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

Percentage Accurate: 82.9% → 96.3%

Time: 13.0s

Alternatives: 14

Speedup: 1.0×

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 82.9% 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: 96.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \frac{\frac{y}{z}}{z \cdot \frac{z + 1}{x}} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z) :precision binary64 (/ (/ y z) (* z (/ (+ z 1.0) x))))

C

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

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
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) / (z * ((z + 1.0d0) / x))
end function

Java

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

Python

[x, y] = sort([x, y])
def code(x, y, z):
	return (y / z) / (z * ((z + 1.0) / x))

Julia

x, y = sort([x, y])
function code(x, y, z)
	return Float64(Float64(y / z) / Float64(z * Float64(Float64(z + 1.0) / x)))
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp = code(x, y, z)
	tmp = (y / z) / (z * ((z + 1.0) / x));
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_, z_] := N[(N[(y / z), $MachinePrecision] / N[(z * N[(N[(z + 1.0), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 84.2%

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

   1. *-commutative 84.2%

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

   2. sqr-neg 84.2%

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

   3. times-frac 87.7%

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

   4. sqr-neg 87.7%

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

3. Simplified 87.7%

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

   1. *-commutative 87.7%

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

   2. clear-num 87.7%

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

   3. associate-/r* 92.9%

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

   4. frac-times 97.3%

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

   5. *-un-lft-identity 97.3%

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

5. Applied egg-rr 97.3%

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

6. Final simplification 97.3%

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

Alternative 2: 94.8% accurate, 0.8× speedup

Math

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

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (or (<= z -1.0) (not (<= z 1.0)))
   (* (/ y z) (/ (/ x z) z))
   (/ (/ y z) (/ z x))))

C

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

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((z <= (-1.0d0)) .or. (.not. (z <= 1.0d0))) then
        tmp = (y / z) * ((x / z) / z)
    else
        tmp = (y / z) / (z / x)
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -1.0) || ~((z <= 1.0)))
		tmp = (y / z) * ((x / z) / z);
	else
		tmp = (y / z) / (z / x);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_, z_] := If[Or[LessEqual[z, -1.0], N[Not[LessEqual[z, 1.0]], $MachinePrecision]], N[(N[(y / z), $MachinePrecision] * N[(N[(x / z), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], N[(N[(y / z), $MachinePrecision] / N[(z / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1 or 1 < z

   1. Initial program 86.0%

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

      1. associate-*l* 86.0%

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

      2. times-frac 93.9%

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

      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 87.7%

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

      2. associate-/r* 86.0%

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

      3. *-commutative 86.0%

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

      4. frac-times 93.9%

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

      5. associate-*l/ 91.3%

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

      6. times-frac 98.2%

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

   5. Applied egg-rr 98.2%

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

   6. Taylor expanded in z around inf 97.7%

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

   if -1 < z < 1

   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. sqr-neg 82.8%

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

      3. times-frac 82.6%

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

      4. sqr-neg 82.6%

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

   3. Simplified 82.6%

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

      1. *-commutative 82.6%

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

      2. clear-num 82.6%

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

      3. associate-/r* 89.1%

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

      4. frac-times 96.7%

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

      5. *-un-lft-identity 96.7%

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

   5. Applied egg-rr 96.7%

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

   6. Taylor expanded in z around 0 95.4%

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

4. Final simplification 96.4%

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

Alternative 3: 94.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -1:\\ \;\;\;\;\frac{y}{z} \cdot \frac{\frac{x}{z}}{z}\\ \mathbf{elif}\;z \leq 1:\\ \;\;\;\;\frac{\frac{y}{z}}{\frac{z}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot \frac{\frac{y}{z}}{z}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= z -1.0)
   (* (/ y z) (/ (/ x z) z))
   (if (<= z 1.0) (/ (/ y z) (/ z x)) (* (/ x z) (/ (/ y z) z)))))

C

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

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
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 <= 1.0d0) then
        tmp = (y / z) / (z / x)
    else
        tmp = (x / z) * ((y / z) / z)
    end if
    code = tmp
end function

Java

assert x < y;
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 <= 1.0) {
		tmp = (y / z) / (z / x);
	} else {
		tmp = (x / z) * ((y / z) / z);
	}
	return tmp;
}

Python

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

Julia

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

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -1.0)
		tmp = (y / z) * ((x / z) / z);
	elseif (z <= 1.0)
		tmp = (y / z) / (z / x);
	else
		tmp = (x / z) * ((y / z) / z);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
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, 1.0], N[(N[(y / z), $MachinePrecision] / N[(z / x), $MachinePrecision]), $MachinePrecision], N[(N[(x / z), $MachinePrecision] * N[(N[(y / z), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -1

