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

Percentage Accurate: 84.0% → 96.9%

Time: 10.6s

Alternatives: 7

Speedup: 1.0×

• 26.3% 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: 84.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 96.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z) {
	return ((y / z) * (x / 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 = ((y / z) * (x / z)) / (z + 1.0d0)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 85.8%

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

   1. associate-/r* 86.9%

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

3. Simplified 86.9%

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

   1. *-commutative 86.9%

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

   2. times-frac 97.6%

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

5. Applied egg-rr 97.6%

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

6. Final simplification 97.6%

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

Alternative 2: 90.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(z + 1\right) \cdot \left(z \cdot z\right)\\ \mathbf{if}\;t_0 \leq -20000000 \lor \neg \left(t_0 \leq 5 \cdot 10^{-117}\right):\\ \;\;\;\;\frac{y \cdot x}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{z}}{\frac{z}{y}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (+ z 1.0) (* z z))))
   (if (or (<= t_0 -20000000.0) (not (<= t_0 5e-117)))
     (/ (* y x) t_0)
     (/ (/ x z) (/ z y)))))

C

double code(double x, double y, double z) {
	double t_0 = (z + 1.0) * (z * z);
	double tmp;
	if ((t_0 <= -20000000.0) || !(t_0 <= 5e-117)) {
		tmp = (y * x) / t_0;
	} else {
		tmp = (x / z) / (z / y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (z + 1.0d0) * (z * z)
    if ((t_0 <= (-20000000.0d0)) .or. (.not. (t_0 <= 5d-117))) then
        tmp = (y * x) / t_0
    else
        tmp = (x / z) / (z / y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = (z + 1.0) * (z * z);
	double tmp;
	if ((t_0 <= -20000000.0) || !(t_0 <= 5e-117)) {
		tmp = (y * x) / t_0;
	} else {
		tmp = (x / z) / (z / y);
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (z + 1.0) * (z * z)
	tmp = 0
	if (t_0 <= -20000000.0) or not (t_0 <= 5e-117):
		tmp = (y * x) / t_0
	else:
		tmp = (x / z) / (z / y)
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(z + 1.0) * Float64(z * z))
	tmp = 0.0
	if ((t_0 <= -20000000.0) || !(t_0 <= 5e-117))
		tmp = Float64(Float64(y * x) / t_0);
	else
		tmp = Float64(Float64(x / z) / Float64(z / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (z + 1.0) * (z * z);
	tmp = 0.0;
	if ((t_0 <= -20000000.0) || ~((t_0 <= 5e-117)))
		tmp = (y * x) / t_0;
	else
		tmp = (x / z) / (z / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(z + 1.0), $MachinePrecision] * N[(z * z), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, -20000000.0], N[Not[LessEqual[t$95$0, 5e-117]], $MachinePrecision]], N[(N[(y * x), $MachinePrecision] / t$95$0), $MachinePrecision], N[(N[(x / z), $MachinePrecision] / N[(z / y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 z z) (+.f64 z 1)) < -2e7 or 5e-117 < (*.f64 (*.f64 z z) (+.f64 z 1))

   1. Initial program 89.7%

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

   if -2e7 < (*.f64 (*.f64 z z) (+.f64 z 1)) < 5e-117

   1. Initial program 80.3%

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

      1. associate-*l* 80.3%

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

      2. times-frac 99.8%

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

      3. *-commutative 99.8%

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

      4. associate-/r/ 90.5%

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

      5. *-commutative 90.5%

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

      6. distribute-lft1-in 90.5%

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

      7. fma-def 90.5%

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

   3. Simplified 90.5%

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

   4. Taylor expanded in z around 0 82.1%

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

      1. unpow2 82.1%

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

      2. associate-*r/ 90.4%

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

   6. Simplified 90.4%

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

      1. associate-/l/ 97.9%

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

      2. un-div-inv 97.9%

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

      3. clear-num 97.9%

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

      4. div-inv 97.8%

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

      5. associate-*r/ 89.7%

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

   8. Applied egg-rr 89.7%

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

      1. *-commutative 89.7%

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

      2. associate-/l* 97.8%

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

      3. associate-*l/ 99.9%

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

      4. div-inv 99.9%

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

   10. Applied egg-rr 99.9%

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

4. Final simplification 93.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(z + 1\right) \cdot \left(z \cdot z\right) \leq -20000000 \lor \neg \left(\left(z + 1\right) \cdot \left(z \cdot z\right) \leq 5 \cdot 10^{-117}\right):\\ \;\;\;\;\frac{y \cdot x}{\left(z + 1\right) \cdot \left(z \cdot z\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{z}}{\frac{z}{y}}\\ \end{array} \]

