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

Percentage Accurate: 83.1% → 96.1%

Time: 9.4s

Alternatives: 15

Speedup: 1.0×

• 25.3% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 83.1% 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: 96.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

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 = ((x / (z / y)) / z) / (z + 1.0d0)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.5%

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

   1. associate-*l* 79.5%

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

   2. times-frac 95.7%

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

   3. associate-/r* 96.4%

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

   4. associate-*r/ 96.5%

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

3. Simplified 96.5%

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

   1. associate-*l/ 96.9%

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

5. Applied egg-rr 96.9%

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

6. Taylor expanded in x around 0 88.1%

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

   1. associate-/l* 97.0%

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

8. Simplified 97.0%

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

9. Final simplification 97.0%

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

Alternative 2: 79.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \cdot y \leq -5000 \lor \neg \left(x \cdot y \leq 5 \cdot 10^{-49}\right):\\ \;\;\;\;x \cdot \frac{y}{z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z} \cdot \frac{x}{z}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (or (<= (* x y) -5000.0) (not (<= (* x y) 5e-49)))
   (* x (/ y (* z z)))
   (* (/ y z) (/ x z))))

C

assert(x < y);
double code(double x, double y, double z) {
	double tmp;
	if (((x * y) <= -5000.0) || !((x * y) <= 5e-49)) {
		tmp = x * (y / (z * z));
	} else {
		tmp = (y / z) * (x / 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 * y) <= (-5000.0d0)) .or. (.not. ((x * y) <= 5d-49))) then
        tmp = x * (y / (z * z))
    else
        tmp = (y / z) * (x / 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 * y) <= -5000.0) || !((x * y) <= 5e-49)) {
		tmp = x * (y / (z * z));
	} else {
		tmp = (y / z) * (x / z);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z):
	tmp = 0
	if ((x * y) <= -5000.0) or not ((x * y) <= 5e-49):
		tmp = x * (y / (z * z))
	else:
		tmp = (y / z) * (x / z)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y, z)
	tmp = 0.0
	if ((Float64(x * y) <= -5000.0) || !(Float64(x * y) <= 5e-49))
		tmp = Float64(x * Float64(y / Float64(z * z)));
	else
		tmp = Float64(Float64(y / z) * Float64(x / z));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (((x * y) <= -5000.0) || ~(((x * y) <= 5e-49)))
		tmp = x * (y / (z * z));
	else
		tmp = (y / z) * (x / 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[Or[LessEqual[N[(x * y), $MachinePrecision], -5000.0], N[Not[LessEqual[N[(x * y), $MachinePrecision], 5e-49]], $MachinePrecision]], N[(x * N[(y / N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y / z), $MachinePrecision] * N[(x / z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x y) < -5e3 or 4.9999999999999999e-49 < (*.f64 x y)

   1. Initial program 86.8%

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

      1. *-commutative 86.8%

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

      2. sqr-neg 86.8%

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

      3. times-frac 94.7%

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

      4. sqr-neg 94.7%

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

   3. Simplified 94.7%

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

   4. Taylor expanded in z around 0 74.6%

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

   if -5e3 < (*.f64 x y) < 4.9999999999999999e-49

   1. Initial program 72.1%

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

      1. *-commutative 72.1%

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

      2. associate-*r/ 75.1%

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

      3. sqr-neg 75.1%

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

      4. associate-*l* 75.1%

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

      5. associate-*l* 75.1%

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

      6. sqr-neg 75.1%

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

      7. associate-*l* 75.1%

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

      8. distribute-lft-in 75.1%

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

      9. fma-def 75.1%

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

      10. *-rgt-identity 75.1%

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

   3. Simplified 75.1%

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

   4. Taylor expanded in z around 0 60.6%

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

      1. unpow2 60.6%

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

      2. times-frac 88.1%

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

   6. Simplified 88.1%

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

4. Final simplification 81.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq -5000 \lor \neg \left(x \cdot y \leq 5 \cdot 10^{-49}\right):\\ \;\;\;\;x \cdot \frac{y}{z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z} \cdot \frac{x}{z}\\ \end{array} \]

Alternative 3: 93.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -4.8 \cdot 10^{-46} \lor \neg \left(z \leq 5.5 \cdot 10^{-131}\right):\\ \;\;\;\;\frac{y}{z \cdot z} \cdot \frac{x}{z + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{\frac{z}{y}}}{z}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (or (<= z -4.8e-46) (not (<= z 5.5e-131)))
   (* (/ y (* z z)) (/ x (+ z 1.0)))
   (/ (/ x (/ z y)) z)))

C

assert(x < y);
double code(double x, double y, double z) {
	double tmp;
	if ((z <= -4.8e-46) || !(z <= 5.5e-131)) {
		tmp = (y / (z * z)) * (x / (z + 1.0));
	} else {
		tmp = (x / (z / y)) / 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 <= (-4.8d-46)) .or. (.not. (z <= 5.5d-131))) then
        tmp = (y / (z * z)) * (x / (z + 1.0d0))
    else
        tmp = (x / (z / y)) / 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 <= -4.8e-46) || !(z <= 5.5e-131)) {
		tmp = (y / (z * z)) * (x / (z + 1.0));
	} else {
		tmp = (x / (z / y)) / z;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z):
	tmp = 0
	if (z <= -4.8e-46) or not (z <= 5.5e-131):
		tmp = (y / (z * z)) * (x / (z + 1.0))
	else:
		tmp = (x / (z / y)) / z
	return tmp