   1. Initial program 93.1%

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

      1. associate-*l* 93.1%

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

      2. times-frac 93.8%

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

      3. associate-/r* 96.9%

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

      4. associate-*r/ 98.4%

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

   3. Simplified 98.4%

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

      1. frac-times 93.1%

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

      2. associate-/r* 93.1%

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

      3. *-commutative 93.1%

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

      4. frac-times 93.8%

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

      5. associate-*l/ 93.2%

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

      6. times-frac 98.9%

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

   5. Applied egg-rr 98.9%

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

   6. Taylor expanded in z around inf 97.9%

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

   if -1 < z < 1

   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. sqr-neg 82.8%

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

      3. times-frac 82.6%

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

      4. sqr-neg 82.6%

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

   3. Simplified 82.6%

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

      1. *-commutative 82.6%

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

      2. clear-num 82.6%

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

      3. associate-/r* 89.1%

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

      4. frac-times 96.7%

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

      5. *-un-lft-identity 96.7%

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

   5. Applied egg-rr 96.7%

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

   6. Taylor expanded in z around 0 95.4%

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

   if 1 < z

   1. Initial program 77.8%

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

      1. *-commutative 77.8%

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

      2. sqr-neg 77.8%

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

      3. times-frac 94.0%

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

      4. sqr-neg 94.0%

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

   3. Simplified 94.0%

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

      1. *-commutative 94.0%

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

      2. clear-num 93.9%

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

      3. associate-/r* 97.9%

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

      4. frac-times 97.4%

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

      5. *-un-lft-identity 97.4%

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

   5. Applied egg-rr 97.4%

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

   6. Taylor expanded in z around inf 94.0%

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

      1. unpow2 94.0%

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

      2. associate-*r/ 97.4%

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

   8. Simplified 97.4%

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

      1. associate-/r* 98.0%

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

      2. div-inv 97.9%

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

      3. clear-num 97.9%

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

   10. Applied egg-rr 97.9%

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

4. Final simplification 96.6%

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

Alternative 4: 95.1% accurate, 0.8× speedup

Math

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

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= z -1.0)
   (* (/ y z) (/ (/ x z) z))
   (if (<= z 0.76) (/ (- (/ x z) x) (/ z y)) (* (/ x z) (/ (/ y z) z)))))

C

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

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
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 <= 0.76d0) then
        tmp = ((x / z) - x) / (z / y)
    else
        tmp = (x / z) * ((y / z) / z)
    end if
    code = tmp
end function

Java

assert x < y;
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 <= 0.76) {
		tmp = ((x / z) - x) / (z / y);
	} else {
		tmp = (x / z) * ((y / z) / z);
	}
	return tmp;
}

Python

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

Julia

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

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -1.0)
		tmp = (y / z) * ((x / z) / z);
	elseif (z <= 0.76)
		tmp = ((x / z) - x) / (z / y);
	else
		tmp = (x / z) * ((y / z) / z);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
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, 0.76], N[(N[(N[(x / z), $MachinePrecision] - x), $MachinePrecision] / N[(z / y), $MachinePrecision]), $MachinePrecision], N[(N[(x / z), $MachinePrecision] * N[(N[(y / z), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -1

   1. Initial program 93.1%

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

      1. associate-*l* 93.1%

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

      2. times-frac 93.8%

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

      3. associate-/r* 96.9%

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

      4. associate-*r/ 98.4%

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

   3. Simplified 98.4%

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

      1. frac-times 93.1%

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

      2. associate-/r* 93.1%

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

      3. *-commutative 93.1%

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

      4. frac-times 93.8%

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

      5. associate-*l/ 93.2%

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

      6. times-frac 98.9%

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

   5. Applied egg-rr 98.9%

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

   6. Taylor expanded in z around inf 97.9%

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

   if -1 < z < 0.76000000000000001

   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 96.4%

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

      3. associate-/r* 96.4%

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

      4. associate-*r/ 96.4%

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

   3. Simplified 96.4%

      \[\leadsto \color{blue}{\frac{\frac{x}{z} \cdot \frac{y}{z}}{z + 1}} \]
   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* 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 82.6%

