Result for (/ (+ (+ (+ P0 (* P1 (* amt 65536))) (* (- (- (* 3.0 (- P1 P0)) V1) (* 2.0 V0)) (* (pow amt 2) 65536))) (* (+ (+ (* 2.0 (- P0 P1)) V0) V1) (* (pow amt 3) 65536))) 65536)

Specification

\[0.85 \leq amt \land amt \leq 1\]

\[\begin{array}{l} \\ \frac{\left(\left(P0 + P1 \cdot \left(amt \cdot 65536\right)\right) + \left(\left(3 \cdot \left(P1 - P0\right) - V1\right) - 2 \cdot V0\right) \cdot \left({amt}^{2} \cdot 65536\right)\right) + \left(\left(2 \cdot \left(P0 - P1\right) + V0\right) + V1\right) \cdot \left({amt}^{3} \cdot 65536\right)}{65536} \end{array} \]

(FPCore (P0 P1 amt V1 V0)
 :precision binary32
 (/
  (+
   (+
    (+ P0 (* P1 (* amt 65536.0)))
    (* (- (- (* 3.0 (- P1 P0)) V1) (* 2.0 V0)) (* (pow amt 2.0) 65536.0)))
   (* (+ (+ (* 2.0 (- P0 P1)) V0) V1) (* (pow amt 3.0) 65536.0)))
  65536.0))

float code(float P0, float P1, float amt, float V1, float V0) {
	return (((P0 + (P1 * (amt * 65536.0f))) + ((((3.0f * (P1 - P0)) - V1) - (2.0f * V0)) * (powf(amt, 2.0f) * 65536.0f))) + ((((2.0f * (P0 - P1)) + V0) + V1) * (powf(amt, 3.0f) * 65536.0f))) / 65536.0f;
}

real(4) function code(p0, p1, amt, v1, v0)
    real(4), intent (in) :: p0
    real(4), intent (in) :: p1
    real(4), intent (in) :: amt
    real(4), intent (in) :: v1
    real(4), intent (in) :: v0
    code = (((p0 + (p1 * (amt * 65536.0e0))) + ((((3.0e0 * (p1 - p0)) - v1) - (2.0e0 * v0)) * ((amt ** 2.0e0) * 65536.0e0))) + ((((2.0e0 * (p0 - p1)) + v0) + v1) * ((amt ** 3.0e0) * 65536.0e0))) / 65536.0e0
end function

function code(P0, P1, amt, V1, V0)
	return Float32(Float32(Float32(Float32(P0 + Float32(P1 * Float32(amt * Float32(65536.0)))) + Float32(Float32(Float32(Float32(Float32(3.0) * Float32(P1 - P0)) - V1) - Float32(Float32(2.0) * V0)) * Float32((amt ^ Float32(2.0)) * Float32(65536.0)))) + Float32(Float32(Float32(Float32(Float32(2.0) * Float32(P0 - P1)) + V0) + V1) * Float32((amt ^ Float32(3.0)) * Float32(65536.0)))) / Float32(65536.0))
end

function tmp = code(P0, P1, amt, V1, V0)
	tmp = (((P0 + (P1 * (amt * single(65536.0)))) + ((((single(3.0) * (P1 - P0)) - V1) - (single(2.0) * V0)) * ((amt ^ single(2.0)) * single(65536.0)))) + ((((single(2.0) * (P0 - P1)) + V0) + V1) * ((amt ^ single(3.0)) * single(65536.0)))) / single(65536.0);
end

\begin{array}{l}

\\
\frac{\left(\left(P0 + P1 \cdot \left(amt \cdot 65536\right)\right) + \left(\left(3 \cdot \left(P1 - P0\right) - V1\right) - 2 \cdot V0\right) \cdot \left({amt}^{2} \cdot 65536\right)\right) + \left(\left(2 \cdot \left(P0 - P1\right) + V0\right) + V1\right) \cdot \left({amt}^{3} \cdot 65536\right)}{65536}
\end{array}

Sampling outcomes in binary32 precision:

Initial Program: 72.7% accurate, 1.0× speedup?

(FPCore (P0 P1 amt V1 V0)
 :precision binary32
 (/
  (+
   (+
    (+ P0 (* P1 (* amt 65536.0)))
    (* (- (- (* 3.0 (- P1 P0)) V1) (* 2.0 V0)) (* (pow amt 2.0) 65536.0)))
   (* (+ (+ (* 2.0 (- P0 P1)) V0) V1) (* (pow amt 3.0) 65536.0)))
  65536.0))

float code(float P0, float P1, float amt, float V1, float V0) {
	return (((P0 + (P1 * (amt * 65536.0f))) + ((((3.0f * (P1 - P0)) - V1) - (2.0f * V0)) * (powf(amt, 2.0f) * 65536.0f))) + ((((2.0f * (P0 - P1)) + V0) + V1) * (powf(amt, 3.0f) * 65536.0f))) / 65536.0f;
}

real(4) function code(p0, p1, amt, v1, v0)
    real(4), intent (in) :: p0
    real(4), intent (in) :: p1
    real(4), intent (in) :: amt
    real(4), intent (in) :: v1
    real(4), intent (in) :: v0
    code = (((p0 + (p1 * (amt * 65536.0e0))) + ((((3.0e0 * (p1 - p0)) - v1) - (2.0e0 * v0)) * ((amt ** 2.0e0) * 65536.0e0))) + ((((2.0e0 * (p0 - p1)) + v0) + v1) * ((amt ** 3.0e0) * 65536.0e0))) / 65536.0e0
end function

function code(P0, P1, amt, V1, V0)
	return Float32(Float32(Float32(Float32(P0 + Float32(P1 * Float32(amt * Float32(65536.0)))) + Float32(Float32(Float32(Float32(Float32(3.0) * Float32(P1 - P0)) - V1) - Float32(Float32(2.0) * V0)) * Float32((amt ^ Float32(2.0)) * Float32(65536.0)))) + Float32(Float32(Float32(Float32(Float32(2.0) * Float32(P0 - P1)) + V0) + V1) * Float32((amt ^ Float32(3.0)) * Float32(65536.0)))) / Float32(65536.0))
end

function tmp = code(P0, P1, amt, V1, V0)
	tmp = (((P0 + (P1 * (amt * single(65536.0)))) + ((((single(3.0) * (P1 - P0)) - V1) - (single(2.0) * V0)) * ((amt ^ single(2.0)) * single(65536.0)))) + ((((single(2.0) * (P0 - P1)) + V0) + V1) * ((amt ^ single(3.0)) * single(65536.0)))) / single(65536.0);
end

\begin{array}{l}

\\
\frac{\left(\left(P0 + P1 \cdot \left(amt \cdot 65536\right)\right) + \left(\left(3 \cdot \left(P1 - P0\right) - V1\right) - 2 \cdot V0\right) \cdot \left({amt}^{2} \cdot 65536\right)\right) + \left(\left(2 \cdot \left(P0 - P1\right) + V0\right) + V1\right) \cdot \left({amt}^{3} \cdot 65536\right)}{65536}
\end{array}

Alternative 1: 97.8% accurate, 5.1× speedup?

\[\begin{array}{l} \\ \mathsf{fma}\left(amt, \mathsf{fma}\left(amt, \mathsf{fma}\left(V0, -2, \mathsf{fma}\left(3, P1 - P0, \mathsf{fma}\left(amt, \mathsf{fma}\left(2, P0 - P1, V0\right), V1 \cdot \left(amt + -1\right)\right)\right)\right), P1\right), P0 \cdot 1.52587890625 \cdot 10^{-5}\right) \end{array} \]

(FPCore (P0 P1 amt V1 V0)
 :precision binary32
 (fma
  amt
  (fma
   amt
   (fma
    V0
    -2.0
    (fma 3.0 (- P1 P0) (fma amt (fma 2.0 (- P0 P1) V0) (* V1 (+ amt -1.0)))))
   P1)
  (* P0 1.52587890625e-5)))

float code(float P0, float P1, float amt, float V1, float V0) {
	return fmaf(amt, fmaf(amt, fmaf(V0, -2.0f, fmaf(3.0f, (P1 - P0), fmaf(amt, fmaf(2.0f, (P0 - P1), V0), (V1 * (amt + -1.0f))))), P1), (P0 * 1.52587890625e-5f));
}

function code(P0, P1, amt, V1, V0)
	return fma(amt, fma(amt, fma(V0, Float32(-2.0), fma(Float32(3.0), Float32(P1 - P0), fma(amt, fma(Float32(2.0), Float32(P0 - P1), V0), Float32(V1 * Float32(amt + Float32(-1.0)))))), P1), Float32(P0 * Float32(1.52587890625e-5)))
end

\begin{array}{l}

\\
\mathsf{fma}\left(amt, \mathsf{fma}\left(amt, \mathsf{fma}\left(V0, -2, \mathsf{fma}\left(3, P1 - P0, \mathsf{fma}\left(amt, \mathsf{fma}\left(2, P0 - P1, V0\right), V1 \cdot \left(amt + -1\right)\right)\right)\right), P1\right), P0 \cdot 1.52587890625 \cdot 10^{-5}\right)
\end{array}

Derivation

Initial program 75.2%
\[\frac{\left(\left(P0 + P1 \cdot \left(amt \cdot 65536\right)\right) + \left(\left(3 \cdot \left(P1 - P0\right) - V1\right) - 2 \cdot V0\right) \cdot \left({amt}^{2} \cdot 65536\right)\right) + \left(\left(2 \cdot \left(P0 - P1\right) + V0\right) + V1\right) \cdot \left({amt}^{3} \cdot 65536\right)}{65536} \]
Add Preprocessing
Taylor expanded in amt around 0
\[\leadsto \color{blue}{\frac{1}{65536} \cdot P0 + amt \cdot \left(P1 + amt \cdot \left(\left(3 \cdot \left(P1 - P0\right) + amt \cdot \left(V0 + \left(V1 + 2 \cdot \left(P0 - P1\right)\right)\right)\right) - \left(V1 + 2 \cdot V0\right)\right)\right)} \]
Step-by-step derivation
1. +-commutativeN/A
  \[\leadsto \color{blue}{amt \cdot \left(P1 + amt \cdot \left(\left(3 \cdot \left(P1 - P0\right) + amt \cdot \left(V0 + \left(V1 + 2 \cdot \left(P0 - P1\right)\right)\right)\right) - \left(V1 + 2 \cdot V0\right)\right)\right) + \frac{1}{65536} \cdot P0} \]
2. lower-fma.f32N/A
  \[\leadsto \color{blue}{\mathsf{fma}\left(amt, P1 + amt \cdot \left(\left(3 \cdot \left(P1 - P0\right) + amt \cdot \left(V0 + \left(V1 + 2 \cdot \left(P0 - P1\right)\right)\right)\right) - \left(V1 + 2 \cdot V0\right)\right), \frac{1}{65536} \cdot P0\right)} \]
Applied rewrites96.5%
\[\leadsto \color{blue}{\mathsf{fma}\left(amt, \mathsf{fma}\left(amt, \mathsf{fma}\left(amt, \mathsf{fma}\left(2, P0 - P1, V1 + V0\right), \mathsf{fma}\left(3, P1 - P0, \mathsf{fma}\left(V0, -2, -V1\right)\right)\right), P1\right), P0 \cdot 1.52587890625 \cdot 10^{-5}\right)} \]
Taylor expanded in V1 around 0
\[\leadsto \mathsf{fma}\left(amt, \mathsf{fma}\left(amt, -2 \cdot V0 + \color{blue}{\left(3 \cdot \left(P1 - P0\right) + \left(V1 \cdot \left(amt - 1\right) + amt \cdot \left(V0 + 2 \cdot \left(P0 - P1\right)\right)\right)\right)}, P1\right), P0 \cdot \frac{1}{65536}\right) \]
Step-by-step derivation
Applied rewrites98.0%
\[\leadsto \mathsf{fma}\left(amt, \mathsf{fma}\left(amt, \mathsf{fma}\left(V0, \color{blue}{-2}, \mathsf{fma}\left(3, P1 - P0, \mathsf{fma}\left(amt, \mathsf{fma}\left(2, P0 - P1, V0\right), V1 \cdot \left(amt + -1\right)\right)\right)\right), P1\right), P0 \cdot 1.52587890625 \cdot 10^{-5}\right) \]
Add Preprocessing

Alternative 2: 84.9% accurate, 4.9× speedup?

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;P1 \leq -500000000:\\ \;\;\;\;\mathsf{fma}\left(amt, \mathsf{fma}\left(amt, \mathsf{fma}\left(V0, -2, \mathsf{fma}\left(amt, \mathsf{fma}\left(2, P0 - P1, V0\right), 3 \cdot \left(P1 - P0\right)\right)\right), P1\right), P0 \cdot 1.52587890625 \cdot 10^{-5}\right)\\ \mathbf{elif}\;P1 \leq 49999999215337470:\\ \;\;\;\;\mathsf{fma}\left(amt \cdot amt, \mathsf{fma}\left(P0, -3, \mathsf{fma}\left(amt, V0 + \mathsf{fma}\left(2, P0, V1\right), V0 \cdot -2\right) - V1\right), P0 \cdot 1.52587890625 \cdot 10^{-5}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(amt, \mathsf{fma}\left(amt, \mathsf{fma}\left(3, P1 - P0, \mathsf{fma}\left(amt, \mathsf{fma}\left(2, P0 - P1, V1\right), -V1\right)\right), P1\right), P0 \cdot 1.52587890625 \cdot 10^{-5}\right)\\ \end{array} \end{array} \]

(FPCore (P0 P1 amt V1 V0)
 :precision binary32
 (if (<= P1 -500000000.0)
   (fma
    amt
    (fma
     amt
     (fma V0 -2.0 (fma amt (fma 2.0 (- P0 P1) V0) (* 3.0 (- P1 P0))))
     P1)
    (* P0 1.52587890625e-5))
   (if (<= P1 49999999215337470.0)
     (fma
      (* amt amt)
      (fma P0 -3.0 (- (fma amt (+ V0 (fma 2.0 P0 V1)) (* V0 -2.0)) V1))
      (* P0 1.52587890625e-5))
     (fma
      amt
      (fma amt (fma 3.0 (- P1 P0) (fma amt (fma 2.0 (- P0 P1) V1) (- V1))) P1)
      (* P0 1.52587890625e-5)))))

float code(float P0, float P1, float amt, float V1, float V0) {
	float tmp;
	if (P1 <= -500000000.0f) {
		tmp = fmaf(amt, fmaf(amt, fmaf(V0, -2.0f, fmaf(amt, fmaf(2.0f, (P0 - P1), V0), (3.0f * (P1 - P0)))), P1), (P0 * 1.52587890625e-5f));
	} else if (P1 <= 49999999215337470.0f) {
		tmp = fmaf((amt * amt), fmaf(P0, -3.0f, (fmaf(amt, (V0 + fmaf(2.0f, P0, V1)), (V0 * -2.0f)) - V1)), (P0 * 1.52587890625e-5f));
	} else {
		tmp = fmaf(amt, fmaf(amt, fmaf(3.0f, (P1 - P0), fmaf(amt, fmaf(2.0f, (P0 - P1), V1), -V1)), P1), (P0 * 1.52587890625e-5f));
	}
	return tmp;
}

function code(P0, P1, amt, V1, V0)
	tmp = Float32(0.0)
	if (P1 <= Float32(-500000000.0))
		tmp = fma(amt, fma(amt, fma(V0, Float32(-2.0), fma(amt, fma(Float32(2.0), Float32(P0 - P1), V0), Float32(Float32(3.0) * Float32(P1 - P0)))), P1), Float32(P0 * Float32(1.52587890625e-5)));
	elseif (P1 <= Float32(49999999215337470.0))
		tmp = fma(Float32(amt * amt), fma(P0, Float32(-3.0), Float32(fma(amt, Float32(V0 + fma(Float32(2.0), P0, V1)), Float32(V0 * Float32(-2.0))) - V1)), Float32(P0 * Float32(1.52587890625e-5)));
	else
		tmp = fma(amt, fma(amt, fma(Float32(3.0), Float32(P1 - P0), fma(amt, fma(Float32(2.0), Float32(P0 - P1), V1), Float32(-V1))), P1), Float32(P0 * Float32(1.52587890625e-5)));
	end
	return tmp
end

\begin{array}{l}

\\
\begin{array}{l}
\mathbf{if}\;P1 \leq -500000000:\\
\;\;\;\;\mathsf{fma}\left(amt, \mathsf{fma}\left(amt, \mathsf{fma}\left(V0, -2, \mathsf{fma}\left(amt, \mathsf{fma}\left(2, P0 - P1, V0\right), 3 \cdot \left(P1 - P0\right)\right)\right), P1\right), P0 \cdot 1.52587890625 \cdot 10^{-5}\right)\\

\mathbf{elif}\;P1 \leq 49999999215337470:\\
\;\;\;\;\mathsf{fma}\left(amt \cdot amt, \mathsf{fma}\left(P0, -3, \mathsf{fma}\left(amt, V0 + \mathsf{fma}\left(2, P0, V1\right), V0 \cdot -2\right) - V1\right), P0 \cdot 1.52587890625 \cdot 10^{-5}\right)\\

\mathbf{else}:\\
\;\;\;\;\mathsf{fma}\left(amt, \mathsf{fma}\left(amt, \mathsf{fma}\left(3, P1 - P0, \mathsf{fma}\left(amt, \mathsf{fma}\left(2, P0 - P1, V1\right), -V1\right)\right), P1\right), P0 \cdot 1.52587890625 \cdot 10^{-5}\right)\\