Julia

x, y = sort([x, y])
function code(x, y, z)
	tmp = 0.0
	if ((z <= -4.8e-46) || !(z <= 5.5e-131))
		tmp = Float64(Float64(y / Float64(z * z)) * Float64(x / Float64(z + 1.0)));
	else
		tmp = Float64(Float64(x / Float64(z / y)) / z);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -4.8e-46) || ~((z <= 5.5e-131)))
		tmp = (y / (z * z)) * (x / (z + 1.0));
	else
		tmp = (x / (z / y)) / 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[Or[LessEqual[z, -4.8e-46], N[Not[LessEqual[z, 5.5e-131]], $MachinePrecision]], N[(N[(y / N[(z * z), $MachinePrecision]), $MachinePrecision] * N[(x / N[(z + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(z / y), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -4.80000000000000027e-46 or 5.4999999999999997e-131 < z

   1. Initial program 87.0%

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

      1. *-commutative 87.0%

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

      2. sqr-neg 87.0%

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

      3. times-frac 95.3%

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

      4. sqr-neg 95.3%

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

   3. Simplified 95.3%

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

   if -4.80000000000000027e-46 < z < 5.4999999999999997e-131

   1. Initial program 65.7%

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

      1. *-commutative 65.7%

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

      2. sqr-neg 65.7%

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

      3. times-frac 66.8%

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

      4. sqr-neg 66.8%

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

   3. Simplified 66.8%

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

   4. Taylor expanded in z around 0 66.8%

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

      1. *-commutative 66.8%

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

      2. associate-/r* 83.0%

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

      3. associate-*r/ 97.1%

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

   6. Applied egg-rr 97.1%

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

   7. Taylor expanded in x around 0 83.7%

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

      1. associate-/l* 97.6%

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

   9. Simplified 97.6%

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

4. Final simplification 96.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -4.8 \cdot 10^{-46} \lor \neg \left(z \leq 5.5 \cdot 10^{-131}\right):\\ \;\;\;\;\frac{y}{z \cdot z} \cdot \frac{x}{z + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{\frac{z}{y}}}{z}\\ \end{array} \]

Alternative 4: 79.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \cdot y \leq -5000:\\ \;\;\;\;\frac{x}{\frac{z \cdot z}{y}}\\ \mathbf{elif}\;x \cdot y \leq 5 \cdot 10^{-49}:\\ \;\;\;\;\frac{y}{z} \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{z \cdot z}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= (* x y) -5000.0)
   (/ x (/ (* z z) y))
   (if (<= (* x y) 5e-49) (* (/ y z) (/ x z)) (* x (/ y (* z z))))))

C

assert(x < y);
double code(double x, double y, double z) {
	double tmp;
	if ((x * y) <= -5000.0) {
		tmp = x / ((z * z) / y);
	} else if ((x * y) <= 5e-49) {
		tmp = (y / z) * (x / z);
	} else {
		tmp = x * (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 ((x * y) <= (-5000.0d0)) then
        tmp = x / ((z * z) / y)
    else if ((x * y) <= 5d-49) then
        tmp = (y / z) * (x / z)
    else
        tmp = x * (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 ((x * y) <= -5000.0) {
		tmp = x / ((z * z) / y);
	} else if ((x * y) <= 5e-49) {
		tmp = (y / z) * (x / z);
	} else {
		tmp = x * (y / (z * z));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z):
	tmp = 0
	if (x * y) <= -5000.0:
		tmp = x / ((z * z) / y)
	elif (x * y) <= 5e-49:
		tmp = (y / z) * (x / z)
	else:
		tmp = x * (y / (z * z))
	return tmp

Julia

x, y = sort([x, y])
function code(x, y, z)
	tmp = 0.0
	if (Float64(x * y) <= -5000.0)
		tmp = Float64(x / Float64(Float64(z * z) / y));
	elseif (Float64(x * y) <= 5e-49)
		tmp = Float64(Float64(y / z) * Float64(x / z));
	else
		tmp = Float64(x * Float64(y / Float64(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 * y) <= -5000.0)
		tmp = x / ((z * z) / y);
	elseif ((x * y) <= 5e-49)
		tmp = (y / z) * (x / z);
	else
		tmp = x * (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[N[(x * y), $MachinePrecision], -5000.0], N[(x / N[(N[(z * z), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], 5e-49], N[(N[(y / z), $MachinePrecision] * N[(x / z), $MachinePrecision]), $MachinePrecision], N[(x * N[(y / N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 x y) < -5e3

   1. Initial program 89.9%

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

      1. *-commutative 89.9%

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

      2. sqr-neg 89.9%

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

      3. times-frac 97.2%

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

      4. sqr-neg 97.2%

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

   3. Simplified 97.2%

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

   4. Taylor expanded in z around 0 80.8%

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

      1. associate-*l/ 76.3%

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

      2. frac-times 71.4%

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

      3. clear-num 71.4%

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

      4. frac-times 76.7%

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

      5. *-un-lft-identity 76.7%

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

   6. Applied egg-rr 76.7%

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

   7. Taylor expanded in z around 0 80.8%

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

      1. unpow2 80.8%

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

   9. Simplified 80.8%

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

   if -5e3 < (*.f64 x y) < 4.9999999999999999e-49

   1. Initial program 72.1%

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

      1. *-commutative 72.1%

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

      2. associate-*r/ 75.1%

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

      3. sqr-neg 75.1%

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

      4. associate-*l* 75.1%

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

      5. associate-*l* 75.1%

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

      6. sqr-neg 75.1%

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

      7. associate-*l* 75.1%

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

      8. distribute-lft-in 75.1%

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

      9. fma-def 75.1%

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

      10. *-rgt-identity 75.1%

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

   3. Simplified 75.1%

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

   4. Taylor expanded in z around 0 60.6%

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

      1. unpow2 60.6%

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

      2. times-frac 88.1%

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

   6. Simplified 88.1%

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

   if 4.9999999999999999e-49 < (*.f64 x y)

