a*x+b*y+c*z+d

Percentage Accurate: 100.0% → 100.0%
Time: 9.1s
Alternatives: 9
Speedup: 1.3×

Specification

?
\[\left(\left(\left(\left(\left(\left(-1 \leq a \land a \leq 1\right) \land \left(-1000 \leq x \land x \leq 1000\right)\right) \land \left(-1 \leq b \land b \leq 1\right)\right) \land \left(-1000 \leq y \land y \leq 1000\right)\right) \land \left(-1 \leq c \land c \leq 1\right)\right) \land \left(-1000 \leq z \land z \leq 1000\right)\right) \land \left(-1000 \leq d \land d \leq 1000\right)\]
\[\begin{array}{l} \\ \left(\left(a \cdot x + b \cdot y\right) + c \cdot z\right) + d \end{array} \]
(FPCore (a x b y c z d)
 :precision binary64
 (+ (+ (+ (* a x) (* b y)) (* c z)) d))
double code(double a, double x, double b, double y, double c, double z, double d) {
	return (((a * x) + (b * y)) + (c * z)) + d;
}

real(8) function code(a, x, b, y, c, z, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: x
    real(8), intent (in) :: b
    real(8), intent (in) :: y
    real(8), intent (in) :: c
    real(8), intent (in) :: z
    real(8), intent (in) :: d
    code = (((a * x) + (b * y)) + (c * z)) + d
end function

public static double code(double a, double x, double b, double y, double c, double z, double d) {
	return (((a * x) + (b * y)) + (c * z)) + d;
}

def code(a, x, b, y, c, z, d):
	return (((a * x) + (b * y)) + (c * z)) + d

function code(a, x, b, y, c, z, d)
	return Float64(Float64(Float64(Float64(a * x) + Float64(b * y)) + Float64(c * z)) + d)
end

function tmp = code(a, x, b, y, c, z, d)
	tmp = (((a * x) + (b * y)) + (c * z)) + d;
end

code[a_, x_, b_, y_, c_, z_, d_] := N[(N[(N[(N[(a * x), $MachinePrecision] + N[(b * y), $MachinePrecision]), $MachinePrecision] + N[(c * z), $MachinePrecision]), $MachinePrecision] + d), $MachinePrecision]

\begin{array}{l}

\\
\left(\left(a \cdot x + b \cdot y\right) + c \cdot z\right) + d
\end{array}

Sampling outcomes in binary64 precision:

Local Percentage Accuracy vs ?

The average percentage accuracy by input value. Horizontal axis shows value of an input variable; the variable is choosen in the title. Vertical axis is accuracy; higher is better. Red represent the original program, while blue represents Herbie's suggestion. These can be toggled with buttons below the plot. The line is an average while dots represent individual samples.

Accuracy vs Speed?

Herbie found 9 alternatives:

AlternativeAccuracySpeedup
The accuracy (vertical axis) and speed (horizontal axis) of each alternatives. Up and to the right is better. The red square shows the initial program, and each blue circle shows an alternative.The line shows the best available speed-accuracy tradeoffs.

Initial Program: 100.0% accurate, 1.0× speedup?

\[\begin{array}{l} \\ \left(\left(a \cdot x + b \cdot y\right) + c \cdot z\right) + d \end{array} \]
(FPCore (a x b y c z d)
 :precision binary64
 (+ (+ (+ (* a x) (* b y)) (* c z)) d))
double code(double a, double x, double b, double y, double c, double z, double d) {
	return (((a * x) + (b * y)) + (c * z)) + d;
}

real(8) function code(a, x, b, y, c, z, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: x
    real(8), intent (in) :: b
    real(8), intent (in) :: y
    real(8), intent (in) :: c
    real(8), intent (in) :: z
    real(8), intent (in) :: d
    code = (((a * x) + (b * y)) + (c * z)) + d
end function

public static double code(double a, double x, double b, double y, double c, double z, double d) {
	return (((a * x) + (b * y)) + (c * z)) + d;
}

def code(a, x, b, y, c, z, d):
	return (((a * x) + (b * y)) + (c * z)) + d

function code(a, x, b, y, c, z, d)
	return Float64(Float64(Float64(Float64(a * x) + Float64(b * y)) + Float64(c * z)) + d)
end

function tmp = code(a, x, b, y, c, z, d)
	tmp = (((a * x) + (b * y)) + (c * z)) + d;
end

code[a_, x_, b_, y_, c_, z_, d_] := N[(N[(N[(N[(a * x), $MachinePrecision] + N[(b * y), $MachinePrecision]), $MachinePrecision] + N[(c * z), $MachinePrecision]), $MachinePrecision] + d), $MachinePrecision]

\begin{array}{l}

\\
\left(\left(a \cdot x + b \cdot y\right) + c \cdot z\right) + d
\end{array}

Alternative 1: 100.0% accurate, 1.3× speedup?

\[\begin{array}{l} \\ \mathsf{fma}\left(y, b, \mathsf{fma}\left(x, a, \mathsf{fma}\left(z, c, d\right)\right)\right) \end{array} \]
(FPCore (a x b y c z d) :precision binary64 (fma y b (fma x a (fma z c d))))
double code(double a, double x, double b, double y, double c, double z, double d) {
	return fma(y, b, fma(x, a, fma(z, c, d)));
}

function code(a, x, b, y, c, z, d)
	return fma(y, b, fma(x, a, fma(z, c, d)))
end

code[a_, x_, b_, y_, c_, z_, d_] := N[(y * b + N[(x * a + N[(z * c + d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

\begin{array}{l}

\\
\mathsf{fma}\left(y, b, \mathsf{fma}\left(x, a, \mathsf{fma}\left(z, c, d\right)\right)\right)
\end{array}
Derivation

  1. Initial program 100.0%

    \[\left(\left(a \cdot x + b \cdot y\right) + c \cdot z\right) + d \]
  2. Add Preprocessing
  3. Step-by-step derivation

    1. lift-+.f64N/A

      \[\leadsto \color{blue}{\left(\left(a \cdot x + b \cdot y\right) + c \cdot z\right) + d} \]

    2. lift-+.f64N/A

      \[\leadsto \color{blue}{\left(\left(a \cdot x + b \cdot y\right) + c \cdot z\right)} + d \]

    3. associate-+l+N/A

      \[\leadsto \color{blue}{\left(a \cdot x + b \cdot y\right) + \left(c \cdot z + d\right)} \]

    4. lift-+.f64N/A

      \[\leadsto \color{blue}{\left(a \cdot x + b \cdot y\right)} + \left(c \cdot z + d\right) \]

    5. +-commutativeN/A

      \[\leadsto \color{blue}{\left(b \cdot y + a \cdot x\right)} + \left(c \cdot z + d\right) \]

    6. associate-+l+N/A

      \[\leadsto \color{blue}{b \cdot y + \left(a \cdot x + \left(c \cdot z + d\right)\right)} \]

    7. lift-*.f64N/A

      \[\leadsto \color{blue}{b \cdot y} + \left(a \cdot x + \left(c \cdot z + d\right)\right) \]

    8. *-commutativeN/A

      \[\leadsto \color{blue}{y \cdot b} + \left(a \cdot x + \left(c \cdot z + d\right)\right) \]

    9. lower-fma.f64N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, b, a \cdot x + \left(c \cdot z + d\right)\right)} \]

    10. lift-*.f64N/A

      \[\leadsto \mathsf{fma}\left(y, b, \color{blue}{a \cdot x} + \left(c \cdot z + d\right)\right) \]

    11. *-commutativeN/A

      \[\leadsto \mathsf{fma}\left(y, b, \color{blue}{x \cdot a} + \left(c \cdot z + d\right)\right) \]

    12. lower-fma.f64N/A

      \[\leadsto \mathsf{fma}\left(y, b, \color{blue}{\mathsf{fma}\left(x, a, c \cdot z + d\right)}\right) \]

    13. lift-*.f64N/A

      \[\leadsto \mathsf{fma}\left(y, b, \mathsf{fma}\left(x, a, \color{blue}{c \cdot z} + d\right)\right) \]

    14. *-commutativeN/A

      \[\leadsto \mathsf{fma}\left(y, b, \mathsf{fma}\left(x, a, \color{blue}{z \cdot c} + d\right)\right) \]

    15. lower-fma.f64100.0

      \[\leadsto \mathsf{fma}\left(y, b, \mathsf{fma}\left(x, a, \color{blue}{\mathsf{fma}\left(z, c, d\right)}\right)\right) \]

  4. Applied rewrites100.0%

    \[\leadsto \color{blue}{\mathsf{fma}\left(y, b, \mathsf{fma}\left(x, a, \mathsf{fma}\left(z, c, d\right)\right)\right)} \]
  5. Add Preprocessing

Alternative 2: 95.7% accurate, 0.7× speedup?

