e24
1. Initial program 42.0%
  \[7.62939453125 \cdot 10^{-6} \cdot \left(\left(\left(2 \cdot u2 + \left(u3 - u1\right) \cdot \left(amt \cdot 65536\right)\right) + \left(\left(\left(2 \cdot u1 - 5 \cdot u2\right) + 4 \cdot u3\right) - u4\right) \cdot \left({amt}^{2} \cdot 65536\right)\right) + \left(\left(\left(3 \cdot u2 - u1\right) - 3 \cdot u3\right) + u4\right) \cdot \left({amt}^{3} \cdot 65536\right)\right) \]
2. Add Preprocessing
3. Taylor expanded in amt around 0
  \[\leadsto \color{blue}{\frac{1}{65536} \cdot u2 + amt \cdot \left(\frac{1}{2} \cdot \left(u3 - u1\right) + amt \cdot \left(\frac{1}{2} \cdot \left(amt \cdot \left(\left(u4 + 3 \cdot u2\right) - \left(u1 + 3 \cdot u3\right)\right)\right) + \frac{1}{2} \cdot \left(\left(2 \cdot u1 + 4 \cdot u3\right) - \left(u4 + 5 \cdot u2\right)\right)\right)\right)} \]
4. Step-by-step derivation
  1. +-commutativeN/A
    \[\leadsto \color{blue}{amt \cdot \left(\frac{1}{2} \cdot \left(u3 - u1\right) + amt \cdot \left(\frac{1}{2} \cdot \left(amt \cdot \left(\left(u4 + 3 \cdot u2\right) - \left(u1 + 3 \cdot u3\right)\right)\right) + \frac{1}{2} \cdot \left(\left(2 \cdot u1 + 4 \cdot u3\right) - \left(u4 + 5 \cdot u2\right)\right)\right)\right) + \frac{1}{65536} \cdot u2} \]
  2. lower-fma.f32N/A
    \[\leadsto \color{blue}{\mathsf{fma}\left(amt, \frac{1}{2} \cdot \left(u3 - u1\right) + amt \cdot \left(\frac{1}{2} \cdot \left(amt \cdot \left(\left(u4 + 3 \cdot u2\right) - \left(u1 + 3 \cdot u3\right)\right)\right) + \frac{1}{2} \cdot \left(\left(2 \cdot u1 + 4 \cdot u3\right) - \left(u4 + 5 \cdot u2\right)\right)\right), \frac{1}{65536} \cdot u2\right)} \]
5. Applied rewrites86.9%
  \[\leadsto \color{blue}{\mathsf{fma}\left(amt, \mathsf{fma}\left(amt, 0.5 \cdot \mathsf{fma}\left(amt, \mathsf{fma}\left(3, u2 - u3, u4\right) - u1, \mathsf{fma}\left(2, u1, \mathsf{fma}\left(u3, 4, u2 \cdot -5\right)\right) - u4\right), \left(u3 - u1\right) \cdot 0.5\right), u2 \cdot 1.52587890625 \cdot 10^{-5}\right)} \]
6. Taylor expanded in u2 around inf
  \[\leadsto u2 \cdot \color{blue}{\left(\frac{1}{65536} + \frac{1}{2} \cdot \left({amt}^{2} \cdot \left(3 \cdot amt - 5\right)\right)\right)} \]
7. Step-by-step derivation
8. Applied rewrites78.3%
  \[\leadsto u2 \cdot \color{blue}{\mathsf{fma}\left(0.5 \cdot \left(amt \cdot amt\right), \mathsf{fma}\left(3, amt, -5\right), 1.52587890625 \cdot 10^{-5}\right)} \]
if -2.00000003e24 < u2 < 1.80000004e19
1. Initial program 80.2%
  \[7.62939453125 \cdot 10^{-6} \cdot \left(\left(\left(2 \cdot u2 + \left(u3 - u1\right) \cdot \left(amt \cdot 65536\right)\right) + \left(\left(\left(2 \cdot u1 - 5 \cdot u2\right) + 4 \cdot u3\right) - u4\right) \cdot \left({amt}^{2} \cdot 65536\right)\right) + \left(\left(\left(3 \cdot u2 - u1\right) - 3 \cdot u3\right) + u4\right) \cdot \left({amt}^{3} \cdot 65536\right)\right) \]
2. Add Preprocessing
3. Taylor expanded in u2 around 0
  \[\leadsto \color{blue}{\frac{1}{131072} \cdot \left(65536 \cdot \left(amt \cdot \left(u3 - u1\right)\right) + \left(65536 \cdot \left({amt}^{2} \cdot \left(\left(2 \cdot u1 + 4 \cdot u3\right) - u4\right)\right) + 65536 \cdot \left({amt}^{3} \cdot \left(u4 - \left(u1 + 3 \cdot u3\right)\right)\right)\right)\right)} \]
4. Step-by-step derivation
  1. +-commutativeN/A
    \[\leadsto \frac{1}{131072} \cdot \color{blue}{\left(\left(65536 \cdot \left({amt}^{2} \cdot \left(\left(2 \cdot u1 + 4 \cdot u3\right) - u4\right)\right) + 65536 \cdot \left({amt}^{3} \cdot \left(u4 - \left(u1 + 3 \cdot u3\right)\right)\right)\right) + 65536 \cdot \left(amt \cdot \left(u3 - u1\right)\right)\right)} \]
  2. distribute-lft-inN/A
    \[\leadsto \color{blue}{\frac{1}{131072} \cdot \left(65536 \cdot \left({amt}^{2} \cdot \left(\left(2 \cdot u1 + 4 \cdot u3\right) - u4\right)\right) + 65536 \cdot \left({amt}^{3} \cdot \left(u4 - \left(u1 + 3 \cdot u3\right)\right)\right)\right) + \frac{1}{131072} \cdot \left(65536 \cdot \left(amt \cdot \left(u3 - u1\right)\right)\right)} \]
  3. distribute-lft-outN/A
    \[\leadsto \frac{1}{131072} \cdot \color{blue}{\left(65536 \cdot \left({amt}^{2} \cdot \left(\left(2 \cdot u1 + 4 \cdot u3\right) - u4\right) + {amt}^{3} \cdot \left(u4 - \left(u1 + 3 \cdot u3\right)\right)\right)\right)} + \frac{1}{131072} \cdot \left(65536 \cdot \left(amt \cdot \left(u3 - u1\right)\right)\right) \]
  4. associate-*r*N/A
    \[\leadsto \color{blue}{\left(\frac{1}{131072} \cdot 65536\right) \cdot \left({amt}^{2} \cdot \left(\left(2 \cdot u1 + 4 \cdot u3\right) - u4\right) + {amt}^{3} \cdot \left(u4 - \left(u1 + 3 \cdot u3\right)\right)\right)} + \frac{1}{131072} \cdot \left(65536 \cdot \left(amt \cdot \left(u3 - u1\right)\right)\right) \]
  5. metadata-evalN/A
    \[\leadsto \color{blue}{\frac{1}{2}} \cdot \left({amt}^{2} \cdot \left(\left(2 \cdot u1 + 4 \cdot u3\right) - u4\right) + {amt}^{3} \cdot \left(u4 - \left(u1 + 3 \cdot u3\right)\right)\right) + \frac{1}{131072} \cdot \left(65536 \cdot \left(amt \cdot \left(u3 - u1\right)\right)\right) \]
  6. associate-*r*N/A
    \[\leadsto \frac{1}{2} \cdot \left({amt}^{2} \cdot \left(\left(2 \cdot u1 + 4 \cdot u3\right) - u4\right) + {amt}^{3} \cdot \left(u4 - \left(u1 + 3 \cdot u3\right)\right)\right) + \color{blue}{\left(\frac{1}{131072} \cdot 65536\right) \cdot \left(amt \cdot \left(u3 - u1\right)\right)} \]
  7. metadata-evalN/A
    \[\leadsto \frac{1}{2} \cdot \left({amt}^{2} \cdot \left(\left(2 \cdot u1 + 4 \cdot u3\right) - u4\right) + {amt}^{3} \cdot \left(u4 - \left(u1 + 3 \cdot u3\right)\right)\right) + \color{blue}{\frac{1}{2}} \cdot \left(amt \cdot \left(u3 - u1\right)\right) \]
  8. *-commutativeN/A
    \[\leadsto \frac{1}{2} \cdot \left({amt}^{2} \cdot \left(\left(2 \cdot u1 + 4 \cdot u3\right) - u4\right) + {amt}^{3} \cdot \left(u4 - \left(u1 + 3 \cdot u3\right)\right)\right) + \color{blue}{\left(amt \cdot \left(u3 - u1\right)\right) \cdot \frac{1}{2}} \]
  9. associate-*r*N/A
    \[\leadsto \frac{1}{2} \cdot \left({amt}^{2} \cdot \left(\left(2 \cdot u1 + 4 \cdot u3\right) - u4\right) + {amt}^{3} \cdot \left(u4 - \left(u1 + 3 \cdot u3\right)\right)\right) + \color{blue}{amt \cdot \left(\left(u3 - u1\right) \cdot \frac{1}{2}\right)} \]
  10. *-commutativeN/A
    \[\leadsto \frac{1}{2} \cdot \left({amt}^{2} \cdot \left(\left(2 \cdot u1 + 4 \cdot u3\right) - u4\right) + {amt}^{3} \cdot \left(u4 - \left(u1 + 3 \cdot u3\right)\right)\right) + amt \cdot \color{blue}{\left(\frac{1}{2} \cdot \left(u3 - u1\right)\right)} \]
5. Applied rewrites80.9%
  \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(amt \cdot \left(amt \cdot amt\right), u4 - \mathsf{fma}\left(3, u3, u1\right), amt \cdot \left(amt \cdot \mathsf{fma}\left(2, u1, u3 \cdot 4 - u4\right)\right)\right), \left(amt \cdot \left(u3 - u1\right)\right) \cdot 0.5\right)} \]
6. Taylor expanded in amt around 0
  \[\leadsto amt \cdot \color{blue}{\left(\frac{1}{2} \cdot \left(u3 - u1\right) + amt \cdot \left(\frac{1}{2} \cdot \left(amt \cdot \left(u4 - \left(u1 + 3 \cdot u3\right)\right)\right) + \frac{1}{2} \cdot \left(\left(2 \cdot u1 + 4 \cdot u3\right) - u4\right)\right)\right)} \]
7. Step-by-step derivation
8. Applied rewrites82.6%
  \[\leadsto amt \cdot \color{blue}{\mathsf{fma}\left(amt, 0.5 \cdot \mathsf{fma}\left(amt, u4 - \mathsf{fma}\left(3, u3, u1\right), \mathsf{fma}\left(2, u1, \mathsf{fma}\left(4, u3, -u4\right)\right)\right), 0.5 \cdot \left(u3 - u1\right)\right)} \]
if 1.80000004e19 < u2
1. Initial program 72.4%
  \[7.62939453125 \cdot 10^{-6} \cdot \left(\left(\left(2 \cdot u2 + \left(u3 - u1\right) \cdot \left(amt \cdot 65536\right)\right) + \left(\left(\left(2 \cdot u1 - 5 \cdot u2\right) + 4 \cdot u3\right) - u4\right) \cdot \left({amt}^{2} \cdot 65536\right)\right) + \left(\left(\left(3 \cdot u2 - u1\right) - 3 \cdot u3\right) + u4\right) \cdot \left({amt}^{3} \cdot 65536\right)\right) \]
2. Add Preprocessing
3. Taylor expanded in amt around 0
  \[\leadsto \color{blue}{\frac{1}{65536} \cdot u2 + amt \cdot \left(\frac{1}{2} \cdot \left(u3 - u1\right) + amt \cdot \left(\frac{1}{2} \cdot \left(amt \cdot \left(\left(u4 + 3 \cdot u2\right) - \left(u1 + 3 \cdot u3\right)\right)\right) + \frac{1}{2} \cdot \left(\left(2 \cdot u1 + 4 \cdot u3\right) - \left(u4 + 5 \cdot u2\right)\right)\right)\right)} \]
4. Step-by-step derivation
  1. +-commutativeN/A
    \[\leadsto \color{blue}{amt \cdot \left(\frac{1}{2} \cdot \left(u3 - u1\right) + amt \cdot \left(\frac{1}{2} \cdot \left(amt \cdot \left(\left(u4 + 3 \cdot u2\right) - \left(u1 + 3 \cdot u3\right)\right)\right) + \frac{1}{2} \cdot \left(\left(2 \cdot u1 + 4 \cdot u3\right) - \left(u4 + 5 \cdot u2\right)\right)\right)\right) + \frac{1}{65536} \cdot u2} \]
  2. lower-fma.f32N/A
    \[\leadsto \color{blue}{\mathsf{fma}\left(amt, \frac{1}{2} \cdot \left(u3 - u1\right) + amt \cdot \left(\frac{1}{2} \cdot \left(amt \cdot \left(\left(u4 + 3 \cdot u2\right) - \left(u1 + 3 \cdot u3\right)\right)\right) + \frac{1}{2} \cdot \left(\left(2 \cdot u1 + 4 \cdot u3\right) - \left(u4 + 5 \cdot u2\right)\right)\right), \frac{1}{65536} \cdot u2\right)} \]
5. Applied rewrites95.5%
  \[\leadsto \color{blue}{\mathsf{fma}\left(amt, \mathsf{fma}\left(amt, 0.5 \cdot \mathsf{fma}\left(amt, \mathsf{fma}\left(3, u2 - u3, u4\right) - u1, \mathsf{fma}\left(2, u1, \mathsf{fma}\left(u3, 4, u2 \cdot -5\right)\right) - u4\right), \left(u3 - u1\right) \cdot 0.5\right), u2 \cdot 1.52587890625 \cdot 10^{-5}\right)} \]
6. Taylor expanded in u2 around inf
  \[\leadsto \mathsf{fma}\left(amt, \mathsf{fma}\left(amt, \frac{1}{2} \cdot \left(u2 \cdot \color{blue}{\left(3 \cdot amt - 5\right)}\right), \left(u3 - u1\right) \cdot \frac{1}{2}\right), u2 \cdot \frac{1}{65536}\right) \]
7. Step-by-step derivation
8. Applied rewrites80.9%
  \[\leadsto \mathsf{fma}\left(amt, \mathsf{fma}\left(amt, 0.5 \cdot \left(u2 \cdot \color{blue}{\mathsf{fma}\left(3, amt, -5\right)}\right), \left(u3 - u1\right) \cdot 0.5\right), u2 \cdot 1.52587890625 \cdot 10^{-5}\right) \]
Recombined 3 regimes into one program.
Final simplification82.0%
\[\leadsto \begin{array}{l} \mathbf{if}\;u2 \leq -2.0000000276968557 \cdot 10^{+24}:\\ \;\;\;\;u2 \cdot \mathsf{fma}\left(0.5 \cdot \left(amt \cdot amt\right), \mathsf{fma}\left(3, amt, -5\right), 1.52587890625 \cdot 10^{-5}\right)\\ \mathbf{elif}\;u2 \leq 18000000404716257000:\\ \;\;\;\;amt \cdot \mathsf{fma}\left(amt, 0.5 \cdot \mathsf{fma}\left(amt, u4 - \mathsf{fma}\left(3, u3, u1\right), \mathsf{fma}\left(2, u1, \mathsf{fma}\left(4, u3, -u4\right)\right)\right), 0.5 \cdot \left(u3 - u1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(amt, \mathsf{fma}\left(amt, 0.5 \cdot \left(u2 \cdot \mathsf{fma}\left(3, amt, -5\right)\right), 0.5 \cdot \left(u3 - u1\right)\right), u2 \cdot 1.52587890625 \cdot 10^{-5}\right)\\ \end{array} \]
Add Preprocessing

Alternative 9: 79.6% accurate, 4.9× speedup?

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u2 \leq -2.0000000276968557 \cdot 10^{+24}:\\ \;\;\;\;u2 \cdot \mathsf{fma}\left(0.5 \cdot \left(amt \cdot amt\right), \mathsf{fma}\left(3, amt, -5\right), 1.52587890625 \cdot 10^{-5}\right)\\ \mathbf{elif}\;u2 \leq 18000000404716257000:\\ \;\;\;\;amt \cdot \left(0.5 \cdot \mathsf{fma}\left(amt, \mathsf{fma}\left(2, u1, \mathsf{fma}\left(amt, \mathsf{fma}\left(-3, u3, u4\right) - u1, u3 \cdot 4\right) - u4\right), u3 - u1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(amt, \mathsf{fma}\left(amt, 0.5 \cdot \left(u2 \cdot \mathsf{fma}\left(3, amt, -5\right)\right), 0.5 \cdot \left(u3 - u1\right)\right), u2 \cdot 1.52587890625 \cdot 10^{-5}\right)\\ \end{array} \end{array} \]

(FPCore (u2 u3 u1 amt u4)
 :precision binary32
 (if (<= u2 -2.0000000276968557e+24)
   (* u2 (fma (* 0.5 (* amt amt)) (fma 3.0 amt -5.0) 1.52587890625e-5))
   (if (<= u2 18000000404716257000.0)
     (*
      amt
      (*
       0.5
       (fma
        amt
        (fma 2.0 u1 (- (fma amt (- (fma -3.0 u3 u4) u1) (* u3 4.0)) u4))
        (- u3 u1))))
     (fma
      amt
      (fma amt (* 0.5 (* u2 (fma 3.0 amt -5.0))) (* 0.5 (- u3 u1)))
      (* u2 1.52587890625e-5)))))

float code(float u2, float u3, float u1, float amt, float u4) {
	float tmp;
	if (u2 <= -2.0000000276968557e+24f) {
		tmp = u2 * fmaf((0.5f * (amt * amt)), fmaf(3.0f, amt, -5.0f), 1.52587890625e-5f);
	} else if (u2 <= 18000000404716257000.0f) {
		tmp = amt * (0.5f * fmaf(amt, fmaf(2.0f, u1, (fmaf(amt, (fmaf(-3.0f, u3, u4) - u1), (u3 * 4.0f)) - u4)), (u3 - u1)));
	} else {
		tmp = fmaf(amt, fmaf(amt, (0.5f * (u2 * fmaf(3.0f, amt, -5.0f))), (0.5f * (u3 - u1))), (u2 * 1.52587890625e-5f));
	}
	return tmp;
}

function code(u2, u3, u1, amt, u4)
	tmp = Float32(0.0)
	if (u2 <= Float32(-2.0000000276968557e+24))
		tmp = Float32(u2 * fma(Float32(Float32(0.5) * Float32(amt * amt)), fma(Float32(3.0), amt, Float32(-5.0)), Float32(1.52587890625e-5)));
	elseif (u2 <= Float32(18000000404716257000.0))
		tmp = Float32(amt * Float32(Float32(0.5) * fma(amt, fma(Float32(2.0), u1, Float32(fma(amt, Float32(fma(Float32(-3.0), u3, u4) - u1), Float32(u3 * Float32(4.0))) - u4)), Float32(u3 - u1))));
	else
		tmp = fma(amt, fma(amt, Float32(Float32(0.5) * Float32(u2 * fma(Float32(3.0), amt, Float32(-5.0)))), Float32(Float32(0.5) * Float32(u3 - u1))), Float32(u2 * Float32(1.52587890625e-5)));
	end
	return tmp
end

\begin{array}{l}

\\
\begin{array}{l}
\mathbf{if}\;u2 \leq -2.0000000276968557 \cdot 10^{+24}:\\
\;\;\;\;u2 \cdot \mathsf{fma}\left(0.5 \cdot \left(amt \cdot amt\right), \mathsf{fma}\left(3, amt, -5\right), 1.52587890625 \cdot 10^{-5}\right)\\

\mathbf{elif}\;u2 \leq 18000000404716257000:\\
\;\;\;\;amt \cdot \left(0.5 \cdot \mathsf{fma}\left(amt, \mathsf{fma}\left(2, u1, \mathsf{fma}\left(amt, \mathsf{fma}\left(-3, u3, u4\right) - u1, u3 \cdot 4\right) - u4\right), u3 - u1\right)\right)\\

\mathbf{else}:\\
\;\;\;\;\mathsf{fma}\left(amt, \mathsf{fma}\left(amt, 0.5 \cdot \left(u2 \cdot \mathsf{fma}\left(3, amt, -5\right)\right), 0.5 \cdot \left(u3 - u1\right)\right), u2 \cdot 1.52587890625 \cdot 10^{-5}\right)\\