   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 91.7%

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

      4. sqr-neg 91.7%

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

   3. Simplified 91.7%

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

   4. Taylor expanded in z around 0 67.0%

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

4. Final simplification 81.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq -5000:\\ \;\;\;\;\frac{x}{\frac{z \cdot z}{y}}\\ \mathbf{elif}\;x \cdot y \leq 5 \cdot 10^{-49}:\\ \;\;\;\;\frac{y}{z} \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{z \cdot z}\\ \end{array} \]

Alternative 5: 92.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -1 \lor \neg \left(z \leq 3.6 \cdot 10^{-9}\right):\\ \;\;\;\;\frac{x}{z} \cdot \frac{y}{z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{\frac{z}{y}}}{z}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (or (<= z -1.0) (not (<= z 3.6e-9)))
   (* (/ x z) (/ y (* z z)))
   (/ (/ x (/ z y)) z)))

C

assert(x < y);
double code(double x, double y, double z) {
	double tmp;
	if ((z <= -1.0) || !(z <= 3.6e-9)) {
		tmp = (x / z) * (y / (z * z));
	} else {
		tmp = (x / (z / y)) / 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)) .or. (.not. (z <= 3.6d-9))) then
        tmp = (x / z) * (y / (z * z))
    else
        tmp = (x / (z / y)) / 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) || !(z <= 3.6e-9)) {
		tmp = (x / z) * (y / (z * z));
	} else {
		tmp = (x / (z / y)) / z;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z):
	tmp = 0
	if (z <= -1.0) or not (z <= 3.6e-9):
		tmp = (x / z) * (y / (z * z))
	else:
		tmp = (x / (z / y)) / z
	return tmp

Julia

x, y = sort([x, y])
function code(x, y, z)
	tmp = 0.0
	if ((z <= -1.0) || !(z <= 3.6e-9))
		tmp = Float64(Float64(x / z) * Float64(y / Float64(z * z)));
	else
		tmp = Float64(Float64(x / Float64(z / y)) / 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) || ~((z <= 3.6e-9)))
		tmp = (x / z) * (y / (z * z));
	else
		tmp = (x / (z / y)) / 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[Or[LessEqual[z, -1.0], N[Not[LessEqual[z, 3.6e-9]], $MachinePrecision]], N[(N[(x / z), $MachinePrecision] * N[(y / N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(z / y), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1 or 3.6e-9 < z

   1. Initial program 85.8%

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

      1. *-commutative 85.8%

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

      2. sqr-neg 85.8%

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

      3. times-frac 94.3%

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

      4. sqr-neg 94.3%

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

   3. Simplified 94.3%

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

   4. Taylor expanded in z around inf 92.5%

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

   if -1 < z < 3.6e-9

   1. Initial program 74.1%

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

      1. *-commutative 74.1%

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

      2. sqr-neg 74.1%

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

      3. times-frac 77.6%

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

      4. sqr-neg 77.6%

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

   3. Simplified 77.6%

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

   4. Taylor expanded in z around 0 77.1%

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

      1. *-commutative 77.1%

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

      2. associate-/r* 87.8%

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

      3. associate-*r/ 96.6%

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

   6. Applied egg-rr 96.6%

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

   7. Taylor expanded in x around 0 85.4%

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

      1. associate-/l* 97.3%

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

   9. Simplified 96.9%

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

4. Final simplification 94.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1 \lor \neg \left(z \leq 3.6 \cdot 10^{-9}\right):\\ \;\;\;\;\frac{x}{z} \cdot \frac{y}{z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{\frac{z}{y}}}{z}\\ \end{array} \]

Alternative 6: 79.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 4.7 \cdot 10^{-178}:\\ \;\;\;\;\frac{x}{z \cdot \frac{z}{y}}\\ \mathbf{elif}\;y \leq 1.9 \cdot 10^{+99}:\\ \;\;\;\;\frac{y}{z} \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{z \cdot z}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y 4.7e-178)
   (/ x (* z (/ z y)))
   (if (<= y 1.9e+99) (* (/ y z) (/ x z)) (* y (/ x (* z z))))))

C

assert(x < y);
double code(double x, double y, double z) {
	double tmp;
	if (y <= 4.7e-178) {
		tmp = x / (z * (z / y));
	} else if (y <= 1.9e+99) {
		tmp = (y / z) * (x / 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 (y <= 4.7d-178) then
        tmp = x / (z * (z / y))
    else if (y <= 1.9d+99) then
        tmp = (y / z) * (x / 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 (y <= 4.7e-178) {
		tmp = x / (z * (z / y));
	} else if (y <= 1.9e+99) {
		tmp = (y / z) * (x / z);
	} else {
		tmp = y * (x / (z * z));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z):
	tmp = 0
	if y <= 4.7e-178:
		tmp = x / (z * (z / y))
	elif y <= 1.9e+99:
		tmp = (y / z) * (x / z)
	else:
		tmp = y * (x / (z * z))
	return tmp