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

      5. associate-*l/ 82.7%

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

      6. times-frac 96.4%

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

   5. Applied egg-rr 96.4%

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

      1. *-commutative 96.4%

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

      2. clear-num 96.4%

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

      3. un-div-inv 97.5%

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

      4. associate-/l/ 97.5%

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

      5. +-commutative 97.5%

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

      6. distribute-lft-in 97.5%

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

      7. *-rgt-identity 97.5%

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

   7. Applied egg-rr 97.5%

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

   8. Taylor expanded in z around 0 97.0%

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

      1. neg-mul-1 97.0%

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

      2. +-commutative 97.0%

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

      3. unsub-neg 97.0%

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

   10. Simplified 97.0%

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

   if 0.76000000000000001 < z

   1. Initial program 77.8%

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

      1. *-commutative 77.8%

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

      2. sqr-neg 77.8%

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

      3. times-frac 94.0%

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

      4. sqr-neg 94.0%

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

   3. Simplified 94.0%

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

      1. *-commutative 94.0%

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

      2. clear-num 93.9%

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

      3. associate-/r* 97.9%

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

      4. frac-times 97.4%

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

      5. *-un-lft-identity 97.4%

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

   5. Applied egg-rr 97.4%

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

   6. Taylor expanded in z around inf 94.0%

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

      1. unpow2 94.0%

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

      2. associate-*r/ 97.4%

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

   8. Simplified 97.4%

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

      1. associate-/r* 98.0%

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

      2. div-inv 97.9%

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

      3. clear-num 97.9%

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

   10. Applied egg-rr 97.9%

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

4. Final simplification 97.4%

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

Alternative 5: 96.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \frac{y}{z} \cdot \frac{\frac{x}{z + 1}}{z} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z) :precision binary64 (* (/ y z) (/ (/ x (+ z 1.0)) z)))

C

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

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
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

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

Python

[x, y] = sort([x, y])
def code(x, y, z):
	return (y / z) * ((x / (z + 1.0)) / z)

Julia

x, y = sort([x, y])
function code(x, y, z)
	return Float64(Float64(y / z) * Float64(Float64(x / Float64(z + 1.0)) / z))
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp = code(x, y, z)
	tmp = (y / z) * ((x / (z + 1.0)) / z);
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
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 84.2%

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

   1. associate-*l* 84.2%

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

   2. times-frac 95.3%

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

   3. associate-/r* 96.9%

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

   4. associate-*r/ 96.9%

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

3. Simplified 96.9%

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

   1. frac-times 85.0%

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

   2. associate-/r* 84.2%

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

   3. *-commutative 84.2%

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

   4. frac-times 87.7%

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

   5. associate-*l/ 86.6%

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

   6. times-frac 97.2%

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

5. Applied egg-rr 97.2%

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

6. Final simplification 97.2%

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

Alternative 6: 79.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -1.5 \cdot 10^{-13}:\\ \;\;\;\;x \cdot \frac{y}{z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{\frac{x}{z}}{z}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= x -1.5e-13) (* x (/ y (* z z))) (* y (/ (/ x z) z))))

C

assert(x < y);
double code(double x, double y, double z) {
	double tmp;
	if (x <= -1.5e-13) {
		tmp = x * (y / (z * z));
	} else {
		tmp = y * ((x / z) / z);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
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 <= (-1.5d-13)) then
        tmp = x * (y / (z * z))
    else
        tmp = y * ((x / z) / z)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -1.5e-13) {
		tmp = x * (y / (z * z));
	} else {
		tmp = y * ((x / z) / z);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z):
	tmp = 0
	if x <= -1.5e-13:
		tmp = x * (y / (z * z))
	else:
		tmp = y * ((x / z) / z)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y, z)
	tmp = 0.0
	if (x <= -1.5e-13)
		tmp = Float64(x * Float64(y / Float64(z * z)));
	else
		tmp = Float64(y * Float64(Float64(x / z) / z));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -1.5e-13)
		tmp = x * (y / (z * z));
	else
		tmp = y * ((x / z) / z);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_, z_] := If[LessEqual[x, -1.5e-13], N[(x * N[(y / N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y * N[(N[(x / z), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.49999999999999992e-13