\end{array}
\end{array}

Derivation

Split input into 3 regimes
if u2 < -2.00000003e24
1. Initial program 42.0%
  \[7.62939453125 \cdot 10^{-6} \cdot \left(\left(\left(2 \cdot u2 + \left(u3 - u1\right) \cdot \left(amt \cdot 65536\right)\right) + \left(\left(\left(2 \cdot u1 - 5 \cdot u2\right) + 4 \cdot u3\right) - u4\right) \cdot \left({amt}^{2} \cdot 65536\right)\right) + \left(\left(\left(3 \cdot u2 - u1\right) - 3 \cdot u3\right) + u4\right) \cdot \left({amt}^{3} \cdot 65536\right)\right) \]
2. Add Preprocessing
3. Taylor expanded in amt around 0
  \[\leadsto \color{blue}{\frac{1}{65536} \cdot u2 + amt \cdot \left(\frac{1}{2} \cdot \left(u3 - u1\right) + amt \cdot \left(\frac{1}{2} \cdot \left(amt \cdot \left(\left(u4 + 3 \cdot u2\right) - \left(u1 + 3 \cdot u3\right)\right)\right) + \frac{1}{2} \cdot \left(\left(2 \cdot u1 + 4 \cdot u3\right) - \left(u4 + 5 \cdot u2\right)\right)\right)\right)} \]
4. Step-by-step derivation
  1. +-commutativeN/A
    \[\leadsto \color{blue}{amt \cdot \left(\frac{1}{2} \cdot \left(u3 - u1\right) + amt \cdot \left(\frac{1}{2} \cdot \left(amt \cdot \left(\left(u4 + 3 \cdot u2\right) - \left(u1 + 3 \cdot u3\right)\right)\right) + \frac{1}{2} \cdot \left(\left(2 \cdot u1 + 4 \cdot u3\right) - \left(u4 + 5 \cdot u2\right)\right)\right)\right) + \frac{1}{65536} \cdot u2} \]
  2. lower-fma.f32N/A
    \[\leadsto \color{blue}{\mathsf{fma}\left(amt, \frac{1}{2} \cdot \left(u3 - u1\right) + amt \cdot \left(\frac{1}{2} \cdot \left(amt \cdot \left(\left(u4 + 3 \cdot u2\right) - \left(u1 + 3 \cdot u3\right)\right)\right) + \frac{1}{2} \cdot \left(\left(2 \cdot u1 + 4 \cdot u3\right) - \left(u4 + 5 \cdot u2\right)\right)\right), \frac{1}{65536} \cdot u2\right)} \]
5. Applied rewrites86.9%
  \[\leadsto \color{blue}{\mathsf{fma}\left(amt, \mathsf{fma}\left(amt, 0.5 \cdot \mathsf{fma}\left(amt, \mathsf{fma}\left(3, u2 - u3, u4\right) - u1, \mathsf{fma}\left(2, u1, \mathsf{fma}\left(u3, 4, u2 \cdot -5\right)\right) - u4\right), \left(u3 - u1\right) \cdot 0.5\right), u2 \cdot 1.52587890625 \cdot 10^{-5}\right)} \]
6. Taylor expanded in u2 around inf
  \[\leadsto u2 \cdot \color{blue}{\left(\frac{1}{65536} + \frac{1}{2} \cdot \left({amt}^{2} \cdot \left(3 \cdot amt - 5\right)\right)\right)} \]
7. Step-by-step derivation
8. Applied rewrites78.3%
  \[\leadsto u2 \cdot \color{blue}{\mathsf{fma}\left(0.5 \cdot \left(amt \cdot amt\right), \mathsf{fma}\left(3, amt, -5\right), 1.52587890625 \cdot 10^{-5}\right)} \]
if -2.00000003e24 < u2 < 1.80000004e19
1. Initial program 80.2%
  \[7.62939453125 \cdot 10^{-6} \cdot \left(\left(\left(2 \cdot u2 + \left(u3 - u1\right) \cdot \left(amt \cdot 65536\right)\right) + \left(\left(\left(2 \cdot u1 - 5 \cdot u2\right) + 4 \cdot u3\right) - u4\right) \cdot \left({amt}^{2} \cdot 65536\right)\right) + \left(\left(\left(3 \cdot u2 - u1\right) - 3 \cdot u3\right) + u4\right) \cdot \left({amt}^{3} \cdot 65536\right)\right) \]
2. Add Preprocessing
3. Taylor expanded in amt around 0
  \[\leadsto \color{blue}{\frac{1}{65536} \cdot u2 + amt \cdot \left(\frac{1}{2} \cdot \left(u3 - u1\right) + amt \cdot \left(\frac{1}{2} \cdot \left(amt \cdot \left(\left(u4 + 3 \cdot u2\right) - \left(u1 + 3 \cdot u3\right)\right)\right) + \frac{1}{2} \cdot \left(\left(2 \cdot u1 + 4 \cdot u3\right) - \left(u4 + 5 \cdot u2\right)\right)\right)\right)} \]
4. Step-by-step derivation
  1. +-commutativeN/A
    \[\leadsto \color{blue}{amt \cdot \left(\frac{1}{2} \cdot \left(u3 - u1\right) + amt \cdot \left(\frac{1}{2} \cdot \left(amt \cdot \left(\left(u4 + 3 \cdot u2\right) - \left(u1 + 3 \cdot u3\right)\right)\right) + \frac{1}{2} \cdot \left(\left(2 \cdot u1 + 4 \cdot u3\right) - \left(u4 + 5 \cdot u2\right)\right)\right)\right) + \frac{1}{65536} \cdot u2} \]
  2. lower-fma.f32N/A
    \[\leadsto \color{blue}{\mathsf{fma}\left(amt, \frac{1}{2} \cdot \left(u3 - u1\right) + amt \cdot \left(\frac{1}{2} \cdot \left(amt \cdot \left(\left(u4 + 3 \cdot u2\right) - \left(u1 + 3 \cdot u3\right)\right)\right) + \frac{1}{2} \cdot \left(\left(2 \cdot u1 + 4 \cdot u3\right) - \left(u4 + 5 \cdot u2\right)\right)\right), \frac{1}{65536} \cdot u2\right)} \]
5. Applied rewrites95.9%
  \[\leadsto \color{blue}{\mathsf{fma}\left(amt, \mathsf{fma}\left(amt, 0.5 \cdot \mathsf{fma}\left(amt, \mathsf{fma}\left(3, u2 - u3, u4\right) - u1, \mathsf{fma}\left(2, u1, \mathsf{fma}\left(u3, 4, u2 \cdot -5\right)\right) - u4\right), \left(u3 - u1\right) \cdot 0.5\right), u2 \cdot 1.52587890625 \cdot 10^{-5}\right)} \]
6. Taylor expanded in u2 around 0
  \[\leadsto amt \cdot \color{blue}{\left(\frac{1}{2} \cdot \left(amt \cdot \left(\left(2 \cdot u1 + \left(4 \cdot u3 + amt \cdot \left(\left(u4 + -3 \cdot u3\right) - u1\right)\right)\right) - u4\right)\right) + \frac{1}{2} \cdot \left(u3 - u1\right)\right)} \]
7. Step-by-step derivation
8. Applied rewrites82.0%
  \[\leadsto amt \cdot \color{blue}{\left(0.5 \cdot \mathsf{fma}\left(amt, \mathsf{fma}\left(2, u1, \mathsf{fma}\left(amt, \mathsf{fma}\left(-3, u3, u4\right) - u1, 4 \cdot u3\right) - u4\right), u3 - u1\right)\right)} \]
if 1.80000004e19 < u2
1. Initial program 72.4%
  \[7.62939453125 \cdot 10^{-6} \cdot \left(\left(\left(2 \cdot u2 + \left(u3 - u1\right) \cdot \left(amt \cdot 65536\right)\right) + \left(\left(\left(2 \cdot u1 - 5 \cdot u2\right) + 4 \cdot u3\right) - u4\right) \cdot \left({amt}^{2} \cdot 65536\right)\right) + \left(\left(\left(3 \cdot u2 - u1\right) - 3 \cdot u3\right) + u4\right) \cdot \left({amt}^{3} \cdot 65536\right)\right) \]
2. Add Preprocessing
3. Taylor expanded in amt around 0
  \[\leadsto \color{blue}{\frac{1}{65536} \cdot u2 + amt \cdot \left(\frac{1}{2} \cdot \left(u3 - u1\right) + amt \cdot \left(\frac{1}{2} \cdot \left(amt \cdot \left(\left(u4 + 3 \cdot u2\right) - \left(u1 + 3 \cdot u3\right)\right)\right) + \frac{1}{2} \cdot \left(\left(2 \cdot u1 + 4 \cdot u3\right) - \left(u4 + 5 \cdot u2\right)\right)\right)\right)} \]
4. Step-by-step derivation
  1. +-commutativeN/A
    \[\leadsto \color{blue}{amt \cdot \left(\frac{1}{2} \cdot \left(u3 - u1\right) + amt \cdot \left(\frac{1}{2} \cdot \left(amt \cdot \left(\left(u4 + 3 \cdot u2\right) - \left(u1 + 3 \cdot u3\right)\right)\right) + \frac{1}{2} \cdot \left(\left(2 \cdot u1 + 4 \cdot u3\right) - \left(u4 + 5 \cdot u2\right)\right)\right)\right) + \frac{1}{65536} \cdot u2} \]
  2. lower-fma.f32N/A
    \[\leadsto \color{blue}{\mathsf{fma}\left(amt, \frac{1}{2} \cdot \left(u3 - u1\right) + amt \cdot \left(\frac{1}{2} \cdot \left(amt \cdot \left(\left(u4 + 3 \cdot u2\right) - \left(u1 + 3 \cdot u3\right)\right)\right) + \frac{1}{2} \cdot \left(\left(2 \cdot u1 + 4 \cdot u3\right) - \left(u4 + 5 \cdot u2\right)\right)\right), \frac{1}{65536} \cdot u2\right)} \]
5. Applied rewrites95.5%
  \[\leadsto \color{blue}{\mathsf{fma}\left(amt, \mathsf{fma}\left(amt, 0.5 \cdot \mathsf{fma}\left(amt, \mathsf{fma}\left(3, u2 - u3, u4\right) - u1, \mathsf{fma}\left(2, u1, \mathsf{fma}\left(u3, 4, u2 \cdot -5\right)\right) - u4\right), \left(u3 - u1\right) \cdot 0.5\right), u2 \cdot 1.52587890625 \cdot 10^{-5}\right)} \]
6. Taylor expanded in u2 around inf
  \[\leadsto \mathsf{fma}\left(amt, \mathsf{fma}\left(amt, \frac{1}{2} \cdot \left(u2 \cdot \color{blue}{\left(3 \cdot amt - 5\right)}\right), \left(u3 - u1\right) \cdot \frac{1}{2}\right), u2 \cdot \frac{1}{65536}\right) \]
7. Step-by-step derivation
8. Applied rewrites80.9%
  \[\leadsto \mathsf{fma}\left(amt, \mathsf{fma}\left(amt, 0.5 \cdot \left(u2 \cdot \color{blue}{\mathsf{fma}\left(3, amt, -5\right)}\right), \left(u3 - u1\right) \cdot 0.5\right), u2 \cdot 1.52587890625 \cdot 10^{-5}\right) \]
Recombined 3 regimes into one program.
Final simplification81.5%
\[\leadsto \begin{array}{l} \mathbf{if}\;u2 \leq -2.0000000276968557 \cdot 10^{+24}:\\ \;\;\;\;u2 \cdot \mathsf{fma}\left(0.5 \cdot \left(amt \cdot amt\right), \mathsf{fma}\left(3, amt, -5\right), 1.52587890625 \cdot 10^{-5}\right)\\ \mathbf{elif}\;u2 \leq 18000000404716257000:\\ \;\;\;\;amt \cdot \left(0.5 \cdot \mathsf{fma}\left(amt, \mathsf{fma}\left(2, u1, \mathsf{fma}\left(amt, \mathsf{fma}\left(-3, u3, u4\right) - u1, u3 \cdot 4\right) - u4\right), u3 - u1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(amt, \mathsf{fma}\left(amt, 0.5 \cdot \left(u2 \cdot \mathsf{fma}\left(3, amt, -5\right)\right), 0.5 \cdot \left(u3 - u1\right)\right), u2 \cdot 1.52587890625 \cdot 10^{-5}\right)\\ \end{array} \]
Add Preprocessing

Alternative 10: 47.5% accurate, 5.5× speedup?

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \left(u3 - u1\right)\\ t_1 := \mathsf{fma}\left(amt, \mathsf{fma}\left(amt, 0.5 \cdot \left(u2 \cdot \mathsf{fma}\left(3, amt, -5\right)\right), t_0\right), u2 \cdot 1.52587890625 \cdot 10^{-5}\right)\\ \mathbf{if}\;u2 \leq -1370:\\ \;\;\;\;t_1\\ \mathbf{elif}\;u2 \leq 2000000:\\ \;\;\;\;\mathsf{fma}\left(amt, \mathsf{fma}\left(amt, 0.5 \cdot \left(u4 \cdot \left(amt + -1\right)\right), t_0\right), u2 \cdot 1.52587890625 \cdot 10^{-5}\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

(FPCore (u2 u3 u1 amt u4)
 :precision binary32
 (let* ((t_0 (* 0.5 (- u3 u1)))
        (t_1
         (fma
          amt
          (fma amt (* 0.5 (* u2 (fma 3.0 amt -5.0))) t_0)
          (* u2 1.52587890625e-5))))
   (if (<= u2 -1370.0)
     t_1
     (if (<= u2 2000000.0)
       (fma
        amt
        (fma amt (* 0.5 (* u4 (+ amt -1.0))) t_0)
        (* u2 1.52587890625e-5))
       t_1))))

float code(float u2, float u3, float u1, float amt, float u4) {
	float t_0 = 0.5f * (u3 - u1);
	float t_1 = fmaf(amt, fmaf(amt, (0.5f * (u2 * fmaf(3.0f, amt, -5.0f))), t_0), (u2 * 1.52587890625e-5f));
	float tmp;
	if (u2 <= -1370.0f) {
		tmp = t_1;
	} else if (u2 <= 2000000.0f) {
		tmp = fmaf(amt, fmaf(amt, (0.5f * (u4 * (amt + -1.0f))), t_0), (u2 * 1.52587890625e-5f));
	} else {
		tmp = t_1;
	}
	return tmp;
}

function code(u2, u3, u1, amt, u4)
	t_0 = Float32(Float32(0.5) * Float32(u3 - u1))
	t_1 = fma(amt, fma(amt, Float32(Float32(0.5) * Float32(u2 * fma(Float32(3.0), amt, Float32(-5.0)))), t_0), Float32(u2 * Float32(1.52587890625e-5)))
	tmp = Float32(0.0)
	if (u2 <= Float32(-1370.0))
		tmp = t_1;
	elseif (u2 <= Float32(2000000.0))
		tmp = fma(amt, fma(amt, Float32(Float32(0.5) * Float32(u4 * Float32(amt + Float32(-1.0)))), t_0), Float32(u2 * Float32(1.52587890625e-5)));
	else
		tmp = t_1;
	end
	return tmp
end

\begin{array}{l}

\\
\begin{array}{l}
t_0 := 0.5 \cdot \left(u3 - u1\right)\\
t_1 := \mathsf{fma}\left(amt, \mathsf{fma}\left(amt, 0.5 \cdot \left(u2 \cdot \mathsf{fma}\left(3, amt, -5\right)\right), t_0\right), u2 \cdot 1.52587890625 \cdot 10^{-5}\right)\\
\mathbf{if}\;u2 \leq -1370:\\
\;\;\;\;t_1\\

\mathbf{elif}\;u2 \leq 2000000:\\
\;\;\;\;\mathsf{fma}\left(amt, \mathsf{fma}\left(amt, 0.5 \cdot \left(u4 \cdot \left(amt + -1\right)\right), t_0\right), u2 \cdot 1.52587890625 \cdot 10^{-5}\right)\\

\mathbf{else}:\\
\;\;\;\;t_1\\


\end{array}
\end{array}

Derivation

Split input into 2 regimes
if u2 < -1370 or 2e6 < u2
1. Initial program 65.6%
  \[7.62939453125 \cdot 10^{-6} \cdot \left(\left(\left(2 \cdot u2 + \left(u3 - u1\right) \cdot \left(amt \cdot 65536\right)\right) + \left(\left(\left(2 \cdot u1 - 5 \cdot u2\right) + 4 \cdot u3\right) - u4\right) \cdot \left({amt}^{2} \cdot 65536\right)\right) + \left(\left(\left(3 \cdot u2 - u1\right) - 3 \cdot u3\right) + u4\right) \cdot \left({amt}^{3} \cdot 65536\right)\right) \]
2. Add Preprocessing
3. Taylor expanded in amt around 0
  \[\leadsto \color{blue}{\frac{1}{65536} \cdot u2 + amt \cdot \left(\frac{1}{2} \cdot \left(u3 - u1\right) + amt \cdot \left(\frac{1}{2} \cdot \left(amt \cdot \left(\left(u4 + 3 \cdot u2\right) - \left(u1 + 3 \cdot u3\right)\right)\right) + \frac{1}{2} \cdot \left(\left(2 \cdot u1 + 4 \cdot u3\right) - \left(u4 + 5 \cdot u2\right)\right)\right)\right)} \]
4. Step-by-step derivation
  1. +-commutativeN/A
    \[\leadsto \color{blue}{amt \cdot \left(\frac{1}{2} \cdot \left(u3 - u1\right) + amt \cdot \left(\frac{1}{2} \cdot \left(amt \cdot \left(\left(u4 + 3 \cdot u2\right) - \left(u1 + 3 \cdot u3\right)\right)\right) + \frac{1}{2} \cdot \left(\left(2 \cdot u1 + 4 \cdot u3\right) - \left(u4 + 5 \cdot u2\right)\right)\right)\right) + \frac{1}{65536} \cdot u2} \]
  2. lower-fma.f32N/A
    \[\leadsto \color{blue}{\mathsf{fma}\left(amt, \frac{1}{2} \cdot \left(u3 - u1\right) + amt \cdot \left(\frac{1}{2} \cdot \left(amt \cdot \left(\left(u4 + 3 \cdot u2\right) - \left(u1 + 3 \cdot u3\right)\right)\right) + \frac{1}{2} \cdot \left(\left(2 \cdot u1 + 4 \cdot u3\right) - \left(u4 + 5 \cdot u2\right)\right)\right), \frac{1}{65536} \cdot u2\right)} \]
5. Applied rewrites92.9%
  \[\leadsto \color{blue}{\mathsf{fma}\left(amt, \mathsf{fma}\left(amt, 0.5 \cdot \mathsf{fma}\left(amt, \mathsf{fma}\left(3, u2 - u3, u4\right) - u1, \mathsf{fma}\left(2, u1, \mathsf{fma}\left(u3, 4, u2 \cdot -5\right)\right) - u4\right), \left(u3 - u1\right) \cdot 0.5\right), u2 \cdot 1.52587890625 \cdot 10^{-5}\right)} \]
6. Taylor expanded in u2 around inf
  \[\leadsto \mathsf{fma}\left(amt, \mathsf{fma}\left(amt, \frac{1}{2} \cdot \left(u2 \cdot \color{blue}{\left(3 \cdot amt - 5\right)}\right), \left(u3 - u1\right) \cdot \frac{1}{2}\right), u2 \cdot \frac{1}{65536}\right) \]
7. Step-by-step derivation
8. Applied rewrites57.2%
  \[\leadsto \mathsf{fma}\left(amt, \mathsf{fma}\left(amt, 0.5 \cdot \left(u2 \cdot \color{blue}{\mathsf{fma}\left(3, amt, -5\right)}\right), \left(u3 - u1\right) \cdot 0.5\right), u2 \cdot 1.52587890625 \cdot 10^{-5}\right) \]
if -1370 < u2 < 2e6
1. Initial program 82.9%
  \[7.62939453125 \cdot 10^{-6} \cdot \left(\left(\left(2 \cdot u2 + \left(u3 - u1\right) \cdot \left(amt \cdot 65536\right)\right) + \left(\left(\left(2 \cdot u1 - 5 \cdot u2\right) + 4 \cdot u3\right) - u4\right) \cdot \left({amt}^{2} \cdot 65536\right)\right) + \left(\left(\left(3 \cdot u2 - u1\right) - 3 \cdot u3\right) + u4\right) \cdot \left({amt}^{3} \cdot 65536\right)\right) \]
2. Add Preprocessing
3. Taylor expanded in amt around 0
  \[\leadsto \color{blue}{\frac{1}{65536} \cdot u2 + amt \cdot \left(\frac{1}{2} \cdot \left(u3 - u1\right) + amt \cdot \left(\frac{1}{2} \cdot \left(amt \cdot \left(\left(u4 + 3 \cdot u2\right) - \left(u1 + 3 \cdot u3\right)\right)\right) + \frac{1}{2} \cdot \left(\left(2 \cdot u1 + 4 \cdot u3\right) - \left(u4 + 5 \cdot u2\right)\right)\right)\right)} \]
4. Step-by-step derivation
  1. +-commutativeN/A
    \[\leadsto \color{blue}{amt \cdot \left(\frac{1}{2} \cdot \left(u3 - u1\right) + amt \cdot \left(\frac{1}{2} \cdot \left(amt \cdot \left(\left(u4 + 3 \cdot u2\right) - \left(u1 + 3 \cdot u3\right)\right)\right) + \frac{1}{2} \cdot \left(\left(2 \cdot u1 + 4 \cdot u3\right) - \left(u4 + 5 \cdot u2\right)\right)\right)\right) + \frac{1}{65536} \cdot u2} \]
  2. lower-fma.f32N/A
    \[\leadsto \color{blue}{\mathsf{fma}\left(amt, \frac{1}{2} \cdot \left(u3 - u1\right) + amt \cdot \left(\frac{1}{2} \cdot \left(amt \cdot \left(\left(u4 + 3 \cdot u2\right) - \left(u1 + 3 \cdot u3\right)\right)\right) + \frac{1}{2} \cdot \left(\left(2 \cdot u1 + 4 \cdot u3\right) - \left(u4 + 5 \cdot u2\right)\right)\right), \frac{1}{65536} \cdot u2\right)} \]
5. Applied rewrites96.4%
  \[\leadsto \color{blue}{\mathsf{fma}\left(amt, \mathsf{fma}\left(amt, 0.5 \cdot \mathsf{fma}\left(amt, \mathsf{fma}\left(3, u2 - u3, u4\right) - u1, \mathsf{fma}\left(2, u1, \mathsf{fma}\left(u3, 4, u2 \cdot -5\right)\right) - u4\right), \left(u3 - u1\right) \cdot 0.5\right), u2 \cdot 1.52587890625 \cdot 10^{-5}\right)} \]
6. Taylor expanded in u4 around inf
  \[\leadsto \mathsf{fma}\left(amt, \mathsf{fma}\left(amt, \frac{1}{2} \cdot \left(u4 \cdot \color{blue}{\left(amt - 1\right)}\right), \left(u3 - u1\right) \cdot \frac{1}{2}\right), u2 \cdot \frac{1}{65536}\right) \]
7. Step-by-step derivation
8. Applied rewrites47.9%
  \[\leadsto \mathsf{fma}\left(amt, \mathsf{fma}\left(amt, 0.5 \cdot \left(u4 \cdot \color{blue}{\left(amt + -1\right)}\right), \left(u3 - u1\right) \cdot 0.5\right), u2 \cdot 1.52587890625 \cdot 10^{-5}\right) \]
Recombined 2 regimes into one program.
Final simplification51.9%
\[\leadsto \begin{array}{l} \mathbf{if}\;u2 \leq -1370:\\ \;\;\;\;\mathsf{fma}\left(amt, \mathsf{fma}\left(amt, 0.5 \cdot \left(u2 \cdot \mathsf{fma}\left(3, amt, -5\right)\right), 0.5 \cdot \left(u3 - u1\right)\right), u2 \cdot 1.52587890625 \cdot 10^{-5}\right)\\ \mathbf{elif}\;u2 \leq 2000000:\\ \;\;\;\;\mathsf{fma}\left(amt, \mathsf{fma}\left(amt, 0.5 \cdot \left(u4 \cdot \left(amt + -1\right)\right), 0.5 \cdot \left(u3 - u1\right)\right), u2 \cdot 1.52587890625 \cdot 10^{-5}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(amt, \mathsf{fma}\left(amt, 0.5 \cdot \left(u2 \cdot \mathsf{fma}\left(3, amt, -5\right)\right), 0.5 \cdot \left(u3 - u1\right)\right), u2 \cdot 1.52587890625 \cdot 10^{-5}\right)\\ \end{array} \]
Add Preprocessing

Alternative 11: 40.3% accurate, 5.8× speedup?