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \cdot x \leq -2 \cdot 10^{-232} \lor \neg \left(a \cdot x \leq 2 \cdot 10^{-284}\right):\\ \;\;\;\;\mathsf{fma}\left(a, x, \mathsf{fma}\left(c, z, d\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, c, \mathsf{fma}\left(y, b, d\right)\right)\\ \end{array} \end{array} \]
(FPCore (a x b y c z d)
 :precision binary64
 (if (or (<= (* a x) -2e-232) (not (<= (* a x) 2e-284)))
   (fma a x (fma c z d))
   (fma z c (fma y b d))))
double code(double a, double x, double b, double y, double c, double z, double d) {
	double tmp;
	if (((a * x) <= -2e-232) || !((a * x) <= 2e-284)) {
		tmp = fma(a, x, fma(c, z, d));
	} else {
		tmp = fma(z, c, fma(y, b, d));
	}
	return tmp;
}

function code(a, x, b, y, c, z, d)
	tmp = 0.0
	if ((Float64(a * x) <= -2e-232) || !(Float64(a * x) <= 2e-284))
		tmp = fma(a, x, fma(c, z, d));
	else
		tmp = fma(z, c, fma(y, b, d));
	end
	return tmp
end

code[a_, x_, b_, y_, c_, z_, d_] := If[Or[LessEqual[N[(a * x), $MachinePrecision], -2e-232], N[Not[LessEqual[N[(a * x), $MachinePrecision], 2e-284]], $MachinePrecision]], N[(a * x + N[(c * z + d), $MachinePrecision]), $MachinePrecision], N[(z * c + N[(y * b + d), $MachinePrecision]), $MachinePrecision]]

\begin{array}{l}

\\
\begin{array}{l}
\mathbf{if}\;a \cdot x \leq -2 \cdot 10^{-232} \lor \neg \left(a \cdot x \leq 2 \cdot 10^{-284}\right):\\
\;\;\;\;\mathsf{fma}\left(a, x, \mathsf{fma}\left(c, z, d\right)\right)\\

\mathbf{else}:\\
\;\;\;\;\mathsf{fma}\left(z, c, \mathsf{fma}\left(y, b, d\right)\right)\\


\end{array}
\end{array}
Derivation
  1. Split input into 2 regimes

  2. if (*.f64 a x) < -2.00000000000000005e-232 or 2.00000000000000007e-284 < (*.f64 a x)

    1. Initial program 99.9%

      \[\left(\left(a \cdot x + b \cdot y\right) + c \cdot z\right) + d \]
    2. Add Preprocessing

    3. Taylor expanded in b around 0

      \[\leadsto \color{blue}{d + \left(a \cdot x + c \cdot z\right)} \]
    4. Step-by-step derivation

      1. associate-+r+N/A

        \[\leadsto \color{blue}{\left(d + a \cdot x\right) + c \cdot z} \]

      2. +-commutativeN/A

        \[\leadsto \color{blue}{c \cdot z + \left(d + a \cdot x\right)} \]

      3. *-commutativeN/A

        \[\leadsto \color{blue}{z \cdot c} + \left(d + a \cdot x\right) \]

      4. lower-fma.f64N/A

        \[\leadsto \color{blue}{\mathsf{fma}\left(z, c, d + a \cdot x\right)} \]

      5. +-commutativeN/A

        \[\leadsto \mathsf{fma}\left(z, c, \color{blue}{a \cdot x + d}\right) \]

      6. *-commutativeN/A

        \[\leadsto \mathsf{fma}\left(z, c, \color{blue}{x \cdot a} + d\right) \]

      7. lower-fma.f6495.4

        \[\leadsto \mathsf{fma}\left(z, c, \color{blue}{\mathsf{fma}\left(x, a, d\right)}\right) \]

    5. Applied rewrites95.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(z, c, \mathsf{fma}\left(x, a, d\right)\right)} \]
    6. Step-by-step derivation

      1. Applied rewrites95.4%

        \[\leadsto \color{blue}{\mathsf{fma}\left(a, x, \mathsf{fma}\left(c, z, d\right)\right)} \]

      if -2.00000000000000005e-232 < (*.f64 a x) < 2.00000000000000007e-284

      1. Initial program 100.0%

        \[\left(\left(a \cdot x + b \cdot y\right) + c \cdot z\right) + d \]
      2. Add Preprocessing

      3. Taylor expanded in a around 0

        \[\leadsto \color{blue}{d + \left(b \cdot y + c \cdot z\right)} \]
      4. Step-by-step derivation

        1. associate-+r+N/A

          \[\leadsto \color{blue}{\left(d + b \cdot y\right) + c \cdot z} \]

        2. +-commutativeN/A

          \[\leadsto \color{blue}{c \cdot z + \left(d + b \cdot y\right)} \]

        3. *-commutativeN/A

          \[\leadsto \color{blue}{z \cdot c} + \left(d + b \cdot y\right) \]

        4. lower-fma.f64N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(z, c, d + b \cdot y\right)} \]

        5. +-commutativeN/A

          \[\leadsto \mathsf{fma}\left(z, c, \color{blue}{b \cdot y + d}\right) \]

        6. *-commutativeN/A

          \[\leadsto \mathsf{fma}\left(z, c, \color{blue}{y \cdot b} + d\right) \]

        7. lower-fma.f6498.9

          \[\leadsto \mathsf{fma}\left(z, c, \color{blue}{\mathsf{fma}\left(y, b, d\right)}\right) \]

      5. Applied rewrites98.9%

        \[\leadsto \color{blue}{\mathsf{fma}\left(z, c, \mathsf{fma}\left(y, b, d\right)\right)} \]
    7. Recombined 2 regimes into one program.

    8. Final simplification97.8%

      \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot x \leq -2 \cdot 10^{-232} \lor \neg \left(a \cdot x \leq 2 \cdot 10^{-284}\right):\\ \;\;\;\;\mathsf{fma}\left(a, x, \mathsf{fma}\left(c, z, d\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, c, \mathsf{fma}\left(y, b, d\right)\right)\\ \end{array} \]
    9. Add Preprocessing

    Alternative 3: 96.2% accurate, 0.7× speedup?

    \[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \cdot z \leq -5 \cdot 10^{-220} \lor \neg \left(c \cdot z \leq 2.5 \cdot 10^{-237}\right):\\ \;\;\;\;\mathsf{fma}\left(a, x, \mathsf{fma}\left(c, z, d\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, b, \mathsf{fma}\left(x, a, d\right)\right)\\ \end{array} \end{array} \]
    (FPCore (a x b y c z d)
 :precision binary64
 (if (or (<= (* c z) -5e-220) (not (<= (* c z) 2.5e-237)))
   (fma a x (fma c z d))
   (fma y b (fma x a d))))
    double code(double a, double x, double b, double y, double c, double z, double d) {
	double tmp;
	if (((c * z) <= -5e-220) || !((c * z) <= 2.5e-237)) {
		tmp = fma(a, x, fma(c, z, d));
	} else {
		tmp = fma(y, b, fma(x, a, d));
	}
	return tmp;
}

    function code(a, x, b, y, c, z, d)
	tmp = 0.0
	if ((Float64(c * z) <= -5e-220) || !(Float64(c * z) <= 2.5e-237))
		tmp = fma(a, x, fma(c, z, d));
	else
		tmp = fma(y, b, fma(x, a, d));
	end
	return tmp
end

    code[a_, x_, b_, y_, c_, z_, d_] := If[Or[LessEqual[N[(c * z), $MachinePrecision], -5e-220], N[Not[LessEqual[N[(c * z), $MachinePrecision], 2.5e-237]], $MachinePrecision]], N[(a * x + N[(c * z + d), $MachinePrecision]), $MachinePrecision], N[(y * b + N[(x * a + d), $MachinePrecision]), $MachinePrecision]]

    \begin{array}{l}

\\
\begin{array}{l}
\mathbf{if}\;c \cdot z \leq -5 \cdot 10^{-220} \lor \neg \left(c \cdot z \leq 2.5 \cdot 10^{-237}\right):\\
\;\;\;\;\mathsf{fma}\left(a, x, \mathsf{fma}\left(c, z, d\right)\right)\\

