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

Percentage Accurate: 83.5% → 95.5%

Time: 11.7s

Alternatives: 15

Speedup: 1.0×

• Unsound rule application detected in e-graph. Results may not be sound.

• 26.4% 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: 83.5% 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: 95.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(z + 1\right) \cdot \left(z \cdot z\right)\\ t_1 := \frac{\frac{y \cdot \frac{x}{z}}{z}}{z}\\ \mathbf{if}\;t_0 \leq -2 \cdot 10^{+32}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_0 \leq 5 \cdot 10^{-291}:\\ \;\;\;\;\frac{x}{z} \cdot \frac{y}{z}\\ \mathbf{elif}\;t_0 \leq 2 \cdot 10^{+124}:\\ \;\;\;\;\frac{y \cdot x}{t_0}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (+ z 1.0) (* z z))) (t_1 (/ (/ (* y (/ x z)) z) z)))
   (if (<= t_0 -2e+32)
     t_1
     (if (<= t_0 5e-291)
       (* (/ x z) (/ y z))
       (if (<= t_0 2e+124) (/ (* y x) t_0) t_1)))))

C

double code(double x, double y, double z) {
	double t_0 = (z + 1.0) * (z * z);
	double t_1 = ((y * (x / z)) / z) / z;
	double tmp;
	if (t_0 <= -2e+32) {
		tmp = t_1;
	} else if (t_0 <= 5e-291) {
		tmp = (x / z) * (y / z);
	} else if (t_0 <= 2e+124) {
		tmp = (y * x) / t_0;
	} else {
		tmp = t_1;
	}
	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) :: t_1
    real(8) :: tmp
    t_0 = (z + 1.0d0) * (z * z)
    t_1 = ((y * (x / z)) / z) / z
    if (t_0 <= (-2d+32)) then
        tmp = t_1
    else if (t_0 <= 5d-291) then
        tmp = (x / z) * (y / z)
    else if (t_0 <= 2d+124) then
        tmp = (y * x) / t_0
    else
        tmp = t_1
    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 t_1 = ((y * (x / z)) / z) / z;
	double tmp;
	if (t_0 <= -2e+32) {
		tmp = t_1;
	} else if (t_0 <= 5e-291) {
		tmp = (x / z) * (y / z);
	} else if (t_0 <= 2e+124) {
		tmp = (y * x) / t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (z + 1.0) * (z * z)
	t_1 = ((y * (x / z)) / z) / z
	tmp = 0
	if t_0 <= -2e+32:
		tmp = t_1
	elif t_0 <= 5e-291:
		tmp = (x / z) * (y / z)
	elif t_0 <= 2e+124:
		tmp = (y * x) / t_0
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(z + 1.0) * Float64(z * z))
	t_1 = Float64(Float64(Float64(y * Float64(x / z)) / z) / z)
	tmp = 0.0
	if (t_0 <= -2e+32)
		tmp = t_1;
	elseif (t_0 <= 5e-291)
		tmp = Float64(Float64(x / z) * Float64(y / z));
	elseif (t_0 <= 2e+124)
		tmp = Float64(Float64(y * x) / t_0);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (z + 1.0) * (z * z);
	t_1 = ((y * (x / z)) / z) / z;
	tmp = 0.0;
	if (t_0 <= -2e+32)
		tmp = t_1;
	elseif (t_0 <= 5e-291)
		tmp = (x / z) * (y / z);
	elseif (t_0 <= 2e+124)
		tmp = (y * x) / t_0;
	else
		tmp = t_1;
	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]}, Block[{t$95$1 = N[(N[(N[(y * N[(x / z), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision] / z), $MachinePrecision]}, If[LessEqual[t$95$0, -2e+32], t$95$1, If[LessEqual[t$95$0, 5e-291], N[(N[(x / z), $MachinePrecision] * N[(y / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 2e+124], N[(N[(y * x), $MachinePrecision] / t$95$0), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 z z) (+.f64 z 1)) < -2.00000000000000011e32 or 1.9999999999999999e124 < (*.f64 (*.f64 z z) (+.f64 z 1))

   1. Initial program 81.7%

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

      1. associate-*l* 81.7%

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

      2. times-frac 92.6%

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

      3. distribute-lft-in 92.6%

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

      4. fma-def 92.6%

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

      5. *-rgt-identity 92.6%

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

   3. Simplified 92.6%

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

      1. *-commutative 92.6%

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

      2. associate-*l/ 92.9%

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

      3. fma-udef 92.9%

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

      4. distribute-lft1-in 92.9%

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

      5. frac-times 96.3%

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

      6. associate-*r/ 98.1%

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

   5. Applied egg-rr 98.1%

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

   6. Taylor expanded in z around inf 88.7%

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

      1. unpow2 88.7%

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

      2. associate-/r* 88.9%

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

      3. associate-*l/ 98.1%

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

      4. *-commutative 98.1%

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

      5. associate-*r/ 97.9%

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

   8. Simplified 97.9%

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

      1. associate-*r/ 98.1%

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

      2. associate-*l/ 98.1%

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

      3. associate-*r/ 99.0%

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

   10. Applied egg-rr 99.0%

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

   if -2.00000000000000011e32 < (*.f64 (*.f64 z z) (+.f64 z 1)) < 5.0000000000000003e-291

   1. Initial program 76.2%

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

      1. associate-*l* 76.2%

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

      2. times-frac 99.8%

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

      3. distribute-lft-in 99.8%

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

      4. fma-def 99.8%

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

      5. *-rgt-identity 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in z around 0 76.2%

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

      1. unpow2 76.2%

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

      2. associate-/l/ 89.4%

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

      3. associate-*r/ 96.0%

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

      4. associate-*l/ 99.8%

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

   6. Simplified 99.8%

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

   if 5.0000000000000003e-291 < (*.f64 (*.f64 z z) (+.f64 z 1)) < 1.9999999999999999e124