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u2 \leq -5000000:\\ \;\;\;\;u2 \cdot \mathsf{fma}\left(0.5 \cdot \left(amt \cdot amt\right), \mathsf{fma}\left(3, amt, -5\right), 1.52587890625 \cdot 10^{-5}\right)\\ \mathbf{elif}\;u2 \leq -1.0000000272452012 \cdot 10^{-27}:\\ \;\;\;\;\left(u3 \cdot amt\right) \cdot \mathsf{fma}\left(amt \cdot 0.5, \mathsf{fma}\left(-3, amt, 4\right), 0.5\right)\\ \mathbf{elif}\;u2 \leq 5000000:\\ \;\;\;\;\mathsf{fma}\left(amt, 0.5 \cdot \left(\left(amt + -1\right) \cdot \left(amt \cdot u4\right)\right), u2 \cdot 1.52587890625 \cdot 10^{-5}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(amt, \left(amt \cdot 0.5\right) \cdot \left(u2 \cdot \mathsf{fma}\left(3, amt, -5\right)\right), u2 \cdot 1.52587890625 \cdot 10^{-5}\right)\\ \end{array} \end{array} \]

(FPCore (u2 u3 u1 amt u4)
 :precision binary32
 (if (<= u2 -5000000.0)
   (* u2 (fma (* 0.5 (* amt amt)) (fma 3.0 amt -5.0) 1.52587890625e-5))
   (if (<= u2 -1.0000000272452012e-27)
     (* (* u3 amt) (fma (* amt 0.5) (fma -3.0 amt 4.0) 0.5))
     (if (<= u2 5000000.0)
       (fma amt (* 0.5 (* (+ amt -1.0) (* amt u4))) (* u2 1.52587890625e-5))
       (fma
        amt
        (* (* amt 0.5) (* u2 (fma 3.0 amt -5.0)))
        (* u2 1.52587890625e-5))))))

float code(float u2, float u3, float u1, float amt, float u4) {
	float tmp;
	if (u2 <= -5000000.0f) {
		tmp = u2 * fmaf((0.5f * (amt * amt)), fmaf(3.0f, amt, -5.0f), 1.52587890625e-5f);
	} else if (u2 <= -1.0000000272452012e-27f) {
		tmp = (u3 * amt) * fmaf((amt * 0.5f), fmaf(-3.0f, amt, 4.0f), 0.5f);
	} else if (u2 <= 5000000.0f) {
		tmp = fmaf(amt, (0.5f * ((amt + -1.0f) * (amt * u4))), (u2 * 1.52587890625e-5f));
	} else {
		tmp = fmaf(amt, ((amt * 0.5f) * (u2 * fmaf(3.0f, amt, -5.0f))), (u2 * 1.52587890625e-5f));
	}
	return tmp;
}

function code(u2, u3, u1, amt, u4)
	tmp = Float32(0.0)
	if (u2 <= Float32(-5000000.0))
		tmp = Float32(u2 * fma(Float32(Float32(0.5) * Float32(amt * amt)), fma(Float32(3.0), amt, Float32(-5.0)), Float32(1.52587890625e-5)));
	elseif (u2 <= Float32(-1.0000000272452012e-27))
		tmp = Float32(Float32(u3 * amt) * fma(Float32(amt * Float32(0.5)), fma(Float32(-3.0), amt, Float32(4.0)), Float32(0.5)));
	elseif (u2 <= Float32(5000000.0))
		tmp = fma(amt, Float32(Float32(0.5) * Float32(Float32(amt + Float32(-1.0)) * Float32(amt * u4))), Float32(u2 * Float32(1.52587890625e-5)));
	else
		tmp = fma(amt, Float32(Float32(amt * Float32(0.5)) * Float32(u2 * fma(Float32(3.0), amt, Float32(-5.0)))), Float32(u2 * Float32(1.52587890625e-5)));
	end
	return tmp
end

\begin{array}{l}

\\
\begin{array}{l}
\mathbf{if}\;u2 \leq -5000000:\\
\;\;\;\;u2 \cdot \mathsf{fma}\left(0.5 \cdot \left(amt \cdot amt\right), \mathsf{fma}\left(3, amt, -5\right), 1.52587890625 \cdot 10^{-5}\right)\\

\mathbf{elif}\;u2 \leq -1.0000000272452012 \cdot 10^{-27}:\\
\;\;\;\;\left(u3 \cdot amt\right) \cdot \mathsf{fma}\left(amt \cdot 0.5, \mathsf{fma}\left(-3, amt, 4\right), 0.5\right)\\

\mathbf{elif}\;u2 \leq 5000000:\\
\;\;\;\;\mathsf{fma}\left(amt, 0.5 \cdot \left(\left(amt + -1\right) \cdot \left(amt \cdot u4\right)\right), u2 \cdot 1.52587890625 \cdot 10^{-5}\right)\\

\mathbf{else}:\\
\;\;\;\;\mathsf{fma}\left(amt, \left(amt \cdot 0.5\right) \cdot \left(u2 \cdot \mathsf{fma}\left(3, amt, -5\right)\right), u2 \cdot 1.52587890625 \cdot 10^{-5}\right)\\


\end{array}
\end{array}

Derivation

Split input into 4 regimes
if u2 < -5e6
1. Initial program 63.3%
  \[7.62939453125 \cdot 10^{-6} \cdot \left(\left(\left(2 \cdot u2 + \left(u3 - u1\right) \cdot \left(amt \cdot 65536\right)\right) + \left(\left(\left(2 \cdot u1 - 5 \cdot u2\right) + 4 \cdot u3\right) - u4\right) \cdot \left({amt}^{2} \cdot 65536\right)\right) + \left(\left(\left(3 \cdot u2 - u1\right) - 3 \cdot u3\right) + u4\right) \cdot \left({amt}^{3} \cdot 65536\right)\right) \]
2. Add Preprocessing
3. Taylor expanded in amt around 0
  \[\leadsto \color{blue}{\frac{1}{65536} \cdot u2 + amt \cdot \left(\frac{1}{2} \cdot \left(u3 - u1\right) + amt \cdot \left(\frac{1}{2} \cdot \left(amt \cdot \left(\left(u4 + 3 \cdot u2\right) - \left(u1 + 3 \cdot u3\right)\right)\right) + \frac{1}{2} \cdot \left(\left(2 \cdot u1 + 4 \cdot u3\right) - \left(u4 + 5 \cdot u2\right)\right)\right)\right)} \]
4. Step-by-step derivation
  1. +-commutativeN/A
    \[\leadsto \color{blue}{amt \cdot \left(\frac{1}{2} \cdot \left(u3 - u1\right) + amt \cdot \left(\frac{1}{2} \cdot \left(amt \cdot \left(\left(u4 + 3 \cdot u2\right) - \left(u1 + 3 \cdot u3\right)\right)\right) + \frac{1}{2} \cdot \left(\left(2 \cdot u1 + 4 \cdot u3\right) - \left(u4 + 5 \cdot u2\right)\right)\right)\right) + \frac{1}{65536} \cdot u2} \]
  2. lower-fma.f32N/A
    \[\leadsto \color{blue}{\mathsf{fma}\left(amt, \frac{1}{2} \cdot \left(u3 - u1\right) + amt \cdot \left(\frac{1}{2} \cdot \left(amt \cdot \left(\left(u4 + 3 \cdot u2\right) - \left(u1 + 3 \cdot u3\right)\right)\right) + \frac{1}{2} \cdot \left(\left(2 \cdot u1 + 4 \cdot u3\right) - \left(u4 + 5 \cdot u2\right)\right)\right), \frac{1}{65536} \cdot u2\right)} \]
5. Applied rewrites90.8%
  \[\leadsto \color{blue}{\mathsf{fma}\left(amt, \mathsf{fma}\left(amt, 0.5 \cdot \mathsf{fma}\left(amt, \mathsf{fma}\left(3, u2 - u3, u4\right) - u1, \mathsf{fma}\left(2, u1, \mathsf{fma}\left(u3, 4, u2 \cdot -5\right)\right) - u4\right), \left(u3 - u1\right) \cdot 0.5\right), u2 \cdot 1.52587890625 \cdot 10^{-5}\right)} \]
6. Taylor expanded in u2 around inf
  \[\leadsto u2 \cdot \color{blue}{\left(\frac{1}{65536} + \frac{1}{2} \cdot \left({amt}^{2} \cdot \left(3 \cdot amt - 5\right)\right)\right)} \]
7. Step-by-step derivation
8. Applied rewrites55.2%
  \[\leadsto u2 \cdot \color{blue}{\mathsf{fma}\left(0.5 \cdot \left(amt \cdot amt\right), \mathsf{fma}\left(3, amt, -5\right), 1.52587890625 \cdot 10^{-5}\right)} \]
if -5e6 < u2 < -1.00000003e-27
1. Initial program 83.1%
  \[7.62939453125 \cdot 10^{-6} \cdot \left(\left(\left(2 \cdot u2 + \left(u3 - u1\right) \cdot \left(amt \cdot 65536\right)\right) + \left(\left(\left(2 \cdot u1 - 5 \cdot u2\right) + 4 \cdot u3\right) - u4\right) \cdot \left({amt}^{2} \cdot 65536\right)\right) + \left(\left(\left(3 \cdot u2 - u1\right) - 3 \cdot u3\right) + u4\right) \cdot \left({amt}^{3} \cdot 65536\right)\right) \]
2. Add Preprocessing
3. Taylor expanded in amt around 0
  \[\leadsto \color{blue}{\frac{1}{65536} \cdot u2 + amt \cdot \left(\frac{1}{2} \cdot \left(u3 - u1\right) + amt \cdot \left(\frac{1}{2} \cdot \left(amt \cdot \left(\left(u4 + 3 \cdot u2\right) - \left(u1 + 3 \cdot u3\right)\right)\right) + \frac{1}{2} \cdot \left(\left(2 \cdot u1 + 4 \cdot u3\right) - \left(u4 + 5 \cdot u2\right)\right)\right)\right)} \]
4. Step-by-step derivation
  1. +-commutativeN/A
    \[\leadsto \color{blue}{amt \cdot \left(\frac{1}{2} \cdot \left(u3 - u1\right) + amt \cdot \left(\frac{1}{2} \cdot \left(amt \cdot \left(\left(u4 + 3 \cdot u2\right) - \left(u1 + 3 \cdot u3\right)\right)\right) + \frac{1}{2} \cdot \left(\left(2 \cdot u1 + 4 \cdot u3\right) - \left(u4 + 5 \cdot u2\right)\right)\right)\right) + \frac{1}{65536} \cdot u2} \]
  2. lower-fma.f32N/A
    \[\leadsto \color{blue}{\mathsf{fma}\left(amt, \frac{1}{2} \cdot \left(u3 - u1\right) + amt \cdot \left(\frac{1}{2} \cdot \left(amt \cdot \left(\left(u4 + 3 \cdot u2\right) - \left(u1 + 3 \cdot u3\right)\right)\right) + \frac{1}{2} \cdot \left(\left(2 \cdot u1 + 4 \cdot u3\right) - \left(u4 + 5 \cdot u2\right)\right)\right), \frac{1}{65536} \cdot u2\right)} \]
5. Applied rewrites96.1%
  \[\leadsto \color{blue}{\mathsf{fma}\left(amt, \mathsf{fma}\left(amt, 0.5 \cdot \mathsf{fma}\left(amt, \mathsf{fma}\left(3, u2 - u3, u4\right) - u1, \mathsf{fma}\left(2, u1, \mathsf{fma}\left(u3, 4, u2 \cdot -5\right)\right) - u4\right), \left(u3 - u1\right) \cdot 0.5\right), u2 \cdot 1.52587890625 \cdot 10^{-5}\right)} \]
6. Taylor expanded in u3 around inf
  \[\leadsto amt \cdot \color{blue}{\left(u3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \left(amt \cdot \left(4 + -3 \cdot amt\right)\right)\right)\right)} \]
7. Step-by-step derivation
8. Applied rewrites40.1%
  \[\leadsto \left(amt \cdot u3\right) \cdot \color{blue}{\mathsf{fma}\left(0.5 \cdot amt, \mathsf{fma}\left(-3, amt, 4\right), 0.5\right)} \]
if -1.00000003e-27 < u2 < 5e6
1. Initial program 82.3%
  \[7.62939453125 \cdot 10^{-6} \cdot \left(\left(\left(2 \cdot u2 + \left(u3 - u1\right) \cdot \left(amt \cdot 65536\right)\right) + \left(\left(\left(2 \cdot u1 - 5 \cdot u2\right) + 4 \cdot u3\right) - u4\right) \cdot \left({amt}^{2} \cdot 65536\right)\right) + \left(\left(\left(3 \cdot u2 - u1\right) - 3 \cdot u3\right) + u4\right) \cdot \left({amt}^{3} \cdot 65536\right)\right) \]
2. Add Preprocessing
3. Taylor expanded in amt around 0
  \[\leadsto \color{blue}{\frac{1}{65536} \cdot u2 + amt \cdot \left(\frac{1}{2} \cdot \left(u3 - u1\right) + amt \cdot \left(\frac{1}{2} \cdot \left(amt \cdot \left(\left(u4 + 3 \cdot u2\right) - \left(u1 + 3 \cdot u3\right)\right)\right) + \frac{1}{2} \cdot \left(\left(2 \cdot u1 + 4 \cdot u3\right) - \left(u4 + 5 \cdot u2\right)\right)\right)\right)} \]
4. Step-by-step derivation
  1. +-commutativeN/A
    \[\leadsto \color{blue}{amt \cdot \left(\frac{1}{2} \cdot \left(u3 - u1\right) + amt \cdot \left(\frac{1}{2} \cdot \left(amt \cdot \left(\left(u4 + 3 \cdot u2\right) - \left(u1 + 3 \cdot u3\right)\right)\right) + \frac{1}{2} \cdot \left(\left(2 \cdot u1 + 4 \cdot u3\right) - \left(u4 + 5 \cdot u2\right)\right)\right)\right) + \frac{1}{65536} \cdot u2} \]
  2. lower-fma.f32N/A
    \[\leadsto \color{blue}{\mathsf{fma}\left(amt, \frac{1}{2} \cdot \left(u3 - u1\right) + amt \cdot \left(\frac{1}{2} \cdot \left(amt \cdot \left(\left(u4 + 3 \cdot u2\right) - \left(u1 + 3 \cdot u3\right)\right)\right) + \frac{1}{2} \cdot \left(\left(2 \cdot u1 + 4 \cdot u3\right) - \left(u4 + 5 \cdot u2\right)\right)\right), \frac{1}{65536} \cdot u2\right)} \]
5. Applied rewrites96.6%
  \[\leadsto \color{blue}{\mathsf{fma}\left(amt, \mathsf{fma}\left(amt, 0.5 \cdot \mathsf{fma}\left(amt, \mathsf{fma}\left(3, u2 - u3, u4\right) - u1, \mathsf{fma}\left(2, u1, \mathsf{fma}\left(u3, 4, u2 \cdot -5\right)\right) - u4\right), \left(u3 - u1\right) \cdot 0.5\right), u2 \cdot 1.52587890625 \cdot 10^{-5}\right)} \]
6. Taylor expanded in u4 around inf
  \[\leadsto \mathsf{fma}\left(amt, \frac{1}{2} \cdot \color{blue}{\left(amt \cdot \left(u4 \cdot \left(amt - 1\right)\right)\right)}, u2 \cdot \frac{1}{65536}\right) \]
7. Step-by-step derivation
8. Applied rewrites45.4%
  \[\leadsto \mathsf{fma}\left(amt, 0.5 \cdot \color{blue}{\left(\left(amt \cdot u4\right) \cdot \left(amt + -1\right)\right)}, u2 \cdot 1.52587890625 \cdot 10^{-5}\right) \]
if 5e6 < u2
1. Initial program 67.5%
  \[7.62939453125 \cdot 10^{-6} \cdot \left(\left(\left(2 \cdot u2 + \left(u3 - u1\right) \cdot \left(amt \cdot 65536\right)\right) + \left(\left(\left(2 \cdot u1 - 5 \cdot u2\right) + 4 \cdot u3\right) - u4\right) \cdot \left({amt}^{2} \cdot 65536\right)\right) + \left(\left(\left(3 \cdot u2 - u1\right) - 3 \cdot u3\right) + u4\right) \cdot \left({amt}^{3} \cdot 65536\right)\right) \]
2. Add Preprocessing
3. Taylor expanded in amt around 0
  \[\leadsto \color{blue}{\frac{1}{65536} \cdot u2 + amt \cdot \left(\frac{1}{2} \cdot \left(u3 - u1\right) + amt \cdot \left(\frac{1}{2} \cdot \left(amt \cdot \left(\left(u4 + 3 \cdot u2\right) - \left(u1 + 3 \cdot u3\right)\right)\right) + \frac{1}{2} \cdot \left(\left(2 \cdot u1 + 4 \cdot u3\right) - \left(u4 + 5 \cdot u2\right)\right)\right)\right)} \]
4. Step-by-step derivation
  1. +-commutativeN/A
    \[\leadsto \color{blue}{amt \cdot \left(\frac{1}{2} \cdot \left(u3 - u1\right) + amt \cdot \left(\frac{1}{2} \cdot \left(amt \cdot \left(\left(u4 + 3 \cdot u2\right) - \left(u1 + 3 \cdot u3\right)\right)\right) + \frac{1}{2} \cdot \left(\left(2 \cdot u1 + 4 \cdot u3\right) - \left(u4 + 5 \cdot u2\right)\right)\right)\right) + \frac{1}{65536} \cdot u2} \]
  2. lower-fma.f32N/A
    \[\leadsto \color{blue}{\mathsf{fma}\left(amt, \frac{1}{2} \cdot \left(u3 - u1\right) + amt \cdot \left(\frac{1}{2} \cdot \left(amt \cdot \left(\left(u4 + 3 \cdot u2\right) - \left(u1 + 3 \cdot u3\right)\right)\right) + \frac{1}{2} \cdot \left(\left(2 \cdot u1 + 4 \cdot u3\right) - \left(u4 + 5 \cdot u2\right)\right)\right), \frac{1}{65536} \cdot u2\right)} \]
5. Applied rewrites95.3%
  \[\leadsto \color{blue}{\mathsf{fma}\left(amt, \mathsf{fma}\left(amt, 0.5 \cdot \mathsf{fma}\left(amt, \mathsf{fma}\left(3, u2 - u3, u4\right) - u1, \mathsf{fma}\left(2, u1, \mathsf{fma}\left(u3, 4, u2 \cdot -5\right)\right) - u4\right), \left(u3 - u1\right) \cdot 0.5\right), u2 \cdot 1.52587890625 \cdot 10^{-5}\right)} \]
6. Taylor expanded in u2 around inf
  \[\leadsto \mathsf{fma}\left(amt, \frac{1}{2} \cdot \color{blue}{\left(amt \cdot \left(u2 \cdot \left(3 \cdot amt - 5\right)\right)\right)}, u2 \cdot \frac{1}{65536}\right) \]
7. Step-by-step derivation
8. Applied rewrites57.0%
  \[\leadsto \mathsf{fma}\left(amt, \left(0.5 \cdot amt\right) \cdot \color{blue}{\left(u2 \cdot \mathsf{fma}\left(3, amt, -5\right)\right)}, u2 \cdot 1.52587890625 \cdot 10^{-5}\right) \]
Recombined 4 regimes into one program.
Final simplification48.3%
\[\leadsto \begin{array}{l} \mathbf{if}\;u2 \leq -5000000:\\ \;\;\;\;u2 \cdot \mathsf{fma}\left(0.5 \cdot \left(amt \cdot amt\right), \mathsf{fma}\left(3, amt, -5\right), 1.52587890625 \cdot 10^{-5}\right)\\ \mathbf{elif}\;u2 \leq -1.0000000272452012 \cdot 10^{-27}:\\ \;\;\;\;\left(u3 \cdot amt\right) \cdot \mathsf{fma}\left(amt \cdot 0.5, \mathsf{fma}\left(-3, amt, 4\right), 0.5\right)\\ \mathbf{elif}\;u2 \leq 5000000:\\ \;\;\;\;\mathsf{fma}\left(amt, 0.5 \cdot \left(\left(amt + -1\right) \cdot \left(amt \cdot u4\right)\right), u2 \cdot 1.52587890625 \cdot 10^{-5}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(amt, \left(amt \cdot 0.5\right) \cdot \left(u2 \cdot \mathsf{fma}\left(3, amt, -5\right)\right), u2 \cdot 1.52587890625 \cdot 10^{-5}\right)\\ \end{array} \]
Add Preprocessing

Alternative 12: 46.0% accurate, 5.8× speedup?