\mathbf{else}:\\
\;\;\;\;\mathsf{fma}\left(y, b, \mathsf{fma}\left(x, a, d\right)\right)\\


\end{array}
\end{array}
    Derivation
    1. Split input into 2 regimes

    2. if (*.f64 c z) < -5.0000000000000002e-220 or 2.5000000000000001e-237 < (*.f64 c z)

      1. Initial program 99.9%

        \[\left(\left(a \cdot x + b \cdot y\right) + c \cdot z\right) + d \]
      2. Add Preprocessing

      3. Taylor expanded in b around 0

        \[\leadsto \color{blue}{d + \left(a \cdot x + c \cdot z\right)} \]
      4. Step-by-step derivation

        1. associate-+r+N/A

          \[\leadsto \color{blue}{\left(d + a \cdot x\right) + c \cdot z} \]

        2. +-commutativeN/A

          \[\leadsto \color{blue}{c \cdot z + \left(d + a \cdot x\right)} \]

        3. *-commutativeN/A

          \[\leadsto \color{blue}{z \cdot c} + \left(d + a \cdot x\right) \]

        4. lower-fma.f64N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(z, c, d + a \cdot x\right)} \]

        5. +-commutativeN/A

          \[\leadsto \mathsf{fma}\left(z, c, \color{blue}{a \cdot x + d}\right) \]

        6. *-commutativeN/A

          \[\leadsto \mathsf{fma}\left(z, c, \color{blue}{x \cdot a} + d\right) \]

        7. lower-fma.f6493.5

          \[\leadsto \mathsf{fma}\left(z, c, \color{blue}{\mathsf{fma}\left(x, a, d\right)}\right) \]

      5. Applied rewrites93.5%

        \[\leadsto \color{blue}{\mathsf{fma}\left(z, c, \mathsf{fma}\left(x, a, d\right)\right)} \]
      6. Step-by-step derivation

        1. Applied rewrites93.5%

          \[\leadsto \color{blue}{\mathsf{fma}\left(a, x, \mathsf{fma}\left(c, z, d\right)\right)} \]

        if -5.0000000000000002e-220 < (*.f64 c z) < 2.5000000000000001e-237

        1. Initial program 100.0%

          \[\left(\left(a \cdot x + b \cdot y\right) + c \cdot z\right) + d \]
        2. Add Preprocessing

        3. Taylor expanded in c around 0

          \[\leadsto \color{blue}{d + \left(a \cdot x + b \cdot y\right)} \]
        4. Step-by-step derivation

          1. associate-+r+N/A

            \[\leadsto \color{blue}{\left(d + a \cdot x\right) + b \cdot y} \]

          2. +-commutativeN/A

            \[\leadsto \color{blue}{b \cdot y + \left(d + a \cdot x\right)} \]

          3. *-commutativeN/A

            \[\leadsto \color{blue}{y \cdot b} + \left(d + a \cdot x\right) \]

          4. lower-fma.f64N/A

            \[\leadsto \color{blue}{\mathsf{fma}\left(y, b, d + a \cdot x\right)} \]

          5. +-commutativeN/A

            \[\leadsto \mathsf{fma}\left(y, b, \color{blue}{a \cdot x + d}\right) \]

          6. *-commutativeN/A

            \[\leadsto \mathsf{fma}\left(y, b, \color{blue}{x \cdot a} + d\right) \]

          7. lower-fma.f6499.5

            \[\leadsto \mathsf{fma}\left(y, b, \color{blue}{\mathsf{fma}\left(x, a, d\right)}\right) \]

        5. Applied rewrites99.5%

          \[\leadsto \color{blue}{\mathsf{fma}\left(y, b, \mathsf{fma}\left(x, a, d\right)\right)} \]
      7. Recombined 2 regimes into one program.

      8. Final simplification97.7%

        \[\leadsto \begin{array}{l} \mathbf{if}\;c \cdot z \leq -5 \cdot 10^{-220} \lor \neg \left(c \cdot z \leq 2.5 \cdot 10^{-237}\right):\\ \;\;\;\;\mathsf{fma}\left(a, x, \mathsf{fma}\left(c, z, d\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, b, \mathsf{fma}\left(x, a, d\right)\right)\\ \end{array} \]
      9. Add Preprocessing

      Alternative 4: 96.2% accurate, 0.7× speedup?

      \[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \cdot z \leq -5 \cdot 10^{-220}:\\ \;\;\;\;\mathsf{fma}\left(a, x, \mathsf{fma}\left(c, z, d\right)\right)\\ \mathbf{elif}\;c \cdot z \leq 2.5 \cdot 10^{-237}:\\ \;\;\;\;\mathsf{fma}\left(y, b, \mathsf{fma}\left(x, a, d\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, c, \mathsf{fma}\left(x, a, d\right)\right)\\ \end{array} \end{array} \]
      (FPCore (a x b y c z d)
 :precision binary64
 (if (<= (* c z) -5e-220)
   (fma a x (fma c z d))
   (if (<= (* c z) 2.5e-237) (fma y b (fma x a d)) (fma z c (fma x a d)))))
      double code(double a, double x, double b, double y, double c, double z, double d) {
	double tmp;
	if ((c * z) <= -5e-220) {
		tmp = fma(a, x, fma(c, z, d));
	} else if ((c * z) <= 2.5e-237) {
		tmp = fma(y, b, fma(x, a, d));
	} else {
		tmp = fma(z, c, fma(x, a, d));
	}
	return tmp;
}

      function code(a, x, b, y, c, z, d)
	tmp = 0.0
	if (Float64(c * z) <= -5e-220)
		tmp = fma(a, x, fma(c, z, d));
	elseif (Float64(c * z) <= 2.5e-237)
		tmp = fma(y, b, fma(x, a, d));
	else
		tmp = fma(z, c, fma(x, a, d));
	end
	return tmp
end

      code[a_, x_, b_, y_, c_, z_, d_] := If[LessEqual[N[(c * z), $MachinePrecision], -5e-220], N[(a * x + N[(c * z + d), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(c * z), $MachinePrecision], 2.5e-237], N[(y * b + N[(x * a + d), $MachinePrecision]), $MachinePrecision], N[(z * c + N[(x * a + d), $MachinePrecision]), $MachinePrecision]]]

      \begin{array}{l}

\\
\begin{array}{l}
\mathbf{if}\;c \cdot z \leq -5 \cdot 10^{-220}:\\
\;\;\;\;\mathsf{fma}\left(a, x, \mathsf{fma}\left(c, z, d\right)\right)\\

\mathbf{elif}\;c \cdot z \leq 2.5 \cdot 10^{-237}:\\
\;\;\;\;\mathsf{fma}\left(y, b, \mathsf{fma}\left(x, a, d\right)\right)\\

\mathbf{else}:\\
\;\;\;\;\mathsf{fma}\left(z, c, \mathsf{fma}\left(x, a, d\right)\right)\\


\end{array}
\end{array}
      Derivation
      1. Split input into 3 regimes

      2. if (*.f64 c z) < -5.0000000000000002e-220

        1. Initial program 100.0%

          \[\left(\left(a \cdot x + b \cdot y\right) + c \cdot z\right) + d \]
        2. Add Preprocessing

        3. Taylor expanded in b around 0

          \[\leadsto \color{blue}{d + \left(a \cdot x + c \cdot z\right)} \]
        4. Step-by-step derivation

          1. associate-+r+N/A

            \[\leadsto \color{blue}{\left(d + a \cdot x\right) + c \cdot z} \]

          2. +-commutativeN/A

            \[\leadsto \color{blue}{c \cdot z + \left(d + a \cdot x\right)} \]

          3. *-commutativeN/A

            \[\leadsto \color{blue}{z \cdot c} + \left(d + a \cdot x\right) \]

          4. lower-fma.f64N/A

            \[\leadsto \color{blue}{\mathsf{fma}\left(z, c, d + a \cdot x\right)} \]

          5. +-commutativeN/A

            \[\leadsto \mathsf{fma}\left(z, c, \color{blue}{a \cdot x + d}\right) \]

