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

Percentage Accurate: 83.3% → 97.3%

Time: 9.1s

Alternatives: 10

Speedup: 1.0×

• 25.3% 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: 83.3% 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: 97.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 80.6%

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

   1. associate-*l/ 82.5%

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

   2. *-commutative 82.5%

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

   3. sqr-neg 82.5%

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

   4. *-commutative 82.5%

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

   5. distribute-rgt1-in 72.3%

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

   6. sqr-neg 72.3%

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

   7. fma-def 82.5%

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

   8. sqr-neg 82.5%

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

   9. cube-unmult 82.5%

      \[\leadsto y \cdot \frac{x}{\mathsf{fma}\left(z, z, \color{blue}{{z}^{3}}\right)} \]

3. Simplified 82.5%

   \[\leadsto \color{blue}{y \cdot \frac{x}{\mathsf{fma}\left(z, z, {z}^{3}\right)}} \]
4. Step-by-step derivation

   1. associate-*r/ 80.6%

      \[\leadsto \color{blue}{\frac{y \cdot x}{\mathsf{fma}\left(z, z, {z}^{3}\right)}} \]

   2. fma-udef 71.2%

      \[\leadsto \frac{y \cdot x}{\color{blue}{z \cdot z + {z}^{3}}} \]

   3. cube-mult 71.2%

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

   4. distribute-rgt1-in 80.6%

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

   5. frac-times 87.5%

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

   6. associate-/r* 92.8%

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

   7. associate-*r/ 97.7%

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

5. Applied egg-rr 97.7%

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

6. Final simplification 97.7%

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

Alternative 2: 95.1% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if z < -1 or 1 < z

   1. Initial program 79.1%

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

      1. frac-times 93.1%

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

      2. associate-*l/ 95.5%

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

      3. times-frac 95.5%

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

   3. Applied egg-rr 95.5%

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

   4. Taylor expanded in z around inf 95.0%

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

   if -1 < z < 1

   1. Initial program 82.1%

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

      1. sqr-neg 82.1%

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

      2. times-frac 82.1%

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

      3. sqr-neg 82.1%

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

   3. Simplified 82.1%

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

   4. Taylor expanded in z around 0 80.6%

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

      1. *-commutative 80.6%

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

      2. associate-/r* 88.8%

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

      3. associate-*r/ 97.4%

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

      4. clear-num 97.2%

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

      5. un-div-inv 97.3%

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

   6. Applied egg-rr 97.3%

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

4. Final simplification 96.2%

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

Alternative 3: 95.2% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if z < -1 or 1 < z

   1. Initial program 79.1%

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

      1. associate-*l/ 82.9%

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

      2. *-commutative 82.9%

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

      3. sqr-neg 82.9%

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

      4. *-commutative 82.9%

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

      5. distribute-rgt1-in 62.0%

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

      6. sqr-neg 62.0%

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

      7. fma-def 82.9%

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

      8. sqr-neg 82.9%

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

      9. cube-unmult 82.9%

         \[\leadsto y \cdot \frac{x}{\mathsf{fma}\left(z, z, \color{blue}{{z}^{3}}\right)} \]

   3. Simplified 82.9%

      \[\leadsto \color{blue}{y \cdot \frac{x}{\mathsf{fma}\left(z, z, {z}^{3}\right)}} \]
   4. Step-by-step derivation

      1. associate-*r/ 79.1%

         \[\leadsto \color{blue}{\frac{y \cdot x}{\mathsf{fma}\left(z, z, {z}^{3}\right)}} \]

      2. *-commutative 79.1%

         \[\leadsto \frac{\color{blue}{x \cdot y}}{\mathsf{fma}\left(z, z, {z}^{3}\right)} \]

      3. fma-udef 59.9%

         \[\leadsto \frac{x \cdot y}{\color{blue}{z \cdot z + {z}^{3}}} \]