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u2 \leq -200000:\\ \;\;\;\;u2 \cdot \mathsf{fma}\left(0.5 \cdot \left(amt \cdot amt\right), \mathsf{fma}\left(3, amt, -5\right), 1.52587890625 \cdot 10^{-5}\right)\\ \mathbf{elif}\;u2 \leq 15000000:\\ \;\;\;\;\mathsf{fma}\left(amt, \mathsf{fma}\left(amt, 0.5 \cdot \left(u4 \cdot \left(amt + -1\right)\right), 0.5 \cdot \left(u3 - u1\right)\right), u2 \cdot 1.52587890625 \cdot 10^{-5}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(amt, \left(amt \cdot 0.5\right) \cdot \left(u2 \cdot \mathsf{fma}\left(3, amt, -5\right)\right), u2 \cdot 1.52587890625 \cdot 10^{-5}\right)\\ \end{array} \end{array} \]

(FPCore (u2 u3 u1 amt u4)
 :precision binary32
 (if (<= u2 -200000.0)
   (* u2 (fma (* 0.5 (* amt amt)) (fma 3.0 amt -5.0) 1.52587890625e-5))
   (if (<= u2 15000000.0)
     (fma
      amt
      (fma amt (* 0.5 (* u4 (+ amt -1.0))) (* 0.5 (- u3 u1)))
      (* u2 1.52587890625e-5))
     (fma
      amt
      (* (* amt 0.5) (* u2 (fma 3.0 amt -5.0)))
      (* u2 1.52587890625e-5)))))

float code(float u2, float u3, float u1, float amt, float u4) {
	float tmp;
	if (u2 <= -200000.0f) {
		tmp = u2 * fmaf((0.5f * (amt * amt)), fmaf(3.0f, amt, -5.0f), 1.52587890625e-5f);
	} else if (u2 <= 15000000.0f) {
		tmp = fmaf(amt, fmaf(amt, (0.5f * (u4 * (amt + -1.0f))), (0.5f * (u3 - u1))), (u2 * 1.52587890625e-5f));
	} else {
		tmp = fmaf(amt, ((amt * 0.5f) * (u2 * fmaf(3.0f, amt, -5.0f))), (u2 * 1.52587890625e-5f));
	}
	return tmp;
}

function code(u2, u3, u1, amt, u4)
	tmp = Float32(0.0)
	if (u2 <= Float32(-200000.0))
		tmp = Float32(u2 * fma(Float32(Float32(0.5) * Float32(amt * amt)), fma(Float32(3.0), amt, Float32(-5.0)), Float32(1.52587890625e-5)));
	elseif (u2 <= Float32(15000000.0))
		tmp = fma(amt, fma(amt, Float32(Float32(0.5) * Float32(u4 * Float32(amt + Float32(-1.0)))), Float32(Float32(0.5) * Float32(u3 - u1))), Float32(u2 * Float32(1.52587890625e-5)));
	else
		tmp = fma(amt, Float32(Float32(amt * Float32(0.5)) * Float32(u2 * fma(Float32(3.0), amt, Float32(-5.0)))), Float32(u2 * Float32(1.52587890625e-5)));
	end
	return tmp
end

\begin{array}{l}

\\
\begin{array}{l}
\mathbf{if}\;u2 \leq -200000:\\
\;\;\;\;u2 \cdot \mathsf{fma}\left(0.5 \cdot \left(amt \cdot amt\right), \mathsf{fma}\left(3, amt, -5\right), 1.52587890625 \cdot 10^{-5}\right)\\

\mathbf{elif}\;u2 \leq 15000000:\\
\;\;\;\;\mathsf{fma}\left(amt, \mathsf{fma}\left(amt, 0.5 \cdot \left(u4 \cdot \left(amt + -1\right)\right), 0.5 \cdot \left(u3 - u1\right)\right), u2 \cdot 1.52587890625 \cdot 10^{-5}\right)\\

\mathbf{else}:\\
\;\;\;\;\mathsf{fma}\left(amt, \left(amt \cdot 0.5\right) \cdot \left(u2 \cdot \mathsf{fma}\left(3, amt, -5\right)\right), u2 \cdot 1.52587890625 \cdot 10^{-5}\right)\\


\end{array}
\end{array}

Derivation

Split input into 3 regimes
if u2 < -2e5
1. Initial program 62.9%
  \[7.62939453125 \cdot 10^{-6} \cdot \left(\left(\left(2 \cdot u2 + \left(u3 - u1\right) \cdot \left(amt \cdot 65536\right)\right) + \left(\left(\left(2 \cdot u1 - 5 \cdot u2\right) + 4 \cdot u3\right) - u4\right) \cdot \left({amt}^{2} \cdot 65536\right)\right) + \left(\left(\left(3 \cdot u2 - u1\right) - 3 \cdot u3\right) + u4\right) \cdot \left({amt}^{3} \cdot 65536\right)\right) \]
2. Add Preprocessing
3. Taylor expanded in amt around 0
  \[\leadsto \color{blue}{\frac{1}{65536} \cdot u2 + amt \cdot \left(\frac{1}{2} \cdot \left(u3 - u1\right) + amt \cdot \left(\frac{1}{2} \cdot \left(amt \cdot \left(\left(u4 + 3 \cdot u2\right) - \left(u1 + 3 \cdot u3\right)\right)\right) + \frac{1}{2} \cdot \left(\left(2 \cdot u1 + 4 \cdot u3\right) - \left(u4 + 5 \cdot u2\right)\right)\right)\right)} \]
4. Step-by-step derivation
  1. +-commutativeN/A
    \[\leadsto \color{blue}{amt \cdot \left(\frac{1}{2} \cdot \left(u3 - u1\right) + amt \cdot \left(\frac{1}{2} \cdot \left(amt \cdot \left(\left(u4 + 3 \cdot u2\right) - \left(u1 + 3 \cdot u3\right)\right)\right) + \frac{1}{2} \cdot \left(\left(2 \cdot u1 + 4 \cdot u3\right) - \left(u4 + 5 \cdot u2\right)\right)\right)\right) + \frac{1}{65536} \cdot u2} \]
  2. lower-fma.f32N/A
    \[\leadsto \color{blue}{\mathsf{fma}\left(amt, \frac{1}{2} \cdot \left(u3 - u1\right) + amt \cdot \left(\frac{1}{2} \cdot \left(amt \cdot \left(\left(u4 + 3 \cdot u2\right) - \left(u1 + 3 \cdot u3\right)\right)\right) + \frac{1}{2} \cdot \left(\left(2 \cdot u1 + 4 \cdot u3\right) - \left(u4 + 5 \cdot u2\right)\right)\right), \frac{1}{65536} \cdot u2\right)} \]
5. Applied rewrites91.1%
  \[\leadsto \color{blue}{\mathsf{fma}\left(amt, \mathsf{fma}\left(amt, 0.5 \cdot \mathsf{fma}\left(amt, \mathsf{fma}\left(3, u2 - u3, u4\right) - u1, \mathsf{fma}\left(2, u1, \mathsf{fma}\left(u3, 4, u2 \cdot -5\right)\right) - u4\right), \left(u3 - u1\right) \cdot 0.5\right), u2 \cdot 1.52587890625 \cdot 10^{-5}\right)} \]
6. Taylor expanded in u2 around inf
  \[\leadsto u2 \cdot \color{blue}{\left(\frac{1}{65536} + \frac{1}{2} \cdot \left({amt}^{2} \cdot \left(3 \cdot amt - 5\right)\right)\right)} \]
7. Step-by-step derivation
8. Applied rewrites54.1%
  \[\leadsto u2 \cdot \color{blue}{\mathsf{fma}\left(0.5 \cdot \left(amt \cdot amt\right), \mathsf{fma}\left(3, amt, -5\right), 1.52587890625 \cdot 10^{-5}\right)} \]
if -2e5 < u2 < 1.5e7
1. Initial program 82.5%
  \[7.62939453125 \cdot 10^{-6} \cdot \left(\left(\left(2 \cdot u2 + \left(u3 - u1\right) \cdot \left(amt \cdot 65536\right)\right) + \left(\left(\left(2 \cdot u1 - 5 \cdot u2\right) + 4 \cdot u3\right) - u4\right) \cdot \left({amt}^{2} \cdot 65536\right)\right) + \left(\left(\left(3 \cdot u2 - u1\right) - 3 \cdot u3\right) + u4\right) \cdot \left({amt}^{3} \cdot 65536\right)\right) \]
2. Add Preprocessing
3. Taylor expanded in amt around 0
  \[\leadsto \color{blue}{\frac{1}{65536} \cdot u2 + amt \cdot \left(\frac{1}{2} \cdot \left(u3 - u1\right) + amt \cdot \left(\frac{1}{2} \cdot \left(amt \cdot \left(\left(u4 + 3 \cdot u2\right) - \left(u1 + 3 \cdot u3\right)\right)\right) + \frac{1}{2} \cdot \left(\left(2 \cdot u1 + 4 \cdot u3\right) - \left(u4 + 5 \cdot u2\right)\right)\right)\right)} \]
4. Step-by-step derivation
  1. +-commutativeN/A
    \[\leadsto \color{blue}{amt \cdot \left(\frac{1}{2} \cdot \left(u3 - u1\right) + amt \cdot \left(\frac{1}{2} \cdot \left(amt \cdot \left(\left(u4 + 3 \cdot u2\right) - \left(u1 + 3 \cdot u3\right)\right)\right) + \frac{1}{2} \cdot \left(\left(2 \cdot u1 + 4 \cdot u3\right) - \left(u4 + 5 \cdot u2\right)\right)\right)\right) + \frac{1}{65536} \cdot u2} \]
  2. lower-fma.f32N/A
    \[\leadsto \color{blue}{\mathsf{fma}\left(amt, \frac{1}{2} \cdot \left(u3 - u1\right) + amt \cdot \left(\frac{1}{2} \cdot \left(amt \cdot \left(\left(u4 + 3 \cdot u2\right) - \left(u1 + 3 \cdot u3\right)\right)\right) + \frac{1}{2} \cdot \left(\left(2 \cdot u1 + 4 \cdot u3\right) - \left(u4 + 5 \cdot u2\right)\right)\right), \frac{1}{65536} \cdot u2\right)} \]
5. Applied rewrites96.1%
  \[\leadsto \color{blue}{\mathsf{fma}\left(amt, \mathsf{fma}\left(amt, 0.5 \cdot \mathsf{fma}\left(amt, \mathsf{fma}\left(3, u2 - u3, u4\right) - u1, \mathsf{fma}\left(2, u1, \mathsf{fma}\left(u3, 4, u2 \cdot -5\right)\right) - u4\right), \left(u3 - u1\right) \cdot 0.5\right), u2 \cdot 1.52587890625 \cdot 10^{-5}\right)} \]
6. Taylor expanded in u4 around inf
  \[\leadsto \mathsf{fma}\left(amt, \mathsf{fma}\left(amt, \frac{1}{2} \cdot \left(u4 \cdot \color{blue}{\left(amt - 1\right)}\right), \left(u3 - u1\right) \cdot \frac{1}{2}\right), u2 \cdot \frac{1}{65536}\right) \]
7. Step-by-step derivation
8. Applied rewrites47.2%
  \[\leadsto \mathsf{fma}\left(amt, \mathsf{fma}\left(amt, 0.5 \cdot \left(u4 \cdot \color{blue}{\left(amt + -1\right)}\right), \left(u3 - u1\right) \cdot 0.5\right), u2 \cdot 1.52587890625 \cdot 10^{-5}\right) \]
if 1.5e7 < u2
1. Initial program 69.0%
  \[7.62939453125 \cdot 10^{-6} \cdot \left(\left(\left(2 \cdot u2 + \left(u3 - u1\right) \cdot \left(amt \cdot 65536\right)\right) + \left(\left(\left(2 \cdot u1 - 5 \cdot u2\right) + 4 \cdot u3\right) - u4\right) \cdot \left({amt}^{2} \cdot 65536\right)\right) + \left(\left(\left(3 \cdot u2 - u1\right) - 3 \cdot u3\right) + u4\right) \cdot \left({amt}^{3} \cdot 65536\right)\right) \]
2. Add Preprocessing
3. Taylor expanded in amt around 0
  \[\leadsto \color{blue}{\frac{1}{65536} \cdot u2 + amt \cdot \left(\frac{1}{2} \cdot \left(u3 - u1\right) + amt \cdot \left(\frac{1}{2} \cdot \left(amt \cdot \left(\left(u4 + 3 \cdot u2\right) - \left(u1 + 3 \cdot u3\right)\right)\right) + \frac{1}{2} \cdot \left(\left(2 \cdot u1 + 4 \cdot u3\right) - \left(u4 + 5 \cdot u2\right)\right)\right)\right)} \]
4. Step-by-step derivation
  1. +-commutativeN/A
    \[\leadsto \color{blue}{amt \cdot \left(\frac{1}{2} \cdot \left(u3 - u1\right) + amt \cdot \left(\frac{1}{2} \cdot \left(amt \cdot \left(\left(u4 + 3 \cdot u2\right) - \left(u1 + 3 \cdot u3\right)\right)\right) + \frac{1}{2} \cdot \left(\left(2 \cdot u1 + 4 \cdot u3\right) - \left(u4 + 5 \cdot u2\right)\right)\right)\right) + \frac{1}{65536} \cdot u2} \]
  2. lower-fma.f32N/A
    \[\leadsto \color{blue}{\mathsf{fma}\left(amt, \frac{1}{2} \cdot \left(u3 - u1\right) + amt \cdot \left(\frac{1}{2} \cdot \left(amt \cdot \left(\left(u4 + 3 \cdot u2\right) - \left(u1 + 3 \cdot u3\right)\right)\right) + \frac{1}{2} \cdot \left(\left(2 \cdot u1 + 4 \cdot u3\right) - \left(u4 + 5 \cdot u2\right)\right)\right), \frac{1}{65536} \cdot u2\right)} \]
5. Applied rewrites96.0%
  \[\leadsto \color{blue}{\mathsf{fma}\left(amt, \mathsf{fma}\left(amt, 0.5 \cdot \mathsf{fma}\left(amt, \mathsf{fma}\left(3, u2 - u3, u4\right) - u1, \mathsf{fma}\left(2, u1, \mathsf{fma}\left(u3, 4, u2 \cdot -5\right)\right) - u4\right), \left(u3 - u1\right) \cdot 0.5\right), u2 \cdot 1.52587890625 \cdot 10^{-5}\right)} \]
6. Taylor expanded in u2 around inf
  \[\leadsto \mathsf{fma}\left(amt, \frac{1}{2} \cdot \color{blue}{\left(amt \cdot \left(u2 \cdot \left(3 \cdot amt - 5\right)\right)\right)}, u2 \cdot \frac{1}{65536}\right) \]
7. Step-by-step derivation
8. Applied rewrites58.1%
  \[\leadsto \mathsf{fma}\left(amt, \left(0.5 \cdot amt\right) \cdot \color{blue}{\left(u2 \cdot \mathsf{fma}\left(3, amt, -5\right)\right)}, u2 \cdot 1.52587890625 \cdot 10^{-5}\right) \]
Recombined 3 regimes into one program.
Final simplification50.8%
\[\leadsto \begin{array}{l} \mathbf{if}\;u2 \leq -200000:\\ \;\;\;\;u2 \cdot \mathsf{fma}\left(0.5 \cdot \left(amt \cdot amt\right), \mathsf{fma}\left(3, amt, -5\right), 1.52587890625 \cdot 10^{-5}\right)\\ \mathbf{elif}\;u2 \leq 15000000:\\ \;\;\;\;\mathsf{fma}\left(amt, \mathsf{fma}\left(amt, 0.5 \cdot \left(u4 \cdot \left(amt + -1\right)\right), 0.5 \cdot \left(u3 - u1\right)\right), u2 \cdot 1.52587890625 \cdot 10^{-5}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(amt, \left(amt \cdot 0.5\right) \cdot \left(u2 \cdot \mathsf{fma}\left(3, amt, -5\right)\right), u2 \cdot 1.52587890625 \cdot 10^{-5}\right)\\ \end{array} \]
Add Preprocessing

Alternative 13: 40.9% accurate, 5.8× speedup?

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u4 \leq -1499999942100385800:\\ \;\;\;\;\left(u4 \cdot 0.5\right) \cdot \mathsf{fma}\left(amt, amt \cdot amt, amt \cdot \left(-amt\right)\right)\\ \mathbf{elif}\;u4 \leq 39999999311872:\\ \;\;\;\;\mathsf{fma}\left(amt, \mathsf{fma}\left(amt, 0.5 \cdot \left(u1 \cdot \left(2 - amt\right)\right), 0.5 \cdot \left(u3 - u1\right)\right), u2 \cdot 1.52587890625 \cdot 10^{-5}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \left(amt \cdot amt\right)\right) \cdot \left(u4 \cdot \left(amt + -1\right)\right)\\ \end{array} \end{array} \]

(FPCore (u2 u3 u1 amt u4)
 :precision binary32
 (if (<= u4 -1499999942100385800.0)
   (* (* u4 0.5) (fma amt (* amt amt) (* amt (- amt))))
   (if (<= u4 39999999311872.0)
     (fma
      amt
      (fma amt (* 0.5 (* u1 (- 2.0 amt))) (* 0.5 (- u3 u1)))
      (* u2 1.52587890625e-5))
     (* (* 0.5 (* amt amt)) (* u4 (+ amt -1.0))))))

float code(float u2, float u3, float u1, float amt, float u4) {
	float tmp;
	if (u4 <= -1499999942100385800.0f) {
		tmp = (u4 * 0.5f) * fmaf(amt, (amt * amt), (amt * -amt));
	} else if (u4 <= 39999999311872.0f) {
		tmp = fmaf(amt, fmaf(amt, (0.5f * (u1 * (2.0f - amt))), (0.5f * (u3 - u1))), (u2 * 1.52587890625e-5f));
	} else {
		tmp = (0.5f * (amt * amt)) * (u4 * (amt + -1.0f));
	}
	return tmp;
}

function code(u2, u3, u1, amt, u4)
	tmp = Float32(0.0)
	if (u4 <= Float32(-1499999942100385800.0))
		tmp = Float32(Float32(u4 * Float32(0.5)) * fma(amt, Float32(amt * amt), Float32(amt * Float32(-amt))));
	elseif (u4 <= Float32(39999999311872.0))
		tmp = fma(amt, fma(amt, Float32(Float32(0.5) * Float32(u1 * Float32(Float32(2.0) - amt))), Float32(Float32(0.5) * Float32(u3 - u1))), Float32(u2 * Float32(1.52587890625e-5)));
	else
		tmp = Float32(Float32(Float32(0.5) * Float32(amt * amt)) * Float32(u4 * Float32(amt + Float32(-1.0))));
	end
	return tmp
end

\begin{array}{l}

\\
\begin{array}{l}
\mathbf{if}\;u4 \leq -1499999942100385800:\\
\;\;\;\;\left(u4 \cdot 0.5\right) \cdot \mathsf{fma}\left(amt, amt \cdot amt, amt \cdot \left(-amt\right)\right)\\

\mathbf{elif}\;u4 \leq 39999999311872:\\
\;\;\;\;\mathsf{fma}\left(amt, \mathsf{fma}\left(amt, 0.5 \cdot \left(u1 \cdot \left(2 - amt\right)\right), 0.5 \cdot \left(u3 - u1\right)\right), u2 \cdot 1.52587890625 \cdot 10^{-5}\right)\\

\mathbf{else}:\\
\;\;\;\;\left(0.5 \cdot \left(amt \cdot amt\right)\right) \cdot \left(u4 \cdot \left(amt + -1\right)\right)\\