Alternative 3: 92.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(z + 1\right) \cdot \left(z \cdot z\right)\\ \mathbf{if}\;t_0 \leq -20000000:\\ \;\;\;\;\frac{y \cdot \frac{x}{z \cdot z}}{z + 1}\\ \mathbf{elif}\;t_0 \leq 5 \cdot 10^{-117}:\\ \;\;\;\;\frac{\frac{x}{z}}{\frac{z}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot x}{t_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (+ z 1.0) (* z z))))
   (if (<= t_0 -20000000.0)
     (/ (* y (/ x (* z z))) (+ z 1.0))
     (if (<= t_0 5e-117) (/ (/ x z) (/ z y)) (/ (* y x) t_0)))))

C

double code(double x, double y, double z) {
	double t_0 = (z + 1.0) * (z * z);
	double tmp;
	if (t_0 <= -20000000.0) {
		tmp = (y * (x / (z * z))) / (z + 1.0);
	} else if (t_0 <= 5e-117) {
		tmp = (x / z) / (z / y);
	} else {
		tmp = (y * x) / 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 = (z + 1.0d0) * (z * z)
    if (t_0 <= (-20000000.0d0)) then
        tmp = (y * (x / (z * z))) / (z + 1.0d0)
    else if (t_0 <= 5d-117) then
        tmp = (x / z) / (z / y)
    else
        tmp = (y * x) / t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = (z + 1.0) * (z * z);
	double tmp;
	if (t_0 <= -20000000.0) {
		tmp = (y * (x / (z * z))) / (z + 1.0);
	} else if (t_0 <= 5e-117) {
		tmp = (x / z) / (z / y);
	} else {
		tmp = (y * x) / t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (z + 1.0) * (z * z)
	tmp = 0
	if t_0 <= -20000000.0:
		tmp = (y * (x / (z * z))) / (z + 1.0)
	elif t_0 <= 5e-117:
		tmp = (x / z) / (z / y)
	else:
		tmp = (y * x) / t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(z + 1.0) * Float64(z * z))
	tmp = 0.0
	if (t_0 <= -20000000.0)
		tmp = Float64(Float64(y * Float64(x / Float64(z * z))) / Float64(z + 1.0));
	elseif (t_0 <= 5e-117)
		tmp = Float64(Float64(x / z) / Float64(z / y));
	else
		tmp = Float64(Float64(y * x) / t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (z + 1.0) * (z * z);
	tmp = 0.0;
	if (t_0 <= -20000000.0)
		tmp = (y * (x / (z * z))) / (z + 1.0);
	elseif (t_0 <= 5e-117)
		tmp = (x / z) / (z / y);
	else
		tmp = (y * x) / t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(z + 1.0), $MachinePrecision] * N[(z * z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -20000000.0], N[(N[(y * N[(x / N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(z + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 5e-117], N[(N[(x / z), $MachinePrecision] / N[(z / y), $MachinePrecision]), $MachinePrecision], N[(N[(y * x), $MachinePrecision] / t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 z z) (+.f64 z 1)) < -2e7