      4. cube-mult 59.8%

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

      5. distribute-rgt1-in 79.1%

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

      6. *-commutative 79.1%

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

      7. frac-times 93.1%

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

      8. associate-*l/ 95.5%

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

      9. associate-/r* 97.0%

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

   5. Applied egg-rr 97.0%

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

   6. Taylor expanded in z around inf 96.5%

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

   if -1 < z < 1

   1. Initial program 82.1%

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

      1. sqr-neg 82.1%

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

      2. times-frac 82.1%

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

      3. sqr-neg 82.1%

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

   3. Simplified 82.1%

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

   4. Taylor expanded in z around 0 80.6%

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

      1. *-commutative 80.6%

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

      2. associate-/r* 88.8%

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

      3. associate-*r/ 97.4%

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

      4. clear-num 97.2%

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

      5. un-div-inv 97.3%

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

   6. Applied egg-rr 97.3%

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

4. Final simplification 96.9%

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

Alternative 4: 95.4% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 3 regimes

2. if z < -1

   1. Initial program 78.3%

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

      1. frac-times 87.6%

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

      2. associate-*l/ 90.5%

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

      3. times-frac 94.9%

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

   3. Applied egg-rr 94.9%

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

   4. Taylor expanded in z around inf 94.8%

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

   if -1 < z < 1

   1. Initial program 82.1%

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

      1. sqr-neg 82.1%

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

      2. times-frac 82.1%

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

      3. sqr-neg 82.1%

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

   3. Simplified 82.1%

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

   4. Taylor expanded in z around 0 80.6%

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

      1. *-commutative 80.6%

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

      2. associate-/r* 88.8%

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

      3. associate-*r/ 97.4%

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

      4. clear-num 97.2%

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

      5. un-div-inv 97.3%

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

   6. Applied egg-rr 97.3%

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

   if 1 < z

   1. Initial program 79.7%

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

      1. associate-*l/ 85.0%

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

      2. *-commutative 85.0%

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

      3. sqr-neg 85.0%

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

      4. *-commutative 85.0%

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

      5. distribute-rgt1-in 84.9%

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

      6. sqr-neg 84.9%

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

      7. fma-def 84.9%

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

      8. sqr-neg 84.9%

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

      9. cube-unmult 85.0%

         \[\leadsto y \cdot \frac{x}{\mathsf{fma}\left(z, z, \color{blue}{{z}^{3}}\right)} \]

   3. Simplified 85.0%

      \[\leadsto \color{blue}{y \cdot \frac{x}{\mathsf{fma}\left(z, z, {z}^{3}\right)}} \]
   4. Step-by-step derivation

      1. associate-*r/ 79.7%

         \[\leadsto \color{blue}{\frac{y \cdot x}{\mathsf{fma}\left(z, z, {z}^{3}\right)}} \]

      2. fma-udef 79.7%

         \[\leadsto \frac{y \cdot x}{\color{blue}{z \cdot z + {z}^{3}}} \]

      3. cube-mult 79.7%

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

      4. distribute-rgt1-in 79.7%

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

      5. frac-times 97.8%

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

      6. associate-/r* 97.7%

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

      7. associate-*r/ 98.8%

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

   5. Applied egg-rr 98.8%

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

   6. Taylor expanded in z around inf 97.9%

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

4. Final simplification 96.9%

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

Alternative 5: 41.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y}{\frac{z}{x}}\\ \mathbf{if}\;z \leq -1:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq -4 \cdot 10^{-310}:\\ \;\;\;\;x \cdot \frac{-y}{z}\\ \mathbf{elif}\;z \leq 2 \cdot 10^{-75}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ y (/ z x))))
   (if (<= z -1.0)
     t_0
     (if (<= z -4e-310)
       (* x (/ (- y) z))
       (if (<= z 2e-75) (/ x (/ z y)) t_0)))))