\end{array}
\end{array}

Derivation

Split input into 3 regimes
if u4 < -1.4999999e18
1. Initial program 59.6%
  \[7.62939453125 \cdot 10^{-6} \cdot \left(\left(\left(2 \cdot u2 + \left(u3 - u1\right) \cdot \left(amt \cdot 65536\right)\right) + \left(\left(\left(2 \cdot u1 - 5 \cdot u2\right) + 4 \cdot u3\right) - u4\right) \cdot \left({amt}^{2} \cdot 65536\right)\right) + \left(\left(\left(3 \cdot u2 - u1\right) - 3 \cdot u3\right) + u4\right) \cdot \left({amt}^{3} \cdot 65536\right)\right) \]
2. Add Preprocessing
3. Taylor expanded in u2 around 0
  \[\leadsto \color{blue}{\frac{1}{131072} \cdot \left(65536 \cdot \left(amt \cdot \left(u3 - u1\right)\right) + \left(65536 \cdot \left({amt}^{2} \cdot \left(\left(2 \cdot u1 + 4 \cdot u3\right) - u4\right)\right) + 65536 \cdot \left({amt}^{3} \cdot \left(u4 - \left(u1 + 3 \cdot u3\right)\right)\right)\right)\right)} \]
4. Step-by-step derivation
  1. +-commutativeN/A
    \[\leadsto \frac{1}{131072} \cdot \color{blue}{\left(\left(65536 \cdot \left({amt}^{2} \cdot \left(\left(2 \cdot u1 + 4 \cdot u3\right) - u4\right)\right) + 65536 \cdot \left({amt}^{3} \cdot \left(u4 - \left(u1 + 3 \cdot u3\right)\right)\right)\right) + 65536 \cdot \left(amt \cdot \left(u3 - u1\right)\right)\right)} \]
  2. distribute-lft-inN/A
    \[\leadsto \color{blue}{\frac{1}{131072} \cdot \left(65536 \cdot \left({amt}^{2} \cdot \left(\left(2 \cdot u1 + 4 \cdot u3\right) - u4\right)\right) + 65536 \cdot \left({amt}^{3} \cdot \left(u4 - \left(u1 + 3 \cdot u3\right)\right)\right)\right) + \frac{1}{131072} \cdot \left(65536 \cdot \left(amt \cdot \left(u3 - u1\right)\right)\right)} \]
  3. distribute-lft-outN/A
    \[\leadsto \frac{1}{131072} \cdot \color{blue}{\left(65536 \cdot \left({amt}^{2} \cdot \left(\left(2 \cdot u1 + 4 \cdot u3\right) - u4\right) + {amt}^{3} \cdot \left(u4 - \left(u1 + 3 \cdot u3\right)\right)\right)\right)} + \frac{1}{131072} \cdot \left(65536 \cdot \left(amt \cdot \left(u3 - u1\right)\right)\right) \]
  4. associate-*r*N/A
    \[\leadsto \color{blue}{\left(\frac{1}{131072} \cdot 65536\right) \cdot \left({amt}^{2} \cdot \left(\left(2 \cdot u1 + 4 \cdot u3\right) - u4\right) + {amt}^{3} \cdot \left(u4 - \left(u1 + 3 \cdot u3\right)\right)\right)} + \frac{1}{131072} \cdot \left(65536 \cdot \left(amt \cdot \left(u3 - u1\right)\right)\right) \]
  5. metadata-evalN/A
    \[\leadsto \color{blue}{\frac{1}{2}} \cdot \left({amt}^{2} \cdot \left(\left(2 \cdot u1 + 4 \cdot u3\right) - u4\right) + {amt}^{3} \cdot \left(u4 - \left(u1 + 3 \cdot u3\right)\right)\right) + \frac{1}{131072} \cdot \left(65536 \cdot \left(amt \cdot \left(u3 - u1\right)\right)\right) \]
  6. associate-*r*N/A
    \[\leadsto \frac{1}{2} \cdot \left({amt}^{2} \cdot \left(\left(2 \cdot u1 + 4 \cdot u3\right) - u4\right) + {amt}^{3} \cdot \left(u4 - \left(u1 + 3 \cdot u3\right)\right)\right) + \color{blue}{\left(\frac{1}{131072} \cdot 65536\right) \cdot \left(amt \cdot \left(u3 - u1\right)\right)} \]
  7. metadata-evalN/A
    \[\leadsto \frac{1}{2} \cdot \left({amt}^{2} \cdot \left(\left(2 \cdot u1 + 4 \cdot u3\right) - u4\right) + {amt}^{3} \cdot \left(u4 - \left(u1 + 3 \cdot u3\right)\right)\right) + \color{blue}{\frac{1}{2}} \cdot \left(amt \cdot \left(u3 - u1\right)\right) \]
  8. *-commutativeN/A
    \[\leadsto \frac{1}{2} \cdot \left({amt}^{2} \cdot \left(\left(2 \cdot u1 + 4 \cdot u3\right) - u4\right) + {amt}^{3} \cdot \left(u4 - \left(u1 + 3 \cdot u3\right)\right)\right) + \color{blue}{\left(amt \cdot \left(u3 - u1\right)\right) \cdot \frac{1}{2}} \]
  9. associate-*r*N/A
    \[\leadsto \frac{1}{2} \cdot \left({amt}^{2} \cdot \left(\left(2 \cdot u1 + 4 \cdot u3\right) - u4\right) + {amt}^{3} \cdot \left(u4 - \left(u1 + 3 \cdot u3\right)\right)\right) + \color{blue}{amt \cdot \left(\left(u3 - u1\right) \cdot \frac{1}{2}\right)} \]
  10. *-commutativeN/A
    \[\leadsto \frac{1}{2} \cdot \left({amt}^{2} \cdot \left(\left(2 \cdot u1 + 4 \cdot u3\right) - u4\right) + {amt}^{3} \cdot \left(u4 - \left(u1 + 3 \cdot u3\right)\right)\right) + amt \cdot \color{blue}{\left(\frac{1}{2} \cdot \left(u3 - u1\right)\right)} \]
5. Applied rewrites83.5%
  \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(amt \cdot \left(amt \cdot amt\right), u4 - \mathsf{fma}\left(3, u3, u1\right), amt \cdot \left(amt \cdot \mathsf{fma}\left(2, u1, u3 \cdot 4 - u4\right)\right)\right), \left(amt \cdot \left(u3 - u1\right)\right) \cdot 0.5\right)} \]
6. Taylor expanded in u4 around inf
  \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(u4 \cdot \left(-1 \cdot {amt}^{2} + {amt}^{3}\right)\right)} \]
7. Step-by-step derivation
8. Applied rewrites68.0%
  \[\leadsto \left(0.5 \cdot u4\right) \cdot \color{blue}{\mathsf{fma}\left(amt, amt \cdot amt, -amt \cdot amt\right)} \]
if -1.4999999e18 < u4 < 3.99999993e13
1. Initial program 77.7%
  \[7.62939453125 \cdot 10^{-6} \cdot \left(\left(\left(2 \cdot u2 + \left(u3 - u1\right) \cdot \left(amt \cdot 65536\right)\right) + \left(\left(\left(2 \cdot u1 - 5 \cdot u2\right) + 4 \cdot u3\right) - u4\right) \cdot \left({amt}^{2} \cdot 65536\right)\right) + \left(\left(\left(3 \cdot u2 - u1\right) - 3 \cdot u3\right) + u4\right) \cdot \left({amt}^{3} \cdot 65536\right)\right) \]
2. Add Preprocessing
3. Taylor expanded in amt around 0
  \[\leadsto \color{blue}{\frac{1}{65536} \cdot u2 + amt \cdot \left(\frac{1}{2} \cdot \left(u3 - u1\right) + amt \cdot \left(\frac{1}{2} \cdot \left(amt \cdot \left(\left(u4 + 3 \cdot u2\right) - \left(u1 + 3 \cdot u3\right)\right)\right) + \frac{1}{2} \cdot \left(\left(2 \cdot u1 + 4 \cdot u3\right) - \left(u4 + 5 \cdot u2\right)\right)\right)\right)} \]
4. Step-by-step derivation
  1. +-commutativeN/A
    \[\leadsto \color{blue}{amt \cdot \left(\frac{1}{2} \cdot \left(u3 - u1\right) + amt \cdot \left(\frac{1}{2} \cdot \left(amt \cdot \left(\left(u4 + 3 \cdot u2\right) - \left(u1 + 3 \cdot u3\right)\right)\right) + \frac{1}{2} \cdot \left(\left(2 \cdot u1 + 4 \cdot u3\right) - \left(u4 + 5 \cdot u2\right)\right)\right)\right) + \frac{1}{65536} \cdot u2} \]
  2. lower-fma.f32N/A
    \[\leadsto \color{blue}{\mathsf{fma}\left(amt, \frac{1}{2} \cdot \left(u3 - u1\right) + amt \cdot \left(\frac{1}{2} \cdot \left(amt \cdot \left(\left(u4 + 3 \cdot u2\right) - \left(u1 + 3 \cdot u3\right)\right)\right) + \frac{1}{2} \cdot \left(\left(2 \cdot u1 + 4 \cdot u3\right) - \left(u4 + 5 \cdot u2\right)\right)\right), \frac{1}{65536} \cdot u2\right)} \]
5. Applied rewrites93.9%
  \[\leadsto \color{blue}{\mathsf{fma}\left(amt, \mathsf{fma}\left(amt, 0.5 \cdot \mathsf{fma}\left(amt, \mathsf{fma}\left(3, u2 - u3, u4\right) - u1, \mathsf{fma}\left(2, u1, \mathsf{fma}\left(u3, 4, u2 \cdot -5\right)\right) - u4\right), \left(u3 - u1\right) \cdot 0.5\right), u2 \cdot 1.52587890625 \cdot 10^{-5}\right)} \]
6. Taylor expanded in u1 around inf
  \[\leadsto \mathsf{fma}\left(amt, \mathsf{fma}\left(amt, \frac{1}{2} \cdot \left(u1 \cdot \color{blue}{\left(2 + -1 \cdot amt\right)}\right), \left(u3 - u1\right) \cdot \frac{1}{2}\right), u2 \cdot \frac{1}{65536}\right) \]
7. Step-by-step derivation
8. Applied rewrites36.0%
  \[\leadsto \mathsf{fma}\left(amt, \mathsf{fma}\left(amt, 0.5 \cdot \left(u1 \cdot \color{blue}{\left(2 - amt\right)}\right), \left(u3 - u1\right) \cdot 0.5\right), u2 \cdot 1.52587890625 \cdot 10^{-5}\right) \]
if 3.99999993e13 < u4
1. Initial program 79.5%
  \[7.62939453125 \cdot 10^{-6} \cdot \left(\left(\left(2 \cdot u2 + \left(u3 - u1\right) \cdot \left(amt \cdot 65536\right)\right) + \left(\left(\left(2 \cdot u1 - 5 \cdot u2\right) + 4 \cdot u3\right) - u4\right) \cdot \left({amt}^{2} \cdot 65536\right)\right) + \left(\left(\left(3 \cdot u2 - u1\right) - 3 \cdot u3\right) + u4\right) \cdot \left({amt}^{3} \cdot 65536\right)\right) \]
2. Add Preprocessing
3. Taylor expanded in amt around 0
  \[\leadsto \color{blue}{\frac{1}{65536} \cdot u2 + amt \cdot \left(\frac{1}{2} \cdot \left(u3 - u1\right) + amt \cdot \left(\frac{1}{2} \cdot \left(amt \cdot \left(\left(u4 + 3 \cdot u2\right) - \left(u1 + 3 \cdot u3\right)\right)\right) + \frac{1}{2} \cdot \left(\left(2 \cdot u1 + 4 \cdot u3\right) - \left(u4 + 5 \cdot u2\right)\right)\right)\right)} \]
4. Step-by-step derivation
  1. +-commutativeN/A
    \[\leadsto \color{blue}{amt \cdot \left(\frac{1}{2} \cdot \left(u3 - u1\right) + amt \cdot \left(\frac{1}{2} \cdot \left(amt \cdot \left(\left(u4 + 3 \cdot u2\right) - \left(u1 + 3 \cdot u3\right)\right)\right) + \frac{1}{2} \cdot \left(\left(2 \cdot u1 + 4 \cdot u3\right) - \left(u4 + 5 \cdot u2\right)\right)\right)\right) + \frac{1}{65536} \cdot u2} \]
  2. lower-fma.f32N/A
    \[\leadsto \color{blue}{\mathsf{fma}\left(amt, \frac{1}{2} \cdot \left(u3 - u1\right) + amt \cdot \left(\frac{1}{2} \cdot \left(amt \cdot \left(\left(u4 + 3 \cdot u2\right) - \left(u1 + 3 \cdot u3\right)\right)\right) + \frac{1}{2} \cdot \left(\left(2 \cdot u1 + 4 \cdot u3\right) - \left(u4 + 5 \cdot u2\right)\right)\right), \frac{1}{65536} \cdot u2\right)} \]
5. Applied rewrites96.5%
  \[\leadsto \color{blue}{\mathsf{fma}\left(amt, \mathsf{fma}\left(amt, 0.5 \cdot \mathsf{fma}\left(amt, \mathsf{fma}\left(3, u2 - u3, u4\right) - u1, \mathsf{fma}\left(2, u1, \mathsf{fma}\left(u3, 4, u2 \cdot -5\right)\right) - u4\right), \left(u3 - u1\right) \cdot 0.5\right), u2 \cdot 1.52587890625 \cdot 10^{-5}\right)} \]
6. Taylor expanded in u4 around inf
  \[\leadsto \frac{1}{2} \cdot \color{blue}{\left({amt}^{2} \cdot \left(u4 \cdot \left(amt - 1\right)\right)\right)} \]
7. Step-by-step derivation
8. Applied rewrites62.4%
  \[\leadsto \left(0.5 \cdot \left(amt \cdot amt\right)\right) \cdot \color{blue}{\left(u4 \cdot \left(amt + -1\right)\right)} \]
Recombined 3 regimes into one program.
Final simplification45.3%
\[\leadsto \begin{array}{l} \mathbf{if}\;u4 \leq -1499999942100385800:\\ \;\;\;\;\left(u4 \cdot 0.5\right) \cdot \mathsf{fma}\left(amt, amt \cdot amt, amt \cdot \left(-amt\right)\right)\\ \mathbf{elif}\;u4 \leq 39999999311872:\\ \;\;\;\;\mathsf{fma}\left(amt, \mathsf{fma}\left(amt, 0.5 \cdot \left(u1 \cdot \left(2 - amt\right)\right), 0.5 \cdot \left(u3 - u1\right)\right), u2 \cdot 1.52587890625 \cdot 10^{-5}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \left(amt \cdot amt\right)\right) \cdot \left(u4 \cdot \left(amt + -1\right)\right)\\ \end{array} \]
Add Preprocessing

Alternative 14: 40.5% accurate, 6.2× speedup?

\[\begin{array}{l} \\ \begin{array}{l} t_0 := u2 \cdot \mathsf{fma}\left(0.5 \cdot \left(amt \cdot amt\right), \mathsf{fma}\left(3, amt, -5\right), 1.52587890625 \cdot 10^{-5}\right)\\ \mathbf{if}\;u2 \leq -5000000:\\ \;\;\;\;t_0\\ \mathbf{elif}\;u2 \leq -1.0000000272452012 \cdot 10^{-27}:\\ \;\;\;\;\left(u3 \cdot amt\right) \cdot \mathsf{fma}\left(amt \cdot 0.5, \mathsf{fma}\left(-3, amt, 4\right), 0.5\right)\\ \mathbf{elif}\;u2 \leq 5000000:\\ \;\;\;\;\mathsf{fma}\left(amt, 0.5 \cdot \left(\left(amt + -1\right) \cdot \left(amt \cdot u4\right)\right), u2 \cdot 1.52587890625 \cdot 10^{-5}\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

(FPCore (u2 u3 u1 amt u4)
 :precision binary32
 (let* ((t_0
         (* u2 (fma (* 0.5 (* amt amt)) (fma 3.0 amt -5.0) 1.52587890625e-5))))
   (if (<= u2 -5000000.0)
     t_0
     (if (<= u2 -1.0000000272452012e-27)
       (* (* u3 amt) (fma (* amt 0.5) (fma -3.0 amt 4.0) 0.5))
       (if (<= u2 5000000.0)
         (fma amt (* 0.5 (* (+ amt -1.0) (* amt u4))) (* u2 1.52587890625e-5))
         t_0)))))

float code(float u2, float u3, float u1, float amt, float u4) {
	float t_0 = u2 * fmaf((0.5f * (amt * amt)), fmaf(3.0f, amt, -5.0f), 1.52587890625e-5f);
	float tmp;
	if (u2 <= -5000000.0f) {
		tmp = t_0;
	} else if (u2 <= -1.0000000272452012e-27f) {
		tmp = (u3 * amt) * fmaf((amt * 0.5f), fmaf(-3.0f, amt, 4.0f), 0.5f);
	} else if (u2 <= 5000000.0f) {
		tmp = fmaf(amt, (0.5f * ((amt + -1.0f) * (amt * u4))), (u2 * 1.52587890625e-5f));
	} else {
		tmp = t_0;
	}
	return tmp;
}

function code(u2, u3, u1, amt, u4)
	t_0 = Float32(u2 * fma(Float32(Float32(0.5) * Float32(amt * amt)), fma(Float32(3.0), amt, Float32(-5.0)), Float32(1.52587890625e-5)))
	tmp = Float32(0.0)
	if (u2 <= Float32(-5000000.0))
		tmp = t_0;
	elseif (u2 <= Float32(-1.0000000272452012e-27))
		tmp = Float32(Float32(u3 * amt) * fma(Float32(amt * Float32(0.5)), fma(Float32(-3.0), amt, Float32(4.0)), Float32(0.5)));
	elseif (u2 <= Float32(5000000.0))
		tmp = fma(amt, Float32(Float32(0.5) * Float32(Float32(amt + Float32(-1.0)) * Float32(amt * u4))), Float32(u2 * Float32(1.52587890625e-5)));
	else
		tmp = t_0;
	end
	return tmp
end

\begin{array}{l}

\\
\begin{array}{l}
t_0 := u2 \cdot \mathsf{fma}\left(0.5 \cdot \left(amt \cdot amt\right), \mathsf{fma}\left(3, amt, -5\right), 1.52587890625 \cdot 10^{-5}\right)\\
\mathbf{if}\;u2 \leq -5000000:\\
\;\;\;\;t_0\\

\mathbf{elif}\;u2 \leq -1.0000000272452012 \cdot 10^{-27}:\\
\;\;\;\;\left(u3 \cdot amt\right) \cdot \mathsf{fma}\left(amt \cdot 0.5, \mathsf{fma}\left(-3, amt, 4\right), 0.5\right)\\

\mathbf{elif}\;u2 \leq 5000000:\\
\;\;\;\;\mathsf{fma}\left(amt, 0.5 \cdot \left(\left(amt + -1\right) \cdot \left(amt \cdot u4\right)\right), u2 \cdot 1.52587890625 \cdot 10^{-5}\right)\\

\mathbf{else}:\\
\;\;\;\;t_0\\


\end{array}
\end{array}

Derivation

Split input into 3 regimes
if u2 < -5e6 or 5e6 < u2
1. Initial program 65.2%
  \[7.62939453125 \cdot 10^{-6} \cdot \left(\left(\left(2 \cdot u2 + \left(u3 - u1\right) \cdot \left(amt \cdot 65536\right)\right) + \left(\left(\left(2 \cdot u1 - 5 \cdot u2\right) + 4 \cdot u3\right) - u4\right) \cdot \left({amt}^{2} \cdot 65536\right)\right) + \left(\left(\left(3 \cdot u2 - u1\right) - 3 \cdot u3\right) + u4\right) \cdot \left({amt}^{3} \cdot 65536\right)\right) \]
2. Add Preprocessing
3. Taylor expanded in amt around 0
  \[\leadsto \color{blue}{\frac{1}{65536} \cdot u2 + amt \cdot \left(\frac{1}{2} \cdot \left(u3 - u1\right) + amt \cdot \left(\frac{1}{2} \cdot \left(amt \cdot \left(\left(u4 + 3 \cdot u2\right) - \left(u1 + 3 \cdot u3\right)\right)\right) + \frac{1}{2} \cdot \left(\left(2 \cdot u1 + 4 \cdot u3\right) - \left(u4 + 5 \cdot u2\right)\right)\right)\right)} \]
4. Step-by-step derivation
  1. +-commutativeN/A
    \[\leadsto \color{blue}{amt \cdot \left(\frac{1}{2} \cdot \left(u3 - u1\right) + amt \cdot \left(\frac{1}{2} \cdot \left(amt \cdot \left(\left(u4 + 3 \cdot u2\right) - \left(u1 + 3 \cdot u3\right)\right)\right) + \frac{1}{2} \cdot \left(\left(2 \cdot u1 + 4 \cdot u3\right) - \left(u4 + 5 \cdot u2\right)\right)\right)\right) + \frac{1}{65536} \cdot u2} \]
  2. lower-fma.f32N/A
    \[\leadsto \color{blue}{\mathsf{fma}\left(amt, \frac{1}{2} \cdot \left(u3 - u1\right) + amt \cdot \left(\frac{1}{2} \cdot \left(amt \cdot \left(\left(u4 + 3 \cdot u2\right) - \left(u1 + 3 \cdot u3\right)\right)\right) + \frac{1}{2} \cdot \left(\left(2 \cdot u1 + 4 \cdot u3\right) - \left(u4 + 5 \cdot u2\right)\right)\right), \frac{1}{65536} \cdot u2\right)} \]
5. Applied rewrites92.8%
  \[\leadsto \color{blue}{\mathsf{fma}\left(amt, \mathsf{fma}\left(amt, 0.5 \cdot \mathsf{fma}\left(amt, \mathsf{fma}\left(3, u2 - u3, u4\right) - u1, \mathsf{fma}\left(2, u1, \mathsf{fma}\left(u3, 4, u2 \cdot -5\right)\right) - u4\right), \left(u3 - u1\right) \cdot 0.5\right), u2 \cdot 1.52587890625 \cdot 10^{-5}\right)} \]
6. Taylor expanded in u2 around inf
  \[\leadsto u2 \cdot \color{blue}{\left(\frac{1}{65536} + \frac{1}{2} \cdot \left({amt}^{2} \cdot \left(3 \cdot amt - 5\right)\right)\right)} \]
7. Step-by-step derivation
8. Applied rewrites56.0%
  \[\leadsto u2 \cdot \color{blue}{\mathsf{fma}\left(0.5 \cdot \left(amt \cdot amt\right), \mathsf{fma}\left(3, amt, -5\right), 1.52587890625 \cdot 10^{-5}\right)} \]
if -5e6 < u2 < -1.00000003e-27
1. Initial program 83.1%
  \[7.62939453125 \cdot 10^{-6} \cdot \left(\left(\left(2 \cdot u2 + \left(u3 - u1\right) \cdot \left(amt \cdot 65536\right)\right) + \left(\left(\left(2 \cdot u1 - 5 \cdot u2\right) + 4 \cdot u3\right) - u4\right) \cdot \left({amt}^{2} \cdot 65536\right)\right) + \left(\left(\left(3 \cdot u2 - u1\right) - 3 \cdot u3\right) + u4\right) \cdot \left({amt}^{3} \cdot 65536\right)\right) \]
2. Add Preprocessing
3. Taylor expanded in amt around 0
  \[\leadsto \color{blue}{\frac{1}{65536} \cdot u2 + amt \cdot \left(\frac{1}{2} \cdot \left(u3 - u1\right) + amt \cdot \left(\frac{1}{2} \cdot \left(amt \cdot \left(\left(u4 + 3 \cdot u2\right) - \left(u1 + 3 \cdot u3\right)\right)\right) + \frac{1}{2} \cdot \left(\left(2 \cdot u1 + 4 \cdot u3\right) - \left(u4 + 5 \cdot u2\right)\right)\right)\right)} \]
4. Step-by-step derivation
  1. +-commutativeN/A
    \[\leadsto \color{blue}{amt \cdot \left(\frac{1}{2} \cdot \left(u3 - u1\right) + amt \cdot \left(\frac{1}{2} \cdot \left(amt \cdot \left(\left(u4 + 3 \cdot u2\right) - \left(u1 + 3 \cdot u3\right)\right)\right) + \frac{1}{2} \cdot \left(\left(2 \cdot u1 + 4 \cdot u3\right) - \left(u4 + 5 \cdot u2\right)\right)\right)\right) + \frac{1}{65536} \cdot u2} \]
  2. lower-fma.f32N/A
    \[\leadsto \color{blue}{\mathsf{fma}\left(amt, \frac{1}{2} \cdot \left(u3 - u1\right) + amt \cdot \left(\frac{1}{2} \cdot \left(amt \cdot \left(\left(u4 + 3 \cdot u2\right) - \left(u1 + 3 \cdot u3\right)\right)\right) + \frac{1}{2} \cdot \left(\left(2 \cdot u1 + 4 \cdot u3\right) - \left(u4 + 5 \cdot u2\right)\right)\right), \frac{1}{65536} \cdot u2\right)} \]
5. Applied rewrites96.1%
  \[\leadsto \color{blue}{\mathsf{fma}\left(amt, \mathsf{fma}\left(amt, 0.5 \cdot \mathsf{fma}\left(amt, \mathsf{fma}\left(3, u2 - u3, u4\right) - u1, \mathsf{fma}\left(2, u1, \mathsf{fma}\left(u3, 4, u2 \cdot -5\right)\right) - u4\right), \left(u3 - u1\right) \cdot 0.5\right), u2 \cdot 1.52587890625 \cdot 10^{-5}\right)} \]
6. Taylor expanded in u3 around inf
  \[\leadsto amt \cdot \color{blue}{\left(u3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \left(amt \cdot \left(4 + -3 \cdot amt\right)\right)\right)\right)} \]
7. Step-by-step derivation
8. Applied rewrites40.1%
  \[\leadsto \left(amt \cdot u3\right) \cdot \color{blue}{\mathsf{fma}\left(0.5 \cdot amt, \mathsf{fma}\left(-3, amt, 4\right), 0.5\right)} \]
if -1.00000003e-27 < u2 < 5e6
1. Initial program 82.3%
  \[7.62939453125 \cdot 10^{-6} \cdot \left(\left(\left(2 \cdot u2 + \left(u3 - u1\right) \cdot \left(amt \cdot 65536\right)\right) + \left(\left(\left(2 \cdot u1 - 5 \cdot u2\right) + 4 \cdot u3\right) - u4\right) \cdot \left({amt}^{2} \cdot 65536\right)\right) + \left(\left(\left(3 \cdot u2 - u1\right) - 3 \cdot u3\right) + u4\right) \cdot \left({amt}^{3} \cdot 65536\right)\right) \]
2. Add Preprocessing
3. Taylor expanded in amt around 0
  \[\leadsto \color{blue}{\frac{1}{65536} \cdot u2 + amt \cdot \left(\frac{1}{2} \cdot \left(u3 - u1\right) + amt \cdot \left(\frac{1}{2} \cdot \left(amt \cdot \left(\left(u4 + 3 \cdot u2\right) - \left(u1 + 3 \cdot u3\right)\right)\right) + \frac{1}{2} \cdot \left(\left(2 \cdot u1 + 4 \cdot u3\right) - \left(u4 + 5 \cdot u2\right)\right)\right)\right)} \]
4. Step-by-step derivation
  1. +-commutativeN/A
    \[\leadsto \color{blue}{amt \cdot \left(\frac{1}{2} \cdot \left(u3 - u1\right) + amt \cdot \left(\frac{1}{2} \cdot \left(amt \cdot \left(\left(u4 + 3 \cdot u2\right) - \left(u1 + 3 \cdot u3\right)\right)\right) + \frac{1}{2} \cdot \left(\left(2 \cdot u1 + 4 \cdot u3\right) - \left(u4 + 5 \cdot u2\right)\right)\right)\right) + \frac{1}{65536} \cdot u2} \]
  2. lower-fma.f32N/A
    \[\leadsto \color{blue}{\mathsf{fma}\left(amt, \frac{1}{2} \cdot \left(u3 - u1\right) + amt \cdot \left(\frac{1}{2} \cdot \left(amt \cdot \left(\left(u4 + 3 \cdot u2\right) - \left(u1 + 3 \cdot u3\right)\right)\right) + \frac{1}{2} \cdot \left(\left(2 \cdot u1 + 4 \cdot u3\right) - \left(u4 + 5 \cdot u2\right)\right)\right), \frac{1}{65536} \cdot u2\right)} \]
5. Applied rewrites96.6%
  \[\leadsto \color{blue}{\mathsf{fma}\left(amt, \mathsf{fma}\left(amt, 0.5 \cdot \mathsf{fma}\left(amt, \mathsf{fma}\left(3, u2 - u3, u4\right) - u1, \mathsf{fma}\left(2, u1, \mathsf{fma}\left(u3, 4, u2 \cdot -5\right)\right) - u4\right), \left(u3 - u1\right) \cdot 0.5\right), u2 \cdot 1.52587890625 \cdot 10^{-5}\right)} \]
6. Taylor expanded in u4 around inf
  \[\leadsto \mathsf{fma}\left(amt, \frac{1}{2} \cdot \color{blue}{\left(amt \cdot \left(u4 \cdot \left(amt - 1\right)\right)\right)}, u2 \cdot \frac{1}{65536}\right) \]
7. Step-by-step derivation
8. Applied rewrites45.4%
  \[\leadsto \mathsf{fma}\left(amt, 0.5 \cdot \color{blue}{\left(\left(amt \cdot u4\right) \cdot \left(amt + -1\right)\right)}, u2 \cdot 1.52587890625 \cdot 10^{-5}\right) \]
Recombined 3 regimes into one program.
Final simplification48.3%
\[\leadsto \begin{array}{l} \mathbf{if}\;u2 \leq -5000000:\\ \;\;\;\;u2 \cdot \mathsf{fma}\left(0.5 \cdot \left(amt \cdot amt\right), \mathsf{fma}\left(3, amt, -5\right), 1.52587890625 \cdot 10^{-5}\right)\\ \mathbf{elif}\;u2 \leq -1.0000000272452012 \cdot 10^{-27}:\\ \;\;\;\;\left(u3 \cdot amt\right) \cdot \mathsf{fma}\left(amt \cdot 0.5, \mathsf{fma}\left(-3, amt, 4\right), 0.5\right)\\ \mathbf{elif}\;u2 \leq 5000000:\\ \;\;\;\;\mathsf{fma}\left(amt, 0.5 \cdot \left(\left(amt + -1\right) \cdot \left(amt \cdot u4\right)\right), u2 \cdot 1.52587890625 \cdot 10^{-5}\right)\\ \mathbf{else}:\\ \;\;\;\;u2 \cdot \mathsf{fma}\left(0.5 \cdot \left(amt \cdot amt\right), \mathsf{fma}\left(3, amt, -5\right), 1.52587890625 \cdot 10^{-5}\right)\\ \end{array} \]
Add Preprocessing

Alternative 15: 40.5% accurate, 6.5× speedup?

