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

Percentage Accurate: 83.5% → 96.9%

Time: 9.9s

Alternatives: 14

Speedup: 1.0×

• 25.8% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 83.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 96.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 82.8%

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

   1. associate-*l* 82.8%

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

   2. times-frac 96.7%

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

   3. associate-/r* 97.4%

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

   4. associate-*r/ 98.7%

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

3. Simplified 98.7%

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

4. Final simplification 98.7%

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

Alternative 2: 92.8% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Python

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

Julia

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

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -1.0) || ~((z <= 0.75)))
		tmp = (y / z) * (x / (z * z));
	else
		tmp = (y / z) * ((x / z) - x);
	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[(y / z), $MachinePrecision] * N[(x / N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y / z), $MachinePrecision] * N[(N[(x / z), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1 or 0.75 < z

   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. associate-*l* 85.5%

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

      2. times-frac 96.0%

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

      3. associate-/r* 97.4%

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

      4. associate-*r/ 99.8%

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

   3. Simplified 99.8%

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

      1. frac-times 91.0%

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

      2. associate-/r* 85.5%

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

      3. *-commutative 85.5%

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

      4. frac-times 95.9%

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

      5. associate-*l/ 97.0%

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

      6. times-frac 97.5%

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

   5. Applied egg-rr 97.5%

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

   6. Taylor expanded in z around inf 95.0%

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

      1. unpow2 95.0%

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

   8. Simplified 95.0%

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

   if -1 < z < 0.75

   1. Initial program 79.9%

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

      1. associate-*l* 79.9%

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

      2. times-frac 97.4%

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

      3. associate-/r* 97.4%

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

      4. associate-*r/ 97.4%

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

   3. Simplified 97.4%

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

      1. frac-times 79.9%

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

      2. associate-/r* 79.9%

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

      3. *-commutative 79.9%

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

      4. frac-times 82.9%

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

      5. associate-*l/ 79.9%

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

      6. times-frac 97.4%

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

   5. Applied egg-rr 97.4%

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

   6. Taylor expanded in z around 0 96.9%

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

      1. neg-mul-1 96.9%

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

      2. +-commutative 96.9%

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

      3. unsub-neg 96.9%

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

   8. Simplified 96.9%

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

4. Final simplification 95.9%

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

Alternative 3: 95.1% accurate, 0.8× speedup

Math

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

FPCore

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

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 = (y / z) * ((x / z) - x);
	}
	return tmp;
}

Fortran

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

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -1.0) || !(z <= 0.76))
		tmp = Float64(Float64(y / z) * Float64(Float64(x / z) / z));
	else
		tmp = Float64(Float64(y / z) * Float64(Float64(x / z) - x));
	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 = (y / z) * ((x / z) - x);
	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[(N[(x / z), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], N[(N[(y / z), $MachinePrecision] * N[(N[(x / z), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1 or 0.76000000000000001 < z