   1. Initial program 96.5%

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

4. Final simplification 98.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(z + 1\right) \cdot \left(z \cdot z\right) \leq -2 \cdot 10^{+32}:\\ \;\;\;\;\frac{\frac{y \cdot \frac{x}{z}}{z}}{z}\\ \mathbf{elif}\;\left(z + 1\right) \cdot \left(z \cdot z\right) \leq 5 \cdot 10^{-291}:\\ \;\;\;\;\frac{x}{z} \cdot \frac{y}{z}\\ \mathbf{elif}\;\left(z + 1\right) \cdot \left(z \cdot z\right) \leq 2 \cdot 10^{+124}:\\ \;\;\;\;\frac{y \cdot x}{\left(z + 1\right) \cdot \left(z \cdot z\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y \cdot \frac{x}{z}}{z}}{z}\\ \end{array} \]

Alternative 2: 41.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -45000000000:\\ \;\;\;\;\frac{y}{\frac{z}{x}}\\ \mathbf{elif}\;z \leq -1 \cdot 10^{-310} \lor \neg \left(z \leq 1.18 \cdot 10^{+139}\right) \land z \leq 1.18 \cdot 10^{+232}:\\ \;\;\;\;\frac{-x}{\frac{z}{y}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -45000000000.0)
   (/ y (/ z x))
   (if (or (<= z -1e-310) (and (not (<= z 1.18e+139)) (<= z 1.18e+232)))
     (/ (- x) (/ z y))
     (* y (/ x z)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -45000000000.0) {
		tmp = y / (z / x);
	} else if ((z <= -1e-310) || (!(z <= 1.18e+139) && (z <= 1.18e+232))) {
		tmp = -x / (z / y);
	} else {
		tmp = y * (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 <= (-45000000000.0d0)) then
        tmp = y / (z / x)
    else if ((z <= (-1d-310)) .or. (.not. (z <= 1.18d+139)) .and. (z <= 1.18d+232)) then
        tmp = -x / (z / y)
    else
        tmp = y * (x / z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -45000000000.0) {
		tmp = y / (z / x);
	} else if ((z <= -1e-310) || (!(z <= 1.18e+139) && (z <= 1.18e+232))) {
		tmp = -x / (z / y);
	} else {
		tmp = y * (x / z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -45000000000.0:
		tmp = y / (z / x)
	elif (z <= -1e-310) or (not (z <= 1.18e+139) and (z <= 1.18e+232)):
		tmp = -x / (z / y)
	else:
		tmp = y * (x / z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -45000000000.0)
		tmp = Float64(y / Float64(z / x));
	elseif ((z <= -1e-310) || (!(z <= 1.18e+139) && (z <= 1.18e+232)))
		tmp = Float64(Float64(-x) / Float64(z / y));
	else
		tmp = Float64(y * Float64(x / z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -45000000000.0)
		tmp = y / (z / x);
	elseif ((z <= -1e-310) || (~((z <= 1.18e+139)) && (z <= 1.18e+232)))
		tmp = -x / (z / y);
	else
		tmp = y * (x / z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -45000000000.0], N[(y / N[(z / x), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[z, -1e-310], And[N[Not[LessEqual[z, 1.18e+139]], $MachinePrecision], LessEqual[z, 1.18e+232]]], N[((-x) / N[(z / y), $MachinePrecision]), $MachinePrecision], N[(y * N[(x / z), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -4.5e10