\end{array}
\end{array}

Derivation

Split input into 3 regimes
if P1 < -5e8
1. Initial program 74.1%
  \[\frac{\left(\left(P0 + P1 \cdot \left(amt \cdot 65536\right)\right) + \left(\left(3 \cdot \left(P1 - P0\right) - V1\right) - 2 \cdot V0\right) \cdot \left({amt}^{2} \cdot 65536\right)\right) + \left(\left(2 \cdot \left(P0 - P1\right) + V0\right) + V1\right) \cdot \left({amt}^{3} \cdot 65536\right)}{65536} \]
2. Add Preprocessing
3. Taylor expanded in amt around 0
  \[\leadsto \color{blue}{\frac{1}{65536} \cdot P0 + amt \cdot \left(P1 + amt \cdot \left(\left(3 \cdot \left(P1 - P0\right) + amt \cdot \left(V0 + \left(V1 + 2 \cdot \left(P0 - P1\right)\right)\right)\right) - \left(V1 + 2 \cdot V0\right)\right)\right)} \]
4. Step-by-step derivation
  1. +-commutativeN/A
    \[\leadsto \color{blue}{amt \cdot \left(P1 + amt \cdot \left(\left(3 \cdot \left(P1 - P0\right) + amt \cdot \left(V0 + \left(V1 + 2 \cdot \left(P0 - P1\right)\right)\right)\right) - \left(V1 + 2 \cdot V0\right)\right)\right) + \frac{1}{65536} \cdot P0} \]
  2. lower-fma.f32N/A
    \[\leadsto \color{blue}{\mathsf{fma}\left(amt, P1 + amt \cdot \left(\left(3 \cdot \left(P1 - P0\right) + amt \cdot \left(V0 + \left(V1 + 2 \cdot \left(P0 - P1\right)\right)\right)\right) - \left(V1 + 2 \cdot V0\right)\right), \frac{1}{65536} \cdot P0\right)} \]
5. Applied rewrites96.2%
  \[\leadsto \color{blue}{\mathsf{fma}\left(amt, \mathsf{fma}\left(amt, \mathsf{fma}\left(amt, \mathsf{fma}\left(2, P0 - P1, V1 + V0\right), \mathsf{fma}\left(3, P1 - P0, \mathsf{fma}\left(V0, -2, -V1\right)\right)\right), P1\right), P0 \cdot 1.52587890625 \cdot 10^{-5}\right)} \]
6. Taylor expanded in V1 around 0
  \[\leadsto \mathsf{fma}\left(amt, \mathsf{fma}\left(amt, -2 \cdot V0 + \color{blue}{\left(3 \cdot \left(P1 - P0\right) + amt \cdot \left(V0 + 2 \cdot \left(P0 - P1\right)\right)\right)}, P1\right), P0 \cdot \frac{1}{65536}\right) \]
7. Step-by-step derivation
8. Applied rewrites86.1%
  \[\leadsto \mathsf{fma}\left(amt, \mathsf{fma}\left(amt, \mathsf{fma}\left(V0, \color{blue}{-2}, \mathsf{fma}\left(amt, \mathsf{fma}\left(2, P0 - P1, V0\right), 3 \cdot \left(P1 - P0\right)\right)\right), P1\right), P0 \cdot 1.52587890625 \cdot 10^{-5}\right) \]
if -5e8 < P1 < 4.99999992e16
1. Initial program 79.4%
  \[\frac{\left(\left(P0 + P1 \cdot \left(amt \cdot 65536\right)\right) + \left(\left(3 \cdot \left(P1 - P0\right) - V1\right) - 2 \cdot V0\right) \cdot \left({amt}^{2} \cdot 65536\right)\right) + \left(\left(2 \cdot \left(P0 - P1\right) + V0\right) + V1\right) \cdot \left({amt}^{3} \cdot 65536\right)}{65536} \]
2. Add Preprocessing
3. Taylor expanded in amt around 0
  \[\leadsto \color{blue}{\frac{1}{65536} \cdot P0 + amt \cdot \left(P1 + amt \cdot \left(\left(3 \cdot \left(P1 - P0\right) + amt \cdot \left(V0 + \left(V1 + 2 \cdot \left(P0 - P1\right)\right)\right)\right) - \left(V1 + 2 \cdot V0\right)\right)\right)} \]
4. Step-by-step derivation
  1. +-commutativeN/A
    \[\leadsto \color{blue}{amt \cdot \left(P1 + amt \cdot \left(\left(3 \cdot \left(P1 - P0\right) + amt \cdot \left(V0 + \left(V1 + 2 \cdot \left(P0 - P1\right)\right)\right)\right) - \left(V1 + 2 \cdot V0\right)\right)\right) + \frac{1}{65536} \cdot P0} \]
  2. lower-fma.f32N/A
    \[\leadsto \color{blue}{\mathsf{fma}\left(amt, P1 + amt \cdot \left(\left(3 \cdot \left(P1 - P0\right) + amt \cdot \left(V0 + \left(V1 + 2 \cdot \left(P0 - P1\right)\right)\right)\right) - \left(V1 + 2 \cdot V0\right)\right), \frac{1}{65536} \cdot P0\right)} \]
5. Applied rewrites96.2%
  \[\leadsto \color{blue}{\mathsf{fma}\left(amt, \mathsf{fma}\left(amt, \mathsf{fma}\left(amt, \mathsf{fma}\left(2, P0 - P1, V1 + V0\right), \mathsf{fma}\left(3, P1 - P0, \mathsf{fma}\left(V0, -2, -V1\right)\right)\right), P1\right), P0 \cdot 1.52587890625 \cdot 10^{-5}\right)} \]
6. Taylor expanded in P1 around 0
  \[\leadsto \frac{1}{65536} \cdot P0 + \color{blue}{{amt}^{2} \cdot \left(\left(-3 \cdot P0 + \left(-2 \cdot V0 + amt \cdot \left(V0 + \left(V1 + 2 \cdot P0\right)\right)\right)\right) - V1\right)} \]
7. Step-by-step derivation
8. Applied rewrites89.7%
  \[\leadsto \mathsf{fma}\left(amt \cdot amt, \color{blue}{\mathsf{fma}\left(P0, -3, \mathsf{fma}\left(amt, V0 + \mathsf{fma}\left(2, P0, V1\right), V0 \cdot -2\right) - V1\right)}, 1.52587890625 \cdot 10^{-5} \cdot P0\right) \]
if 4.99999992e16 < P1
1. Initial program 50.6%
  \[\frac{\left(\left(P0 + P1 \cdot \left(amt \cdot 65536\right)\right) + \left(\left(3 \cdot \left(P1 - P0\right) - V1\right) - 2 \cdot V0\right) \cdot \left({amt}^{2} \cdot 65536\right)\right) + \left(\left(2 \cdot \left(P0 - P1\right) + V0\right) + V1\right) \cdot \left({amt}^{3} \cdot 65536\right)}{65536} \]
2. Add Preprocessing
3. Taylor expanded in amt around 0
  \[\leadsto \color{blue}{\frac{1}{65536} \cdot P0 + amt \cdot \left(P1 + amt \cdot \left(\left(3 \cdot \left(P1 - P0\right) + amt \cdot \left(V0 + \left(V1 + 2 \cdot \left(P0 - P1\right)\right)\right)\right) - \left(V1 + 2 \cdot V0\right)\right)\right)} \]
4. Step-by-step derivation
  1. +-commutativeN/A
    \[\leadsto \color{blue}{amt \cdot \left(P1 + amt \cdot \left(\left(3 \cdot \left(P1 - P0\right) + amt \cdot \left(V0 + \left(V1 + 2 \cdot \left(P0 - P1\right)\right)\right)\right) - \left(V1 + 2 \cdot V0\right)\right)\right) + \frac{1}{65536} \cdot P0} \]
  2. lower-fma.f32N/A
    \[\leadsto \color{blue}{\mathsf{fma}\left(amt, P1 + amt \cdot \left(\left(3 \cdot \left(P1 - P0\right) + amt \cdot \left(V0 + \left(V1 + 2 \cdot \left(P0 - P1\right)\right)\right)\right) - \left(V1 + 2 \cdot V0\right)\right), \frac{1}{65536} \cdot P0\right)} \]
5. Applied rewrites98.3%
  \[\leadsto \color{blue}{\mathsf{fma}\left(amt, \mathsf{fma}\left(amt, \mathsf{fma}\left(amt, \mathsf{fma}\left(2, P0 - P1, V1 + V0\right), \mathsf{fma}\left(3, P1 - P0, \mathsf{fma}\left(V0, -2, -V1\right)\right)\right), P1\right), P0 \cdot 1.52587890625 \cdot 10^{-5}\right)} \]
6. Taylor expanded in V0 around 0
  \[\leadsto \mathsf{fma}\left(amt, \mathsf{fma}\left(amt, \left(3 \cdot \left(P1 - P0\right) + amt \cdot \left(V1 + 2 \cdot \left(P0 - P1\right)\right)\right) - \color{blue}{V1}, P1\right), P0 \cdot \frac{1}{65536}\right) \]
7. Step-by-step derivation
8. Applied rewrites93.5%
  \[\leadsto \mathsf{fma}\left(amt, \mathsf{fma}\left(amt, \mathsf{fma}\left(3, \color{blue}{P1 - P0}, \mathsf{fma}\left(amt, \mathsf{fma}\left(2, P0 - P1, V1\right), -V1\right)\right), P1\right), P0 \cdot 1.52587890625 \cdot 10^{-5}\right) \]
Recombined 3 regimes into one program.
Final simplification89.5%
\[\leadsto \begin{array}{l} \mathbf{if}\;P1 \leq -500000000:\\ \;\;\;\;\mathsf{fma}\left(amt, \mathsf{fma}\left(amt, \mathsf{fma}\left(V0, -2, \mathsf{fma}\left(amt, \mathsf{fma}\left(2, P0 - P1, V0\right), 3 \cdot \left(P1 - P0\right)\right)\right), P1\right), P0 \cdot 1.52587890625 \cdot 10^{-5}\right)\\ \mathbf{elif}\;P1 \leq 49999999215337470:\\ \;\;\;\;\mathsf{fma}\left(amt \cdot amt, \mathsf{fma}\left(P0, -3, \mathsf{fma}\left(amt, V0 + \mathsf{fma}\left(2, P0, V1\right), V0 \cdot -2\right) - V1\right), P0 \cdot 1.52587890625 \cdot 10^{-5}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(amt, \mathsf{fma}\left(amt, \mathsf{fma}\left(3, P1 - P0, \mathsf{fma}\left(amt, \mathsf{fma}\left(2, P0 - P1, V1\right), -V1\right)\right), P1\right), P0 \cdot 1.52587890625 \cdot 10^{-5}\right)\\ \end{array} \]
Add Preprocessing

Alternative 3: 84.9% accurate, 4.9× speedup?

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;P1 \leq -4999999913984:\\ \;\;\;\;amt \cdot \mathsf{fma}\left(amt, \mathsf{fma}\left(V0, -2, \mathsf{fma}\left(amt, V0 + \mathsf{fma}\left(P1, -2, V1\right), P1 \cdot 3\right) - V1\right), P1\right)\\ \mathbf{elif}\;P1 \leq 49999999215337470:\\ \;\;\;\;\mathsf{fma}\left(amt \cdot amt, \mathsf{fma}\left(P0, -3, \mathsf{fma}\left(amt, V0 + \mathsf{fma}\left(2, P0, V1\right), V0 \cdot -2\right) - V1\right), P0 \cdot 1.52587890625 \cdot 10^{-5}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(amt, \mathsf{fma}\left(amt, \mathsf{fma}\left(3, P1 - P0, \mathsf{fma}\left(amt, \mathsf{fma}\left(2, P0 - P1, V1\right), -V1\right)\right), P1\right), P0 \cdot 1.52587890625 \cdot 10^{-5}\right)\\ \end{array} \end{array} \]

(FPCore (P0 P1 amt V1 V0)
 :precision binary32
 (if (<= P1 -4999999913984.0)
   (*
    amt
    (fma
     amt
     (fma V0 -2.0 (- (fma amt (+ V0 (fma P1 -2.0 V1)) (* P1 3.0)) V1))
     P1))
   (if (<= P1 49999999215337470.0)
     (fma
      (* amt amt)
      (fma P0 -3.0 (- (fma amt (+ V0 (fma 2.0 P0 V1)) (* V0 -2.0)) V1))
      (* P0 1.52587890625e-5))
     (fma
      amt
      (fma amt (fma 3.0 (- P1 P0) (fma amt (fma 2.0 (- P0 P1) V1) (- V1))) P1)
      (* P0 1.52587890625e-5)))))

float code(float P0, float P1, float amt, float V1, float V0) {
	float tmp;
	if (P1 <= -4999999913984.0f) {
		tmp = amt * fmaf(amt, fmaf(V0, -2.0f, (fmaf(amt, (V0 + fmaf(P1, -2.0f, V1)), (P1 * 3.0f)) - V1)), P1);
	} else if (P1 <= 49999999215337470.0f) {
		tmp = fmaf((amt * amt), fmaf(P0, -3.0f, (fmaf(amt, (V0 + fmaf(2.0f, P0, V1)), (V0 * -2.0f)) - V1)), (P0 * 1.52587890625e-5f));
	} else {
		tmp = fmaf(amt, fmaf(amt, fmaf(3.0f, (P1 - P0), fmaf(amt, fmaf(2.0f, (P0 - P1), V1), -V1)), P1), (P0 * 1.52587890625e-5f));
	}
	return tmp;
}

function code(P0, P1, amt, V1, V0)
	tmp = Float32(0.0)
	if (P1 <= Float32(-4999999913984.0))
		tmp = Float32(amt * fma(amt, fma(V0, Float32(-2.0), Float32(fma(amt, Float32(V0 + fma(P1, Float32(-2.0), V1)), Float32(P1 * Float32(3.0))) - V1)), P1));
	elseif (P1 <= Float32(49999999215337470.0))
		tmp = fma(Float32(amt * amt), fma(P0, Float32(-3.0), Float32(fma(amt, Float32(V0 + fma(Float32(2.0), P0, V1)), Float32(V0 * Float32(-2.0))) - V1)), Float32(P0 * Float32(1.52587890625e-5)));
	else
		tmp = fma(amt, fma(amt, fma(Float32(3.0), Float32(P1 - P0), fma(amt, fma(Float32(2.0), Float32(P0 - P1), V1), Float32(-V1))), P1), Float32(P0 * Float32(1.52587890625e-5)));
	end
	return tmp
end

\begin{array}{l}

\\
\begin{array}{l}
\mathbf{if}\;P1 \leq -4999999913984:\\
\;\;\;\;amt \cdot \mathsf{fma}\left(amt, \mathsf{fma}\left(V0, -2, \mathsf{fma}\left(amt, V0 + \mathsf{fma}\left(P1, -2, V1\right), P1 \cdot 3\right) - V1\right), P1\right)\\

\mathbf{elif}\;P1 \leq 49999999215337470:\\
\;\;\;\;\mathsf{fma}\left(amt \cdot amt, \mathsf{fma}\left(P0, -3, \mathsf{fma}\left(amt, V0 + \mathsf{fma}\left(2, P0, V1\right), V0 \cdot -2\right) - V1\right), P0 \cdot 1.52587890625 \cdot 10^{-5}\right)\\

\mathbf{else}:\\
\;\;\;\;\mathsf{fma}\left(amt, \mathsf{fma}\left(amt, \mathsf{fma}\left(3, P1 - P0, \mathsf{fma}\left(amt, \mathsf{fma}\left(2, P0 - P1, V1\right), -V1\right)\right), P1\right), P0 \cdot 1.52587890625 \cdot 10^{-5}\right)\\


\end{array}
\end{array}

Derivation

Split input into 3 regimes
if P1 < -4999999910000
1. Initial program 67.7%
  \[\frac{\left(\left(P0 + P1 \cdot \left(amt \cdot 65536\right)\right) + \left(\left(3 \cdot \left(P1 - P0\right) - V1\right) - 2 \cdot V0\right) \cdot \left({amt}^{2} \cdot 65536\right)\right) + \left(\left(2 \cdot \left(P0 - P1\right) + V0\right) + V1\right) \cdot \left({amt}^{3} \cdot 65536\right)}{65536} \]
2. Add Preprocessing
3. Taylor expanded in amt around 0
  \[\leadsto \color{blue}{\frac{1}{65536} \cdot P0 + amt \cdot \left(P1 + amt \cdot \left(\left(3 \cdot \left(P1 - P0\right) + amt \cdot \left(V0 + \left(V1 + 2 \cdot \left(P0 - P1\right)\right)\right)\right) - \left(V1 + 2 \cdot V0\right)\right)\right)} \]
4. Step-by-step derivation
  1. +-commutativeN/A
    \[\leadsto \color{blue}{amt \cdot \left(P1 + amt \cdot \left(\left(3 \cdot \left(P1 - P0\right) + amt \cdot \left(V0 + \left(V1 + 2 \cdot \left(P0 - P1\right)\right)\right)\right) - \left(V1 + 2 \cdot V0\right)\right)\right) + \frac{1}{65536} \cdot P0} \]
  2. lower-fma.f32N/A
    \[\leadsto \color{blue}{\mathsf{fma}\left(amt, P1 + amt \cdot \left(\left(3 \cdot \left(P1 - P0\right) + amt \cdot \left(V0 + \left(V1 + 2 \cdot \left(P0 - P1\right)\right)\right)\right) - \left(V1 + 2 \cdot V0\right)\right), \frac{1}{65536} \cdot P0\right)} \]
5. Applied rewrites95.8%
  \[\leadsto \color{blue}{\mathsf{fma}\left(amt, \mathsf{fma}\left(amt, \mathsf{fma}\left(amt, \mathsf{fma}\left(2, P0 - P1, V1 + V0\right), \mathsf{fma}\left(3, P1 - P0, \mathsf{fma}\left(V0, -2, -V1\right)\right)\right), P1\right), P0 \cdot 1.52587890625 \cdot 10^{-5}\right)} \]
6. Taylor expanded in P0 around 0
  \[\leadsto amt \cdot \color{blue}{\left(P1 + amt \cdot \left(\left(-2 \cdot V0 + \left(3 \cdot P1 + amt \cdot \left(V0 + \left(V1 + -2 \cdot P1\right)\right)\right)\right) - V1\right)\right)} \]
7. Step-by-step derivation
8. Applied rewrites83.8%
  \[\leadsto amt \cdot \color{blue}{\mathsf{fma}\left(amt, \mathsf{fma}\left(V0, -2, \mathsf{fma}\left(amt, V0 + \mathsf{fma}\left(P1, -2, V1\right), P1 \cdot 3\right) - V1\right), P1\right)} \]
if -4999999910000 < P1 < 4.99999992e16
1. Initial program 80.2%
  \[\frac{\left(\left(P0 + P1 \cdot \left(amt \cdot 65536\right)\right) + \left(\left(3 \cdot \left(P1 - P0\right) - V1\right) - 2 \cdot V0\right) \cdot \left({amt}^{2} \cdot 65536\right)\right) + \left(\left(2 \cdot \left(P0 - P1\right) + V0\right) + V1\right) \cdot \left({amt}^{3} \cdot 65536\right)}{65536} \]
2. Add Preprocessing
3. Taylor expanded in amt around 0
  \[\leadsto \color{blue}{\frac{1}{65536} \cdot P0 + amt \cdot \left(P1 + amt \cdot \left(\left(3 \cdot \left(P1 - P0\right) + amt \cdot \left(V0 + \left(V1 + 2 \cdot \left(P0 - P1\right)\right)\right)\right) - \left(V1 + 2 \cdot V0\right)\right)\right)} \]
4. Step-by-step derivation
  1. +-commutativeN/A
    \[\leadsto \color{blue}{amt \cdot \left(P1 + amt \cdot \left(\left(3 \cdot \left(P1 - P0\right) + amt \cdot \left(V0 + \left(V1 + 2 \cdot \left(P0 - P1\right)\right)\right)\right) - \left(V1 + 2 \cdot V0\right)\right)\right) + \frac{1}{65536} \cdot P0} \]
  2. lower-fma.f32N/A
    \[\leadsto \color{blue}{\mathsf{fma}\left(amt, P1 + amt \cdot \left(\left(3 \cdot \left(P1 - P0\right) + amt \cdot \left(V0 + \left(V1 + 2 \cdot \left(P0 - P1\right)\right)\right)\right) - \left(V1 + 2 \cdot V0\right)\right), \frac{1}{65536} \cdot P0\right)} \]
5. Applied rewrites96.3%
  \[\leadsto \color{blue}{\mathsf{fma}\left(amt, \mathsf{fma}\left(amt, \mathsf{fma}\left(amt, \mathsf{fma}\left(2, P0 - P1, V1 + V0\right), \mathsf{fma}\left(3, P1 - P0, \mathsf{fma}\left(V0, -2, -V1\right)\right)\right), P1\right), P0 \cdot 1.52587890625 \cdot 10^{-5}\right)} \]
6. Taylor expanded in P1 around 0
  \[\leadsto \frac{1}{65536} \cdot P0 + \color{blue}{{amt}^{2} \cdot \left(\left(-3 \cdot P0 + \left(-2 \cdot V0 + amt \cdot \left(V0 + \left(V1 + 2 \cdot P0\right)\right)\right)\right) - V1\right)} \]
7. Step-by-step derivation
8. Applied rewrites89.0%
  \[\leadsto \mathsf{fma}\left(amt \cdot amt, \color{blue}{\mathsf{fma}\left(P0, -3, \mathsf{fma}\left(amt, V0 + \mathsf{fma}\left(2, P0, V1\right), V0 \cdot -2\right) - V1\right)}, 1.52587890625 \cdot 10^{-5} \cdot P0\right) \]
if 4.99999992e16 < P1
1. Initial program 50.6%
  \[\frac{\left(\left(P0 + P1 \cdot \left(amt \cdot 65536\right)\right) + \left(\left(3 \cdot \left(P1 - P0\right) - V1\right) - 2 \cdot V0\right) \cdot \left({amt}^{2} \cdot 65536\right)\right) + \left(\left(2 \cdot \left(P0 - P1\right) + V0\right) + V1\right) \cdot \left({amt}^{3} \cdot 65536\right)}{65536} \]
2. Add Preprocessing
3. Taylor expanded in amt around 0
  \[\leadsto \color{blue}{\frac{1}{65536} \cdot P0 + amt \cdot \left(P1 + amt \cdot \left(\left(3 \cdot \left(P1 - P0\right) + amt \cdot \left(V0 + \left(V1 + 2 \cdot \left(P0 - P1\right)\right)\right)\right) - \left(V1 + 2 \cdot V0\right)\right)\right)} \]
4. Step-by-step derivation
  1. +-commutativeN/A
    \[\leadsto \color{blue}{amt \cdot \left(P1 + amt \cdot \left(\left(3 \cdot \left(P1 - P0\right) + amt \cdot \left(V0 + \left(V1 + 2 \cdot \left(P0 - P1\right)\right)\right)\right) - \left(V1 + 2 \cdot V0\right)\right)\right) + \frac{1}{65536} \cdot P0} \]
  2. lower-fma.f32N/A
    \[\leadsto \color{blue}{\mathsf{fma}\left(amt, P1 + amt \cdot \left(\left(3 \cdot \left(P1 - P0\right) + amt \cdot \left(V0 + \left(V1 + 2 \cdot \left(P0 - P1\right)\right)\right)\right) - \left(V1 + 2 \cdot V0\right)\right), \frac{1}{65536} \cdot P0\right)} \]
5. Applied rewrites98.3%
  \[\leadsto \color{blue}{\mathsf{fma}\left(amt, \mathsf{fma}\left(amt, \mathsf{fma}\left(amt, \mathsf{fma}\left(2, P0 - P1, V1 + V0\right), \mathsf{fma}\left(3, P1 - P0, \mathsf{fma}\left(V0, -2, -V1\right)\right)\right), P1\right), P0 \cdot 1.52587890625 \cdot 10^{-5}\right)} \]
6. Taylor expanded in V0 around 0
  \[\leadsto \mathsf{fma}\left(amt, \mathsf{fma}\left(amt, \left(3 \cdot \left(P1 - P0\right) + amt \cdot \left(V1 + 2 \cdot \left(P0 - P1\right)\right)\right) - \color{blue}{V1}, P1\right), P0 \cdot \frac{1}{65536}\right) \]
7. Step-by-step derivation
8. Applied rewrites93.5%
  \[\leadsto \mathsf{fma}\left(amt, \mathsf{fma}\left(amt, \mathsf{fma}\left(3, \color{blue}{P1 - P0}, \mathsf{fma}\left(amt, \mathsf{fma}\left(2, P0 - P1, V1\right), -V1\right)\right), P1\right), P0 \cdot 1.52587890625 \cdot 10^{-5}\right) \]
Recombined 3 regimes into one program.
Final simplification88.9%
\[\leadsto \begin{array}{l} \mathbf{if}\;P1 \leq -4999999913984:\\ \;\;\;\;amt \cdot \mathsf{fma}\left(amt, \mathsf{fma}\left(V0, -2, \mathsf{fma}\left(amt, V0 + \mathsf{fma}\left(P1, -2, V1\right), P1 \cdot 3\right) - V1\right), P1\right)\\ \mathbf{elif}\;P1 \leq 49999999215337470:\\ \;\;\;\;\mathsf{fma}\left(amt \cdot amt, \mathsf{fma}\left(P0, -3, \mathsf{fma}\left(amt, V0 + \mathsf{fma}\left(2, P0, V1\right), V0 \cdot -2\right) - V1\right), P0 \cdot 1.52587890625 \cdot 10^{-5}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(amt, \mathsf{fma}\left(amt, \mathsf{fma}\left(3, P1 - P0, \mathsf{fma}\left(amt, \mathsf{fma}\left(2, P0 - P1, V1\right), -V1\right)\right), P1\right), P0 \cdot 1.52587890625 \cdot 10^{-5}\right)\\ \end{array} \]
Add Preprocessing

Alternative 4: 85.8% accurate, 5.1× speedup?

\[\begin{array}{l} \\ \begin{array}{l} t_0 := amt \cdot \mathsf{fma}\left(amt, \mathsf{fma}\left(V0, -2, \mathsf{fma}\left(amt, V0 + \mathsf{fma}\left(P1, -2, V1\right), P1 \cdot 3\right) - V1\right), P1\right)\\ \mathbf{if}\;V0 \leq -2.0000000400817547 \cdot 10^{+20}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;V0 \leq 39999999311872:\\ \;\;\;\;\mathsf{fma}\left(amt, \mathsf{fma}\left(amt, \mathsf{fma}\left(3, P1 - P0, \mathsf{fma}\left(amt, \mathsf{fma}\left(2, P0 - P1, V1\right), -V1\right)\right), P1\right), P0 \cdot 1.52587890625 \cdot 10^{-5}\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