   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. associate-*l* 85.5%

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

      2. times-frac 96.0%

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

      3. associate-/r* 97.4%

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

      4. associate-*r/ 99.8%

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

   3. Simplified 99.8%

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

      1. frac-times 91.0%

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

      2. associate-/r* 85.5%

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

      3. *-commutative 85.5%

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

      4. frac-times 95.9%

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

      5. associate-*l/ 97.0%

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

      6. times-frac 97.5%

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

   5. Applied egg-rr 97.5%

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

   6. Taylor expanded in z around inf 96.4%

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

   if -1 < z < 0.76000000000000001

   1. Initial program 79.9%

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

      1. associate-*l* 79.9%

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

      2. times-frac 97.4%

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

      3. associate-/r* 97.4%

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

      4. associate-*r/ 97.4%

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

   3. Simplified 97.4%

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

      1. frac-times 79.9%

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

      2. associate-/r* 79.9%

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

      3. *-commutative 79.9%

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

      4. frac-times 82.9%

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

      5. associate-*l/ 79.9%

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

      6. times-frac 97.4%

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

   5. Applied egg-rr 97.4%

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

   6. Taylor expanded in z around 0 96.9%

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

      1. neg-mul-1 96.9%

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

      2. +-commutative 96.9%

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

      3. unsub-neg 96.9%

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

   8. Simplified 96.9%

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

4. Final simplification 96.6%

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

Alternative 4: 80.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1:\\ \;\;\;\;y \cdot \frac{x}{z \cdot \left(-z\right)}\\ \mathbf{elif}\;z \leq 0.62:\\ \;\;\;\;\frac{y}{z} \cdot \left(\frac{x}{z} - x\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{z \cdot z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -1.0)
   (* y (/ x (* z (- z))))
   (if (<= z 0.62) (* (/ y z) (- (/ x z) x)) (* x (/ y (* z z))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -1.0) {
		tmp = y * (x / (z * -z));
	} else if (z <= 0.62) {
		tmp = (y / z) * ((x / z) - x);
	} else {
		tmp = x * (y / (z * z));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (z <= (-1.0d0)) then
        tmp = y * (x / (z * -z))
    else if (z <= 0.62d0) then
        tmp = (y / z) * ((x / z) - x)
    else
        tmp = x * (y / (z * z))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -1.0) {
		tmp = y * (x / (z * -z));
	} else if (z <= 0.62) {
		tmp = (y / z) * ((x / z) - x);
	} else {
		tmp = x * (y / (z * z));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -1.0:
		tmp = y * (x / (z * -z))
	elif z <= 0.62:
		tmp = (y / z) * ((x / z) - x)
	else:
		tmp = x * (y / (z * z))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -1.0)
		tmp = Float64(y * Float64(x / Float64(z * Float64(-z))));
	elseif (z <= 0.62)
		tmp = Float64(Float64(y / z) * Float64(Float64(x / z) - x));
	else
		tmp = Float64(x * Float64(y / Float64(z * z)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -1.0)
		tmp = y * (x / (z * -z));
	elseif (z <= 0.62)
		tmp = (y / z) * ((x / z) - x);
	else
		tmp = x * (y / (z * z));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -1.0], N[(y * N[(x / N[(z * (-z)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 0.62], N[(N[(y / z), $MachinePrecision] * N[(N[(x / z), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision], N[(x * N[(y / N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -1

   1. Initial program 86.6%

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

      1. *-commutative 86.6%

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

      2. associate-*r/ 92.7%

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

      3. sqr-neg 92.7%

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

      4. associate-*l* 92.7%

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

      5. associate-*l* 92.7%

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

      6. sqr-neg 92.7%

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

      7. associate-*l* 92.7%

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

      8. distribute-lft-in 92.7%

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

      9. fma-def 92.7%

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

      10. *-rgt-identity 92.7%

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

   3. Simplified 92.7%

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

   4. Taylor expanded in z around 0 62.1%

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

      1. unpow2 62.1%

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

   6. Simplified 62.1%

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

      1. associate-*r/ 59.0%

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

      2. *-commutative 59.0%

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

      3. times-frac 53.0%

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

      4. clear-num 53.0%

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

      5. frac-2neg 53.0%

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

      6. frac-times 57.9%

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

      7. *-un-lft-identity 57.9%

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

      8. add-sqr-sqrt 23.4%

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

      9. sqrt-unprod 56.8%

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

      10. sqr-neg 56.8%

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

      11. sqrt-unprod 36.4%

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

      12. add-sqr-sqrt 63.7%

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

   8. Applied egg-rr 63.7%

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

      1. associate-*l/ 68.9%

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

      2. associate-/l* 64.8%

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

      3. *-rgt-identity 64.8%

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

      4. associate-*r/ 64.8%

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

      5. associate-*l* 68.0%

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

      6. associate-*r/ 68.0%

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

      7. *-rgt-identity 68.0%

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

   10. Simplified 68.0%

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

   if -1 < z < 0.619999999999999996

   1. Initial program 79.9%

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

      1. associate-*l* 79.9%

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

      2. times-frac 97.4%

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

      3. associate-/r* 97.4%

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

      4. associate-*r/ 97.4%

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

   3. Simplified 97.4%

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

      1. frac-times 79.9%

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

      2. associate-/r* 79.9%

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

      3. *-commutative 79.9%

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

      4. frac-times 82.9%

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

      5. associate-*l/ 79.9%

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

      6. times-frac 97.4%

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

   5. Applied egg-rr 97.4%

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

   6. Taylor expanded in z around 0 96.9%

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

      1. neg-mul-1 96.9%

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

      2. +-commutative 96.9%

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

      3. unsub-neg 96.9%

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

   8. Simplified 96.9%

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

   if 0.619999999999999996 < z

   1. Initial program 84.3%

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

      1. *-commutative 84.3%

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

      2. sqr-neg 84.3%

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

      3. times-frac 94.8%

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

      4. sqr-neg 94.8%

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

   3. Simplified 94.8%

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

   4. Taylor expanded in z around 0 66.9%

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

4. Final simplification 81.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1:\\ \;\;\;\;y \cdot \frac{x}{z \cdot \left(-z\right)}\\ \mathbf{elif}\;z \leq 0.62:\\ \;\;\;\;\frac{y}{z} \cdot \left(\frac{x}{z} - x\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{z \cdot z}\\ \end{array} \]

Alternative 5: 94.0% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 3 regimes