          6. *-commutativeN/A

            \[\leadsto \mathsf{fma}\left(z, c, \color{blue}{x \cdot a} + d\right) \]

          7. lower-fma.f6492.9

            \[\leadsto \mathsf{fma}\left(z, c, \color{blue}{\mathsf{fma}\left(x, a, d\right)}\right) \]

        5. Applied rewrites92.9%

          \[\leadsto \color{blue}{\mathsf{fma}\left(z, c, \mathsf{fma}\left(x, a, d\right)\right)} \]
        6. Step-by-step derivation

          1. Applied rewrites92.9%

            \[\leadsto \color{blue}{\mathsf{fma}\left(a, x, \mathsf{fma}\left(c, z, d\right)\right)} \]

          if -5.0000000000000002e-220 < (*.f64 c z) < 2.5000000000000001e-237

          1. Initial program 100.0%

            \[\left(\left(a \cdot x + b \cdot y\right) + c \cdot z\right) + d \]
          2. Add Preprocessing

          3. Taylor expanded in c around 0

            \[\leadsto \color{blue}{d + \left(a \cdot x + b \cdot y\right)} \]
          4. Step-by-step derivation

            1. associate-+r+N/A

              \[\leadsto \color{blue}{\left(d + a \cdot x\right) + b \cdot y} \]

            2. +-commutativeN/A

              \[\leadsto \color{blue}{b \cdot y + \left(d + a \cdot x\right)} \]

            3. *-commutativeN/A

              \[\leadsto \color{blue}{y \cdot b} + \left(d + a \cdot x\right) \]

            4. lower-fma.f64N/A

              \[\leadsto \color{blue}{\mathsf{fma}\left(y, b, d + a \cdot x\right)} \]

            5. +-commutativeN/A

              \[\leadsto \mathsf{fma}\left(y, b, \color{blue}{a \cdot x + d}\right) \]

            6. *-commutativeN/A

              \[\leadsto \mathsf{fma}\left(y, b, \color{blue}{x \cdot a} + d\right) \]

            7. lower-fma.f6499.5

              \[\leadsto \mathsf{fma}\left(y, b, \color{blue}{\mathsf{fma}\left(x, a, d\right)}\right) \]

          5. Applied rewrites99.5%

            \[\leadsto \color{blue}{\mathsf{fma}\left(y, b, \mathsf{fma}\left(x, a, d\right)\right)} \]

          if 2.5000000000000001e-237 < (*.f64 c z)

          1. Initial program 99.9%

            \[\left(\left(a \cdot x + b \cdot y\right) + c \cdot z\right) + d \]
          2. Add Preprocessing

          3. Taylor expanded in b around 0

            \[\leadsto \color{blue}{d + \left(a \cdot x + c \cdot z\right)} \]
          4. Step-by-step derivation

            1. associate-+r+N/A

              \[\leadsto \color{blue}{\left(d + a \cdot x\right) + c \cdot z} \]

            2. +-commutativeN/A

              \[\leadsto \color{blue}{c \cdot z + \left(d + a \cdot x\right)} \]

            3. *-commutativeN/A

              \[\leadsto \color{blue}{z \cdot c} + \left(d + a \cdot x\right) \]

            4. lower-fma.f64N/A

              \[\leadsto \color{blue}{\mathsf{fma}\left(z, c, d + a \cdot x\right)} \]

            5. +-commutativeN/A

              \[\leadsto \mathsf{fma}\left(z, c, \color{blue}{a \cdot x + d}\right) \]

            6. *-commutativeN/A

              \[\leadsto \mathsf{fma}\left(z, c, \color{blue}{x \cdot a} + d\right) \]

            7. lower-fma.f6493.8

              \[\leadsto \mathsf{fma}\left(z, c, \color{blue}{\mathsf{fma}\left(x, a, d\right)}\right) \]

          5. Applied rewrites93.8%

            \[\leadsto \color{blue}{\mathsf{fma}\left(z, c, \mathsf{fma}\left(x, a, d\right)\right)} \]
        7. Recombined 3 regimes into one program.

        8. Final simplification97.7%

          \[\leadsto \begin{array}{l} \mathbf{if}\;c \cdot z \leq -5 \cdot 10^{-220}:\\ \;\;\;\;\mathsf{fma}\left(a, x, \mathsf{fma}\left(c, z, d\right)\right)\\ \mathbf{elif}\;c \cdot z \leq 2.5 \cdot 10^{-237}:\\ \;\;\;\;\mathsf{fma}\left(y, b, \mathsf{fma}\left(x, a, d\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, c, \mathsf{fma}\left(x, a, d\right)\right)\\ \end{array} \]
        9. Add Preprocessing

        Alternative 5: 77.0% accurate, 0.9× speedup?

        \[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \cdot x \leq -5 \cdot 10^{-119} \lor \neg \left(a \cdot x \leq 5 \cdot 10^{-74}\right):\\ \;\;\;\;a \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b, y, d\right)\\ \end{array} \end{array} \]
        (FPCore (a x b y c z d)
 :precision binary64
 (if (or (<= (* a x) -5e-119) (not (<= (* a x) 5e-74))) (* a x) (fma b y d)))
        double code(double a, double x, double b, double y, double c, double z, double d) {
	double tmp;
	if (((a * x) <= -5e-119) || !((a * x) <= 5e-74)) {
		tmp = a * x;
	} else {
		tmp = fma(b, y, d);
	}
	return tmp;
}

        function code(a, x, b, y, c, z, d)
	tmp = 0.0
	if ((Float64(a * x) <= -5e-119) || !(Float64(a * x) <= 5e-74))
		tmp = Float64(a * x);
	else
		tmp = fma(b, y, d);
	end
	return tmp
end

        code[a_, x_, b_, y_, c_, z_, d_] := If[Or[LessEqual[N[(a * x), $MachinePrecision], -5e-119], N[Not[LessEqual[N[(a * x), $MachinePrecision], 5e-74]], $MachinePrecision]], N[(a * x), $MachinePrecision], N[(b * y + d), $MachinePrecision]]

        \begin{array}{l}

\\
\begin{array}{l}
\mathbf{if}\;a \cdot x \leq -5 \cdot 10^{-119} \lor \neg \left(a \cdot x \leq 5 \cdot 10^{-74}\right):\\
\;\;\;\;a \cdot x\\

\mathbf{else}:\\
\;\;\;\;\mathsf{fma}\left(b, y, d\right)\\


\end{array}
\end{array}
        Derivation
        1. Split input into 2 regimes

        2. if (*.f64 a x) < -4.99999999999999993e-119 or 4.99999999999999998e-74 < (*.f64 a x)

          1. Initial program 100.0%

            \[\left(\left(a \cdot x + b \cdot y\right) + c \cdot z\right) + d \]
          2. Add Preprocessing
          3. Step-by-step derivation

            1. lift-+.f64N/A

              \[\leadsto \color{blue}{\left(\left(a \cdot x + b \cdot y\right) + c \cdot z\right) + d} \]

            2. lift-+.f64N/A

              \[\leadsto \color{blue}{\left(\left(a \cdot x + b \cdot y\right) + c \cdot z\right)} + d \]

            3. associate-+l+N/A

              \[\leadsto \color{blue}{\left(a \cdot x + b \cdot y\right) + \left(c \cdot z + d\right)} \]

            4. lift-+.f64N/A

              \[\leadsto \color{blue}{\left(a \cdot x + b \cdot y\right)} + \left(c \cdot z + d\right) \]

            5. +-commutativeN/A

              \[\leadsto \color{blue}{\left(b \cdot y + a \cdot x\right)} + \left(c \cdot z + d\right) \]

            6. associate-+l+N/A

              \[\leadsto \color{blue}{b \cdot y + \left(a \cdot x + \left(c \cdot z + d\right)\right)} \]

            7. lift-*.f64N/A

              \[\leadsto \color{blue}{b \cdot y} + \left(a \cdot x + \left(c \cdot z + d\right)\right) \]

            8. *-commutativeN/A

              \[\leadsto \color{blue}{y \cdot b} + \left(a \cdot x + \left(c \cdot z + d\right)\right) \]

            9. lower-fma.f64N/A

              \[\leadsto \color{blue}{\mathsf{fma}\left(y, b, a \cdot x + \left(c \cdot z + d\right)\right)} \]

            10. lift-*.f64N/A

              \[\leadsto \mathsf{fma}\left(y, b, \color{blue}{a \cdot x} + \left(c \cdot z + d\right)\right) \]