   1. Initial program 84.0%

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

      1. associate-/r* 85.9%

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

   3. Simplified 85.9%

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

      1. associate-*l/ 94.7%

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

   5. Applied egg-rr 94.7%

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

   if -2e7 < (*.f64 (*.f64 z z) (+.f64 z 1)) < 5e-117

   1. Initial program 80.3%

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

      1. associate-*l* 80.3%

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

      2. times-frac 99.8%

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

      3. *-commutative 99.8%

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

      4. associate-/r/ 90.5%

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

      5. *-commutative 90.5%

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

      6. distribute-lft1-in 90.5%

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

      7. fma-def 90.5%

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

   3. Simplified 90.5%

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

   4. Taylor expanded in z around 0 82.1%

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

      1. unpow2 82.1%

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

      2. associate-*r/ 90.4%

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

   6. Simplified 90.4%

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

      1. associate-/l/ 97.9%

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

      2. un-div-inv 97.9%

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

      3. clear-num 97.9%

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

      4. div-inv 97.8%

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

      5. associate-*r/ 89.7%

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

   8. Applied egg-rr 89.7%

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

      1. *-commutative 89.7%

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

      2. associate-/l* 97.8%

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

      3. associate-*l/ 99.9%

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

      4. div-inv 99.9%

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

   10. Applied egg-rr 99.9%

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

   if 5e-117 < (*.f64 (*.f64 z z) (+.f64 z 1))

   1. Initial program 93.8%

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

4. Final simplification 96.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(z + 1\right) \cdot \left(z \cdot z\right) \leq -20000000:\\ \;\;\;\;\frac{y \cdot \frac{x}{z \cdot z}}{z + 1}\\ \mathbf{elif}\;\left(z + 1\right) \cdot \left(z \cdot z\right) \leq 5 \cdot 10^{-117}:\\ \;\;\;\;\frac{\frac{x}{z}}{\frac{z}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot x}{\left(z + 1\right) \cdot \left(z \cdot z\right)}\\ \end{array} \]

Alternative 4: 77.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \cdot x \leq -1 \cdot 10^{-9}:\\ \;\;\;\;\frac{-x}{\left(z \cdot z\right) \cdot \frac{-1}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{z}}{\frac{z}{y}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (* y x) -1e-9) (/ (- x) (* (* z z) (/ -1.0 y))) (/ (/ x z) (/ z y))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y * x) <= -1e-9) {
		tmp = -x / ((z * z) * (-1.0 / y));
	} else {
		tmp = (x / z) / (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 ((y * x) <= (-1d-9)) then
        tmp = -x / ((z * z) * ((-1.0d0) / y))
    else
        tmp = (x / z) / (z / y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((y * x) <= -1e-9) {
		tmp = -x / ((z * z) * (-1.0 / y));
	} else {
		tmp = (x / z) / (z / y);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y * x) <= -1e-9:
		tmp = -x / ((z * z) * (-1.0 / y))
	else:
		tmp = (x / z) / (z / y)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(y * x) <= -1e-9)
		tmp = Float64(Float64(-x) / Float64(Float64(z * z) * Float64(-1.0 / y)));
	else
		tmp = Float64(Float64(x / z) / Float64(z / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((y * x) <= -1e-9)
		tmp = -x / ((z * z) * (-1.0 / y));
	else
		tmp = (x / z) / (z / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(y * x), $MachinePrecision], -1e-9], N[((-x) / N[(N[(z * z), $MachinePrecision] * N[(-1.0 / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / z), $MachinePrecision] / N[(z / y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x y) < -1.00000000000000006e-9

