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

Percentage Accurate: 83.0% → 96.8%

Time: 9.7s

Alternatives: 10

Speedup: 1.0×

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 83.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 96.8% 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 85.6%

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

   1. associate-*r/ 86.6%

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

   2. sqr-neg 86.6%

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

   3. *-commutative 86.6%

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

   4. distribute-rgt1-in 76.4%

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

   5. sqr-neg 76.4%

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

   6. fma-def 86.6%

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

   7. sqr-neg 86.6%

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

   8. cube-unmult 86.6%

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

3. Simplified 86.6%

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

   1. associate-*r/ 85.7%

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

   2. fma-udef 76.7%

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

   3. cube-mult 76.6%

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

   4. distribute-rgt1-in 85.6%

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

   5. *-commutative 85.6%

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

   6. frac-times 88.1%

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

   7. *-commutative 88.1%

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

   8. associate-/r* 93.7%

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

   9. 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: 93.3% accurate, 0.8× speedup

Math

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

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -1.0) (not (<= z 0.76)))
   (* (/ y z) (/ x (* z z)))
   (* (/ x z) (- (/ y z) y))))

C

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

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((z <= (-1.0d0)) .or. (.not. (z <= 0.76d0))) then
        tmp = (y / z) * (x / (z * z))
    else
        tmp = (x / z) * ((y / z) - y)
    end if
    code = tmp
end function

Java

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

Python

def code(x, y, z):
	tmp = 0
	if (z <= -1.0) or not (z <= 0.76):
		tmp = (y / z) * (x / (z * z))
	else:
		tmp = (x / z) * ((y / z) - y)
	return tmp

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -1.0], N[Not[LessEqual[z, 0.76]], $MachinePrecision]], N[(N[(y / z), $MachinePrecision] * N[(x / N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / z), $MachinePrecision] * N[(N[(y / z), $MachinePrecision] - y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1 or 0.76000000000000001 < z

   1. Initial program 89.6%

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

      1. sqr-neg 89.6%

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

      2. times-frac 93.9%

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

      3. sqr-neg 93.9%

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

   3. Simplified 93.9%

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

   4. Taylor expanded in z around inf 92.2%

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

   if -1 < z < 0.76000000000000001

   1. Initial program 81.3%

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

      1. frac-times 81.8%

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

      2. associate-*l/ 81.3%

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

      3. times-frac 97.6%

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

   3. Applied egg-rr 97.6%

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

   4. Taylor expanded in z around 0 97.1%

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

      1. +-commutative 97.1%

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

      2. mul-1-neg 97.1%

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

      3. unsub-neg 97.1%

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

   6. Simplified 97.1%

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

4. Final simplification 94.6%

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

Alternative 3: 95.8% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if z < -1 or 0.75 < z

   1. Initial program 89.6%

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

      1. associate-*r/ 92.1%

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

      2. sqr-neg 92.1%

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

      3. *-commutative 92.1%

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

      4. distribute-rgt1-in 72.5%

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

      5. sqr-neg 72.5%

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

      6. fma-def 92.1%

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

      7. sqr-neg 92.1%

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

      8. cube-unmult 92.1%

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

   3. Simplified 92.1%

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

      1. associate-*r/ 89.6%

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

      2. fma-udef 72.3%

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

      3. cube-mult 72.3%

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

      4. distribute-rgt1-in 89.6%

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

      5. *-commutative 89.6%

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

      6. frac-times 93.9%

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

      7. *-commutative 93.9%

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

      8. associate-/r* 97.2%

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

      9. associate-*r/ 97.8%

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

   5. Applied egg-rr 97.8%

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

   6. Taylor expanded in z around inf 95.9%

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

   if -1 < z < 0.75

   1. Initial program 81.3%

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

      1. frac-times 81.8%

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

      2. associate-*l/ 81.3%

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

      3. times-frac 97.6%

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

   3. Applied egg-rr 97.6%

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

   4. Taylor expanded in z around 0 97.1%

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

      1. +-commutative 97.1%

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

      2. mul-1-neg 97.1%

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

      3. unsub-neg 97.1%

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

   6. Simplified 97.1%

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

4. Final simplification 96.5%

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

Alternative 4: 95.4% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -1.0)
		tmp = Float64(Float64(x / z) / Float64(z * Float64(z / y)));
	elseif (z <= 0.76)
		tmp = Float64(Float64(x / z) * Float64(Float64(y / z) - y));
	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) / (z * (z / y));
	elseif (z <= 0.76)
		tmp = (x / z) * ((y / z) - y);
	else
		tmp = ((x / z) * (y / z)) / z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -1.0], N[(N[(x / z), $MachinePrecision] / N[(z * N[(z / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 0.76], N[(N[(x / z), $MachinePrecision] * N[(N[(y / z), $MachinePrecision] - y), $MachinePrecision]), $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 88.5%