(FPCore (P0 P1 amt V1 V0)
 :precision binary32
 (let* ((t_0
         (*
          amt
          (fma
           amt
           (fma V0 -2.0 (- (fma amt (+ V0 (fma P1 -2.0 V1)) (* P1 3.0)) V1))
           P1))))
   (if (<= V0 -2.0000000400817547e+20)
     t_0
     (if (<= V0 39999999311872.0)
       (fma
        amt
        (fma
         amt
         (fma 3.0 (- P1 P0) (fma amt (fma 2.0 (- P0 P1) V1) (- V1)))
         P1)
        (* P0 1.52587890625e-5))
       t_0))))

float code(float P0, float P1, float amt, float V1, float V0) {
	float t_0 = amt * fmaf(amt, fmaf(V0, -2.0f, (fmaf(amt, (V0 + fmaf(P1, -2.0f, V1)), (P1 * 3.0f)) - V1)), P1);
	float tmp;
	if (V0 <= -2.0000000400817547e+20f) {
		tmp = t_0;
	} else if (V0 <= 39999999311872.0f) {
		tmp = fmaf(amt, fmaf(amt, fmaf(3.0f, (P1 - P0), fmaf(amt, fmaf(2.0f, (P0 - P1), V1), -V1)), P1), (P0 * 1.52587890625e-5f));
	} else {
		tmp = t_0;
	}
	return tmp;
}

function code(P0, P1, amt, V1, V0)
	t_0 = Float32(amt * fma(amt, fma(V0, Float32(-2.0), Float32(fma(amt, Float32(V0 + fma(P1, Float32(-2.0), V1)), Float32(P1 * Float32(3.0))) - V1)), P1))
	tmp = Float32(0.0)
	if (V0 <= Float32(-2.0000000400817547e+20))
		tmp = t_0;
	elseif (V0 <= Float32(39999999311872.0))
		tmp = fma(amt, fma(amt, fma(Float32(3.0), Float32(P1 - P0), fma(amt, fma(Float32(2.0), Float32(P0 - P1), V1), Float32(-V1))), P1), Float32(P0 * Float32(1.52587890625e-5)));
	else
		tmp = t_0;
	end
	return tmp
end

\begin{array}{l}

\\
\begin{array}{l}
t_0 := amt \cdot \mathsf{fma}\left(amt, \mathsf{fma}\left(V0, -2, \mathsf{fma}\left(amt, V0 + \mathsf{fma}\left(P1, -2, V1\right), P1 \cdot 3\right) - V1\right), P1\right)\\
\mathbf{if}\;V0 \leq -2.0000000400817547 \cdot 10^{+20}:\\
\;\;\;\;t_0\\

\mathbf{elif}\;V0 \leq 39999999311872:\\
\;\;\;\;\mathsf{fma}\left(amt, \mathsf{fma}\left(amt, \mathsf{fma}\left(3, P1 - P0, \mathsf{fma}\left(amt, \mathsf{fma}\left(2, P0 - P1, V1\right), -V1\right)\right), P1\right), P0 \cdot 1.52587890625 \cdot 10^{-5}\right)\\

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


\end{array}
\end{array}

Derivation

Split input into 2 regimes
if V0 < -2.00000004e20 or 3.99999993e13 < V0
1. Initial program 68.4%
  \[\frac{\left(\left(P0 + P1 \cdot \left(amt \cdot 65536\right)\right) + \left(\left(3 \cdot \left(P1 - P0\right) - V1\right) - 2 \cdot V0\right) \cdot \left({amt}^{2} \cdot 65536\right)\right) + \left(\left(2 \cdot \left(P0 - P1\right) + V0\right) + V1\right) \cdot \left({amt}^{3} \cdot 65536\right)}{65536} \]
2. Add Preprocessing
3. Taylor expanded in amt around 0
  \[\leadsto \color{blue}{\frac{1}{65536} \cdot P0 + amt \cdot \left(P1 + amt \cdot \left(\left(3 \cdot \left(P1 - P0\right) + amt \cdot \left(V0 + \left(V1 + 2 \cdot \left(P0 - P1\right)\right)\right)\right) - \left(V1 + 2 \cdot V0\right)\right)\right)} \]
4. Step-by-step derivation
  1. +-commutativeN/A
    \[\leadsto \color{blue}{amt \cdot \left(P1 + amt \cdot \left(\left(3 \cdot \left(P1 - P0\right) + amt \cdot \left(V0 + \left(V1 + 2 \cdot \left(P0 - P1\right)\right)\right)\right) - \left(V1 + 2 \cdot V0\right)\right)\right) + \frac{1}{65536} \cdot P0} \]
  2. lower-fma.f32N/A
    \[\leadsto \color{blue}{\mathsf{fma}\left(amt, P1 + amt \cdot \left(\left(3 \cdot \left(P1 - P0\right) + amt \cdot \left(V0 + \left(V1 + 2 \cdot \left(P0 - P1\right)\right)\right)\right) - \left(V1 + 2 \cdot V0\right)\right), \frac{1}{65536} \cdot P0\right)} \]
5. Applied rewrites97.7%
  \[\leadsto \color{blue}{\mathsf{fma}\left(amt, \mathsf{fma}\left(amt, \mathsf{fma}\left(amt, \mathsf{fma}\left(2, P0 - P1, V1 + V0\right), \mathsf{fma}\left(3, P1 - P0, \mathsf{fma}\left(V0, -2, -V1\right)\right)\right), P1\right), P0 \cdot 1.52587890625 \cdot 10^{-5}\right)} \]
6. Taylor expanded in P0 around 0
  \[\leadsto amt \cdot \color{blue}{\left(P1 + amt \cdot \left(\left(-2 \cdot V0 + \left(3 \cdot P1 + amt \cdot \left(V0 + \left(V1 + -2 \cdot P1\right)\right)\right)\right) - V1\right)\right)} \]
7. Step-by-step derivation
8. Applied rewrites86.6%
  \[\leadsto amt \cdot \color{blue}{\mathsf{fma}\left(amt, \mathsf{fma}\left(V0, -2, \mathsf{fma}\left(amt, V0 + \mathsf{fma}\left(P1, -2, V1\right), P1 \cdot 3\right) - V1\right), P1\right)} \]
if -2.00000004e20 < V0 < 3.99999993e13
1. Initial program 78.2%
  \[\frac{\left(\left(P0 + P1 \cdot \left(amt \cdot 65536\right)\right) + \left(\left(3 \cdot \left(P1 - P0\right) - V1\right) - 2 \cdot V0\right) \cdot \left({amt}^{2} \cdot 65536\right)\right) + \left(\left(2 \cdot \left(P0 - P1\right) + V0\right) + V1\right) \cdot \left({amt}^{3} \cdot 65536\right)}{65536} \]
2. Add Preprocessing
3. Taylor expanded in amt around 0
  \[\leadsto \color{blue}{\frac{1}{65536} \cdot P0 + amt \cdot \left(P1 + amt \cdot \left(\left(3 \cdot \left(P1 - P0\right) + amt \cdot \left(V0 + \left(V1 + 2 \cdot \left(P0 - P1\right)\right)\right)\right) - \left(V1 + 2 \cdot V0\right)\right)\right)} \]
4. Step-by-step derivation
  1. +-commutativeN/A
    \[\leadsto \color{blue}{amt \cdot \left(P1 + amt \cdot \left(\left(3 \cdot \left(P1 - P0\right) + amt \cdot \left(V0 + \left(V1 + 2 \cdot \left(P0 - P1\right)\right)\right)\right) - \left(V1 + 2 \cdot V0\right)\right)\right) + \frac{1}{65536} \cdot P0} \]
  2. lower-fma.f32N/A
    \[\leadsto \color{blue}{\mathsf{fma}\left(amt, P1 + amt \cdot \left(\left(3 \cdot \left(P1 - P0\right) + amt \cdot \left(V0 + \left(V1 + 2 \cdot \left(P0 - P1\right)\right)\right)\right) - \left(V1 + 2 \cdot V0\right)\right), \frac{1}{65536} \cdot P0\right)} \]
5. Applied rewrites95.9%
  \[\leadsto \color{blue}{\mathsf{fma}\left(amt, \mathsf{fma}\left(amt, \mathsf{fma}\left(amt, \mathsf{fma}\left(2, P0 - P1, V1 + V0\right), \mathsf{fma}\left(3, P1 - P0, \mathsf{fma}\left(V0, -2, -V1\right)\right)\right), P1\right), P0 \cdot 1.52587890625 \cdot 10^{-5}\right)} \]
6. Taylor expanded in V0 around 0
  \[\leadsto \mathsf{fma}\left(amt, \mathsf{fma}\left(amt, \left(3 \cdot \left(P1 - P0\right) + amt \cdot \left(V1 + 2 \cdot \left(P0 - P1\right)\right)\right) - \color{blue}{V1}, P1\right), P0 \cdot \frac{1}{65536}\right) \]
7. Step-by-step derivation
8. Applied rewrites87.6%
  \[\leadsto \mathsf{fma}\left(amt, \mathsf{fma}\left(amt, \mathsf{fma}\left(3, \color{blue}{P1 - P0}, \mathsf{fma}\left(amt, \mathsf{fma}\left(2, P0 - P1, V1\right), -V1\right)\right), P1\right), P0 \cdot 1.52587890625 \cdot 10^{-5}\right) \]
Recombined 2 regimes into one program.
Add Preprocessing

Alternative 5: 80.3% accurate, 5.4× speedup?

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(amt, \mathsf{fma}\left(amt, P0 \cdot \mathsf{fma}\left(amt, 2, -3\right), P1\right), P0 \cdot 1.52587890625 \cdot 10^{-5}\right)\\ \mathbf{if}\;P0 \leq -1.0000000200408773 \cdot 10^{+20}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;P0 \leq 40000001090256900:\\ \;\;\;\;amt \cdot \mathsf{fma}\left(amt, \mathsf{fma}\left(V0, -2, \mathsf{fma}\left(amt, V0 + \mathsf{fma}\left(P1, -2, V1\right), P1 \cdot 3\right) - V1\right), P1\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

(FPCore (P0 P1 amt V1 V0)
 :precision binary32
 (let* ((t_0
         (fma
          amt
          (fma amt (* P0 (fma amt 2.0 -3.0)) P1)
          (* P0 1.52587890625e-5))))
   (if (<= P0 -1.0000000200408773e+20)
     t_0
     (if (<= P0 40000001090256900.0)
       (*
        amt
        (fma
         amt
         (fma V0 -2.0 (- (fma amt (+ V0 (fma P1 -2.0 V1)) (* P1 3.0)) V1))
         P1))
       t_0))))

float code(float P0, float P1, float amt, float V1, float V0) {
	float t_0 = fmaf(amt, fmaf(amt, (P0 * fmaf(amt, 2.0f, -3.0f)), P1), (P0 * 1.52587890625e-5f));
	float tmp;
	if (P0 <= -1.0000000200408773e+20f) {
		tmp = t_0;
	} else if (P0 <= 40000001090256900.0f) {
		tmp = amt * fmaf(amt, fmaf(V0, -2.0f, (fmaf(amt, (V0 + fmaf(P1, -2.0f, V1)), (P1 * 3.0f)) - V1)), P1);
	} else {
		tmp = t_0;
	}
	return tmp;
}

function code(P0, P1, amt, V1, V0)
	t_0 = fma(amt, fma(amt, Float32(P0 * fma(amt, Float32(2.0), Float32(-3.0))), P1), Float32(P0 * Float32(1.52587890625e-5)))
	tmp = Float32(0.0)
	if (P0 <= Float32(-1.0000000200408773e+20))
		tmp = t_0;
	elseif (P0 <= Float32(40000001090256900.0))
		tmp = Float32(amt * fma(amt, fma(V0, Float32(-2.0), Float32(fma(amt, Float32(V0 + fma(P1, Float32(-2.0), V1)), Float32(P1 * Float32(3.0))) - V1)), P1));
	else
		tmp = t_0;
	end
	return tmp
end

\begin{array}{l}

\\
\begin{array}{l}
t_0 := \mathsf{fma}\left(amt, \mathsf{fma}\left(amt, P0 \cdot \mathsf{fma}\left(amt, 2, -3\right), P1\right), P0 \cdot 1.52587890625 \cdot 10^{-5}\right)\\
\mathbf{if}\;P0 \leq -1.0000000200408773 \cdot 10^{+20}:\\
\;\;\;\;t_0\\

\mathbf{elif}\;P0 \leq 40000001090256900:\\
\;\;\;\;amt \cdot \mathsf{fma}\left(amt, \mathsf{fma}\left(V0, -2, \mathsf{fma}\left(amt, V0 + \mathsf{fma}\left(P1, -2, V1\right), P1 \cdot 3\right) - V1\right), P1\right)\\

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


\end{array}
\end{array}

Derivation

Split input into 2 regimes
if P0 < -1.00000002e20 or 4.00000011e16 < P0
1. Initial program 61.0%
  \[\frac{\left(\left(P0 + P1 \cdot \left(amt \cdot 65536\right)\right) + \left(\left(3 \cdot \left(P1 - P0\right) - V1\right) - 2 \cdot V0\right) \cdot \left({amt}^{2} \cdot 65536\right)\right) + \left(\left(2 \cdot \left(P0 - P1\right) + V0\right) + V1\right) \cdot \left({amt}^{3} \cdot 65536\right)}{65536} \]
2. Add Preprocessing
3. Taylor expanded in amt around 0
  \[\leadsto \color{blue}{\frac{1}{65536} \cdot P0 + amt \cdot \left(P1 + amt \cdot \left(\left(3 \cdot \left(P1 - P0\right) + amt \cdot \left(V0 + \left(V1 + 2 \cdot \left(P0 - P1\right)\right)\right)\right) - \left(V1 + 2 \cdot V0\right)\right)\right)} \]
4. Step-by-step derivation
  1. +-commutativeN/A
    \[\leadsto \color{blue}{amt \cdot \left(P1 + amt \cdot \left(\left(3 \cdot \left(P1 - P0\right) + amt \cdot \left(V0 + \left(V1 + 2 \cdot \left(P0 - P1\right)\right)\right)\right) - \left(V1 + 2 \cdot V0\right)\right)\right) + \frac{1}{65536} \cdot P0} \]
  2. lower-fma.f32N/A
    \[\leadsto \color{blue}{\mathsf{fma}\left(amt, P1 + amt \cdot \left(\left(3 \cdot \left(P1 - P0\right) + amt \cdot \left(V0 + \left(V1 + 2 \cdot \left(P0 - P1\right)\right)\right)\right) - \left(V1 + 2 \cdot V0\right)\right), \frac{1}{65536} \cdot P0\right)} \]
5. Applied rewrites93.9%
  \[\leadsto \color{blue}{\mathsf{fma}\left(amt, \mathsf{fma}\left(amt, \mathsf{fma}\left(amt, \mathsf{fma}\left(2, P0 - P1, V1 + V0\right), \mathsf{fma}\left(3, P1 - P0, \mathsf{fma}\left(V0, -2, -V1\right)\right)\right), P1\right), P0 \cdot 1.52587890625 \cdot 10^{-5}\right)} \]
6. Taylor expanded in P0 around inf
  \[\leadsto \mathsf{fma}\left(amt, \mathsf{fma}\left(amt, P0 \cdot \color{blue}{\left(2 \cdot amt - 3\right)}, P1\right), P0 \cdot \frac{1}{65536}\right) \]
7. Step-by-step derivation
8. Applied rewrites73.0%
  \[\leadsto \mathsf{fma}\left(amt, \mathsf{fma}\left(amt, P0 \cdot \color{blue}{\mathsf{fma}\left(amt, 2, -3\right)}, P1\right), P0 \cdot 1.52587890625 \cdot 10^{-5}\right) \]
if -1.00000002e20 < P0 < 4.00000011e16
1. Initial program 80.9%
  \[\frac{\left(\left(P0 + P1 \cdot \left(amt \cdot 65536\right)\right) + \left(\left(3 \cdot \left(P1 - P0\right) - V1\right) - 2 \cdot V0\right) \cdot \left({amt}^{2} \cdot 65536\right)\right) + \left(\left(2 \cdot \left(P0 - P1\right) + V0\right) + V1\right) \cdot \left({amt}^{3} \cdot 65536\right)}{65536} \]
2. Add Preprocessing
3. Taylor expanded in amt around 0
  \[\leadsto \color{blue}{\frac{1}{65536} \cdot P0 + amt \cdot \left(P1 + amt \cdot \left(\left(3 \cdot \left(P1 - P0\right) + amt \cdot \left(V0 + \left(V1 + 2 \cdot \left(P0 - P1\right)\right)\right)\right) - \left(V1 + 2 \cdot V0\right)\right)\right)} \]
4. Step-by-step derivation
  1. +-commutativeN/A
    \[\leadsto \color{blue}{amt \cdot \left(P1 + amt \cdot \left(\left(3 \cdot \left(P1 - P0\right) + amt \cdot \left(V0 + \left(V1 + 2 \cdot \left(P0 - P1\right)\right)\right)\right) - \left(V1 + 2 \cdot V0\right)\right)\right) + \frac{1}{65536} \cdot P0} \]
  2. lower-fma.f32N/A
    \[\leadsto \color{blue}{\mathsf{fma}\left(amt, P1 + amt \cdot \left(\left(3 \cdot \left(P1 - P0\right) + amt \cdot \left(V0 + \left(V1 + 2 \cdot \left(P0 - P1\right)\right)\right)\right) - \left(V1 + 2 \cdot V0\right)\right), \frac{1}{65536} \cdot P0\right)} \]
5. Applied rewrites97.5%
  \[\leadsto \color{blue}{\mathsf{fma}\left(amt, \mathsf{fma}\left(amt, \mathsf{fma}\left(amt, \mathsf{fma}\left(2, P0 - P1, V1 + V0\right), \mathsf{fma}\left(3, P1 - P0, \mathsf{fma}\left(V0, -2, -V1\right)\right)\right), P1\right), P0 \cdot 1.52587890625 \cdot 10^{-5}\right)} \]
6. Taylor expanded in P0 around 0
  \[\leadsto amt \cdot \color{blue}{\left(P1 + amt \cdot \left(\left(-2 \cdot V0 + \left(3 \cdot P1 + amt \cdot \left(V0 + \left(V1 + -2 \cdot P1\right)\right)\right)\right) - V1\right)\right)} \]
7. Step-by-step derivation
8. Applied rewrites84.0%
  \[\leadsto amt \cdot \color{blue}{\mathsf{fma}\left(amt, \mathsf{fma}\left(V0, -2, \mathsf{fma}\left(amt, V0 + \mathsf{fma}\left(P1, -2, V1\right), P1 \cdot 3\right) - V1\right), P1\right)} \]
Recombined 2 regimes into one program.
Add Preprocessing

Alternative 6: 96.4% accurate, 5.4× speedup?

\[\begin{array}{l} \\ \mathsf{fma}\left(amt, \mathsf{fma}\left(amt, \mathsf{fma}\left(amt, \mathsf{fma}\left(2, P0 - P1, V1 + V0\right), \mathsf{fma}\left(3, P1 - P0, \mathsf{fma}\left(V0, -2, -V1\right)\right)\right), P1\right), P0 \cdot 1.52587890625 \cdot 10^{-5}\right) \end{array} \]

(FPCore (P0 P1 amt V1 V0)
 :precision binary32
 (fma
  amt
  (fma
   amt
   (fma
    amt
    (fma 2.0 (- P0 P1) (+ V1 V0))
    (fma 3.0 (- P1 P0) (fma V0 -2.0 (- V1))))
   P1)
  (* P0 1.52587890625e-5)))

float code(float P0, float P1, float amt, float V1, float V0) {
	return fmaf(amt, fmaf(amt, fmaf(amt, fmaf(2.0f, (P0 - P1), (V1 + V0)), fmaf(3.0f, (P1 - P0), fmaf(V0, -2.0f, -V1))), P1), (P0 * 1.52587890625e-5f));
}

function code(P0, P1, amt, V1, V0)
	return fma(amt, fma(amt, fma(amt, fma(Float32(2.0), Float32(P0 - P1), Float32(V1 + V0)), fma(Float32(3.0), Float32(P1 - P0), fma(V0, Float32(-2.0), Float32(-V1)))), P1), Float32(P0 * Float32(1.52587890625e-5)))
end

\begin{array}{l}

\\
\mathsf{fma}\left(amt, \mathsf{fma}\left(amt, \mathsf{fma}\left(amt, \mathsf{fma}\left(2, P0 - P1, V1 + V0\right), \mathsf{fma}\left(3, P1 - P0, \mathsf{fma}\left(V0, -2, -V1\right)\right)\right), P1\right), P0 \cdot 1.52587890625 \cdot 10^{-5}\right)
\end{array}

Derivation

Initial program 75.2%
\[\frac{\left(\left(P0 + P1 \cdot \left(amt \cdot 65536\right)\right) + \left(\left(3 \cdot \left(P1 - P0\right) - V1\right) - 2 \cdot V0\right) \cdot \left({amt}^{2} \cdot 65536\right)\right) + \left(\left(2 \cdot \left(P0 - P1\right) + V0\right) + V1\right) \cdot \left({amt}^{3} \cdot 65536\right)}{65536} \]
Add Preprocessing
Taylor expanded in amt around 0
\[\leadsto \color{blue}{\frac{1}{65536} \cdot P0 + amt \cdot \left(P1 + amt \cdot \left(\left(3 \cdot \left(P1 - P0\right) + amt \cdot \left(V0 + \left(V1 + 2 \cdot \left(P0 - P1\right)\right)\right)\right) - \left(V1 + 2 \cdot V0\right)\right)\right)} \]
Step-by-step derivation
1. +-commutativeN/A
  \[\leadsto \color{blue}{amt \cdot \left(P1 + amt \cdot \left(\left(3 \cdot \left(P1 - P0\right) + amt \cdot \left(V0 + \left(V1 + 2 \cdot \left(P0 - P1\right)\right)\right)\right) - \left(V1 + 2 \cdot V0\right)\right)\right) + \frac{1}{65536} \cdot P0} \]
2. lower-fma.f32N/A
  \[\leadsto \color{blue}{\mathsf{fma}\left(amt, P1 + amt \cdot \left(\left(3 \cdot \left(P1 - P0\right) + amt \cdot \left(V0 + \left(V1 + 2 \cdot \left(P0 - P1\right)\right)\right)\right) - \left(V1 + 2 \cdot V0\right)\right), \frac{1}{65536} \cdot P0\right)} \]
Applied rewrites96.5%
\[\leadsto \color{blue}{\mathsf{fma}\left(amt, \mathsf{fma}\left(amt, \mathsf{fma}\left(amt, \mathsf{fma}\left(2, P0 - P1, V1 + V0\right), \mathsf{fma}\left(3, P1 - P0, \mathsf{fma}\left(V0, -2, -V1\right)\right)\right), P1\right), P0 \cdot 1.52587890625 \cdot 10^{-5}\right)} \]
Add Preprocessing