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \left(amt \cdot amt\right)\\ t_1 := u2 \cdot \mathsf{fma}\left(t_0, \mathsf{fma}\left(3, amt, -5\right), 1.52587890625 \cdot 10^{-5}\right)\\ \mathbf{if}\;u2 \leq -5000000:\\ \;\;\;\;t_1\\ \mathbf{elif}\;u2 \leq -1.0000000272452012 \cdot 10^{-27}:\\ \;\;\;\;\left(u3 \cdot amt\right) \cdot \mathsf{fma}\left(amt \cdot 0.5, \mathsf{fma}\left(-3, amt, 4\right), 0.5\right)\\ \mathbf{elif}\;u2 \leq 5000000:\\ \;\;\;\;t_0 \cdot \left(u4 \cdot \left(amt + -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

(FPCore (u2 u3 u1 amt u4)
 :precision binary32
 (let* ((t_0 (* 0.5 (* amt amt)))
        (t_1 (* u2 (fma t_0 (fma 3.0 amt -5.0) 1.52587890625e-5))))
   (if (<= u2 -5000000.0)
     t_1
     (if (<= u2 -1.0000000272452012e-27)
       (* (* u3 amt) (fma (* amt 0.5) (fma -3.0 amt 4.0) 0.5))
       (if (<= u2 5000000.0) (* t_0 (* u4 (+ amt -1.0))) t_1)))))

float code(float u2, float u3, float u1, float amt, float u4) {
	float t_0 = 0.5f * (amt * amt);
	float t_1 = u2 * fmaf(t_0, fmaf(3.0f, amt, -5.0f), 1.52587890625e-5f);
	float tmp;
	if (u2 <= -5000000.0f) {
		tmp = t_1;
	} else if (u2 <= -1.0000000272452012e-27f) {
		tmp = (u3 * amt) * fmaf((amt * 0.5f), fmaf(-3.0f, amt, 4.0f), 0.5f);
	} else if (u2 <= 5000000.0f) {
		tmp = t_0 * (u4 * (amt + -1.0f));
	} else {
		tmp = t_1;
	}
	return tmp;
}

function code(u2, u3, u1, amt, u4)
	t_0 = Float32(Float32(0.5) * Float32(amt * amt))
	t_1 = Float32(u2 * fma(t_0, fma(Float32(3.0), amt, Float32(-5.0)), Float32(1.52587890625e-5)))
	tmp = Float32(0.0)
	if (u2 <= Float32(-5000000.0))
		tmp = t_1;
	elseif (u2 <= Float32(-1.0000000272452012e-27))
		tmp = Float32(Float32(u3 * amt) * fma(Float32(amt * Float32(0.5)), fma(Float32(-3.0), amt, Float32(4.0)), Float32(0.5)));
	elseif (u2 <= Float32(5000000.0))
		tmp = Float32(t_0 * Float32(u4 * Float32(amt + Float32(-1.0))));
	else
		tmp = t_1;
	end
	return tmp
end

\begin{array}{l}

\\
\begin{array}{l}
t_0 := 0.5 \cdot \left(amt \cdot amt\right)\\
t_1 := u2 \cdot \mathsf{fma}\left(t_0, \mathsf{fma}\left(3, amt, -5\right), 1.52587890625 \cdot 10^{-5}\right)\\
\mathbf{if}\;u2 \leq -5000000:\\
\;\;\;\;t_1\\

\mathbf{elif}\;u2 \leq -1.0000000272452012 \cdot 10^{-27}:\\
\;\;\;\;\left(u3 \cdot amt\right) \cdot \mathsf{fma}\left(amt \cdot 0.5, \mathsf{fma}\left(-3, amt, 4\right), 0.5\right)\\

\mathbf{elif}\;u2 \leq 5000000:\\
\;\;\;\;t_0 \cdot \left(u4 \cdot \left(amt + -1\right)\right)\\

\mathbf{else}:\\
\;\;\;\;t_1\\


\end{array}
\end{array}

Derivation

Split input into 3 regimes
if u2 < -5e6 or 5e6 < u2
1. Initial program 65.2%
  \[7.62939453125 \cdot 10^{-6} \cdot \left(\left(\left(2 \cdot u2 + \left(u3 - u1\right) \cdot \left(amt \cdot 65536\right)\right) + \left(\left(\left(2 \cdot u1 - 5 \cdot u2\right) + 4 \cdot u3\right) - u4\right) \cdot \left({amt}^{2} \cdot 65536\right)\right) + \left(\left(\left(3 \cdot u2 - u1\right) - 3 \cdot u3\right) + u4\right) \cdot \left({amt}^{3} \cdot 65536\right)\right) \]
2. Add Preprocessing
3. Taylor expanded in amt around 0
  \[\leadsto \color{blue}{\frac{1}{65536} \cdot u2 + amt \cdot \left(\frac{1}{2} \cdot \left(u3 - u1\right) + amt \cdot \left(\frac{1}{2} \cdot \left(amt \cdot \left(\left(u4 + 3 \cdot u2\right) - \left(u1 + 3 \cdot u3\right)\right)\right) + \frac{1}{2} \cdot \left(\left(2 \cdot u1 + 4 \cdot u3\right) - \left(u4 + 5 \cdot u2\right)\right)\right)\right)} \]
4. Step-by-step derivation
  1. +-commutativeN/A
    \[\leadsto \color{blue}{amt \cdot \left(\frac{1}{2} \cdot \left(u3 - u1\right) + amt \cdot \left(\frac{1}{2} \cdot \left(amt \cdot \left(\left(u4 + 3 \cdot u2\right) - \left(u1 + 3 \cdot u3\right)\right)\right) + \frac{1}{2} \cdot \left(\left(2 \cdot u1 + 4 \cdot u3\right) - \left(u4 + 5 \cdot u2\right)\right)\right)\right) + \frac{1}{65536} \cdot u2} \]
  2. lower-fma.f32N/A
    \[\leadsto \color{blue}{\mathsf{fma}\left(amt, \frac{1}{2} \cdot \left(u3 - u1\right) + amt \cdot \left(\frac{1}{2} \cdot \left(amt \cdot \left(\left(u4 + 3 \cdot u2\right) - \left(u1 + 3 \cdot u3\right)\right)\right) + \frac{1}{2} \cdot \left(\left(2 \cdot u1 + 4 \cdot u3\right) - \left(u4 + 5 \cdot u2\right)\right)\right), \frac{1}{65536} \cdot u2\right)} \]
5. Applied rewrites92.8%
  \[\leadsto \color{blue}{\mathsf{fma}\left(amt, \mathsf{fma}\left(amt, 0.5 \cdot \mathsf{fma}\left(amt, \mathsf{fma}\left(3, u2 - u3, u4\right) - u1, \mathsf{fma}\left(2, u1, \mathsf{fma}\left(u3, 4, u2 \cdot -5\right)\right) - u4\right), \left(u3 - u1\right) \cdot 0.5\right), u2 \cdot 1.52587890625 \cdot 10^{-5}\right)} \]
6. Taylor expanded in u2 around inf
  \[\leadsto u2 \cdot \color{blue}{\left(\frac{1}{65536} + \frac{1}{2} \cdot \left({amt}^{2} \cdot \left(3 \cdot amt - 5\right)\right)\right)} \]
7. Step-by-step derivation
8. Applied rewrites56.0%
  \[\leadsto u2 \cdot \color{blue}{\mathsf{fma}\left(0.5 \cdot \left(amt \cdot amt\right), \mathsf{fma}\left(3, amt, -5\right), 1.52587890625 \cdot 10^{-5}\right)} \]
if -5e6 < u2 < -1.00000003e-27
1. Initial program 83.1%
  \[7.62939453125 \cdot 10^{-6} \cdot \left(\left(\left(2 \cdot u2 + \left(u3 - u1\right) \cdot \left(amt \cdot 65536\right)\right) + \left(\left(\left(2 \cdot u1 - 5 \cdot u2\right) + 4 \cdot u3\right) - u4\right) \cdot \left({amt}^{2} \cdot 65536\right)\right) + \left(\left(\left(3 \cdot u2 - u1\right) - 3 \cdot u3\right) + u4\right) \cdot \left({amt}^{3} \cdot 65536\right)\right) \]
2. Add Preprocessing
3. Taylor expanded in amt around 0
  \[\leadsto \color{blue}{\frac{1}{65536} \cdot u2 + amt \cdot \left(\frac{1}{2} \cdot \left(u3 - u1\right) + amt \cdot \left(\frac{1}{2} \cdot \left(amt \cdot \left(\left(u4 + 3 \cdot u2\right) - \left(u1 + 3 \cdot u3\right)\right)\right) + \frac{1}{2} \cdot \left(\left(2 \cdot u1 + 4 \cdot u3\right) - \left(u4 + 5 \cdot u2\right)\right)\right)\right)} \]
4. Step-by-step derivation
  1. +-commutativeN/A
    \[\leadsto \color{blue}{amt \cdot \left(\frac{1}{2} \cdot \left(u3 - u1\right) + amt \cdot \left(\frac{1}{2} \cdot \left(amt \cdot \left(\left(u4 + 3 \cdot u2\right) - \left(u1 + 3 \cdot u3\right)\right)\right) + \frac{1}{2} \cdot \left(\left(2 \cdot u1 + 4 \cdot u3\right) - \left(u4 + 5 \cdot u2\right)\right)\right)\right) + \frac{1}{65536} \cdot u2} \]
  2. lower-fma.f32N/A
    \[\leadsto \color{blue}{\mathsf{fma}\left(amt, \frac{1}{2} \cdot \left(u3 - u1\right) + amt \cdot \left(\frac{1}{2} \cdot \left(amt \cdot \left(\left(u4 + 3 \cdot u2\right) - \left(u1 + 3 \cdot u3\right)\right)\right) + \frac{1}{2} \cdot \left(\left(2 \cdot u1 + 4 \cdot u3\right) - \left(u4 + 5 \cdot u2\right)\right)\right), \frac{1}{65536} \cdot u2\right)} \]
5. Applied rewrites96.1%
  \[\leadsto \color{blue}{\mathsf{fma}\left(amt, \mathsf{fma}\left(amt, 0.5 \cdot \mathsf{fma}\left(amt, \mathsf{fma}\left(3, u2 - u3, u4\right) - u1, \mathsf{fma}\left(2, u1, \mathsf{fma}\left(u3, 4, u2 \cdot -5\right)\right) - u4\right), \left(u3 - u1\right) \cdot 0.5\right), u2 \cdot 1.52587890625 \cdot 10^{-5}\right)} \]
6. Taylor expanded in u3 around inf
  \[\leadsto amt \cdot \color{blue}{\left(u3 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \left(amt \cdot \left(4 + -3 \cdot amt\right)\right)\right)\right)} \]
7. Step-by-step derivation
8. Applied rewrites40.1%
  \[\leadsto \left(amt \cdot u3\right) \cdot \color{blue}{\mathsf{fma}\left(0.5 \cdot amt, \mathsf{fma}\left(-3, amt, 4\right), 0.5\right)} \]
if -1.00000003e-27 < u2 < 5e6
1. Initial program 82.3%
  \[7.62939453125 \cdot 10^{-6} \cdot \left(\left(\left(2 \cdot u2 + \left(u3 - u1\right) \cdot \left(amt \cdot 65536\right)\right) + \left(\left(\left(2 \cdot u1 - 5 \cdot u2\right) + 4 \cdot u3\right) - u4\right) \cdot \left({amt}^{2} \cdot 65536\right)\right) + \left(\left(\left(3 \cdot u2 - u1\right) - 3 \cdot u3\right) + u4\right) \cdot \left({amt}^{3} \cdot 65536\right)\right) \]
2. Add Preprocessing
3. Taylor expanded in amt around 0
  \[\leadsto \color{blue}{\frac{1}{65536} \cdot u2 + amt \cdot \left(\frac{1}{2} \cdot \left(u3 - u1\right) + amt \cdot \left(\frac{1}{2} \cdot \left(amt \cdot \left(\left(u4 + 3 \cdot u2\right) - \left(u1 + 3 \cdot u3\right)\right)\right) + \frac{1}{2} \cdot \left(\left(2 \cdot u1 + 4 \cdot u3\right) - \left(u4 + 5 \cdot u2\right)\right)\right)\right)} \]
4. Step-by-step derivation
  1. +-commutativeN/A
    \[\leadsto \color{blue}{amt \cdot \left(\frac{1}{2} \cdot \left(u3 - u1\right) + amt \cdot \left(\frac{1}{2} \cdot \left(amt \cdot \left(\left(u4 + 3 \cdot u2\right) - \left(u1 + 3 \cdot u3\right)\right)\right) + \frac{1}{2} \cdot \left(\left(2 \cdot u1 + 4 \cdot u3\right) - \left(u4 + 5 \cdot u2\right)\right)\right)\right) + \frac{1}{65536} \cdot u2} \]
  2. lower-fma.f32N/A
    \[\leadsto \color{blue}{\mathsf{fma}\left(amt, \frac{1}{2} \cdot \left(u3 - u1\right) + amt \cdot \left(\frac{1}{2} \cdot \left(amt \cdot \left(\left(u4 + 3 \cdot u2\right) - \left(u1 + 3 \cdot u3\right)\right)\right) + \frac{1}{2} \cdot \left(\left(2 \cdot u1 + 4 \cdot u3\right) - \left(u4 + 5 \cdot u2\right)\right)\right), \frac{1}{65536} \cdot u2\right)} \]
5. Applied rewrites96.6%
  \[\leadsto \color{blue}{\mathsf{fma}\left(amt, \mathsf{fma}\left(amt, 0.5 \cdot \mathsf{fma}\left(amt, \mathsf{fma}\left(3, u2 - u3, u4\right) - u1, \mathsf{fma}\left(2, u1, \mathsf{fma}\left(u3, 4, u2 \cdot -5\right)\right) - u4\right), \left(u3 - u1\right) \cdot 0.5\right), u2 \cdot 1.52587890625 \cdot 10^{-5}\right)} \]
6. Taylor expanded in u4 around inf
  \[\leadsto \frac{1}{2} \cdot \color{blue}{\left({amt}^{2} \cdot \left(u4 \cdot \left(amt - 1\right)\right)\right)} \]
7. Step-by-step derivation
8. Applied rewrites45.4%
  \[\leadsto \left(0.5 \cdot \left(amt \cdot amt\right)\right) \cdot \color{blue}{\left(u4 \cdot \left(amt + -1\right)\right)} \]
Recombined 3 regimes into one program.
Final simplification48.3%
\[\leadsto \begin{array}{l} \mathbf{if}\;u2 \leq -5000000:\\ \;\;\;\;u2 \cdot \mathsf{fma}\left(0.5 \cdot \left(amt \cdot amt\right), \mathsf{fma}\left(3, amt, -5\right), 1.52587890625 \cdot 10^{-5}\right)\\ \mathbf{elif}\;u2 \leq -1.0000000272452012 \cdot 10^{-27}:\\ \;\;\;\;\left(u3 \cdot amt\right) \cdot \mathsf{fma}\left(amt \cdot 0.5, \mathsf{fma}\left(-3, amt, 4\right), 0.5\right)\\ \mathbf{elif}\;u2 \leq 5000000:\\ \;\;\;\;\left(0.5 \cdot \left(amt \cdot amt\right)\right) \cdot \left(u4 \cdot \left(amt + -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;u2 \cdot \mathsf{fma}\left(0.5 \cdot \left(amt \cdot amt\right), \mathsf{fma}\left(3, amt, -5\right), 1.52587890625 \cdot 10^{-5}\right)\\ \end{array} \]
Add Preprocessing

Alternative 16: 39.5% accurate, 7.4× speedup?

(FPCore (u2 u3 u1 amt u4)
 :precision binary32
 (let* ((t_0 (* 0.5 (* amt amt)))
        (t_1 (* u2 (fma t_0 (fma 3.0 amt -5.0) 1.52587890625e-5))))
   (if (<= u2 -1370.0)
     t_1
     (if (<= u2 5000000.0) (* t_0 (* u4 (+ amt -1.0))) t_1))))

float code(float u2, float u3, float u1, float amt, float u4) {
	float t_0 = 0.5f * (amt * amt);
	float t_1 = u2 * fmaf(t_0, fmaf(3.0f, amt, -5.0f), 1.52587890625e-5f);
	float tmp;
	if (u2 <= -1370.0f) {
		tmp = t_1;
	} else if (u2 <= 5000000.0f) {
		tmp = t_0 * (u4 * (amt + -1.0f));
	} else {
		tmp = t_1;
	}
	return tmp;
}

function code(u2, u3, u1, amt, u4)
	t_0 = Float32(Float32(0.5) * Float32(amt * amt))
	t_1 = Float32(u2 * fma(t_0, fma(Float32(3.0), amt, Float32(-5.0)), Float32(1.52587890625e-5)))
	tmp = Float32(0.0)
	if (u2 <= Float32(-1370.0))
		tmp = t_1;
	elseif (u2 <= Float32(5000000.0))
		tmp = Float32(t_0 * Float32(u4 * Float32(amt + Float32(-1.0))));
	else
		tmp = t_1;
	end
	return tmp
end

\begin{array}{l}

\\
\begin{array}{l}
t_0 := 0.5 \cdot \left(amt \cdot amt\right)\\
t_1 := u2 \cdot \mathsf{fma}\left(t_0, \mathsf{fma}\left(3, amt, -5\right), 1.52587890625 \cdot 10^{-5}\right)\\
\mathbf{if}\;u2 \leq -1370:\\
\;\;\;\;t_1\\

\mathbf{elif}\;u2 \leq 5000000:\\
\;\;\;\;t_0 \cdot \left(u4 \cdot \left(amt + -1\right)\right)\\