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

      1. sqr-neg 88.5%

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

      2. times-frac 95.9%

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

      3. sqr-neg 95.9%

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

   3. Simplified 95.9%

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

   4. Taylor expanded in z around inf 95.4%

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

      1. associate-/r* 97.0%

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

      2. clear-num 97.0%

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

      3. frac-times 97.2%

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

      4. *-commutative 97.2%

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

      5. *-un-lft-identity 97.2%

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

   6. Applied egg-rr 97.2%

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

   if -1 < z < 0.76000000000000001

   1. Initial program 81.3%

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

      1. frac-times 81.8%

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

      2. associate-*l/ 81.3%

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

      3. times-frac 97.6%

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

   3. Applied egg-rr 97.6%

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

   4. Taylor expanded in z around 0 97.1%

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

      1. +-commutative 97.1%

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

      2. mul-1-neg 97.1%

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

      3. unsub-neg 97.1%

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

   6. Simplified 97.1%

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

   if 0.76000000000000001 < z

   1. Initial program 90.5%

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

      1. associate-*r/ 91.0%

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

      2. sqr-neg 91.0%

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

      3. *-commutative 91.0%

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

      4. distribute-rgt1-in 91.0%

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

      5. sqr-neg 91.0%

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

      6. fma-def 91.0%

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

      7. sqr-neg 91.0%

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

      8. cube-unmult 91.0%

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

   3. Simplified 91.0%

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

      1. associate-*r/ 90.5%

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

      2. fma-udef 90.5%

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

      3. cube-mult 90.5%

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

      4. distribute-rgt1-in 90.5%

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

      5. *-commutative 90.5%

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

      6. frac-times 92.2%

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

      7. *-commutative 92.2%

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

      8. associate-/r* 97.0%

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

      9. associate-*r/ 98.7%

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

   5. Applied egg-rr 98.7%

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

   6. Taylor expanded in z around inf 96.0%

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

4. Final simplification 96.8%

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

Alternative 5: 95.6% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -1.0)
		tmp = Float64(Float64(Float64(x * Float64(y / z)) / z) / z);
	elseif (z <= 0.76)
		tmp = Float64(Float64(x / z) * Float64(Float64(y / z) - y));
	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 * (y / z)) / z) / z;
	elseif (z <= 0.76)
		tmp = (x / z) * ((y / z) - y);
	else
		tmp = ((x / z) * (y / z)) / z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -1.0], N[(N[(N[(x * N[(y / z), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision] / z), $MachinePrecision], If[LessEqual[z, 0.76], N[(N[(x / z), $MachinePrecision] * N[(N[(y / z), $MachinePrecision] - y), $MachinePrecision]), $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 88.5%