2. if z < -1

   1. Initial program 86.6%

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

      1. associate-*l* 86.7%

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

      2. times-frac 97.0%

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

      3. associate-/r* 98.5%

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

      4. associate-*r/ 99.9%

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

   3. Simplified 99.9%

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

      1. frac-times 89.5%

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

      2. associate-/r* 86.6%

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

      3. *-commutative 86.6%

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

      4. frac-times 97.0%

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

      5. associate-*l/ 95.6%

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

      6. times-frac 98.3%

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

   5. Applied egg-rr 98.3%

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

   6. Taylor expanded in z around inf 97.9%

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

   if -1 < z < 0.75

   1. Initial program 79.9%

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

      1. associate-*l* 79.9%

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

      2. times-frac 97.4%

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

      3. associate-/r* 97.4%

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

      4. associate-*r/ 97.4%

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

   3. Simplified 97.4%

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

      1. frac-times 79.9%

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

      2. associate-/r* 79.9%

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

      3. *-commutative 79.9%

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

      4. frac-times 82.9%

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

      5. associate-*l/ 79.9%

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

      6. times-frac 97.4%

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

   5. Applied egg-rr 97.4%

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

   6. Taylor expanded in z around 0 96.9%

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

      1. neg-mul-1 96.9%

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

      2. +-commutative 96.9%

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

      3. unsub-neg 96.9%

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

   8. Simplified 96.9%

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

   if 0.75 < z

   1. Initial program 84.3%

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

      1. associate-*l* 84.3%

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

      2. times-frac 95.0%

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

      3. associate-/r* 96.4%

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

      4. associate-*r/ 99.8%

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

   3. Simplified 99.8%

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

      1. frac-times 92.5%

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

      2. associate-/r* 84.3%

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

      3. *-commutative 84.3%

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

      4. frac-times 94.8%

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

      5. associate-*l/ 98.3%

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

      6. times-frac 96.6%

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

   5. Applied egg-rr 96.6%

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

   6. Taylor expanded in z around inf 93.5%

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

      1. unpow2 93.5%

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

   8. Simplified 93.5%

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

      1. associate-*r/ 96.6%

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

      2. frac-2neg 96.6%

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

      3. distribute-lft-neg-out 96.6%

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

      4. distribute-frac-neg 96.6%

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

      5. *-commutative 96.6%

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

      6. add-sqr-sqrt 57.7%

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

      7. sqrt-unprod 69.7%

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

      8. sqr-neg 69.7%

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

      9. sqrt-unprod 24.7%

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

      10. add-sqr-sqrt 65.7%

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

      11. distribute-rgt-neg-out 65.7%

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

      12. add-sqr-sqrt 0.0%

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

      13. sqrt-unprod 81.4%

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

      14. swap-sqr 81.4%

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

      15. sqr-neg 81.4%

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

      16. sqrt-unprod 96.6%

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

      17. add-sqr-sqrt 96.6%

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

   10. Applied egg-rr 96.6%

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

4. Final simplification 97.1%

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

Alternative 6: 76.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.15 \cdot 10^{+19}:\\ \;\;\;\;x \cdot \frac{y}{z \cdot z}\\ \mathbf{elif}\;x \leq -5 \cdot 10^{-301}:\\ \;\;\;\;\frac{x}{z} \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z \cdot \frac{z}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -3.15e+19)
   (* x (/ y (* z z)))
   (if (<= x -5e-301) (* (/ x z) (/ y z)) (/ y (* z (/ z x))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -3.15e+19) {
		tmp = x * (y / (z * z));
	} else if (x <= -5e-301) {
		tmp = (x / z) * (y / z);
	} else {
		tmp = y / (z * (z / x));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x <= (-3.15d+19)) then
        tmp = x * (y / (z * z))
    else if (x <= (-5d-301)) then
        tmp = (x / z) * (y / z)
    else
        tmp = y / (z * (z / x))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -3.15e+19) {
		tmp = x * (y / (z * z));
	} else if (x <= -5e-301) {
		tmp = (x / z) * (y / z);
	} else {
		tmp = y / (z * (z / x));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -3.15e+19:
		tmp = x * (y / (z * z))
	elif x <= -5e-301:
		tmp = (x / z) * (y / z)
	else:
		tmp = y / (z * (z / x))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -3.15e+19)
		tmp = Float64(x * Float64(y / Float64(z * z)));
	elseif (x <= -5e-301)
		tmp = Float64(Float64(x / z) * Float64(y / z));
	else
		tmp = Float64(y / Float64(z * Float64(z / x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -3.15e+19)
		tmp = x * (y / (z * z));
	elseif (x <= -5e-301)
		tmp = (x / z) * (y / z);
	else
		tmp = y / (z * (z / x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -3.15e+19], N[(x * N[(y / N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -5e-301], N[(N[(x / z), $MachinePrecision] * N[(y / z), $MachinePrecision]), $MachinePrecision], N[(y / N[(z * N[(z / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -3.15e19