   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-*r/ 87.3%

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

      2. associate-*l* 87.3%

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

      3. distribute-lft-in 87.3%

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

      4. fma-def 87.3%

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

      5. *-rgt-identity 87.3%

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

   3. Simplified 87.3%

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

   4. Taylor expanded in z around 0 40.6%

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

      1. +-commutative 40.6%

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

      2. mul-1-neg 40.6%

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

      3. unsub-neg 40.6%

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

      4. unpow2 40.6%

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

      5. associate-/r* 40.6%

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

   6. Simplified 40.6%

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

   7. Taylor expanded in z around inf 34.3%

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

      1. associate-*r/ 34.3%

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

      2. neg-mul-1 34.3%

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

      3. distribute-lft-neg-in 34.3%

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

      4. associate-*l/ 40.6%

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

      5. *-commutative 40.6%

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

   9. Simplified 40.6%

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

       1. add-sqr-sqrt 18.1%

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

       2. sqrt-unprod 40.4%

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

       3. sqr-neg 40.4%

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

       4. sqrt-unprod 22.8%

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

       5. add-sqr-sqrt 43.3%

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

       6. associate-*r/ 36.9%

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

       7. *-commutative 36.9%

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

       8. associate-/l* 44.7%

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

   11. Applied egg-rr 44.7%

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

   if -4.5e10 < z < -9.999999999999969e-311 or 1.18e139 < z < 1.18e232

   1. Initial program 86.1%

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

      1. associate-*r/ 80.0%

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

      2. associate-*l* 80.0%

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

      3. distribute-lft-in 80.1%

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

      4. fma-def 80.1%

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

      5. *-rgt-identity 80.1%

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

   3. Simplified 80.1%

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

   4. Taylor expanded in z around 0 74.3%

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

      1. +-commutative 74.3%

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

      2. mul-1-neg 74.3%

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

      3. unsub-neg 74.3%

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

      4. unpow2 74.3%

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

      5. associate-/r* 78.3%

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

   6. Simplified 78.3%

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

   7. Taylor expanded in z around inf 39.4%

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

      1. associate-*r/ 39.4%

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

      2. neg-mul-1 39.4%

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

      3. distribute-lft-neg-in 39.4%

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

      4. associate-*l/ 46.6%

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

      5. *-commutative 46.6%

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

   9. Simplified 46.6%

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

   10. Taylor expanded in x around 0 39.4%

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

       1. mul-1-neg 39.4%

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

       2. *-commutative 39.4%

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

       3. associate-*l/ 43.4%

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

       4. associate-/r/ 47.6%

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

   12. Simplified 47.6%

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

   if -9.999999999999969e-311 < z < 1.18e139 or 1.18e232 < z

   1. Initial program 84.5%

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

      1. associate-*r/ 85.2%

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

      2. associate-*l* 85.2%

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

      3. distribute-lft-in 85.2%

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

      4. fma-def 85.2%

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

      5. *-rgt-identity 85.2%

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

   3. Simplified 85.2%

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

   4. Taylor expanded in z around 0 48.8%

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

      1. +-commutative 48.8%

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

      2. mul-1-neg 48.8%

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

      3. unsub-neg 48.8%

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

      4. unpow2 48.8%

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

      5. associate-/r* 53.1%

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

   6. Simplified 53.1%

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

   7. Taylor expanded in z around inf 16.6%

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

      1. associate-*r/ 16.6%

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

      2. neg-mul-1 16.6%

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

      3. distribute-lft-neg-in 16.6%

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

      4. associate-*l/ 17.5%

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

      5. *-commutative 17.5%

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

   9. Simplified 17.5%

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

       1. add-sqr-sqrt 9.7%

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

       2. sqrt-unprod 30.8%

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

       3. sqr-neg 30.8%

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

       4. sqrt-unprod 21.9%

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

       5. add-sqr-sqrt 43.4%

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

       6. associate-*r/ 39.7%

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

       7. *-commutative 39.7%

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

       8. associate-/l* 43.7%

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

   11. Applied egg-rr 43.7%

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

       1. associate-/l* 39.7%

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

       2. *-commutative 39.7%

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

       3. associate-*l/ 42.8%

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

   13. Applied egg-rr 42.8%

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

4. Final simplification 45.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -45000000000:\\ \;\;\;\;\frac{y}{\frac{z}{x}}\\ \mathbf{elif}\;z \leq -1 \cdot 10^{-310} \lor \neg \left(z \leq 1.18 \cdot 10^{+139}\right) \land z \leq 1.18 \cdot 10^{+232}:\\ \;\;\;\;\frac{-x}{\frac{z}{y}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \end{array} \]

Alternative 3: 92.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -45000000000 \lor \neg \left(z \leq 1\right):\\ \;\;\;\;\frac{x}{z} \cdot \frac{y}{z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot \frac{x}{z}}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -45000000000.0) (not (<= z 1.0)))
   (* (/ x z) (/ y (* z z)))
   (/ (* y (/ x z)) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -45000000000.0) || !(z <= 1.0)) {
		tmp = (x / z) * (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 ((z <= (-45000000000.0d0)) .or. (.not. (z <= 1.0d0))) then
        tmp = (x / z) * (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 ((z <= -45000000000.0) || !(z <= 1.0)) {
		tmp = (x / z) * (y / (z * z));
	} else {
		tmp = (y * (x / z)) / z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -45000000000.0) or not (z <= 1.0):
		tmp = (x / z) * (y / (z * z))
	else:
		tmp = (y * (x / z)) / z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -45000000000.0) || !(z <= 1.0))
		tmp = Float64(Float64(x / z) * Float64(y / Float64(z * z)));
	else
		tmp = Float64(Float64(y * Float64(x / z)) / z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -45000000000.0) || ~((z <= 1.0)))
		tmp = (x / z) * (y / (z * z));
	else
		tmp = (y * (x / z)) / z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -45000000000.0], N[Not[LessEqual[z, 1.0]], $MachinePrecision]], N[(N[(x / z), $MachinePrecision] * N[(y / N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y * N[(x / z), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -4.5e10 or 1 < z

   1. Initial program 83.7%

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

      1. associate-*l* 83.7%

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

      2. times-frac 91.9%

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

      3. distribute-lft-in 91.9%

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

      4. fma-def 91.9%

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

      5. *-rgt-identity 91.9%

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

   3. Simplified 91.9%

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

   4. Taylor expanded in z around inf 90.8%

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

      1. unpow2 90.8%

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

   6. Simplified 90.8%

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

   if -4.5e10 < z < 1

   1. Initial program 85.4%

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

      1. times-frac 84.9%

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

   3. Simplified 84.9%

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

   4. Taylor expanded in z around 0 83.4%

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

      1. *-commutative 83.4%

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

      2. associate-/r* 91.3%

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

      3. associate-*r/ 93.5%

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

   6. Applied egg-rr 93.5%

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

4. Final simplification 92.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -45000000000 \lor \neg \left(z \leq 1\right):\\ \;\;\;\;\frac{x}{z} \cdot \frac{y}{z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot \frac{x}{z}}{z}\\ \end{array} \]

Alternative 4: 94.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -45000000000 \lor \neg \left(z \leq 1\right):\\ \;\;\;\;\frac{x}{z} \cdot \frac{\frac{y}{z}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot \frac{x}{z}}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -45000000000.0) (not (<= z 1.0)))
   (* (/ x z) (/ (/ y z) z))
   (/ (* y (/ x z)) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -45000000000.0) || !(z <= 1.0)) {
		tmp = (x / z) * ((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 ((z <= (-45000000000.0d0)) .or. (.not. (z <= 1.0d0))) then
        tmp = (x / z) * ((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 ((z <= -45000000000.0) || !(z <= 1.0)) {
		tmp = (x / z) * ((y / z) / z);
	} else {
		tmp = (y * (x / z)) / z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -45000000000.0) or not (z <= 1.0):
		tmp = (x / z) * ((y / z) / z)
	else:
		tmp = (y * (x / z)) / z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -45000000000.0) || !(z <= 1.0))
		tmp = Float64(Float64(x / z) * Float64(Float64(y / z) / z));
	else
		tmp = Float64(Float64(y * Float64(x / z)) / z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -45000000000.0) || ~((z <= 1.0)))
		tmp = (x / z) * ((y / z) / z);
	else
		tmp = (y * (x / z)) / z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -45000000000.0], N[Not[LessEqual[z, 1.0]], $MachinePrecision]], N[(N[(x / z), $MachinePrecision] * N[(N[(y / z), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], N[(N[(y * N[(x / z), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -4.5e10 or 1 < z

   1. Initial program 83.7%

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

      1. associate-*l* 83.7%

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

      2. times-frac 91.9%

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

      3. distribute-lft-in 91.9%

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

      4. fma-def 91.9%

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

      5. *-rgt-identity 91.9%

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

   3. Simplified 91.9%

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

   4. Taylor expanded in z around inf 90.8%

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

      1. unpow2 90.8%

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

      2. associate-/r* 96.3%

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

   6. Simplified 96.3%

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

   if -4.5e10 < z < 1

   1. Initial program 85.4%

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

      1. times-frac 84.9%

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

   3. Simplified 84.9%

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

   4. Taylor expanded in z around 0 83.4%

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

      1. *-commutative 83.4%

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

      2. associate-/r* 91.3%

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

      3. associate-*r/ 93.5%

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

   6. Applied egg-rr 93.5%

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

4. Final simplification 94.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -45000000000 \lor \neg \left(z \leq 1\right):\\ \;\;\;\;\frac{x}{z} \cdot \frac{\frac{y}{z}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot \frac{x}{z}}{z}\\ \end{array} \]