\mathbf{else}:\\
\;\;\;\;t_1\\


\end{array}
\end{array}

Derivation

Split input into 2 regimes
if u2 < -1370 or 5e6 < u2
1. Initial program 65.4%
  \[7.62939453125 \cdot 10^{-6} \cdot \left(\left(\left(2 \cdot u2 + \left(u3 - u1\right) \cdot \left(amt \cdot 65536\right)\right) + \left(\left(\left(2 \cdot u1 - 5 \cdot u2\right) + 4 \cdot u3\right) - u4\right) \cdot \left({amt}^{2} \cdot 65536\right)\right) + \left(\left(\left(3 \cdot u2 - u1\right) - 3 \cdot u3\right) + u4\right) \cdot \left({amt}^{3} \cdot 65536\right)\right) \]
2. Add Preprocessing
3. Taylor expanded in amt around 0
  \[\leadsto \color{blue}{\frac{1}{65536} \cdot u2 + amt \cdot \left(\frac{1}{2} \cdot \left(u3 - u1\right) + amt \cdot \left(\frac{1}{2} \cdot \left(amt \cdot \left(\left(u4 + 3 \cdot u2\right) - \left(u1 + 3 \cdot u3\right)\right)\right) + \frac{1}{2} \cdot \left(\left(2 \cdot u1 + 4 \cdot u3\right) - \left(u4 + 5 \cdot u2\right)\right)\right)\right)} \]
4. Step-by-step derivation
  1. +-commutativeN/A
    \[\leadsto \color{blue}{amt \cdot \left(\frac{1}{2} \cdot \left(u3 - u1\right) + amt \cdot \left(\frac{1}{2} \cdot \left(amt \cdot \left(\left(u4 + 3 \cdot u2\right) - \left(u1 + 3 \cdot u3\right)\right)\right) + \frac{1}{2} \cdot \left(\left(2 \cdot u1 + 4 \cdot u3\right) - \left(u4 + 5 \cdot u2\right)\right)\right)\right) + \frac{1}{65536} \cdot u2} \]
  2. lower-fma.f32N/A
    \[\leadsto \color{blue}{\mathsf{fma}\left(amt, \frac{1}{2} \cdot \left(u3 - u1\right) + amt \cdot \left(\frac{1}{2} \cdot \left(amt \cdot \left(\left(u4 + 3 \cdot u2\right) - \left(u1 + 3 \cdot u3\right)\right)\right) + \frac{1}{2} \cdot \left(\left(2 \cdot u1 + 4 \cdot u3\right) - \left(u4 + 5 \cdot u2\right)\right)\right), \frac{1}{65536} \cdot u2\right)} \]
5. Applied rewrites92.9%
  \[\leadsto \color{blue}{\mathsf{fma}\left(amt, \mathsf{fma}\left(amt, 0.5 \cdot \mathsf{fma}\left(amt, \mathsf{fma}\left(3, u2 - u3, u4\right) - u1, \mathsf{fma}\left(2, u1, \mathsf{fma}\left(u3, 4, u2 \cdot -5\right)\right) - u4\right), \left(u3 - u1\right) \cdot 0.5\right), u2 \cdot 1.52587890625 \cdot 10^{-5}\right)} \]
6. Taylor expanded in u2 around inf
  \[\leadsto u2 \cdot \color{blue}{\left(\frac{1}{65536} + \frac{1}{2} \cdot \left({amt}^{2} \cdot \left(3 \cdot amt - 5\right)\right)\right)} \]
7. Step-by-step derivation
8. Applied rewrites54.7%
  \[\leadsto u2 \cdot \color{blue}{\mathsf{fma}\left(0.5 \cdot \left(amt \cdot amt\right), \mathsf{fma}\left(3, amt, -5\right), 1.52587890625 \cdot 10^{-5}\right)} \]
if -1370 < u2 < 5e6
1. Initial program 82.9%
  \[7.62939453125 \cdot 10^{-6} \cdot \left(\left(\left(2 \cdot u2 + \left(u3 - u1\right) \cdot \left(amt \cdot 65536\right)\right) + \left(\left(\left(2 \cdot u1 - 5 \cdot u2\right) + 4 \cdot u3\right) - u4\right) \cdot \left({amt}^{2} \cdot 65536\right)\right) + \left(\left(\left(3 \cdot u2 - u1\right) - 3 \cdot u3\right) + u4\right) \cdot \left({amt}^{3} \cdot 65536\right)\right) \]
2. Add Preprocessing
3. Taylor expanded in amt around 0
  \[\leadsto \color{blue}{\frac{1}{65536} \cdot u2 + amt \cdot \left(\frac{1}{2} \cdot \left(u3 - u1\right) + amt \cdot \left(\frac{1}{2} \cdot \left(amt \cdot \left(\left(u4 + 3 \cdot u2\right) - \left(u1 + 3 \cdot u3\right)\right)\right) + \frac{1}{2} \cdot \left(\left(2 \cdot u1 + 4 \cdot u3\right) - \left(u4 + 5 \cdot u2\right)\right)\right)\right)} \]
4. Step-by-step derivation
  1. +-commutativeN/A
    \[\leadsto \color{blue}{amt \cdot \left(\frac{1}{2} \cdot \left(u3 - u1\right) + amt \cdot \left(\frac{1}{2} \cdot \left(amt \cdot \left(\left(u4 + 3 \cdot u2\right) - \left(u1 + 3 \cdot u3\right)\right)\right) + \frac{1}{2} \cdot \left(\left(2 \cdot u1 + 4 \cdot u3\right) - \left(u4 + 5 \cdot u2\right)\right)\right)\right) + \frac{1}{65536} \cdot u2} \]
  2. lower-fma.f32N/A
    \[\leadsto \color{blue}{\mathsf{fma}\left(amt, \frac{1}{2} \cdot \left(u3 - u1\right) + amt \cdot \left(\frac{1}{2} \cdot \left(amt \cdot \left(\left(u4 + 3 \cdot u2\right) - \left(u1 + 3 \cdot u3\right)\right)\right) + \frac{1}{2} \cdot \left(\left(2 \cdot u1 + 4 \cdot u3\right) - \left(u4 + 5 \cdot u2\right)\right)\right), \frac{1}{65536} \cdot u2\right)} \]
5. Applied rewrites96.4%
  \[\leadsto \color{blue}{\mathsf{fma}\left(amt, \mathsf{fma}\left(amt, 0.5 \cdot \mathsf{fma}\left(amt, \mathsf{fma}\left(3, u2 - u3, u4\right) - u1, \mathsf{fma}\left(2, u1, \mathsf{fma}\left(u3, 4, u2 \cdot -5\right)\right) - u4\right), \left(u3 - u1\right) \cdot 0.5\right), u2 \cdot 1.52587890625 \cdot 10^{-5}\right)} \]
6. Taylor expanded in u4 around inf
  \[\leadsto \frac{1}{2} \cdot \color{blue}{\left({amt}^{2} \cdot \left(u4 \cdot \left(amt - 1\right)\right)\right)} \]
7. Step-by-step derivation
8. Applied rewrites37.6%
  \[\leadsto \left(0.5 \cdot \left(amt \cdot amt\right)\right) \cdot \color{blue}{\left(u4 \cdot \left(amt + -1\right)\right)} \]
Recombined 2 regimes into one program.
Add Preprocessing

Alternative 17: 37.0% accurate, 7.6× speedup?

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(0.5 \cdot \left(amt \cdot amt\right)\right) \cdot \left(u4 \cdot \left(amt + -1\right)\right)\\ \mathbf{if}\;u4 \leq -50000000188416:\\ \;\;\;\;t_0\\ \mathbf{elif}\;u4 \leq 39999999311872:\\ \;\;\;\;amt \cdot \left(\left(-u1\right) \cdot \mathsf{fma}\left(0.5, amt \cdot \left(amt + -2\right), 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

(FPCore (u2 u3 u1 amt u4)
 :precision binary32
 (let* ((t_0 (* (* 0.5 (* amt amt)) (* u4 (+ amt -1.0)))))
   (if (<= u4 -50000000188416.0)
     t_0
     (if (<= u4 39999999311872.0)
       (* amt (* (- u1) (fma 0.5 (* amt (+ amt -2.0)) 0.5)))
       t_0))))

float code(float u2, float u3, float u1, float amt, float u4) {
	float t_0 = (0.5f * (amt * amt)) * (u4 * (amt + -1.0f));
	float tmp;
	if (u4 <= -50000000188416.0f) {
		tmp = t_0;
	} else if (u4 <= 39999999311872.0f) {
		tmp = amt * (-u1 * fmaf(0.5f, (amt * (amt + -2.0f)), 0.5f));
	} else {
		tmp = t_0;
	}
	return tmp;
}

function code(u2, u3, u1, amt, u4)
	t_0 = Float32(Float32(Float32(0.5) * Float32(amt * amt)) * Float32(u4 * Float32(amt + Float32(-1.0))))
	tmp = Float32(0.0)
	if (u4 <= Float32(-50000000188416.0))
		tmp = t_0;
	elseif (u4 <= Float32(39999999311872.0))
		tmp = Float32(amt * Float32(Float32(-u1) * fma(Float32(0.5), Float32(amt * Float32(amt + Float32(-2.0))), Float32(0.5))));
	else
		tmp = t_0;
	end
	return tmp
end

\begin{array}{l}

\\
\begin{array}{l}
t_0 := \left(0.5 \cdot \left(amt \cdot amt\right)\right) \cdot \left(u4 \cdot \left(amt + -1\right)\right)\\
\mathbf{if}\;u4 \leq -50000000188416:\\
\;\;\;\;t_0\\

\mathbf{elif}\;u4 \leq 39999999311872:\\
\;\;\;\;amt \cdot \left(\left(-u1\right) \cdot \mathsf{fma}\left(0.5, amt \cdot \left(amt + -2\right), 0.5\right)\right)\\

\mathbf{else}:\\
\;\;\;\;t_0\\


\end{array}
\end{array}

Derivation

Split input into 2 regimes
if u4 < -5.00000002e13 or 3.99999993e13 < u4
1. Initial program 71.2%
  \[7.62939453125 \cdot 10^{-6} \cdot \left(\left(\left(2 \cdot u2 + \left(u3 - u1\right) \cdot \left(amt \cdot 65536\right)\right) + \left(\left(\left(2 \cdot u1 - 5 \cdot u2\right) + 4 \cdot u3\right) - u4\right) \cdot \left({amt}^{2} \cdot 65536\right)\right) + \left(\left(\left(3 \cdot u2 - u1\right) - 3 \cdot u3\right) + u4\right) \cdot \left({amt}^{3} \cdot 65536\right)\right) \]
2. Add Preprocessing
3. Taylor expanded in amt around 0
  \[\leadsto \color{blue}{\frac{1}{65536} \cdot u2 + amt \cdot \left(\frac{1}{2} \cdot \left(u3 - u1\right) + amt \cdot \left(\frac{1}{2} \cdot \left(amt \cdot \left(\left(u4 + 3 \cdot u2\right) - \left(u1 + 3 \cdot u3\right)\right)\right) + \frac{1}{2} \cdot \left(\left(2 \cdot u1 + 4 \cdot u3\right) - \left(u4 + 5 \cdot u2\right)\right)\right)\right)} \]
4. Step-by-step derivation
  1. +-commutativeN/A
    \[\leadsto \color{blue}{amt \cdot \left(\frac{1}{2} \cdot \left(u3 - u1\right) + amt \cdot \left(\frac{1}{2} \cdot \left(amt \cdot \left(\left(u4 + 3 \cdot u2\right) - \left(u1 + 3 \cdot u3\right)\right)\right) + \frac{1}{2} \cdot \left(\left(2 \cdot u1 + 4 \cdot u3\right) - \left(u4 + 5 \cdot u2\right)\right)\right)\right) + \frac{1}{65536} \cdot u2} \]
  2. lower-fma.f32N/A
    \[\leadsto \color{blue}{\mathsf{fma}\left(amt, \frac{1}{2} \cdot \left(u3 - u1\right) + amt \cdot \left(\frac{1}{2} \cdot \left(amt \cdot \left(\left(u4 + 3 \cdot u2\right) - \left(u1 + 3 \cdot u3\right)\right)\right) + \frac{1}{2} \cdot \left(\left(2 \cdot u1 + 4 \cdot u3\right) - \left(u4 + 5 \cdot u2\right)\right)\right), \frac{1}{65536} \cdot u2\right)} \]
5. Applied rewrites96.0%
  \[\leadsto \color{blue}{\mathsf{fma}\left(amt, \mathsf{fma}\left(amt, 0.5 \cdot \mathsf{fma}\left(amt, \mathsf{fma}\left(3, u2 - u3, u4\right) - u1, \mathsf{fma}\left(2, u1, \mathsf{fma}\left(u3, 4, u2 \cdot -5\right)\right) - u4\right), \left(u3 - u1\right) \cdot 0.5\right), u2 \cdot 1.52587890625 \cdot 10^{-5}\right)} \]
6. Taylor expanded in u4 around inf
  \[\leadsto \frac{1}{2} \cdot \color{blue}{\left({amt}^{2} \cdot \left(u4 \cdot \left(amt - 1\right)\right)\right)} \]
7. Step-by-step derivation
8. Applied rewrites59.6%
  \[\leadsto \left(0.5 \cdot \left(amt \cdot amt\right)\right) \cdot \color{blue}{\left(u4 \cdot \left(amt + -1\right)\right)} \]
if -5.00000002e13 < u4 < 3.99999993e13
1. Initial program 77.9%
  \[7.62939453125 \cdot 10^{-6} \cdot \left(\left(\left(2 \cdot u2 + \left(u3 - u1\right) \cdot \left(amt \cdot 65536\right)\right) + \left(\left(\left(2 \cdot u1 - 5 \cdot u2\right) + 4 \cdot u3\right) - u4\right) \cdot \left({amt}^{2} \cdot 65536\right)\right) + \left(\left(\left(3 \cdot u2 - u1\right) - 3 \cdot u3\right) + u4\right) \cdot \left({amt}^{3} \cdot 65536\right)\right) \]
2. Add Preprocessing
3. Taylor expanded in amt around 0
  \[\leadsto \color{blue}{\frac{1}{65536} \cdot u2 + amt \cdot \left(\frac{1}{2} \cdot \left(u3 - u1\right) + amt \cdot \left(\frac{1}{2} \cdot \left(amt \cdot \left(\left(u4 + 3 \cdot u2\right) - \left(u1 + 3 \cdot u3\right)\right)\right) + \frac{1}{2} \cdot \left(\left(2 \cdot u1 + 4 \cdot u3\right) - \left(u4 + 5 \cdot u2\right)\right)\right)\right)} \]
4. Step-by-step derivation
  1. +-commutativeN/A
    \[\leadsto \color{blue}{amt \cdot \left(\frac{1}{2} \cdot \left(u3 - u1\right) + amt \cdot \left(\frac{1}{2} \cdot \left(amt \cdot \left(\left(u4 + 3 \cdot u2\right) - \left(u1 + 3 \cdot u3\right)\right)\right) + \frac{1}{2} \cdot \left(\left(2 \cdot u1 + 4 \cdot u3\right) - \left(u4 + 5 \cdot u2\right)\right)\right)\right) + \frac{1}{65536} \cdot u2} \]
  2. lower-fma.f32N/A
    \[\leadsto \color{blue}{\mathsf{fma}\left(amt, \frac{1}{2} \cdot \left(u3 - u1\right) + amt \cdot \left(\frac{1}{2} \cdot \left(amt \cdot \left(\left(u4 + 3 \cdot u2\right) - \left(u1 + 3 \cdot u3\right)\right)\right) + \frac{1}{2} \cdot \left(\left(2 \cdot u1 + 4 \cdot u3\right) - \left(u4 + 5 \cdot u2\right)\right)\right), \frac{1}{65536} \cdot u2\right)} \]
5. Applied rewrites94.3%
  \[\leadsto \color{blue}{\mathsf{fma}\left(amt, \mathsf{fma}\left(amt, 0.5 \cdot \mathsf{fma}\left(amt, \mathsf{fma}\left(3, u2 - u3, u4\right) - u1, \mathsf{fma}\left(2, u1, \mathsf{fma}\left(u3, 4, u2 \cdot -5\right)\right) - u4\right), \left(u3 - u1\right) \cdot 0.5\right), u2 \cdot 1.52587890625 \cdot 10^{-5}\right)} \]
6. Taylor expanded in u1 around -inf
  \[\leadsto -1 \cdot \color{blue}{\left(amt \cdot \left(u1 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \left(amt \cdot \left(amt - 2\right)\right)\right)\right)\right)} \]
7. Step-by-step derivation
8. Applied rewrites31.9%
  \[\leadsto \left(-amt\right) \cdot \color{blue}{\left(u1 \cdot \mathsf{fma}\left(0.5, amt \cdot \left(amt + -2\right), 0.5\right)\right)} \]
Recombined 2 regimes into one program.
Final simplification41.8%
\[\leadsto \begin{array}{l} \mathbf{if}\;u4 \leq -50000000188416:\\ \;\;\;\;\left(0.5 \cdot \left(amt \cdot amt\right)\right) \cdot \left(u4 \cdot \left(amt + -1\right)\right)\\ \mathbf{elif}\;u4 \leq 39999999311872:\\ \;\;\;\;amt \cdot \left(\left(-u1\right) \cdot \mathsf{fma}\left(0.5, amt \cdot \left(amt + -2\right), 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \left(amt \cdot amt\right)\right) \cdot \left(u4 \cdot \left(amt + -1\right)\right)\\ \end{array} \]
Add Preprocessing

Alternative 18: 37.0% accurate, 8.0× speedup?

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(0.5 \cdot \left(amt \cdot amt\right)\right) \cdot \left(u4 \cdot \left(amt + -1\right)\right)\\ \mathbf{if}\;u4 \leq -50000000188416:\\ \;\;\;\;t_0\\ \mathbf{elif}\;u4 \leq 39999999311872:\\ \;\;\;\;\left(u1 \cdot amt\right) \cdot \mathsf{fma}\left(0.5, amt \cdot \left(2 - amt\right), -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

(FPCore (u2 u3 u1 amt u4)
 :precision binary32
 (let* ((t_0 (* (* 0.5 (* amt amt)) (* u4 (+ amt -1.0)))))
   (if (<= u4 -50000000188416.0)
     t_0
     (if (<= u4 39999999311872.0)
       (* (* u1 amt) (fma 0.5 (* amt (- 2.0 amt)) -0.5))
       t_0))))

float code(float u2, float u3, float u1, float amt, float u4) {
	float t_0 = (0.5f * (amt * amt)) * (u4 * (amt + -1.0f));
	float tmp;
	if (u4 <= -50000000188416.0f) {
		tmp = t_0;
	} else if (u4 <= 39999999311872.0f) {
		tmp = (u1 * amt) * fmaf(0.5f, (amt * (2.0f - amt)), -0.5f);
	} else {
		tmp = t_0;
	}
	return tmp;
}

function code(u2, u3, u1, amt, u4)
	t_0 = Float32(Float32(Float32(0.5) * Float32(amt * amt)) * Float32(u4 * Float32(amt + Float32(-1.0))))
	tmp = Float32(0.0)
	if (u4 <= Float32(-50000000188416.0))
		tmp = t_0;
	elseif (u4 <= Float32(39999999311872.0))
		tmp = Float32(Float32(u1 * amt) * fma(Float32(0.5), Float32(amt * Float32(Float32(2.0) - amt)), Float32(-0.5)));
	else
		tmp = t_0;
	end
	return tmp
end

\begin{array}{l}

\\
\begin{array}{l}
t_0 := \left(0.5 \cdot \left(amt \cdot amt\right)\right) \cdot \left(u4 \cdot \left(amt + -1\right)\right)\\
\mathbf{if}\;u4 \leq -50000000188416:\\
\;\;\;\;t_0\\

\mathbf{elif}\;u4 \leq 39999999311872:\\
\;\;\;\;\left(u1 \cdot amt\right) \cdot \mathsf{fma}\left(0.5, amt \cdot \left(2 - amt\right), -0.5\right)\\

\mathbf{else}:\\
\;\;\;\;t_0\\


\end{array}
\end{array}

Derivation

Split input into 2 regimes
if u4 < -5.00000002e13 or 3.99999993e13 < u4
1. Initial program 71.2%
  \[7.62939453125 \cdot 10^{-6} \cdot \left(\left(\left(2 \cdot u2 + \left(u3 - u1\right) \cdot \left(amt \cdot 65536\right)\right) + \left(\left(\left(2 \cdot u1 - 5 \cdot u2\right) + 4 \cdot u3\right) - u4\right) \cdot \left({amt}^{2} \cdot 65536\right)\right) + \left(\left(\left(3 \cdot u2 - u1\right) - 3 \cdot u3\right) + u4\right) \cdot \left({amt}^{3} \cdot 65536\right)\right) \]
2. Add Preprocessing
3. Taylor expanded in amt around 0
  \[\leadsto \color{blue}{\frac{1}{65536} \cdot u2 + amt \cdot \left(\frac{1}{2} \cdot \left(u3 - u1\right) + amt \cdot \left(\frac{1}{2} \cdot \left(amt \cdot \left(\left(u4 + 3 \cdot u2\right) - \left(u1 + 3 \cdot u3\right)\right)\right) + \frac{1}{2} \cdot \left(\left(2 \cdot u1 + 4 \cdot u3\right) - \left(u4 + 5 \cdot u2\right)\right)\right)\right)} \]
4. Step-by-step derivation
  1. +-commutativeN/A
    \[\leadsto \color{blue}{amt \cdot \left(\frac{1}{2} \cdot \left(u3 - u1\right) + amt \cdot \left(\frac{1}{2} \cdot \left(amt \cdot \left(\left(u4 + 3 \cdot u2\right) - \left(u1 + 3 \cdot u3\right)\right)\right) + \frac{1}{2} \cdot \left(\left(2 \cdot u1 + 4 \cdot u3\right) - \left(u4 + 5 \cdot u2\right)\right)\right)\right) + \frac{1}{65536} \cdot u2} \]
  2. lower-fma.f32N/A
    \[\leadsto \color{blue}{\mathsf{fma}\left(amt, \frac{1}{2} \cdot \left(u3 - u1\right) + amt \cdot \left(\frac{1}{2} \cdot \left(amt \cdot \left(\left(u4 + 3 \cdot u2\right) - \left(u1 + 3 \cdot u3\right)\right)\right) + \frac{1}{2} \cdot \left(\left(2 \cdot u1 + 4 \cdot u3\right) - \left(u4 + 5 \cdot u2\right)\right)\right), \frac{1}{65536} \cdot u2\right)} \]
5. Applied rewrites96.0%
  \[\leadsto \color{blue}{\mathsf{fma}\left(amt, \mathsf{fma}\left(amt, 0.5 \cdot \mathsf{fma}\left(amt, \mathsf{fma}\left(3, u2 - u3, u4\right) - u1, \mathsf{fma}\left(2, u1, \mathsf{fma}\left(u3, 4, u2 \cdot -5\right)\right) - u4\right), \left(u3 - u1\right) \cdot 0.5\right), u2 \cdot 1.52587890625 \cdot 10^{-5}\right)} \]
6. Taylor expanded in u4 around inf
  \[\leadsto \frac{1}{2} \cdot \color{blue}{\left({amt}^{2} \cdot \left(u4 \cdot \left(amt - 1\right)\right)\right)} \]
7. Step-by-step derivation
8. Applied rewrites59.6%
  \[\leadsto \left(0.5 \cdot \left(amt \cdot amt\right)\right) \cdot \color{blue}{\left(u4 \cdot \left(amt + -1\right)\right)} \]
if -5.00000002e13 < u4 < 3.99999993e13
1. Initial program 77.9%
  \[7.62939453125 \cdot 10^{-6} \cdot \left(\left(\left(2 \cdot u2 + \left(u3 - u1\right) \cdot \left(amt \cdot 65536\right)\right) + \left(\left(\left(2 \cdot u1 - 5 \cdot u2\right) + 4 \cdot u3\right) - u4\right) \cdot \left({amt}^{2} \cdot 65536\right)\right) + \left(\left(\left(3 \cdot u2 - u1\right) - 3 \cdot u3\right) + u4\right) \cdot \left({amt}^{3} \cdot 65536\right)\right) \]
2. Add Preprocessing
3. Taylor expanded in amt around 0
  \[\leadsto \color{blue}{\frac{1}{65536} \cdot u2 + amt \cdot \left(\frac{1}{2} \cdot \left(u3 - u1\right) + amt \cdot \left(\frac{1}{2} \cdot \left(amt \cdot \left(\left(u4 + 3 \cdot u2\right) - \left(u1 + 3 \cdot u3\right)\right)\right) + \frac{1}{2} \cdot \left(\left(2 \cdot u1 + 4 \cdot u3\right) - \left(u4 + 5 \cdot u2\right)\right)\right)\right)} \]
4. Step-by-step derivation
  1. +-commutativeN/A
    \[\leadsto \color{blue}{amt \cdot \left(\frac{1}{2} \cdot \left(u3 - u1\right) + amt \cdot \left(\frac{1}{2} \cdot \left(amt \cdot \left(\left(u4 + 3 \cdot u2\right) - \left(u1 + 3 \cdot u3\right)\right)\right) + \frac{1}{2} \cdot \left(\left(2 \cdot u1 + 4 \cdot u3\right) - \left(u4 + 5 \cdot u2\right)\right)\right)\right) + \frac{1}{65536} \cdot u2} \]
  2. lower-fma.f32N/A
    \[\leadsto \color{blue}{\mathsf{fma}\left(amt, \frac{1}{2} \cdot \left(u3 - u1\right) + amt \cdot \left(\frac{1}{2} \cdot \left(amt \cdot \left(\left(u4 + 3 \cdot u2\right) - \left(u1 + 3 \cdot u3\right)\right)\right) + \frac{1}{2} \cdot \left(\left(2 \cdot u1 + 4 \cdot u3\right) - \left(u4 + 5 \cdot u2\right)\right)\right), \frac{1}{65536} \cdot u2\right)} \]
5. Applied rewrites94.3%
  \[\leadsto \color{blue}{\mathsf{fma}\left(amt, \mathsf{fma}\left(amt, 0.5 \cdot \mathsf{fma}\left(amt, \mathsf{fma}\left(3, u2 - u3, u4\right) - u1, \mathsf{fma}\left(2, u1, \mathsf{fma}\left(u3, 4, u2 \cdot -5\right)\right) - u4\right), \left(u3 - u1\right) \cdot 0.5\right), u2 \cdot 1.52587890625 \cdot 10^{-5}\right)} \]
6. Taylor expanded in u1 around inf
  \[\leadsto amt \cdot \color{blue}{\left(u1 \cdot \left(\frac{1}{2} \cdot \left(amt \cdot \left(2 + -1 \cdot amt\right)\right) - \frac{1}{2}\right)\right)} \]
7. Step-by-step derivation
8. Applied rewrites31.9%
  \[\leadsto \left(amt \cdot u1\right) \cdot \color{blue}{\mathsf{fma}\left(0.5, amt \cdot \left(2 - amt\right), -0.5\right)} \]
Recombined 2 regimes into one program.
Final simplification41.7%
\[\leadsto \begin{array}{l} \mathbf{if}\;u4 \leq -50000000188416:\\ \;\;\;\;\left(0.5 \cdot \left(amt \cdot amt\right)\right) \cdot \left(u4 \cdot \left(amt + -1\right)\right)\\ \mathbf{elif}\;u4 \leq 39999999311872:\\ \;\;\;\;\left(u1 \cdot amt\right) \cdot \mathsf{fma}\left(0.5, amt \cdot \left(2 - amt\right), -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \left(amt \cdot amt\right)\right) \cdot \left(u4 \cdot \left(amt + -1\right)\right)\\ \end{array} \]
Add Preprocessing