   1. Initial program 87.7%

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

      1. *-commutative 87.7%

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

      2. sqr-neg 87.7%

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

      3. times-frac 96.6%

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

      4. sqr-neg 96.6%

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

   3. Simplified 96.6%

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

   4. Taylor expanded in z around 0 78.5%

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

   if -3.15e19 < x < -5.00000000000000013e-301

   1. Initial program 81.5%

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

      1. associate-*l* 81.4%

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

      2. times-frac 99.7%

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

      3. associate-/r* 99.8%

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

      4. associate-*r/ 99.8%

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

   3. Simplified 99.8%

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

      1. frac-times 82.7%

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

      2. associate-/r* 81.5%

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

      3. *-commutative 81.5%

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

      4. frac-times 87.5%

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

      5. associate-*l/ 82.8%

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

      6. times-frac 96.7%

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

   5. Applied egg-rr 96.7%

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

   6. Taylor expanded in z around 0 87.5%

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

   if -5.00000000000000013e-301 < x

   1. Initial program 81.1%

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

      1. *-commutative 81.1%

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

      2. associate-*r/ 82.4%

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

      3. sqr-neg 82.4%

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

      4. associate-*l* 82.4%

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

      5. associate-*l* 82.4%

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

      6. sqr-neg 82.4%

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

      7. associate-*l* 82.4%

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

      8. distribute-lft-in 82.4%

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

      9. fma-def 82.4%

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

      10. *-rgt-identity 82.4%

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

   3. Simplified 82.4%

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

   4. Taylor expanded in z around 0 66.6%

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

      1. unpow2 66.6%

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

   6. Simplified 66.6%

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

      1. associate-*r/ 66.1%

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

      2. *-commutative 66.1%

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

      3. times-frac 70.6%

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

      4. clear-num 70.7%

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

      5. frac-times 73.6%

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

      6. *-un-lft-identity 73.6%

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

   8. Applied egg-rr 73.6%

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

4. Final simplification 78.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.15 \cdot 10^{+19}:\\ \;\;\;\;x \cdot \frac{y}{z \cdot z}\\ \mathbf{elif}\;x \leq -5 \cdot 10^{-301}:\\ \;\;\;\;\frac{x}{z} \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z \cdot \frac{z}{x}}\\ \end{array} \]

Alternative 7: 76.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2200000000:\\ \;\;\;\;x \cdot \frac{y}{z \cdot z}\\ \mathbf{elif}\;x \leq -7 \cdot 10^{-301}:\\ \;\;\;\;\frac{\frac{y}{z}}{\frac{z}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z \cdot \frac{z}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -2200000000.0)
   (* x (/ y (* z z)))
   (if (<= x -7e-301) (/ (/ y z) (/ z x)) (/ y (* z (/ z x))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -2200000000.0) {
		tmp = x * (y / (z * z));
	} else if (x <= -7e-301) {
		tmp = (y / z) / (z / x);
	} else {
		tmp = y / (z * (z / x));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x <= (-2200000000.0d0)) then
        tmp = x * (y / (z * z))
    else if (x <= (-7d-301)) then
        tmp = (y / z) / (z / x)
    else
        tmp = y / (z * (z / x))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -2200000000.0) {
		tmp = x * (y / (z * z));
	} else if (x <= -7e-301) {
		tmp = (y / z) / (z / x);
	} else {
		tmp = y / (z * (z / x));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -2200000000.0:
		tmp = x * (y / (z * z))
	elif x <= -7e-301:
		tmp = (y / z) / (z / x)
	else:
		tmp = y / (z * (z / x))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -2200000000.0)
		tmp = Float64(x * Float64(y / Float64(z * z)));
	elseif (x <= -7e-301)
		tmp = Float64(Float64(y / z) / Float64(z / x));
	else
		tmp = Float64(y / Float64(z * Float64(z / x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -2200000000.0)
		tmp = x * (y / (z * z));
	elseif (x <= -7e-301)
		tmp = (y / z) / (z / x);
	else
		tmp = y / (z * (z / x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -2200000000.0], N[(x * N[(y / N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -7e-301], N[(N[(y / z), $MachinePrecision] / N[(z / x), $MachinePrecision]), $MachinePrecision], N[(y / N[(z * N[(z / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.2e9