   1. Initial program 87.1%

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

      1. associate-*l* 87.1%

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

      2. times-frac 92.1%

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

      3. *-commutative 92.1%

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

      4. associate-/r/ 91.6%

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

      5. *-commutative 91.6%

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

      6. distribute-lft1-in 91.6%

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

      7. fma-def 91.6%

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

   3. Simplified 91.6%

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

   4. Taylor expanded in z around 0 63.2%

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

      1. unpow2 63.2%

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

      2. associate-*r/ 53.5%

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

   6. Simplified 53.5%

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

      1. associate-/l/ 49.8%

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

      2. un-div-inv 48.6%

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

      3. clear-num 48.6%

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

      4. div-inv 48.5%

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

      5. associate-*r/ 51.1%

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

   8. Applied egg-rr 51.1%

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

      1. *-commutative 51.1%

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

      2. associate-/l* 52.1%

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

      3. associate-*l/ 49.5%

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

      4. div-inv 49.5%

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

      5. frac-2neg 49.5%

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

      6. div-inv 49.5%

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

      7. associate-/l* 64.4%

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

      8. neg-mul-1 64.4%

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

      9. metadata-eval 64.4%

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

      10. associate-/r* 64.4%

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

      11. metadata-eval 64.4%

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

      12. metadata-eval 64.4%

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

   10. Applied egg-rr 64.4%

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

       1. clear-num 64.4%

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

       2. associate-/l/ 64.4%

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

       3. associate-/r* 64.4%

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

       4. associate-/r/ 64.4%

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

       5. metadata-eval 64.4%

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

       6. div-inv 64.4%

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

       7. times-frac 68.3%

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

       8. associate-/r/ 68.3%

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

       9. /-rgt-identity 68.3%

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

   12. Applied egg-rr 68.3%

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

   if -1.00000000000000006e-9 < (*.f64 x y)

   1. Initial program 85.3%

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

      1. associate-*l* 85.3%

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

      2. times-frac 97.9%

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

      3. *-commutative 97.9%

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

      4. associate-/r/ 92.1%

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

      5. *-commutative 92.1%

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

      6. distribute-lft1-in 92.1%

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

      7. fma-def 92.1%

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

   3. Simplified 92.1%

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

   4. Taylor expanded in z around 0 75.5%

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

      1. unpow2 75.5%

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

      2. associate-*r/ 80.0%

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

   6. Simplified 80.0%

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

      1. associate-/l/ 83.8%

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

      2. un-div-inv 83.8%

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

      3. clear-num 83.8%

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

      4. div-inv 83.8%

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

      5. associate-*r/ 78.6%

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

   8. Applied egg-rr 78.6%

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

      1. *-commutative 78.6%

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

      2. associate-/l* 83.6%

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

      3. associate-*l/ 85.3%

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

      4. div-inv 85.3%

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

   10. Applied egg-rr 85.3%

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

4. Final simplification 80.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot x \leq -1 \cdot 10^{-9}:\\ \;\;\;\;\frac{-x}{\left(z \cdot z\right) \cdot \frac{-1}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{z}}{\frac{z}{y}}\\ \end{array} \]

Alternative 5: 76.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \cdot x \leq -2 \cdot 10^{-14}:\\ \;\;\;\;x \cdot \frac{y}{z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z} \cdot \frac{x}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (* y x) -2e-14) (* x (/ y (* z z))) (* (/ y z) (/ x z))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y * x) <= -2e-14) {
		tmp = x * (y / (z * z));
	} 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 * x) <= (-2d-14)) then
        tmp = x * (y / (z * z))
    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 * x) <= -2e-14) {
		tmp = x * (y / (z * z));
	} else {
		tmp = (y / z) * (x / z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y * x) <= -2e-14:
		tmp = x * (y / (z * z))
	else:
		tmp = (y / z) * (x / z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(y * x) <= -2e-14)
		tmp = Float64(x * Float64(y / Float64(z * z)));
	else
		tmp = Float64(Float64(y / z) * Float64(x / z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((y * x) <= -2e-14)
		tmp = x * (y / (z * z));
	else
		tmp = (y / z) * (x / z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(y * x), $MachinePrecision], -2e-14], N[(x * N[(y / N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y / z), $MachinePrecision] * N[(x / z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x y) < -2e-14

   1. Initial program 87.3%

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

      1. associate-*r/ 90.2%

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

      2. associate-*l* 90.2%

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

      3. associate-/r* 93.0%

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

      4. *-commutative 93.0%

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

      5. distribute-lft1-in 93.0%

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

      6. fma-def 93.0%

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

   3. Simplified 93.0%

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

   4. Taylor expanded in z around 0 68.9%

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

      1. unpow2 68.9%

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

   6. Simplified 68.9%

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

   if -2e-14 < (*.f64 x y)