Alternative 5: 94.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y \cdot \frac{x}{z}}{z}\\ \mathbf{if}\;z \leq -45000000000 \lor \neg \left(z \leq 1\right):\\ \;\;\;\;\frac{t_0}{z}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (* y (/ x z)) z)))
   (if (or (<= z -45000000000.0) (not (<= z 1.0))) (/ t_0 z) t_0)))

C

double code(double x, double y, double z) {
	double t_0 = (y * (x / z)) / z;
	double tmp;
	if ((z <= -45000000000.0) || !(z <= 1.0)) {
		tmp = t_0 / z;
	} else {
		tmp = t_0;
	}
	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 = (y * (x / z)) / z
    if ((z <= (-45000000000.0d0)) .or. (.not. (z <= 1.0d0))) then
        tmp = t_0 / z
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = (y * (x / z)) / z;
	double tmp;
	if ((z <= -45000000000.0) || !(z <= 1.0)) {
		tmp = t_0 / z;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (y * (x / z)) / z
	tmp = 0
	if (z <= -45000000000.0) or not (z <= 1.0):
		tmp = t_0 / z
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(y * Float64(x / z)) / z)
	tmp = 0.0
	if ((z <= -45000000000.0) || !(z <= 1.0))
		tmp = Float64(t_0 / z);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (y * (x / z)) / z;
	tmp = 0.0;
	if ((z <= -45000000000.0) || ~((z <= 1.0)))
		tmp = t_0 / z;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(y * N[(x / z), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]}, If[Or[LessEqual[z, -45000000000.0], N[Not[LessEqual[z, 1.0]], $MachinePrecision]], N[(t$95$0 / z), $MachinePrecision], t$95$0]]

Derivation

1. Split input into 2 regimes

2. if z < -4.5e10 or 1 < z

   1. Initial program 83.7%

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

      1. associate-*l* 83.7%

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

      2. times-frac 91.9%

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

      3. distribute-lft-in 91.9%

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

      4. fma-def 91.9%

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

      5. *-rgt-identity 91.9%

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

   3. Simplified 91.9%

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

      1. *-commutative 91.9%

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

      2. associate-*l/ 92.2%

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

      3. fma-udef 92.2%

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

      4. distribute-lft1-in 92.2%

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

      5. frac-times 95.1%

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

      6. associate-*r/ 96.9%

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

   5. Applied egg-rr 96.9%

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

   6. Taylor expanded in z around inf 88.7%

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

      1. unpow2 88.7%

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

      2. associate-/r* 89.0%

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

      3. associate-*l/ 97.1%

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

      4. *-commutative 97.1%

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

      5. associate-*r/ 96.9%

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

   8. Simplified 96.9%

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

      1. associate-*r/ 97.1%

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

      2. associate-*l/ 95.7%

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

      3. associate-*r/ 96.5%

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

   10. Applied egg-rr 96.5%

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

   if -4.5e10 < z < 1

   1. Initial program 85.4%

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

      1. times-frac 84.9%

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

   3. Simplified 84.9%

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

   4. Taylor expanded in z around 0 83.4%

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

      1. *-commutative 83.4%

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

      2. associate-/r* 91.3%

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

      3. associate-*r/ 93.5%

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

   6. Applied egg-rr 93.5%

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

4. Final simplification 94.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -45000000000 \lor \neg \left(z \leq 1\right):\\ \;\;\;\;\frac{\frac{y \cdot \frac{x}{z}}{z}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot \frac{x}{z}}{z}\\ \end{array} \]

Alternative 6: 94.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\frac{y}{z}}{z}\\ \mathbf{if}\;z \leq -45000000000:\\ \;\;\;\;\frac{x}{z} \cdot t_0\\ \mathbf{elif}\;z \leq 1:\\ \;\;\;\;\frac{y \cdot \frac{x}{z}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot t_0}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (/ y z) z)))
   (if (<= z -45000000000.0)
     (* (/ x z) t_0)
     (if (<= z 1.0) (/ (* y (/ x z)) z) (/ (* x t_0) z)))))

C

double code(double x, double y, double z) {
	double t_0 = (y / z) / z;
	double tmp;
	if (z <= -45000000000.0) {
		tmp = (x / z) * t_0;
	} else if (z <= 1.0) {
		tmp = (y * (x / z)) / z;
	} else {
		tmp = (x * t_0) / 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) :: t_0
    real(8) :: tmp
    t_0 = (y / z) / z
    if (z <= (-45000000000.0d0)) then
        tmp = (x / z) * t_0
    else if (z <= 1.0d0) then
        tmp = (y * (x / z)) / z
    else
        tmp = (x * t_0) / z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = (y / z) / z;
	double tmp;
	if (z <= -45000000000.0) {
		tmp = (x / z) * t_0;
	} else if (z <= 1.0) {
		tmp = (y * (x / z)) / z;
	} else {
		tmp = (x * t_0) / z;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (y / z) / z
	tmp = 0
	if z <= -45000000000.0:
		tmp = (x / z) * t_0
	elif z <= 1.0:
		tmp = (y * (x / z)) / z
	else:
		tmp = (x * t_0) / z
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(y / z) / z)
	tmp = 0.0
	if (z <= -45000000000.0)
		tmp = Float64(Float64(x / z) * t_0);
	elseif (z <= 1.0)
		tmp = Float64(Float64(y * Float64(x / z)) / z);
	else
		tmp = Float64(Float64(x * t_0) / z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (y / z) / z;
	tmp = 0.0;
	if (z <= -45000000000.0)
		tmp = (x / z) * t_0;
	elseif (z <= 1.0)
		tmp = (y * (x / z)) / z;
	else
		tmp = (x * t_0) / z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(y / z), $MachinePrecision] / z), $MachinePrecision]}, If[LessEqual[z, -45000000000.0], N[(N[(x / z), $MachinePrecision] * t$95$0), $MachinePrecision], If[LessEqual[z, 1.0], N[(N[(y * N[(x / z), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], N[(N[(x * t$95$0), $MachinePrecision] / z), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if z < -4.5e10