   1. Initial program 85.2%

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

      1. *-commutative 85.2%

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

      2. sqr-neg 85.2%

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

      3. times-frac 93.8%

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

      4. sqr-neg 93.8%

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

   3. Simplified 93.8%

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

   4. Taylor expanded in z around 0 76.7%

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

   if -1.49999999999999992e-13 < x

   1. Initial program 83.8%

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

      1. *-commutative 83.8%

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

      2. sqr-neg 83.8%

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

      3. times-frac 85.4%

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

      4. sqr-neg 85.4%

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

   3. Simplified 85.4%

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

      1. *-commutative 85.4%

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

      2. clear-num 85.3%

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

      3. associate-/r* 91.0%

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

      4. frac-times 96.6%

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

      5. *-un-lft-identity 96.6%

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

   5. Applied egg-rr 96.6%

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

   6. Taylor expanded in z around 0 74.3%

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

      1. unpow2 74.3%

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

      2. associate-/r* 78.7%

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

      3. associate-*r/ 83.9%

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

      4. associate-*r/ 81.1%

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

   8. Simplified 81.1%

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

4. Final simplification 79.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.5 \cdot 10^{-13}:\\ \;\;\;\;x \cdot \frac{y}{z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{\frac{x}{z}}{z}\\ \end{array} \]

Alternative 7: 79.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 4.3 \cdot 10^{-72}:\\ \;\;\;\;\frac{x}{z \cdot \frac{z}{y}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{\frac{x}{z}}{z}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y 4.3e-72) (/ x (* z (/ z y))) (* y (/ (/ x z) z))))

C

assert(x < y);
double code(double x, double y, double z) {
	double tmp;
	if (y <= 4.3e-72) {
		tmp = x / (z * (z / y));
	} else {
		tmp = y * ((x / z) / z);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
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 <= 4.3d-72) then
        tmp = x / (z * (z / y))
    else
        tmp = y * ((x / z) / z)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 4.3e-72) {
		tmp = x / (z * (z / y));
	} else {
		tmp = y * ((x / z) / z);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z):
	tmp = 0
	if y <= 4.3e-72:
		tmp = x / (z * (z / y))
	else:
		tmp = y * ((x / z) / z)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y, z)
	tmp = 0.0
	if (y <= 4.3e-72)
		tmp = Float64(x / Float64(z * Float64(z / y)));
	else
		tmp = Float64(y * Float64(Float64(x / z) / z));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 4.3e-72)
		tmp = x / (z * (z / y));
	else
		tmp = y * ((x / z) / z);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_, z_] := If[LessEqual[y, 4.3e-72], N[(x / N[(z * N[(z / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y * N[(N[(x / z), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 4.2999999999999999e-72

   1. Initial program 83.9%

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

      1. *-commutative 83.9%

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

      2. associate-*r/ 85.8%

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

      3. sqr-neg 85.8%

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

      4. associate-*l* 85.8%

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

      5. associate-*l* 85.8%

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

      6. sqr-neg 85.8%

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

      7. associate-*l* 85.8%

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

      8. distribute-lft-in 85.8%

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

      9. fma-def 85.8%

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

      10. *-rgt-identity 85.8%

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

   3. Simplified 85.8%

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

   4. Taylor expanded in z around 0 72.7%

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

      1. unpow2 72.7%

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

   6. Simplified 72.7%

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

      1. associate-*r/ 71.3%

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

      2. frac-times 78.2%

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

      3. clear-num 78.2%

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

      4. frac-times 80.0%

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

      5. *-un-lft-identity 80.0%

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

   8. Applied egg-rr 80.0%

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

   if 4.2999999999999999e-72 < y

   1. Initial program 85.2%

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

      1. *-commutative 85.2%

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

      2. sqr-neg 85.2%

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

      3. times-frac 90.0%

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

      4. sqr-neg 90.0%

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

   3. Simplified 90.0%

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

      1. *-commutative 90.0%

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

      2. clear-num 90.0%

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

      3. associate-/r* 93.6%

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

      4. frac-times 97.7%

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

      5. *-un-lft-identity 97.7%

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

   5. Applied egg-rr 97.7%

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

   6. Taylor expanded in z around 0 77.9%

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

      1. unpow2 77.9%

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

      2. associate-/r* 79.3%

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

      3. associate-*r/ 81.3%

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

      4. associate-*r/ 84.9%

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

   8. Simplified 84.9%

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

4. Final simplification 81.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 4.3 \cdot 10^{-72}:\\ \;\;\;\;\frac{x}{z \cdot \frac{z}{y}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{\frac{x}{z}}{z}\\ \end{array} \]

Alternative 8: 80.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 4.3 \cdot 10^{-74}:\\ \;\;\;\;\frac{x}{z \cdot \frac{z}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z \cdot \frac{z}{x}}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y 4.3e-74) (/ x (* z (/ z y))) (/ y (* z (/ z x)))))

