Statistics.Distribution.Beta:$cvariance from math-functions-0.1.5.2

Percentage Accurate: 83.0% → 95.8%

Time: 12.0s

Alternatives: 17

Speedup: 1.0×

• 25.4% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ \frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (/ (* x y) (* (* z z) (+ z 1.0))))

C

double code(double x, double y, double z) {
	return (x * y) / ((z * z) * (z + 1.0));
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = (x * y) / ((z * z) * (z + 1.0d0))
end function

Java

public static double code(double x, double y, double z) {
	return (x * y) / ((z * z) * (z + 1.0));
}

Python

def code(x, y, z):
	return (x * y) / ((z * z) * (z + 1.0))

Julia

function code(x, y, z)
	return Float64(Float64(x * y) / Float64(Float64(z * z) * Float64(z + 1.0)))
end

MATLAB

function tmp = code(x, y, z)
	tmp = (x * y) / ((z * z) * (z + 1.0));
end

Wolfram

code[x_, y_, z_] := N[(N[(x * y), $MachinePrecision] / N[(N[(z * z), $MachinePrecision] * N[(z + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 83.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (/ (* x y) (* (* z z) (+ z 1.0))))

C

double code(double x, double y, double z) {
	return (x * y) / ((z * z) * (z + 1.0));
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = (x * y) / ((z * z) * (z + 1.0d0))
end function

Java

public static double code(double x, double y, double z) {
	return (x * y) / ((z * z) * (z + 1.0));
}

Python

def code(x, y, z):
	return (x * y) / ((z * z) * (z + 1.0))

Julia

function code(x, y, z)
	return Float64(Float64(x * y) / Float64(Float64(z * z) * Float64(z + 1.0)))
end

MATLAB

function tmp = code(x, y, z)
	tmp = (x * y) / ((z * z) * (z + 1.0));
end

Wolfram

code[x_, y_, z_] := N[(N[(x * y), $MachinePrecision] / N[(N[(z * z), $MachinePrecision] * N[(z + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 95.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(z + 1\right) \cdot \left(z \cdot z\right)\\ \mathbf{if}\;t_0 \leq -1 \cdot 10^{+20}:\\ \;\;\;\;\frac{\frac{y}{z}}{z \cdot \frac{z}{x}}\\ \mathbf{elif}\;t_0 \leq 5 \cdot 10^{-153}:\\ \;\;\;\;\frac{\frac{x}{z}}{\frac{z}{y}}\\ \mathbf{elif}\;t_0 \leq 2 \cdot 10^{+140}:\\ \;\;\;\;\frac{x \cdot y}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{z \cdot \left(\frac{z}{x} \cdot \frac{z}{y}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (+ z 1.0) (* z z))))
   (if (<= t_0 -1e+20)
     (/ (/ y z) (* z (/ z x)))
     (if (<= t_0 5e-153)
       (/ (/ x z) (/ z y))
       (if (<= t_0 2e+140)
         (/ (* x y) t_0)
         (/ 1.0 (* z (* (/ z x) (/ z y)))))))))

C

double code(double x, double y, double z) {
	double t_0 = (z + 1.0) * (z * z);
	double tmp;
	if (t_0 <= -1e+20) {
		tmp = (y / z) / (z * (z / x));
	} else if (t_0 <= 5e-153) {
		tmp = (x / z) / (z / y);
	} else if (t_0 <= 2e+140) {
		tmp = (x * y) / t_0;
	} else {
		tmp = 1.0 / (z * ((z / x) * (z / y)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (z + 1.0d0) * (z * z)
    if (t_0 <= (-1d+20)) then
        tmp = (y / z) / (z * (z / x))
    else if (t_0 <= 5d-153) then
        tmp = (x / z) / (z / y)
    else if (t_0 <= 2d+140) then
        tmp = (x * y) / t_0
    else
        tmp = 1.0d0 / (z * ((z / x) * (z / y)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = (z + 1.0) * (z * z);
	double tmp;
	if (t_0 <= -1e+20) {
		tmp = (y / z) / (z * (z / x));
	} else if (t_0 <= 5e-153) {
		tmp = (x / z) / (z / y);
	} else if (t_0 <= 2e+140) {
		tmp = (x * y) / t_0;
	} else {
		tmp = 1.0 / (z * ((z / x) * (z / y)));
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (z + 1.0) * (z * z)
	tmp = 0
	if t_0 <= -1e+20:
		tmp = (y / z) / (z * (z / x))
	elif t_0 <= 5e-153:
		tmp = (x / z) / (z / y)
	elif t_0 <= 2e+140:
		tmp = (x * y) / t_0
	else:
		tmp = 1.0 / (z * ((z / x) * (z / y)))
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(z + 1.0) * Float64(z * z))
	tmp = 0.0
	if (t_0 <= -1e+20)
		tmp = Float64(Float64(y / z) / Float64(z * Float64(z / x)));
	elseif (t_0 <= 5e-153)
		tmp = Float64(Float64(x / z) / Float64(z / y));
	elseif (t_0 <= 2e+140)
		tmp = Float64(Float64(x * y) / t_0);
	else
		tmp = Float64(1.0 / Float64(z * Float64(Float64(z / x) * Float64(z / y))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (z + 1.0) * (z * z);
	tmp = 0.0;
	if (t_0 <= -1e+20)
		tmp = (y / z) / (z * (z / x));
	elseif (t_0 <= 5e-153)
		tmp = (x / z) / (z / y);
	elseif (t_0 <= 2e+140)
		tmp = (x * y) / t_0;
	else
		tmp = 1.0 / (z * ((z / x) * (z / y)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(z + 1.0), $MachinePrecision] * N[(z * z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -1e+20], N[(N[(y / z), $MachinePrecision] / N[(z * N[(z / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 5e-153], N[(N[(x / z), $MachinePrecision] / N[(z / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 2e+140], N[(N[(x * y), $MachinePrecision] / t$95$0), $MachinePrecision], N[(1.0 / N[(z * N[(N[(z / x), $MachinePrecision] * N[(z / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 (*.f64 z z) (+.f64 z 1)) < -1e20

   1. Initial program 81.6%

      \[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \]
   2. Step-by-step derivation

      1. associate-*l* 81.6%

         \[\leadsto \frac{x \cdot y}{\color{blue}{z \cdot \left(z \cdot \left(z + 1\right)\right)}} \]

      2. times-frac 93.6%

         \[\leadsto \color{blue}{\frac{x}{z} \cdot \frac{y}{z \cdot \left(z + 1\right)}} \]

      3. distribute-lft-in 93.6%

         \[\leadsto \frac{x}{z} \cdot \frac{y}{\color{blue}{z \cdot z + z \cdot 1}} \]

      4. fma-def 93.6%

         \[\leadsto \frac{x}{z} \cdot \frac{y}{\color{blue}{\mathsf{fma}\left(z, z, z \cdot 1\right)}} \]

      5. *-rgt-identity 93.6%

         \[\leadsto \frac{x}{z} \cdot \frac{y}{\mathsf{fma}\left(z, z, \color{blue}{z}\right)} \]

   3. Simplified 93.6%

      \[\leadsto \color{blue}{\frac{x}{z} \cdot \frac{y}{\mathsf{fma}\left(z, z, z\right)}} \]

   4. Taylor expanded in z around inf 92.9%

      \[\leadsto \frac{x}{z} \cdot \color{blue}{\frac{y}{{z}^{2}}} \]
   5. Step-by-step derivation

      1. unpow2 92.9%

         \[\leadsto \frac{x}{z} \cdot \frac{y}{\color{blue}{z \cdot z}} \]

      2. associate-/r* 96.9%

         \[\leadsto \frac{x}{z} \cdot \color{blue}{\frac{\frac{y}{z}}{z}} \]

   6. Simplified 96.9%

      \[\leadsto \frac{x}{z} \cdot \color{blue}{\frac{\frac{y}{z}}{z}} \]
   7. Step-by-step derivation

      1. clear-num 96.9%

         \[\leadsto \color{blue}{\frac{1}{\frac{z}{x}}} \cdot \frac{\frac{y}{z}}{z} \]

      2. frac-times 99.1%

         \[\leadsto \color{blue}{\frac{1 \cdot \frac{y}{z}}{\frac{z}{x} \cdot z}} \]

      3. *-un-lft-identity 99.1%

         \[\leadsto \frac{\color{blue}{\frac{y}{z}}}{\frac{z}{x} \cdot z} \]

   8. Applied egg-rr 99.1%

      \[\leadsto \color{blue}{\frac{\frac{y}{z}}{\frac{z}{x} \cdot z}} \]

   if -1e20 < (*.f64 (*.f64 z z) (+.f64 z 1)) < 5.00000000000000033e-153

   1. Initial program 82.0%

      \[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \]
   2. Step-by-step derivation

      1. associate-*l* 82.0%

         \[\leadsto \frac{x \cdot y}{\color{blue}{z \cdot \left(z \cdot \left(z + 1\right)\right)}} \]

      2. times-frac 98.8%

         \[\leadsto \color{blue}{\frac{x}{z} \cdot \frac{y}{z \cdot \left(z + 1\right)}} \]

      3. distribute-lft-in 98.8%

         \[\leadsto \frac{x}{z} \cdot \frac{y}{\color{blue}{z \cdot z + z \cdot 1}} \]

      4. fma-def 98.8%

         \[\leadsto \frac{x}{z} \cdot \frac{y}{\color{blue}{\mathsf{fma}\left(z, z, z \cdot 1\right)}} \]

      5. *-rgt-identity 98.8%

         \[\leadsto \frac{x}{z} \cdot \frac{y}{\mathsf{fma}\left(z, z, \color{blue}{z}\right)} \]

   3. Simplified 98.8%

      \[\leadsto \color{blue}{\frac{x}{z} \cdot \frac{y}{\mathsf{fma}\left(z, z, z\right)}} \]

   4. Taylor expanded in z around 0 98.8%

      \[\leadsto \frac{x}{z} \cdot \color{blue}{\frac{y}{z}} \]
   5. Step-by-step derivation

      1. clear-num 98.8%

         \[\leadsto \frac{x}{z} \cdot \color{blue}{\frac{1}{\frac{z}{y}}} \]

      2. un-div-inv 98.9%

         \[\leadsto \color{blue}{\frac{\frac{x}{z}}{\frac{z}{y}}} \]

   6. Applied egg-rr 98.9%

      \[\leadsto \color{blue}{\frac{\frac{x}{z}}{\frac{z}{y}}} \]

   if 5.00000000000000033e-153 < (*.f64 (*.f64 z z) (+.f64 z 1)) < 2.00000000000000012e140

   1. Initial program 99.6%

      \[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \]

   if 2.00000000000000012e140 < (*.f64 (*.f64 z z) (+.f64 z 1))

   1. Initial program 86.4%

      \[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \]
   2. Step-by-step derivation

      1. /-rgt-identity 86.4%

         \[\leadsto \frac{\color{blue}{\frac{x \cdot y}{1}}}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \]

      2. associate-/l* 86.4%

         \[\leadsto \frac{\color{blue}{\frac{x}{\frac{1}{y}}}}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \]

      3. associate-/l/ 84.7%

         \[\leadsto \color{blue}{\frac{x}{\left(\left(z \cdot z\right) \cdot \left(z + 1\right)\right) \cdot \frac{1}{y}}} \]

      4. associate-*l* 86.9%

         \[\leadsto \frac{x}{\color{blue}{\left(z \cdot z\right) \cdot \left(\left(z + 1\right) \cdot \frac{1}{y}\right)}} \]

      5. associate-*r/ 87.0%

         \[\leadsto \frac{x}{\left(z \cdot z\right) \cdot \color{blue}{\frac{\left(z + 1\right) \cdot 1}{y}}} \]

      6. *-rgt-identity 87.0%

         \[\leadsto \frac{x}{\left(z \cdot z\right) \cdot \frac{\color{blue}{z + 1}}{y}} \]

      7. associate-*l* 89.1%

         \[\leadsto \frac{x}{\color{blue}{z \cdot \left(z \cdot \frac{z + 1}{y}\right)}} \]

      8. associate-*r/ 87.0%

         \[\leadsto \frac{x}{z \cdot \color{blue}{\frac{z \cdot \left(z + 1\right)}{y}}} \]

      9. distribute-lft-in 87.0%

         \[\leadsto \frac{x}{z \cdot \frac{\color{blue}{z \cdot z + z \cdot 1}}{y}} \]

      10. fma-def 87.0%

          \[\leadsto \frac{x}{z \cdot \frac{\color{blue}{\mathsf{fma}\left(z, z, z \cdot 1\right)}}{y}} \]

      11. *-rgt-identity 87.0%

          \[\leadsto \frac{x}{z \cdot \frac{\mathsf{fma}\left(z, z, \color{blue}{z}\right)}{y}} \]

   3. Simplified 87.0%

      \[\leadsto \color{blue}{\frac{x}{z \cdot \frac{\mathsf{fma}\left(z, z, z\right)}{y}}} \]

   4. Taylor expanded in z around inf 87.0%

      \[\leadsto \frac{x}{z \cdot \color{blue}{\frac{{z}^{2}}{y}}} \]
   5. Step-by-step derivation

      1. unpow2 87.0%

         \[\leadsto \frac{x}{z \cdot \frac{\color{blue}{z \cdot z}}{y}} \]

   6. Simplified 87.0%

      \[\leadsto \frac{x}{z \cdot \color{blue}{\frac{z \cdot z}{y}}} \]
   7. Step-by-step derivation

      1. associate-/r* 89.0%

         \[\leadsto \color{blue}{\frac{\frac{x}{z}}{\frac{z \cdot z}{y}}} \]

      2. add-cube-cbrt 88.8%

         \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{\frac{x}{z}} \cdot \sqrt[3]{\frac{x}{z}}\right) \cdot \sqrt[3]{\frac{x}{z}}}}{\frac{z \cdot z}{y}} \]

      3. *-un-lft-identity 88.8%

         \[\leadsto \frac{\left(\sqrt[3]{\frac{x}{z}} \cdot \sqrt[3]{\frac{x}{z}}\right) \cdot \sqrt[3]{\frac{x}{z}}}{\color{blue}{1 \cdot \frac{z \cdot z}{y}}} \]

      4. times-frac 88.9%

         \[\leadsto \color{blue}{\frac{\sqrt[3]{\frac{x}{z}} \cdot \sqrt[3]{\frac{x}{z}}}{1} \cdot \frac{\sqrt[3]{\frac{x}{z}}}{\frac{z \cdot z}{y}}} \]

      5. pow2 88.9%

         \[\leadsto \frac{\color{blue}{{\left(\sqrt[3]{\frac{x}{z}}\right)}^{2}}}{1} \cdot \frac{\sqrt[3]{\frac{x}{z}}}{\frac{z \cdot z}{y}} \]

      6. div-inv 88.9%

         \[\leadsto \frac{{\left(\sqrt[3]{\frac{x}{z}}\right)}^{2}}{1} \cdot \frac{\sqrt[3]{\frac{x}{z}}}{\color{blue}{\left(z \cdot z\right) \cdot \frac{1}{y}}} \]

      7. associate-*l* 95.3%

         \[\leadsto \frac{{\left(\sqrt[3]{\frac{x}{z}}\right)}^{2}}{1} \cdot \frac{\sqrt[3]{\frac{x}{z}}}{\color{blue}{z \cdot \left(z \cdot \frac{1}{y}\right)}} \]

      8. div-inv 95.3%

         \[\leadsto \frac{{\left(\sqrt[3]{\frac{x}{z}}\right)}^{2}}{1} \cdot \frac{\sqrt[3]{\frac{x}{z}}}{z \cdot \color{blue}{\frac{z}{y}}} \]

   8. Applied egg-rr 95.3%

      \[\leadsto \color{blue}{\frac{{\left(\sqrt[3]{\frac{x}{z}}\right)}^{2}}{1} \cdot \frac{\sqrt[3]{\frac{x}{z}}}{z \cdot \frac{z}{y}}} \]
   9. Step-by-step derivation

      1. times-frac 95.2%

         \[\leadsto \color{blue}{\frac{{\left(\sqrt[3]{\frac{x}{z}}\right)}^{2} \cdot \sqrt[3]{\frac{x}{z}}}{1 \cdot \left(z \cdot \frac{z}{y}\right)}} \]