   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 93.3%

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

      3. distribute-lft-in 93.3%

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

      4. fma-def 93.3%

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

      5. *-rgt-identity 93.3%

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

   3. Simplified 93.3%

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

   4. Taylor expanded in z around inf 93.3%

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

      1. unpow2 93.3%

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

      2. associate-/r* 97.9%

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

   6. Simplified 97.9%

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

   if -4.5e10 < z < 1

   1. Initial program 85.4%

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

      1. times-frac 84.9%

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

   3. Simplified 84.9%

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

   4. Taylor expanded in z around 0 83.4%

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

      1. *-commutative 83.4%

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

      2. associate-/r* 91.3%

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

      3. associate-*r/ 93.5%

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

   6. Applied egg-rr 93.5%

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

   if 1 < z

   1. Initial program 85.0%

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

      1. associate-*l* 85.0%

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

      2. times-frac 90.6%

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

      3. distribute-lft-in 90.6%

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

      4. fma-def 90.6%

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

      5. *-rgt-identity 90.6%

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

   3. Simplified 90.6%

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

      1. *-commutative 90.6%

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

      2. associate-*l/ 90.6%

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

      3. fma-udef 90.6%

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

      4. distribute-lft1-in 90.6%

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

      5. frac-times 93.9%

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

      6. associate-*r/ 96.9%

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

   5. Applied egg-rr 96.9%

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

   6. Taylor expanded in z around inf 88.9%

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

      1. unpow2 88.9%

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

      2. associate-/r* 89.3%

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

      3. associate-*l/ 97.4%

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

      4. *-commutative 97.4%

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

      5. associate-*r/ 97.4%

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

   8. Simplified 97.4%

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

4. Final simplification 95.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -45000000000:\\ \;\;\;\;\frac{x}{z} \cdot \frac{\frac{y}{z}}{z}\\ \mathbf{elif}\;z \leq 1:\\ \;\;\;\;\frac{y \cdot \frac{x}{z}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \frac{\frac{y}{z}}{z}}{z}\\ \end{array} \]

Alternative 7: 97.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 4 \cdot 10^{+132}:\\ \;\;\;\;\frac{\frac{x}{z} \cdot \frac{y}{z}}{z + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{z} \cdot \frac{y}{z + 1}}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 4e+132)
   (/ (* (/ x z) (/ y z)) (+ z 1.0))
   (/ (* (/ x z) (/ y (+ z 1.0))) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 4e+132) {
		tmp = ((x / z) * (y / z)) / (z + 1.0);
	} else {
		tmp = ((x / z) * (y / (z + 1.0))) / 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 <= 4d+132) then
        tmp = ((x / z) * (y / z)) / (z + 1.0d0)
    else
        tmp = ((x / z) * (y / (z + 1.0d0))) / z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 4e+132) {
		tmp = ((x / z) * (y / z)) / (z + 1.0);
	} else {
		tmp = ((x / z) * (y / (z + 1.0))) / z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 4e+132:
		tmp = ((x / z) * (y / z)) / (z + 1.0)
	else:
		tmp = ((x / z) * (y / (z + 1.0))) / z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 4e+132)
		tmp = Float64(Float64(Float64(x / z) * Float64(y / z)) / Float64(z + 1.0));
	else
		tmp = Float64(Float64(Float64(x / z) * Float64(y / Float64(z + 1.0))) / z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 4e+132)
		tmp = ((x / z) * (y / z)) / (z + 1.0);
	else
		tmp = ((x / z) * (y / (z + 1.0))) / z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 4e+132], N[(N[(N[(x / z), $MachinePrecision] * N[(y / z), $MachinePrecision]), $MachinePrecision] / N[(z + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x / z), $MachinePrecision] * N[(y / N[(z + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 3.99999999999999996e132

   1. Initial program 84.3%

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

      1. associate-*l* 84.3%

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

      2. times-frac 96.2%

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

      3. distribute-lft-in 96.2%

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

      4. fma-def 96.2%

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

      5. *-rgt-identity 96.2%

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

   3. Simplified 96.2%

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

      1. fma-udef 96.2%

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

      2. *-rgt-identity 96.2%

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

      3. distribute-lft-in 96.2%

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

      4. times-frac 84.3%

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

      5. associate-*l* 84.3%

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

      6. associate-/r* 87.2%

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

      7. times-frac 97.2%

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

   5. Applied egg-rr 97.2%

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

   if 3.99999999999999996e132 < y

   1. Initial program 86.5%

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

      1. associate-*l* 86.5%

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

      2. times-frac 78.0%

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

      3. distribute-lft-in 78.0%

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

      4. fma-def 78.0%

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

      5. *-rgt-identity 78.0%

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

   3. Simplified 78.0%

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

      1. *-commutative 78.0%

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

      2. associate-*l/ 88.1%

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

      3. fma-udef 88.1%

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

      4. distribute-lft1-in 88.1%

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

      5. frac-times 93.1%

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

      6. associate-*r/ 97.9%

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

   5. Applied egg-rr 97.9%

      \[\leadsto \color{blue}{\frac{\frac{y}{z + 1} \cdot \frac{x}{z}}{z}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 97.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 4 \cdot 10^{+132}:\\ \;\;\;\;\frac{\frac{x}{z} \cdot \frac{y}{z}}{z + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{z} \cdot \frac{y}{z + 1}}{z}\\ \end{array} \]