Julia

x, y = sort([x, y])
function code(x, y, z)
	tmp = 0.0
	if (y <= 4.7e-178)
		tmp = Float64(x / Float64(z * Float64(z / y)));
	elseif (y <= 1.9e+99)
		tmp = Float64(Float64(y / z) * Float64(x / z));
	else
		tmp = Float64(y * Float64(x / Float64(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 <= 4.7e-178)
		tmp = x / (z * (z / y));
	elseif (y <= 1.9e+99)
		tmp = (y / z) * (x / 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[y, 4.7e-178], N[(x / N[(z * N[(z / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.9e+99], N[(N[(y / z), $MachinePrecision] * N[(x / z), $MachinePrecision]), $MachinePrecision], N[(y * N[(x / N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 4.69999999999999999e-178

   1. Initial program 78.0%

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

      1. *-commutative 78.0%

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

      2. sqr-neg 78.0%

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

      3. times-frac 83.1%

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

      4. sqr-neg 83.1%

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

   3. Simplified 83.1%

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

   4. Taylor expanded in z around 0 71.2%

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

      1. associate-*l/ 65.6%

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

      2. frac-times 80.6%

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

      3. clear-num 80.7%

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

      4. frac-times 80.5%

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

      5. *-un-lft-identity 80.5%

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

   6. Applied egg-rr 80.5%

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

   if 4.69999999999999999e-178 < y < 1.9e99

   1. Initial program 82.1%

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

      1. *-commutative 82.1%

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

      2. associate-*r/ 80.3%

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

      3. sqr-neg 80.3%

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

      4. associate-*l* 80.3%

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

      5. associate-*l* 80.3%

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

      6. sqr-neg 80.3%

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

      7. associate-*l* 80.3%

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

      8. distribute-lft-in 80.3%

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

      9. fma-def 80.3%

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

      10. *-rgt-identity 80.3%

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

   3. Simplified 80.3%

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

   4. Taylor expanded in z around 0 61.7%

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

      1. unpow2 61.7%

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

      2. times-frac 75.5%

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

   6. Simplified 75.5%

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

   if 1.9e99 < y

   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. *-commutative 81.5%

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

      2. associate-*r/ 83.8%

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

      3. sqr-neg 83.8%

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

      4. associate-*l* 83.8%

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

      5. associate-*l* 83.8%

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

      6. sqr-neg 83.8%

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

      7. associate-*l* 83.8%

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

      8. distribute-lft-in 83.8%

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

      9. fma-def 83.8%

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

      10. *-rgt-identity 83.8%

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

   3. Simplified 83.8%

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

   4. Taylor expanded in z around 0 68.8%

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

      1. unpow2 68.8%

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

      2. *-rgt-identity 68.8%

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

      3. times-frac 78.2%

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

      4. /-rgt-identity 78.2%

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

   6. Simplified 78.2%

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

4. Final simplification 79.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 4.7 \cdot 10^{-178}:\\ \;\;\;\;\frac{x}{z \cdot \frac{z}{y}}\\ \mathbf{elif}\;y \leq 1.9 \cdot 10^{+99}:\\ \;\;\;\;\frac{y}{z} \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{z \cdot z}\\ \end{array} \]

Alternative 7: 96.7% accurate, 1.0× speedup

Math

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

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)) (+ z 1.0)))

C

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

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)) / (z + 1.0d0)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.5%

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

   1. associate-*l* 79.5%

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

   2. times-frac 95.7%

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

   3. associate-/r* 96.4%

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

   4. associate-*r/ 96.5%

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

3. Simplified 96.5%

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

4. Final simplification 96.5%

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

Alternative 8: 96.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

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 = ((x * (y / z)) / z) / (z + 1.0d0)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.5%

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

   1. associate-*l* 79.5%

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

   2. times-frac 95.7%

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

   3. associate-/r* 96.4%

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

   4. associate-*r/ 96.5%

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

3. Simplified 96.5%

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

   1. associate-*l/ 96.9%

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

5. Applied egg-rr 96.9%

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

6. Final simplification 96.9%

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

Alternative 9: 40.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -1:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{elif}\;z \leq -1 \cdot 10^{-309}:\\ \;\;\;\;\frac{y}{z} \cdot \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \end{array} \end{array} \]

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 (/ x z))
   (if (<= z -1e-309) (* (/ y z) (- x)) (* x (/ y z)))))

C

assert(x < y);
double code(double x, double y, double z) {
	double tmp;
	if (z <= -1.0) {
		tmp = y * (x / z);
	} else if (z <= -1e-309) {
		tmp = (y / z) * -x;
	} else {
		tmp = x * (y / 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 * (x / z)
    else if (z <= (-1d-309)) then
        tmp = (y / z) * -x
    else
        tmp = x * (y / 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 * (x / z);
	} else if (z <= -1e-309) {
		tmp = (y / z) * -x;
	} else {
		tmp = x * (y / z);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z):
	tmp = 0
	if z <= -1.0:
		tmp = y * (x / z)
	elif z <= -1e-309:
		tmp = (y / z) * -x
	else:
		tmp = x * (y / z)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y, z)
	tmp = 0.0
	if (z <= -1.0)
		tmp = Float64(y * Float64(x / z));
	elseif (z <= -1e-309)
		tmp = Float64(Float64(y / z) * Float64(-x));
	else
		tmp = Float64(x * Float64(y / 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 * (x / z);
	elseif (z <= -1e-309)
		tmp = (y / z) * -x;
	else
		tmp = x * (y / 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[(y * N[(x / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, -1e-309], N[(N[(y / z), $MachinePrecision] * (-x)), $MachinePrecision], N[(x * N[(y / z), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -1