   1. Initial program 87.9%

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

      1. *-commutative 87.9%

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

      2. sqr-neg 87.9%

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

      3. times-frac 96.7%

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

      4. sqr-neg 96.7%

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

   3. Simplified 96.7%

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

   4. Taylor expanded in z around 0 77.4%

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

   if -2.2e9 < x < -6.99999999999999984e-301

   1. Initial program 81.2%

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

      1. *-commutative 81.2%

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

      2. associate-*r/ 82.8%

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

      3. sqr-neg 82.8%

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

      4. associate-*l* 82.8%

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

      5. associate-*l* 82.8%

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

      6. sqr-neg 82.8%

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

      7. associate-*l* 82.8%

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

      8. distribute-lft-in 82.8%

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

      9. fma-def 82.8%

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

      10. *-rgt-identity 82.8%

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

   3. Simplified 82.8%

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

   4. Taylor expanded in z around 0 78.8%

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

      1. unpow2 78.8%

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

   6. Simplified 78.8%

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

      1. associate-*r/ 73.0%

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

      2. frac-times 88.7%

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

      3. clear-num 88.6%

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

      4. un-div-inv 88.7%

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

   8. Applied egg-rr 88.7%

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

   if -6.99999999999999984e-301 < x

   1. Initial program 81.1%

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

      1. *-commutative 81.1%

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

      2. associate-*r/ 82.4%

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

      3. sqr-neg 82.4%

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

      4. associate-*l* 82.4%

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

      5. associate-*l* 82.4%

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

      6. sqr-neg 82.4%

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

      7. associate-*l* 82.4%

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

      8. distribute-lft-in 82.4%

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

      9. fma-def 82.4%

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

      10. *-rgt-identity 82.4%

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

   3. Simplified 82.4%

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

   4. Taylor expanded in z around 0 66.6%

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

      1. unpow2 66.6%

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

   6. Simplified 66.6%

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

      1. associate-*r/ 66.1%

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

      2. *-commutative 66.1%

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

      3. times-frac 70.6%

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

      4. clear-num 70.7%

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

      5. frac-times 73.6%

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

      6. *-un-lft-identity 73.6%

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

   8. Applied egg-rr 73.6%

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

4. Final simplification 78.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2200000000:\\ \;\;\;\;x \cdot \frac{y}{z \cdot z}\\ \mathbf{elif}\;x \leq -7 \cdot 10^{-301}:\\ \;\;\;\;\frac{\frac{y}{z}}{\frac{z}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z \cdot \frac{z}{x}}\\ \end{array} \]

Alternative 8: 78.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1:\\ \;\;\;\;y \cdot \frac{x}{z \cdot \left(-z\right)}\\ \mathbf{elif}\;z \leq 2.55 \cdot 10^{-144}:\\ \;\;\;\;\frac{\frac{y}{z}}{\frac{z}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z \cdot \frac{z}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -1.0)
   (* y (/ x (* z (- z))))
   (if (<= z 2.55e-144) (/ (/ y z) (/ z x)) (/ y (* z (/ z x))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -1.0) {
		tmp = y * (x / (z * -z));
	} else if (z <= 2.55e-144) {
		tmp = (y / z) / (z / x);
	} else {
		tmp = y / (z * (z / x));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (z <= (-1.0d0)) then
        tmp = y * (x / (z * -z))
    else if (z <= 2.55d-144) then
        tmp = (y / z) / (z / x)
    else
        tmp = y / (z * (z / x))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -1.0) {
		tmp = y * (x / (z * -z));
	} else if (z <= 2.55e-144) {
		tmp = (y / z) / (z / x);
	} else {
		tmp = y / (z * (z / x));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -1.0:
		tmp = y * (x / (z * -z))
	elif z <= 2.55e-144:
		tmp = (y / z) / (z / x)
	else:
		tmp = y / (z * (z / x))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -1.0)
		tmp = Float64(y * Float64(x / Float64(z * Float64(-z))));
	elseif (z <= 2.55e-144)
		tmp = Float64(Float64(y / z) / Float64(z / x));
	else
		tmp = Float64(y / Float64(z * Float64(z / x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -1.0)
		tmp = y * (x / (z * -z));
	elseif (z <= 2.55e-144)
		tmp = (y / z) / (z / x);
	else
		tmp = y / (z * (z / x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -1.0], N[(y * N[(x / N[(z * (-z)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 2.55e-144], N[(N[(y / z), $MachinePrecision] / N[(z / x), $MachinePrecision]), $MachinePrecision], N[(y / N[(z * N[(z / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -1