      2. *-lft-identity 95.2%

         \[\leadsto \frac{{\left(\sqrt[3]{\frac{x}{z}}\right)}^{2} \cdot \sqrt[3]{\frac{x}{z}}}{\color{blue}{z \cdot \frac{z}{y}}} \]

      3. unpow2 95.2%

         \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{\frac{x}{z}} \cdot \sqrt[3]{\frac{x}{z}}\right)} \cdot \sqrt[3]{\frac{x}{z}}}{z \cdot \frac{z}{y}} \]

      4. rem-3cbrt-lft 95.5%

         \[\leadsto \frac{\color{blue}{\frac{x}{z}}}{z \cdot \frac{z}{y}} \]

   10. Simplified 95.5%

       \[\leadsto \color{blue}{\frac{\frac{x}{z}}{z \cdot \frac{z}{y}}} \]
   11. Step-by-step derivation

       1. associate-/r* 97.8%

          \[\leadsto \color{blue}{\frac{\frac{\frac{x}{z}}{z}}{\frac{z}{y}}} \]

       2. associate-/r/ 93.4%

          \[\leadsto \color{blue}{\frac{\frac{\frac{x}{z}}{z}}{z} \cdot y} \]

       3. associate-/l/ 91.2%

          \[\leadsto \frac{\color{blue}{\frac{x}{z \cdot z}}}{z} \cdot y \]

   12. Applied egg-rr 91.2%

       \[\leadsto \color{blue}{\frac{\frac{x}{z \cdot z}}{z} \cdot y} \]
   13. Step-by-step derivation

       1. associate-*l/ 91.3%

          \[\leadsto \color{blue}{\frac{\frac{x}{z \cdot z} \cdot y}{z}} \]

       2. associate-/r* 97.8%

          \[\leadsto \frac{\color{blue}{\frac{\frac{x}{z}}{z}} \cdot y}{z} \]

       3. associate-/r/ 99.9%

          \[\leadsto \frac{\color{blue}{\frac{\frac{x}{z}}{\frac{z}{y}}}}{z} \]

       4. clear-num 99.8%

          \[\leadsto \frac{\color{blue}{\frac{1}{\frac{\frac{z}{y}}{\frac{x}{z}}}}}{z} \]

       5. associate-/l/ 99.9%

          \[\leadsto \color{blue}{\frac{1}{z \cdot \frac{\frac{z}{y}}{\frac{x}{z}}}} \]

       6. div-inv 99.8%

          \[\leadsto \frac{1}{z \cdot \color{blue}{\left(\frac{z}{y} \cdot \frac{1}{\frac{x}{z}}\right)}} \]

       7. clear-num 99.8%

          \[\leadsto \frac{1}{z \cdot \left(\frac{z}{y} \cdot \color{blue}{\frac{z}{x}}\right)} \]

   14. Applied egg-rr 99.8%

       \[\leadsto \color{blue}{\frac{1}{z \cdot \left(\frac{z}{y} \cdot \frac{z}{x}\right)}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(z + 1\right) \cdot \left(z \cdot z\right) \leq -1 \cdot 10^{+20}:\\ \;\;\;\;\frac{\frac{y}{z}}{z \cdot \frac{z}{x}}\\ \mathbf{elif}\;\left(z + 1\right) \cdot \left(z \cdot z\right) \leq 5 \cdot 10^{-153}:\\ \;\;\;\;\frac{\frac{x}{z}}{\frac{z}{y}}\\ \mathbf{elif}\;\left(z + 1\right) \cdot \left(z \cdot z\right) \leq 2 \cdot 10^{+140}:\\ \;\;\;\;\frac{x \cdot y}{\left(z + 1\right) \cdot \left(z \cdot z\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{z \cdot \left(\frac{z}{x} \cdot \frac{z}{y}\right)}\\ \end{array} \]

Alternative 2: 94.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1:\\ \;\;\;\;\frac{\frac{y}{z}}{z \cdot \frac{z}{x}}\\ \mathbf{elif}\;z \leq 0.75:\\ \;\;\;\;\frac{x}{z} \cdot \left(\frac{y}{z} - y\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{z \cdot \left(\frac{z}{x} \cdot \frac{z}{y}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -1.0)
   (/ (/ y z) (* z (/ z x)))
   (if (<= z 0.75)
     (* (/ x z) (- (/ y z) y))
     (/ 1.0 (* z (* (/ z x) (/ z y)))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -1.0) {
		tmp = (y / z) / (z * (z / x));
	} else if (z <= 0.75) {
		tmp = (x / z) * ((y / z) - y);
	} else {
		tmp = 1.0 / (z * ((z / x) * (z / y)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (z <= (-1.0d0)) then
        tmp = (y / z) / (z * (z / x))
    else if (z <= 0.75d0) then
        tmp = (x / z) * ((y / z) - y)
    else
        tmp = 1.0d0 / (z * ((z / x) * (z / y)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -1.0) {
		tmp = (y / z) / (z * (z / x));
	} else if (z <= 0.75) {
		tmp = (x / z) * ((y / z) - y);
	} else {
		tmp = 1.0 / (z * ((z / x) * (z / y)));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -1.0:
		tmp = (y / z) / (z * (z / x))
	elif z <= 0.75:
		tmp = (x / z) * ((y / z) - y)
	else:
		tmp = 1.0 / (z * ((z / x) * (z / y)))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -1.0)
		tmp = Float64(Float64(y / z) / Float64(z * Float64(z / x)));
	elseif (z <= 0.75)
		tmp = Float64(Float64(x / z) * Float64(Float64(y / z) - y));
	else
		tmp = Float64(1.0 / Float64(z * Float64(Float64(z / x) * Float64(z / y))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -1.0)
		tmp = (y / z) / (z * (z / x));
	elseif (z <= 0.75)
		tmp = (x / z) * ((y / z) - y);
	else
		tmp = 1.0 / (z * ((z / x) * (z / y)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -1.0], N[(N[(y / z), $MachinePrecision] / N[(z * N[(z / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 0.75], N[(N[(x / z), $MachinePrecision] * N[(N[(y / z), $MachinePrecision] - y), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(z * N[(N[(z / x), $MachinePrecision] * N[(z / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -1

   1. Initial program 81.6%

      \[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \]
   2. Step-by-step derivation

      1. associate-*l* 81.6%

         \[\leadsto \frac{x \cdot y}{\color{blue}{z \cdot \left(z \cdot \left(z + 1\right)\right)}} \]

      2. times-frac 93.6%

         \[\leadsto \color{blue}{\frac{x}{z} \cdot \frac{y}{z \cdot \left(z + 1\right)}} \]

      3. distribute-lft-in 93.6%

         \[\leadsto \frac{x}{z} \cdot \frac{y}{\color{blue}{z \cdot z + z \cdot 1}} \]

      4. fma-def 93.6%

         \[\leadsto \frac{x}{z} \cdot \frac{y}{\color{blue}{\mathsf{fma}\left(z, z, z \cdot 1\right)}} \]

      5. *-rgt-identity 93.6%

         \[\leadsto \frac{x}{z} \cdot \frac{y}{\mathsf{fma}\left(z, z, \color{blue}{z}\right)} \]

   3. Simplified 93.6%

      \[\leadsto \color{blue}{\frac{x}{z} \cdot \frac{y}{\mathsf{fma}\left(z, z, z\right)}} \]

   4. Taylor expanded in z around inf 92.9%

      \[\leadsto \frac{x}{z} \cdot \color{blue}{\frac{y}{{z}^{2}}} \]
   5. Step-by-step derivation

      1. unpow2 92.9%

         \[\leadsto \frac{x}{z} \cdot \frac{y}{\color{blue}{z \cdot z}} \]

      2. associate-/r* 96.9%

         \[\leadsto \frac{x}{z} \cdot \color{blue}{\frac{\frac{y}{z}}{z}} \]

   6. Simplified 96.9%

      \[\leadsto \frac{x}{z} \cdot \color{blue}{\frac{\frac{y}{z}}{z}} \]
   7. Step-by-step derivation

      1. clear-num 96.9%

         \[\leadsto \color{blue}{\frac{1}{\frac{z}{x}}} \cdot \frac{\frac{y}{z}}{z} \]

      2. frac-times 99.1%

         \[\leadsto \color{blue}{\frac{1 \cdot \frac{y}{z}}{\frac{z}{x} \cdot z}} \]

      3. *-un-lft-identity 99.1%

         \[\leadsto \frac{\color{blue}{\frac{y}{z}}}{\frac{z}{x} \cdot z} \]

   8. Applied egg-rr 99.1%

      \[\leadsto \color{blue}{\frac{\frac{y}{z}}{\frac{z}{x} \cdot z}} \]

   if -1 < z < 0.75

   1. Initial program 86.7%

      \[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \]
   2. Step-by-step derivation

      1. associate-*l* 86.8%

         \[\leadsto \frac{x \cdot y}{\color{blue}{z \cdot \left(z \cdot \left(z + 1\right)\right)}} \]

      2. times-frac 97.6%

         \[\leadsto \color{blue}{\frac{x}{z} \cdot \frac{y}{z \cdot \left(z + 1\right)}} \]

      3. distribute-lft-in 97.6%

         \[\leadsto \frac{x}{z} \cdot \frac{y}{\color{blue}{z \cdot z + z \cdot 1}} \]

      4. fma-def 97.6%

         \[\leadsto \frac{x}{z} \cdot \frac{y}{\color{blue}{\mathsf{fma}\left(z, z, z \cdot 1\right)}} \]

      5. *-rgt-identity 97.6%

         \[\leadsto \frac{x}{z} \cdot \frac{y}{\mathsf{fma}\left(z, z, \color{blue}{z}\right)} \]

   3. Simplified 97.6%

      \[\leadsto \color{blue}{\frac{x}{z} \cdot \frac{y}{\mathsf{fma}\left(z, z, z\right)}} \]

   4. Taylor expanded in z around 0 97.6%

      \[\leadsto \frac{x}{z} \cdot \color{blue}{\left(-1 \cdot y + \frac{y}{z}\right)} \]
   5. Step-by-step derivation

      1. neg-mul-1 97.6%

         \[\leadsto \frac{x}{z} \cdot \left(\color{blue}{\left(-y\right)} + \frac{y}{z}\right) \]

      2. +-commutative 97.6%

         \[\leadsto \frac{x}{z} \cdot \color{blue}{\left(\frac{y}{z} + \left(-y\right)\right)} \]

      3. unsub-neg 97.6%

         \[\leadsto \frac{x}{z} \cdot \color{blue}{\left(\frac{y}{z} - y\right)} \]

   6. Simplified 97.6%

      \[\leadsto \frac{x}{z} \cdot \color{blue}{\left(\frac{y}{z} - y\right)} \]

   if 0.75 < z

   1. Initial program 88.9%

      \[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \]
   2. Step-by-step derivation

      1. /-rgt-identity 88.9%

         \[\leadsto \frac{\color{blue}{\frac{x \cdot y}{1}}}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \]

      2. associate-/l* 88.9%

         \[\leadsto \frac{\color{blue}{\frac{x}{\frac{1}{y}}}}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \]

      3. associate-/l/ 87.5%

         \[\leadsto \color{blue}{\frac{x}{\left(\left(z \cdot z\right) \cdot \left(z + 1\right)\right) \cdot \frac{1}{y}}} \]

      4. associate-*l* 89.4%

         \[\leadsto \frac{x}{\color{blue}{\left(z \cdot z\right) \cdot \left(\left(z + 1\right) \cdot \frac{1}{y}\right)}} \]

      5. associate-*r/ 89.4%

         \[\leadsto \frac{x}{\left(z \cdot z\right) \cdot \color{blue}{\frac{\left(z + 1\right) \cdot 1}{y}}} \]

      6. *-rgt-identity 89.4%

         \[\leadsto \frac{x}{\left(z \cdot z\right) \cdot \frac{\color{blue}{z + 1}}{y}} \]

      7. associate-*l* 91.1%

         \[\leadsto \frac{x}{\color{blue}{z \cdot \left(z \cdot \frac{z + 1}{y}\right)}} \]

      8. associate-*r/ 89.4%

         \[\leadsto \frac{x}{z \cdot \color{blue}{\frac{z \cdot \left(z + 1\right)}{y}}} \]

      9. distribute-lft-in 89.4%

         \[\leadsto \frac{x}{z \cdot \frac{\color{blue}{z \cdot z + z \cdot 1}}{y}} \]

      10. fma-def 89.4%

          \[\leadsto \frac{x}{z \cdot \frac{\color{blue}{\mathsf{fma}\left(z, z, z \cdot 1\right)}}{y}} \]

      11. *-rgt-identity 89.4%

          \[\leadsto \frac{x}{z \cdot \frac{\mathsf{fma}\left(z, z, \color{blue}{z}\right)}{y}} \]

   3. Simplified 89.4%

      \[\leadsto \color{blue}{\frac{x}{z \cdot \frac{\mathsf{fma}\left(z, z, z\right)}{y}}} \]

   4. Taylor expanded in z around inf 87.1%

      \[\leadsto \frac{x}{z \cdot \color{blue}{\frac{{z}^{2}}{y}}} \]
   5. Step-by-step derivation

      1. unpow2 87.1%

         \[\leadsto \frac{x}{z \cdot \frac{\color{blue}{z \cdot z}}{y}} \]

   6. Simplified 87.1%

      \[\leadsto \frac{x}{z \cdot \color{blue}{\frac{z \cdot z}{y}}} \]
   7. Step-by-step derivation

      1. associate-/r* 88.1%

         \[\leadsto \color{blue}{\frac{\frac{x}{z}}{\frac{z \cdot z}{y}}} \]

      2. add-cube-cbrt 87.9%

         \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{\frac{x}{z}} \cdot \sqrt[3]{\frac{x}{z}}\right) \cdot \sqrt[3]{\frac{x}{z}}}}{\frac{z \cdot z}{y}} \]

      3. *-un-lft-identity 87.9%

         \[\leadsto \frac{\left(\sqrt[3]{\frac{x}{z}} \cdot \sqrt[3]{\frac{x}{z}}\right) \cdot \sqrt[3]{\frac{x}{z}}}{\color{blue}{1 \cdot \frac{z \cdot z}{y}}} \]

      4. times-frac 87.9%

         \[\leadsto \color{blue}{\frac{\sqrt[3]{\frac{x}{z}} \cdot \sqrt[3]{\frac{x}{z}}}{1} \cdot \frac{\sqrt[3]{\frac{x}{z}}}{\frac{z \cdot z}{y}}} \]

      5. pow2 87.9%

         \[\leadsto \frac{\color{blue}{{\left(\sqrt[3]{\frac{x}{z}}\right)}^{2}}}{1} \cdot \frac{\sqrt[3]{\frac{x}{z}}}{\frac{z \cdot z}{y}} \]

      6. div-inv 88.0%

         \[\leadsto \frac{{\left(\sqrt[3]{\frac{x}{z}}\right)}^{2}}{1} \cdot \frac{\sqrt[3]{\frac{x}{z}}}{\color{blue}{\left(z \cdot z\right) \cdot \frac{1}{y}}} \]

      7. associate-*l* 93.1%

         \[\leadsto \frac{{\left(\sqrt[3]{\frac{x}{z}}\right)}^{2}}{1} \cdot \frac{\sqrt[3]{\frac{x}{z}}}{\color{blue}{z \cdot \left(z \cdot \frac{1}{y}\right)}} \]

      8. div-inv 93.1%

         \[\leadsto \frac{{\left(\sqrt[3]{\frac{x}{z}}\right)}^{2}}{1} \cdot \frac{\sqrt[3]{\frac{x}{z}}}{z \cdot \color{blue}{\frac{z}{y}}} \]

   8. Applied egg-rr 93.1%

      \[\leadsto \color{blue}{\frac{{\left(\sqrt[3]{\frac{x}{z}}\right)}^{2}}{1} \cdot \frac{\sqrt[3]{\frac{x}{z}}}{z \cdot \frac{z}{y}}} \]
   9. Step-by-step derivation