   1. Initial program 88.2%

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

      1. *-commutative 88.2%

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

      2. associate-*r/ 88.5%

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

      3. sqr-neg 88.5%

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

      4. associate-*l* 88.5%

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

      5. associate-*l* 88.5%

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

      6. sqr-neg 88.5%

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

      7. associate-*l* 88.5%

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

      8. distribute-lft-in 88.5%

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

      9. fma-def 88.5%

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

      10. *-rgt-identity 88.5%

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

   3. Simplified 88.5%

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

   4. Taylor expanded in z around 0 34.2%

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

      1. +-commutative 34.2%

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

      2. unpow2 34.2%

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

      3. times-frac 34.3%

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

      4. mul-1-neg 34.3%

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

      5. associate-*r/ 40.0%

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

      6. distribute-lft-neg-in 40.0%

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

      7. distribute-rgt-out 40.0%

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

   6. Simplified 40.0%

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

   7. Taylor expanded in z around inf 34.3%

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

      1. mul-1-neg 34.3%

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

      2. associate-*r/ 40.0%

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

      3. distribute-lft-neg-in 40.0%

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

   9. Simplified 40.0%

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

       1. add-sqr-sqrt 15.8%

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

       2. sqrt-unprod 39.9%

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

       3. sqr-neg 39.9%

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

       4. sqrt-unprod 26.3%

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

       5. add-sqr-sqrt 44.3%

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

       6. associate-*r/ 38.6%

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

       7. associate-/l* 44.3%

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

   11. Applied egg-rr 44.3%

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

       1. associate-/r/ 45.7%

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

   13. Applied egg-rr 45.7%

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

   if -1 < z < -1.000000000000002e-309

   1. Initial program 72.6%

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

      1. *-commutative 72.6%

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

      2. associate-*r/ 77.7%

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

      3. sqr-neg 77.7%

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

      4. associate-*l* 77.7%

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

      5. associate-*l* 77.7%

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

      6. sqr-neg 77.7%

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

      7. associate-*l* 77.7%

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

      8. distribute-lft-in 77.7%

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

      9. fma-def 77.7%

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

      10. *-rgt-identity 77.7%

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

   3. Simplified 77.7%

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

   4. Taylor expanded in z around 0 72.6%

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

      1. +-commutative 72.6%

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

      2. unpow2 72.6%

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

      3. times-frac 95.0%

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

      4. mul-1-neg 95.0%

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

      5. associate-*r/ 95.0%

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

      6. distribute-lft-neg-in 95.0%

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

      7. distribute-rgt-out 95.0%

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

   6. Simplified 95.0%

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

   7. Taylor expanded in z around inf 32.7%

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

      1. mul-1-neg 32.7%

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

      2. associate-*r/ 32.8%

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

      3. distribute-lft-neg-in 32.8%

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

   9. Simplified 32.8%

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

   if -1.000000000000002e-309 < z

   1. Initial program 78.1%

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

      1. *-commutative 78.1%

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

      2. associate-*r/ 81.9%

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

      3. sqr-neg 81.9%

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

      4. associate-*l* 82.0%

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

      5. associate-*l* 81.9%

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

      6. sqr-neg 81.9%

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

      7. associate-*l* 82.0%

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

      8. distribute-lft-in 82.0%

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

      9. fma-def 82.0%

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

      10. *-rgt-identity 82.0%

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

   3. Simplified 82.0%

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

   4. Taylor expanded in z around 0 38.6%

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

      1. +-commutative 38.6%

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

      2. unpow2 38.6%

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

      3. times-frac 52.7%

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

      4. mul-1-neg 52.7%

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

      5. associate-*r/ 54.1%

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

      6. distribute-lft-neg-in 54.1%

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

      7. distribute-rgt-out 75.1%

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

   6. Simplified 75.1%

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

   7. Taylor expanded in z around inf 13.2%

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

      1. mul-1-neg 13.2%

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

      2. associate-*r/ 16.2%

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

      3. distribute-lft-neg-in 16.2%

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

   9. Simplified 16.2%

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

       1. add-sqr-sqrt 10.6%

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

       2. sqrt-unprod 30.2%

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

       3. sqr-neg 30.2%

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

       4. sqrt-unprod 20.2%

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

       5. add-sqr-sqrt 39.3%

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

       6. associate-*r/ 34.8%

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

       7. associate-/l* 38.6%

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

   11. Applied egg-rr 38.6%

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

       1. div-inv 39.3%

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

       2. clear-num 39.3%

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

       3. *-commutative 39.3%

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

   13. Applied egg-rr 39.3%

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

4. Final simplification 39.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{elif}\;z \leq -1 \cdot 10^{-309}:\\ \;\;\;\;\frac{y}{z} \cdot \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \end{array} \]

Alternative 10: 77.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 1.85 \cdot 10^{+99}:\\ \;\;\;\;\frac{y}{z} \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{z \cdot z}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y 1.85e+99) (* (/ y z) (/ x z)) (* y (/ x (* z z)))))