   1. Initial program 86.6%

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

      1. *-commutative 86.6%

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

      2. associate-*r/ 92.7%

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

      3. sqr-neg 92.7%

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

      4. associate-*l* 92.7%

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

      5. associate-*l* 92.7%

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

      6. sqr-neg 92.7%

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

      7. associate-*l* 92.7%

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

      8. distribute-lft-in 92.7%

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

      9. fma-def 92.7%

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

      10. *-rgt-identity 92.7%

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

   3. Simplified 92.7%

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

   4. Taylor expanded in z around 0 62.1%

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

      1. unpow2 62.1%

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

   6. Simplified 62.1%

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

      1. associate-*r/ 59.0%

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

      2. *-commutative 59.0%

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

      3. times-frac 53.0%

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

      4. clear-num 53.0%

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

      5. frac-2neg 53.0%

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

      6. frac-times 57.9%

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

      7. *-un-lft-identity 57.9%

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

      8. add-sqr-sqrt 23.4%

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

      9. sqrt-unprod 56.8%

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

      10. sqr-neg 56.8%

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

      11. sqrt-unprod 36.4%

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

      12. add-sqr-sqrt 63.7%

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

   8. Applied egg-rr 63.7%

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

      1. associate-*l/ 68.9%

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

      2. associate-/l* 64.8%

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

      3. *-rgt-identity 64.8%

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

      4. associate-*r/ 64.8%

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

      5. associate-*l* 68.0%

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

      6. associate-*r/ 68.0%

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

      7. *-rgt-identity 68.0%

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

   10. Simplified 68.0%

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

   if -1 < z < 2.55e-144

   1. Initial program 78.2%

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

      1. *-commutative 78.2%

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

      2. associate-*r/ 77.9%

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

      3. sqr-neg 77.9%

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

      4. associate-*l* 77.9%

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

      5. associate-*l* 77.9%

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

      6. sqr-neg 77.9%

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

      7. associate-*l* 77.9%

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

      8. distribute-lft-in 77.9%

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

      9. fma-def 77.9%

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

      10. *-rgt-identity 77.9%

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

   3. Simplified 77.9%

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

   4. Taylor expanded in z around 0 76.6%

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

      1. unpow2 76.6%

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

   6. Simplified 76.6%

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

      1. associate-*r/ 76.9%

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

      2. frac-times 96.4%

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

      3. clear-num 96.4%

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

      4. un-div-inv 96.4%

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

   8. Applied egg-rr 96.4%

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

   if 2.55e-144 < z

   1. Initial program 84.8%

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

      1. *-commutative 84.8%

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

      2. associate-*r/ 86.8%

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

      3. sqr-neg 86.8%

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

      4. associate-*l* 86.8%

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

      5. associate-*l* 86.8%

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

      6. sqr-neg 86.8%

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

      7. associate-*l* 86.8%

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

      8. distribute-lft-in 86.8%

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

      9. fma-def 86.8%

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

      10. *-rgt-identity 86.8%

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

   3. Simplified 86.8%

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

   4. Taylor expanded in z around 0 72.6%

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

      1. unpow2 72.6%

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

   6. Simplified 72.6%

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

      1. associate-*r/ 69.6%

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

      2. *-commutative 69.6%

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

      3. times-frac 68.7%

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

      4. clear-num 68.7%

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

      5. frac-times 72.6%

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

      6. *-un-lft-identity 72.6%

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

   8. Applied egg-rr 72.6%

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

4. Final simplification 80.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1:\\ \;\;\;\;y \cdot \frac{x}{z \cdot \left(-z\right)}\\ \mathbf{elif}\;z \leq 2.55 \cdot 10^{-144}:\\ \;\;\;\;\frac{\frac{y}{z}}{\frac{z}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z \cdot \frac{z}{x}}\\ \end{array} \]

Alternative 9: 96.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 82.8%

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

   1. associate-*l* 82.8%

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

   2. times-frac 96.7%

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

   3. associate-/r* 97.4%

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

   4. associate-*r/ 98.7%

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

3. Simplified 98.7%

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

   1. frac-times 85.7%

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

   2. associate-/r* 82.8%

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

   3. *-commutative 82.8%

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

   4. frac-times 89.6%

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

   5. associate-*l/ 88.8%

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

   6. times-frac 97.4%

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

5. Applied egg-rr 97.4%

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

6. Final simplification 97.4%

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

Alternative 10: 41.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1 \lor \neg \left(z \leq -2 \cdot 10^{-310}\right):\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z} \cdot \left(-x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -1.0) (not (<= z -2e-310))) (/ x (/ z y)) (* (/ y z) (- x))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -1.0) || !(z <= -2e-310)) {
		tmp = x / (z / y);
	} else {
		tmp = (y / z) * -x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((z <= (-1.0d0)) .or. (.not. (z <= (-2d-310)))) then
        tmp = x / (z / y)
    else
        tmp = (y / z) * -x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((z <= -1.0) || !(z <= -2e-310)) {
		tmp = x / (z / y);
	} else {
		tmp = (y / z) * -x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -1.0) or not (z <= -2e-310):
		tmp = x / (z / y)
	else:
		tmp = (y / z) * -x
	return tmp