Alternative 7: 96.4% accurate, 5.4× speedup?

\[\begin{array}{l} \\ \mathsf{fma}\left(P0, 1.52587890625 \cdot 10^{-5}, amt \cdot \mathsf{fma}\left(amt, \mathsf{fma}\left(amt, V0 + \mathsf{fma}\left(2, P0 - P1, V1\right), \mathsf{fma}\left(3, P1 - P0, V0 \cdot -2\right) - V1\right), P1\right)\right) \end{array} \]

(FPCore (P0 P1 amt V1 V0)
 :precision binary32
 (fma
  P0
  1.52587890625e-5
  (*
   amt
   (fma
    amt
    (fma
     amt
     (+ V0 (fma 2.0 (- P0 P1) V1))
     (- (fma 3.0 (- P1 P0) (* V0 -2.0)) V1))
    P1))))

float code(float P0, float P1, float amt, float V1, float V0) {
	return fmaf(P0, 1.52587890625e-5f, (amt * fmaf(amt, fmaf(amt, (V0 + fmaf(2.0f, (P0 - P1), V1)), (fmaf(3.0f, (P1 - P0), (V0 * -2.0f)) - V1)), P1)));
}

function code(P0, P1, amt, V1, V0)
	return fma(P0, Float32(1.52587890625e-5), Float32(amt * fma(amt, fma(amt, Float32(V0 + fma(Float32(2.0), Float32(P0 - P1), V1)), Float32(fma(Float32(3.0), Float32(P1 - P0), Float32(V0 * Float32(-2.0))) - V1)), P1)))
end

\begin{array}{l}

\\
\mathsf{fma}\left(P0, 1.52587890625 \cdot 10^{-5}, amt \cdot \mathsf{fma}\left(amt, \mathsf{fma}\left(amt, V0 + \mathsf{fma}\left(2, P0 - P1, V1\right), \mathsf{fma}\left(3, P1 - P0, V0 \cdot -2\right) - V1\right), P1\right)\right)
\end{array}

Derivation

Initial program 75.2%
\[\frac{\left(\left(P0 + P1 \cdot \left(amt \cdot 65536\right)\right) + \left(\left(3 \cdot \left(P1 - P0\right) - V1\right) - 2 \cdot V0\right) \cdot \left({amt}^{2} \cdot 65536\right)\right) + \left(\left(2 \cdot \left(P0 - P1\right) + V0\right) + V1\right) \cdot \left({amt}^{3} \cdot 65536\right)}{65536} \]
Add Preprocessing
Taylor expanded in amt around 0
\[\leadsto \color{blue}{\frac{1}{65536} \cdot P0 + amt \cdot \left(P1 + amt \cdot \left(\left(3 \cdot \left(P1 - P0\right) + amt \cdot \left(V0 + \left(V1 + 2 \cdot \left(P0 - P1\right)\right)\right)\right) - \left(V1 + 2 \cdot V0\right)\right)\right)} \]
Step-by-step derivation
1. +-commutativeN/A
  \[\leadsto \color{blue}{amt \cdot \left(P1 + amt \cdot \left(\left(3 \cdot \left(P1 - P0\right) + amt \cdot \left(V0 + \left(V1 + 2 \cdot \left(P0 - P1\right)\right)\right)\right) - \left(V1 + 2 \cdot V0\right)\right)\right) + \frac{1}{65536} \cdot P0} \]
2. lower-fma.f32N/A
  \[\leadsto \color{blue}{\mathsf{fma}\left(amt, P1 + amt \cdot \left(\left(3 \cdot \left(P1 - P0\right) + amt \cdot \left(V0 + \left(V1 + 2 \cdot \left(P0 - P1\right)\right)\right)\right) - \left(V1 + 2 \cdot V0\right)\right), \frac{1}{65536} \cdot P0\right)} \]
Applied rewrites96.5%
\[\leadsto \color{blue}{\mathsf{fma}\left(amt, \mathsf{fma}\left(amt, \mathsf{fma}\left(amt, \mathsf{fma}\left(2, P0 - P1, V1 + V0\right), \mathsf{fma}\left(3, P1 - P0, \mathsf{fma}\left(V0, -2, -V1\right)\right)\right), P1\right), P0 \cdot 1.52587890625 \cdot 10^{-5}\right)} \]
Step-by-step derivation
Applied rewrites96.3%
\[\leadsto \mathsf{fma}\left(P0, \color{blue}{1.52587890625 \cdot 10^{-5}}, amt \cdot \mathsf{fma}\left(amt, \mathsf{fma}\left(amt, V0 + \mathsf{fma}\left(2, P0 - P1, V1\right), \mathsf{fma}\left(3, P1 - P0, V0 \cdot -2\right) - V1\right), P1\right)\right) \]
Add Preprocessing

Alternative 8: 45.1% accurate, 6.1× speedup?

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(amt, \mathsf{fma}\left(amt, P0 \cdot \mathsf{fma}\left(amt, 2, -3\right), P1\right), P0 \cdot 1.52587890625 \cdot 10^{-5}\right)\\ \mathbf{if}\;P0 \leq -1000000000:\\ \;\;\;\;t_0\\ \mathbf{elif}\;P0 \leq -5.000000156871975 \cdot 10^{-23}:\\ \;\;\;\;\mathsf{fma}\left(amt, \mathsf{fma}\left(amt, V1 \cdot \left(amt + -1\right), P1\right), P0 \cdot 1.52587890625 \cdot 10^{-5}\right)\\ \mathbf{elif}\;P0 \leq 2000000:\\ \;\;\;\;\mathsf{fma}\left(amt, \mathsf{fma}\left(amt, V0 \cdot \left(amt + -2\right), P1\right), P0 \cdot 1.52587890625 \cdot 10^{-5}\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

(FPCore (P0 P1 amt V1 V0)
 :precision binary32
 (let* ((t_0
         (fma
          amt
          (fma amt (* P0 (fma amt 2.0 -3.0)) P1)
          (* P0 1.52587890625e-5))))
   (if (<= P0 -1000000000.0)
     t_0
     (if (<= P0 -5.000000156871975e-23)
       (fma amt (fma amt (* V1 (+ amt -1.0)) P1) (* P0 1.52587890625e-5))
       (if (<= P0 2000000.0)
         (fma amt (fma amt (* V0 (+ amt -2.0)) P1) (* P0 1.52587890625e-5))
         t_0)))))

float code(float P0, float P1, float amt, float V1, float V0) {
	float t_0 = fmaf(amt, fmaf(amt, (P0 * fmaf(amt, 2.0f, -3.0f)), P1), (P0 * 1.52587890625e-5f));
	float tmp;
	if (P0 <= -1000000000.0f) {
		tmp = t_0;
	} else if (P0 <= -5.000000156871975e-23f) {
		tmp = fmaf(amt, fmaf(amt, (V1 * (amt + -1.0f)), P1), (P0 * 1.52587890625e-5f));
	} else if (P0 <= 2000000.0f) {
		tmp = fmaf(amt, fmaf(amt, (V0 * (amt + -2.0f)), P1), (P0 * 1.52587890625e-5f));
	} else {
		tmp = t_0;
	}
	return tmp;
}

function code(P0, P1, amt, V1, V0)
	t_0 = fma(amt, fma(amt, Float32(P0 * fma(amt, Float32(2.0), Float32(-3.0))), P1), Float32(P0 * Float32(1.52587890625e-5)))
	tmp = Float32(0.0)
	if (P0 <= Float32(-1000000000.0))
		tmp = t_0;
	elseif (P0 <= Float32(-5.000000156871975e-23))
		tmp = fma(amt, fma(amt, Float32(V1 * Float32(amt + Float32(-1.0))), P1), Float32(P0 * Float32(1.52587890625e-5)));
	elseif (P0 <= Float32(2000000.0))
		tmp = fma(amt, fma(amt, Float32(V0 * Float32(amt + Float32(-2.0))), P1), Float32(P0 * Float32(1.52587890625e-5)));
	else
		tmp = t_0;
	end
	return tmp
end

\begin{array}{l}

\\
\begin{array}{l}
t_0 := \mathsf{fma}\left(amt, \mathsf{fma}\left(amt, P0 \cdot \mathsf{fma}\left(amt, 2, -3\right), P1\right), P0 \cdot 1.52587890625 \cdot 10^{-5}\right)\\
\mathbf{if}\;P0 \leq -1000000000:\\
\;\;\;\;t_0\\

\mathbf{elif}\;P0 \leq -5.000000156871975 \cdot 10^{-23}:\\
\;\;\;\;\mathsf{fma}\left(amt, \mathsf{fma}\left(amt, V1 \cdot \left(amt + -1\right), P1\right), P0 \cdot 1.52587890625 \cdot 10^{-5}\right)\\

\mathbf{elif}\;P0 \leq 2000000:\\
\;\;\;\;\mathsf{fma}\left(amt, \mathsf{fma}\left(amt, V0 \cdot \left(amt + -2\right), P1\right), P0 \cdot 1.52587890625 \cdot 10^{-5}\right)\\

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


\end{array}
\end{array}

Derivation

Split input into 3 regimes
if P0 < -1e9 or 2e6 < P0
1. Initial program 68.0%
  \[\frac{\left(\left(P0 + P1 \cdot \left(amt \cdot 65536\right)\right) + \left(\left(3 \cdot \left(P1 - P0\right) - V1\right) - 2 \cdot V0\right) \cdot \left({amt}^{2} \cdot 65536\right)\right) + \left(\left(2 \cdot \left(P0 - P1\right) + V0\right) + V1\right) \cdot \left({amt}^{3} \cdot 65536\right)}{65536} \]
2. Add Preprocessing
3. Taylor expanded in amt around 0
  \[\leadsto \color{blue}{\frac{1}{65536} \cdot P0 + amt \cdot \left(P1 + amt \cdot \left(\left(3 \cdot \left(P1 - P0\right) + amt \cdot \left(V0 + \left(V1 + 2 \cdot \left(P0 - P1\right)\right)\right)\right) - \left(V1 + 2 \cdot V0\right)\right)\right)} \]
4. Step-by-step derivation
  1. +-commutativeN/A
    \[\leadsto \color{blue}{amt \cdot \left(P1 + amt \cdot \left(\left(3 \cdot \left(P1 - P0\right) + amt \cdot \left(V0 + \left(V1 + 2 \cdot \left(P0 - P1\right)\right)\right)\right) - \left(V1 + 2 \cdot V0\right)\right)\right) + \frac{1}{65536} \cdot P0} \]
  2. lower-fma.f32N/A
    \[\leadsto \color{blue}{\mathsf{fma}\left(amt, P1 + amt \cdot \left(\left(3 \cdot \left(P1 - P0\right) + amt \cdot \left(V0 + \left(V1 + 2 \cdot \left(P0 - P1\right)\right)\right)\right) - \left(V1 + 2 \cdot V0\right)\right), \frac{1}{65536} \cdot P0\right)} \]
5. Applied rewrites94.9%
  \[\leadsto \color{blue}{\mathsf{fma}\left(amt, \mathsf{fma}\left(amt, \mathsf{fma}\left(amt, \mathsf{fma}\left(2, P0 - P1, V1 + V0\right), \mathsf{fma}\left(3, P1 - P0, \mathsf{fma}\left(V0, -2, -V1\right)\right)\right), P1\right), P0 \cdot 1.52587890625 \cdot 10^{-5}\right)} \]
6. Taylor expanded in P0 around inf
  \[\leadsto \mathsf{fma}\left(amt, \mathsf{fma}\left(amt, P0 \cdot \color{blue}{\left(2 \cdot amt - 3\right)}, P1\right), P0 \cdot \frac{1}{65536}\right) \]
7. Step-by-step derivation
8. Applied rewrites64.7%
  \[\leadsto \mathsf{fma}\left(amt, \mathsf{fma}\left(amt, P0 \cdot \color{blue}{\mathsf{fma}\left(amt, 2, -3\right)}, P1\right), P0 \cdot 1.52587890625 \cdot 10^{-5}\right) \]
if -1e9 < P0 < -5.00000016e-23
1. Initial program 80.5%
  \[\frac{\left(\left(P0 + P1 \cdot \left(amt \cdot 65536\right)\right) + \left(\left(3 \cdot \left(P1 - P0\right) - V1\right) - 2 \cdot V0\right) \cdot \left({amt}^{2} \cdot 65536\right)\right) + \left(\left(2 \cdot \left(P0 - P1\right) + V0\right) + V1\right) \cdot \left({amt}^{3} \cdot 65536\right)}{65536} \]
2. Add Preprocessing
3. Taylor expanded in amt around 0
  \[\leadsto \color{blue}{\frac{1}{65536} \cdot P0 + amt \cdot \left(P1 + amt \cdot \left(\left(3 \cdot \left(P1 - P0\right) + amt \cdot \left(V0 + \left(V1 + 2 \cdot \left(P0 - P1\right)\right)\right)\right) - \left(V1 + 2 \cdot V0\right)\right)\right)} \]
4. Step-by-step derivation
  1. +-commutativeN/A
    \[\leadsto \color{blue}{amt \cdot \left(P1 + amt \cdot \left(\left(3 \cdot \left(P1 - P0\right) + amt \cdot \left(V0 + \left(V1 + 2 \cdot \left(P0 - P1\right)\right)\right)\right) - \left(V1 + 2 \cdot V0\right)\right)\right) + \frac{1}{65536} \cdot P0} \]
  2. lower-fma.f32N/A
    \[\leadsto \color{blue}{\mathsf{fma}\left(amt, P1 + amt \cdot \left(\left(3 \cdot \left(P1 - P0\right) + amt \cdot \left(V0 + \left(V1 + 2 \cdot \left(P0 - P1\right)\right)\right)\right) - \left(V1 + 2 \cdot V0\right)\right), \frac{1}{65536} \cdot P0\right)} \]
5. Applied rewrites97.5%
  \[\leadsto \color{blue}{\mathsf{fma}\left(amt, \mathsf{fma}\left(amt, \mathsf{fma}\left(amt, \mathsf{fma}\left(2, P0 - P1, V1 + V0\right), \mathsf{fma}\left(3, P1 - P0, \mathsf{fma}\left(V0, -2, -V1\right)\right)\right), P1\right), P0 \cdot 1.52587890625 \cdot 10^{-5}\right)} \]
6. Taylor expanded in V1 around inf
  \[\leadsto \mathsf{fma}\left(amt, \mathsf{fma}\left(amt, V1 \cdot \color{blue}{\left(amt - 1\right)}, P1\right), P0 \cdot \frac{1}{65536}\right) \]
7. Step-by-step derivation
8. Applied rewrites53.0%
  \[\leadsto \mathsf{fma}\left(amt, \mathsf{fma}\left(amt, V1 \cdot \color{blue}{\left(amt + -1\right)}, P1\right), P0 \cdot 1.52587890625 \cdot 10^{-5}\right) \]
if -5.00000016e-23 < P0 < 2e6
1. Initial program 80.1%
  \[\frac{\left(\left(P0 + P1 \cdot \left(amt \cdot 65536\right)\right) + \left(\left(3 \cdot \left(P1 - P0\right) - V1\right) - 2 \cdot V0\right) \cdot \left({amt}^{2} \cdot 65536\right)\right) + \left(\left(2 \cdot \left(P0 - P1\right) + V0\right) + V1\right) \cdot \left({amt}^{3} \cdot 65536\right)}{65536} \]
2. Add Preprocessing
3. Taylor expanded in amt around 0
  \[\leadsto \color{blue}{\frac{1}{65536} \cdot P0 + amt \cdot \left(P1 + amt \cdot \left(\left(3 \cdot \left(P1 - P0\right) + amt \cdot \left(V0 + \left(V1 + 2 \cdot \left(P0 - P1\right)\right)\right)\right) - \left(V1 + 2 \cdot V0\right)\right)\right)} \]
4. Step-by-step derivation
  1. +-commutativeN/A
    \[\leadsto \color{blue}{amt \cdot \left(P1 + amt \cdot \left(\left(3 \cdot \left(P1 - P0\right) + amt \cdot \left(V0 + \left(V1 + 2 \cdot \left(P0 - P1\right)\right)\right)\right) - \left(V1 + 2 \cdot V0\right)\right)\right) + \frac{1}{65536} \cdot P0} \]
  2. lower-fma.f32N/A
    \[\leadsto \color{blue}{\mathsf{fma}\left(amt, P1 + amt \cdot \left(\left(3 \cdot \left(P1 - P0\right) + amt \cdot \left(V0 + \left(V1 + 2 \cdot \left(P0 - P1\right)\right)\right)\right) - \left(V1 + 2 \cdot V0\right)\right), \frac{1}{65536} \cdot P0\right)} \]
5. Applied rewrites97.5%
  \[\leadsto \color{blue}{\mathsf{fma}\left(amt, \mathsf{fma}\left(amt, \mathsf{fma}\left(amt, \mathsf{fma}\left(2, P0 - P1, V1 + V0\right), \mathsf{fma}\left(3, P1 - P0, \mathsf{fma}\left(V0, -2, -V1\right)\right)\right), P1\right), P0 \cdot 1.52587890625 \cdot 10^{-5}\right)} \]
6. Taylor expanded in V0 around inf
  \[\leadsto \mathsf{fma}\left(amt, \mathsf{fma}\left(amt, V0 \cdot \color{blue}{\left(amt - 2\right)}, P1\right), P0 \cdot \frac{1}{65536}\right) \]
7. Step-by-step derivation
8. Applied rewrites49.8%
  \[\leadsto \mathsf{fma}\left(amt, \mathsf{fma}\left(amt, V0 \cdot \color{blue}{\left(amt + -2\right)}, P1\right), P0 \cdot 1.52587890625 \cdot 10^{-5}\right) \]
Recombined 3 regimes into one program.
Add Preprocessing

Alternative 9: 43.6% accurate, 6.5× speedup?

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;P0 \leq -1000000000:\\ \;\;\;\;\mathsf{fma}\left(amt, P0 \cdot \left(amt \cdot \mathsf{fma}\left(amt, 2, -3\right)\right), P0 \cdot 1.52587890625 \cdot 10^{-5}\right)\\ \mathbf{elif}\;P0 \leq -5.000000156871975 \cdot 10^{-23}:\\ \;\;\;\;\mathsf{fma}\left(amt, \mathsf{fma}\left(amt, V1 \cdot \left(amt + -1\right), P1\right), P0 \cdot 1.52587890625 \cdot 10^{-5}\right)\\ \mathbf{elif}\;P0 \leq 2000000:\\ \;\;\;\;\mathsf{fma}\left(amt, \mathsf{fma}\left(amt, V0 \cdot \left(amt + -2\right), P1\right), P0 \cdot 1.52587890625 \cdot 10^{-5}\right)\\ \mathbf{else}:\\ \;\;\;\;P0 \cdot \mathsf{fma}\left(amt \cdot amt, \mathsf{fma}\left(amt, 2, -3\right), 1.52587890625 \cdot 10^{-5}\right)\\ \end{array} \end{array} \]

(FPCore (P0 P1 amt V1 V0)
 :precision binary32
 (if (<= P0 -1000000000.0)
   (fma amt (* P0 (* amt (fma amt 2.0 -3.0))) (* P0 1.52587890625e-5))
   (if (<= P0 -5.000000156871975e-23)
     (fma amt (fma amt (* V1 (+ amt -1.0)) P1) (* P0 1.52587890625e-5))
     (if (<= P0 2000000.0)
       (fma amt (fma amt (* V0 (+ amt -2.0)) P1) (* P0 1.52587890625e-5))
       (* P0 (fma (* amt amt) (fma amt 2.0 -3.0) 1.52587890625e-5))))))

float code(float P0, float P1, float amt, float V1, float V0) {
	float tmp;
	if (P0 <= -1000000000.0f) {
		tmp = fmaf(amt, (P0 * (amt * fmaf(amt, 2.0f, -3.0f))), (P0 * 1.52587890625e-5f));
	} else if (P0 <= -5.000000156871975e-23f) {
		tmp = fmaf(amt, fmaf(amt, (V1 * (amt + -1.0f)), P1), (P0 * 1.52587890625e-5f));
	} else if (P0 <= 2000000.0f) {
		tmp = fmaf(amt, fmaf(amt, (V0 * (amt + -2.0f)), P1), (P0 * 1.52587890625e-5f));
	} else {
		tmp = P0 * fmaf((amt * amt), fmaf(amt, 2.0f, -3.0f), 1.52587890625e-5f);
	}
	return tmp;
}

function code(P0, P1, amt, V1, V0)
	tmp = Float32(0.0)
	if (P0 <= Float32(-1000000000.0))
		tmp = fma(amt, Float32(P0 * Float32(amt * fma(amt, Float32(2.0), Float32(-3.0)))), Float32(P0 * Float32(1.52587890625e-5)));
	elseif (P0 <= Float32(-5.000000156871975e-23))
		tmp = fma(amt, fma(amt, Float32(V1 * Float32(amt + Float32(-1.0))), P1), Float32(P0 * Float32(1.52587890625e-5)));
	elseif (P0 <= Float32(2000000.0))
		tmp = fma(amt, fma(amt, Float32(V0 * Float32(amt + Float32(-2.0))), P1), Float32(P0 * Float32(1.52587890625e-5)));
	else
		tmp = Float32(P0 * fma(Float32(amt * amt), fma(amt, Float32(2.0), Float32(-3.0)), Float32(1.52587890625e-5)));
	end
	return tmp
end

\begin{array}{l}

\\
\begin{array}{l}
\mathbf{if}\;P0 \leq -1000000000:\\
\;\;\;\;\mathsf{fma}\left(amt, P0 \cdot \left(amt \cdot \mathsf{fma}\left(amt, 2, -3\right)\right), P0 \cdot 1.52587890625 \cdot 10^{-5}\right)\\

\mathbf{elif}\;P0 \leq -5.000000156871975 \cdot 10^{-23}:\\
\;\;\;\;\mathsf{fma}\left(amt, \mathsf{fma}\left(amt, V1 \cdot \left(amt + -1\right), P1\right), P0 \cdot 1.52587890625 \cdot 10^{-5}\right)\\

\mathbf{elif}\;P0 \leq 2000000:\\
\;\;\;\;\mathsf{fma}\left(amt, \mathsf{fma}\left(amt, V0 \cdot \left(amt + -2\right), P1\right), P0 \cdot 1.52587890625 \cdot 10^{-5}\right)\\