Alternative 8: 41.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1 \cdot 10^{-310}:\\ \;\;\;\;\frac{-y}{\frac{z}{x}}\\ \mathbf{elif}\;z \leq 3.65 \cdot 10^{+139} \lor \neg \left(z \leq 1.8 \cdot 10^{+232}\right):\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{-x}{\frac{z}{y}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -1e-310)
   (/ (- y) (/ z x))
   (if (or (<= z 3.65e+139) (not (<= z 1.8e+232)))
     (* y (/ x z))
     (/ (- x) (/ z y)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -1e-310) {
		tmp = -y / (z / x);
	} else if ((z <= 3.65e+139) || !(z <= 1.8e+232)) {
		tmp = y * (x / z);
	} else {
		tmp = -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 <= (-1d-310)) then
        tmp = -y / (z / x)
    else if ((z <= 3.65d+139) .or. (.not. (z <= 1.8d+232))) then
        tmp = y * (x / z)
    else
        tmp = -x / (z / y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -1e-310) {
		tmp = -y / (z / x);
	} else if ((z <= 3.65e+139) || !(z <= 1.8e+232)) {
		tmp = y * (x / z);
	} else {
		tmp = -x / (z / y);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -1e-310:
		tmp = -y / (z / x)
	elif (z <= 3.65e+139) or not (z <= 1.8e+232):
		tmp = y * (x / z)
	else:
		tmp = -x / (z / y)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -1e-310)
		tmp = Float64(Float64(-y) / Float64(z / x));
	elseif ((z <= 3.65e+139) || !(z <= 1.8e+232))
		tmp = Float64(y * Float64(x / z));
	else
		tmp = Float64(Float64(-x) / Float64(z / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -1e-310)
		tmp = -y / (z / x);
	elseif ((z <= 3.65e+139) || ~((z <= 1.8e+232)))
		tmp = y * (x / z);
	else
		tmp = -x / (z / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -1e-310], N[((-y) / N[(z / x), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[z, 3.65e+139], N[Not[LessEqual[z, 1.8e+232]], $MachinePrecision]], N[(y * N[(x / z), $MachinePrecision]), $MachinePrecision], N[((-x) / N[(z / y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -9.999999999999969e-311

   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-*r/ 83.6%

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

      2. associate-*l* 83.6%

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

      3. distribute-lft-in 83.6%

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

      4. fma-def 83.6%

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

      5. *-rgt-identity 83.6%

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

   3. Simplified 83.6%

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

   4. Taylor expanded in z around 0 63.0%

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

      1. +-commutative 63.0%

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

      2. mul-1-neg 63.0%

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

      3. unsub-neg 63.0%

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

      4. unpow2 63.0%

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

      5. associate-/r* 65.8%

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

   6. Simplified 65.8%

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

   7. Taylor expanded in z around inf 37.3%

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

      1. associate-*r/ 37.3%

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

      2. neg-mul-1 37.3%

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

      3. distribute-lft-neg-in 37.3%

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

      4. associate-*l/ 44.1%

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

      5. *-commutative 44.1%

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

   9. Simplified 44.1%

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

       1. *-commutative 44.1%

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

       2. distribute-frac-neg 44.1%

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

       3. distribute-lft-neg-out 44.1%

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

       4. associate-*l/ 37.3%

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

       5. associate-/l* 42.7%

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

   11. Applied egg-rr 42.7%

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

   if -9.999999999999969e-311 < z < 3.64999999999999989e139 or 1.79999999999999996e232 < z

   1. Initial program 84.5%

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

      1. associate-*r/ 85.2%

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

      2. associate-*l* 85.2%

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

      3. distribute-lft-in 85.2%

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

      4. fma-def 85.2%

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

      5. *-rgt-identity 85.2%

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

   3. Simplified 85.2%

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

   4. Taylor expanded in z around 0 48.8%

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

      1. +-commutative 48.8%

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

      2. mul-1-neg 48.8%

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

      3. unsub-neg 48.8%

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

      4. unpow2 48.8%

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

      5. associate-/r* 53.1%

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

   6. Simplified 53.1%

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

   7. Taylor expanded in z around inf 16.6%

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

      1. associate-*r/ 16.6%

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

      2. neg-mul-1 16.6%

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

      3. distribute-lft-neg-in 16.6%

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

      4. associate-*l/ 17.5%

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

      5. *-commutative 17.5%

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

   9. Simplified 17.5%

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

       1. add-sqr-sqrt 9.7%

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

       2. sqrt-unprod 30.8%

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

       3. sqr-neg 30.8%

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

       4. sqrt-unprod 21.9%

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

       5. add-sqr-sqrt 43.4%

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

       6. associate-*r/ 39.7%

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

       7. *-commutative 39.7%

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

       8. associate-/l* 43.7%

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

   11. Applied egg-rr 43.7%

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

       1. associate-/l* 39.7%

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

       2. *-commutative 39.7%

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

       3. associate-*l/ 42.8%

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

   13. Applied egg-rr 42.8%

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

   if 3.64999999999999989e139 < z < 1.79999999999999996e232

   1. Initial program 75.0%

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

      1. associate-*r/ 76.6%

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

      2. associate-*l* 76.6%

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

      3. distribute-lft-in 76.6%

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

      4. fma-def 76.6%

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

      5. *-rgt-identity 76.6%

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

   3. Simplified 76.6%

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

   4. Taylor expanded in z around 0 46.2%

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

      1. +-commutative 46.2%

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

      2. mul-1-neg 46.2%

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

      3. unsub-neg 46.2%

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

      4. unpow2 46.2%

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

      5. associate-/r* 46.2%

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

   6. Simplified 46.2%

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

   7. Taylor expanded in z around inf 38.4%

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

      1. associate-*r/ 38.4%

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

      2. neg-mul-1 38.4%

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

      3. distribute-lft-neg-in 38.4%

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

      4. associate-*l/ 46.2%

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

      5. *-commutative 46.2%

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

   9. Simplified 46.2%

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

   10. Taylor expanded in x around 0 38.4%

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

       1. mul-1-neg 38.4%

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

       2. *-commutative 38.4%

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

       3. associate-*l/ 44.5%

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

       4. associate-/r/ 52.1%

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

   12. Simplified 52.1%

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

4. Final simplification 43.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1 \cdot 10^{-310}:\\ \;\;\;\;\frac{-y}{\frac{z}{x}}\\ \mathbf{elif}\;z \leq 3.65 \cdot 10^{+139} \lor \neg \left(z \leq 1.8 \cdot 10^{+232}\right):\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{-x}{\frac{z}{y}}\\ \end{array} \]