C

assert(x < y);
double code(double x, double y, double z) {
	double tmp;
	if (y <= 1.85e+99) {
		tmp = (y / z) * (x / 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 (y <= 1.85d+99) then
        tmp = (y / z) * (x / 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 (y <= 1.85e+99) {
		tmp = (y / z) * (x / z);
	} else {
		tmp = y * (x / (z * z));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z):
	tmp = 0
	if y <= 1.85e+99:
		tmp = (y / z) * (x / z)
	else:
		tmp = y * (x / (z * z))
	return tmp

Julia

x, y = sort([x, y])
function code(x, y, z)
	tmp = 0.0
	if (y <= 1.85e+99)
		tmp = Float64(Float64(y / z) * Float64(x / z));
	else
		tmp = Float64(y * Float64(x / Float64(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 <= 1.85e+99)
		tmp = (y / z) * (x / 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[y, 1.85e+99], N[(N[(y / z), $MachinePrecision] * N[(x / z), $MachinePrecision]), $MachinePrecision], N[(y * N[(x / N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.85000000000000005e99

   1. Initial program 78.9%

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

      1. *-commutative 78.9%

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

      2. associate-*r/ 82.3%

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

      3. sqr-neg 82.3%

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

      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* 82.3%

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

      6. sqr-neg 82.3%

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

      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 64.7%

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

      1. unpow2 64.7%

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

      2. times-frac 79.4%

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

   6. Simplified 79.4%

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

   if 1.85000000000000005e99 < y

   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. *-commutative 81.5%

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

      2. associate-*r/ 83.8%

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

      3. sqr-neg 83.8%

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

      4. associate-*l* 83.8%

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

      5. associate-*l* 83.8%

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

      6. sqr-neg 83.8%

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

      7. associate-*l* 83.8%

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

      8. distribute-lft-in 83.8%

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

      9. fma-def 83.8%

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

      10. *-rgt-identity 83.8%

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

   3. Simplified 83.8%

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

   4. Taylor expanded in z around 0 68.8%

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

      1. unpow2 68.8%

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

      2. *-rgt-identity 68.8%

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

      3. times-frac 78.2%

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

      4. /-rgt-identity 78.2%

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

   6. Simplified 78.2%

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

4. Final simplification 79.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.85 \cdot 10^{+99}:\\ \;\;\;\;\frac{y}{z} \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{z \cdot z}\\ \end{array} \]

Alternative 11: 80.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 10^{-90}:\\ \;\;\;\;\frac{x}{z \cdot \frac{z}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z \cdot \frac{z}{x}}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y 1e-90) (/ x (* z (/ z y))) (/ y (* z (/ z x)))))

C

assert(x < y);
double code(double x, double y, double z) {
	double tmp;
	if (y <= 1e-90) {
		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 <= 1d-90) 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 <= 1e-90) {
		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 <= 1e-90:
		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 <= 1e-90)
		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 <= 1e-90)
		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, 1e-90], 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 < 9.99999999999999995e-91

   1. Initial program 79.6%

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

      1. *-commutative 79.6%

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

      2. sqr-neg 79.6%

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

      3. times-frac 84.3%

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

      4. sqr-neg 84.3%

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

   3. Simplified 84.3%

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

   4. Taylor expanded in z around 0 71.2%

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

      1. associate-*l/ 66.0%

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

      2. frac-times 79.9%

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

      3. clear-num 79.9%

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

      4. frac-times 79.8%

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

      5. *-un-lft-identity 79.8%

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

   6. Applied egg-rr 79.8%

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

   if 9.99999999999999995e-91 < y

   1. Initial program 79.2%

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

      1. *-commutative 79.2%

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

      2. associate-*r/ 81.7%

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

      3. sqr-neg 81.7%

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

      4. associate-*l* 81.7%

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

      5. associate-*l* 81.7%

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

      6. sqr-neg 81.7%

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

      7. associate-*l* 81.7%

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

      8. distribute-lft-in 81.7%

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

      9. fma-def 81.7%

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

      10. *-rgt-identity 81.7%

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

   3. Simplified 81.7%

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

   4. Taylor expanded in z around 0 64.7%

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

      1. unpow2 64.7%

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

      2. times-frac 70.3%

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

   6. Simplified 70.3%

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

      1. clear-num 70.3%

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

      2. frac-times 78.9%

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

      3. *-un-lft-identity 78.9%

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

   8. Applied egg-rr 78.9%

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

4. Final simplification 79.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 10^{-90}:\\ \;\;\;\;\frac{x}{z \cdot \frac{z}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z \cdot \frac{z}{x}}\\ \end{array} \]

Alternative 12: 39.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -1 \cdot 10^{-309}:\\ \;\;\;\;y \cdot \frac{-x}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= z -1e-309) (* y (/ (- x) z)) (* x (/ y z))))

C

assert(x < y);
double code(double x, double y, double z) {
	double tmp;
	if (z <= -1e-309) {
		tmp = y * (-x / z);
	} else {
		tmp = x * (y / 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 <= (-1d-309)) then
        tmp = y * (-x / z)
    else
        tmp = x * (y / 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 <= -1e-309) {
		tmp = y * (-x / z);
	} else {
		tmp = x * (y / z);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z):
	tmp = 0
	if z <= -1e-309:
		tmp = y * (-x / z)
	else:
		tmp = x * (y / z)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y, z)
	tmp = 0.0
	if (z <= -1e-309)
		tmp = Float64(y * Float64(Float64(-x) / z));
	else
		tmp = Float64(x * Float64(y / z));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -1e-309)
		tmp = y * (-x / z);
	else
		tmp = x * (y / 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, -1e-309], N[(y * N[((-x) / z), $MachinePrecision]), $MachinePrecision], N[(x * N[(y / z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.000000000000002e-309