\mathbf{else}:\\
\;\;\;\;P0 \cdot \mathsf{fma}\left(amt \cdot amt, \mathsf{fma}\left(amt, 2, -3\right), 1.52587890625 \cdot 10^{-5}\right)\\


\end{array}
\end{array}

Derivation

Split input into 4 regimes
if P0 < -1e9
1. Initial program 61.0%
  \[\frac{\left(\left(P0 + P1 \cdot \left(amt \cdot 65536\right)\right) + \left(\left(3 \cdot \left(P1 - P0\right) - V1\right) - 2 \cdot V0\right) \cdot \left({amt}^{2} \cdot 65536\right)\right) + \left(\left(2 \cdot \left(P0 - P1\right) + V0\right) + V1\right) \cdot \left({amt}^{3} \cdot 65536\right)}{65536} \]
2. Add Preprocessing
3. Taylor expanded in amt around 0
  \[\leadsto \color{blue}{\frac{1}{65536} \cdot P0 + amt \cdot \left(P1 + amt \cdot \left(\left(3 \cdot \left(P1 - P0\right) + amt \cdot \left(V0 + \left(V1 + 2 \cdot \left(P0 - P1\right)\right)\right)\right) - \left(V1 + 2 \cdot V0\right)\right)\right)} \]
4. Step-by-step derivation
  1. +-commutativeN/A
    \[\leadsto \color{blue}{amt \cdot \left(P1 + amt \cdot \left(\left(3 \cdot \left(P1 - P0\right) + amt \cdot \left(V0 + \left(V1 + 2 \cdot \left(P0 - P1\right)\right)\right)\right) - \left(V1 + 2 \cdot V0\right)\right)\right) + \frac{1}{65536} \cdot P0} \]
  2. lower-fma.f32N/A
    \[\leadsto \color{blue}{\mathsf{fma}\left(amt, P1 + amt \cdot \left(\left(3 \cdot \left(P1 - P0\right) + amt \cdot \left(V0 + \left(V1 + 2 \cdot \left(P0 - P1\right)\right)\right)\right) - \left(V1 + 2 \cdot V0\right)\right), \frac{1}{65536} \cdot P0\right)} \]
5. Applied rewrites92.4%
  \[\leadsto \color{blue}{\mathsf{fma}\left(amt, \mathsf{fma}\left(amt, \mathsf{fma}\left(amt, \mathsf{fma}\left(2, P0 - P1, V1 + V0\right), \mathsf{fma}\left(3, P1 - P0, \mathsf{fma}\left(V0, -2, -V1\right)\right)\right), P1\right), P0 \cdot 1.52587890625 \cdot 10^{-5}\right)} \]
6. Taylor expanded in P0 around inf
  \[\leadsto \mathsf{fma}\left(amt, P0 \cdot \color{blue}{\left(amt \cdot \left(2 \cdot amt - 3\right)\right)}, P0 \cdot \frac{1}{65536}\right) \]
7. Step-by-step derivation
8. Applied rewrites60.6%
  \[\leadsto \mathsf{fma}\left(amt, P0 \cdot \color{blue}{\left(amt \cdot \mathsf{fma}\left(amt, 2, -3\right)\right)}, P0 \cdot 1.52587890625 \cdot 10^{-5}\right) \]
if -1e9 < P0 < -5.00000016e-23
1. Initial program 80.5%
  \[\frac{\left(\left(P0 + P1 \cdot \left(amt \cdot 65536\right)\right) + \left(\left(3 \cdot \left(P1 - P0\right) - V1\right) - 2 \cdot V0\right) \cdot \left({amt}^{2} \cdot 65536\right)\right) + \left(\left(2 \cdot \left(P0 - P1\right) + V0\right) + V1\right) \cdot \left({amt}^{3} \cdot 65536\right)}{65536} \]
2. Add Preprocessing
3. Taylor expanded in amt around 0
  \[\leadsto \color{blue}{\frac{1}{65536} \cdot P0 + amt \cdot \left(P1 + amt \cdot \left(\left(3 \cdot \left(P1 - P0\right) + amt \cdot \left(V0 + \left(V1 + 2 \cdot \left(P0 - P1\right)\right)\right)\right) - \left(V1 + 2 \cdot V0\right)\right)\right)} \]
4. Step-by-step derivation
  1. +-commutativeN/A
    \[\leadsto \color{blue}{amt \cdot \left(P1 + amt \cdot \left(\left(3 \cdot \left(P1 - P0\right) + amt \cdot \left(V0 + \left(V1 + 2 \cdot \left(P0 - P1\right)\right)\right)\right) - \left(V1 + 2 \cdot V0\right)\right)\right) + \frac{1}{65536} \cdot P0} \]
  2. lower-fma.f32N/A
    \[\leadsto \color{blue}{\mathsf{fma}\left(amt, P1 + amt \cdot \left(\left(3 \cdot \left(P1 - P0\right) + amt \cdot \left(V0 + \left(V1 + 2 \cdot \left(P0 - P1\right)\right)\right)\right) - \left(V1 + 2 \cdot V0\right)\right), \frac{1}{65536} \cdot P0\right)} \]
5. Applied rewrites97.5%
  \[\leadsto \color{blue}{\mathsf{fma}\left(amt, \mathsf{fma}\left(amt, \mathsf{fma}\left(amt, \mathsf{fma}\left(2, P0 - P1, V1 + V0\right), \mathsf{fma}\left(3, P1 - P0, \mathsf{fma}\left(V0, -2, -V1\right)\right)\right), P1\right), P0 \cdot 1.52587890625 \cdot 10^{-5}\right)} \]
6. Taylor expanded in V1 around inf
  \[\leadsto \mathsf{fma}\left(amt, \mathsf{fma}\left(amt, V1 \cdot \color{blue}{\left(amt - 1\right)}, P1\right), P0 \cdot \frac{1}{65536}\right) \]
7. Step-by-step derivation
8. Applied rewrites53.0%
  \[\leadsto \mathsf{fma}\left(amt, \mathsf{fma}\left(amt, V1 \cdot \color{blue}{\left(amt + -1\right)}, P1\right), P0 \cdot 1.52587890625 \cdot 10^{-5}\right) \]
if -5.00000016e-23 < P0 < 2e6
1. Initial program 80.1%
  \[\frac{\left(\left(P0 + P1 \cdot \left(amt \cdot 65536\right)\right) + \left(\left(3 \cdot \left(P1 - P0\right) - V1\right) - 2 \cdot V0\right) \cdot \left({amt}^{2} \cdot 65536\right)\right) + \left(\left(2 \cdot \left(P0 - P1\right) + V0\right) + V1\right) \cdot \left({amt}^{3} \cdot 65536\right)}{65536} \]
2. Add Preprocessing
3. Taylor expanded in amt around 0
  \[\leadsto \color{blue}{\frac{1}{65536} \cdot P0 + amt \cdot \left(P1 + amt \cdot \left(\left(3 \cdot \left(P1 - P0\right) + amt \cdot \left(V0 + \left(V1 + 2 \cdot \left(P0 - P1\right)\right)\right)\right) - \left(V1 + 2 \cdot V0\right)\right)\right)} \]
4. Step-by-step derivation
  1. +-commutativeN/A
    \[\leadsto \color{blue}{amt \cdot \left(P1 + amt \cdot \left(\left(3 \cdot \left(P1 - P0\right) + amt \cdot \left(V0 + \left(V1 + 2 \cdot \left(P0 - P1\right)\right)\right)\right) - \left(V1 + 2 \cdot V0\right)\right)\right) + \frac{1}{65536} \cdot P0} \]
  2. lower-fma.f32N/A
    \[\leadsto \color{blue}{\mathsf{fma}\left(amt, P1 + amt \cdot \left(\left(3 \cdot \left(P1 - P0\right) + amt \cdot \left(V0 + \left(V1 + 2 \cdot \left(P0 - P1\right)\right)\right)\right) - \left(V1 + 2 \cdot V0\right)\right), \frac{1}{65536} \cdot P0\right)} \]
5. Applied rewrites97.5%
  \[\leadsto \color{blue}{\mathsf{fma}\left(amt, \mathsf{fma}\left(amt, \mathsf{fma}\left(amt, \mathsf{fma}\left(2, P0 - P1, V1 + V0\right), \mathsf{fma}\left(3, P1 - P0, \mathsf{fma}\left(V0, -2, -V1\right)\right)\right), P1\right), P0 \cdot 1.52587890625 \cdot 10^{-5}\right)} \]
6. Taylor expanded in V0 around inf
  \[\leadsto \mathsf{fma}\left(amt, \mathsf{fma}\left(amt, V0 \cdot \color{blue}{\left(amt - 2\right)}, P1\right), P0 \cdot \frac{1}{65536}\right) \]
7. Step-by-step derivation
8. Applied rewrites49.8%
  \[\leadsto \mathsf{fma}\left(amt, \mathsf{fma}\left(amt, V0 \cdot \color{blue}{\left(amt + -2\right)}, P1\right), P0 \cdot 1.52587890625 \cdot 10^{-5}\right) \]
if 2e6 < P0
1. Initial program 75.3%
  \[\frac{\left(\left(P0 + P1 \cdot \left(amt \cdot 65536\right)\right) + \left(\left(3 \cdot \left(P1 - P0\right) - V1\right) - 2 \cdot V0\right) \cdot \left({amt}^{2} \cdot 65536\right)\right) + \left(\left(2 \cdot \left(P0 - P1\right) + V0\right) + V1\right) \cdot \left({amt}^{3} \cdot 65536\right)}{65536} \]
2. Add Preprocessing
3. Taylor expanded in amt around 0
  \[\leadsto \color{blue}{\frac{1}{65536} \cdot P0 + amt \cdot \left(P1 + amt \cdot \left(\left(3 \cdot \left(P1 - P0\right) + amt \cdot \left(V0 + \left(V1 + 2 \cdot \left(P0 - P1\right)\right)\right)\right) - \left(V1 + 2 \cdot V0\right)\right)\right)} \]
4. Step-by-step derivation
  1. +-commutativeN/A
    \[\leadsto \color{blue}{amt \cdot \left(P1 + amt \cdot \left(\left(3 \cdot \left(P1 - P0\right) + amt \cdot \left(V0 + \left(V1 + 2 \cdot \left(P0 - P1\right)\right)\right)\right) - \left(V1 + 2 \cdot V0\right)\right)\right) + \frac{1}{65536} \cdot P0} \]
  2. lower-fma.f32N/A
    \[\leadsto \color{blue}{\mathsf{fma}\left(amt, P1 + amt \cdot \left(\left(3 \cdot \left(P1 - P0\right) + amt \cdot \left(V0 + \left(V1 + 2 \cdot \left(P0 - P1\right)\right)\right)\right) - \left(V1 + 2 \cdot V0\right)\right), \frac{1}{65536} \cdot P0\right)} \]
5. Applied rewrites97.6%
  \[\leadsto \color{blue}{\mathsf{fma}\left(amt, \mathsf{fma}\left(amt, \mathsf{fma}\left(amt, \mathsf{fma}\left(2, P0 - P1, V1 + V0\right), \mathsf{fma}\left(3, P1 - P0, \mathsf{fma}\left(V0, -2, -V1\right)\right)\right), P1\right), P0 \cdot 1.52587890625 \cdot 10^{-5}\right)} \]
6. Taylor expanded in P0 around inf
  \[\leadsto P0 \cdot \color{blue}{\left(\frac{1}{65536} + {amt}^{2} \cdot \left(2 \cdot amt - 3\right)\right)} \]
7. Step-by-step derivation
8. Applied rewrites62.8%
  \[\leadsto P0 \cdot \color{blue}{\mathsf{fma}\left(amt \cdot amt, \mathsf{fma}\left(amt, 2, -3\right), 1.52587890625 \cdot 10^{-5}\right)} \]
Recombined 4 regimes into one program.
Add Preprocessing

Alternative 10: 43.7% accurate, 6.5× speedup?

\[\begin{array}{l} \\ \begin{array}{l} t_0 := P0 \cdot \mathsf{fma}\left(amt \cdot amt, \mathsf{fma}\left(amt, 2, -3\right), 1.52587890625 \cdot 10^{-5}\right)\\ \mathbf{if}\;P0 \leq -1000000000:\\ \;\;\;\;t_0\\ \mathbf{elif}\;P0 \leq -5.000000156871975 \cdot 10^{-23}:\\ \;\;\;\;\mathsf{fma}\left(amt, \mathsf{fma}\left(amt, V1 \cdot \left(amt + -1\right), P1\right), P0 \cdot 1.52587890625 \cdot 10^{-5}\right)\\ \mathbf{elif}\;P0 \leq 2000000:\\ \;\;\;\;\mathsf{fma}\left(amt, \mathsf{fma}\left(amt, V0 \cdot \left(amt + -2\right), P1\right), P0 \cdot 1.52587890625 \cdot 10^{-5}\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

(FPCore (P0 P1 amt V1 V0)
 :precision binary32
 (let* ((t_0 (* P0 (fma (* amt amt) (fma amt 2.0 -3.0) 1.52587890625e-5))))
   (if (<= P0 -1000000000.0)
     t_0
     (if (<= P0 -5.000000156871975e-23)
       (fma amt (fma amt (* V1 (+ amt -1.0)) P1) (* P0 1.52587890625e-5))
       (if (<= P0 2000000.0)
         (fma amt (fma amt (* V0 (+ amt -2.0)) P1) (* P0 1.52587890625e-5))
         t_0)))))

float code(float P0, float P1, float amt, float V1, float V0) {
	float t_0 = P0 * fmaf((amt * amt), fmaf(amt, 2.0f, -3.0f), 1.52587890625e-5f);
	float tmp;
	if (P0 <= -1000000000.0f) {
		tmp = t_0;
	} else if (P0 <= -5.000000156871975e-23f) {
		tmp = fmaf(amt, fmaf(amt, (V1 * (amt + -1.0f)), P1), (P0 * 1.52587890625e-5f));
	} else if (P0 <= 2000000.0f) {
		tmp = fmaf(amt, fmaf(amt, (V0 * (amt + -2.0f)), P1), (P0 * 1.52587890625e-5f));
	} else {
		tmp = t_0;
	}
	return tmp;
}

function code(P0, P1, amt, V1, V0)
	t_0 = Float32(P0 * fma(Float32(amt * amt), fma(amt, Float32(2.0), Float32(-3.0)), Float32(1.52587890625e-5)))
	tmp = Float32(0.0)
	if (P0 <= Float32(-1000000000.0))
		tmp = t_0;
	elseif (P0 <= Float32(-5.000000156871975e-23))
		tmp = fma(amt, fma(amt, Float32(V1 * Float32(amt + Float32(-1.0))), P1), Float32(P0 * Float32(1.52587890625e-5)));
	elseif (P0 <= Float32(2000000.0))
		tmp = fma(amt, fma(amt, Float32(V0 * Float32(amt + Float32(-2.0))), P1), Float32(P0 * Float32(1.52587890625e-5)));
	else
		tmp = t_0;
	end
	return tmp
end

\begin{array}{l}

\\
\begin{array}{l}
t_0 := P0 \cdot \mathsf{fma}\left(amt \cdot amt, \mathsf{fma}\left(amt, 2, -3\right), 1.52587890625 \cdot 10^{-5}\right)\\
\mathbf{if}\;P0 \leq -1000000000:\\
\;\;\;\;t_0\\

\mathbf{elif}\;P0 \leq -5.000000156871975 \cdot 10^{-23}:\\
\;\;\;\;\mathsf{fma}\left(amt, \mathsf{fma}\left(amt, V1 \cdot \left(amt + -1\right), P1\right), P0 \cdot 1.52587890625 \cdot 10^{-5}\right)\\

\mathbf{elif}\;P0 \leq 2000000:\\
\;\;\;\;\mathsf{fma}\left(amt, \mathsf{fma}\left(amt, V0 \cdot \left(amt + -2\right), P1\right), P0 \cdot 1.52587890625 \cdot 10^{-5}\right)\\

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


\end{array}
\end{array}

Derivation

Split input into 3 regimes
if P0 < -1e9 or 2e6 < P0
1. Initial program 68.0%
  \[\frac{\left(\left(P0 + P1 \cdot \left(amt \cdot 65536\right)\right) + \left(\left(3 \cdot \left(P1 - P0\right) - V1\right) - 2 \cdot V0\right) \cdot \left({amt}^{2} \cdot 65536\right)\right) + \left(\left(2 \cdot \left(P0 - P1\right) + V0\right) + V1\right) \cdot \left({amt}^{3} \cdot 65536\right)}{65536} \]
2. Add Preprocessing
3. Taylor expanded in amt around 0
  \[\leadsto \color{blue}{\frac{1}{65536} \cdot P0 + amt \cdot \left(P1 + amt \cdot \left(\left(3 \cdot \left(P1 - P0\right) + amt \cdot \left(V0 + \left(V1 + 2 \cdot \left(P0 - P1\right)\right)\right)\right) - \left(V1 + 2 \cdot V0\right)\right)\right)} \]
4. Step-by-step derivation
  1. +-commutativeN/A
    \[\leadsto \color{blue}{amt \cdot \left(P1 + amt \cdot \left(\left(3 \cdot \left(P1 - P0\right) + amt \cdot \left(V0 + \left(V1 + 2 \cdot \left(P0 - P1\right)\right)\right)\right) - \left(V1 + 2 \cdot V0\right)\right)\right) + \frac{1}{65536} \cdot P0} \]
  2. lower-fma.f32N/A
    \[\leadsto \color{blue}{\mathsf{fma}\left(amt, P1 + amt \cdot \left(\left(3 \cdot \left(P1 - P0\right) + amt \cdot \left(V0 + \left(V1 + 2 \cdot \left(P0 - P1\right)\right)\right)\right) - \left(V1 + 2 \cdot V0\right)\right), \frac{1}{65536} \cdot P0\right)} \]
5. Applied rewrites94.9%
  \[\leadsto \color{blue}{\mathsf{fma}\left(amt, \mathsf{fma}\left(amt, \mathsf{fma}\left(amt, \mathsf{fma}\left(2, P0 - P1, V1 + V0\right), \mathsf{fma}\left(3, P1 - P0, \mathsf{fma}\left(V0, -2, -V1\right)\right)\right), P1\right), P0 \cdot 1.52587890625 \cdot 10^{-5}\right)} \]
6. Taylor expanded in P0 around inf
  \[\leadsto P0 \cdot \color{blue}{\left(\frac{1}{65536} + {amt}^{2} \cdot \left(2 \cdot amt - 3\right)\right)} \]
7. Step-by-step derivation
8. Applied rewrites61.6%
  \[\leadsto P0 \cdot \color{blue}{\mathsf{fma}\left(amt \cdot amt, \mathsf{fma}\left(amt, 2, -3\right), 1.52587890625 \cdot 10^{-5}\right)} \]
if -1e9 < P0 < -5.00000016e-23
1. Initial program 80.5%
  \[\frac{\left(\left(P0 + P1 \cdot \left(amt \cdot 65536\right)\right) + \left(\left(3 \cdot \left(P1 - P0\right) - V1\right) - 2 \cdot V0\right) \cdot \left({amt}^{2} \cdot 65536\right)\right) + \left(\left(2 \cdot \left(P0 - P1\right) + V0\right) + V1\right) \cdot \left({amt}^{3} \cdot 65536\right)}{65536} \]
2. Add Preprocessing
3. Taylor expanded in amt around 0
  \[\leadsto \color{blue}{\frac{1}{65536} \cdot P0 + amt \cdot \left(P1 + amt \cdot \left(\left(3 \cdot \left(P1 - P0\right) + amt \cdot \left(V0 + \left(V1 + 2 \cdot \left(P0 - P1\right)\right)\right)\right) - \left(V1 + 2 \cdot V0\right)\right)\right)} \]
4. Step-by-step derivation
  1. +-commutativeN/A
    \[\leadsto \color{blue}{amt \cdot \left(P1 + amt \cdot \left(\left(3 \cdot \left(P1 - P0\right) + amt \cdot \left(V0 + \left(V1 + 2 \cdot \left(P0 - P1\right)\right)\right)\right) - \left(V1 + 2 \cdot V0\right)\right)\right) + \frac{1}{65536} \cdot P0} \]
  2. lower-fma.f32N/A
    \[\leadsto \color{blue}{\mathsf{fma}\left(amt, P1 + amt \cdot \left(\left(3 \cdot \left(P1 - P0\right) + amt \cdot \left(V0 + \left(V1 + 2 \cdot \left(P0 - P1\right)\right)\right)\right) - \left(V1 + 2 \cdot V0\right)\right), \frac{1}{65536} \cdot P0\right)} \]
5. Applied rewrites97.5%
  \[\leadsto \color{blue}{\mathsf{fma}\left(amt, \mathsf{fma}\left(amt, \mathsf{fma}\left(amt, \mathsf{fma}\left(2, P0 - P1, V1 + V0\right), \mathsf{fma}\left(3, P1 - P0, \mathsf{fma}\left(V0, -2, -V1\right)\right)\right), P1\right), P0 \cdot 1.52587890625 \cdot 10^{-5}\right)} \]
6. Taylor expanded in V1 around inf
  \[\leadsto \mathsf{fma}\left(amt, \mathsf{fma}\left(amt, V1 \cdot \color{blue}{\left(amt - 1\right)}, P1\right), P0 \cdot \frac{1}{65536}\right) \]
7. Step-by-step derivation
8. Applied rewrites53.0%
  \[\leadsto \mathsf{fma}\left(amt, \mathsf{fma}\left(amt, V1 \cdot \color{blue}{\left(amt + -1\right)}, P1\right), P0 \cdot 1.52587890625 \cdot 10^{-5}\right) \]
if -5.00000016e-23 < P0 < 2e6
1. Initial program 80.1%
  \[\frac{\left(\left(P0 + P1 \cdot \left(amt \cdot 65536\right)\right) + \left(\left(3 \cdot \left(P1 - P0\right) - V1\right) - 2 \cdot V0\right) \cdot \left({amt}^{2} \cdot 65536\right)\right) + \left(\left(2 \cdot \left(P0 - P1\right) + V0\right) + V1\right) \cdot \left({amt}^{3} \cdot 65536\right)}{65536} \]
2. Add Preprocessing
3. Taylor expanded in amt around 0
  \[\leadsto \color{blue}{\frac{1}{65536} \cdot P0 + amt \cdot \left(P1 + amt \cdot \left(\left(3 \cdot \left(P1 - P0\right) + amt \cdot \left(V0 + \left(V1 + 2 \cdot \left(P0 - P1\right)\right)\right)\right) - \left(V1 + 2 \cdot V0\right)\right)\right)} \]
4. Step-by-step derivation
  1. +-commutativeN/A
    \[\leadsto \color{blue}{amt \cdot \left(P1 + amt \cdot \left(\left(3 \cdot \left(P1 - P0\right) + amt \cdot \left(V0 + \left(V1 + 2 \cdot \left(P0 - P1\right)\right)\right)\right) - \left(V1 + 2 \cdot V0\right)\right)\right) + \frac{1}{65536} \cdot P0} \]
  2. lower-fma.f32N/A
    \[\leadsto \color{blue}{\mathsf{fma}\left(amt, P1 + amt \cdot \left(\left(3 \cdot \left(P1 - P0\right) + amt \cdot \left(V0 + \left(V1 + 2 \cdot \left(P0 - P1\right)\right)\right)\right) - \left(V1 + 2 \cdot V0\right)\right), \frac{1}{65536} \cdot P0\right)} \]
5. Applied rewrites97.5%
  \[\leadsto \color{blue}{\mathsf{fma}\left(amt, \mathsf{fma}\left(amt, \mathsf{fma}\left(amt, \mathsf{fma}\left(2, P0 - P1, V1 + V0\right), \mathsf{fma}\left(3, P1 - P0, \mathsf{fma}\left(V0, -2, -V1\right)\right)\right), P1\right), P0 \cdot 1.52587890625 \cdot 10^{-5}\right)} \]
6. Taylor expanded in V0 around inf
  \[\leadsto \mathsf{fma}\left(amt, \mathsf{fma}\left(amt, V0 \cdot \color{blue}{\left(amt - 2\right)}, P1\right), P0 \cdot \frac{1}{65536}\right) \]
7. Step-by-step derivation
8. Applied rewrites49.8%
  \[\leadsto \mathsf{fma}\left(amt, \mathsf{fma}\left(amt, V0 \cdot \color{blue}{\left(amt + -2\right)}, P1\right), P0 \cdot 1.52587890625 \cdot 10^{-5}\right) \]
Recombined 3 regimes into one program.
Add Preprocessing