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

      1. associate-*r/ 93.5%

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

      2. sqr-neg 93.5%

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

      3. *-commutative 93.5%

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

      4. distribute-rgt1-in 50.0%

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

      5. sqr-neg 50.0%

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

      6. fma-def 93.4%

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

      7. sqr-neg 93.4%

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

      8. cube-unmult 93.4%

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

   3. Simplified 93.4%

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

      1. associate-*r/ 88.5%

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

      2. fma-udef 50.2%

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

      3. cube-mult 50.1%

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

      4. distribute-rgt1-in 88.5%

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

      5. *-commutative 88.5%

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

      6. frac-times 95.9%

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

      7. associate-*l/ 95.1%

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

      8. associate-/r* 96.7%

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

   5. Applied egg-rr 96.7%

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

   6. Taylor expanded in z around inf 95.7%

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

   if -1 < z < 0.76000000000000001

   1. Initial program 81.3%

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

      1. frac-times 81.8%

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

      2. associate-*l/ 81.3%

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

      3. times-frac 97.6%

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

   3. Applied egg-rr 97.6%

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

   4. Taylor expanded in z around 0 97.1%

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

      1. +-commutative 97.1%

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

      2. mul-1-neg 97.1%

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

      3. unsub-neg 97.1%

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

   6. Simplified 97.1%

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

   if 0.76000000000000001 < z

   1. Initial program 90.5%

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

      1. associate-*r/ 91.0%

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

      2. sqr-neg 91.0%

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

      3. *-commutative 91.0%

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

      4. distribute-rgt1-in 91.0%

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

      5. sqr-neg 91.0%

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

      6. fma-def 91.0%

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

      7. sqr-neg 91.0%

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

      8. cube-unmult 91.0%

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

   3. Simplified 91.0%

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

      1. associate-*r/ 90.5%

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

      2. fma-udef 90.5%

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

      3. cube-mult 90.5%

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

      4. distribute-rgt1-in 90.5%

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

      5. *-commutative 90.5%

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

      6. frac-times 92.2%

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

      7. *-commutative 92.2%

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

      8. associate-/r* 97.0%

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

      9. associate-*r/ 98.7%

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

   5. Applied egg-rr 98.7%

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

   6. Taylor expanded in z around inf 96.0%

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

4. Final simplification 96.5%

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

Alternative 6: 95.5% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -1.0)
		tmp = Float64(Float64(Float64(x / Float64(z / y)) / z) / z);
	elseif (z <= 0.76)
		tmp = Float64(Float64(x / z) * Float64(Float64(y / z) - y));
	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 <= 0.76)
		tmp = (x / z) * ((y / z) - y);
	else
		tmp = ((x / z) * (y / z)) / z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -1.0], N[(N[(N[(x / N[(z / y), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision] / z), $MachinePrecision], If[LessEqual[z, 0.76], N[(N[(x / z), $MachinePrecision] * N[(N[(y / z), $MachinePrecision] - y), $MachinePrecision]), $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 88.5%