   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. *-commutative 80.8%

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

      2. associate-*r/ 83.4%

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

      3. sqr-neg 83.4%

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

      4. associate-*l* 83.4%

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

      5. associate-*l* 83.4%

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

      6. sqr-neg 83.4%

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

      7. associate-*l* 83.4%

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

      8. distribute-lft-in 83.4%

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

      9. fma-def 83.4%

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

      10. *-rgt-identity 83.4%

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

   3. Simplified 83.4%

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

   4. Taylor expanded in z around 0 52.4%

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

      1. +-commutative 52.4%

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

      2. unpow2 52.4%

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

      3. times-frac 63.0%

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

      4. mul-1-neg 63.0%

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

      5. associate-*r/ 66.0%

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

      6. distribute-lft-neg-in 66.0%

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

      7. distribute-rgt-out 66.0%

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

   6. Simplified 66.0%

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

   7. Taylor expanded in z around inf 33.5%

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

      1. mul-1-neg 33.5%

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

      2. associate-*r/ 36.6%

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

      3. distribute-lft-neg-in 36.6%

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

   9. Simplified 36.6%

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

   10. Taylor expanded in x around 0 33.5%

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

       1. mul-1-neg 33.5%

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

       2. associate-*l/ 41.0%

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

       3. distribute-lft-neg-out 41.0%

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

       4. *-commutative 41.0%

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

       5. distribute-frac-neg 41.0%

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

   12. Simplified 41.0%

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

   if -1.000000000000002e-309 < z

   1. Initial program 78.1%

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

      1. *-commutative 78.1%

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

      2. associate-*r/ 81.9%

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

      3. sqr-neg 81.9%

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

      4. associate-*l* 82.0%

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

      5. associate-*l* 81.9%

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

      6. sqr-neg 81.9%

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

      7. associate-*l* 82.0%

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

      8. distribute-lft-in 82.0%

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

      9. fma-def 82.0%

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

      10. *-rgt-identity 82.0%

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

   3. Simplified 82.0%

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

   4. Taylor expanded in z around 0 38.6%

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

      1. +-commutative 38.6%

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

      2. unpow2 38.6%

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

      3. times-frac 52.7%

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

      4. mul-1-neg 52.7%

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

      5. associate-*r/ 54.1%

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

      6. distribute-lft-neg-in 54.1%

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

      7. distribute-rgt-out 75.1%

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

   6. Simplified 75.1%

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

   7. Taylor expanded in z around inf 13.2%

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

      1. mul-1-neg 13.2%

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

      2. associate-*r/ 16.2%

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

      3. distribute-lft-neg-in 16.2%

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

   9. Simplified 16.2%

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

       1. add-sqr-sqrt 10.6%

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

       2. sqrt-unprod 30.2%

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

       3. sqr-neg 30.2%

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

       4. sqrt-unprod 20.2%

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

       5. add-sqr-sqrt 39.3%

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

       6. associate-*r/ 34.8%

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

       7. associate-/l* 38.6%

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

   11. Applied egg-rr 38.6%

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

       1. div-inv 39.3%

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

       2. clear-num 39.3%

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

       3. *-commutative 39.3%

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

   13. Applied egg-rr 39.3%

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

4. Final simplification 40.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1 \cdot 10^{-309}:\\ \;\;\;\;y \cdot \frac{-x}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \end{array} \]

Alternative 13: 32.3% accurate, 1.6× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 10^{+171}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y 1e+171) (* x (/ y z)) (* y (/ x z))))

C

assert(x < y);
double code(double x, double y, double z) {
	double tmp;
	if (y <= 1e+171) {
		tmp = x * (y / z);
	} else {
		tmp = y * (x / 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 <= 1d+171) then
        tmp = x * (y / z)
    else
        tmp = y * (x / 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 <= 1e+171) {
		tmp = x * (y / z);
	} else {
		tmp = y * (x / z);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z):
	tmp = 0
	if y <= 1e+171:
		tmp = x * (y / z)
	else:
		tmp = y * (x / z)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y, z)
	tmp = 0.0
	if (y <= 1e+171)
		tmp = Float64(x * Float64(y / z));
	else
		tmp = Float64(y * Float64(x / z));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 1e+171)
		tmp = x * (y / z);
	else
		tmp = y * (x / 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, 1e+171], N[(x * N[(y / z), $MachinePrecision]), $MachinePrecision], N[(y * N[(x / z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 9.99999999999999954e170