      1. times-frac 93.0%

         \[\leadsto \color{blue}{\frac{{\left(\sqrt[3]{\frac{x}{z}}\right)}^{2} \cdot \sqrt[3]{\frac{x}{z}}}{1 \cdot \left(z \cdot \frac{z}{y}\right)}} \]

      2. *-lft-identity 93.0%

         \[\leadsto \frac{{\left(\sqrt[3]{\frac{x}{z}}\right)}^{2} \cdot \sqrt[3]{\frac{x}{z}}}{\color{blue}{z \cdot \frac{z}{y}}} \]

      3. unpow2 93.0%

         \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{\frac{x}{z}} \cdot \sqrt[3]{\frac{x}{z}}\right)} \cdot \sqrt[3]{\frac{x}{z}}}{z \cdot \frac{z}{y}} \]

      4. rem-3cbrt-lft 93.4%

         \[\leadsto \frac{\color{blue}{\frac{x}{z}}}{z \cdot \frac{z}{y}} \]

   10. Simplified 93.4%

       \[\leadsto \color{blue}{\frac{\frac{x}{z}}{z \cdot \frac{z}{y}}} \]
   11. Step-by-step derivation

       1. associate-/r* 94.1%

          \[\leadsto \color{blue}{\frac{\frac{\frac{x}{z}}{z}}{\frac{z}{y}}} \]

       2. associate-/r/ 90.5%

          \[\leadsto \color{blue}{\frac{\frac{\frac{x}{z}}{z}}{z} \cdot y} \]

       3. associate-/l/ 88.8%

          \[\leadsto \frac{\color{blue}{\frac{x}{z \cdot z}}}{z} \cdot y \]

   12. Applied egg-rr 88.8%

       \[\leadsto \color{blue}{\frac{\frac{x}{z \cdot z}}{z} \cdot y} \]
   13. Step-by-step derivation

       1. associate-*l/ 88.9%

          \[\leadsto \color{blue}{\frac{\frac{x}{z \cdot z} \cdot y}{z}} \]

       2. associate-/r* 94.1%

          \[\leadsto \frac{\color{blue}{\frac{\frac{x}{z}}{z}} \cdot y}{z} \]

       3. associate-/r/ 96.9%

          \[\leadsto \frac{\color{blue}{\frac{\frac{x}{z}}{\frac{z}{y}}}}{z} \]

       4. clear-num 96.8%

          \[\leadsto \frac{\color{blue}{\frac{1}{\frac{\frac{z}{y}}{\frac{x}{z}}}}}{z} \]

       5. associate-/l/ 96.9%

          \[\leadsto \color{blue}{\frac{1}{z \cdot \frac{\frac{z}{y}}{\frac{x}{z}}}} \]

       6. div-inv 95.8%

          \[\leadsto \frac{1}{z \cdot \color{blue}{\left(\frac{z}{y} \cdot \frac{1}{\frac{x}{z}}\right)}} \]

       7. clear-num 95.8%

          \[\leadsto \frac{1}{z \cdot \left(\frac{z}{y} \cdot \color{blue}{\frac{z}{x}}\right)} \]

   14. Applied egg-rr 95.8%

       \[\leadsto \color{blue}{\frac{1}{z \cdot \left(\frac{z}{y} \cdot \frac{z}{x}\right)}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 97.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1:\\ \;\;\;\;\frac{\frac{y}{z}}{z \cdot \frac{z}{x}}\\ \mathbf{elif}\;z \leq 0.75:\\ \;\;\;\;\frac{x}{z} \cdot \left(\frac{y}{z} - y\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{z \cdot \left(\frac{z}{x} \cdot \frac{z}{y}\right)}\\ \end{array} \]

Alternative 3: 93.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1 \lor \neg \left(z \leq 0.75\right):\\ \;\;\;\;\frac{x}{z} \cdot \frac{y}{z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot \left(\frac{y}{z} - y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -1.0) (not (<= z 0.75)))
   (* (/ x z) (/ y (* z z)))
   (* (/ x z) (- (/ y z) y))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -1.0) || !(z <= 0.75)) {
		tmp = (x / z) * (y / (z * z));
	} else {
		tmp = (x / z) * ((y / z) - y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((z <= (-1.0d0)) .or. (.not. (z <= 0.75d0))) then
        tmp = (x / z) * (y / (z * z))
    else
        tmp = (x / z) * ((y / z) - y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((z <= -1.0) || !(z <= 0.75)) {
		tmp = (x / z) * (y / (z * z));
	} else {
		tmp = (x / z) * ((y / z) - y);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -1.0) or not (z <= 0.75):
		tmp = (x / z) * (y / (z * z))
	else:
		tmp = (x / z) * ((y / z) - y)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -1.0) || !(z <= 0.75))
		tmp = Float64(Float64(x / z) * Float64(y / Float64(z * z)));
	else
		tmp = Float64(Float64(x / z) * Float64(Float64(y / z) - y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -1.0) || ~((z <= 0.75)))
		tmp = (x / z) * (y / (z * z));
	else
		tmp = (x / z) * ((y / z) - y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -1.0], N[Not[LessEqual[z, 0.75]], $MachinePrecision]], N[(N[(x / z), $MachinePrecision] * N[(y / N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / z), $MachinePrecision] * N[(N[(y / z), $MachinePrecision] - y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1 or 0.75 < z

   1. Initial program 84.8%

      \[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \]
   2. Step-by-step derivation

      1. associate-*l* 84.8%

         \[\leadsto \frac{x \cdot y}{\color{blue}{z \cdot \left(z \cdot \left(z + 1\right)\right)}} \]

      2. times-frac 92.2%

         \[\leadsto \color{blue}{\frac{x}{z} \cdot \frac{y}{z \cdot \left(z + 1\right)}} \]

      3. distribute-lft-in 92.2%

         \[\leadsto \frac{x}{z} \cdot \frac{y}{\color{blue}{z \cdot z + z \cdot 1}} \]

      4. fma-def 92.2%

         \[\leadsto \frac{x}{z} \cdot \frac{y}{\color{blue}{\mathsf{fma}\left(z, z, z \cdot 1\right)}} \]

      5. *-rgt-identity 92.2%

         \[\leadsto \frac{x}{z} \cdot \frac{y}{\mathsf{fma}\left(z, z, \color{blue}{z}\right)} \]

   3. Simplified 92.2%

      \[\leadsto \color{blue}{\frac{x}{z} \cdot \frac{y}{\mathsf{fma}\left(z, z, z\right)}} \]

   4. Taylor expanded in z around inf 90.9%

      \[\leadsto \frac{x}{z} \cdot \color{blue}{\frac{y}{{z}^{2}}} \]
   5. Step-by-step derivation

      1. unpow2 90.9%

         \[\leadsto \frac{x}{z} \cdot \frac{y}{\color{blue}{z \cdot z}} \]

   6. Simplified 90.9%

      \[\leadsto \frac{x}{z} \cdot \color{blue}{\frac{y}{z \cdot z}} \]

   if -1 < z < 0.75

   1. Initial program 86.7%

      \[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \]
   2. Step-by-step derivation

      1. associate-*l* 86.8%

         \[\leadsto \frac{x \cdot y}{\color{blue}{z \cdot \left(z \cdot \left(z + 1\right)\right)}} \]

      2. times-frac 97.6%

         \[\leadsto \color{blue}{\frac{x}{z} \cdot \frac{y}{z \cdot \left(z + 1\right)}} \]

      3. distribute-lft-in 97.6%

         \[\leadsto \frac{x}{z} \cdot \frac{y}{\color{blue}{z \cdot z + z \cdot 1}} \]

      4. fma-def 97.6%

         \[\leadsto \frac{x}{z} \cdot \frac{y}{\color{blue}{\mathsf{fma}\left(z, z, z \cdot 1\right)}} \]

      5. *-rgt-identity 97.6%

         \[\leadsto \frac{x}{z} \cdot \frac{y}{\mathsf{fma}\left(z, z, \color{blue}{z}\right)} \]

   3. Simplified 97.6%

      \[\leadsto \color{blue}{\frac{x}{z} \cdot \frac{y}{\mathsf{fma}\left(z, z, z\right)}} \]

   4. Taylor expanded in z around 0 97.6%

      \[\leadsto \frac{x}{z} \cdot \color{blue}{\left(-1 \cdot y + \frac{y}{z}\right)} \]
   5. Step-by-step derivation

      1. neg-mul-1 97.6%

         \[\leadsto \frac{x}{z} \cdot \left(\color{blue}{\left(-y\right)} + \frac{y}{z}\right) \]

      2. +-commutative 97.6%

         \[\leadsto \frac{x}{z} \cdot \color{blue}{\left(\frac{y}{z} + \left(-y\right)\right)} \]

      3. unsub-neg 97.6%

         \[\leadsto \frac{x}{z} \cdot \color{blue}{\left(\frac{y}{z} - y\right)} \]

   6. Simplified 97.6%

      \[\leadsto \frac{x}{z} \cdot \color{blue}{\left(\frac{y}{z} - y\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 94.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1 \lor \neg \left(z \leq 0.75\right):\\ \;\;\;\;\frac{x}{z} \cdot \frac{y}{z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot \left(\frac{y}{z} - y\right)\\ \end{array} \]

Alternative 4: 95.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1 \lor \neg \left(z \leq 0.75\right):\\ \;\;\;\;\frac{x}{z} \cdot \frac{\frac{y}{z}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot \left(\frac{y}{z} - y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -1.0) (not (<= z 0.75)))
   (* (/ x z) (/ (/ y z) z))
   (* (/ x z) (- (/ y z) y))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -1.0) || !(z <= 0.75)) {
		tmp = (x / z) * ((y / z) / z);
	} else {
		tmp = (x / z) * ((y / z) - y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((z <= (-1.0d0)) .or. (.not. (z <= 0.75d0))) then
        tmp = (x / z) * ((y / z) / z)
    else
        tmp = (x / z) * ((y / z) - y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((z <= -1.0) || !(z <= 0.75)) {
		tmp = (x / z) * ((y / z) / z);
	} else {
		tmp = (x / z) * ((y / z) - y);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -1.0) or not (z <= 0.75):
		tmp = (x / z) * ((y / z) / z)
	else:
		tmp = (x / z) * ((y / z) - y)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -1.0) || !(z <= 0.75))
		tmp = Float64(Float64(x / z) * Float64(Float64(y / z) / z));
	else
		tmp = Float64(Float64(x / z) * Float64(Float64(y / z) - y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -1.0) || ~((z <= 0.75)))
		tmp = (x / z) * ((y / z) / z);
	else
		tmp = (x / z) * ((y / z) - y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -1.0], N[Not[LessEqual[z, 0.75]], $MachinePrecision]], N[(N[(x / z), $MachinePrecision] * N[(N[(y / z), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], N[(N[(x / z), $MachinePrecision] * N[(N[(y / z), $MachinePrecision] - y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1 or 0.75 < z

   1. Initial program 84.8%

      \[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \]
   2. Step-by-step derivation

      1. associate-*l* 84.8%

         \[\leadsto \frac{x \cdot y}{\color{blue}{z \cdot \left(z \cdot \left(z + 1\right)\right)}} \]

      2. times-frac 92.2%

         \[\leadsto \color{blue}{\frac{x}{z} \cdot \frac{y}{z \cdot \left(z + 1\right)}} \]

      3. distribute-lft-in 92.2%

         \[\leadsto \frac{x}{z} \cdot \frac{y}{\color{blue}{z \cdot z + z \cdot 1}} \]

      4. fma-def 92.2%

         \[\leadsto \frac{x}{z} \cdot \frac{y}{\color{blue}{\mathsf{fma}\left(z, z, z \cdot 1\right)}} \]

      5. *-rgt-identity 92.2%

         \[\leadsto \frac{x}{z} \cdot \frac{y}{\mathsf{fma}\left(z, z, \color{blue}{z}\right)} \]

   3. Simplified 92.2%

      \[\leadsto \color{blue}{\frac{x}{z} \cdot \frac{y}{\mathsf{fma}\left(z, z, z\right)}} \]

   4. Taylor expanded in z around inf 90.9%

      \[\leadsto \frac{x}{z} \cdot \color{blue}{\frac{y}{{z}^{2}}} \]
   5. Step-by-step derivation

      1. unpow2 90.9%

         \[\leadsto \frac{x}{z} \cdot \frac{y}{\color{blue}{z \cdot z}} \]

      2. associate-/r* 95.4%

         \[\leadsto \frac{x}{z} \cdot \color{blue}{\frac{\frac{y}{z}}{z}} \]

   6. Simplified 95.4%

      \[\leadsto \frac{x}{z} \cdot \color{blue}{\frac{\frac{y}{z}}{z}} \]

   if -1 < z < 0.75

   1. Initial program 86.7%

      \[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \]
   2. Step-by-step derivation

      1. associate-*l* 86.8%

         \[\leadsto \frac{x \cdot y}{\color{blue}{z \cdot \left(z \cdot \left(z + 1\right)\right)}} \]

      2. times-frac 97.6%

         \[\leadsto \color{blue}{\frac{x}{z} \cdot \frac{y}{z \cdot \left(z + 1\right)}} \]

      3. distribute-lft-in 97.6%

         \[\leadsto \frac{x}{z} \cdot \frac{y}{\color{blue}{z \cdot z + z \cdot 1}} \]

      4. fma-def 97.6%

         \[\leadsto \frac{x}{z} \cdot \frac{y}{\color{blue}{\mathsf{fma}\left(z, z, z \cdot 1\right)}} \]

      5. *-rgt-identity 97.6%

         \[\leadsto \frac{x}{z} \cdot \frac{y}{\mathsf{fma}\left(z, z, \color{blue}{z}\right)} \]

   3. Simplified 97.6%

      \[\leadsto \color{blue}{\frac{x}{z} \cdot \frac{y}{\mathsf{fma}\left(z, z, z\right)}} \]

   4. Taylor expanded in z around 0 97.6%

      \[\leadsto \frac{x}{z} \cdot \color{blue}{\left(-1 \cdot y + \frac{y}{z}\right)} \]
   5. Step-by-step derivation

      1. neg-mul-1 97.6%

         \[\leadsto \frac{x}{z} \cdot \left(\color{blue}{\left(-y\right)} + \frac{y}{z}\right) \]

      2. +-commutative 97.6%

         \[\leadsto \frac{x}{z} \cdot \color{blue}{\left(\frac{y}{z} + \left(-y\right)\right)} \]

      3. unsub-neg 97.6%

         \[\leadsto \frac{x}{z} \cdot \color{blue}{\left(\frac{y}{z} - y\right)} \]

   6. Simplified 97.6%

      \[\leadsto \frac{x}{z} \cdot \color{blue}{\left(\frac{y}{z} - y\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 96.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1 \lor \neg \left(z \leq 0.75\right):\\ \;\;\;\;\frac{x}{z} \cdot \frac{\frac{y}{z}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot \left(\frac{y}{z} - y\right)\\ \end{array} \]

Alternative 5: 94.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1:\\ \;\;\;\;\frac{y}{z} \cdot \frac{\frac{x}{z}}{z}\\ \mathbf{elif}\;z \leq 0.76:\\ \;\;\;\;\frac{x}{z} \cdot \left(\frac{y}{z} - y\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot \frac{\frac{y}{z}}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -1.0)
   (* (/ y z) (/ (/ x z) z))
   (if (<= z 0.76) (* (/ x z) (- (/ y z) y)) (* (/ x z) (/ (/ y z) z)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -1.0) {
		tmp = (y / z) * ((x / z) / z);
	} else if (z <= 0.76) {
		tmp = (x / z) * ((y / z) - y);
	} else {
		tmp = (x / z) * ((y / z) / z);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (z <= (-1.0d0)) then
        tmp = (y / z) * ((x / z) / z)
    else if (z <= 0.76d0) then
        tmp = (x / z) * ((y / z) - y)
    else
        tmp = (x / z) * ((y / z) / z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -1.0) {
		tmp = (y / z) * ((x / z) / z);
	} else if (z <= 0.76) {
		tmp = (x / z) * ((y / z) - y);
	} else {
		tmp = (x / z) * ((y / z) / z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -1.0:
		tmp = (y / z) * ((x / z) / z)
	elif z <= 0.76:
		tmp = (x / z) * ((y / z) - y)
	else:
		tmp = (x / z) * ((y / z) / z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -1.0)
		tmp = Float64(Float64(y / z) * Float64(Float64(x / z) / z));
	elseif (z <= 0.76)
		tmp = Float64(Float64(x / z) * Float64(Float64(y / z) - y));
	else
		tmp = Float64(Float64(x / z) * Float64(Float64(y / z) / z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -1.0)
		tmp = (y / z) * ((x / z) / z);
	elseif (z <= 0.76)
		tmp = (x / z) * ((y / z) - y);
	else
		tmp = (x / z) * ((y / z) / z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -1.0], N[(N[(y / z), $MachinePrecision] * N[(N[(x / z), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 0.76], N[(N[(x / z), $MachinePrecision] * N[(N[(y / z), $MachinePrecision] - y), $MachinePrecision]), $MachinePrecision], N[(N[(x / z), $MachinePrecision] * N[(N[(y / z), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -1