            11. *-commutativeN/A

              \[\leadsto \mathsf{fma}\left(y, b, \color{blue}{x \cdot a} + \left(c \cdot z + d\right)\right) \]

            12. lower-fma.f64N/A

              \[\leadsto \mathsf{fma}\left(y, b, \color{blue}{\mathsf{fma}\left(x, a, c \cdot z + d\right)}\right) \]

            13. lift-*.f64N/A

              \[\leadsto \mathsf{fma}\left(y, b, \mathsf{fma}\left(x, a, \color{blue}{c \cdot z} + d\right)\right) \]

            14. *-commutativeN/A

              \[\leadsto \mathsf{fma}\left(y, b, \mathsf{fma}\left(x, a, \color{blue}{z \cdot c} + d\right)\right) \]

            15. lower-fma.f64100.0

              \[\leadsto \mathsf{fma}\left(y, b, \mathsf{fma}\left(x, a, \color{blue}{\mathsf{fma}\left(z, c, d\right)}\right)\right) \]

          4. Applied rewrites100.0%

            \[\leadsto \color{blue}{\mathsf{fma}\left(y, b, \mathsf{fma}\left(x, a, \mathsf{fma}\left(z, c, d\right)\right)\right)} \]

          5. Taylor expanded in a around inf

            \[\leadsto \color{blue}{a \cdot x} \]
          6. Step-by-step derivation

            1. lower-*.f6484.4

              \[\leadsto \color{blue}{a \cdot x} \]

          7. Applied rewrites84.4%

            \[\leadsto \color{blue}{a \cdot x} \]

          if -4.99999999999999993e-119 < (*.f64 a x) < 4.99999999999999998e-74

          1. Initial program 100.0%

            \[\left(\left(a \cdot x + b \cdot y\right) + c \cdot z\right) + d \]
          2. Add Preprocessing

          3. Taylor expanded in c around 0

            \[\leadsto \color{blue}{d + \left(a \cdot x + b \cdot y\right)} \]
          4. Step-by-step derivation

            1. associate-+r+N/A

              \[\leadsto \color{blue}{\left(d + a \cdot x\right) + b \cdot y} \]

            2. +-commutativeN/A

              \[\leadsto \color{blue}{b \cdot y + \left(d + a \cdot x\right)} \]

            3. *-commutativeN/A

              \[\leadsto \color{blue}{y \cdot b} + \left(d + a \cdot x\right) \]

            4. lower-fma.f64N/A

              \[\leadsto \color{blue}{\mathsf{fma}\left(y, b, d + a \cdot x\right)} \]

            5. +-commutativeN/A

              \[\leadsto \mathsf{fma}\left(y, b, \color{blue}{a \cdot x + d}\right) \]

            6. *-commutativeN/A

              \[\leadsto \mathsf{fma}\left(y, b, \color{blue}{x \cdot a} + d\right) \]

            7. lower-fma.f6486.1

              \[\leadsto \mathsf{fma}\left(y, b, \color{blue}{\mathsf{fma}\left(x, a, d\right)}\right) \]

          5. Applied rewrites86.1%

            \[\leadsto \color{blue}{\mathsf{fma}\left(y, b, \mathsf{fma}\left(x, a, d\right)\right)} \]

          6. Taylor expanded in a around 0

            \[\leadsto d + \color{blue}{b \cdot y} \]
          7. Step-by-step derivation

            1. Applied rewrites81.0%

              \[\leadsto \mathsf{fma}\left(b, \color{blue}{y}, d\right) \]
          8. Recombined 2 regimes into one program.

          9. Final simplification81.2%

            \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot x \leq -5 \cdot 10^{-119} \lor \neg \left(a \cdot x \leq 5 \cdot 10^{-74}\right):\\ \;\;\;\;a \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b, y, d\right)\\ \end{array} \]
          10. Add Preprocessing

          Alternative 6: 23.9% accurate, 0.9× speedup?

          \[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \cdot z \leq -5 \cdot 10^{-220} \lor \neg \left(c \cdot z \leq 5 \cdot 10^{-237}\right):\\ \;\;\;\;z \cdot c\\ \mathbf{else}:\\ \;\;\;\;b \cdot y\\ \end{array} \end{array} \]
          (FPCore (a x b y c z d)
 :precision binary64
 (if (or (<= (* c z) -5e-220) (not (<= (* c z) 5e-237))) (* z c) (* b y)))
          double code(double a, double x, double b, double y, double c, double z, double d) {
	double tmp;
	if (((c * z) <= -5e-220) || !((c * z) <= 5e-237)) {
		tmp = z * c;
	} else {
		tmp = b * y;
	}
	return tmp;
}

          real(8) function code(a, x, b, y, c, z, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: x
    real(8), intent (in) :: b
    real(8), intent (in) :: y
    real(8), intent (in) :: c
    real(8), intent (in) :: z
    real(8), intent (in) :: d
    real(8) :: tmp
    if (((c * z) <= (-5d-220)) .or. (.not. ((c * z) <= 5d-237))) then
        tmp = z * c
    else
        tmp = b * y
    end if
    code = tmp
end function

          public static double code(double a, double x, double b, double y, double c, double z, double d) {
	double tmp;
	if (((c * z) <= -5e-220) || !((c * z) <= 5e-237)) {
		tmp = z * c;
	} else {
		tmp = b * y;
	}
	return tmp;
}

          def code(a, x, b, y, c, z, d):
	tmp = 0
	if ((c * z) <= -5e-220) or not ((c * z) <= 5e-237):
		tmp = z * c
	else:
		tmp = b * y
	return tmp

          function code(a, x, b, y, c, z, d)
	tmp = 0.0
	if ((Float64(c * z) <= -5e-220) || !(Float64(c * z) <= 5e-237))
		tmp = Float64(z * c);
	else
		tmp = Float64(b * y);
	end
	return tmp
end

          function tmp_2 = code(a, x, b, y, c, z, d)
	tmp = 0.0;
	if (((c * z) <= -5e-220) || ~(((c * z) <= 5e-237)))
		tmp = z * c;
	else
		tmp = b * y;
	end
	tmp_2 = tmp;
end

          code[a_, x_, b_, y_, c_, z_, d_] := If[Or[LessEqual[N[(c * z), $MachinePrecision], -5e-220], N[Not[LessEqual[N[(c * z), $MachinePrecision], 5e-237]], $MachinePrecision]], N[(z * c), $MachinePrecision], N[(b * y), $MachinePrecision]]

          \begin{array}{l}

\\
\begin{array}{l}
\mathbf{if}\;c \cdot z \leq -5 \cdot 10^{-220} \lor \neg \left(c \cdot z \leq 5 \cdot 10^{-237}\right):\\
\;\;\;\;z \cdot c\\

\mathbf{else}:\\
\;\;\;\;b \cdot y\\


\end{array}
\end{array}
          Derivation
          1. Split input into 2 regimes

          2. if (*.f64 c z) < -5.0000000000000002e-220 or 5.0000000000000002e-237 < (*.f64 c z)

            1. Initial program 99.9%

              \[\left(\left(a \cdot x + b \cdot y\right) + c \cdot z\right) + d \]
            2. Add Preprocessing

            3. Taylor expanded in c around inf

              \[\leadsto \color{blue}{c \cdot z} \]
            4. Step-by-step derivation

              1. *-commutativeN/A

                \[\leadsto \color{blue}{z \cdot c} \]

              2. lower-*.f6442.0

                \[\leadsto \color{blue}{z \cdot c} \]

            5. Applied rewrites42.0%

              \[\leadsto \color{blue}{z \cdot c} \]

            if -5.0000000000000002e-220 < (*.f64 c z) < 5.0000000000000002e-237

            1. Initial program 100.0%

              \[\left(\left(a \cdot x + b \cdot y\right) + c \cdot z\right) + d \]
            2. Add Preprocessing
            3. Step-by-step derivation

              1. lift-+.f64N/A

                \[\leadsto \color{blue}{\left(\left(a \cdot x + b \cdot y\right) + c \cdot z\right) + d} \]

              2. lift-+.f64N/A

                \[\leadsto \color{blue}{\left(\left(a \cdot x + b \cdot y\right) + c \cdot z\right)} + d \]

              3. associate-+l+N/A

                \[\leadsto \color{blue}{\left(a \cdot x + b \cdot y\right) + \left(c \cdot z + d\right)} \]