C

double code(double x, double y, double z) {
	double t_0 = y / (z / x);
	double tmp;
	if (z <= -1.0) {
		tmp = t_0;
	} else if (z <= -4e-310) {
		tmp = x * (-y / z);
	} else if (z <= 2e-75) {
		tmp = x / (z / y);
	} 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 / (z / x)
    if (z <= (-1.0d0)) then
        tmp = t_0
    else if (z <= (-4d-310)) then
        tmp = x * (-y / z)
    else if (z <= 2d-75) then
        tmp = x / (z / y)
    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 / (z / x);
	double tmp;
	if (z <= -1.0) {
		tmp = t_0;
	} else if (z <= -4e-310) {
		tmp = x * (-y / z);
	} else if (z <= 2e-75) {
		tmp = x / (z / y);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = y / (z / x)
	tmp = 0
	if z <= -1.0:
		tmp = t_0
	elif z <= -4e-310:
		tmp = x * (-y / z)
	elif z <= 2e-75:
		tmp = x / (z / y)
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(y / Float64(z / x))
	tmp = 0.0
	if (z <= -1.0)
		tmp = t_0;
	elseif (z <= -4e-310)
		tmp = Float64(x * Float64(Float64(-y) / z));
	elseif (z <= 2e-75)
		tmp = Float64(x / Float64(z / y));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = y / (z / x);
	tmp = 0.0;
	if (z <= -1.0)
		tmp = t_0;
	elseif (z <= -4e-310)
		tmp = x * (-y / z);
	elseif (z <= 2e-75)
		tmp = x / (z / y);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(y / N[(z / x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -1.0], t$95$0, If[LessEqual[z, -4e-310], N[(x * N[((-y) / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 2e-75], N[(x / N[(z / y), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if z < -1 or 1.9999999999999999e-75 < z

   1. Initial program 81.0%

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

      1. associate-*l/ 83.8%

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

      2. *-commutative 83.8%

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

      3. sqr-neg 83.8%

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

      4. *-commutative 83.8%

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

      5. distribute-rgt1-in 64.8%

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

      6. sqr-neg 64.8%

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

      7. fma-def 83.8%

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

      8. sqr-neg 83.8%

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

      9. cube-unmult 83.8%

         \[\leadsto y \cdot \frac{x}{\mathsf{fma}\left(z, z, \color{blue}{{z}^{3}}\right)} \]

   3. Simplified 83.8%

      \[\leadsto \color{blue}{y \cdot \frac{x}{\mathsf{fma}\left(z, z, {z}^{3}\right)}} \]
   4. Step-by-step derivation

      1. associate-*r/ 81.0%

         \[\leadsto \color{blue}{\frac{y \cdot x}{\mathsf{fma}\left(z, z, {z}^{3}\right)}} \]

      2. fma-udef 63.6%

         \[\leadsto \frac{y \cdot x}{\color{blue}{z \cdot z + {z}^{3}}} \]

      3. cube-mult 63.6%

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

      4. distribute-rgt1-in 81.0%

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

      5. frac-times 93.0%

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

      6. associate-/r* 95.1%

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

      7. associate-*r/ 96.8%

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

   5. Applied egg-rr 96.8%

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

   6. Taylor expanded in z around 0 33.6%

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

      1. mul-1-neg 33.6%

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

      2. distribute-rgt-neg-out 33.6%

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

      3. +-commutative 33.6%

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

      4. associate-*r/ 33.0%

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

      5. distribute-lft-out 34.4%

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

      6. unsub-neg 34.4%

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

   8. Simplified 34.4%

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

   9. Taylor expanded in z around inf 27.6%

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

       1. mul-1-neg 27.6%

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

       2. associate-*r/ 32.4%

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

       3. distribute-rgt-neg-in 32.4%

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

       4. distribute-frac-neg 32.4%

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

   11. Simplified 32.4%

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

       1. *-commutative 32.4%

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

       2. add-sqr-sqrt 13.9%

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

       3. sqrt-unprod 35.3%

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

       4. sqr-neg 35.3%

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

       5. sqrt-unprod 20.9%

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

       6. add-sqr-sqrt 36.1%

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

       7. associate-/r/ 34.1%

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

   13. Applied egg-rr 34.1%

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

   if -1 < z < -3.999999999999988e-310

   1. Initial program 83.0%

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

      1. associate-*l/ 86.0%

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

      2. *-commutative 86.0%

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

      3. sqr-neg 86.0%

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

      4. *-commutative 86.0%

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

      5. distribute-rgt1-in 86.0%

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

      6. sqr-neg 86.0%

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

      7. fma-def 86.0%

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

      8. sqr-neg 86.0%

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

      9. cube-unmult 86.0%

         \[\leadsto y \cdot \frac{x}{\mathsf{fma}\left(z, z, \color{blue}{{z}^{3}}\right)} \]