   1. Initial program 81.6%

      \[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \]
   2. Step-by-step derivation

      1. /-rgt-identity 81.6%

         \[\leadsto \frac{\color{blue}{\frac{x \cdot y}{1}}}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \]

      2. associate-/l* 81.7%

         \[\leadsto \frac{\color{blue}{\frac{x}{\frac{1}{y}}}}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \]

      3. associate-/l/ 88.8%

         \[\leadsto \color{blue}{\frac{x}{\left(\left(z \cdot z\right) \cdot \left(z + 1\right)\right) \cdot \frac{1}{y}}} \]

      4. associate-*l* 91.5%

         \[\leadsto \frac{x}{\color{blue}{\left(z \cdot z\right) \cdot \left(\left(z + 1\right) \cdot \frac{1}{y}\right)}} \]

      5. associate-*r/ 91.5%

         \[\leadsto \frac{x}{\left(z \cdot z\right) \cdot \color{blue}{\frac{\left(z + 1\right) \cdot 1}{y}}} \]

      6. *-rgt-identity 91.5%

         \[\leadsto \frac{x}{\left(z \cdot z\right) \cdot \frac{\color{blue}{z + 1}}{y}} \]

      7. associate-*l* 91.5%

         \[\leadsto \frac{x}{\color{blue}{z \cdot \left(z \cdot \frac{z + 1}{y}\right)}} \]

      8. associate-*r/ 91.6%

         \[\leadsto \frac{x}{z \cdot \color{blue}{\frac{z \cdot \left(z + 1\right)}{y}}} \]

      9. distribute-lft-in 91.6%

         \[\leadsto \frac{x}{z \cdot \frac{\color{blue}{z \cdot z + z \cdot 1}}{y}} \]

      10. fma-def 91.6%

          \[\leadsto \frac{x}{z \cdot \frac{\color{blue}{\mathsf{fma}\left(z, z, z \cdot 1\right)}}{y}} \]

      11. *-rgt-identity 91.6%

          \[\leadsto \frac{x}{z \cdot \frac{\mathsf{fma}\left(z, z, \color{blue}{z}\right)}{y}} \]

   3. Simplified 91.6%

      \[\leadsto \color{blue}{\frac{x}{z \cdot \frac{\mathsf{fma}\left(z, z, z\right)}{y}}} \]

   4. Taylor expanded in z around inf 90.9%

      \[\leadsto \frac{x}{z \cdot \color{blue}{\frac{{z}^{2}}{y}}} \]
   5. Step-by-step derivation

      1. unpow2 90.9%

         \[\leadsto \frac{x}{z \cdot \frac{\color{blue}{z \cdot z}}{y}} \]

   6. Simplified 90.9%

      \[\leadsto \frac{x}{z \cdot \color{blue}{\frac{z \cdot z}{y}}} \]
   7. Step-by-step derivation

      1. associate-/r* 93.0%

         \[\leadsto \color{blue}{\frac{\frac{x}{z}}{\frac{z \cdot z}{y}}} \]

      2. associate-/l* 96.6%

         \[\leadsto \frac{\frac{x}{z}}{\color{blue}{\frac{z}{\frac{y}{z}}}} \]

      3. associate-/r/ 99.0%

         \[\leadsto \color{blue}{\frac{\frac{x}{z}}{z} \cdot \frac{y}{z}} \]

   8. Applied egg-rr 99.0%

      \[\leadsto \color{blue}{\frac{\frac{x}{z}}{z} \cdot \frac{y}{z}} \]

   if -1 < z < 0.76000000000000001

   1. Initial program 86.7%

      \[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \]
   2. Step-by-step derivation

      1. associate-*l* 86.8%

         \[\leadsto \frac{x \cdot y}{\color{blue}{z \cdot \left(z \cdot \left(z + 1\right)\right)}} \]

      2. times-frac 97.6%

         \[\leadsto \color{blue}{\frac{x}{z} \cdot \frac{y}{z \cdot \left(z + 1\right)}} \]

      3. distribute-lft-in 97.6%

         \[\leadsto \frac{x}{z} \cdot \frac{y}{\color{blue}{z \cdot z + z \cdot 1}} \]

      4. fma-def 97.6%

         \[\leadsto \frac{x}{z} \cdot \frac{y}{\color{blue}{\mathsf{fma}\left(z, z, z \cdot 1\right)}} \]

      5. *-rgt-identity 97.6%

         \[\leadsto \frac{x}{z} \cdot \frac{y}{\mathsf{fma}\left(z, z, \color{blue}{z}\right)} \]

   3. Simplified 97.6%

      \[\leadsto \color{blue}{\frac{x}{z} \cdot \frac{y}{\mathsf{fma}\left(z, z, z\right)}} \]

   4. Taylor expanded in z around 0 97.6%

      \[\leadsto \frac{x}{z} \cdot \color{blue}{\left(-1 \cdot y + \frac{y}{z}\right)} \]
   5. Step-by-step derivation

      1. neg-mul-1 97.6%

         \[\leadsto \frac{x}{z} \cdot \left(\color{blue}{\left(-y\right)} + \frac{y}{z}\right) \]

      2. +-commutative 97.6%

         \[\leadsto \frac{x}{z} \cdot \color{blue}{\left(\frac{y}{z} + \left(-y\right)\right)} \]

      3. unsub-neg 97.6%

         \[\leadsto \frac{x}{z} \cdot \color{blue}{\left(\frac{y}{z} - y\right)} \]

   6. Simplified 97.6%

      \[\leadsto \frac{x}{z} \cdot \color{blue}{\left(\frac{y}{z} - y\right)} \]

   if 0.76000000000000001 < z

   1. Initial program 88.9%

      \[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \]
   2. Step-by-step derivation

      1. associate-*l* 88.9%

         \[\leadsto \frac{x \cdot y}{\color{blue}{z \cdot \left(z \cdot \left(z + 1\right)\right)}} \]

      2. times-frac 90.4%

         \[\leadsto \color{blue}{\frac{x}{z} \cdot \frac{y}{z \cdot \left(z + 1\right)}} \]

      3. distribute-lft-in 90.4%

         \[\leadsto \frac{x}{z} \cdot \frac{y}{\color{blue}{z \cdot z + z \cdot 1}} \]

      4. fma-def 90.4%

         \[\leadsto \frac{x}{z} \cdot \frac{y}{\color{blue}{\mathsf{fma}\left(z, z, z \cdot 1\right)}} \]

      5. *-rgt-identity 90.4%

         \[\leadsto \frac{x}{z} \cdot \frac{y}{\mathsf{fma}\left(z, z, \color{blue}{z}\right)} \]

   3. Simplified 90.4%

      \[\leadsto \color{blue}{\frac{x}{z} \cdot \frac{y}{\mathsf{fma}\left(z, z, z\right)}} \]

   4. Taylor expanded in z around inf 88.2%

      \[\leadsto \frac{x}{z} \cdot \color{blue}{\frac{y}{{z}^{2}}} \]
   5. Step-by-step derivation

      1. unpow2 88.2%

         \[\leadsto \frac{x}{z} \cdot \frac{y}{\color{blue}{z \cdot z}} \]

      2. associate-/r* 93.4%

         \[\leadsto \frac{x}{z} \cdot \color{blue}{\frac{\frac{y}{z}}{z}} \]

   6. Simplified 93.4%

      \[\leadsto \frac{x}{z} \cdot \color{blue}{\frac{\frac{y}{z}}{z}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 97.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1:\\ \;\;\;\;\frac{y}{z} \cdot \frac{\frac{x}{z}}{z}\\ \mathbf{elif}\;z \leq 0.76:\\ \;\;\;\;\frac{x}{z} \cdot \left(\frac{y}{z} - y\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot \frac{\frac{y}{z}}{z}\\ \end{array} \]

Alternative 6: 94.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1:\\ \;\;\;\;\frac{\frac{y}{z}}{z \cdot \frac{z}{x}}\\ \mathbf{elif}\;z \leq 0.75:\\ \;\;\;\;\frac{x}{z} \cdot \left(\frac{y}{z} - y\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot \frac{\frac{y}{z}}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -1.0)
   (/ (/ y z) (* z (/ z x)))
   (if (<= z 0.75) (* (/ x z) (- (/ y z) y)) (* (/ x z) (/ (/ y z) z)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -1.0) {
		tmp = (y / z) / (z * (z / x));
	} else if (z <= 0.75) {
		tmp = (x / z) * ((y / z) - y);
	} else {
		tmp = (x / z) * ((y / z) / z);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (z <= (-1.0d0)) then
        tmp = (y / z) / (z * (z / x))
    else if (z <= 0.75d0) then
        tmp = (x / z) * ((y / z) - y)
    else
        tmp = (x / z) * ((y / z) / z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -1.0) {
		tmp = (y / z) / (z * (z / x));
	} else if (z <= 0.75) {
		tmp = (x / z) * ((y / z) - y);
	} else {
		tmp = (x / z) * ((y / z) / z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -1.0:
		tmp = (y / z) / (z * (z / x))
	elif z <= 0.75:
		tmp = (x / z) * ((y / z) - y)
	else:
		tmp = (x / z) * ((y / z) / z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -1.0)
		tmp = Float64(Float64(y / z) / Float64(z * Float64(z / x)));
	elseif (z <= 0.75)
		tmp = Float64(Float64(x / z) * Float64(Float64(y / z) - y));
	else
		tmp = Float64(Float64(x / z) * Float64(Float64(y / z) / z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -1.0)
		tmp = (y / z) / (z * (z / x));
	elseif (z <= 0.75)
		tmp = (x / z) * ((y / z) - y);
	else
		tmp = (x / z) * ((y / z) / z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -1.0], N[(N[(y / z), $MachinePrecision] / N[(z * N[(z / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 0.75], N[(N[(x / z), $MachinePrecision] * N[(N[(y / z), $MachinePrecision] - y), $MachinePrecision]), $MachinePrecision], N[(N[(x / z), $MachinePrecision] * N[(N[(y / z), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -1

   1. Initial program 81.6%

      \[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \]
   2. Step-by-step derivation

      1. associate-*l* 81.6%

         \[\leadsto \frac{x \cdot y}{\color{blue}{z \cdot \left(z \cdot \left(z + 1\right)\right)}} \]

      2. times-frac 93.6%

         \[\leadsto \color{blue}{\frac{x}{z} \cdot \frac{y}{z \cdot \left(z + 1\right)}} \]

      3. distribute-lft-in 93.6%

         \[\leadsto \frac{x}{z} \cdot \frac{y}{\color{blue}{z \cdot z + z \cdot 1}} \]

      4. fma-def 93.6%

         \[\leadsto \frac{x}{z} \cdot \frac{y}{\color{blue}{\mathsf{fma}\left(z, z, z \cdot 1\right)}} \]

      5. *-rgt-identity 93.6%

         \[\leadsto \frac{x}{z} \cdot \frac{y}{\mathsf{fma}\left(z, z, \color{blue}{z}\right)} \]

   3. Simplified 93.6%

      \[\leadsto \color{blue}{\frac{x}{z} \cdot \frac{y}{\mathsf{fma}\left(z, z, z\right)}} \]

   4. Taylor expanded in z around inf 92.9%

      \[\leadsto \frac{x}{z} \cdot \color{blue}{\frac{y}{{z}^{2}}} \]
   5. Step-by-step derivation

      1. unpow2 92.9%

         \[\leadsto \frac{x}{z} \cdot \frac{y}{\color{blue}{z \cdot z}} \]

      2. associate-/r* 96.9%

         \[\leadsto \frac{x}{z} \cdot \color{blue}{\frac{\frac{y}{z}}{z}} \]

   6. Simplified 96.9%

      \[\leadsto \frac{x}{z} \cdot \color{blue}{\frac{\frac{y}{z}}{z}} \]
   7. Step-by-step derivation

      1. clear-num 96.9%

         \[\leadsto \color{blue}{\frac{1}{\frac{z}{x}}} \cdot \frac{\frac{y}{z}}{z} \]

      2. frac-times 99.1%

         \[\leadsto \color{blue}{\frac{1 \cdot \frac{y}{z}}{\frac{z}{x} \cdot z}} \]

      3. *-un-lft-identity 99.1%

         \[\leadsto \frac{\color{blue}{\frac{y}{z}}}{\frac{z}{x} \cdot z} \]

   8. Applied egg-rr 99.1%

      \[\leadsto \color{blue}{\frac{\frac{y}{z}}{\frac{z}{x} \cdot z}} \]

   if -1 < z < 0.75

   1. Initial program 86.7%

      \[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \]
   2. Step-by-step derivation

      1. associate-*l* 86.8%

         \[\leadsto \frac{x \cdot y}{\color{blue}{z \cdot \left(z \cdot \left(z + 1\right)\right)}} \]

      2. times-frac 97.6%

         \[\leadsto \color{blue}{\frac{x}{z} \cdot \frac{y}{z \cdot \left(z + 1\right)}} \]

      3. distribute-lft-in 97.6%

         \[\leadsto \frac{x}{z} \cdot \frac{y}{\color{blue}{z \cdot z + z \cdot 1}} \]

      4. fma-def 97.6%

         \[\leadsto \frac{x}{z} \cdot \frac{y}{\color{blue}{\mathsf{fma}\left(z, z, z \cdot 1\right)}} \]

      5. *-rgt-identity 97.6%

         \[\leadsto \frac{x}{z} \cdot \frac{y}{\mathsf{fma}\left(z, z, \color{blue}{z}\right)} \]

   3. Simplified 97.6%

      \[\leadsto \color{blue}{\frac{x}{z} \cdot \frac{y}{\mathsf{fma}\left(z, z, z\right)}} \]

   4. Taylor expanded in z around 0 97.6%

      \[\leadsto \frac{x}{z} \cdot \color{blue}{\left(-1 \cdot y + \frac{y}{z}\right)} \]
   5. Step-by-step derivation

      1. neg-mul-1 97.6%

         \[\leadsto \frac{x}{z} \cdot \left(\color{blue}{\left(-y\right)} + \frac{y}{z}\right) \]

      2. +-commutative 97.6%

         \[\leadsto \frac{x}{z} \cdot \color{blue}{\left(\frac{y}{z} + \left(-y\right)\right)} \]

      3. unsub-neg 97.6%

         \[\leadsto \frac{x}{z} \cdot \color{blue}{\left(\frac{y}{z} - y\right)} \]

   6. Simplified 97.6%

      \[\leadsto \frac{x}{z} \cdot \color{blue}{\left(\frac{y}{z} - y\right)} \]

   if 0.75 < z

   1. Initial program 88.9%

      \[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \]
   2. Step-by-step derivation

      1. associate-*l* 88.9%

         \[\leadsto \frac{x \cdot y}{\color{blue}{z \cdot \left(z \cdot \left(z + 1\right)\right)}} \]

      2. times-frac 90.4%

         \[\leadsto \color{blue}{\frac{x}{z} \cdot \frac{y}{z \cdot \left(z + 1\right)}} \]

      3. distribute-lft-in 90.4%

         \[\leadsto \frac{x}{z} \cdot \frac{y}{\color{blue}{z \cdot z + z \cdot 1}} \]

      4. fma-def 90.4%

         \[\leadsto \frac{x}{z} \cdot \frac{y}{\color{blue}{\mathsf{fma}\left(z, z, z \cdot 1\right)}} \]