              4. lift-+.f64N/A

                \[\leadsto \color{blue}{\left(a \cdot x + b \cdot y\right)} + \left(c \cdot z + d\right) \]

              5. +-commutativeN/A

                \[\leadsto \color{blue}{\left(b \cdot y + a \cdot x\right)} + \left(c \cdot z + d\right) \]

              6. associate-+l+N/A

                \[\leadsto \color{blue}{b \cdot y + \left(a \cdot x + \left(c \cdot z + d\right)\right)} \]

              7. lift-*.f64N/A

                \[\leadsto \color{blue}{b \cdot y} + \left(a \cdot x + \left(c \cdot z + d\right)\right) \]

              8. *-commutativeN/A

                \[\leadsto \color{blue}{y \cdot b} + \left(a \cdot x + \left(c \cdot z + d\right)\right) \]

              9. lower-fma.f64N/A

                \[\leadsto \color{blue}{\mathsf{fma}\left(y, b, a \cdot x + \left(c \cdot z + d\right)\right)} \]

              10. lift-*.f64N/A

                \[\leadsto \mathsf{fma}\left(y, b, \color{blue}{a \cdot x} + \left(c \cdot z + d\right)\right) \]

              11. *-commutativeN/A

                \[\leadsto \mathsf{fma}\left(y, b, \color{blue}{x \cdot a} + \left(c \cdot z + d\right)\right) \]

              12. lower-fma.f64N/A

                \[\leadsto \mathsf{fma}\left(y, b, \color{blue}{\mathsf{fma}\left(x, a, c \cdot z + d\right)}\right) \]

              13. lift-*.f64N/A

                \[\leadsto \mathsf{fma}\left(y, b, \mathsf{fma}\left(x, a, \color{blue}{c \cdot z} + d\right)\right) \]

              14. *-commutativeN/A

                \[\leadsto \mathsf{fma}\left(y, b, \mathsf{fma}\left(x, a, \color{blue}{z \cdot c} + d\right)\right) \]

              15. lower-fma.f64100.0

                \[\leadsto \mathsf{fma}\left(y, b, \mathsf{fma}\left(x, a, \color{blue}{\mathsf{fma}\left(z, c, d\right)}\right)\right) \]

            4. Applied rewrites100.0%

              \[\leadsto \color{blue}{\mathsf{fma}\left(y, b, \mathsf{fma}\left(x, a, \mathsf{fma}\left(z, c, d\right)\right)\right)} \]

            5. Taylor expanded in b around inf

              \[\leadsto \color{blue}{b \cdot y} \]
            6. Step-by-step derivation

              1. lower-*.f6418.4

                \[\leadsto \color{blue}{b \cdot y} \]

            7. Applied rewrites18.4%

              \[\leadsto \color{blue}{b \cdot y} \]
          3. Recombined 2 regimes into one program.

          4. Final simplification25.3%

            \[\leadsto \begin{array}{l} \mathbf{if}\;c \cdot z \leq -5 \cdot 10^{-220} \lor \neg \left(c \cdot z \leq 5 \cdot 10^{-237}\right):\\ \;\;\;\;z \cdot c\\ \mathbf{else}:\\ \;\;\;\;b \cdot y\\ \end{array} \]
          5. Add Preprocessing

          Alternative 7: 23.3% accurate, 0.9× speedup?

          \[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \cdot x \leq -6 \cdot 10^{-233} \lor \neg \left(a \cdot x \leq 7 \cdot 10^{-271}\right):\\ \;\;\;\;a \cdot x\\ \mathbf{else}:\\ \;\;\;\;b \cdot y\\ \end{array} \end{array} \]
          (FPCore (a x b y c z d)
 :precision binary64
 (if (or (<= (* a x) -6e-233) (not (<= (* a x) 7e-271))) (* a x) (* b y)))
          double code(double a, double x, double b, double y, double c, double z, double d) {
	double tmp;
	if (((a * x) <= -6e-233) || !((a * x) <= 7e-271)) {
		tmp = a * x;
	} else {
		tmp = b * y;
	}
	return tmp;
}

          real(8) function code(a, x, b, y, c, z, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: x
    real(8), intent (in) :: b
    real(8), intent (in) :: y
    real(8), intent (in) :: c
    real(8), intent (in) :: z
    real(8), intent (in) :: d
    real(8) :: tmp
    if (((a * x) <= (-6d-233)) .or. (.not. ((a * x) <= 7d-271))) then
        tmp = a * x
    else
        tmp = b * y
    end if
    code = tmp
end function

          public static double code(double a, double x, double b, double y, double c, double z, double d) {
	double tmp;
	if (((a * x) <= -6e-233) || !((a * x) <= 7e-271)) {
		tmp = a * x;
	} else {
		tmp = b * y;
	}
	return tmp;
}

          def code(a, x, b, y, c, z, d):
	tmp = 0
	if ((a * x) <= -6e-233) or not ((a * x) <= 7e-271):
		tmp = a * x
	else:
		tmp = b * y
	return tmp

          function code(a, x, b, y, c, z, d)
	tmp = 0.0
	if ((Float64(a * x) <= -6e-233) || !(Float64(a * x) <= 7e-271))
		tmp = Float64(a * x);
	else
		tmp = Float64(b * y);
	end
	return tmp
end

          function tmp_2 = code(a, x, b, y, c, z, d)
	tmp = 0.0;
	if (((a * x) <= -6e-233) || ~(((a * x) <= 7e-271)))
		tmp = a * x;
	else
		tmp = b * y;
	end
	tmp_2 = tmp;
end

          code[a_, x_, b_, y_, c_, z_, d_] := If[Or[LessEqual[N[(a * x), $MachinePrecision], -6e-233], N[Not[LessEqual[N[(a * x), $MachinePrecision], 7e-271]], $MachinePrecision]], N[(a * x), $MachinePrecision], N[(b * y), $MachinePrecision]]

          \begin{array}{l}

\\
\begin{array}{l}
\mathbf{if}\;a \cdot x \leq -6 \cdot 10^{-233} \lor \neg \left(a \cdot x \leq 7 \cdot 10^{-271}\right):\\
\;\;\;\;a \cdot x\\

\mathbf{else}:\\
\;\;\;\;b \cdot y\\


\end{array}
\end{array}
          Derivation
          1. Split input into 2 regimes

          2. if (*.f64 a x) < -5.99999999999999997e-233 or 6.9999999999999999e-271 < (*.f64 a x)

            1. Initial program 99.9%

              \[\left(\left(a \cdot x + b \cdot y\right) + c \cdot z\right) + d \]
            2. Add Preprocessing
            3. Step-by-step derivation

              1. lift-+.f64N/A

                \[\leadsto \color{blue}{\left(\left(a \cdot x + b \cdot y\right) + c \cdot z\right) + d} \]

              2. lift-+.f64N/A

                \[\leadsto \color{blue}{\left(\left(a \cdot x + b \cdot y\right) + c \cdot z\right)} + d \]

              3. associate-+l+N/A

                \[\leadsto \color{blue}{\left(a \cdot x + b \cdot y\right) + \left(c \cdot z + d\right)} \]

              4. lift-+.f64N/A

                \[\leadsto \color{blue}{\left(a \cdot x + b \cdot y\right)} + \left(c \cdot z + d\right) \]

              5. +-commutativeN/A

                \[\leadsto \color{blue}{\left(b \cdot y + a \cdot x\right)} + \left(c \cdot z + d\right) \]

              6. associate-+l+N/A

                \[\leadsto \color{blue}{b \cdot y + \left(a \cdot x + \left(c \cdot z + d\right)\right)} \]

              7. lift-*.f64N/A

                \[\leadsto \color{blue}{b \cdot y} + \left(a \cdot x + \left(c \cdot z + d\right)\right) \]

              8. *-commutativeN/A

                \[\leadsto \color{blue}{y \cdot b} + \left(a \cdot x + \left(c \cdot z + d\right)\right) \]

              9. lower-fma.f64N/A

                \[\leadsto \color{blue}{\mathsf{fma}\left(y, b, a \cdot x + \left(c \cdot z + d\right)\right)} \]

              10. lift-*.f64N/A

                \[\leadsto \mathsf{fma}\left(y, b, \color{blue}{a \cdot x} + \left(c \cdot z + d\right)\right) \]