C

assert(x < y);
double code(double x, double y, double z) {
	double tmp;
	if (y <= 4.3e-74) {
		tmp = x / (z * (z / y));
	} else {
		tmp = y / (z * (z / x));
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
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 <= 4.3d-74) then
        tmp = x / (z * (z / y))
    else
        tmp = y / (z * (z / x))
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 4.3e-74) {
		tmp = x / (z * (z / y));
	} else {
		tmp = y / (z * (z / x));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z):
	tmp = 0
	if y <= 4.3e-74:
		tmp = x / (z * (z / y))
	else:
		tmp = y / (z * (z / x))
	return tmp

Julia

x, y = sort([x, y])
function code(x, y, z)
	tmp = 0.0
	if (y <= 4.3e-74)
		tmp = Float64(x / Float64(z * Float64(z / y)));
	else
		tmp = Float64(y / Float64(z * Float64(z / x)));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 4.3e-74)
		tmp = x / (z * (z / y));
	else
		tmp = y / (z * (z / x));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_, z_] := If[LessEqual[y, 4.3e-74], N[(x / N[(z * N[(z / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y / N[(z * N[(z / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 4.29999999999999972e-74

   1. Initial program 84.0%

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

      1. *-commutative 84.0%

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

      2. associate-*r/ 85.8%

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

      3. sqr-neg 85.8%

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

      4. associate-*l* 85.8%

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

      5. associate-*l* 85.8%

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

      6. sqr-neg 85.8%

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

      7. associate-*l* 85.8%

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

      8. distribute-lft-in 85.8%

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

      9. fma-def 85.8%

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

      10. *-rgt-identity 85.8%

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

   3. Simplified 85.8%

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

   4. Taylor expanded in z around 0 73.1%

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

      1. unpow2 73.1%

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

   6. Simplified 73.1%

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

      1. associate-*r/ 71.7%

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

      2. frac-times 79.0%

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

      3. clear-num 78.9%

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

      4. frac-times 80.5%

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

      5. *-un-lft-identity 80.5%

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

   8. Applied egg-rr 80.5%

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

   if 4.29999999999999972e-74 < y

   1. Initial program 85.0%

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

      1. *-commutative 85.0%

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

      2. associate-*r/ 88.0%

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

      3. sqr-neg 88.0%

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

      4. associate-*l* 88.0%

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

      5. associate-*l* 88.0%

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

      6. sqr-neg 88.0%

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

      7. associate-*l* 88.0%

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

      8. distribute-lft-in 88.0%

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

      9. fma-def 88.0%

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

      10. *-rgt-identity 88.0%

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

   3. Simplified 88.0%

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

   4. Taylor expanded in z around 0 78.9%

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

      1. unpow2 78.9%

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

   6. Simplified 78.9%

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

      1. associate-*r/ 76.5%

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

      2. *-commutative 76.5%

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

      3. times-frac 77.4%

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

      4. clear-num 77.4%

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

      5. frac-times 82.4%

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

      6. *-un-lft-identity 82.4%

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

   8. Applied egg-rr 82.4%

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

4. Final simplification 81.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 4.3 \cdot 10^{-74}:\\ \;\;\;\;\frac{x}{z \cdot \frac{z}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z \cdot \frac{z}{x}}\\ \end{array} \]

Alternative 9: 78.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -5.2 \cdot 10^{+74}:\\ \;\;\;\;x \cdot \frac{y}{z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{z}}{\frac{z}{y}}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= x -5.2e+74) (* x (/ y (* z z))) (/ (/ x z) (/ z y))))

C

assert(x < y);
double code(double x, double y, double z) {
	double tmp;
	if (x <= -5.2e+74) {
		tmp = x * (y / (z * z));
	} else {
		tmp = (x / z) / (z / y);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
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 <= (-5.2d+74)) then
        tmp = x * (y / (z * z))
    else
        tmp = (x / z) / (z / y)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -5.2e+74) {
		tmp = x * (y / (z * z));
	} else {
		tmp = (x / z) / (z / y);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z):
	tmp = 0
	if x <= -5.2e+74:
		tmp = x * (y / (z * z))
	else:
		tmp = (x / z) / (z / y)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y, z)
	tmp = 0.0
	if (x <= -5.2e+74)
		tmp = Float64(x * Float64(y / Float64(z * z)));
	else
		tmp = Float64(Float64(x / z) / Float64(z / y));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -5.2e+74)
		tmp = x * (y / (z * z));
	else
		tmp = (x / z) / (z / y);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_, z_] := If[LessEqual[x, -5.2e+74], N[(x * N[(y / N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / z), $MachinePrecision] / N[(z / y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -5.2000000000000001e74