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

      1. associate-*r/ 93.5%

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

      2. sqr-neg 93.5%

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

      3. *-commutative 93.5%

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

      4. distribute-rgt1-in 50.0%

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

      5. sqr-neg 50.0%

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

      6. fma-def 93.4%

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

      7. sqr-neg 93.4%

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

      8. cube-unmult 93.4%

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

   3. Simplified 93.4%

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

      1. associate-*r/ 88.5%

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

      2. fma-udef 50.2%

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

      3. cube-mult 50.1%

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

      4. distribute-rgt1-in 88.5%

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

      5. *-commutative 88.5%

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

      6. frac-times 95.9%

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

      7. associate-*l/ 95.1%

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

      8. associate-/r* 96.7%

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

   5. Applied egg-rr 96.7%

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

   6. Taylor expanded in z around inf 95.7%

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

      1. associate-*r/ 89.1%

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

      2. associate-/l* 95.7%

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

   8. Applied egg-rr 95.7%

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

   if -1 < z < 0.76000000000000001

   1. Initial program 81.3%

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

      1. frac-times 81.8%

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

      2. associate-*l/ 81.3%

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

      3. times-frac 97.6%

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

   3. Applied egg-rr 97.6%

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

   4. Taylor expanded in z around 0 97.1%

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

      1. +-commutative 97.1%

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

      2. mul-1-neg 97.1%

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

      3. unsub-neg 97.1%

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

   6. Simplified 97.1%

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

   if 0.76000000000000001 < z

   1. Initial program 90.5%

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

      1. associate-*r/ 91.0%

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

      2. sqr-neg 91.0%

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

      3. *-commutative 91.0%

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

      4. distribute-rgt1-in 91.0%

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

      5. sqr-neg 91.0%

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

      6. fma-def 91.0%

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

      7. sqr-neg 91.0%

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

      8. cube-unmult 91.0%

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

   3. Simplified 91.0%

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

      1. associate-*r/ 90.5%

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

      2. fma-udef 90.5%

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

      3. cube-mult 90.5%

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

      4. distribute-rgt1-in 90.5%

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

      5. *-commutative 90.5%

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

      6. frac-times 92.2%

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

      7. *-commutative 92.2%

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

      8. associate-/r* 97.0%

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

      9. associate-*r/ 98.7%

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

   5. Applied egg-rr 98.7%

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

   6. Taylor expanded in z around inf 96.0%

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

4. Final simplification 96.5%

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

Alternative 7: 77.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.8 \cdot 10^{-222}:\\ \;\;\;\;\frac{x}{z \cdot \frac{z}{y}}\\ \mathbf{elif}\;y \leq 13000:\\ \;\;\;\;\frac{x}{z} \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{z \cdot z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 1.8e-222)
   (/ x (* z (/ z y)))
   (if (<= y 13000.0) (* (/ x z) (/ y z)) (* y (/ x (* z z))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 1.8e-222) {
		tmp = x / (z * (z / y));
	} else if (y <= 13000.0) {
		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 <= 1.8d-222) then
        tmp = x / (z * (z / y))
    else if (y <= 13000.0d0) 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 <= 1.8e-222) {
		tmp = x / (z * (z / y));
	} else if (y <= 13000.0) {
		tmp = (x / z) * (y / z);
	} else {
		tmp = y * (x / (z * z));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 1.8e-222:
		tmp = x / (z * (z / y))
	elif y <= 13000.0:
		tmp = (x / z) * (y / z)
	else:
		tmp = y * (x / (z * z))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 1.8e-222)
		tmp = Float64(x / Float64(z * Float64(z / y)));
	elseif (y <= 13000.0)
		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 <= 1.8e-222)
		tmp = x / (z * (z / y));
	elseif (y <= 13000.0)
		tmp = (x / z) * (y / z);
	else
		tmp = y * (x / (z * z));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 1.8e-222], N[(x / N[(z * N[(z / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 13000.0], 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 3 regimes

2. if y < 1.79999999999999987e-222

   1. Initial program 85.5%

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

      1. frac-times 87.0%

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

      2. associate-*l/ 86.8%

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

      3. times-frac 97.0%

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

   3. Applied egg-rr 97.0%

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

   4. Taylor expanded in z around 0 77.9%

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

      1. associate-/r/ 77.2%

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

      2. div-inv 77.2%

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

      3. clear-num 77.2%

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

   6. Applied egg-rr 77.2%

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

   if 1.79999999999999987e-222 < y < 13000

   1. Initial program 86.9%

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

      1. frac-times 87.7%

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

      2. associate-*l/ 86.9%

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

      3. times-frac 99.7%

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

   3. Applied egg-rr 99.7%

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

   4. Taylor expanded in z around 0 84.3%

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

   if 13000 < 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. sqr-neg 85.2%

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

      2. times-frac 91.3%

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

      3. sqr-neg 91.3%

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

   3. Simplified 91.3%

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

   4. Taylor expanded in z around 0 72.3%

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

4. Final simplification 77.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.8 \cdot 10^{-222}:\\ \;\;\;\;\frac{x}{z \cdot \frac{z}{y}}\\ \mathbf{elif}\;y \leq 13000:\\ \;\;\;\;\frac{x}{z} \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{z \cdot z}\\ \end{array} \]

Alternative 8: 96.2% 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 85.6%

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

   1. frac-times 88.1%

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

   2. associate-*l/ 87.3%

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

   3. times-frac 97.3%

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

3. Applied egg-rr 97.3%

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

4. Final simplification 97.3%

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

Alternative 9: 77.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 23000:\\ \;\;\;\;\frac{x}{z} \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{z \cdot z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 23000.0) (* (/ x z) (/ y z)) (* y (/ x (* z z)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 23000.0) {
		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 <= 23000.0d0) 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 <= 23000.0) {
		tmp = (x / z) * (y / z);
	} else {
		tmp = y * (x / (z * z));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 23000.0:
		tmp = (x / z) * (y / z)
	else:
		tmp = y * (x / (z * z))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 23000.0)
		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 <= 23000.0)
		tmp = (x / z) * (y / z);
	else
		tmp = y * (x / (z * z));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 23000.0], N[(N[(x / z), $MachinePrecision] * N[(y / z), $MachinePrecision]), $MachinePrecision], N[(y * N[(x / N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 23000

   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. frac-times 87.1%

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

      2. associate-*l/ 86.8%

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

      3. times-frac 97.5%

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

   3. Applied egg-rr 97.5%

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

   4. Taylor expanded in z around 0 79.1%

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

   if 23000 < 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. sqr-neg 85.2%

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

      2. times-frac 91.3%

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

      3. sqr-neg 91.3%

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

   3. Simplified 91.3%

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

   4. Taylor expanded in z around 0 72.3%

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

4. Final simplification 77.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 23000:\\ \;\;\;\;\frac{x}{z} \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{z \cdot z}\\ \end{array} \]

Alternative 10: 74.7% 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 85.6%

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

   1. frac-times 88.1%

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

   2. associate-*l/ 87.3%

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

   3. times-frac 97.3%

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

3. Applied egg-rr 97.3%

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

4. Taylor expanded in z around 0 74.6%

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

5. Final simplification 74.6%

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

Developer target: 95.2% 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 2023320 
(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))))