Alternative 9: 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 84.6%

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

   1. associate-*l* 84.6%

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

   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. Step-by-step derivation

   1. fma-udef 93.6%

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

   2. *-rgt-identity 93.6%

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

   3. distribute-lft-in 93.6%

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

   4. times-frac 84.6%

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

   5. associate-*l* 84.6%

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

   6. associate-/r* 87.5%

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

   7. times-frac 95.9%

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

5. Applied egg-rr 95.9%

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

6. Final simplification 95.9%

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

Alternative 10: 76.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 4.5 \cdot 10^{+65}:\\ \;\;\;\;\frac{x}{z} \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{z \cdot z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 4.5e+65) (* (/ x z) (/ y z)) (* y (/ x (* z z)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 4.5e+65) {
		tmp = (x / z) * (y / 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 <= 4.5d+65) then
        tmp = (x / z) * (y / 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 <= 4.5e+65) {
		tmp = (x / z) * (y / z);
	} else {
		tmp = y * (x / (z * z));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 4.5e+65:
		tmp = (x / z) * (y / z)
	else:
		tmp = y * (x / (z * z))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 4.5e+65)
		tmp = Float64(Float64(x / z) * Float64(y / 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 <= 4.5e+65)
		tmp = (x / z) * (y / z);
	else
		tmp = y * (x / (z * z));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 4.5e+65], N[(N[(x / z), $MachinePrecision] * N[(y / z), $MachinePrecision]), $MachinePrecision], N[(y * N[(x / N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 4.5e65

   1. Initial program 84.4%

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

      1. associate-*l* 84.4%

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

      2. times-frac 96.0%

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

      3. distribute-lft-in 96.1%

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

      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 72.2%

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

      1. unpow2 72.2%

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

      2. associate-/l/ 73.8%

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

      3. associate-*r/ 75.5%

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

      4. associate-*l/ 77.8%

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

   6. Simplified 77.8%

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

   if 4.5e65 < y

   1. Initial program 85.6%

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

      1. times-frac 86.3%

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

   3. Simplified 86.3%

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

   4. Taylor expanded in z around 0 80.7%

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

4. Final simplification 78.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 4.5 \cdot 10^{+65}:\\ \;\;\;\;\frac{x}{z} \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{z \cdot z}\\ \end{array} \]

Alternative 11: 77.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.85 \cdot 10^{-78}:\\ \;\;\;\;\frac{x}{z \cdot \frac{z}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z \cdot \frac{z}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 1.85e-78) (/ x (* z (/ z y))) (/ y (* z (/ z x)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 1.85e-78) {
		tmp = x / (z * (z / y));
	} 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 (y <= 1.85d-78) then
        tmp = x / (z * (z / y))
    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 (y <= 1.85e-78) {
		tmp = x / (z * (z / y));
	} else {
		tmp = y / (z * (z / x));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 1.85e-78:
		tmp = x / (z * (z / y))
	else:
		tmp = y / (z * (z / x))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 1.85e-78)
		tmp = Float64(x / Float64(z * Float64(z / y)));
	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 (y <= 1.85e-78)
		tmp = x / (z * (z / y));
	else
		tmp = y / (z * (z / x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 1.85e-78], N[(x / N[(z * N[(z / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y / N[(z * N[(z / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.85000000000000003e-78

   1. Initial program 85.6%

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

      1. /-rgt-identity 85.6%

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

      2. associate-/l* 85.5%

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

      3. associate-/l/ 87.4%

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

      4. associate-*l* 89.0%

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

      5. associate-*r/ 89.1%

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

      6. *-rgt-identity 89.1%

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

      7. associate-*l* 93.7%

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

      8. associate-*r/ 93.7%

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

      9. distribute-lft-in 93.7%

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

      10. fma-def 93.7%

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

      11. *-rgt-identity 93.7%

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

   3. Simplified 93.7%

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

   4. Taylor expanded in z around 0 78.7%

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

   if 1.85000000000000003e-78 < y

   1. Initial program 82.7%

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

      1. associate-*l* 82.7%

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

      2. times-frac 90.2%

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

      3. distribute-lft-in 90.2%

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

      4. fma-def 90.2%

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

      5. *-rgt-identity 90.2%

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

   3. Simplified 90.2%

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

   4. Taylor expanded in z around 0 72.4%

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

      1. unpow2 72.4%

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

      2. associate-/l* 75.0%

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

      3. associate-/l* 80.5%

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

   6. Simplified 80.5%

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

      1. associate-/r/ 80.4%

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

   8. Applied egg-rr 80.4%

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

4. Final simplification 79.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.85 \cdot 10^{-78}:\\ \;\;\;\;\frac{x}{z \cdot \frac{z}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z \cdot \frac{z}{x}}\\ \end{array} \]

Alternative 12: 77.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2.4 \cdot 10^{-78}:\\ \;\;\;\;\frac{x}{z \cdot \frac{z}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{z}{\frac{x}{z}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 2.4e-78) (/ x (* z (/ z y))) (/ y (/ z (/ x z)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 2.4e-78) {
		tmp = x / (z * (z / y));
	} else {
		tmp = y / (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 (y <= 2.4d-78) then
        tmp = x / (z * (z / y))
    else
        tmp = y / (z / (x / z))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 2.4e-78) {
		tmp = x / (z * (z / y));
	} else {
		tmp = y / (z / (x / z));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 2.4e-78:
		tmp = x / (z * (z / y))
	else:
		tmp = y / (z / (x / z))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 2.4e-78)
		tmp = Float64(x / Float64(z * Float64(z / y)));
	else
		tmp = Float64(y / Float64(z / Float64(x / z)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 2.4e-78)
		tmp = x / (z * (z / y));
	else
		tmp = y / (z / (x / z));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 2.4e-78], N[(x / N[(z * N[(z / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y / N[(z / N[(x / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2.4e-78

   1. Initial program 85.6%

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

      1. /-rgt-identity 85.6%

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

      2. associate-/l* 85.5%

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

      3. associate-/l/ 87.4%

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

      4. associate-*l* 89.0%

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

      5. associate-*r/ 89.1%

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

      6. *-rgt-identity 89.1%

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

      7. associate-*l* 93.7%

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

      8. associate-*r/ 93.7%

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

      9. distribute-lft-in 93.7%

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

      10. fma-def 93.7%

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

      11. *-rgt-identity 93.7%

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

   3. Simplified 93.7%

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

   4. Taylor expanded in z around 0 78.7%

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

   if 2.4e-78 < y

   1. Initial program 82.7%

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

      1. associate-*l* 82.7%

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

      2. times-frac 90.2%

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

      3. distribute-lft-in 90.2%

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

      4. fma-def 90.2%

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

      5. *-rgt-identity 90.2%

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

   3. Simplified 90.2%

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

   4. Taylor expanded in z around 0 72.4%

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

      1. unpow2 72.4%

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

      2. associate-/l* 75.0%

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

      3. associate-/l* 80.5%

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

   6. Simplified 80.5%

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

4. Final simplification 79.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.4 \cdot 10^{-78}:\\ \;\;\;\;\frac{x}{z \cdot \frac{z}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{z}{\frac{x}{z}}}\\ \end{array} \]