   1. Initial program 83.1%

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

      1. *-commutative 83.1%

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

      2. sqr-neg 83.1%

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

      3. times-frac 95.1%

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

      4. sqr-neg 95.1%

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

   3. Simplified 95.1%

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

   4. Taylor expanded in z around 0 78.5%

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

   if -5.2000000000000001e74 < x

   1. Initial program 84.5%

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

      1. *-commutative 84.5%

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

      2. associate-*r/ 85.8%

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

      3. sqr-neg 85.8%

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

      4. associate-*l* 85.8%

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

      5. associate-*l* 85.8%

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

      6. sqr-neg 85.8%

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

      7. associate-*l* 85.8%

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

      8. distribute-lft-in 85.8%

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

      9. fma-def 85.8%

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

      10. *-rgt-identity 85.8%

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

   3. Simplified 85.8%

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

   4. Taylor expanded in z around 0 76.1%

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

      1. unpow2 76.1%

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

   6. Simplified 76.1%

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

      1. associate-*r/ 73.6%

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

      2. *-commutative 73.6%

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

      3. times-frac 82.8%

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

      4. clear-num 82.8%

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

      5. un-div-inv 83.6%

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

   8. Applied egg-rr 83.6%

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

4. Final simplification 82.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.2 \cdot 10^{+74}:\\ \;\;\;\;x \cdot \frac{y}{z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{z}}{\frac{z}{y}}\\ \end{array} \]

Alternative 10: 40.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -1 \cdot 10^{-310}:\\ \;\;\;\;y \cdot \frac{-x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z} \cdot x\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= z -1e-310) (* y (/ (- x) z)) (* (/ y z) x)))

C

assert(x < y);
double code(double x, double y, double z) {
	double tmp;
	if (z <= -1e-310) {
		tmp = y * (-x / z);
	} else {
		tmp = (y / z) * x;
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (z <= (-1d-310)) then
        tmp = y * (-x / z)
    else
        tmp = (y / z) * x
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -1e-310) {
		tmp = y * (-x / z);
	} else {
		tmp = (y / z) * x;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z):
	tmp = 0
	if z <= -1e-310:
		tmp = y * (-x / z)
	else:
		tmp = (y / z) * x
	return tmp

Julia

x, y = sort([x, y])
function code(x, y, z)
	tmp = 0.0
	if (z <= -1e-310)
		tmp = Float64(y * Float64(Float64(-x) / z));
	else
		tmp = Float64(Float64(y / z) * x);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -1e-310)
		tmp = y * (-x / z);
	else
		tmp = (y / z) * x;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_, z_] := If[LessEqual[z, -1e-310], N[(y * N[((-x) / z), $MachinePrecision]), $MachinePrecision], N[(N[(y / z), $MachinePrecision] * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -9.999999999999969e-311