Alternative 19: 25.2% accurate, 12.4× speedup?

\[\begin{array}{l} \\ \left(0.5 \cdot \left(amt \cdot amt\right)\right) \cdot \left(u4 \cdot \left(amt + -1\right)\right) \end{array} \]

(FPCore (u2 u3 u1 amt u4)
 :precision binary32
 (* (* 0.5 (* amt amt)) (* u4 (+ amt -1.0))))

float code(float u2, float u3, float u1, float amt, float u4) {
	return (0.5f * (amt * amt)) * (u4 * (amt + -1.0f));
}

real(4) function code(u2, u3, u1, amt, u4)
    real(4), intent (in) :: u2
    real(4), intent (in) :: u3
    real(4), intent (in) :: u1
    real(4), intent (in) :: amt
    real(4), intent (in) :: u4
    code = (0.5e0 * (amt * amt)) * (u4 * (amt + (-1.0e0)))
end function

function code(u2, u3, u1, amt, u4)
	return Float32(Float32(Float32(0.5) * Float32(amt * amt)) * Float32(u4 * Float32(amt + Float32(-1.0))))
end

function tmp = code(u2, u3, u1, amt, u4)
	tmp = (single(0.5) * (amt * amt)) * (u4 * (amt + single(-1.0)));
end

\begin{array}{l}

\\
\left(0.5 \cdot \left(amt \cdot amt\right)\right) \cdot \left(u4 \cdot \left(amt + -1\right)\right)
\end{array}

Derivation

Initial program 75.5%
\[7.62939453125 \cdot 10^{-6} \cdot \left(\left(\left(2 \cdot u2 + \left(u3 - u1\right) \cdot \left(amt \cdot 65536\right)\right) + \left(\left(\left(2 \cdot u1 - 5 \cdot u2\right) + 4 \cdot u3\right) - u4\right) \cdot \left({amt}^{2} \cdot 65536\right)\right) + \left(\left(\left(3 \cdot u2 - u1\right) - 3 \cdot u3\right) + u4\right) \cdot \left({amt}^{3} \cdot 65536\right)\right) \]
Add Preprocessing
Taylor expanded in amt around 0
\[\leadsto \color{blue}{\frac{1}{65536} \cdot u2 + amt \cdot \left(\frac{1}{2} \cdot \left(u3 - u1\right) + amt \cdot \left(\frac{1}{2} \cdot \left(amt \cdot \left(\left(u4 + 3 \cdot u2\right) - \left(u1 + 3 \cdot u3\right)\right)\right) + \frac{1}{2} \cdot \left(\left(2 \cdot u1 + 4 \cdot u3\right) - \left(u4 + 5 \cdot u2\right)\right)\right)\right)} \]
Step-by-step derivation
1. +-commutativeN/A
  \[\leadsto \color{blue}{amt \cdot \left(\frac{1}{2} \cdot \left(u3 - u1\right) + amt \cdot \left(\frac{1}{2} \cdot \left(amt \cdot \left(\left(u4 + 3 \cdot u2\right) - \left(u1 + 3 \cdot u3\right)\right)\right) + \frac{1}{2} \cdot \left(\left(2 \cdot u1 + 4 \cdot u3\right) - \left(u4 + 5 \cdot u2\right)\right)\right)\right) + \frac{1}{65536} \cdot u2} \]
2. lower-fma.f32N/A
  \[\leadsto \color{blue}{\mathsf{fma}\left(amt, \frac{1}{2} \cdot \left(u3 - u1\right) + amt \cdot \left(\frac{1}{2} \cdot \left(amt \cdot \left(\left(u4 + 3 \cdot u2\right) - \left(u1 + 3 \cdot u3\right)\right)\right) + \frac{1}{2} \cdot \left(\left(2 \cdot u1 + 4 \cdot u3\right) - \left(u4 + 5 \cdot u2\right)\right)\right), \frac{1}{65536} \cdot u2\right)} \]
Applied rewrites94.9%
\[\leadsto \color{blue}{\mathsf{fma}\left(amt, \mathsf{fma}\left(amt, 0.5 \cdot \mathsf{fma}\left(amt, \mathsf{fma}\left(3, u2 - u3, u4\right) - u1, \mathsf{fma}\left(2, u1, \mathsf{fma}\left(u3, 4, u2 \cdot -5\right)\right) - u4\right), \left(u3 - u1\right) \cdot 0.5\right), u2 \cdot 1.52587890625 \cdot 10^{-5}\right)} \]
Taylor expanded in u4 around inf
\[\leadsto \frac{1}{2} \cdot \color{blue}{\left({amt}^{2} \cdot \left(u4 \cdot \left(amt - 1\right)\right)\right)} \]
Step-by-step derivation
Applied rewrites27.5%
\[\leadsto \left(0.5 \cdot \left(amt \cdot amt\right)\right) \cdot \color{blue}{\left(u4 \cdot \left(amt + -1\right)\right)} \]
Add Preprocessing

Alternative 20: 15.7% accurate, 21.3× speedup?

\[\begin{array}{l} \\ \left(u3 - u1\right) \cdot \left(amt \cdot 0.5\right) \end{array} \]

(FPCore (u2 u3 u1 amt u4) :precision binary32 (* (- u3 u1) (* amt 0.5)))

float code(float u2, float u3, float u1, float amt, float u4) {
	return (u3 - u1) * (amt * 0.5f);
}

real(4) function code(u2, u3, u1, amt, u4)
    real(4), intent (in) :: u2
    real(4), intent (in) :: u3
    real(4), intent (in) :: u1
    real(4), intent (in) :: amt
    real(4), intent (in) :: u4
    code = (u3 - u1) * (amt * 0.5e0)
end function

function code(u2, u3, u1, amt, u4)
	return Float32(Float32(u3 - u1) * Float32(amt * Float32(0.5)))
end

function tmp = code(u2, u3, u1, amt, u4)
	tmp = (u3 - u1) * (amt * single(0.5));
end

\begin{array}{l}

\\
\left(u3 - u1\right) \cdot \left(amt \cdot 0.5\right)
\end{array}

Derivation

Initial program 75.5%
\[7.62939453125 \cdot 10^{-6} \cdot \left(\left(\left(2 \cdot u2 + \left(u3 - u1\right) \cdot \left(amt \cdot 65536\right)\right) + \left(\left(\left(2 \cdot u1 - 5 \cdot u2\right) + 4 \cdot u3\right) - u4\right) \cdot \left({amt}^{2} \cdot 65536\right)\right) + \left(\left(\left(3 \cdot u2 - u1\right) - 3 \cdot u3\right) + u4\right) \cdot \left({amt}^{3} \cdot 65536\right)\right) \]
Add Preprocessing
Taylor expanded in u2 around 0
\[\leadsto \color{blue}{\frac{1}{131072} \cdot \left(65536 \cdot \left(amt \cdot \left(u3 - u1\right)\right) + \left(65536 \cdot \left({amt}^{2} \cdot \left(\left(2 \cdot u1 + 4 \cdot u3\right) - u4\right)\right) + 65536 \cdot \left({amt}^{3} \cdot \left(u4 - \left(u1 + 3 \cdot u3\right)\right)\right)\right)\right)} \]
Step-by-step derivation
1. +-commutativeN/A
  \[\leadsto \frac{1}{131072} \cdot \color{blue}{\left(\left(65536 \cdot \left({amt}^{2} \cdot \left(\left(2 \cdot u1 + 4 \cdot u3\right) - u4\right)\right) + 65536 \cdot \left({amt}^{3} \cdot \left(u4 - \left(u1 + 3 \cdot u3\right)\right)\right)\right) + 65536 \cdot \left(amt \cdot \left(u3 - u1\right)\right)\right)} \]
2. distribute-lft-inN/A
  \[\leadsto \color{blue}{\frac{1}{131072} \cdot \left(65536 \cdot \left({amt}^{2} \cdot \left(\left(2 \cdot u1 + 4 \cdot u3\right) - u4\right)\right) + 65536 \cdot \left({amt}^{3} \cdot \left(u4 - \left(u1 + 3 \cdot u3\right)\right)\right)\right) + \frac{1}{131072} \cdot \left(65536 \cdot \left(amt \cdot \left(u3 - u1\right)\right)\right)} \]
3. distribute-lft-outN/A
  \[\leadsto \frac{1}{131072} \cdot \color{blue}{\left(65536 \cdot \left({amt}^{2} \cdot \left(\left(2 \cdot u1 + 4 \cdot u3\right) - u4\right) + {amt}^{3} \cdot \left(u4 - \left(u1 + 3 \cdot u3\right)\right)\right)\right)} + \frac{1}{131072} \cdot \left(65536 \cdot \left(amt \cdot \left(u3 - u1\right)\right)\right) \]
4. associate-*r*N/A
  \[\leadsto \color{blue}{\left(\frac{1}{131072} \cdot 65536\right) \cdot \left({amt}^{2} \cdot \left(\left(2 \cdot u1 + 4 \cdot u3\right) - u4\right) + {amt}^{3} \cdot \left(u4 - \left(u1 + 3 \cdot u3\right)\right)\right)} + \frac{1}{131072} \cdot \left(65536 \cdot \left(amt \cdot \left(u3 - u1\right)\right)\right) \]
5. metadata-evalN/A
  \[\leadsto \color{blue}{\frac{1}{2}} \cdot \left({amt}^{2} \cdot \left(\left(2 \cdot u1 + 4 \cdot u3\right) - u4\right) + {amt}^{3} \cdot \left(u4 - \left(u1 + 3 \cdot u3\right)\right)\right) + \frac{1}{131072} \cdot \left(65536 \cdot \left(amt \cdot \left(u3 - u1\right)\right)\right) \]
6. associate-*r*N/A
  \[\leadsto \frac{1}{2} \cdot \left({amt}^{2} \cdot \left(\left(2 \cdot u1 + 4 \cdot u3\right) - u4\right) + {amt}^{3} \cdot \left(u4 - \left(u1 + 3 \cdot u3\right)\right)\right) + \color{blue}{\left(\frac{1}{131072} \cdot 65536\right) \cdot \left(amt \cdot \left(u3 - u1\right)\right)} \]
7. metadata-evalN/A
  \[\leadsto \frac{1}{2} \cdot \left({amt}^{2} \cdot \left(\left(2 \cdot u1 + 4 \cdot u3\right) - u4\right) + {amt}^{3} \cdot \left(u4 - \left(u1 + 3 \cdot u3\right)\right)\right) + \color{blue}{\frac{1}{2}} \cdot \left(amt \cdot \left(u3 - u1\right)\right) \]
8. *-commutativeN/A
  \[\leadsto \frac{1}{2} \cdot \left({amt}^{2} \cdot \left(\left(2 \cdot u1 + 4 \cdot u3\right) - u4\right) + {amt}^{3} \cdot \left(u4 - \left(u1 + 3 \cdot u3\right)\right)\right) + \color{blue}{\left(amt \cdot \left(u3 - u1\right)\right) \cdot \frac{1}{2}} \]
9. associate-*r*N/A
  \[\leadsto \frac{1}{2} \cdot \left({amt}^{2} \cdot \left(\left(2 \cdot u1 + 4 \cdot u3\right) - u4\right) + {amt}^{3} \cdot \left(u4 - \left(u1 + 3 \cdot u3\right)\right)\right) + \color{blue}{amt \cdot \left(\left(u3 - u1\right) \cdot \frac{1}{2}\right)} \]
10. *-commutativeN/A
  \[\leadsto \frac{1}{2} \cdot \left({amt}^{2} \cdot \left(\left(2 \cdot u1 + 4 \cdot u3\right) - u4\right) + {amt}^{3} \cdot \left(u4 - \left(u1 + 3 \cdot u3\right)\right)\right) + amt \cdot \color{blue}{\left(\frac{1}{2} \cdot \left(u3 - u1\right)\right)} \]
Applied rewrites67.4%
\[\leadsto \color{blue}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(amt \cdot \left(amt \cdot amt\right), u4 - \mathsf{fma}\left(3, u3, u1\right), amt \cdot \left(amt \cdot \mathsf{fma}\left(2, u1, u3 \cdot 4 - u4\right)\right)\right), \left(amt \cdot \left(u3 - u1\right)\right) \cdot 0.5\right)} \]
Taylor expanded in amt around 0
\[\leadsto \frac{1}{2} \cdot \color{blue}{\left(amt \cdot \left(u3 - u1\right)\right)} \]
Step-by-step derivation
Applied rewrites14.9%
\[\leadsto \left(0.5 \cdot amt\right) \cdot \color{blue}{\left(u3 - u1\right)} \]
Final simplification14.9%
\[\leadsto \left(u3 - u1\right) \cdot \left(amt \cdot 0.5\right) \]
Add Preprocessing

Specification

Local Percentage Accuracy vs ?

Accuracy vs Speed?

Initial Program: 71.7% accurate, 1.0× speedupMathFPCoreCFortranJuliaMATLABTeX?

Alternative 1: 95.5% accurate, 3.9× speedupMathFPCoreCJuliaTeX?

Alternative 2: 95.2% accurate, 4.0× speedupMathFPCoreCJuliaTeX?

Alternative 3: 95.0% accurate, 4.1× speedupMathFPCoreCJuliaTeX?

Alternative 4: 94.9% accurate, 4.1× speedupMathFPCoreCJuliaTeX?

Alternative 5: 85.1% accurate, 4.3× speedupMathFPCoreCJuliaTeX?

if u3 < -2 or 4e4 < u3

if -2 < u3 < 4e4

Alternative 6: 94.3% accurate, 4.3× speedupMathFPCoreCJuliaTeX?

Alternative 7: 85.1% accurate, 4.5× speedupMathFPCoreCJuliaTeX?

if u2 < -5e3 or 1.99999997e17 < u2

if -5e3 < u2 < 1.99999997e17

Alternative 8: 79.8% accurate, 4.5× speedupMathFPCoreCJuliaTeX?

if u2 < -2.00000003e24

if -2.00000003e24 < u2 < 1.80000004e19

if 1.80000004e19 < u2

Alternative 9: 79.6% accurate, 4.9× speedupMathFPCoreCJuliaTeX?

if u2 < -2.00000003e24

if -2.00000003e24 < u2 < 1.80000004e19

if 1.80000004e19 < u2

Alternative 10: 47.5% accurate, 5.5× speedupMathFPCoreCJuliaTeX?

if u2 < -1370 or 2e6 < u2

if -1370 < u2 < 2e6

Alternative 11: 40.3% accurate, 5.8× speedupMathFPCoreCJuliaTeX?

if u2 < -5e6

if -5e6 < u2 < -1.00000003e-27

if -1.00000003e-27 < u2 < 5e6

if 5e6 < u2

Alternative 12: 46.0% accurate, 5.8× speedupMathFPCoreCJuliaTeX?

if u2 < -2e5

if -2e5 < u2 < 1.5e7

if 1.5e7 < u2

Alternative 13: 40.9% accurate, 5.8× speedupMathFPCoreCJuliaTeX?

if u4 < -1.4999999e18

if -1.4999999e18 < u4 < 3.99999993e13

if 3.99999993e13 < u4

Alternative 14: 40.5% accurate, 6.2× speedupMathFPCoreCJuliaTeX?

if u2 < -5e6 or 5e6 < u2

if -5e6 < u2 < -1.00000003e-27

if -1.00000003e-27 < u2 < 5e6

Alternative 15: 40.5% accurate, 6.5× speedupMathFPCoreCJuliaTeX?

if u2 < -5e6 or 5e6 < u2

if -5e6 < u2 < -1.00000003e-27

if -1.00000003e-27 < u2 < 5e6

Alternative 16: 39.5% accurate, 7.4× speedupMathFPCoreCJuliaTeX?

if u2 < -1370 or 5e6 < u2

if -1370 < u2 < 5e6

Alternative 17: 37.0% accurate, 7.6× speedupMathFPCoreCJuliaTeX?

if u4 < -5.00000002e13 or 3.99999993e13 < u4

if -5.00000002e13 < u4 < 3.99999993e13

Alternative 18: 37.0% accurate, 8.0× speedupMathFPCoreCJuliaTeX?

if u4 < -5.00000002e13 or 3.99999993e13 < u4

if -5.00000002e13 < u4 < 3.99999993e13

Alternative 19: 25.2% accurate, 12.4× speedupMathFPCoreCFortranJuliaMATLABTeX?

Alternative 20: 15.7% accurate, 21.3× speedupMathFPCoreCFortranJuliaMATLABTeX?

Alternative 21: 4.1% accurate, 49.7× speedupMathFPCoreCFortranJuliaMATLABTeX?

Reproduce

Initial Program: 71.7% accurate, 1.0× speedup?

Alternative 1: 95.5% accurate, 3.9× speedup?

Alternative 2: 95.2% accurate, 4.0× speedup?

Alternative 3: 95.0% accurate, 4.1× speedup?

Alternative 4: 94.9% accurate, 4.1× speedup?

Alternative 5: 85.1% accurate, 4.3× speedup?

`if u3 < -2 or 4e4 < u3`

`if -2 < u3 < 4e4`

Alternative 6: 94.3% accurate, 4.3× speedup?

Alternative 7: 85.1% accurate, 4.5× speedup?

`if u2 < -5e3 or 1.99999997e17 < u2`

`if -5e3 < u2 < 1.99999997e17`

Alternative 8: 79.8% accurate, 4.5× speedup?

`if u2 < -2.00000003e24`

`if -2.00000003e24 < u2 < 1.80000004e19`

`if 1.80000004e19 < u2`

Alternative 9: 79.6% accurate, 4.9× speedup?

`if u2 < -2.00000003e24`

`if -2.00000003e24 < u2 < 1.80000004e19`

`if 1.80000004e19 < u2`

Alternative 10: 47.5% accurate, 5.5× speedup?

`if u2 < -1370 or 2e6 < u2`

`if -1370 < u2 < 2e6`

Alternative 11: 40.3% accurate, 5.8× speedup?

`if u2 < -5e6`

`if -5e6 < u2 < -1.00000003e-27`

`if -1.00000003e-27 < u2 < 5e6`

`if 5e6 < u2`

Alternative 12: 46.0% accurate, 5.8× speedup?

`if u2 < -2e5`

`if -2e5 < u2 < 1.5e7`

`if 1.5e7 < u2`

Alternative 13: 40.9% accurate, 5.8× speedup?

`if u4 < -1.4999999e18`

`if -1.4999999e18 < u4 < 3.99999993e13`

`if 3.99999993e13 < u4`

Alternative 14: 40.5% accurate, 6.2× speedup?

`if u2 < -5e6 or 5e6 < u2`

`if -5e6 < u2 < -1.00000003e-27`

`if -1.00000003e-27 < u2 < 5e6`

Alternative 15: 40.5% accurate, 6.5× speedup?

`if u2 < -5e6 or 5e6 < u2`

`if -5e6 < u2 < -1.00000003e-27`

`if -1.00000003e-27 < u2 < 5e6`

Alternative 16: 39.5% accurate, 7.4× speedup?

`if u2 < -1370 or 5e6 < u2`

`if -1370 < u2 < 5e6`

Alternative 17: 37.0% accurate, 7.6× speedup?

`if u4 < -5.00000002e13 or 3.99999993e13 < u4`

`if -5.00000002e13 < u4 < 3.99999993e13`

Alternative 18: 37.0% accurate, 8.0× speedup?

`if u4 < -5.00000002e13 or 3.99999993e13 < u4`

`if -5.00000002e13 < u4 < 3.99999993e13`

Alternative 19: 25.2% accurate, 12.4× speedup?

Alternative 20: 15.7% accurate, 21.3× speedup?

Alternative 21: 4.1% accurate, 49.7× speedup?