Alternative 11: 42.3% accurate, 6.5× speedup?

\[\begin{array}{l} \\ \begin{array}{l} t_0 := P0 \cdot \mathsf{fma}\left(amt \cdot amt, \mathsf{fma}\left(amt, 2, -3\right), 1.52587890625 \cdot 10^{-5}\right)\\ \mathbf{if}\;P0 \leq -1000000000:\\ \;\;\;\;t_0\\ \mathbf{elif}\;P0 \leq -5.000000156871975 \cdot 10^{-23}:\\ \;\;\;\;V1 \cdot \left(\left(amt + -1\right) \cdot \left(amt \cdot amt\right)\right)\\ \mathbf{elif}\;P0 \leq 2000000:\\ \;\;\;\;\mathsf{fma}\left(amt, \mathsf{fma}\left(amt, V0 \cdot \left(amt + -2\right), P1\right), P0 \cdot 1.52587890625 \cdot 10^{-5}\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

(FPCore (P0 P1 amt V1 V0)
 :precision binary32
 (let* ((t_0 (* P0 (fma (* amt amt) (fma amt 2.0 -3.0) 1.52587890625e-5))))
   (if (<= P0 -1000000000.0)
     t_0
     (if (<= P0 -5.000000156871975e-23)
       (* V1 (* (+ amt -1.0) (* amt amt)))
       (if (<= P0 2000000.0)
         (fma amt (fma amt (* V0 (+ amt -2.0)) P1) (* P0 1.52587890625e-5))
         t_0)))))

float code(float P0, float P1, float amt, float V1, float V0) {
	float t_0 = P0 * fmaf((amt * amt), fmaf(amt, 2.0f, -3.0f), 1.52587890625e-5f);
	float tmp;
	if (P0 <= -1000000000.0f) {
		tmp = t_0;
	} else if (P0 <= -5.000000156871975e-23f) {
		tmp = V1 * ((amt + -1.0f) * (amt * amt));
	} else if (P0 <= 2000000.0f) {
		tmp = fmaf(amt, fmaf(amt, (V0 * (amt + -2.0f)), P1), (P0 * 1.52587890625e-5f));
	} else {
		tmp = t_0;
	}
	return tmp;
}

function code(P0, P1, amt, V1, V0)
	t_0 = Float32(P0 * fma(Float32(amt * amt), fma(amt, Float32(2.0), Float32(-3.0)), Float32(1.52587890625e-5)))
	tmp = Float32(0.0)
	if (P0 <= Float32(-1000000000.0))
		tmp = t_0;
	elseif (P0 <= Float32(-5.000000156871975e-23))
		tmp = Float32(V1 * Float32(Float32(amt + Float32(-1.0)) * Float32(amt * amt)));
	elseif (P0 <= Float32(2000000.0))
		tmp = fma(amt, fma(amt, Float32(V0 * Float32(amt + Float32(-2.0))), P1), Float32(P0 * Float32(1.52587890625e-5)));
	else
		tmp = t_0;
	end
	return tmp
end

\begin{array}{l}

\\
\begin{array}{l}
t_0 := P0 \cdot \mathsf{fma}\left(amt \cdot amt, \mathsf{fma}\left(amt, 2, -3\right), 1.52587890625 \cdot 10^{-5}\right)\\
\mathbf{if}\;P0 \leq -1000000000:\\
\;\;\;\;t_0\\

\mathbf{elif}\;P0 \leq -5.000000156871975 \cdot 10^{-23}:\\
\;\;\;\;V1 \cdot \left(\left(amt + -1\right) \cdot \left(amt \cdot amt\right)\right)\\

\mathbf{elif}\;P0 \leq 2000000:\\
\;\;\;\;\mathsf{fma}\left(amt, \mathsf{fma}\left(amt, V0 \cdot \left(amt + -2\right), P1\right), P0 \cdot 1.52587890625 \cdot 10^{-5}\right)\\

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


\end{array}
\end{array}

Derivation

Split input into 3 regimes
if P0 < -1e9 or 2e6 < P0
1. Initial program 68.0%
  \[\frac{\left(\left(P0 + P1 \cdot \left(amt \cdot 65536\right)\right) + \left(\left(3 \cdot \left(P1 - P0\right) - V1\right) - 2 \cdot V0\right) \cdot \left({amt}^{2} \cdot 65536\right)\right) + \left(\left(2 \cdot \left(P0 - P1\right) + V0\right) + V1\right) \cdot \left({amt}^{3} \cdot 65536\right)}{65536} \]
2. Add Preprocessing
3. Taylor expanded in amt around 0
  \[\leadsto \color{blue}{\frac{1}{65536} \cdot P0 + amt \cdot \left(P1 + amt \cdot \left(\left(3 \cdot \left(P1 - P0\right) + amt \cdot \left(V0 + \left(V1 + 2 \cdot \left(P0 - P1\right)\right)\right)\right) - \left(V1 + 2 \cdot V0\right)\right)\right)} \]
4. Step-by-step derivation
  1. +-commutativeN/A
    \[\leadsto \color{blue}{amt \cdot \left(P1 + amt \cdot \left(\left(3 \cdot \left(P1 - P0\right) + amt \cdot \left(V0 + \left(V1 + 2 \cdot \left(P0 - P1\right)\right)\right)\right) - \left(V1 + 2 \cdot V0\right)\right)\right) + \frac{1}{65536} \cdot P0} \]
  2. lower-fma.f32N/A
    \[\leadsto \color{blue}{\mathsf{fma}\left(amt, P1 + amt \cdot \left(\left(3 \cdot \left(P1 - P0\right) + amt \cdot \left(V0 + \left(V1 + 2 \cdot \left(P0 - P1\right)\right)\right)\right) - \left(V1 + 2 \cdot V0\right)\right), \frac{1}{65536} \cdot P0\right)} \]
5. Applied rewrites94.9%
  \[\leadsto \color{blue}{\mathsf{fma}\left(amt, \mathsf{fma}\left(amt, \mathsf{fma}\left(amt, \mathsf{fma}\left(2, P0 - P1, V1 + V0\right), \mathsf{fma}\left(3, P1 - P0, \mathsf{fma}\left(V0, -2, -V1\right)\right)\right), P1\right), P0 \cdot 1.52587890625 \cdot 10^{-5}\right)} \]
6. Taylor expanded in P0 around inf
  \[\leadsto P0 \cdot \color{blue}{\left(\frac{1}{65536} + {amt}^{2} \cdot \left(2 \cdot amt - 3\right)\right)} \]
7. Step-by-step derivation
8. Applied rewrites61.6%
  \[\leadsto P0 \cdot \color{blue}{\mathsf{fma}\left(amt \cdot amt, \mathsf{fma}\left(amt, 2, -3\right), 1.52587890625 \cdot 10^{-5}\right)} \]
if -1e9 < P0 < -5.00000016e-23
1. Initial program 80.5%
  \[\frac{\left(\left(P0 + P1 \cdot \left(amt \cdot 65536\right)\right) + \left(\left(3 \cdot \left(P1 - P0\right) - V1\right) - 2 \cdot V0\right) \cdot \left({amt}^{2} \cdot 65536\right)\right) + \left(\left(2 \cdot \left(P0 - P1\right) + V0\right) + V1\right) \cdot \left({amt}^{3} \cdot 65536\right)}{65536} \]
2. Add Preprocessing
3. Taylor expanded in amt around 0
  \[\leadsto \color{blue}{\frac{1}{65536} \cdot P0 + amt \cdot \left(P1 + amt \cdot \left(\left(3 \cdot \left(P1 - P0\right) + amt \cdot \left(V0 + \left(V1 + 2 \cdot \left(P0 - P1\right)\right)\right)\right) - \left(V1 + 2 \cdot V0\right)\right)\right)} \]
4. Step-by-step derivation
  1. +-commutativeN/A
    \[\leadsto \color{blue}{amt \cdot \left(P1 + amt \cdot \left(\left(3 \cdot \left(P1 - P0\right) + amt \cdot \left(V0 + \left(V1 + 2 \cdot \left(P0 - P1\right)\right)\right)\right) - \left(V1 + 2 \cdot V0\right)\right)\right) + \frac{1}{65536} \cdot P0} \]
  2. lower-fma.f32N/A
    \[\leadsto \color{blue}{\mathsf{fma}\left(amt, P1 + amt \cdot \left(\left(3 \cdot \left(P1 - P0\right) + amt \cdot \left(V0 + \left(V1 + 2 \cdot \left(P0 - P1\right)\right)\right)\right) - \left(V1 + 2 \cdot V0\right)\right), \frac{1}{65536} \cdot P0\right)} \]
5. Applied rewrites97.5%
  \[\leadsto \color{blue}{\mathsf{fma}\left(amt, \mathsf{fma}\left(amt, \mathsf{fma}\left(amt, \mathsf{fma}\left(2, P0 - P1, V1 + V0\right), \mathsf{fma}\left(3, P1 - P0, \mathsf{fma}\left(V0, -2, -V1\right)\right)\right), P1\right), P0 \cdot 1.52587890625 \cdot 10^{-5}\right)} \]
6. Taylor expanded in V1 around inf
  \[\leadsto V1 \cdot \color{blue}{\left({amt}^{2} \cdot \left(amt - 1\right)\right)} \]
7. Step-by-step derivation
8. Applied rewrites45.8%
  \[\leadsto V1 \cdot \color{blue}{\left(\left(amt \cdot amt\right) \cdot \left(amt + -1\right)\right)} \]
if -5.00000016e-23 < P0 < 2e6
1. Initial program 80.1%
  \[\frac{\left(\left(P0 + P1 \cdot \left(amt \cdot 65536\right)\right) + \left(\left(3 \cdot \left(P1 - P0\right) - V1\right) - 2 \cdot V0\right) \cdot \left({amt}^{2} \cdot 65536\right)\right) + \left(\left(2 \cdot \left(P0 - P1\right) + V0\right) + V1\right) \cdot \left({amt}^{3} \cdot 65536\right)}{65536} \]
2. Add Preprocessing
3. Taylor expanded in amt around 0
  \[\leadsto \color{blue}{\frac{1}{65536} \cdot P0 + amt \cdot \left(P1 + amt \cdot \left(\left(3 \cdot \left(P1 - P0\right) + amt \cdot \left(V0 + \left(V1 + 2 \cdot \left(P0 - P1\right)\right)\right)\right) - \left(V1 + 2 \cdot V0\right)\right)\right)} \]
4. Step-by-step derivation
  1. +-commutativeN/A
    \[\leadsto \color{blue}{amt \cdot \left(P1 + amt \cdot \left(\left(3 \cdot \left(P1 - P0\right) + amt \cdot \left(V0 + \left(V1 + 2 \cdot \left(P0 - P1\right)\right)\right)\right) - \left(V1 + 2 \cdot V0\right)\right)\right) + \frac{1}{65536} \cdot P0} \]
  2. lower-fma.f32N/A
    \[\leadsto \color{blue}{\mathsf{fma}\left(amt, P1 + amt \cdot \left(\left(3 \cdot \left(P1 - P0\right) + amt \cdot \left(V0 + \left(V1 + 2 \cdot \left(P0 - P1\right)\right)\right)\right) - \left(V1 + 2 \cdot V0\right)\right), \frac{1}{65536} \cdot P0\right)} \]
5. Applied rewrites97.5%
  \[\leadsto \color{blue}{\mathsf{fma}\left(amt, \mathsf{fma}\left(amt, \mathsf{fma}\left(amt, \mathsf{fma}\left(2, P0 - P1, V1 + V0\right), \mathsf{fma}\left(3, P1 - P0, \mathsf{fma}\left(V0, -2, -V1\right)\right)\right), P1\right), P0 \cdot 1.52587890625 \cdot 10^{-5}\right)} \]
6. Taylor expanded in V0 around inf
  \[\leadsto \mathsf{fma}\left(amt, \mathsf{fma}\left(amt, V0 \cdot \color{blue}{\left(amt - 2\right)}, P1\right), P0 \cdot \frac{1}{65536}\right) \]
7. Step-by-step derivation
8. Applied rewrites49.8%
  \[\leadsto \mathsf{fma}\left(amt, \mathsf{fma}\left(amt, V0 \cdot \color{blue}{\left(amt + -2\right)}, P1\right), P0 \cdot 1.52587890625 \cdot 10^{-5}\right) \]
Recombined 3 regimes into one program.
Final simplification53.8%
\[\leadsto \begin{array}{l} \mathbf{if}\;P0 \leq -1000000000:\\ \;\;\;\;P0 \cdot \mathsf{fma}\left(amt \cdot amt, \mathsf{fma}\left(amt, 2, -3\right), 1.52587890625 \cdot 10^{-5}\right)\\ \mathbf{elif}\;P0 \leq -5.000000156871975 \cdot 10^{-23}:\\ \;\;\;\;V1 \cdot \left(\left(amt + -1\right) \cdot \left(amt \cdot amt\right)\right)\\ \mathbf{elif}\;P0 \leq 2000000:\\ \;\;\;\;\mathsf{fma}\left(amt, \mathsf{fma}\left(amt, V0 \cdot \left(amt + -2\right), P1\right), P0 \cdot 1.52587890625 \cdot 10^{-5}\right)\\ \mathbf{else}:\\ \;\;\;\;P0 \cdot \mathsf{fma}\left(amt \cdot amt, \mathsf{fma}\left(amt, 2, -3\right), 1.52587890625 \cdot 10^{-5}\right)\\ \end{array} \]
Add Preprocessing

Alternative 12: 39.3% accurate, 6.6× speedup?

\[\begin{array}{l} \\ \begin{array}{l} t_0 := P0 \cdot \mathsf{fma}\left(amt \cdot amt, \mathsf{fma}\left(amt, 2, -3\right), 1.52587890625 \cdot 10^{-5}\right)\\ \mathbf{if}\;P0 \leq -1000000000:\\ \;\;\;\;t_0\\ \mathbf{elif}\;P0 \leq -5.000000156871975 \cdot 10^{-23}:\\ \;\;\;\;V1 \cdot \left(\left(amt + -1\right) \cdot \left(amt \cdot amt\right)\right)\\ \mathbf{elif}\;P0 \leq 2000000:\\ \;\;\;\;\mathsf{fma}\left(amt, V0 \cdot \left(amt \cdot \left(amt + -2\right)\right), P0 \cdot 1.52587890625 \cdot 10^{-5}\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

(FPCore (P0 P1 amt V1 V0)
 :precision binary32
 (let* ((t_0 (* P0 (fma (* amt amt) (fma amt 2.0 -3.0) 1.52587890625e-5))))
   (if (<= P0 -1000000000.0)
     t_0
     (if (<= P0 -5.000000156871975e-23)
       (* V1 (* (+ amt -1.0) (* amt amt)))
       (if (<= P0 2000000.0)
         (fma amt (* V0 (* amt (+ amt -2.0))) (* P0 1.52587890625e-5))
         t_0)))))

float code(float P0, float P1, float amt, float V1, float V0) {
	float t_0 = P0 * fmaf((amt * amt), fmaf(amt, 2.0f, -3.0f), 1.52587890625e-5f);
	float tmp;
	if (P0 <= -1000000000.0f) {
		tmp = t_0;
	} else if (P0 <= -5.000000156871975e-23f) {
		tmp = V1 * ((amt + -1.0f) * (amt * amt));
	} else if (P0 <= 2000000.0f) {
		tmp = fmaf(amt, (V0 * (amt * (amt + -2.0f))), (P0 * 1.52587890625e-5f));
	} else {
		tmp = t_0;
	}
	return tmp;
}

function code(P0, P1, amt, V1, V0)
	t_0 = Float32(P0 * fma(Float32(amt * amt), fma(amt, Float32(2.0), Float32(-3.0)), Float32(1.52587890625e-5)))
	tmp = Float32(0.0)
	if (P0 <= Float32(-1000000000.0))
		tmp = t_0;
	elseif (P0 <= Float32(-5.000000156871975e-23))
		tmp = Float32(V1 * Float32(Float32(amt + Float32(-1.0)) * Float32(amt * amt)));
	elseif (P0 <= Float32(2000000.0))
		tmp = fma(amt, Float32(V0 * Float32(amt * Float32(amt + Float32(-2.0)))), Float32(P0 * Float32(1.52587890625e-5)));
	else
		tmp = t_0;
	end
	return tmp
end

\begin{array}{l}

\\
\begin{array}{l}
t_0 := P0 \cdot \mathsf{fma}\left(amt \cdot amt, \mathsf{fma}\left(amt, 2, -3\right), 1.52587890625 \cdot 10^{-5}\right)\\
\mathbf{if}\;P0 \leq -1000000000:\\
\;\;\;\;t_0\\

\mathbf{elif}\;P0 \leq -5.000000156871975 \cdot 10^{-23}:\\
\;\;\;\;V1 \cdot \left(\left(amt + -1\right) \cdot \left(amt \cdot amt\right)\right)\\

\mathbf{elif}\;P0 \leq 2000000:\\
\;\;\;\;\mathsf{fma}\left(amt, V0 \cdot \left(amt \cdot \left(amt + -2\right)\right), P0 \cdot 1.52587890625 \cdot 10^{-5}\right)\\