      5. *-rgt-identity 90.4%

         \[\leadsto \frac{x}{z} \cdot \frac{y}{\mathsf{fma}\left(z, z, \color{blue}{z}\right)} \]

   3. Simplified 90.4%

      \[\leadsto \color{blue}{\frac{x}{z} \cdot \frac{y}{\mathsf{fma}\left(z, z, z\right)}} \]

   4. Taylor expanded in z around inf 88.2%

      \[\leadsto \frac{x}{z} \cdot \color{blue}{\frac{y}{{z}^{2}}} \]
   5. Step-by-step derivation

      1. unpow2 88.2%

         \[\leadsto \frac{x}{z} \cdot \frac{y}{\color{blue}{z \cdot z}} \]

      2. associate-/r* 93.4%

         \[\leadsto \frac{x}{z} \cdot \color{blue}{\frac{\frac{y}{z}}{z}} \]

   6. Simplified 93.4%

      \[\leadsto \frac{x}{z} \cdot \color{blue}{\frac{\frac{y}{z}}{z}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 97.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1:\\ \;\;\;\;\frac{\frac{y}{z}}{z \cdot \frac{z}{x}}\\ \mathbf{elif}\;z \leq 0.75:\\ \;\;\;\;\frac{x}{z} \cdot \left(\frac{y}{z} - y\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot \frac{\frac{y}{z}}{z}\\ \end{array} \]

Alternative 7: 96.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{x}{z} \cdot \frac{y}{z}}{z + 1} \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (/ (* (/ x z) (/ y z)) (+ z 1.0)))

C

double code(double x, double y, double z) {
	return ((x / z) * (y / z)) / (z + 1.0);
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = ((x / z) * (y / z)) / (z + 1.0d0)
end function

Java

public static double code(double x, double y, double z) {
	return ((x / z) * (y / z)) / (z + 1.0);
}

Python

def code(x, y, z):
	return ((x / z) * (y / z)) / (z + 1.0)

Julia

function code(x, y, z)
	return Float64(Float64(Float64(x / z) * Float64(y / z)) / Float64(z + 1.0))
end

MATLAB

function tmp = code(x, y, z)
	tmp = ((x / z) * (y / z)) / (z + 1.0);
end

Wolfram

code[x_, y_, z_] := N[(N[(N[(x / z), $MachinePrecision] * N[(y / z), $MachinePrecision]), $MachinePrecision] / N[(z + 1.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 85.8%

   \[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \]
2. Step-by-step derivation

   1. associate-*l* 85.8%

      \[\leadsto \frac{x \cdot y}{\color{blue}{z \cdot \left(z \cdot \left(z + 1\right)\right)}} \]

   2. times-frac 95.0%

      \[\leadsto \color{blue}{\frac{x}{z} \cdot \frac{y}{z \cdot \left(z + 1\right)}} \]

   3. distribute-lft-in 95.0%

      \[\leadsto \frac{x}{z} \cdot \frac{y}{\color{blue}{z \cdot z + z \cdot 1}} \]

   4. fma-def 95.0%

      \[\leadsto \frac{x}{z} \cdot \frac{y}{\color{blue}{\mathsf{fma}\left(z, z, z \cdot 1\right)}} \]

   5. *-rgt-identity 95.0%

      \[\leadsto \frac{x}{z} \cdot \frac{y}{\mathsf{fma}\left(z, z, \color{blue}{z}\right)} \]

3. Simplified 95.0%

   \[\leadsto \color{blue}{\frac{x}{z} \cdot \frac{y}{\mathsf{fma}\left(z, z, z\right)}} \]
4. Step-by-step derivation

   1. fma-udef 95.0%

      \[\leadsto \frac{x}{z} \cdot \frac{y}{\color{blue}{z \cdot z + z}} \]

   2. *-rgt-identity 95.0%

      \[\leadsto \frac{x}{z} \cdot \frac{y}{z \cdot z + \color{blue}{z \cdot 1}} \]

   3. distribute-lft-in 95.0%

      \[\leadsto \frac{x}{z} \cdot \frac{y}{\color{blue}{z \cdot \left(z + 1\right)}} \]

   4. times-frac 85.8%

      \[\leadsto \color{blue}{\frac{x \cdot y}{z \cdot \left(z \cdot \left(z + 1\right)\right)}} \]

   5. associate-*l* 85.8%

      \[\leadsto \frac{x \cdot y}{\color{blue}{\left(z \cdot z\right) \cdot \left(z + 1\right)}} \]

   6. associate-/r* 86.8%

      \[\leadsto \color{blue}{\frac{\frac{x \cdot y}{z \cdot z}}{z + 1}} \]

   7. times-frac 97.8%

      \[\leadsto \frac{\color{blue}{\frac{x}{z} \cdot \frac{y}{z}}}{z + 1} \]

5. Applied egg-rr 97.8%

   \[\leadsto \color{blue}{\frac{\frac{x}{z} \cdot \frac{y}{z}}{z + 1}} \]

6. Final simplification 97.8%

   \[\leadsto \frac{\frac{x}{z} \cdot \frac{y}{z}}{z + 1} \]

Alternative 8: 75.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.2 \cdot 10^{-157}:\\ \;\;\;\;x \cdot \frac{\frac{y}{z}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot \frac{y}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -4.2e-157) (* x (/ (/ y z) z)) (* (/ x z) (/ y z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -4.2e-157) {
		tmp = x * ((y / z) / z);
	} else {
		tmp = (x / z) * (y / z);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x <= (-4.2d-157)) then
        tmp = x * ((y / z) / z)
    else
        tmp = (x / z) * (y / z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -4.2e-157) {
		tmp = x * ((y / z) / z);
	} else {
		tmp = (x / z) * (y / z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -4.2e-157:
		tmp = x * ((y / z) / z)
	else:
		tmp = (x / z) * (y / z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -4.2e-157)
		tmp = Float64(x * Float64(Float64(y / z) / z));
	else
		tmp = Float64(Float64(x / z) * Float64(y / z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -4.2e-157)
		tmp = x * ((y / z) / z);
	else
		tmp = (x / z) * (y / z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -4.2e-157], N[(x * N[(N[(y / z), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], N[(N[(x / z), $MachinePrecision] * N[(y / z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -4.2e-157

   1. Initial program 82.8%

      \[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \]
   2. Step-by-step derivation

      1. associate-*l* 82.8%

         \[\leadsto \frac{x \cdot y}{\color{blue}{z \cdot \left(z \cdot \left(z + 1\right)\right)}} \]

      2. times-frac 93.5%

         \[\leadsto \color{blue}{\frac{x}{z} \cdot \frac{y}{z \cdot \left(z + 1\right)}} \]

      3. distribute-lft-in 93.4%

         \[\leadsto \frac{x}{z} \cdot \frac{y}{\color{blue}{z \cdot z + z \cdot 1}} \]

      4. fma-def 93.4%

         \[\leadsto \frac{x}{z} \cdot \frac{y}{\color{blue}{\mathsf{fma}\left(z, z, z \cdot 1\right)}} \]

      5. *-rgt-identity 93.4%

         \[\leadsto \frac{x}{z} \cdot \frac{y}{\mathsf{fma}\left(z, z, \color{blue}{z}\right)} \]

   3. Simplified 93.4%

      \[\leadsto \color{blue}{\frac{x}{z} \cdot \frac{y}{\mathsf{fma}\left(z, z, z\right)}} \]

   4. Taylor expanded in z around 0 70.1%

      \[\leadsto \frac{x}{z} \cdot \color{blue}{\frac{y}{z}} \]

   5. Taylor expanded in x around 0 73.5%

      \[\leadsto \color{blue}{\frac{y \cdot x}{{z}^{2}}} \]
   6. Step-by-step derivation

      1. *-commutative 73.5%

         \[\leadsto \frac{\color{blue}{x \cdot y}}{{z}^{2}} \]

      2. unpow2 73.5%

         \[\leadsto \frac{x \cdot y}{\color{blue}{z \cdot z}} \]

      3. associate-*r/ 74.9%

         \[\leadsto \color{blue}{x \cdot \frac{y}{z \cdot z}} \]

      4. associate-/r* 75.4%

         \[\leadsto x \cdot \color{blue}{\frac{\frac{y}{z}}{z}} \]

   7. Simplified 75.4%

      \[\leadsto \color{blue}{x \cdot \frac{\frac{y}{z}}{z}} \]

   if -4.2e-157 < x

   1. Initial program 87.6%

      \[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \]
   2. Step-by-step derivation

      1. associate-*l* 87.6%

         \[\leadsto \frac{x \cdot y}{\color{blue}{z \cdot \left(z \cdot \left(z + 1\right)\right)}} \]

      2. times-frac 96.0%

         \[\leadsto \color{blue}{\frac{x}{z} \cdot \frac{y}{z \cdot \left(z + 1\right)}} \]

      3. distribute-lft-in 96.0%

         \[\leadsto \frac{x}{z} \cdot \frac{y}{\color{blue}{z \cdot z + z \cdot 1}} \]

      4. fma-def 96.0%

         \[\leadsto \frac{x}{z} \cdot \frac{y}{\color{blue}{\mathsf{fma}\left(z, z, z \cdot 1\right)}} \]

      5. *-rgt-identity 96.0%

         \[\leadsto \frac{x}{z} \cdot \frac{y}{\mathsf{fma}\left(z, z, \color{blue}{z}\right)} \]

   3. Simplified 96.0%

      \[\leadsto \color{blue}{\frac{x}{z} \cdot \frac{y}{\mathsf{fma}\left(z, z, z\right)}} \]

   4. Taylor expanded in z around 0 81.3%

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

4. Final simplification 79.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.2 \cdot 10^{-157}:\\ \;\;\;\;x \cdot \frac{\frac{y}{z}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot \frac{y}{z}\\ \end{array} \]

Alternative 9: 76.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2 \cdot 10^{+43}:\\ \;\;\;\;x \cdot \frac{\frac{y}{z}}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{z \cdot z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 2e+43) (* x (/ (/ y z) z)) (* y (/ x (* z z)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 2e+43) {
		tmp = x * ((y / z) / z);
	} else {
		tmp = y * (x / (z * z));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= 2d+43) then
        tmp = x * ((y / z) / z)
    else
        tmp = y * (x / (z * z))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 2e+43) {
		tmp = x * ((y / z) / z);
	} else {
		tmp = y * (x / (z * z));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 2e+43:
		tmp = x * ((y / z) / z)
	else:
		tmp = y * (x / (z * z))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 2e+43)
		tmp = Float64(x * Float64(Float64(y / z) / z));
	else
		tmp = Float64(y * Float64(x / Float64(z * z)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 2e+43)
		tmp = x * ((y / z) / z);
	else
		tmp = y * (x / (z * z));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 2e+43], N[(x * N[(N[(y / z), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], N[(y * N[(x / N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2.00000000000000003e43

   1. Initial program 86.7%

      \[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \]
   2. Step-by-step derivation

      1. associate-*l* 86.7%

         \[\leadsto \frac{x \cdot y}{\color{blue}{z \cdot \left(z \cdot \left(z + 1\right)\right)}} \]

      2. times-frac 97.2%

         \[\leadsto \color{blue}{\frac{x}{z} \cdot \frac{y}{z \cdot \left(z + 1\right)}} \]

      3. distribute-lft-in 97.2%

         \[\leadsto \frac{x}{z} \cdot \frac{y}{\color{blue}{z \cdot z + z \cdot 1}} \]

      4. fma-def 97.2%

         \[\leadsto \frac{x}{z} \cdot \frac{y}{\color{blue}{\mathsf{fma}\left(z, z, z \cdot 1\right)}} \]

      5. *-rgt-identity 97.2%

         \[\leadsto \frac{x}{z} \cdot \frac{y}{\mathsf{fma}\left(z, z, \color{blue}{z}\right)} \]

   3. Simplified 97.2%

      \[\leadsto \color{blue}{\frac{x}{z} \cdot \frac{y}{\mathsf{fma}\left(z, z, z\right)}} \]

   4. Taylor expanded in z around 0 79.6%

      \[\leadsto \frac{x}{z} \cdot \color{blue}{\frac{y}{z}} \]

   5. Taylor expanded in x around 0 76.0%

      \[\leadsto \color{blue}{\frac{y \cdot x}{{z}^{2}}} \]
   6. Step-by-step derivation

      1. *-commutative 76.0%

         \[\leadsto \frac{\color{blue}{x \cdot y}}{{z}^{2}} \]

      2. unpow2 76.0%

         \[\leadsto \frac{x \cdot y}{\color{blue}{z \cdot z}} \]

      3. associate-*r/ 75.2%

         \[\leadsto \color{blue}{x \cdot \frac{y}{z \cdot z}} \]

      4. associate-/r* 78.9%

         \[\leadsto x \cdot \color{blue}{\frac{\frac{y}{z}}{z}} \]

   7. Simplified 78.9%

      \[\leadsto \color{blue}{x \cdot \frac{\frac{y}{z}}{z}} \]

   if 2.00000000000000003e43 < y

   1. Initial program 82.4%

      \[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \]
   2. Step-by-step derivation

      1. times-frac 89.0%

         \[\leadsto \color{blue}{\frac{x}{z \cdot z} \cdot \frac{y}{z + 1}} \]

   3. Simplified 89.0%

      \[\leadsto \color{blue}{\frac{x}{z \cdot z} \cdot \frac{y}{z + 1}} \]

   4. Taylor expanded in z around 0 76.1%

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

4. Final simplification 78.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2 \cdot 10^{+43}:\\ \;\;\;\;x \cdot \frac{\frac{y}{z}}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{z \cdot z}\\ \end{array} \]

Alternative 10: 75.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.02 \cdot 10^{-103}:\\ \;\;\;\;x \cdot \frac{y}{z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot \frac{y}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -1.02e-103) (* x (/ y (* z z))) (* (/ x z) (/ y z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -1.02e-103) {
		tmp = x * (y / (z * z));
	} else {
		tmp = (x / z) * (y / z);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x <= (-1.02d-103)) then
        tmp = x * (y / (z * z))
    else
        tmp = (x / z) * (y / z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -1.02e-103) {
		tmp = x * (y / (z * z));
	} else {
		tmp = (x / z) * (y / z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -1.02e-103:
		tmp = x * (y / (z * z))
	else:
		tmp = (x / z) * (y / z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -1.02e-103)
		tmp = Float64(x * Float64(y / Float64(z * z)));
	else
		tmp = Float64(Float64(x / z) * Float64(y / z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -1.02e-103)
		tmp = x * (y / (z * z));
	else
		tmp = (x / z) * (y / z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -1.02e-103], N[(x * N[(y / N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / z), $MachinePrecision] * N[(y / z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.01999999999999998e-103

   1. Initial program 84.8%

      \[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \]
   2. Step-by-step derivation

      1. associate-*l* 84.8%

         \[\leadsto \frac{x \cdot y}{\color{blue}{z \cdot \left(z \cdot \left(z + 1\right)\right)}} \]

      2. times-frac 93.0%

         \[\leadsto \color{blue}{\frac{x}{z} \cdot \frac{y}{z \cdot \left(z + 1\right)}} \]

      3. distribute-lft-in 93.0%

         \[\leadsto \frac{x}{z} \cdot \frac{y}{\color{blue}{z \cdot z + z \cdot 1}} \]

      4. fma-def 93.0%

         \[\leadsto \frac{x}{z} \cdot \frac{y}{\color{blue}{\mathsf{fma}\left(z, z, z \cdot 1\right)}} \]

      5. *-rgt-identity 93.0%

         \[\leadsto \frac{x}{z} \cdot \frac{y}{\mathsf{fma}\left(z, z, \color{blue}{z}\right)} \]

   3. Simplified 93.0%

      \[\leadsto \color{blue}{\frac{x}{z} \cdot \frac{y}{\mathsf{fma}\left(z, z, z\right)}} \]

   4. Taylor expanded in z around 0 74.8%

      \[\leadsto \color{blue}{\frac{y \cdot x}{{z}^{2}}} \]
   5. Step-by-step derivation

      1. unpow2 74.8%

         \[\leadsto \frac{y \cdot x}{\color{blue}{z \cdot z}} \]