              11. *-commutativeN/A

                \[\leadsto \mathsf{fma}\left(y, b, \color{blue}{x \cdot a} + \left(c \cdot z + d\right)\right) \]

              12. lower-fma.f64N/A

                \[\leadsto \mathsf{fma}\left(y, b, \color{blue}{\mathsf{fma}\left(x, a, c \cdot z + d\right)}\right) \]

              13. lift-*.f64N/A

                \[\leadsto \mathsf{fma}\left(y, b, \mathsf{fma}\left(x, a, \color{blue}{c \cdot z} + d\right)\right) \]

              14. *-commutativeN/A

                \[\leadsto \mathsf{fma}\left(y, b, \mathsf{fma}\left(x, a, \color{blue}{z \cdot c} + d\right)\right) \]

              15. lower-fma.f64100.0

                \[\leadsto \mathsf{fma}\left(y, b, \mathsf{fma}\left(x, a, \color{blue}{\mathsf{fma}\left(z, c, d\right)}\right)\right) \]

            4. Applied rewrites100.0%

              \[\leadsto \color{blue}{\mathsf{fma}\left(y, b, \mathsf{fma}\left(x, a, \mathsf{fma}\left(z, c, d\right)\right)\right)} \]

            5. Taylor expanded in a around inf

              \[\leadsto \color{blue}{a \cdot x} \]
            6. Step-by-step derivation

              1. lower-*.f6431.2

                \[\leadsto \color{blue}{a \cdot x} \]

            7. Applied rewrites31.2%

              \[\leadsto \color{blue}{a \cdot x} \]

            if -5.99999999999999997e-233 < (*.f64 a x) < 6.9999999999999999e-271

            1. Initial program 100.0%

              \[\left(\left(a \cdot x + b \cdot y\right) + c \cdot z\right) + d \]
            2. Add Preprocessing
            3. Step-by-step derivation

              1. lift-+.f64N/A

                \[\leadsto \color{blue}{\left(\left(a \cdot x + b \cdot y\right) + c \cdot z\right) + d} \]

              2. lift-+.f64N/A

                \[\leadsto \color{blue}{\left(\left(a \cdot x + b \cdot y\right) + c \cdot z\right)} + d \]

              3. associate-+l+N/A

                \[\leadsto \color{blue}{\left(a \cdot x + b \cdot y\right) + \left(c \cdot z + d\right)} \]

              4. lift-+.f64N/A

                \[\leadsto \color{blue}{\left(a \cdot x + b \cdot y\right)} + \left(c \cdot z + d\right) \]

              5. +-commutativeN/A

                \[\leadsto \color{blue}{\left(b \cdot y + a \cdot x\right)} + \left(c \cdot z + d\right) \]

              6. associate-+l+N/A

                \[\leadsto \color{blue}{b \cdot y + \left(a \cdot x + \left(c \cdot z + d\right)\right)} \]

              7. lift-*.f64N/A

                \[\leadsto \color{blue}{b \cdot y} + \left(a \cdot x + \left(c \cdot z + d\right)\right) \]

              8. *-commutativeN/A

                \[\leadsto \color{blue}{y \cdot b} + \left(a \cdot x + \left(c \cdot z + d\right)\right) \]

              9. lower-fma.f64N/A

                \[\leadsto \color{blue}{\mathsf{fma}\left(y, b, a \cdot x + \left(c \cdot z + d\right)\right)} \]

              10. lift-*.f64N/A

                \[\leadsto \mathsf{fma}\left(y, b, \color{blue}{a \cdot x} + \left(c \cdot z + d\right)\right) \]

              11. *-commutativeN/A

                \[\leadsto \mathsf{fma}\left(y, b, \color{blue}{x \cdot a} + \left(c \cdot z + d\right)\right) \]

              12. lower-fma.f64N/A

                \[\leadsto \mathsf{fma}\left(y, b, \color{blue}{\mathsf{fma}\left(x, a, c \cdot z + d\right)}\right) \]

              13. lift-*.f64N/A

                \[\leadsto \mathsf{fma}\left(y, b, \mathsf{fma}\left(x, a, \color{blue}{c \cdot z} + d\right)\right) \]

              14. *-commutativeN/A

                \[\leadsto \mathsf{fma}\left(y, b, \mathsf{fma}\left(x, a, \color{blue}{z \cdot c} + d\right)\right) \]

              15. lower-fma.f64100.0

                \[\leadsto \mathsf{fma}\left(y, b, \mathsf{fma}\left(x, a, \color{blue}{\mathsf{fma}\left(z, c, d\right)}\right)\right) \]

            4. Applied rewrites100.0%

              \[\leadsto \color{blue}{\mathsf{fma}\left(y, b, \mathsf{fma}\left(x, a, \mathsf{fma}\left(z, c, d\right)\right)\right)} \]

            5. Taylor expanded in b around inf

              \[\leadsto \color{blue}{b \cdot y} \]
            6. Step-by-step derivation

              1. lower-*.f6418.7

                \[\leadsto \color{blue}{b \cdot y} \]

            7. Applied rewrites18.7%

              \[\leadsto \color{blue}{b \cdot y} \]
          3. Recombined 2 regimes into one program.

          4. Final simplification22.6%

            \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot x \leq -6 \cdot 10^{-233} \lor \neg \left(a \cdot x \leq 7 \cdot 10^{-271}\right):\\ \;\;\;\;a \cdot x\\ \mathbf{else}:\\ \;\;\;\;b \cdot y\\ \end{array} \]
          5. Add Preprocessing

          Alternative 8: 90.3% accurate, 1.0× speedup?

          \[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1 \cdot 10^{-54} \lor \neg \left(b \leq 1.5 \cdot 10^{-96}\right):\\ \;\;\;\;\mathsf{fma}\left(b, y, d\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, x, \mathsf{fma}\left(c, z, d\right)\right)\\ \end{array} \end{array} \]
          (FPCore (a x b y c z d)
 :precision binary64
 (if (or (<= b -1e-54) (not (<= b 1.5e-96)))
   (fma b y d)
   (fma a x (fma c z d))))
          double code(double a, double x, double b, double y, double c, double z, double d) {
	double tmp;
	if ((b <= -1e-54) || !(b <= 1.5e-96)) {
		tmp = fma(b, y, d);
	} else {
		tmp = fma(a, x, fma(c, z, d));
	}
	return tmp;
}

          function code(a, x, b, y, c, z, d)
	tmp = 0.0
	if ((b <= -1e-54) || !(b <= 1.5e-96))
		tmp = fma(b, y, d);
	else
		tmp = fma(a, x, fma(c, z, d));
	end
	return tmp
end

          code[a_, x_, b_, y_, c_, z_, d_] := If[Or[LessEqual[b, -1e-54], N[Not[LessEqual[b, 1.5e-96]], $MachinePrecision]], N[(b * y + d), $MachinePrecision], N[(a * x + N[(c * z + d), $MachinePrecision]), $MachinePrecision]]

          \begin{array}{l}

\\
\begin{array}{l}
\mathbf{if}\;b \leq -1 \cdot 10^{-54} \lor \neg \left(b \leq 1.5 \cdot 10^{-96}\right):\\
\;\;\;\;\mathsf{fma}\left(b, y, d\right)\\

\mathbf{else}:\\
\;\;\;\;\mathsf{fma}\left(a, x, \mathsf{fma}\left(c, z, d\right)\right)\\


\end{array}
\end{array}
          Derivation
          1. Split input into 2 regimes

          2. if b < -1e-54 or 1.5e-96 < b

            1. Initial program 100.0%

              \[\left(\left(a \cdot x + b \cdot y\right) + c \cdot z\right) + d \]
            2. Add Preprocessing

            3. Taylor expanded in c around 0

              \[\leadsto \color{blue}{d + \left(a \cdot x + b \cdot y\right)} \]
            4. Step-by-step derivation

              1. associate-+r+N/A

                \[\leadsto \color{blue}{\left(d + a \cdot x\right) + b \cdot y} \]

              2. +-commutativeN/A

                \[\leadsto \color{blue}{b \cdot y + \left(d + a \cdot x\right)} \]

              3. *-commutativeN/A

                \[\leadsto \color{blue}{y \cdot b} + \left(d + a \cdot x\right) \]

              4. lower-fma.f64N/A

                \[\leadsto \color{blue}{\mathsf{fma}\left(y, b, d + a \cdot x\right)} \]