Alternative 13: 32.6% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 7.5 \cdot 10^{-174}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{z}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 7.5e-174) (* x (/ y z)) (/ y (/ z x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 7.5e-174) {
		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 (y <= 7.5d-174) 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 (y <= 7.5e-174) {
		tmp = x * (y / z);
	} else {
		tmp = y / (z / x);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 7.5e-174:
		tmp = x * (y / z)
	else:
		tmp = y / (z / x)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 7.5e-174)
		tmp = Float64(x * 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 (y <= 7.5e-174)
		tmp = x * (y / z);
	else
		tmp = y / (z / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 7.5e-174], N[(x * N[(y / z), $MachinePrecision]), $MachinePrecision], N[(y / N[(z / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 7.5000000000000003e-174

   1. Initial program 86.0%

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

      1. associate-*r/ 88.1%

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

      2. associate-*l* 88.1%

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

      3. distribute-lft-in 88.1%

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

      4. fma-def 88.1%

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

      5. *-rgt-identity 88.1%

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

   3. Simplified 88.1%

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

   4. Taylor expanded in z around 0 58.0%

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

      1. +-commutative 58.0%

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

      2. mul-1-neg 58.0%

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

      3. unsub-neg 58.0%

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

      4. unpow2 58.0%

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

      5. associate-/r* 60.9%

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

   6. Simplified 60.9%

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

   7. Taylor expanded in z around inf 30.2%

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

      1. associate-*r/ 30.2%

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

      2. neg-mul-1 30.2%

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

      3. distribute-lft-neg-in 30.2%

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

      4. associate-*l/ 36.7%

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

      5. *-commutative 36.7%

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

   9. Simplified 36.7%

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

       1. add-sqr-sqrt 30.9%

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

       2. sqrt-unprod 37.1%

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

       3. sqr-neg 37.1%

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

       4. sqrt-unprod 5.7%

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

       5. add-sqr-sqrt 32.2%

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

       6. associate-*r/ 26.3%

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

       7. *-commutative 26.3%

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

       8. associate-/l* 30.7%

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

   11. Applied egg-rr 30.7%

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

       1. associate-/r/ 32.2%

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

   13. Applied egg-rr 32.2%

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

   if 7.5000000000000003e-174 < y

   1. Initial program 82.6%

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

      1. associate-*r/ 77.7%

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

      2. associate-*l* 77.7%

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

      3. distribute-lft-in 77.7%

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

      4. fma-def 77.7%

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

      5. *-rgt-identity 77.7%

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

   3. Simplified 77.7%

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

   4. Taylor expanded in z around 0 54.0%

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

      1. +-commutative 54.0%

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

      2. mul-1-neg 54.0%

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

      3. unsub-neg 54.0%

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

      4. unpow2 54.0%

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

      5. associate-/r* 57.6%

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

   6. Simplified 57.6%

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

   7. Taylor expanded in z around inf 27.7%

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

      1. associate-*r/ 27.7%

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

      2. neg-mul-1 27.7%

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

      3. distribute-lft-neg-in 27.7%

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

      4. associate-*l/ 29.6%

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

      5. *-commutative 29.6%

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

   9. Simplified 29.6%

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

       1. add-sqr-sqrt 0.0%

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

       2. sqrt-unprod 28.1%

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

       3. sqr-neg 28.1%

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

       4. sqrt-unprod 29.9%

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

       5. add-sqr-sqrt 29.9%

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

       6. associate-*r/ 29.9%

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

       7. *-commutative 29.9%

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

       8. associate-/l* 33.7%

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

   11. Applied egg-rr 33.7%

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

4. Final simplification 32.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 7.5 \cdot 10^{-174}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{z}{x}}\\ \end{array} \]

Alternative 14: 72.7% accurate, 1.6× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z) {
	return y * (x / (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 = y * (x / (z * z))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 84.6%

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

   1. times-frac 87.4%

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

3. Simplified 87.4%

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

4. Taylor expanded in z around 0 74.6%

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

5. Final simplification 74.6%

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

Alternative 15: 31.4% accurate, 2.2× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z) {
	return y * (x / 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 = y * (x / z)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 84.6%

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

   1. associate-*r/ 83.8%

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

   2. associate-*l* 83.8%

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

   3. distribute-lft-in 83.8%

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

   4. fma-def 83.8%

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

   5. *-rgt-identity 83.8%

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

3. Simplified 83.8%

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

4. Taylor expanded in z around 0 56.4%

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

   1. +-commutative 56.4%

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

   2. mul-1-neg 56.4%

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

   3. unsub-neg 56.4%

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

   4. unpow2 56.4%

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

   5. associate-/r* 59.5%

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

6. Simplified 59.5%

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

7. Taylor expanded in z around inf 29.2%

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

   1. associate-*r/ 29.2%

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

   2. neg-mul-1 29.2%

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

   3. distribute-lft-neg-in 29.2%

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

   4. associate-*l/ 33.7%

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

   5. *-commutative 33.7%

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

9. Simplified 33.7%

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

    1. add-sqr-sqrt 18.0%

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

    2. sqrt-unprod 33.3%

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

    3. sqr-neg 33.3%

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

    4. sqrt-unprod 15.8%

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

    5. add-sqr-sqrt 31.2%

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

    6. associate-*r/ 27.8%

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

    7. *-commutative 27.8%

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

    8. associate-/l* 31.9%

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

11. Applied egg-rr 31.9%

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

    1. associate-/l* 27.8%

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

    2. *-commutative 27.8%

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

    3. associate-*l/ 30.8%

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

13. Applied egg-rr 30.8%

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

14. Final simplification 30.8%

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

Developer target: 95.4% 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 2023174 
(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))))