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -1.0], N[Not[LessEqual[z, -2e-310]], $MachinePrecision]], N[(x / N[(z / y), $MachinePrecision]), $MachinePrecision], N[(N[(y / z), $MachinePrecision] * (-x)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1 or -1.999999999999994e-310 < z

   1. Initial program 82.7%

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

      1. associate-*l* 82.7%

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

      2. times-frac 96.1%

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

      3. associate-/r* 97.2%

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

      4. associate-*r/ 98.8%

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

   3. Simplified 98.8%

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

   4. Taylor expanded in z around 0 37.5%

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

      1. *-commutative 37.5%

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

      2. unpow2 37.5%

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

      3. times-frac 43.6%

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

      4. associate-*r/ 43.6%

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

      5. *-commutative 43.6%

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

      6. neg-mul-1 43.6%

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

      7. distribute-rgt-neg-out 43.6%

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

      8. associate-*l/ 44.5%

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

      9. distribute-lft-out 54.5%

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

   6. Simplified 54.5%

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

   7. Taylor expanded in z around inf 23.7%

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

      1. mul-1-neg 23.7%

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

      2. associate-*l/ 27.9%

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

      3. *-commutative 27.9%

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

      4. distribute-rgt-neg-in 27.9%

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

      5. distribute-neg-frac 27.9%

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

   9. Simplified 27.9%

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

       1. add-sqr-sqrt 14.4%

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

       2. sqrt-unprod 35.7%

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

       3. sqr-neg 35.7%

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

       4. sqrt-unprod 21.0%

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

       5. add-sqr-sqrt 41.7%

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

       6. clear-num 42.7%

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

       7. div-inv 42.7%

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

   11. Applied egg-rr 42.7%

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

   if -1 < z < -1.999999999999994e-310

   1. Initial program 83.2%

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

      1. associate-*l* 83.2%

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

      2. times-frac 98.2%

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

      3. associate-/r* 98.2%

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

      4. associate-*r/ 98.2%

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

   3. Simplified 98.2%

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

   4. Taylor expanded in z around 0 82.7%

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

      1. *-commutative 82.7%

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

      2. unpow2 82.7%

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

      3. times-frac 97.7%

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

      4. associate-*r/ 97.7%

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

      5. *-commutative 97.7%

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

      6. neg-mul-1 97.7%

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

      7. distribute-rgt-neg-out 97.7%

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

      8. associate-*l/ 97.7%

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

      9. distribute-lft-out 97.7%

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

   6. Simplified 97.7%

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

   7. Taylor expanded in z around inf 30.6%

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

      1. mul-1-neg 30.6%

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

      2. associate-*l/ 36.3%

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

      3. *-commutative 36.3%

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

      4. distribute-rgt-neg-in 36.3%

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

      5. distribute-neg-frac 36.3%

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

   9. Simplified 36.3%

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

4. Final simplification 41.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1 \lor \neg \left(z \leq -2 \cdot 10^{-310}\right):\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z} \cdot \left(-x\right)\\ \end{array} \]

Alternative 11: 76.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 4.2 \cdot 10^{+87}:\\ \;\;\;\;\frac{x}{z} \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{z \cdot z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 4.2e+87) (* (/ x z) (/ y z)) (* y (/ x (* z z)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 4.2e+87) {
		tmp = (x / z) * (y / z);
	} else {
		tmp = y * (x / (z * z));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= 4.2d+87) then
        tmp = (x / z) * (y / z)
    else
        tmp = y * (x / (z * z))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 4.2e+87) {
		tmp = (x / z) * (y / z);
	} else {
		tmp = y * (x / (z * z));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 4.2e+87:
		tmp = (x / z) * (y / z)
	else:
		tmp = y * (x / (z * z))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 4.2e+87)
		tmp = Float64(Float64(x / z) * Float64(y / z));
	else
		tmp = Float64(y * Float64(x / Float64(z * z)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 4.2e+87)
		tmp = (x / z) * (y / z);
	else
		tmp = y * (x / (z * z));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 4.2e+87], N[(N[(x / z), $MachinePrecision] * N[(y / z), $MachinePrecision]), $MachinePrecision], N[(y * N[(x / N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 4.2e87