              5. +-commutativeN/A

                \[\leadsto \mathsf{fma}\left(y, b, \color{blue}{a \cdot x + d}\right) \]

              6. *-commutativeN/A

                \[\leadsto \mathsf{fma}\left(y, b, \color{blue}{x \cdot a} + d\right) \]

              7. lower-fma.f6489.4

                \[\leadsto \mathsf{fma}\left(y, b, \color{blue}{\mathsf{fma}\left(x, a, d\right)}\right) \]

            5. Applied rewrites89.4%

              \[\leadsto \color{blue}{\mathsf{fma}\left(y, b, \mathsf{fma}\left(x, a, d\right)\right)} \]

            6. Taylor expanded in a around 0

              \[\leadsto d + \color{blue}{b \cdot y} \]
            7. Step-by-step derivation

              1. Applied rewrites78.5%

                \[\leadsto \mathsf{fma}\left(b, \color{blue}{y}, d\right) \]

              if -1e-54 < b < 1.5e-96

              1. Initial program 100.0%

                \[\left(\left(a \cdot x + b \cdot y\right) + c \cdot z\right) + d \]
              2. Add Preprocessing

              3. Taylor expanded in b around 0

                \[\leadsto \color{blue}{d + \left(a \cdot x + c \cdot z\right)} \]
              4. Step-by-step derivation

                1. associate-+r+N/A

                  \[\leadsto \color{blue}{\left(d + a \cdot x\right) + c \cdot z} \]

                2. +-commutativeN/A

                  \[\leadsto \color{blue}{c \cdot z + \left(d + a \cdot x\right)} \]

                3. *-commutativeN/A

                  \[\leadsto \color{blue}{z \cdot c} + \left(d + a \cdot x\right) \]

                4. lower-fma.f64N/A

                  \[\leadsto \color{blue}{\mathsf{fma}\left(z, c, d + a \cdot x\right)} \]

                5. +-commutativeN/A

                  \[\leadsto \mathsf{fma}\left(z, c, \color{blue}{a \cdot x + d}\right) \]

                6. *-commutativeN/A

                  \[\leadsto \mathsf{fma}\left(z, c, \color{blue}{x \cdot a} + d\right) \]

                7. lower-fma.f6494.3

                  \[\leadsto \mathsf{fma}\left(z, c, \color{blue}{\mathsf{fma}\left(x, a, d\right)}\right) \]

              5. Applied rewrites94.3%

                \[\leadsto \color{blue}{\mathsf{fma}\left(z, c, \mathsf{fma}\left(x, a, d\right)\right)} \]
              6. Step-by-step derivation

                1. Applied rewrites94.3%

                  \[\leadsto \color{blue}{\mathsf{fma}\left(a, x, \mathsf{fma}\left(c, z, d\right)\right)} \]
              7. Recombined 2 regimes into one program.

              8. Final simplification90.5%

                \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1 \cdot 10^{-54} \lor \neg \left(b \leq 1.5 \cdot 10^{-96}\right):\\ \;\;\;\;\mathsf{fma}\left(b, y, d\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, x, \mathsf{fma}\left(c, z, d\right)\right)\\ \end{array} \]
              9. Add Preprocessing

              Alternative 9: 15.8% accurate, 4.2× speedup?

              \[\begin{array}{l} \\ a \cdot x \end{array} \]
              (FPCore (a x b y c z d) :precision binary64 (* a x))
              double code(double a, double x, double b, double y, double c, double z, double d) {
	return a * x;
}

              real(8) function code(a, x, b, y, c, z, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: x
    real(8), intent (in) :: b
    real(8), intent (in) :: y
    real(8), intent (in) :: c
    real(8), intent (in) :: z
    real(8), intent (in) :: d
    code = a * x
end function

              public static double code(double a, double x, double b, double y, double c, double z, double d) {
	return a * x;
}

              def code(a, x, b, y, c, z, d):
	return a * x

              function code(a, x, b, y, c, z, d)
	return Float64(a * x)
end

              function tmp = code(a, x, b, y, c, z, d)
	tmp = a * x;
end

              code[a_, x_, b_, y_, c_, z_, d_] := N[(a * x), $MachinePrecision]

              \begin{array}{l}

\\
a \cdot x
\end{array}
              Derivation

              1. Initial program 100.0%

                \[\left(\left(a \cdot x + b \cdot y\right) + c \cdot z\right) + d \]
              2. Add Preprocessing
              3. Step-by-step derivation

                1. lift-+.f64N/A

                  \[\leadsto \color{blue}{\left(\left(a \cdot x + b \cdot y\right) + c \cdot z\right) + d} \]

                2. lift-+.f64N/A

                  \[\leadsto \color{blue}{\left(\left(a \cdot x + b \cdot y\right) + c \cdot z\right)} + d \]

                3. associate-+l+N/A

                  \[\leadsto \color{blue}{\left(a \cdot x + b \cdot y\right) + \left(c \cdot z + d\right)} \]

                4. lift-+.f64N/A

                  \[\leadsto \color{blue}{\left(a \cdot x + b \cdot y\right)} + \left(c \cdot z + d\right) \]

                5. +-commutativeN/A

                  \[\leadsto \color{blue}{\left(b \cdot y + a \cdot x\right)} + \left(c \cdot z + d\right) \]

                6. associate-+l+N/A

                  \[\leadsto \color{blue}{b \cdot y + \left(a \cdot x + \left(c \cdot z + d\right)\right)} \]

                7. lift-*.f64N/A

                  \[\leadsto \color{blue}{b \cdot y} + \left(a \cdot x + \left(c \cdot z + d\right)\right) \]

                8. *-commutativeN/A

                  \[\leadsto \color{blue}{y \cdot b} + \left(a \cdot x + \left(c \cdot z + d\right)\right) \]

                9. lower-fma.f64N/A

                  \[\leadsto \color{blue}{\mathsf{fma}\left(y, b, a \cdot x + \left(c \cdot z + d\right)\right)} \]

                10. lift-*.f64N/A

                  \[\leadsto \mathsf{fma}\left(y, b, \color{blue}{a \cdot x} + \left(c \cdot z + d\right)\right) \]

                11. *-commutativeN/A

                  \[\leadsto \mathsf{fma}\left(y, b, \color{blue}{x \cdot a} + \left(c \cdot z + d\right)\right) \]

                12. lower-fma.f64N/A

                  \[\leadsto \mathsf{fma}\left(y, b, \color{blue}{\mathsf{fma}\left(x, a, c \cdot z + d\right)}\right) \]

                13. lift-*.f64N/A

                  \[\leadsto \mathsf{fma}\left(y, b, \mathsf{fma}\left(x, a, \color{blue}{c \cdot z} + d\right)\right) \]

                14. *-commutativeN/A

                  \[\leadsto \mathsf{fma}\left(y, b, \mathsf{fma}\left(x, a, \color{blue}{z \cdot c} + d\right)\right) \]

                15. lower-fma.f64100.0

                  \[\leadsto \mathsf{fma}\left(y, b, \mathsf{fma}\left(x, a, \color{blue}{\mathsf{fma}\left(z, c, d\right)}\right)\right) \]

              4. Applied rewrites100.0%

                \[\leadsto \color{blue}{\mathsf{fma}\left(y, b, \mathsf{fma}\left(x, a, \mathsf{fma}\left(z, c, d\right)\right)\right)} \]

              5. Taylor expanded in a around inf

                \[\leadsto \color{blue}{a \cdot x} \]
              6. Step-by-step derivation

                1. lower-*.f6413.3

                  \[\leadsto \color{blue}{a \cdot x} \]

              7. Applied rewrites13.3%

                \[\leadsto \color{blue}{a \cdot x} \]
              8. Add Preprocessing

              Reproduce

              ?
              herbie shell --seed 1 
(FPCore (a x b y c z d)
  :name "a*x+b*y+c*z+d"
  :precision binary64
  :pre (and (and (and (and (and (and (and (<= -1.0 a) (<= a 1.0)) (and (<= -1000.0 x) (<= x 1000.0))) (and (<= -1.0 b) (<= b 1.0))) (and (<= -1000.0 y) (<= y 1000.0))) (and (<= -1.0 c) (<= c 1.0))) (and (<= -1000.0 z) (<= z 1000.0))) (and (<= -1000.0 d) (<= d 1000.0)))
  (+ (+ (+ (* a x) (* b y)) (* c z)) d))