      2. associate-/l* 77.2%

         \[\leadsto \color{blue}{\frac{y}{\frac{z \cdot z}{x}}} \]

      3. associate-/l* 70.6%

         \[\leadsto \frac{y}{\color{blue}{\frac{z}{\frac{x}{z}}}} \]

      4. associate-/r/ 70.6%

         \[\leadsto \frac{y}{\color{blue}{\frac{z}{x} \cdot z}} \]

   6. Simplified 70.6%

      \[\leadsto \color{blue}{\frac{y}{\frac{z}{x} \cdot z}} \]
   7. Step-by-step derivation

      1. associate-*l/ 77.2%

         \[\leadsto \frac{y}{\color{blue}{\frac{z \cdot z}{x}}} \]

      2. associate-/r/ 75.1%

         \[\leadsto \color{blue}{\frac{y}{z \cdot z} \cdot x} \]

   8. Applied egg-rr 75.1%

      \[\leadsto \color{blue}{\frac{y}{z \cdot z} \cdot x} \]

   if -1.01999999999999998e-103 < x

   1. Initial program 86.4%

      \[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \]
   2. Step-by-step derivation

      1. associate-*l* 86.4%

         \[\leadsto \frac{x \cdot y}{\color{blue}{z \cdot \left(z \cdot \left(z + 1\right)\right)}} \]

      2. times-frac 96.1%

         \[\leadsto \color{blue}{\frac{x}{z} \cdot \frac{y}{z \cdot \left(z + 1\right)}} \]

      3. distribute-lft-in 96.1%

         \[\leadsto \frac{x}{z} \cdot \frac{y}{\color{blue}{z \cdot z + z \cdot 1}} \]

      4. fma-def 96.1%

         \[\leadsto \frac{x}{z} \cdot \frac{y}{\color{blue}{\mathsf{fma}\left(z, z, z \cdot 1\right)}} \]

      5. *-rgt-identity 96.1%

         \[\leadsto \frac{x}{z} \cdot \frac{y}{\mathsf{fma}\left(z, z, \color{blue}{z}\right)} \]

   3. Simplified 96.1%

      \[\leadsto \color{blue}{\frac{x}{z} \cdot \frac{y}{\mathsf{fma}\left(z, z, z\right)}} \]

   4. Taylor expanded in z around 0 82.0%

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

4. Final simplification 79.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.02 \cdot 10^{-103}:\\ \;\;\;\;x \cdot \frac{y}{z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot \frac{y}{z}\\ \end{array} \]

Alternative 11: 76.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -7 \cdot 10^{-28}:\\ \;\;\;\;x \cdot \frac{y}{z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z \cdot \frac{z}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -7e-28) (* x (/ y (* z z))) (/ y (* z (/ z x)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -7e-28) {
		tmp = x * (y / (z * z));
	} else {
		tmp = y / (z * (z / x));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x <= (-7d-28)) then
        tmp = x * (y / (z * z))
    else
        tmp = y / (z * (z / x))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -7e-28) {
		tmp = x * (y / (z * z));
	} else {
		tmp = y / (z * (z / x));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -7e-28:
		tmp = x * (y / (z * z))
	else:
		tmp = y / (z * (z / x))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -7e-28)
		tmp = Float64(x * Float64(y / Float64(z * z)));
	else
		tmp = Float64(y / Float64(z * Float64(z / x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -7e-28)
		tmp = x * (y / (z * z));
	else
		tmp = y / (z * (z / x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -7e-28], N[(x * N[(y / N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y / N[(z * N[(z / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -6.9999999999999999e-28

   1. Initial program 82.3%

      \[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \]
   2. Step-by-step derivation

      1. associate-*l* 82.3%

         \[\leadsto \frac{x \cdot y}{\color{blue}{z \cdot \left(z \cdot \left(z + 1\right)\right)}} \]

      2. times-frac 91.9%

         \[\leadsto \color{blue}{\frac{x}{z} \cdot \frac{y}{z \cdot \left(z + 1\right)}} \]

      3. distribute-lft-in 91.8%

         \[\leadsto \frac{x}{z} \cdot \frac{y}{\color{blue}{z \cdot z + z \cdot 1}} \]

      4. fma-def 91.8%

         \[\leadsto \frac{x}{z} \cdot \frac{y}{\color{blue}{\mathsf{fma}\left(z, z, z \cdot 1\right)}} \]

      5. *-rgt-identity 91.8%

         \[\leadsto \frac{x}{z} \cdot \frac{y}{\mathsf{fma}\left(z, z, \color{blue}{z}\right)} \]

   3. Simplified 91.8%

      \[\leadsto \color{blue}{\frac{x}{z} \cdot \frac{y}{\mathsf{fma}\left(z, z, z\right)}} \]

   4. Taylor expanded in z around 0 75.9%

      \[\leadsto \color{blue}{\frac{y \cdot x}{{z}^{2}}} \]
   5. Step-by-step derivation

      1. unpow2 75.9%

         \[\leadsto \frac{y \cdot x}{\color{blue}{z \cdot z}} \]

      2. associate-/l* 77.5%

         \[\leadsto \color{blue}{\frac{y}{\frac{z \cdot z}{x}}} \]

      3. associate-/l* 69.9%

         \[\leadsto \frac{y}{\color{blue}{\frac{z}{\frac{x}{z}}}} \]

      4. associate-/r/ 69.9%

         \[\leadsto \frac{y}{\color{blue}{\frac{z}{x} \cdot z}} \]

   6. Simplified 69.9%

      \[\leadsto \color{blue}{\frac{y}{\frac{z}{x} \cdot z}} \]
   7. Step-by-step derivation

      1. associate-*l/ 77.5%

         \[\leadsto \frac{y}{\color{blue}{\frac{z \cdot z}{x}}} \]

      2. associate-/r/ 77.6%

         \[\leadsto \color{blue}{\frac{y}{z \cdot z} \cdot x} \]

   8. Applied egg-rr 77.6%

      \[\leadsto \color{blue}{\frac{y}{z \cdot z} \cdot x} \]

   if -6.9999999999999999e-28 < x

   1. Initial program 87.3%

      \[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \]
   2. Step-by-step derivation

      1. associate-*l* 87.3%

         \[\leadsto \frac{x \cdot y}{\color{blue}{z \cdot \left(z \cdot \left(z + 1\right)\right)}} \]

      2. times-frac 96.4%

         \[\leadsto \color{blue}{\frac{x}{z} \cdot \frac{y}{z \cdot \left(z + 1\right)}} \]

      3. distribute-lft-in 96.4%

         \[\leadsto \frac{x}{z} \cdot \frac{y}{\color{blue}{z \cdot z + z \cdot 1}} \]

      4. fma-def 96.4%

         \[\leadsto \frac{x}{z} \cdot \frac{y}{\color{blue}{\mathsf{fma}\left(z, z, z \cdot 1\right)}} \]

      5. *-rgt-identity 96.4%

         \[\leadsto \frac{x}{z} \cdot \frac{y}{\mathsf{fma}\left(z, z, \color{blue}{z}\right)} \]

   3. Simplified 96.4%

      \[\leadsto \color{blue}{\frac{x}{z} \cdot \frac{y}{\mathsf{fma}\left(z, z, z\right)}} \]

   4. Taylor expanded in z around 0 74.9%

      \[\leadsto \color{blue}{\frac{y \cdot x}{{z}^{2}}} \]
   5. Step-by-step derivation

      1. unpow2 74.9%

         \[\leadsto \frac{y \cdot x}{\color{blue}{z \cdot z}} \]

      2. associate-/l* 79.3%

         \[\leadsto \color{blue}{\frac{y}{\frac{z \cdot z}{x}}} \]

      3. associate-/l* 83.9%

         \[\leadsto \frac{y}{\color{blue}{\frac{z}{\frac{x}{z}}}} \]

      4. associate-/r/ 83.9%

         \[\leadsto \frac{y}{\color{blue}{\frac{z}{x} \cdot z}} \]

   6. Simplified 83.9%

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

4. Final simplification 82.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -7 \cdot 10^{-28}:\\ \;\;\;\;x \cdot \frac{y}{z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z \cdot \frac{z}{x}}\\ \end{array} \]

Alternative 12: 39.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2 \cdot 10^{-310}:\\ \;\;\;\;x \cdot \left(-\frac{y}{z}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{z}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -2e-310) (* x (- (/ y z))) (/ y (/ z x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -2e-310) {
		tmp = x * -(y / z);
	} else {
		tmp = y / (z / x);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (z <= (-2d-310)) then
        tmp = x * -(y / z)
    else
        tmp = y / (z / x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -2e-310) {
		tmp = x * -(y / z);
	} else {
		tmp = y / (z / x);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -2e-310:
		tmp = x * -(y / z)
	else:
		tmp = y / (z / x)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -2e-310)
		tmp = Float64(x * Float64(-Float64(y / z)));
	else
		tmp = Float64(y / Float64(z / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -2e-310)
		tmp = x * -(y / z);
	else
		tmp = y / (z / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -2e-310], N[(x * (-N[(y / z), $MachinePrecision])), $MachinePrecision], N[(y / N[(z / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.999999999999994e-310

   1. Initial program 84.1%

      \[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \]
   2. Step-by-step derivation

      1. associate-*l* 84.1%

         \[\leadsto \frac{x \cdot y}{\color{blue}{z \cdot \left(z \cdot \left(z + 1\right)\right)}} \]

      2. times-frac 95.9%

         \[\leadsto \color{blue}{\frac{x}{z} \cdot \frac{y}{z \cdot \left(z + 1\right)}} \]

      3. distribute-lft-in 95.9%

         \[\leadsto \frac{x}{z} \cdot \frac{y}{\color{blue}{z \cdot z + z \cdot 1}} \]

      4. fma-def 95.9%

         \[\leadsto \frac{x}{z} \cdot \frac{y}{\color{blue}{\mathsf{fma}\left(z, z, z \cdot 1\right)}} \]

      5. *-rgt-identity 95.9%

         \[\leadsto \frac{x}{z} \cdot \frac{y}{\mathsf{fma}\left(z, z, \color{blue}{z}\right)} \]

   3. Simplified 95.9%

      \[\leadsto \color{blue}{\frac{x}{z} \cdot \frac{y}{\mathsf{fma}\left(z, z, z\right)}} \]

   4. Taylor expanded in z around 0 68.6%

      \[\leadsto \frac{x}{z} \cdot \color{blue}{\left(-1 \cdot y + \frac{y}{z}\right)} \]
   5. Step-by-step derivation

      1. neg-mul-1 68.6%

         \[\leadsto \frac{x}{z} \cdot \left(\color{blue}{\left(-y\right)} + \frac{y}{z}\right) \]

      2. +-commutative 68.6%

         \[\leadsto \frac{x}{z} \cdot \color{blue}{\left(\frac{y}{z} + \left(-y\right)\right)} \]

      3. unsub-neg 68.6%

         \[\leadsto \frac{x}{z} \cdot \color{blue}{\left(\frac{y}{z} - y\right)} \]

   6. Simplified 68.6%

      \[\leadsto \frac{x}{z} \cdot \color{blue}{\left(\frac{y}{z} - y\right)} \]

   7. Taylor expanded in z around inf 36.4%

      \[\leadsto \color{blue}{-1 \cdot \frac{y \cdot x}{z}} \]
   8. Step-by-step derivation

      1. mul-1-neg 36.4%

         \[\leadsto \color{blue}{-\frac{y \cdot x}{z}} \]

      2. *-commutative 36.4%

         \[\leadsto -\frac{\color{blue}{x \cdot y}}{z} \]

      3. associate-*r/ 41.3%

         \[\leadsto -\color{blue}{x \cdot \frac{y}{z}} \]

      4. distribute-rgt-neg-in 41.3%

         \[\leadsto \color{blue}{x \cdot \left(-\frac{y}{z}\right)} \]

   9. Simplified 41.3%

      \[\leadsto \color{blue}{x \cdot \left(-\frac{y}{z}\right)} \]

   if -1.999999999999994e-310 < z

   1. Initial program 87.7%

      \[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \]
   2. Step-by-step derivation

      1. associate-*l* 87.8%

         \[\leadsto \frac{x \cdot y}{\color{blue}{z \cdot \left(z \cdot \left(z + 1\right)\right)}} \]

      2. times-frac 94.0%

         \[\leadsto \color{blue}{\frac{x}{z} \cdot \frac{y}{z \cdot \left(z + 1\right)}} \]

      3. distribute-lft-in 94.0%

         \[\leadsto \frac{x}{z} \cdot \frac{y}{\color{blue}{z \cdot z + z \cdot 1}} \]

      4. fma-def 94.0%

         \[\leadsto \frac{x}{z} \cdot \frac{y}{\color{blue}{\mathsf{fma}\left(z, z, z \cdot 1\right)}} \]

      5. *-rgt-identity 94.0%

         \[\leadsto \frac{x}{z} \cdot \frac{y}{\mathsf{fma}\left(z, z, \color{blue}{z}\right)} \]

   3. Simplified 94.0%

      \[\leadsto \color{blue}{\frac{x}{z} \cdot \frac{y}{\mathsf{fma}\left(z, z, z\right)}} \]

   4. Taylor expanded in z around 0 70.9%

      \[\leadsto \frac{x}{z} \cdot \color{blue}{\left(-1 \cdot y + \frac{y}{z}\right)} \]
   5. Step-by-step derivation

      1. neg-mul-1 70.9%

         \[\leadsto \frac{x}{z} \cdot \left(\color{blue}{\left(-y\right)} + \frac{y}{z}\right) \]

      2. +-commutative 70.9%

         \[\leadsto \frac{x}{z} \cdot \color{blue}{\left(\frac{y}{z} + \left(-y\right)\right)} \]

      3. unsub-neg 70.9%

         \[\leadsto \frac{x}{z} \cdot \color{blue}{\left(\frac{y}{z} - y\right)} \]

   6. Simplified 70.9%

      \[\leadsto \frac{x}{z} \cdot \color{blue}{\left(\frac{y}{z} - y\right)} \]

   7. Taylor expanded in z around inf 15.9%

      \[\leadsto \color{blue}{-1 \cdot \frac{y \cdot x}{z}} \]
   8. Step-by-step derivation

      1. mul-1-neg 15.9%

         \[\leadsto \color{blue}{-\frac{y \cdot x}{z}} \]

      2. *-commutative 15.9%

         \[\leadsto -\frac{\color{blue}{x \cdot y}}{z} \]

      3. associate-*r/ 17.4%

         \[\leadsto -\color{blue}{x \cdot \frac{y}{z}} \]

      4. distribute-rgt-neg-in 17.4%

         \[\leadsto \color{blue}{x \cdot \left(-\frac{y}{z}\right)} \]

   9. Simplified 17.4%

      \[\leadsto \color{blue}{x \cdot \left(-\frac{y}{z}\right)} \]
   10. Step-by-step derivation

       1. expm1-log1p-u 17.3%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(x \cdot \left(-\frac{y}{z}\right)\right)\right)} \]

       2. expm1-udef 29.2%

          \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(x \cdot \left(-\frac{y}{z}\right)\right)} - 1} \]

       3. add-sqr-sqrt 16.3%

          \[\leadsto e^{\mathsf{log1p}\left(x \cdot \color{blue}{\left(\sqrt{-\frac{y}{z}} \cdot \sqrt{-\frac{y}{z}}\right)}\right)} - 1 \]

       4. sqrt-unprod 42.0%

          \[\leadsto e^{\mathsf{log1p}\left(x \cdot \color{blue}{\sqrt{\left(-\frac{y}{z}\right) \cdot \left(-\frac{y}{z}\right)}}\right)} - 1 \]

       5. sqr-neg 42.0%

          \[\leadsto e^{\mathsf{log1p}\left(x \cdot \sqrt{\color{blue}{\frac{y}{z} \cdot \frac{y}{z}}}\right)} - 1 \]

       6. sqrt-unprod 27.6%

          \[\leadsto e^{\mathsf{log1p}\left(x \cdot \color{blue}{\left(\sqrt{\frac{y}{z}} \cdot \sqrt{\frac{y}{z}}\right)}\right)} - 1 \]