   3. Simplified 86.0%

      \[\leadsto \color{blue}{y \cdot \frac{x}{\mathsf{fma}\left(z, z, {z}^{3}\right)}} \]
   4. Step-by-step derivation

      1. associate-*r/ 83.0%

         \[\leadsto \color{blue}{\frac{y \cdot x}{\mathsf{fma}\left(z, z, {z}^{3}\right)}} \]

      2. fma-udef 83.0%

         \[\leadsto \frac{y \cdot x}{\color{blue}{z \cdot z + {z}^{3}}} \]

      3. cube-mult 83.0%

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

      4. distribute-rgt1-in 83.0%

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

      5. frac-times 86.0%

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

      6. associate-/r* 95.7%

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

      7. associate-*r/ 99.5%

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

   5. Applied egg-rr 99.5%

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

   6. Taylor expanded in z around 0 90.8%

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

      1. mul-1-neg 90.8%

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

      2. distribute-rgt-neg-out 90.8%

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

      3. +-commutative 90.8%

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

      4. associate-*r/ 96.9%

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

      5. distribute-lft-out 96.9%

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

      6. unsub-neg 96.9%

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

   8. Simplified 96.9%

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

   9. Taylor expanded in z around inf 43.4%

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

       1. mul-1-neg 43.4%

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

       2. associate-*r/ 47.4%

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

       3. distribute-rgt-neg-in 47.4%

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

       4. distribute-frac-neg 47.4%

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

   11. Simplified 47.4%

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

   if -3.999999999999988e-310 < z < 1.9999999999999999e-75

   1. Initial program 75.8%

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

      1. associate-*l/ 73.3%

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

      2. *-commutative 73.3%

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

      3. sqr-neg 73.3%

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

      4. *-commutative 73.3%

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

      5. distribute-rgt1-in 73.3%

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

      6. sqr-neg 73.3%

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

      7. fma-def 73.3%

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

      8. sqr-neg 73.3%

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

      9. cube-unmult 73.3%

         \[\leadsto y \cdot \frac{x}{\mathsf{fma}\left(z, z, \color{blue}{{z}^{3}}\right)} \]

   3. Simplified 73.3%

      \[\leadsto \color{blue}{y \cdot \frac{x}{\mathsf{fma}\left(z, z, {z}^{3}\right)}} \]
   4. Step-by-step derivation

      1. associate-*r/ 75.8%

         \[\leadsto \color{blue}{\frac{y \cdot x}{\mathsf{fma}\left(z, z, {z}^{3}\right)}} \]

      2. fma-udef 75.8%

         \[\leadsto \frac{y \cdot x}{\color{blue}{z \cdot z + {z}^{3}}} \]