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


\end{array}
\end{array}

Derivation

Split input into 3 regimes
if P0 < -1e9 or 2e6 < P0
1. Initial program 68.0%
  \[\frac{\left(\left(P0 + P1 \cdot \left(amt \cdot 65536\right)\right) + \left(\left(3 \cdot \left(P1 - P0\right) - V1\right) - 2 \cdot V0\right) \cdot \left({amt}^{2} \cdot 65536\right)\right) + \left(\left(2 \cdot \left(P0 - P1\right) + V0\right) + V1\right) \cdot \left({amt}^{3} \cdot 65536\right)}{65536} \]
2. Add Preprocessing
3. Taylor expanded in amt around 0
  \[\leadsto \color{blue}{\frac{1}{65536} \cdot P0 + amt \cdot \left(P1 + amt \cdot \left(\left(3 \cdot \left(P1 - P0\right) + amt \cdot \left(V0 + \left(V1 + 2 \cdot \left(P0 - P1\right)\right)\right)\right) - \left(V1 + 2 \cdot V0\right)\right)\right)} \]
4. Step-by-step derivation
  1. +-commutativeN/A
    \[\leadsto \color{blue}{amt \cdot \left(P1 + amt \cdot \left(\left(3 \cdot \left(P1 - P0\right) + amt \cdot \left(V0 + \left(V1 + 2 \cdot \left(P0 - P1\right)\right)\right)\right) - \left(V1 + 2 \cdot V0\right)\right)\right) + \frac{1}{65536} \cdot P0} \]
  2. lower-fma.f32N/A
    \[\leadsto \color{blue}{\mathsf{fma}\left(amt, P1 + amt \cdot \left(\left(3 \cdot \left(P1 - P0\right) + amt \cdot \left(V0 + \left(V1 + 2 \cdot \left(P0 - P1\right)\right)\right)\right) - \left(V1 + 2 \cdot V0\right)\right), \frac{1}{65536} \cdot P0\right)} \]
5. Applied rewrites94.9%
  \[\leadsto \color{blue}{\mathsf{fma}\left(amt, \mathsf{fma}\left(amt, \mathsf{fma}\left(amt, \mathsf{fma}\left(2, P0 - P1, V1 + V0\right), \mathsf{fma}\left(3, P1 - P0, \mathsf{fma}\left(V0, -2, -V1\right)\right)\right), P1\right), P0 \cdot 1.52587890625 \cdot 10^{-5}\right)} \]
6. Taylor expanded in P0 around inf
  \[\leadsto P0 \cdot \color{blue}{\left(\frac{1}{65536} + {amt}^{2} \cdot \left(2 \cdot amt - 3\right)\right)} \]
7. Step-by-step derivation
8. Applied rewrites61.6%
  \[\leadsto P0 \cdot \color{blue}{\mathsf{fma}\left(amt \cdot amt, \mathsf{fma}\left(amt, 2, -3\right), 1.52587890625 \cdot 10^{-5}\right)} \]
if -1e9 < P0 < -5.00000016e-23
1. Initial program 80.5%
  \[\frac{\left(\left(P0 + P1 \cdot \left(amt \cdot 65536\right)\right) + \left(\left(3 \cdot \left(P1 - P0\right) - V1\right) - 2 \cdot V0\right) \cdot \left({amt}^{2} \cdot 65536\right)\right) + \left(\left(2 \cdot \left(P0 - P1\right) + V0\right) + V1\right) \cdot \left({amt}^{3} \cdot 65536\right)}{65536} \]
2. Add Preprocessing
3. Taylor expanded in amt around 0
  \[\leadsto \color{blue}{\frac{1}{65536} \cdot P0 + amt \cdot \left(P1 + amt \cdot \left(\left(3 \cdot \left(P1 - P0\right) + amt \cdot \left(V0 + \left(V1 + 2 \cdot \left(P0 - P1\right)\right)\right)\right) - \left(V1 + 2 \cdot V0\right)\right)\right)} \]
4. Step-by-step derivation
  1. +-commutativeN/A
    \[\leadsto \color{blue}{amt \cdot \left(P1 + amt \cdot \left(\left(3 \cdot \left(P1 - P0\right) + amt \cdot \left(V0 + \left(V1 + 2 \cdot \left(P0 - P1\right)\right)\right)\right) - \left(V1 + 2 \cdot V0\right)\right)\right) + \frac{1}{65536} \cdot P0} \]
  2. lower-fma.f32N/A
    \[\leadsto \color{blue}{\mathsf{fma}\left(amt, P1 + amt \cdot \left(\left(3 \cdot \left(P1 - P0\right) + amt \cdot \left(V0 + \left(V1 + 2 \cdot \left(P0 - P1\right)\right)\right)\right) - \left(V1 + 2 \cdot V0\right)\right), \frac{1}{65536} \cdot P0\right)} \]
5. Applied rewrites97.5%
  \[\leadsto \color{blue}{\mathsf{fma}\left(amt, \mathsf{fma}\left(amt, \mathsf{fma}\left(amt, \mathsf{fma}\left(2, P0 - P1, V1 + V0\right), \mathsf{fma}\left(3, P1 - P0, \mathsf{fma}\left(V0, -2, -V1\right)\right)\right), P1\right), P0 \cdot 1.52587890625 \cdot 10^{-5}\right)} \]
6. Taylor expanded in V1 around inf
  \[\leadsto V1 \cdot \color{blue}{\left({amt}^{2} \cdot \left(amt - 1\right)\right)} \]
7. Step-by-step derivation
8. Applied rewrites45.8%
  \[\leadsto V1 \cdot \color{blue}{\left(\left(amt \cdot amt\right) \cdot \left(amt + -1\right)\right)} \]
if -5.00000016e-23 < P0 < 2e6
1. Initial program 80.1%
  \[\frac{\left(\left(P0 + P1 \cdot \left(amt \cdot 65536\right)\right) + \left(\left(3 \cdot \left(P1 - P0\right) - V1\right) - 2 \cdot V0\right) \cdot \left({amt}^{2} \cdot 65536\right)\right) + \left(\left(2 \cdot \left(P0 - P1\right) + V0\right) + V1\right) \cdot \left({amt}^{3} \cdot 65536\right)}{65536} \]
2. Add Preprocessing
3. Taylor expanded in amt around 0
  \[\leadsto \color{blue}{\frac{1}{65536} \cdot P0 + amt \cdot \left(P1 + amt \cdot \left(\left(3 \cdot \left(P1 - P0\right) + amt \cdot \left(V0 + \left(V1 + 2 \cdot \left(P0 - P1\right)\right)\right)\right) - \left(V1 + 2 \cdot V0\right)\right)\right)} \]
4. Step-by-step derivation
  1. +-commutativeN/A
    \[\leadsto \color{blue}{amt \cdot \left(P1 + amt \cdot \left(\left(3 \cdot \left(P1 - P0\right) + amt \cdot \left(V0 + \left(V1 + 2 \cdot \left(P0 - P1\right)\right)\right)\right) - \left(V1 + 2 \cdot V0\right)\right)\right) + \frac{1}{65536} \cdot P0} \]
  2. lower-fma.f32N/A
    \[\leadsto \color{blue}{\mathsf{fma}\left(amt, P1 + amt \cdot \left(\left(3 \cdot \left(P1 - P0\right) + amt \cdot \left(V0 + \left(V1 + 2 \cdot \left(P0 - P1\right)\right)\right)\right) - \left(V1 + 2 \cdot V0\right)\right), \frac{1}{65536} \cdot P0\right)} \]
5. Applied rewrites97.5%
  \[\leadsto \color{blue}{\mathsf{fma}\left(amt, \mathsf{fma}\left(amt, \mathsf{fma}\left(amt, \mathsf{fma}\left(2, P0 - P1, V1 + V0\right), \mathsf{fma}\left(3, P1 - P0, \mathsf{fma}\left(V0, -2, -V1\right)\right)\right), P1\right), P0 \cdot 1.52587890625 \cdot 10^{-5}\right)} \]
6. Taylor expanded in V0 around inf
  \[\leadsto \mathsf{fma}\left(amt, V0 \cdot \color{blue}{\left(amt \cdot \left(amt - 2\right)\right)}, P0 \cdot \frac{1}{65536}\right) \]
7. Step-by-step derivation
8. Applied rewrites44.9%
  \[\leadsto \mathsf{fma}\left(amt, V0 \cdot \color{blue}{\left(amt \cdot \left(amt + -2\right)\right)}, P0 \cdot 1.52587890625 \cdot 10^{-5}\right) \]
Recombined 3 regimes into one program.
Final simplification51.9%
\[\leadsto \begin{array}{l} \mathbf{if}\;P0 \leq -1000000000:\\ \;\;\;\;P0 \cdot \mathsf{fma}\left(amt \cdot amt, \mathsf{fma}\left(amt, 2, -3\right), 1.52587890625 \cdot 10^{-5}\right)\\ \mathbf{elif}\;P0 \leq -5.000000156871975 \cdot 10^{-23}:\\ \;\;\;\;V1 \cdot \left(\left(amt + -1\right) \cdot \left(amt \cdot amt\right)\right)\\ \mathbf{elif}\;P0 \leq 2000000:\\ \;\;\;\;\mathsf{fma}\left(amt, V0 \cdot \left(amt \cdot \left(amt + -2\right)\right), P0 \cdot 1.52587890625 \cdot 10^{-5}\right)\\ \mathbf{else}:\\ \;\;\;\;P0 \cdot \mathsf{fma}\left(amt \cdot amt, \mathsf{fma}\left(amt, 2, -3\right), 1.52587890625 \cdot 10^{-5}\right)\\ \end{array} \]
Add Preprocessing

Alternative 13: 39.3% accurate, 7.0× speedup?

\[\begin{array}{l} \\ \begin{array}{l} t_0 := P0 \cdot \mathsf{fma}\left(amt \cdot amt, \mathsf{fma}\left(amt, 2, -3\right), 1.52587890625 \cdot 10^{-5}\right)\\ \mathbf{if}\;P0 \leq -1000000000:\\ \;\;\;\;t_0\\ \mathbf{elif}\;P0 \leq -5.000000156871975 \cdot 10^{-23}:\\ \;\;\;\;V1 \cdot \left(\left(amt + -1\right) \cdot \left(amt \cdot amt\right)\right)\\ \mathbf{elif}\;P0 \leq 2000000:\\ \;\;\;\;V0 \cdot \left(\left(amt \cdot amt\right) \cdot \left(amt + -2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

(FPCore (P0 P1 amt V1 V0)
 :precision binary32
 (let* ((t_0 (* P0 (fma (* amt amt) (fma amt 2.0 -3.0) 1.52587890625e-5))))
   (if (<= P0 -1000000000.0)
     t_0
     (if (<= P0 -5.000000156871975e-23)
       (* V1 (* (+ amt -1.0) (* amt amt)))
       (if (<= P0 2000000.0) (* V0 (* (* amt amt) (+ amt -2.0))) t_0)))))

float code(float P0, float P1, float amt, float V1, float V0) {
	float t_0 = P0 * fmaf((amt * amt), fmaf(amt, 2.0f, -3.0f), 1.52587890625e-5f);
	float tmp;
	if (P0 <= -1000000000.0f) {
		tmp = t_0;
	} else if (P0 <= -5.000000156871975e-23f) {
		tmp = V1 * ((amt + -1.0f) * (amt * amt));
	} else if (P0 <= 2000000.0f) {
		tmp = V0 * ((amt * amt) * (amt + -2.0f));
	} else {
		tmp = t_0;
	}
	return tmp;
}

function code(P0, P1, amt, V1, V0)
	t_0 = Float32(P0 * fma(Float32(amt * amt), fma(amt, Float32(2.0), Float32(-3.0)), Float32(1.52587890625e-5)))
	tmp = Float32(0.0)
	if (P0 <= Float32(-1000000000.0))
		tmp = t_0;
	elseif (P0 <= Float32(-5.000000156871975e-23))
		tmp = Float32(V1 * Float32(Float32(amt + Float32(-1.0)) * Float32(amt * amt)));
	elseif (P0 <= Float32(2000000.0))
		tmp = Float32(V0 * Float32(Float32(amt * amt) * Float32(amt + Float32(-2.0))));
	else
		tmp = t_0;
	end
	return tmp
end

\begin{array}{l}

\\
\begin{array}{l}
t_0 := P0 \cdot \mathsf{fma}\left(amt \cdot amt, \mathsf{fma}\left(amt, 2, -3\right), 1.52587890625 \cdot 10^{-5}\right)\\
\mathbf{if}\;P0 \leq -1000000000:\\
\;\;\;\;t_0\\

\mathbf{elif}\;P0 \leq -5.000000156871975 \cdot 10^{-23}:\\
\;\;\;\;V1 \cdot \left(\left(amt + -1\right) \cdot \left(amt \cdot amt\right)\right)\\

\mathbf{elif}\;P0 \leq 2000000:\\
\;\;\;\;V0 \cdot \left(\left(amt \cdot amt\right) \cdot \left(amt + -2\right)\right)\\

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


\end{array}
\end{array}

Derivation

Split input into 3 regimes
if P0 < -1e9 or 2e6 < P0
1. Initial program 68.0%
  \[\frac{\left(\left(P0 + P1 \cdot \left(amt \cdot 65536\right)\right) + \left(\left(3 \cdot \left(P1 - P0\right) - V1\right) - 2 \cdot V0\right) \cdot \left({amt}^{2} \cdot 65536\right)\right) + \left(\left(2 \cdot \left(P0 - P1\right) + V0\right) + V1\right) \cdot \left({amt}^{3} \cdot 65536\right)}{65536} \]
2. Add Preprocessing
3. Taylor expanded in amt around 0
  \[\leadsto \color{blue}{\frac{1}{65536} \cdot P0 + amt \cdot \left(P1 + amt \cdot \left(\left(3 \cdot \left(P1 - P0\right) + amt \cdot \left(V0 + \left(V1 + 2 \cdot \left(P0 - P1\right)\right)\right)\right) - \left(V1 + 2 \cdot V0\right)\right)\right)} \]
4. Step-by-step derivation
  1. +-commutativeN/A
    \[\leadsto \color{blue}{amt \cdot \left(P1 + amt \cdot \left(\left(3 \cdot \left(P1 - P0\right) + amt \cdot \left(V0 + \left(V1 + 2 \cdot \left(P0 - P1\right)\right)\right)\right) - \left(V1 + 2 \cdot V0\right)\right)\right) + \frac{1}{65536} \cdot P0} \]
  2. lower-fma.f32N/A
    \[\leadsto \color{blue}{\mathsf{fma}\left(amt, P1 + amt \cdot \left(\left(3 \cdot \left(P1 - P0\right) + amt \cdot \left(V0 + \left(V1 + 2 \cdot \left(P0 - P1\right)\right)\right)\right) - \left(V1 + 2 \cdot V0\right)\right), \frac{1}{65536} \cdot P0\right)} \]
5. Applied rewrites94.9%
  \[\leadsto \color{blue}{\mathsf{fma}\left(amt, \mathsf{fma}\left(amt, \mathsf{fma}\left(amt, \mathsf{fma}\left(2, P0 - P1, V1 + V0\right), \mathsf{fma}\left(3, P1 - P0, \mathsf{fma}\left(V0, -2, -V1\right)\right)\right), P1\right), P0 \cdot 1.52587890625 \cdot 10^{-5}\right)} \]
6. Taylor expanded in P0 around inf
  \[\leadsto P0 \cdot \color{blue}{\left(\frac{1}{65536} + {amt}^{2} \cdot \left(2 \cdot amt - 3\right)\right)} \]
7. Step-by-step derivation
8. Applied rewrites61.6%
  \[\leadsto P0 \cdot \color{blue}{\mathsf{fma}\left(amt \cdot amt, \mathsf{fma}\left(amt, 2, -3\right), 1.52587890625 \cdot 10^{-5}\right)} \]
if -1e9 < P0 < -5.00000016e-23
1. Initial program 80.5%
  \[\frac{\left(\left(P0 + P1 \cdot \left(amt \cdot 65536\right)\right) + \left(\left(3 \cdot \left(P1 - P0\right) - V1\right) - 2 \cdot V0\right) \cdot \left({amt}^{2} \cdot 65536\right)\right) + \left(\left(2 \cdot \left(P0 - P1\right) + V0\right) + V1\right) \cdot \left({amt}^{3} \cdot 65536\right)}{65536} \]
2. Add Preprocessing
3. Taylor expanded in amt around 0
  \[\leadsto \color{blue}{\frac{1}{65536} \cdot P0 + amt \cdot \left(P1 + amt \cdot \left(\left(3 \cdot \left(P1 - P0\right) + amt \cdot \left(V0 + \left(V1 + 2 \cdot \left(P0 - P1\right)\right)\right)\right) - \left(V1 + 2 \cdot V0\right)\right)\right)} \]
4. Step-by-step derivation
  1. +-commutativeN/A
    \[\leadsto \color{blue}{amt \cdot \left(P1 + amt \cdot \left(\left(3 \cdot \left(P1 - P0\right) + amt \cdot \left(V0 + \left(V1 + 2 \cdot \left(P0 - P1\right)\right)\right)\right) - \left(V1 + 2 \cdot V0\right)\right)\right) + \frac{1}{65536} \cdot P0} \]
  2. lower-fma.f32N/A
    \[\leadsto \color{blue}{\mathsf{fma}\left(amt, P1 + amt \cdot \left(\left(3 \cdot \left(P1 - P0\right) + amt \cdot \left(V0 + \left(V1 + 2 \cdot \left(P0 - P1\right)\right)\right)\right) - \left(V1 + 2 \cdot V0\right)\right), \frac{1}{65536} \cdot P0\right)} \]
5. Applied rewrites97.5%
  \[\leadsto \color{blue}{\mathsf{fma}\left(amt, \mathsf{fma}\left(amt, \mathsf{fma}\left(amt, \mathsf{fma}\left(2, P0 - P1, V1 + V0\right), \mathsf{fma}\left(3, P1 - P0, \mathsf{fma}\left(V0, -2, -V1\right)\right)\right), P1\right), P0 \cdot 1.52587890625 \cdot 10^{-5}\right)} \]
6. Taylor expanded in V1 around inf
  \[\leadsto V1 \cdot \color{blue}{\left({amt}^{2} \cdot \left(amt - 1\right)\right)} \]
7. Step-by-step derivation
8. Applied rewrites45.8%
  \[\leadsto V1 \cdot \color{blue}{\left(\left(amt \cdot amt\right) \cdot \left(amt + -1\right)\right)} \]
if -5.00000016e-23 < P0 < 2e6
1. Initial program 80.1%
  \[\frac{\left(\left(P0 + P1 \cdot \left(amt \cdot 65536\right)\right) + \left(\left(3 \cdot \left(P1 - P0\right) - V1\right) - 2 \cdot V0\right) \cdot \left({amt}^{2} \cdot 65536\right)\right) + \left(\left(2 \cdot \left(P0 - P1\right) + V0\right) + V1\right) \cdot \left({amt}^{3} \cdot 65536\right)}{65536} \]
2. Add Preprocessing
3. Taylor expanded in amt around 0
  \[\leadsto \color{blue}{\frac{1}{65536} \cdot P0 + amt \cdot \left(P1 + amt \cdot \left(\left(3 \cdot \left(P1 - P0\right) + amt \cdot \left(V0 + \left(V1 + 2 \cdot \left(P0 - P1\right)\right)\right)\right) - \left(V1 + 2 \cdot V0\right)\right)\right)} \]
4. Step-by-step derivation
  1. +-commutativeN/A
    \[\leadsto \color{blue}{amt \cdot \left(P1 + amt \cdot \left(\left(3 \cdot \left(P1 - P0\right) + amt \cdot \left(V0 + \left(V1 + 2 \cdot \left(P0 - P1\right)\right)\right)\right) - \left(V1 + 2 \cdot V0\right)\right)\right) + \frac{1}{65536} \cdot P0} \]
  2. lower-fma.f32N/A
    \[\leadsto \color{blue}{\mathsf{fma}\left(amt, P1 + amt \cdot \left(\left(3 \cdot \left(P1 - P0\right) + amt \cdot \left(V0 + \left(V1 + 2 \cdot \left(P0 - P1\right)\right)\right)\right) - \left(V1 + 2 \cdot V0\right)\right), \frac{1}{65536} \cdot P0\right)} \]
5. Applied rewrites97.5%
  \[\leadsto \color{blue}{\mathsf{fma}\left(amt, \mathsf{fma}\left(amt, \mathsf{fma}\left(amt, \mathsf{fma}\left(2, P0 - P1, V1 + V0\right), \mathsf{fma}\left(3, P1 - P0, \mathsf{fma}\left(V0, -2, -V1\right)\right)\right), P1\right), P0 \cdot 1.52587890625 \cdot 10^{-5}\right)} \]
6. Taylor expanded in V0 around inf
  \[\leadsto V0 \cdot \color{blue}{\left({amt}^{2} \cdot \left(amt - 2\right)\right)} \]
7. Step-by-step derivation
8. Applied rewrites44.4%
  \[\leadsto V0 \cdot \color{blue}{\left(\left(amt \cdot amt\right) \cdot \left(amt + -2\right)\right)} \]
Recombined 3 regimes into one program.
Final simplification51.7%
\[\leadsto \begin{array}{l} \mathbf{if}\;P0 \leq -1000000000:\\ \;\;\;\;P0 \cdot \mathsf{fma}\left(amt \cdot amt, \mathsf{fma}\left(amt, 2, -3\right), 1.52587890625 \cdot 10^{-5}\right)\\ \mathbf{elif}\;P0 \leq -5.000000156871975 \cdot 10^{-23}:\\ \;\;\;\;V1 \cdot \left(\left(amt + -1\right) \cdot \left(amt \cdot amt\right)\right)\\ \mathbf{elif}\;P0 \leq 2000000:\\ \;\;\;\;V0 \cdot \left(\left(amt \cdot amt\right) \cdot \left(amt + -2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;P0 \cdot \mathsf{fma}\left(amt \cdot amt, \mathsf{fma}\left(amt, 2, -3\right), 1.52587890625 \cdot 10^{-5}\right)\\ \end{array} \]
Add Preprocessing

Alternative 14: 38.7% accurate, 9.2× speedup?

\[\begin{array}{l} \\ \begin{array}{l} t_0 := V0 \cdot \left(\left(amt \cdot amt\right) \cdot \left(amt + -2\right)\right)\\ \mathbf{if}\;V0 \leq -20:\\ \;\;\;\;t_0\\ \mathbf{elif}\;V0 \leq 200:\\ \;\;\;\;V1 \cdot \left(\left(amt + -1\right) \cdot \left(amt \cdot amt\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

(FPCore (P0 P1 amt V1 V0)
 :precision binary32
 (let* ((t_0 (* V0 (* (* amt amt) (+ amt -2.0)))))
   (if (<= V0 -20.0)
     t_0
     (if (<= V0 200.0) (* V1 (* (+ amt -1.0) (* amt amt))) t_0))))

float code(float P0, float P1, float amt, float V1, float V0) {
	float t_0 = V0 * ((amt * amt) * (amt + -2.0f));
	float tmp;
	if (V0 <= -20.0f) {
		tmp = t_0;
	} else if (V0 <= 200.0f) {
		tmp = V1 * ((amt + -1.0f) * (amt * amt));
	} else {
		tmp = t_0;
	}
	return tmp;
}

real(4) function code(p0, p1, amt, v1, v0)
    real(4), intent (in) :: p0
    real(4), intent (in) :: p1
    real(4), intent (in) :: amt
    real(4), intent (in) :: v1
    real(4), intent (in) :: v0
    real(4) :: t_0
    real(4) :: tmp
    t_0 = v0 * ((amt * amt) * (amt + (-2.0e0)))
    if (v0 <= (-20.0e0)) then
        tmp = t_0
    else if (v0 <= 200.0e0) then
        tmp = v1 * ((amt + (-1.0e0)) * (amt * amt))
    else
        tmp = t_0
    end if
    code = tmp
end function

function code(P0, P1, amt, V1, V0)
	t_0 = Float32(V0 * Float32(Float32(amt * amt) * Float32(amt + Float32(-2.0))))
	tmp = Float32(0.0)
	if (V0 <= Float32(-20.0))
		tmp = t_0;
	elseif (V0 <= Float32(200.0))
		tmp = Float32(V1 * Float32(Float32(amt + Float32(-1.0)) * Float32(amt * amt)));
	else
		tmp = t_0;
	end
	return tmp
end

function tmp_2 = code(P0, P1, amt, V1, V0)
	t_0 = V0 * ((amt * amt) * (amt + single(-2.0)));
	tmp = single(0.0);
	if (V0 <= single(-20.0))
		tmp = t_0;
	elseif (V0 <= single(200.0))
		tmp = V1 * ((amt + single(-1.0)) * (amt * amt));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

\begin{array}{l}

\\
\begin{array}{l}
t_0 := V0 \cdot \left(\left(amt \cdot amt\right) \cdot \left(amt + -2\right)\right)\\
\mathbf{if}\;V0 \leq -20:\\
\;\;\;\;t_0\\

\mathbf{elif}\;V0 \leq 200:\\
\;\;\;\;V1 \cdot \left(\left(amt + -1\right) \cdot \left(amt \cdot amt\right)\right)\\