       7. add-sqr-sqrt 44.4%

          \[\leadsto e^{\mathsf{log1p}\left(x \cdot \color{blue}{\frac{y}{z}}\right)} - 1 \]

       8. clear-num 44.4%

          \[\leadsto e^{\mathsf{log1p}\left(x \cdot \color{blue}{\frac{1}{\frac{z}{y}}}\right)} - 1 \]

       9. un-div-inv 44.4%

          \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{x}{\frac{z}{y}}}\right)} - 1 \]

   11. Applied egg-rr 44.4%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{x}{\frac{z}{y}}\right)} - 1} \]
   12. Step-by-step derivation

       1. expm1-def 32.8%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{x}{\frac{z}{y}}\right)\right)} \]

       2. expm1-log1p 43.6%

          \[\leadsto \color{blue}{\frac{x}{\frac{z}{y}}} \]

       3. associate-/r/ 43.5%

          \[\leadsto \color{blue}{\frac{x}{z} \cdot y} \]

       4. *-commutative 43.5%

          \[\leadsto \color{blue}{y \cdot \frac{x}{z}} \]

   13. Simplified 43.5%

       \[\leadsto \color{blue}{y \cdot \frac{x}{z}} \]
   14. Step-by-step derivation

       1. clear-num 43.5%

          \[\leadsto y \cdot \color{blue}{\frac{1}{\frac{z}{x}}} \]

       2. un-div-inv 42.7%

          \[\leadsto \color{blue}{\frac{y}{\frac{z}{x}}} \]

   15. Applied egg-rr 42.7%

       \[\leadsto \color{blue}{\frac{y}{\frac{z}{x}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 42.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2 \cdot 10^{-310}:\\ \;\;\;\;x \cdot \left(-\frac{y}{z}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{z}{x}}\\ \end{array} \]

Alternative 13: 39.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2 \cdot 10^{-310}:\\ \;\;\;\;-\frac{x}{z} \cdot y\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{z}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -2e-310) (- (* (/ x z) y)) (/ y (/ z x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -2e-310) {
		tmp = -((x / z) * y);
	} else {
		tmp = y / (z / x);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (z <= (-2d-310)) then
        tmp = -((x / z) * y)
    else
        tmp = y / (z / x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -2e-310) {
		tmp = -((x / z) * y);
	} else {
		tmp = y / (z / x);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -2e-310:
		tmp = -((x / z) * y)
	else:
		tmp = y / (z / x)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -2e-310)
		tmp = Float64(-Float64(Float64(x / z) * y));
	else
		tmp = Float64(y / Float64(z / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -2e-310)
		tmp = -((x / z) * y);
	else
		tmp = y / (z / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -2e-310], (-N[(N[(x / z), $MachinePrecision] * y), $MachinePrecision]), N[(y / N[(z / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.999999999999994e-310

   1. Initial program 84.1%

      \[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \]
   2. Step-by-step derivation

      1. associate-*l* 84.1%

         \[\leadsto \frac{x \cdot y}{\color{blue}{z \cdot \left(z \cdot \left(z + 1\right)\right)}} \]

      2. times-frac 95.9%

         \[\leadsto \color{blue}{\frac{x}{z} \cdot \frac{y}{z \cdot \left(z + 1\right)}} \]

      3. distribute-lft-in 95.9%

         \[\leadsto \frac{x}{z} \cdot \frac{y}{\color{blue}{z \cdot z + z \cdot 1}} \]

      4. fma-def 95.9%

         \[\leadsto \frac{x}{z} \cdot \frac{y}{\color{blue}{\mathsf{fma}\left(z, z, z \cdot 1\right)}} \]

      5. *-rgt-identity 95.9%

         \[\leadsto \frac{x}{z} \cdot \frac{y}{\mathsf{fma}\left(z, z, \color{blue}{z}\right)} \]

   3. Simplified 95.9%

      \[\leadsto \color{blue}{\frac{x}{z} \cdot \frac{y}{\mathsf{fma}\left(z, z, z\right)}} \]

   4. Taylor expanded in z around 0 68.6%

      \[\leadsto \frac{x}{z} \cdot \color{blue}{\left(-1 \cdot y + \frac{y}{z}\right)} \]
   5. Step-by-step derivation

      1. neg-mul-1 68.6%

         \[\leadsto \frac{x}{z} \cdot \left(\color{blue}{\left(-y\right)} + \frac{y}{z}\right) \]

      2. +-commutative 68.6%

         \[\leadsto \frac{x}{z} \cdot \color{blue}{\left(\frac{y}{z} + \left(-y\right)\right)} \]

      3. unsub-neg 68.6%

         \[\leadsto \frac{x}{z} \cdot \color{blue}{\left(\frac{y}{z} - y\right)} \]

   6. Simplified 68.6%

      \[\leadsto \frac{x}{z} \cdot \color{blue}{\left(\frac{y}{z} - y\right)} \]

   7. Taylor expanded in z around inf 42.6%

      \[\leadsto \frac{x}{z} \cdot \color{blue}{\left(-1 \cdot y\right)} \]
   8. Step-by-step derivation

      1. neg-mul-1 42.6%

         \[\leadsto \frac{x}{z} \cdot \color{blue}{\left(-y\right)} \]

   9. Simplified 42.6%

      \[\leadsto \frac{x}{z} \cdot \color{blue}{\left(-y\right)} \]

   if -1.999999999999994e-310 < z

   1. Initial program 87.7%

      \[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \]
   2. Step-by-step derivation

      1. associate-*l* 87.8%

         \[\leadsto \frac{x \cdot y}{\color{blue}{z \cdot \left(z \cdot \left(z + 1\right)\right)}} \]

      2. times-frac 94.0%

         \[\leadsto \color{blue}{\frac{x}{z} \cdot \frac{y}{z \cdot \left(z + 1\right)}} \]

      3. distribute-lft-in 94.0%

         \[\leadsto \frac{x}{z} \cdot \frac{y}{\color{blue}{z \cdot z + z \cdot 1}} \]

      4. fma-def 94.0%

         \[\leadsto \frac{x}{z} \cdot \frac{y}{\color{blue}{\mathsf{fma}\left(z, z, z \cdot 1\right)}} \]

      5. *-rgt-identity 94.0%

         \[\leadsto \frac{x}{z} \cdot \frac{y}{\mathsf{fma}\left(z, z, \color{blue}{z}\right)} \]

   3. Simplified 94.0%

      \[\leadsto \color{blue}{\frac{x}{z} \cdot \frac{y}{\mathsf{fma}\left(z, z, z\right)}} \]

   4. Taylor expanded in z around 0 70.9%

      \[\leadsto \frac{x}{z} \cdot \color{blue}{\left(-1 \cdot y + \frac{y}{z}\right)} \]
   5. Step-by-step derivation

      1. neg-mul-1 70.9%

         \[\leadsto \frac{x}{z} \cdot \left(\color{blue}{\left(-y\right)} + \frac{y}{z}\right) \]

      2. +-commutative 70.9%

         \[\leadsto \frac{x}{z} \cdot \color{blue}{\left(\frac{y}{z} + \left(-y\right)\right)} \]

      3. unsub-neg 70.9%

         \[\leadsto \frac{x}{z} \cdot \color{blue}{\left(\frac{y}{z} - y\right)} \]

   6. Simplified 70.9%

      \[\leadsto \frac{x}{z} \cdot \color{blue}{\left(\frac{y}{z} - y\right)} \]

   7. Taylor expanded in z around inf 15.9%

      \[\leadsto \color{blue}{-1 \cdot \frac{y \cdot x}{z}} \]
   8. Step-by-step derivation

      1. mul-1-neg 15.9%

         \[\leadsto \color{blue}{-\frac{y \cdot x}{z}} \]

      2. *-commutative 15.9%

         \[\leadsto -\frac{\color{blue}{x \cdot y}}{z} \]

      3. associate-*r/ 17.4%

         \[\leadsto -\color{blue}{x \cdot \frac{y}{z}} \]

      4. distribute-rgt-neg-in 17.4%

         \[\leadsto \color{blue}{x \cdot \left(-\frac{y}{z}\right)} \]

   9. Simplified 17.4%

      \[\leadsto \color{blue}{x \cdot \left(-\frac{y}{z}\right)} \]
   10. Step-by-step derivation

       1. expm1-log1p-u 17.3%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(x \cdot \left(-\frac{y}{z}\right)\right)\right)} \]

       2. expm1-udef 29.2%

          \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(x \cdot \left(-\frac{y}{z}\right)\right)} - 1} \]

       3. add-sqr-sqrt 16.3%

          \[\leadsto e^{\mathsf{log1p}\left(x \cdot \color{blue}{\left(\sqrt{-\frac{y}{z}} \cdot \sqrt{-\frac{y}{z}}\right)}\right)} - 1 \]

       4. sqrt-unprod 42.0%

          \[\leadsto e^{\mathsf{log1p}\left(x \cdot \color{blue}{\sqrt{\left(-\frac{y}{z}\right) \cdot \left(-\frac{y}{z}\right)}}\right)} - 1 \]

       5. sqr-neg 42.0%

          \[\leadsto e^{\mathsf{log1p}\left(x \cdot \sqrt{\color{blue}{\frac{y}{z} \cdot \frac{y}{z}}}\right)} - 1 \]

       6. sqrt-unprod 27.6%

          \[\leadsto e^{\mathsf{log1p}\left(x \cdot \color{blue}{\left(\sqrt{\frac{y}{z}} \cdot \sqrt{\frac{y}{z}}\right)}\right)} - 1 \]

       7. add-sqr-sqrt 44.4%

          \[\leadsto e^{\mathsf{log1p}\left(x \cdot \color{blue}{\frac{y}{z}}\right)} - 1 \]

       8. clear-num 44.4%

          \[\leadsto e^{\mathsf{log1p}\left(x \cdot \color{blue}{\frac{1}{\frac{z}{y}}}\right)} - 1 \]

       9. un-div-inv 44.4%

          \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{x}{\frac{z}{y}}}\right)} - 1 \]

   11. Applied egg-rr 44.4%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{x}{\frac{z}{y}}\right)} - 1} \]
   12. Step-by-step derivation

       1. expm1-def 32.8%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{x}{\frac{z}{y}}\right)\right)} \]

       2. expm1-log1p 43.6%

          \[\leadsto \color{blue}{\frac{x}{\frac{z}{y}}} \]

       3. associate-/r/ 43.5%

          \[\leadsto \color{blue}{\frac{x}{z} \cdot y} \]

       4. *-commutative 43.5%

          \[\leadsto \color{blue}{y \cdot \frac{x}{z}} \]

   13. Simplified 43.5%

       \[\leadsto \color{blue}{y \cdot \frac{x}{z}} \]
   14. Step-by-step derivation

       1. clear-num 43.5%

          \[\leadsto y \cdot \color{blue}{\frac{1}{\frac{z}{x}}} \]

       2. un-div-inv 42.7%

          \[\leadsto \color{blue}{\frac{y}{\frac{z}{x}}} \]

   15. Applied egg-rr 42.7%

       \[\leadsto \color{blue}{\frac{y}{\frac{z}{x}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 42.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2 \cdot 10^{-310}:\\ \;\;\;\;-\frac{x}{z} \cdot y\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{z}{x}}\\ \end{array} \]

Alternative 14: 73.6% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ x \cdot \frac{\frac{y}{z}}{z} \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (* x (/ (/ y z) z)))

C

double code(double x, double y, double z) {
	return x * ((y / z) / z);
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = x * ((y / z) / z)
end function

Java

public static double code(double x, double y, double z) {
	return x * ((y / z) / z);
}

Python

def code(x, y, z):
	return x * ((y / z) / z)

Julia

function code(x, y, z)
	return Float64(x * Float64(Float64(y / z) / z))
end

MATLAB

function tmp = code(x, y, z)
	tmp = x * ((y / z) / z);
end

Wolfram

code[x_, y_, z_] := N[(x * N[(N[(y / z), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 85.8%

   \[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \]
2. Step-by-step derivation

   1. associate-*l* 85.8%

      \[\leadsto \frac{x \cdot y}{\color{blue}{z \cdot \left(z \cdot \left(z + 1\right)\right)}} \]

   2. times-frac 95.0%

      \[\leadsto \color{blue}{\frac{x}{z} \cdot \frac{y}{z \cdot \left(z + 1\right)}} \]

   3. distribute-lft-in 95.0%

      \[\leadsto \frac{x}{z} \cdot \frac{y}{\color{blue}{z \cdot z + z \cdot 1}} \]

   4. fma-def 95.0%

      \[\leadsto \frac{x}{z} \cdot \frac{y}{\color{blue}{\mathsf{fma}\left(z, z, z \cdot 1\right)}} \]

   5. *-rgt-identity 95.0%

      \[\leadsto \frac{x}{z} \cdot \frac{y}{\mathsf{fma}\left(z, z, \color{blue}{z}\right)} \]

3. Simplified 95.0%

   \[\leadsto \color{blue}{\frac{x}{z} \cdot \frac{y}{\mathsf{fma}\left(z, z, z\right)}} \]

4. Taylor expanded in z around 0 77.0%

   \[\leadsto \frac{x}{z} \cdot \color{blue}{\frac{y}{z}} \]

5. Taylor expanded in x around 0 75.2%

   \[\leadsto \color{blue}{\frac{y \cdot x}{{z}^{2}}} \]
6. Step-by-step derivation

   1. *-commutative 75.2%

      \[\leadsto \frac{\color{blue}{x \cdot y}}{{z}^{2}} \]

   2. unpow2 75.2%

      \[\leadsto \frac{x \cdot y}{\color{blue}{z \cdot z}} \]

   3. associate-*r/ 73.6%

      \[\leadsto \color{blue}{x \cdot \frac{y}{z \cdot z}} \]

   4. associate-/r* 76.1%

      \[\leadsto x \cdot \color{blue}{\frac{\frac{y}{z}}{z}} \]

7. Simplified 76.1%

   \[\leadsto \color{blue}{x \cdot \frac{\frac{y}{z}}{z}} \]

8. Final simplification 76.1%

   \[\leadsto x \cdot \frac{\frac{y}{z}}{z} \]

Alternative 15: 30.1% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \frac{x}{z} \cdot y \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (* (/ x z) y))

C

double code(double x, double y, double z) {
	return (x / z) * y;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = (x / z) * y
end function

Java

public static double code(double x, double y, double z) {
	return (x / z) * y;
}

Python

def code(x, y, z):
	return (x / z) * y

Julia

function code(x, y, z)
	return Float64(Float64(x / z) * y)
end

MATLAB

function tmp = code(x, y, z)
	tmp = (x / z) * y;
end

Wolfram

code[x_, y_, z_] := N[(N[(x / z), $MachinePrecision] * y), $MachinePrecision]

Derivation

1. Initial program 85.8%

   \[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \]
2. Step-by-step derivation

   1. associate-*l* 85.8%

      \[\leadsto \frac{x \cdot y}{\color{blue}{z \cdot \left(z \cdot \left(z + 1\right)\right)}} \]

   2. times-frac 95.0%

      \[\leadsto \color{blue}{\frac{x}{z} \cdot \frac{y}{z \cdot \left(z + 1\right)}} \]

   3. distribute-lft-in 95.0%

      \[\leadsto \frac{x}{z} \cdot \frac{y}{\color{blue}{z \cdot z + z \cdot 1}} \]

   4. fma-def 95.0%

      \[\leadsto \frac{x}{z} \cdot \frac{y}{\color{blue}{\mathsf{fma}\left(z, z, z \cdot 1\right)}} \]

   5. *-rgt-identity 95.0%

      \[\leadsto \frac{x}{z} \cdot \frac{y}{\mathsf{fma}\left(z, z, \color{blue}{z}\right)} \]

3. Simplified 95.0%

   \[\leadsto \color{blue}{\frac{x}{z} \cdot \frac{y}{\mathsf{fma}\left(z, z, z\right)}} \]

4. Taylor expanded in z around 0 69.7%

   \[\leadsto \frac{x}{z} \cdot \color{blue}{\left(-1 \cdot y + \frac{y}{z}\right)} \]
5. Step-by-step derivation

   1. neg-mul-1 69.7%

      \[\leadsto \frac{x}{z} \cdot \left(\color{blue}{\left(-y\right)} + \frac{y}{z}\right) \]