      3. cube-mult 75.8%

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

      4. distribute-rgt1-in 75.8%

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

      5. frac-times 73.3%

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

      6. associate-/r* 81.3%

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

      7. associate-*r/ 97.7%

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

   5. Applied egg-rr 97.7%

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

   6. Taylor expanded in z around 0 72.1%

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

      1. mul-1-neg 72.1%

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

      2. distribute-rgt-neg-out 72.1%

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

      3. +-commutative 72.1%

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

      4. associate-*r/ 86.8%

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

      5. distribute-lft-out 97.6%

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

      6. unsub-neg 97.6%

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

   8. Simplified 97.6%

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

   9. Taylor expanded in z around inf 2.9%

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

       1. mul-1-neg 2.9%

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

       2. associate-*r/ 2.8%

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

       3. distribute-rgt-neg-in 2.8%

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

       4. distribute-frac-neg 2.8%

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

   11. Simplified 2.8%

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

       1. clear-num 2.8%

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

       2. un-div-inv 2.8%

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

       3. add-sqr-sqrt 0.3%

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

       4. sqrt-unprod 23.5%

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

       5. sqr-neg 23.5%

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

       6. sqrt-unprod 21.2%

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

       7. add-sqr-sqrt 40.2%

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

   13. Applied egg-rr 40.2%

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

4. Final simplification 38.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1:\\ \;\;\;\;\frac{y}{\frac{z}{x}}\\ \mathbf{elif}\;z \leq -4 \cdot 10^{-310}:\\ \;\;\;\;x \cdot \frac{-y}{z}\\ \mathbf{elif}\;z \leq 2 \cdot 10^{-75}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{z}{x}}\\ \end{array} \]

Alternative 6: 76.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \cdot x \leq 10^{-69}:\\ \;\;\;\;\frac{x}{z} \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{z \cdot z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (* y x) 1e-69) (* (/ x z) (/ y z)) (* y (/ x (* z z)))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y * x) <= 1e-69) {
		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 * x) <= 1d-69) 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 * x) <= 1e-69) {
		tmp = (x / z) * (y / z);
	} else {
		tmp = y * (x / (z * z));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y * x) <= 1e-69:
		tmp = (x / z) * (y / z)
	else:
		tmp = y * (x / (z * z))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(y * x) <= 1e-69)
		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 * x) <= 1e-69)
		tmp = (x / z) * (y / z);
	else
		tmp = y * (x / (z * z));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(y * x), $MachinePrecision], 1e-69], 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 (*.f64 x y) < 9.9999999999999996e-70

   1. Initial program 80.8%

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

      1. frac-times 83.7%

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

      2. associate-*l/ 84.1%

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

      3. times-frac 97.1%

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

   3. Applied egg-rr 97.1%

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

   4. Taylor expanded in z around 0 81.5%

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

   if 9.9999999999999996e-70 < (*.f64 x y)

   1. Initial program 80.2%

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

      1. sqr-neg 80.2%

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

      2. times-frac 95.7%

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

      3. sqr-neg 95.7%

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

   3. Simplified 95.7%

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

   4. Taylor expanded in z around 0 71.5%

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

4. Final simplification 78.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot x \leq 10^{-69}:\\ \;\;\;\;\frac{x}{z} \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{z \cdot z}\\ \end{array} \]

Alternative 7: 96.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 80.6%

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

   1. frac-times 87.5%

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

   2. associate-*l/ 88.6%

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

   3. times-frac 96.6%

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

3. Applied egg-rr 96.6%

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

4. Final simplification 96.6%

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

Alternative 8: 74.4% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 80.6%

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

   1. frac-times 87.5%

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

   2. associate-*l/ 88.6%

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

   3. times-frac 96.6%

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

3. Applied egg-rr 96.6%

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

4. Taylor expanded in z around 0 73.3%

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

5. Final simplification 73.3%

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

Alternative 9: 31.9% accurate, 2.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 80.6%

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

   1. associate-*l/ 82.5%

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

   2. *-commutative 82.5%

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

   3. sqr-neg 82.5%

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

   4. *-commutative 82.5%

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

   5. distribute-rgt1-in 72.3%

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

   6. sqr-neg 72.3%

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

   7. fma-def 82.5%

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

   8. sqr-neg 82.5%

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

   9. cube-unmult 82.5%

      \[\leadsto y \cdot \frac{x}{\mathsf{fma}\left(z, z, \color{blue}{{z}^{3}}\right)} \]

3. Simplified 82.5%

   \[\leadsto \color{blue}{y \cdot \frac{x}{\mathsf{fma}\left(z, z, {z}^{3}\right)}} \]
4. Step-by-step derivation

   1. associate-*r/ 80.6%

      \[\leadsto \color{blue}{\frac{y \cdot x}{\mathsf{fma}\left(z, z, {z}^{3}\right)}} \]