   1. Initial program 79.8%

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

      1. *-commutative 79.8%

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

      2. associate-*r/ 82.6%

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

      3. sqr-neg 82.6%

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

      4. associate-*l* 82.6%

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

      5. associate-*l* 82.6%

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

      6. sqr-neg 82.6%

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

      7. associate-*l* 82.6%

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

      8. distribute-lft-in 82.6%

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

      9. fma-def 82.6%

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

      10. *-rgt-identity 82.6%

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

   3. Simplified 82.6%

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

   4. Taylor expanded in z around 0 46.8%

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

      1. +-commutative 46.8%

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

      2. unpow2 46.8%

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

      3. times-frac 61.5%

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

      4. mul-1-neg 61.5%

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

      5. associate-*r/ 64.0%

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

      6. distribute-lft-neg-in 64.0%

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

      7. distribute-rgt-out 73.9%

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

   6. Simplified 73.9%

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

   7. Taylor expanded in z around inf 23.4%

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

      1. mul-1-neg 23.4%

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

      2. associate-*r/ 26.9%

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

      3. distribute-lft-neg-in 26.9%

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

   9. Simplified 26.9%

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

       1. add-sqr-sqrt 13.3%

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

       2. sqrt-unprod 30.0%

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

       3. sqr-neg 30.0%

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

       4. sqrt-unprod 18.7%

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

       5. add-sqr-sqrt 34.3%

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

       6. associate-*r/ 30.0%

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

       7. associate-/l* 33.9%

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

   11. Applied egg-rr 33.9%

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

       1. div-inv 34.3%

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

       2. clear-num 34.3%

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

       3. *-commutative 34.3%

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

   13. Applied egg-rr 34.3%

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

   if 9.99999999999999954e170 < y

   1. Initial program 77.2%

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

      1. *-commutative 77.2%

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

      2. associate-*r/ 83.2%

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

      3. sqr-neg 83.2%

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

      4. associate-*l* 83.2%

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

      5. associate-*l* 83.2%

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

      6. sqr-neg 83.2%

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

      7. associate-*l* 83.2%

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

      8. distribute-lft-in 83.2%

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

      9. fma-def 83.2%

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

      10. *-rgt-identity 83.2%

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

   3. Simplified 83.2%

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

   4. Taylor expanded in z around 0 36.4%

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

      1. +-commutative 36.4%

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

      2. unpow2 36.4%

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

      3. times-frac 33.7%

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

      4. mul-1-neg 33.7%

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

      5. associate-*r/ 33.8%

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

      6. distribute-lft-neg-in 33.8%

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

      7. distribute-rgt-out 48.5%

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

   6. Simplified 48.5%

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

   7. Taylor expanded in z around inf 22.5%

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

      1. mul-1-neg 22.5%

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

      2. associate-*r/ 22.5%

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

      3. distribute-lft-neg-in 22.5%

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

   9. Simplified 22.5%

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

       1. add-sqr-sqrt 12.9%

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

       2. sqrt-unprod 34.4%

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

       3. sqr-neg 34.4%

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

       4. sqrt-unprod 13.1%

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

       5. add-sqr-sqrt 20.1%

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

       6. associate-*r/ 20.0%

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

       7. associate-/l* 20.1%

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

   11. Applied egg-rr 20.1%

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

       1. associate-/r/ 34.2%

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

   13. Applied egg-rr 34.2%

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

4. Final simplification 34.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 10^{+171}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \end{array} \]

Alternative 14: 74.4% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Derivation

1. Initial program 79.5%

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

   1. *-commutative 79.5%

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

   2. associate-*r/ 82.6%

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

   3. sqr-neg 82.6%

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

   4. associate-*l* 82.7%

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

   5. associate-*l* 82.6%

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

   6. sqr-neg 82.6%

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

   7. associate-*l* 82.7%

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

   8. distribute-lft-in 82.7%

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

   9. fma-def 82.7%

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

   10. *-rgt-identity 82.7%

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

3. Simplified 82.7%

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

4. Taylor expanded in z around 0 65.5%

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

   1. unpow2 65.5%

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

   2. times-frac 76.6%

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

6. Simplified 76.6%

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

7. Final simplification 76.6%

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

Alternative 15: 30.7% 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 79.5%

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

   1. *-commutative 79.5%

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

   2. associate-*r/ 82.6%

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

   3. sqr-neg 82.6%

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

   4. associate-*l* 82.7%

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

   5. associate-*l* 82.6%

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

   6. sqr-neg 82.6%

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

   7. associate-*l* 82.7%

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

   8. distribute-lft-in 82.7%

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

   9. fma-def 82.7%

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

   10. *-rgt-identity 82.7%

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

3. Simplified 82.7%

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

4. Taylor expanded in z around 0 45.4%

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

   1. +-commutative 45.4%

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

   2. unpow2 45.4%

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

   3. times-frac 57.8%

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

   4. mul-1-neg 57.8%

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

   5. associate-*r/ 60.0%

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

   6. distribute-lft-neg-in 60.0%

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

   7. distribute-rgt-out 70.5%

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

6. Simplified 70.5%

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

7. Taylor expanded in z around inf 23.3%

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

   1. mul-1-neg 23.3%

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

   2. associate-*r/ 26.3%

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

   3. distribute-lft-neg-in 26.3%

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

9. Simplified 26.3%

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

    1. add-sqr-sqrt 13.2%

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

    2. sqrt-unprod 30.6%

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

    3. sqr-neg 30.6%

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

    4. sqrt-unprod 17.9%

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

    5. add-sqr-sqrt 32.4%

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

    6. associate-*r/ 28.7%

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

    7. associate-/l* 32.1%

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

11. Applied egg-rr 32.1%

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

    1. associate-/r/ 32.8%

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

13. Applied egg-rr 32.8%

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

14. Final simplification 32.8%

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

Developer target: 95.4% 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 2023293 
(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))))