   2. +-commutative 69.7%

      \[\leadsto \frac{x}{z} \cdot \color{blue}{\left(\frac{y}{z} + \left(-y\right)\right)} \]

   3. unsub-neg 69.7%

      \[\leadsto \frac{x}{z} \cdot \color{blue}{\left(\frac{y}{z} - y\right)} \]

6. Simplified 69.7%

   \[\leadsto \frac{x}{z} \cdot \color{blue}{\left(\frac{y}{z} - y\right)} \]

7. Taylor expanded in z around inf 26.8%

   \[\leadsto \color{blue}{-1 \cdot \frac{y \cdot x}{z}} \]
8. Step-by-step derivation

   1. mul-1-neg 26.8%

      \[\leadsto \color{blue}{-\frac{y \cdot x}{z}} \]

   2. *-commutative 26.8%

      \[\leadsto -\frac{\color{blue}{x \cdot y}}{z} \]

   3. associate-*r/ 30.1%

      \[\leadsto -\color{blue}{x \cdot \frac{y}{z}} \]

   4. distribute-rgt-neg-in 30.1%

      \[\leadsto \color{blue}{x \cdot \left(-\frac{y}{z}\right)} \]

9. Simplified 30.1%

   \[\leadsto \color{blue}{x \cdot \left(-\frac{y}{z}\right)} \]
10. Step-by-step derivation

    1. expm1-log1p-u 26.9%

       \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(x \cdot \left(-\frac{y}{z}\right)\right)\right)} \]

    2. expm1-udef 37.3%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(x \cdot \left(-\frac{y}{z}\right)\right)} - 1} \]

    3. add-sqr-sqrt 23.3%

       \[\leadsto e^{\mathsf{log1p}\left(x \cdot \color{blue}{\left(\sqrt{-\frac{y}{z}} \cdot \sqrt{-\frac{y}{z}}\right)}\right)} - 1 \]

    4. sqrt-unprod 39.2%

       \[\leadsto e^{\mathsf{log1p}\left(x \cdot \color{blue}{\sqrt{\left(-\frac{y}{z}\right) \cdot \left(-\frac{y}{z}\right)}}\right)} - 1 \]

    5. sqr-neg 39.2%

       \[\leadsto e^{\mathsf{log1p}\left(x \cdot \sqrt{\color{blue}{\frac{y}{z} \cdot \frac{y}{z}}}\right)} - 1 \]

    6. sqrt-unprod 23.4%

       \[\leadsto e^{\mathsf{log1p}\left(x \cdot \color{blue}{\left(\sqrt{\frac{y}{z}} \cdot \sqrt{\frac{y}{z}}\right)}\right)} - 1 \]

    7. add-sqr-sqrt 37.6%

       \[\leadsto e^{\mathsf{log1p}\left(x \cdot \color{blue}{\frac{y}{z}}\right)} - 1 \]

    8. clear-num 37.6%

       \[\leadsto e^{\mathsf{log1p}\left(x \cdot \color{blue}{\frac{1}{\frac{z}{y}}}\right)} - 1 \]

    9. un-div-inv 37.6%

       \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{x}{\frac{z}{y}}}\right)} - 1 \]

11. Applied egg-rr 37.6%

    \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{x}{\frac{z}{y}}\right)} - 1} \]
12. Step-by-step derivation

    1. expm1-def 27.5%

       \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{x}{\frac{z}{y}}\right)\right)} \]

    2. expm1-log1p 32.8%

       \[\leadsto \color{blue}{\frac{x}{\frac{z}{y}}} \]

    3. associate-/r/ 32.0%

       \[\leadsto \color{blue}{\frac{x}{z} \cdot y} \]

    4. *-commutative 32.0%

       \[\leadsto \color{blue}{y \cdot \frac{x}{z}} \]

13. Simplified 32.0%

    \[\leadsto \color{blue}{y \cdot \frac{x}{z}} \]

14. Final simplification 32.0%

    \[\leadsto \frac{x}{z} \cdot y \]

Alternative 16: 30.7% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ x \cdot \frac{y}{z} \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (* x (/ y z)))

C

double code(double x, double y, double z) {
	return x * (y / z);
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = x * (y / z)
end function

Java

public static double code(double x, double y, double z) {
	return x * (y / z);
}

Python

def code(x, y, z):
	return x * (y / z)

Julia

function code(x, y, z)
	return Float64(x * Float64(y / z))
end

MATLAB

function tmp = code(x, y, z)
	tmp = x * (y / z);
end

Wolfram

code[x_, y_, z_] := N[(x * N[(y / z), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 85.8%

   \[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \]
2. Step-by-step derivation

   1. associate-*l* 85.8%

      \[\leadsto \frac{x \cdot y}{\color{blue}{z \cdot \left(z \cdot \left(z + 1\right)\right)}} \]

   2. times-frac 95.0%

      \[\leadsto \color{blue}{\frac{x}{z} \cdot \frac{y}{z \cdot \left(z + 1\right)}} \]

   3. distribute-lft-in 95.0%

      \[\leadsto \frac{x}{z} \cdot \frac{y}{\color{blue}{z \cdot z + z \cdot 1}} \]

   4. fma-def 95.0%

      \[\leadsto \frac{x}{z} \cdot \frac{y}{\color{blue}{\mathsf{fma}\left(z, z, z \cdot 1\right)}} \]

   5. *-rgt-identity 95.0%

      \[\leadsto \frac{x}{z} \cdot \frac{y}{\mathsf{fma}\left(z, z, \color{blue}{z}\right)} \]

3. Simplified 95.0%

   \[\leadsto \color{blue}{\frac{x}{z} \cdot \frac{y}{\mathsf{fma}\left(z, z, z\right)}} \]

4. Taylor expanded in z around 0 69.7%

   \[\leadsto \frac{x}{z} \cdot \color{blue}{\left(-1 \cdot y + \frac{y}{z}\right)} \]
5. Step-by-step derivation

   1. neg-mul-1 69.7%

      \[\leadsto \frac{x}{z} \cdot \left(\color{blue}{\left(-y\right)} + \frac{y}{z}\right) \]

   2. +-commutative 69.7%

      \[\leadsto \frac{x}{z} \cdot \color{blue}{\left(\frac{y}{z} + \left(-y\right)\right)} \]

   3. unsub-neg 69.7%

      \[\leadsto \frac{x}{z} \cdot \color{blue}{\left(\frac{y}{z} - y\right)} \]

6. Simplified 69.7%

   \[\leadsto \frac{x}{z} \cdot \color{blue}{\left(\frac{y}{z} - y\right)} \]

7. Taylor expanded in z around inf 26.8%

   \[\leadsto \color{blue}{-1 \cdot \frac{y \cdot x}{z}} \]
8. Step-by-step derivation

   1. mul-1-neg 26.8%

      \[\leadsto \color{blue}{-\frac{y \cdot x}{z}} \]

   2. *-commutative 26.8%

      \[\leadsto -\frac{\color{blue}{x \cdot y}}{z} \]

   3. associate-*r/ 30.1%

      \[\leadsto -\color{blue}{x \cdot \frac{y}{z}} \]

   4. distribute-rgt-neg-in 30.1%

      \[\leadsto \color{blue}{x \cdot \left(-\frac{y}{z}\right)} \]

9. Simplified 30.1%

   \[\leadsto \color{blue}{x \cdot \left(-\frac{y}{z}\right)} \]
10. Step-by-step derivation

    1. *-commutative 30.1%

       \[\leadsto \color{blue}{\left(-\frac{y}{z}\right) \cdot x} \]

    2. distribute-neg-frac 30.1%

       \[\leadsto \color{blue}{\frac{-y}{z}} \cdot x \]

    3. associate-*l/ 26.8%

       \[\leadsto \color{blue}{\frac{\left(-y\right) \cdot x}{z}} \]

11. Applied egg-rr 26.8%

    \[\leadsto \color{blue}{\frac{\left(-y\right) \cdot x}{z}} \]
12. Step-by-step derivation

    1. associate-/l* 31.8%

       \[\leadsto \color{blue}{\frac{-y}{\frac{z}{x}}} \]

    2. associate-/r/ 30.1%

       \[\leadsto \color{blue}{\frac{-y}{z} \cdot x} \]

    3. add-sqr-sqrt 14.2%

       \[\leadsto \frac{\color{blue}{\sqrt{-y} \cdot \sqrt{-y}}}{z} \cdot x \]

    4. sqrt-unprod 31.5%

       \[\leadsto \frac{\color{blue}{\sqrt{\left(-y\right) \cdot \left(-y\right)}}}{z} \cdot x \]

    5. sqr-neg 31.5%

       \[\leadsto \frac{\sqrt{\color{blue}{y \cdot y}}}{z} \cdot x \]

    6. sqrt-unprod 17.3%

       \[\leadsto \frac{\color{blue}{\sqrt{y} \cdot \sqrt{y}}}{z} \cdot x \]

    7. add-sqr-sqrt 32.5%

       \[\leadsto \frac{\color{blue}{y}}{z} \cdot x \]

13. Applied egg-rr 32.5%

    \[\leadsto \color{blue}{\frac{y}{z} \cdot x} \]

14. Final simplification 32.5%

    \[\leadsto x \cdot \frac{y}{z} \]

Alternative 17: 31.0% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \frac{x}{\frac{z}{y}} \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (/ x (/ z y)))

C

double code(double x, double y, double z) {
	return x / (z / y);
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = x / (z / y)
end function

Java

public static double code(double x, double y, double z) {
	return x / (z / y);
}

Python

def code(x, y, z):
	return x / (z / y)

Julia

function code(x, y, z)
	return Float64(x / Float64(z / y))
end

MATLAB

function tmp = code(x, y, z)
	tmp = x / (z / y);
end

Wolfram

code[x_, y_, z_] := N[(x / N[(z / y), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 85.8%

   \[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \]
2. Step-by-step derivation

   1. associate-*l* 85.8%

      \[\leadsto \frac{x \cdot y}{\color{blue}{z \cdot \left(z \cdot \left(z + 1\right)\right)}} \]

   2. times-frac 95.0%

      \[\leadsto \color{blue}{\frac{x}{z} \cdot \frac{y}{z \cdot \left(z + 1\right)}} \]

   3. distribute-lft-in 95.0%

      \[\leadsto \frac{x}{z} \cdot \frac{y}{\color{blue}{z \cdot z + z \cdot 1}} \]

   4. fma-def 95.0%

      \[\leadsto \frac{x}{z} \cdot \frac{y}{\color{blue}{\mathsf{fma}\left(z, z, z \cdot 1\right)}} \]

   5. *-rgt-identity 95.0%

      \[\leadsto \frac{x}{z} \cdot \frac{y}{\mathsf{fma}\left(z, z, \color{blue}{z}\right)} \]

3. Simplified 95.0%

   \[\leadsto \color{blue}{\frac{x}{z} \cdot \frac{y}{\mathsf{fma}\left(z, z, z\right)}} \]

4. Taylor expanded in z around 0 69.7%

   \[\leadsto \frac{x}{z} \cdot \color{blue}{\left(-1 \cdot y + \frac{y}{z}\right)} \]
5. Step-by-step derivation

   1. neg-mul-1 69.7%

      \[\leadsto \frac{x}{z} \cdot \left(\color{blue}{\left(-y\right)} + \frac{y}{z}\right) \]

   2. +-commutative 69.7%

      \[\leadsto \frac{x}{z} \cdot \color{blue}{\left(\frac{y}{z} + \left(-y\right)\right)} \]

   3. unsub-neg 69.7%

      \[\leadsto \frac{x}{z} \cdot \color{blue}{\left(\frac{y}{z} - y\right)} \]

6. Simplified 69.7%

   \[\leadsto \frac{x}{z} \cdot \color{blue}{\left(\frac{y}{z} - y\right)} \]

7. Taylor expanded in z around inf 26.8%

   \[\leadsto \color{blue}{-1 \cdot \frac{y \cdot x}{z}} \]
8. Step-by-step derivation

   1. mul-1-neg 26.8%

      \[\leadsto \color{blue}{-\frac{y \cdot x}{z}} \]

   2. *-commutative 26.8%

      \[\leadsto -\frac{\color{blue}{x \cdot y}}{z} \]

   3. associate-*r/ 30.1%

      \[\leadsto -\color{blue}{x \cdot \frac{y}{z}} \]

   4. distribute-rgt-neg-in 30.1%

      \[\leadsto \color{blue}{x \cdot \left(-\frac{y}{z}\right)} \]

9. Simplified 30.1%

   \[\leadsto \color{blue}{x \cdot \left(-\frac{y}{z}\right)} \]
10. Step-by-step derivation

    1. add-sqr-sqrt 20.6%

       \[\leadsto x \cdot \color{blue}{\left(\sqrt{-\frac{y}{z}} \cdot \sqrt{-\frac{y}{z}}\right)} \]

    2. sqrt-unprod 38.3%

       \[\leadsto x \cdot \color{blue}{\sqrt{\left(-\frac{y}{z}\right) \cdot \left(-\frac{y}{z}\right)}} \]

    3. sqr-neg 38.3%

       \[\leadsto x \cdot \sqrt{\color{blue}{\frac{y}{z} \cdot \frac{y}{z}}} \]

    4. sqrt-unprod 20.1%

       \[\leadsto x \cdot \color{blue}{\left(\sqrt{\frac{y}{z}} \cdot \sqrt{\frac{y}{z}}\right)} \]

    5. add-sqr-sqrt 32.5%

       \[\leadsto x \cdot \color{blue}{\frac{y}{z}} \]

    6. clear-num 32.8%

       \[\leadsto x \cdot \color{blue}{\frac{1}{\frac{z}{y}}} \]

    7. un-div-inv 32.8%

       \[\leadsto \color{blue}{\frac{x}{\frac{z}{y}}} \]

11. Applied egg-rr 32.8%

    \[\leadsto \color{blue}{\frac{x}{\frac{z}{y}}} \]

12. Final simplification 32.8%

    \[\leadsto \frac{x}{\frac{z}{y}} \]

Developer target: 95.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z < 249.6182814532307:\\ \;\;\;\;\frac{y \cdot \frac{x}{z}}{z + z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{y}{z}}{1 + z} \cdot x}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (< z 249.6182814532307)
   (/ (* y (/ x z)) (+ z (* z z)))
   (/ (* (/ (/ y z) (+ 1.0 z)) x) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if (z < 249.6182814532307) {
		tmp = (y * (x / z)) / (z + (z * z));
	} else {
		tmp = (((y / z) / (1.0 + z)) * x) / z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (z < 249.6182814532307d0) then
        tmp = (y * (x / z)) / (z + (z * z))
    else
        tmp = (((y / z) / (1.0d0 + z)) * x) / z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z < 249.6182814532307) {
		tmp = (y * (x / z)) / (z + (z * z));
	} else {
		tmp = (((y / z) / (1.0 + z)) * x) / z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z < 249.6182814532307:
		tmp = (y * (x / z)) / (z + (z * z))
	else:
		tmp = (((y / z) / (1.0 + z)) * x) / z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z < 249.6182814532307)
		tmp = Float64(Float64(y * Float64(x / z)) / Float64(z + Float64(z * z)));
	else
		tmp = Float64(Float64(Float64(Float64(y / z) / Float64(1.0 + z)) * x) / z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z < 249.6182814532307)
		tmp = (y * (x / z)) / (z + (z * z));
	else
		tmp = (((y / z) / (1.0 + z)) * x) / z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Less[z, 249.6182814532307], N[(N[(y * N[(x / z), $MachinePrecision]), $MachinePrecision] / N[(z + N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(y / z), $MachinePrecision] / N[(1.0 + z), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision] / z), $MachinePrecision]]

Reproduce

herbie shell --seed 2023240 
(FPCore (x y z)
  :name "Statistics.Distribution.Beta:$cvariance from math-functions-0.1.5.2"
  :precision binary64

  :herbie-target
  (if (< z 249.6182814532307) (/ (* y (/ x z)) (+ z (* z z))) (/ (* (/ (/ y z) (+ 1.0 z)) x) z))

  (/ (* x y) (* (* z z) (+ z 1.0))))