   2. fma-udef 71.2%

      \[\leadsto \frac{y \cdot x}{\color{blue}{z \cdot z + {z}^{3}}} \]

   3. cube-mult 71.2%

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

   4. distribute-rgt1-in 80.6%

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

   5. frac-times 87.5%

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

   6. associate-/r* 92.8%

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

   7. associate-*r/ 97.7%

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

5. Applied egg-rr 97.7%

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

6. Taylor expanded in z around 0 56.6%

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

   1. mul-1-neg 56.6%

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

   2. distribute-rgt-neg-out 56.6%

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

   3. +-commutative 56.6%

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

   4. associate-*r/ 60.6%

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

   5. distribute-lft-out 63.3%

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

   6. unsub-neg 63.3%

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

8. Simplified 63.3%

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

9. Taylor expanded in z around inf 27.6%

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

    1. mul-1-neg 27.6%

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

    2. associate-*r/ 31.3%

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

    3. distribute-rgt-neg-in 31.3%

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

    4. distribute-frac-neg 31.3%

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

11. Simplified 31.3%

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

    1. clear-num 32.4%

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

    2. un-div-inv 31.7%

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

    3. add-sqr-sqrt 14.9%

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

    4. sqrt-unprod 31.1%

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

    5. sqr-neg 31.1%

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

    6. sqrt-unprod 15.9%

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

    7. add-sqr-sqrt 28.8%

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

13. Applied egg-rr 28.8%

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

14. Final simplification 28.8%

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

Alternative 10: 31.9% accurate, 2.2× speedup

Math

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

FPCore

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

C

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

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)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 80.6%

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

   1. associate-*l/ 82.5%

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

   2. *-commutative 82.5%

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

   3. sqr-neg 82.5%

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

   4. *-commutative 82.5%

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

   5. distribute-rgt1-in 72.3%

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

   6. sqr-neg 72.3%

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

   7. fma-def 82.5%

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

   8. sqr-neg 82.5%

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

   9. cube-unmult 82.5%

      \[\leadsto y \cdot \frac{x}{\mathsf{fma}\left(z, z, \color{blue}{{z}^{3}}\right)} \]

3. Simplified 82.5%

   \[\leadsto \color{blue}{y \cdot \frac{x}{\mathsf{fma}\left(z, z, {z}^{3}\right)}} \]
4. Step-by-step derivation

   1. associate-*r/ 80.6%

      \[\leadsto \color{blue}{\frac{y \cdot x}{\mathsf{fma}\left(z, z, {z}^{3}\right)}} \]

   2. fma-udef 71.2%

      \[\leadsto \frac{y \cdot x}{\color{blue}{z \cdot z + {z}^{3}}} \]

   3. cube-mult 71.2%

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

   4. distribute-rgt1-in 80.6%

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

   5. frac-times 87.5%

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

   6. associate-/r* 92.8%

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

   7. associate-*r/ 97.7%

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

5. Applied egg-rr 97.7%

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

6. Taylor expanded in z around 0 56.6%

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

   1. mul-1-neg 56.6%

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

   2. distribute-rgt-neg-out 56.6%

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

   3. +-commutative 56.6%

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

   4. associate-*r/ 60.6%

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

   5. distribute-lft-out 63.3%

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

   6. unsub-neg 63.3%

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

8. Simplified 63.3%

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

9. Taylor expanded in z around inf 27.6%

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

    1. mul-1-neg 27.6%

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

    2. associate-*r/ 31.3%

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

    3. distribute-rgt-neg-in 31.3%

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

    4. distribute-frac-neg 31.3%

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

11. Simplified 31.3%

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

    1. *-commutative 31.3%

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

    2. add-sqr-sqrt 14.6%

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

    3. sqrt-unprod 30.7%

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

    4. sqr-neg 30.7%

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

    5. sqrt-unprod 15.6%

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

    6. add-sqr-sqrt 27.7%

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

    7. associate-/r/ 27.3%

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

13. Applied egg-rr 27.3%

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

14. Final simplification 27.3%

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

Developer target: 95.7% 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 2023322 
(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))))