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


\end{array}
\end{array}

Derivation

Split input into 2 regimes
if V0 < -20 or 200 < V0
1. Initial program 75.1%
  \[\frac{\left(\left(P0 + P1 \cdot \left(amt \cdot 65536\right)\right) + \left(\left(3 \cdot \left(P1 - P0\right) - V1\right) - 2 \cdot V0\right) \cdot \left({amt}^{2} \cdot 65536\right)\right) + \left(\left(2 \cdot \left(P0 - P1\right) + V0\right) + V1\right) \cdot \left({amt}^{3} \cdot 65536\right)}{65536} \]
2. Add Preprocessing
3. Taylor expanded in amt around 0
  \[\leadsto \color{blue}{\frac{1}{65536} \cdot P0 + amt \cdot \left(P1 + amt \cdot \left(\left(3 \cdot \left(P1 - P0\right) + amt \cdot \left(V0 + \left(V1 + 2 \cdot \left(P0 - P1\right)\right)\right)\right) - \left(V1 + 2 \cdot V0\right)\right)\right)} \]
4. Step-by-step derivation
  1. +-commutativeN/A
    \[\leadsto \color{blue}{amt \cdot \left(P1 + amt \cdot \left(\left(3 \cdot \left(P1 - P0\right) + amt \cdot \left(V0 + \left(V1 + 2 \cdot \left(P0 - P1\right)\right)\right)\right) - \left(V1 + 2 \cdot V0\right)\right)\right) + \frac{1}{65536} \cdot P0} \]
  2. lower-fma.f32N/A
    \[\leadsto \color{blue}{\mathsf{fma}\left(amt, P1 + amt \cdot \left(\left(3 \cdot \left(P1 - P0\right) + amt \cdot \left(V0 + \left(V1 + 2 \cdot \left(P0 - P1\right)\right)\right)\right) - \left(V1 + 2 \cdot V0\right)\right), \frac{1}{65536} \cdot P0\right)} \]
5. Applied rewrites96.8%
  \[\leadsto \color{blue}{\mathsf{fma}\left(amt, \mathsf{fma}\left(amt, \mathsf{fma}\left(amt, \mathsf{fma}\left(2, P0 - P1, V1 + V0\right), \mathsf{fma}\left(3, P1 - P0, \mathsf{fma}\left(V0, -2, -V1\right)\right)\right), P1\right), P0 \cdot 1.52587890625 \cdot 10^{-5}\right)} \]
6. Taylor expanded in V0 around inf
  \[\leadsto V0 \cdot \color{blue}{\left({amt}^{2} \cdot \left(amt - 2\right)\right)} \]
7. Step-by-step derivation
8. Applied rewrites45.6%
  \[\leadsto V0 \cdot \color{blue}{\left(\left(amt \cdot amt\right) \cdot \left(amt + -2\right)\right)} \]
if -20 < V0 < 200
1. Initial program 75.3%
  \[\frac{\left(\left(P0 + P1 \cdot \left(amt \cdot 65536\right)\right) + \left(\left(3 \cdot \left(P1 - P0\right) - V1\right) - 2 \cdot V0\right) \cdot \left({amt}^{2} \cdot 65536\right)\right) + \left(\left(2 \cdot \left(P0 - P1\right) + V0\right) + V1\right) \cdot \left({amt}^{3} \cdot 65536\right)}{65536} \]
2. Add Preprocessing
3. Taylor expanded in amt around 0
  \[\leadsto \color{blue}{\frac{1}{65536} \cdot P0 + amt \cdot \left(P1 + amt \cdot \left(\left(3 \cdot \left(P1 - P0\right) + amt \cdot \left(V0 + \left(V1 + 2 \cdot \left(P0 - P1\right)\right)\right)\right) - \left(V1 + 2 \cdot V0\right)\right)\right)} \]
4. Step-by-step derivation
  1. +-commutativeN/A
    \[\leadsto \color{blue}{amt \cdot \left(P1 + amt \cdot \left(\left(3 \cdot \left(P1 - P0\right) + amt \cdot \left(V0 + \left(V1 + 2 \cdot \left(P0 - P1\right)\right)\right)\right) - \left(V1 + 2 \cdot V0\right)\right)\right) + \frac{1}{65536} \cdot P0} \]
  2. lower-fma.f32N/A
    \[\leadsto \color{blue}{\mathsf{fma}\left(amt, P1 + amt \cdot \left(\left(3 \cdot \left(P1 - P0\right) + amt \cdot \left(V0 + \left(V1 + 2 \cdot \left(P0 - P1\right)\right)\right)\right) - \left(V1 + 2 \cdot V0\right)\right), \frac{1}{65536} \cdot P0\right)} \]
5. Applied rewrites96.1%
  \[\leadsto \color{blue}{\mathsf{fma}\left(amt, \mathsf{fma}\left(amt, \mathsf{fma}\left(amt, \mathsf{fma}\left(2, P0 - P1, V1 + V0\right), \mathsf{fma}\left(3, P1 - P0, \mathsf{fma}\left(V0, -2, -V1\right)\right)\right), P1\right), P0 \cdot 1.52587890625 \cdot 10^{-5}\right)} \]
6. Taylor expanded in V1 around inf
  \[\leadsto V1 \cdot \color{blue}{\left({amt}^{2} \cdot \left(amt - 1\right)\right)} \]
7. Step-by-step derivation
8. Applied rewrites36.2%
  \[\leadsto V1 \cdot \color{blue}{\left(\left(amt \cdot amt\right) \cdot \left(amt + -1\right)\right)} \]
Recombined 2 regimes into one program.
Final simplification40.9%
\[\leadsto \begin{array}{l} \mathbf{if}\;V0 \leq -20:\\ \;\;\;\;V0 \cdot \left(\left(amt \cdot amt\right) \cdot \left(amt + -2\right)\right)\\ \mathbf{elif}\;V0 \leq 200:\\ \;\;\;\;V1 \cdot \left(\left(amt + -1\right) \cdot \left(amt \cdot amt\right)\right)\\ \mathbf{else}:\\ \;\;\;\;V0 \cdot \left(\left(amt \cdot amt\right) \cdot \left(amt + -2\right)\right)\\ \end{array} \]
Add Preprocessing

Alternative 15: 26.7% accurate, 15.1× speedup?

\[\begin{array}{l} \\ V0 \cdot \left(\left(amt \cdot amt\right) \cdot \left(amt + -2\right)\right) \end{array} \]

(FPCore (P0 P1 amt V1 V0)
 :precision binary32
 (* V0 (* (* amt amt) (+ amt -2.0))))

float code(float P0, float P1, float amt, float V1, float V0) {
	return V0 * ((amt * amt) * (amt + -2.0f));
}

real(4) function code(p0, p1, amt, v1, v0)
    real(4), intent (in) :: p0
    real(4), intent (in) :: p1
    real(4), intent (in) :: amt
    real(4), intent (in) :: v1
    real(4), intent (in) :: v0
    code = v0 * ((amt * amt) * (amt + (-2.0e0)))
end function

function code(P0, P1, amt, V1, V0)
	return Float32(V0 * Float32(Float32(amt * amt) * Float32(amt + Float32(-2.0))))
end

function tmp = code(P0, P1, amt, V1, V0)
	tmp = V0 * ((amt * amt) * (amt + single(-2.0)));
end

\begin{array}{l}

\\
V0 \cdot \left(\left(amt \cdot amt\right) \cdot \left(amt + -2\right)\right)
\end{array}

Derivation

Initial program 75.2%
\[\frac{\left(\left(P0 + P1 \cdot \left(amt \cdot 65536\right)\right) + \left(\left(3 \cdot \left(P1 - P0\right) - V1\right) - 2 \cdot V0\right) \cdot \left({amt}^{2} \cdot 65536\right)\right) + \left(\left(2 \cdot \left(P0 - P1\right) + V0\right) + V1\right) \cdot \left({amt}^{3} \cdot 65536\right)}{65536} \]
Add Preprocessing
Taylor expanded in amt around 0
\[\leadsto \color{blue}{\frac{1}{65536} \cdot P0 + amt \cdot \left(P1 + amt \cdot \left(\left(3 \cdot \left(P1 - P0\right) + amt \cdot \left(V0 + \left(V1 + 2 \cdot \left(P0 - P1\right)\right)\right)\right) - \left(V1 + 2 \cdot V0\right)\right)\right)} \]
Step-by-step derivation
1. +-commutativeN/A
  \[\leadsto \color{blue}{amt \cdot \left(P1 + amt \cdot \left(\left(3 \cdot \left(P1 - P0\right) + amt \cdot \left(V0 + \left(V1 + 2 \cdot \left(P0 - P1\right)\right)\right)\right) - \left(V1 + 2 \cdot V0\right)\right)\right) + \frac{1}{65536} \cdot P0} \]
2. lower-fma.f32N/A
  \[\leadsto \color{blue}{\mathsf{fma}\left(amt, P1 + amt \cdot \left(\left(3 \cdot \left(P1 - P0\right) + amt \cdot \left(V0 + \left(V1 + 2 \cdot \left(P0 - P1\right)\right)\right)\right) - \left(V1 + 2 \cdot V0\right)\right), \frac{1}{65536} \cdot P0\right)} \]
Applied rewrites96.5%
\[\leadsto \color{blue}{\mathsf{fma}\left(amt, \mathsf{fma}\left(amt, \mathsf{fma}\left(amt, \mathsf{fma}\left(2, P0 - P1, V1 + V0\right), \mathsf{fma}\left(3, P1 - P0, \mathsf{fma}\left(V0, -2, -V1\right)\right)\right), P1\right), P0 \cdot 1.52587890625 \cdot 10^{-5}\right)} \]
Taylor expanded in V0 around inf
\[\leadsto V0 \cdot \color{blue}{\left({amt}^{2} \cdot \left(amt - 2\right)\right)} \]
Step-by-step derivation
Applied rewrites27.0%
\[\leadsto V0 \cdot \color{blue}{\left(\left(amt \cdot amt\right) \cdot \left(amt + -2\right)\right)} \]
Add Preprocessing

Alternative 16: 12.0% accurate, 16.8× speedup?

\[\begin{array}{l} \\ amt \cdot \mathsf{fma}\left(P1 \cdot amt, 3, P1\right) \end{array} \]

(FPCore (P0 P1 amt V1 V0) :precision binary32 (* amt (fma (* P1 amt) 3.0 P1)))

float code(float P0, float P1, float amt, float V1, float V0) {
	return amt * fmaf((P1 * amt), 3.0f, P1);
}

function code(P0, P1, amt, V1, V0)
	return Float32(amt * fma(Float32(P1 * amt), Float32(3.0), P1))
end

\begin{array}{l}

\\
amt \cdot \mathsf{fma}\left(P1 \cdot amt, 3, P1\right)
\end{array}

Derivation

Initial program 75.2%
\[\frac{\left(\left(P0 + P1 \cdot \left(amt \cdot 65536\right)\right) + \left(\left(3 \cdot \left(P1 - P0\right) - V1\right) - 2 \cdot V0\right) \cdot \left({amt}^{2} \cdot 65536\right)\right) + \left(\left(2 \cdot \left(P0 - P1\right) + V0\right) + V1\right) \cdot \left({amt}^{3} \cdot 65536\right)}{65536} \]
Add Preprocessing
Taylor expanded in P1 around -inf
\[\leadsto \color{blue}{\frac{-1}{65536} \cdot \left(P1 \cdot \left(-196608 \cdot {amt}^{2} + \left(-65536 \cdot amt + 131072 \cdot {amt}^{3}\right)\right)\right)} \]
Step-by-step derivation
1. *-commutativeN/A
  \[\leadsto \color{blue}{\left(P1 \cdot \left(-196608 \cdot {amt}^{2} + \left(-65536 \cdot amt + 131072 \cdot {amt}^{3}\right)\right)\right) \cdot \frac{-1}{65536}} \]
2. associate-*l*N/A
  \[\leadsto \color{blue}{P1 \cdot \left(\left(-196608 \cdot {amt}^{2} + \left(-65536 \cdot amt + 131072 \cdot {amt}^{3}\right)\right) \cdot \frac{-1}{65536}\right)} \]
3. lower-*.f32N/A
  \[\leadsto \color{blue}{P1 \cdot \left(\left(-196608 \cdot {amt}^{2} + \left(-65536 \cdot amt + 131072 \cdot {amt}^{3}\right)\right) \cdot \frac{-1}{65536}\right)} \]
4. lower-*.f32N/A
  \[\leadsto P1 \cdot \color{blue}{\left(\left(-196608 \cdot {amt}^{2} + \left(-65536 \cdot amt + 131072 \cdot {amt}^{3}\right)\right) \cdot \frac{-1}{65536}\right)} \]
Applied rewrites22.0%
\[\leadsto \color{blue}{P1 \cdot \left(\mathsf{fma}\left(amt, \left(amt \cdot amt\right) \cdot 131072, \mathsf{fma}\left(amt, -65536, \left(amt \cdot amt\right) \cdot -196608\right)\right) \cdot -1.52587890625 \cdot 10^{-5}\right)} \]
Taylor expanded in amt around 0
\[\leadsto amt \cdot \color{blue}{\left(P1 + 3 \cdot \left(P1 \cdot amt\right)\right)} \]
Step-by-step derivation
Applied rewrites10.8%
\[\leadsto amt \cdot \color{blue}{\mathsf{fma}\left(P1 \cdot amt, 3, P1\right)} \]
Add Preprocessing

Alternative 17: 11.5% accurate, 47.7× speedup?

\[\begin{array}{l} \\ P1 \cdot amt \end{array} \]

(FPCore (P0 P1 amt V1 V0) :precision binary32 (* P1 amt))

float code(float P0, float P1, float amt, float V1, float V0) {
	return P1 * amt;
}

real(4) function code(p0, p1, amt, v1, v0)
    real(4), intent (in) :: p0
    real(4), intent (in) :: p1
    real(4), intent (in) :: amt
    real(4), intent (in) :: v1
    real(4), intent (in) :: v0
    code = p1 * amt
end function

function code(P0, P1, amt, V1, V0)
	return Float32(P1 * amt)
end

function tmp = code(P0, P1, amt, V1, V0)
	tmp = P1 * amt;
end

\begin{array}{l}

\\
P1 \cdot amt
\end{array}

Derivation

Initial program 75.2%
\[\frac{\left(\left(P0 + P1 \cdot \left(amt \cdot 65536\right)\right) + \left(\left(3 \cdot \left(P1 - P0\right) - V1\right) - 2 \cdot V0\right) \cdot \left({amt}^{2} \cdot 65536\right)\right) + \left(\left(2 \cdot \left(P0 - P1\right) + V0\right) + V1\right) \cdot \left({amt}^{3} \cdot 65536\right)}{65536} \]
Add Preprocessing
Taylor expanded in P1 around -inf
\[\leadsto \color{blue}{\frac{-1}{65536} \cdot \left(P1 \cdot \left(-196608 \cdot {amt}^{2} + \left(-65536 \cdot amt + 131072 \cdot {amt}^{3}\right)\right)\right)} \]
Step-by-step derivation
1. *-commutativeN/A
  \[\leadsto \color{blue}{\left(P1 \cdot \left(-196608 \cdot {amt}^{2} + \left(-65536 \cdot amt + 131072 \cdot {amt}^{3}\right)\right)\right) \cdot \frac{-1}{65536}} \]
2. associate-*l*N/A
  \[\leadsto \color{blue}{P1 \cdot \left(\left(-196608 \cdot {amt}^{2} + \left(-65536 \cdot amt + 131072 \cdot {amt}^{3}\right)\right) \cdot \frac{-1}{65536}\right)} \]
3. lower-*.f32N/A
  \[\leadsto \color{blue}{P1 \cdot \left(\left(-196608 \cdot {amt}^{2} + \left(-65536 \cdot amt + 131072 \cdot {amt}^{3}\right)\right) \cdot \frac{-1}{65536}\right)} \]
4. lower-*.f32N/A
  \[\leadsto P1 \cdot \color{blue}{\left(\left(-196608 \cdot {amt}^{2} + \left(-65536 \cdot amt + 131072 \cdot {amt}^{3}\right)\right) \cdot \frac{-1}{65536}\right)} \]
Applied rewrites22.0%
\[\leadsto \color{blue}{P1 \cdot \left(\mathsf{fma}\left(amt, \left(amt \cdot amt\right) \cdot 131072, \mathsf{fma}\left(amt, -65536, \left(amt \cdot amt\right) \cdot -196608\right)\right) \cdot -1.52587890625 \cdot 10^{-5}\right)} \]
Taylor expanded in amt around 0
\[\leadsto P1 \cdot \color{blue}{amt} \]
Step-by-step derivation
Applied rewrites10.6%
\[\leadsto P1 \cdot \color{blue}{amt} \]
Add Preprocessing

Specification

Local Percentage Accuracy vs ?

Accuracy vs Speed?

Initial Program: 72.7% accurate, 1.0× speedupMathFPCoreCFortranJuliaMATLABTeX?

Alternative 1: 97.8% accurate, 5.1× speedupMathFPCoreCJuliaTeX?

Alternative 2: 84.9% accurate, 4.9× speedupMathFPCoreCJuliaTeX?

if P1 < -5e8

if -5e8 < P1 < 4.99999992e16

if 4.99999992e16 < P1

Alternative 3: 84.9% accurate, 4.9× speedupMathFPCoreCJuliaTeX?

if P1 < -4999999910000

if -4999999910000 < P1 < 4.99999992e16

if 4.99999992e16 < P1

Alternative 4: 85.8% accurate, 5.1× speedupMathFPCoreCJuliaTeX?

if V0 < -2.00000004e20 or 3.99999993e13 < V0

if -2.00000004e20 < V0 < 3.99999993e13

Alternative 5: 80.3% accurate, 5.4× speedupMathFPCoreCJuliaTeX?

if P0 < -1.00000002e20 or 4.00000011e16 < P0

if -1.00000002e20 < P0 < 4.00000011e16

Alternative 6: 96.4% accurate, 5.4× speedupMathFPCoreCJuliaTeX?

Alternative 7: 96.4% accurate, 5.4× speedupMathFPCoreCJuliaTeX?

Alternative 8: 45.1% accurate, 6.1× speedupMathFPCoreCJuliaTeX?

if P0 < -1e9 or 2e6 < P0

if -1e9 < P0 < -5.00000016e-23

if -5.00000016e-23 < P0 < 2e6

Alternative 9: 43.6% accurate, 6.5× speedupMathFPCoreCJuliaTeX?

if P0 < -1e9

if -1e9 < P0 < -5.00000016e-23

if -5.00000016e-23 < P0 < 2e6

if 2e6 < P0

Alternative 10: 43.7% accurate, 6.5× speedupMathFPCoreCJuliaTeX?

if P0 < -1e9 or 2e6 < P0

if -1e9 < P0 < -5.00000016e-23

if -5.00000016e-23 < P0 < 2e6

Alternative 11: 42.3% accurate, 6.5× speedupMathFPCoreCJuliaTeX?

if P0 < -1e9 or 2e6 < P0

if -1e9 < P0 < -5.00000016e-23

if -5.00000016e-23 < P0 < 2e6

Alternative 12: 39.3% accurate, 6.6× speedupMathFPCoreCJuliaTeX?

if P0 < -1e9 or 2e6 < P0

if -1e9 < P0 < -5.00000016e-23

if -5.00000016e-23 < P0 < 2e6

Alternative 13: 39.3% accurate, 7.0× speedupMathFPCoreCJuliaTeX?

if P0 < -1e9 or 2e6 < P0

if -1e9 < P0 < -5.00000016e-23

if -5.00000016e-23 < P0 < 2e6

Alternative 14: 38.7% accurate, 9.2× speedupMathFPCoreCFortranJuliaMATLABTeX?

if V0 < -20 or 200 < V0

if -20 < V0 < 200

Alternative 15: 26.7% accurate, 15.1× speedupMathFPCoreCFortranJuliaMATLABTeX?

Alternative 16: 12.0% accurate, 16.8× speedupMathFPCoreCJuliaTeX?

Alternative 17: 11.5% accurate, 47.7× speedupMathFPCoreCFortranJuliaMATLABTeX?

Reproduce

Initial Program: 72.7% accurate, 1.0× speedup?

Alternative 1: 97.8% accurate, 5.1× speedup?

Alternative 2: 84.9% accurate, 4.9× speedup?

`if P1 < -5e8`

`if -5e8 < P1 < 4.99999992e16`

`if 4.99999992e16 < P1`

Alternative 3: 84.9% accurate, 4.9× speedup?

`if P1 < -4999999910000`

`if -4999999910000 < P1 < 4.99999992e16`

`if 4.99999992e16 < P1`

Alternative 4: 85.8% accurate, 5.1× speedup?

`if V0 < -2.00000004e20 or 3.99999993e13 < V0`

`if -2.00000004e20 < V0 < 3.99999993e13`

Alternative 5: 80.3% accurate, 5.4× speedup?

`if P0 < -1.00000002e20 or 4.00000011e16 < P0`

`if -1.00000002e20 < P0 < 4.00000011e16`

Alternative 6: 96.4% accurate, 5.4× speedup?

Alternative 7: 96.4% accurate, 5.4× speedup?

Alternative 8: 45.1% accurate, 6.1× speedup?

`if P0 < -1e9 or 2e6 < P0`

`if -1e9 < P0 < -5.00000016e-23`

`if -5.00000016e-23 < P0 < 2e6`

Alternative 9: 43.6% accurate, 6.5× speedup?

`if P0 < -1e9`

`if -1e9 < P0 < -5.00000016e-23`

`if -5.00000016e-23 < P0 < 2e6`

`if 2e6 < P0`

Alternative 10: 43.7% accurate, 6.5× speedup?

`if P0 < -1e9 or 2e6 < P0`

`if -1e9 < P0 < -5.00000016e-23`

`if -5.00000016e-23 < P0 < 2e6`

Alternative 11: 42.3% accurate, 6.5× speedup?

`if P0 < -1e9 or 2e6 < P0`

`if -1e9 < P0 < -5.00000016e-23`

`if -5.00000016e-23 < P0 < 2e6`

Alternative 12: 39.3% accurate, 6.6× speedup?

`if P0 < -1e9 or 2e6 < P0`

`if -1e9 < P0 < -5.00000016e-23`

`if -5.00000016e-23 < P0 < 2e6`

Alternative 13: 39.3% accurate, 7.0× speedup?

`if P0 < -1e9 or 2e6 < P0`

`if -1e9 < P0 < -5.00000016e-23`

`if -5.00000016e-23 < P0 < 2e6`

Alternative 14: 38.7% accurate, 9.2× speedup?

`if V0 < -20 or 200 < V0`

`if -20 < V0 < 200`

Alternative 15: 26.7% accurate, 15.1× speedup?

Alternative 16: 12.0% accurate, 16.8× speedup?

Alternative 17: 11.5% accurate, 47.7× speedup?