   1. Initial program 87.1%

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

      1. associate-*l* 87.1%

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

      2. times-frac 95.1%

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

      3. associate-/r* 96.5%

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

      4. associate-*r/ 97.1%

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

   3. Simplified 97.1%

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

   4. Taylor expanded in z around 0 62.8%

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

      1. *-commutative 62.8%

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

      2. unpow2 62.8%

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

      3. times-frac 70.4%

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

      4. associate-*r/ 70.4%

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

      5. *-commutative 70.4%

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

      6. neg-mul-1 70.4%

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

      7. distribute-rgt-neg-out 70.4%

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

      8. associate-*l/ 73.2%

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

      9. distribute-lft-out 73.2%

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

   6. Simplified 73.2%

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

   7. Taylor expanded in z around inf 40.4%

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

      1. mul-1-neg 40.4%

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

   9. Simplified 40.4%

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

   if -9.999999999999969e-311 < z

   1. Initial program 80.8%

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

      1. associate-*l* 80.8%

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

      2. times-frac 95.5%

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

      3. associate-/r* 97.4%

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

      4. associate-*r/ 96.6%

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

   3. Simplified 96.6%

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

   4. Taylor expanded in z around 0 40.7%

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

      1. *-commutative 40.7%

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

      2. unpow2 40.7%

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

      3. times-frac 48.1%

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

      4. associate-*r/ 48.1%

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

      5. *-commutative 48.1%

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

      6. neg-mul-1 48.1%

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

      7. distribute-rgt-neg-out 48.1%

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

      8. associate-*l/ 47.6%

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

      9. distribute-lft-out 66.6%

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

   6. Simplified 66.6%

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

   7. Taylor expanded in z around inf 16.4%

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

      1. mul-1-neg 16.4%

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

   9. Simplified 16.4%

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

       1. clear-num 16.4%

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

       2. add-sqr-sqrt 7.2%

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

       3. sqrt-unprod 23.5%

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

       4. sqr-neg 23.5%

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

       5. sqrt-unprod 17.2%

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

       6. add-sqr-sqrt 37.6%

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

       7. associate-*l/ 37.6%

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

       8. *-un-lft-identity 37.6%

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

   11. Applied egg-rr 37.6%

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

       1. associate-/r/ 34.3%

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

   13. Applied egg-rr 34.3%

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

4. Final simplification 37.6%

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

Alternative 11: 72.4% accurate, 1.6× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ y \cdot \frac{x}{z \cdot z} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z) :precision binary64 (* y (/ x (* z z))))

C

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

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
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

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

Python

[x, y] = sort([x, y])
def code(x, y, z):
	return y * (x / (z * z))

Julia

x, y = sort([x, y])
function code(x, y, z)
	return Float64(y * Float64(x / Float64(z * z)))
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp = code(x, y, z)
	tmp = y * (x / (z * z));
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_, z_] := N[(y * N[(x / N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 84.2%

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

   1. *-commutative 84.2%

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

   2. associate-*r/ 86.4%

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

   3. sqr-neg 86.4%

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

   4. associate-*l* 86.4%

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

   5. associate-*l* 86.4%

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

   6. sqr-neg 86.4%

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

   7. associate-*l* 86.4%

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

   8. distribute-lft-in 86.4%

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

   9. fma-def 86.4%

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

   10. *-rgt-identity 86.4%

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

3. Simplified 86.4%

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

4. Taylor expanded in z around 0 74.5%

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

   1. unpow2 74.5%

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

6. Simplified 74.5%

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

7. Final simplification 74.5%

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

Alternative 12: 74.3% accurate, 1.6× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ y \cdot \frac{\frac{x}{z}}{z} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z) :precision binary64 (* y (/ (/ x z) z)))

C

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

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
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

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

Python

[x, y] = sort([x, y])
def code(x, y, z):
	return y * ((x / z) / z)

Julia

x, y = sort([x, y])
function code(x, y, z)
	return Float64(y * Float64(Float64(x / z) / z))
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp = code(x, y, z)
	tmp = y * ((x / z) / z);
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_, z_] := N[(y * N[(N[(x / z), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 84.2%

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

   1. *-commutative 84.2%

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

   2. sqr-neg 84.2%

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

   3. times-frac 87.7%

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

   4. sqr-neg 87.7%

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

3. Simplified 87.7%

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

   1. *-commutative 87.7%

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

   2. clear-num 87.7%

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

   3. associate-/r* 92.9%

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

   4. frac-times 97.3%

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

   5. *-un-lft-identity 97.3%

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

5. Applied egg-rr 97.3%

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

6. Taylor expanded in z around 0 72.8%

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

   1. unpow2 72.8%

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

   2. associate-/r* 74.6%

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

   3. associate-*r/ 78.0%

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

   4. associate-*r/ 76.0%

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

8. Simplified 76.0%

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

9. Final simplification 76.0%

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

Alternative 13: 31.4% accurate, 2.2× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ y \cdot \frac{x}{z} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z) :precision binary64 (* y (/ x z)))

C

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

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = y * (x / z)
end function

Java

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

Python

[x, y] = sort([x, y])
def code(x, y, z):
	return y * (x / z)

Julia

x, y = sort([x, y])
function code(x, y, z)
	return Float64(y * Float64(x / z))
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp = code(x, y, z)
	tmp = y * (x / z);
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_, z_] := N[(y * N[(x / z), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 84.2%