   1. Initial program 83.4%

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

      1. associate-*l* 83.4%

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

      2. times-frac 97.3%

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

      3. associate-/r* 97.8%

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

      4. associate-*r/ 99.3%

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

   3. Simplified 99.3%

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

      1. frac-times 85.9%

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

      2. associate-/r* 83.4%

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

      3. *-commutative 83.4%

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

      4. frac-times 88.8%

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

      5. associate-*l/ 87.8%

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

      6. times-frac 98.3%

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

   5. Applied egg-rr 98.3%

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

   6. Taylor expanded in z around 0 77.8%

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

   if 4.2e87 < y

   1. Initial program 80.0%

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

      1. *-commutative 80.0%

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

      2. associate-*r/ 84.8%

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

      3. sqr-neg 84.8%

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

      4. associate-*l* 84.9%

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

      5. associate-*l* 84.8%

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

      6. sqr-neg 84.8%

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

      7. associate-*l* 84.9%

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

      8. distribute-lft-in 84.9%

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

      9. fma-def 84.9%

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

      10. *-rgt-identity 84.9%

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

   3. Simplified 84.9%

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

   4. Taylor expanded in z around 0 76.0%

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

      1. unpow2 76.0%

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

   6. Simplified 76.0%

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

4. Final simplification 77.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 4.2 \cdot 10^{+87}:\\ \;\;\;\;\frac{x}{z} \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{z \cdot z}\\ \end{array} \]

Alternative 12: 72.6% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 82.8%

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

   1. *-commutative 82.8%

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

   2. associate-*r/ 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. associate-*l* 85.0%

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

   5. associate-*l* 85.0%

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

   6. sqr-neg 85.0%

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

   7. associate-*l* 85.0%

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

   8. distribute-lft-in 85.0%

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

   9. fma-def 85.0%

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

   10. *-rgt-identity 85.0%

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

3. Simplified 85.0%

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

4. Taylor expanded in z around 0 71.4%

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

   1. unpow2 71.4%

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

6. Simplified 71.4%

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

7. Final simplification 71.4%

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

Alternative 13: 31.8% accurate, 2.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 82.8%

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

   1. associate-*l* 82.8%

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

   2. times-frac 96.7%

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

   3. associate-/r* 97.4%

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

   4. associate-*r/ 98.7%

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

3. Simplified 98.7%

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

4. Taylor expanded in z around 0 49.2%

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

   1. *-commutative 49.2%

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

   2. unpow2 49.2%

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

   3. times-frac 57.6%

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

   4. associate-*r/ 57.6%

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

   5. *-commutative 57.6%

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

   6. neg-mul-1 57.6%

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

   7. distribute-rgt-neg-out 57.6%

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

   8. associate-*l/ 58.2%

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

   9. distribute-lft-out 65.6%

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

6. Simplified 65.6%

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

7. Taylor expanded in z around inf 25.4%

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

   1. mul-1-neg 25.4%

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

   2. associate-*l/ 30.1%

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

   3. *-commutative 30.1%

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

   4. distribute-rgt-neg-in 30.1%

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

   5. distribute-neg-frac 30.1%

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

9. Simplified 30.1%

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

    1. expm1-log1p-u 27.5%

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

    2. expm1-udef 39.7%

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

    3. add-sqr-sqrt 19.7%

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

    4. sqrt-unprod 38.4%

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

    5. sqr-neg 38.4%

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

    6. sqrt-unprod 20.7%

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

    7. add-sqr-sqrt 41.1%

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

11. Applied egg-rr 41.1%

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

    1. expm1-def 28.9%

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

    2. expm1-log1p 32.4%

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

    3. associate-*r/ 27.7%

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

    4. associate-*l/ 31.3%

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

    5. *-commutative 31.3%

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

13. Simplified 31.3%

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

14. Final simplification 31.3%

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

Alternative 14: 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 82.8%

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

   1. associate-*l* 82.8%

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

   2. times-frac 96.7%

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

   3. associate-/r* 97.4%

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

   4. associate-*r/ 98.7%

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

3. Simplified 98.7%

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

4. Taylor expanded in z around 0 49.2%

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

   1. *-commutative 49.2%

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

   2. unpow2 49.2%

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

   3. times-frac 57.6%

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

   4. associate-*r/ 57.6%

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

   5. *-commutative 57.6%

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

   6. neg-mul-1 57.6%

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

   7. distribute-rgt-neg-out 57.6%

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

   8. associate-*l/ 58.2%

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

   9. distribute-lft-out 65.6%

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

6. Simplified 65.6%

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

7. Taylor expanded in z around inf 25.4%

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

   1. mul-1-neg 25.4%

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

   2. associate-*l/ 30.1%

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

   3. *-commutative 30.1%

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

   4. distribute-rgt-neg-in 30.1%

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

   5. distribute-neg-frac 30.1%

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

9. Simplified 30.1%

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

    1. add-sqr-sqrt 14.8%

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

    2. sqrt-unprod 31.9%

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

    3. sqr-neg 31.9%

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

    4. sqrt-unprod 16.9%

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

    5. add-sqr-sqrt 32.4%

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

    6. clear-num 33.1%

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

    7. div-inv 33.1%

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

11. Applied egg-rr 33.1%

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

12. Final simplification 33.1%

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

Developer target: 95.1% 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 2023268 
(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))))