   1. Initial program 85.2%

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

      1. associate-*l* 85.2%

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

      2. times-frac 97.9%

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

      3. *-commutative 97.9%

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

      4. associate-/r/ 92.0%

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

      5. *-commutative 92.0%

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

      6. distribute-lft1-in 92.0%

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

      7. fma-def 92.0%

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

   3. Simplified 92.0%

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

   4. Taylor expanded in z around 0 75.4%

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

      1. unpow2 75.4%

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

      2. associate-*r/ 79.9%

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

   6. Simplified 79.9%

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

      1. associate-*r/ 75.4%

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

      2. associate-/l* 73.6%

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

      3. frac-times 85.2%

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

   8. Applied egg-rr 85.2%

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

4. Final simplification 80.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot x \leq -2 \cdot 10^{-14}:\\ \;\;\;\;x \cdot \frac{y}{z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z} \cdot \frac{x}{z}\\ \end{array} \]

Alternative 6: 77.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \cdot x \leq -1 \cdot 10^{-9}:\\ \;\;\;\;x \cdot \frac{y}{z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{z}}{\frac{z}{y}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (* y x) -1e-9) (* x (/ y (* z z))) (/ (/ x z) (/ z y))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y * x) <= -1e-9) {
		tmp = x * (y / (z * z));
	} else {
		tmp = (x / z) / (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 ((y * x) <= (-1d-9)) then
        tmp = x * (y / (z * z))
    else
        tmp = (x / z) / (z / y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((y * x) <= -1e-9) {
		tmp = x * (y / (z * z));
	} else {
		tmp = (x / z) / (z / y);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y * x) <= -1e-9:
		tmp = x * (y / (z * z))
	else:
		tmp = (x / z) / (z / y)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(y * x) <= -1e-9)
		tmp = Float64(x * Float64(y / Float64(z * z)));
	else
		tmp = Float64(Float64(x / z) / Float64(z / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((y * x) <= -1e-9)
		tmp = x * (y / (z * z));
	else
		tmp = (x / z) / (z / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(y * x), $MachinePrecision], -1e-9], N[(x * N[(y / N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / z), $MachinePrecision] / N[(z / y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x y) < -1.00000000000000006e-9

   1. Initial program 87.1%

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

      1. associate-*r/ 90.0%

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

      2. associate-*l* 90.0%

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

      3. associate-/r* 92.9%

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

      4. *-commutative 92.9%

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

      5. distribute-lft1-in 92.9%

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

      6. fma-def 92.9%

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

   3. Simplified 92.9%

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

   4. Taylor expanded in z around 0 68.5%

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

      1. unpow2 68.5%

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

   6. Simplified 68.5%

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

   if -1.00000000000000006e-9 < (*.f64 x y)

   1. Initial program 85.3%

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

      1. associate-*l* 85.3%

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

      2. times-frac 97.9%

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

      3. *-commutative 97.9%

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

      4. associate-/r/ 92.1%

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

      5. *-commutative 92.1%

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

      6. distribute-lft1-in 92.1%

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

      7. fma-def 92.1%

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

   3. Simplified 92.1%

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

   4. Taylor expanded in z around 0 75.5%

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

      1. unpow2 75.5%

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

      2. associate-*r/ 80.0%

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

   6. Simplified 80.0%

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

      1. associate-/l/ 83.8%

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

      2. un-div-inv 83.8%

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

      3. clear-num 83.8%

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

      4. div-inv 83.8%

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

      5. associate-*r/ 78.6%

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

   8. Applied egg-rr 78.6%

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

      1. *-commutative 78.6%

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

      2. associate-/l* 83.6%

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

      3. associate-*l/ 85.3%

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

      4. div-inv 85.3%

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

   10. Applied egg-rr 85.3%

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

4. Final simplification 80.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot x \leq -1 \cdot 10^{-9}:\\ \;\;\;\;x \cdot \frac{y}{z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{z}}{\frac{z}{y}}\\ \end{array} \]

Alternative 7: 73.1% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 85.8%

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

   1. associate-*r/ 87.8%

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

   2. associate-*l* 87.8%

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

   3. associate-/r* 94.0%

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

   4. *-commutative 94.0%

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

   5. distribute-lft1-in 94.0%

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

   6. fma-def 94.0%

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

3. Simplified 94.0%

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

4. Taylor expanded in z around 0 74.3%

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

   1. unpow2 74.3%

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

6. Simplified 74.3%

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

7. Final simplification 74.3%

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

Developer target: 95.3% 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 2023297 
(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))))