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

   1. associate-*l* 84.2%

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

   2. times-frac 95.3%

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

   3. associate-/r* 96.9%

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

   4. associate-*r/ 96.9%

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

3. Simplified 96.9%

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

4. Taylor expanded in z around 0 52.7%

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

   1. *-commutative 52.7%

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

   2. unpow2 52.7%

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

   3. times-frac 60.2%

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

   4. associate-*r/ 60.2%

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

   5. *-commutative 60.2%

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

   6. neg-mul-1 60.2%

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

   7. distribute-rgt-neg-out 60.2%

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

   8. associate-*l/ 61.5%

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

   9. distribute-lft-out 70.2%

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

6. Simplified 70.2%

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

7. Taylor expanded in z around inf 29.4%

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

   1. mul-1-neg 29.4%

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

9. Simplified 29.4%

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

    1. clear-num 29.4%

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

    2. add-sqr-sqrt 15.9%

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

    3. sqrt-unprod 29.4%

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

    4. sqr-neg 29.4%

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

    5. sqrt-unprod 13.4%

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

    6. add-sqr-sqrt 29.8%

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

    7. associate-*l/ 29.8%

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

    8. *-un-lft-identity 29.8%

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

11. Applied egg-rr 29.8%

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

    1. associate-/l* 24.6%

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

    2. *-commutative 24.6%

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

    3. associate-*l/ 29.8%

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

13. Applied egg-rr 29.8%

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

14. Final simplification 29.8%

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

Alternative 14: 31.2% accurate, 2.2× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \frac{y}{z} \cdot x \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z) :precision binary64 (* (/ y z) x))

C

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

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = (y / z) * x
end function

Java

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

Python

[x, y] = sort([x, y])
def code(x, y, z):
	return (y / z) * x

Julia

x, y = sort([x, y])
function code(x, y, z)
	return Float64(Float64(y / z) * x)
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp = code(x, y, z)
	tmp = (y / z) * x;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_, z_] := N[(N[(y / z), $MachinePrecision] * x), $MachinePrecision]

Derivation

1. Initial program 84.2%

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

   1. associate-*l* 84.2%

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

   2. times-frac 95.3%

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

   3. associate-/r* 96.9%

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

   4. associate-*r/ 96.9%

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

3. Simplified 96.9%

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

4. Taylor expanded in z around 0 52.7%

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

   1. *-commutative 52.7%

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

   2. unpow2 52.7%

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

   3. times-frac 60.2%

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

   4. associate-*r/ 60.2%

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

   5. *-commutative 60.2%

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

   6. neg-mul-1 60.2%

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

   7. distribute-rgt-neg-out 60.2%

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

   8. associate-*l/ 61.5%

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

   9. distribute-lft-out 70.2%

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

6. Simplified 70.2%

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

7. Taylor expanded in z around inf 29.4%

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

   1. mul-1-neg 29.4%

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

9. Simplified 29.4%

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

    1. clear-num 29.4%

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

    2. add-sqr-sqrt 15.9%

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

    3. sqrt-unprod 29.4%

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

    4. sqr-neg 29.4%

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

    5. sqrt-unprod 13.4%

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

    6. add-sqr-sqrt 29.8%

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

    7. associate-*l/ 29.8%

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

    8. *-un-lft-identity 29.8%

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

11. Applied egg-rr 29.8%

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

    1. associate-/r/ 28.9%

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

13. Applied egg-rr 28.9%

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

14. Final simplification 28.9%

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

Developer target: 95.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z < 249.6182814532307:\\ \;\;\;\;\frac{y \cdot \frac{x}{z}}{z + z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{y}{z}}{1 + z} \cdot x}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (< z 249.6182814532307)
   (/ (* y (/ x z)) (+ z (* z z)))
   (/ (* (/ (/ y z) (+ 1.0 z)) x) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if (z < 249.6182814532307) {
		tmp = (y * (x / z)) / (z + (z * z));
	} else {
		tmp = (((y / z) / (1.0 + z)) * x) / z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (z < 249.6182814532307d0) then
        tmp = (y * (x / z)) / (z + (z * z))
    else
        tmp = (((y / z) / (1.0d0 + z)) * x) / z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z < 249.6182814532307) {
		tmp = (y * (x / z)) / (z + (z * z));
	} else {
		tmp = (((y / z) / (1.0 + z)) * x) / z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z < 249.6182814532307:
		tmp = (y * (x / z)) / (z + (z * z))
	else:
		tmp = (((y / z) / (1.0 + z)) * x) / z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z < 249.6182814532307)
		tmp = Float64(Float64(y * Float64(x / z)) / Float64(z + Float64(z * z)));
	else
		tmp = Float64(Float64(Float64(Float64(y / z) / Float64(1.0 + z)) * x) / z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z < 249.6182814532307)
		tmp = (y * (x / z)) / (z + (z * z));
	else
		tmp = (((y / z) / (1.0 + z)) * x) / z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Less[z, 249.6182814532307], N[(N[(y * N[(x / z), $MachinePrecision]), $MachinePrecision] / N[(z + N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(y / z), $MachinePrecision] / N[(1.0 + z), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision] / z), $MachinePrecision]]

Reproduce

herbie shell --seed 2023279 
(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))))