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

Percentage Accurate: 83.3% → 97.0%

Time: 10.4s

Alternatives: 16

Speedup: 1.0×

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 83.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 97.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Derivation

1. Initial program 81.4%

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

   1. associate-*l* 81.4%

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

   2. times-frac 94.6%

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

   3. associate-/r* 97.3%

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

   4. associate-*r/ 98.6%

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

3. Simplified 98.6%

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

4. Taylor expanded in x around 0 83.2%

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

   1. unpow2 83.2%

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

   2. associate-/r* 87.9%

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

   3. associate-*r/ 96.5%

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

   4. *-commutative 96.5%

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

   5. associate-*l/ 92.5%

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

   6. associate-/r/ 98.6%

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

6. Simplified 98.6%

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

7. Final simplification 98.6%

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

Alternative 2: 93.3% accurate, 0.8× speedup

Math

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

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 0.76)))
   (* (/ x z) (/ y (* z z)))
   (* (/ x z) (- (/ y z) y))))

C

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

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((z <= (-1.0d0)) .or. (.not. (z <= 0.76d0))) then
        tmp = (x / z) * (y / (z * z))
    else
        tmp = (x / z) * ((y / z) - y)
    end if
    code = tmp
end function

Java

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

Python

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

Julia

x, y = sort([x, y])
function code(x, y, z)
	tmp = 0.0
	if ((z <= -1.0) || !(z <= 0.76))
		tmp = Float64(Float64(x / z) * Float64(y / Float64(z * z)));
	else
		tmp = Float64(Float64(x / z) * Float64(Float64(y / z) - y));
	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 <= 0.76)))
		tmp = (x / z) * (y / (z * z));
	else
		tmp = (x / z) * ((y / z) - y);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_, z_] := If[Or[LessEqual[z, -1.0], N[Not[LessEqual[z, 0.76]], $MachinePrecision]], N[(N[(x / z), $MachinePrecision] * N[(y / N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / z), $MachinePrecision] * N[(N[(y / z), $MachinePrecision] - y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1 or 0.76000000000000001 < z

   1. Initial program 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. sqr-neg 82.1%

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

      3. times-frac 91.9%

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

      4. sqr-neg 91.9%

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

   3. Simplified 91.9%

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

   4. Taylor expanded in z around inf 90.0%

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

   if -1 < z < 0.76000000000000001

   1. Initial program 80.8%

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

      1. associate-*l* 80.8%

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

      2. times-frac 97.4%

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

      3. associate-/r* 97.4%

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

      4. associate-*r/ 97.4%

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

   3. Simplified 97.4%

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

   4. Taylor expanded in x around 0 80.8%

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

      1. unpow2 80.8%

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

      2. associate-/r* 88.6%

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

      3. associate-*r/ 96.1%

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

      4. *-commutative 96.1%

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

      5. associate-*l/ 87.7%

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

      6. associate-/r/ 97.5%

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

   6. Simplified 97.5%

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

   7. Taylor expanded in z around 0 68.4%

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

      1. +-commutative 68.4%

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

      2. unpow2 68.4%

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

      3. associate-/l* 66.0%

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

      4. *-rgt-identity 66.0%

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

      5. associate-*r/ 64.8%

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

      6. mul-1-neg 64.8%

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

      7. associate-*l/ 62.6%

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

      8. fma-def 62.6%

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

      9. associate-/r/ 62.2%

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

      10. associate-*l/ 62.6%

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

      11. *-lft-identity 62.6%

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

      12. associate-/r* 73.0%

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

      13. fma-neg 73.0%

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

      14. associate-*r/ 81.3%

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

      15. associate-*l/ 82.7%

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

      16. distribute-lft-out-- 94.7%

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

   9. Simplified 94.7%

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

4. Final simplification 92.3%

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

Alternative 3: 93.4% accurate, 0.8× speedup

Math

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

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 0.76)))
   (* (/ x z) (/ y (* z z)))
   (/ (* x (- (/ y z) y)) z)))

C

assert(x < y);
double code(double x, double y, double z) {
	double tmp;
	if ((z <= -1.0) || !(z <= 0.76)) {
		tmp = (x / z) * (y / (z * z));
	} else {
		tmp = (x * ((y / 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 <= 0.76d0))) then
        tmp = (x / z) * (y / (z * z))
    else
        tmp = (x * ((y / 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 <= 0.76)) {
		tmp = (x / z) * (y / (z * z));
	} else {
		tmp = (x * ((y / 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 <= 0.76):
		tmp = (x / z) * (y / (z * z))
	else:
		tmp = (x * ((y / z) - y)) / z
	return tmp

Julia

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

Derivation

1. Split input into 2 regimes

2. if z < -1 or 0.76000000000000001 < z

   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. sqr-neg 82.1%

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

      3. times-frac 91.9%

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

      4. sqr-neg 91.9%

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

   3. Simplified 91.9%

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

   4. Taylor expanded in z around inf 90.0%

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

   if -1 < z < 0.76000000000000001

   1. Initial program 80.8%

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

      1. associate-*l* 80.8%

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

      2. times-frac 97.4%

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

      3. associate-/r* 97.4%

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

      4. associate-*r/ 97.4%

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

   3. Simplified 97.4%

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

   4. Taylor expanded in z around 0 68.4%

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

      1. +-commutative 68.4%

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

      2. unpow2 68.4%

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

      3. times-frac 84.8%

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

      4. mul-1-neg 84.8%

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

      5. associate-*l/ 82.7%

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

      6. distribute-rgt-neg-in 82.7%

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

      7. distribute-lft-out 94.7%

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

   6. Simplified 94.7%

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

   7. Taylor expanded in x around 0 93.3%

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

4. Final simplification 91.6%

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

Alternative 4: 95.2% accurate, 0.8× speedup

Math

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

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 0.76)))
   (/ (/ x z) (* z (/ z y)))
   (/ (* x (- (/ y z) y)) z)))

C

assert(x < y);
double code(double x, double y, double z) {
	double tmp;
	if ((z <= -1.0) || !(z <= 0.76)) {
		tmp = (x / z) / (z * (z / y));
	} else {
		tmp = (x * ((y / 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 <= 0.76d0))) then
        tmp = (x / z) / (z * (z / y))
    else
        tmp = (x * ((y / 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 <= 0.76)) {
		tmp = (x / z) / (z * (z / y));
	} else {
		tmp = (x * ((y / 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 <= 0.76):
		tmp = (x / z) / (z * (z / y))
	else:
		tmp = (x * ((y / z) - y)) / z
	return tmp

Julia

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

Derivation

1. Split input into 2 regimes

2. if z < -1 or 0.76000000000000001 < z

   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. sqr-neg 82.1%

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

      3. times-frac 91.9%

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

      4. sqr-neg 91.9%

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

   3. Simplified 91.9%

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

      1. frac-times 82.1%

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

      2. *-commutative 82.1%

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

      3. associate-/r* 85.5%

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

      4. frac-times 99.7%

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

      5. associate-/l* 97.0%

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

      6. div-inv 96.9%

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

      7. clear-num 97.0%

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

   5. Applied egg-rr 97.0%

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

   6. Taylor expanded in z around inf 90.0%

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

      1. unpow2 90.0%

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

      2. associate-*r/ 95.1%

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

   8. Simplified 95.1%

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

   if -1 < z < 0.76000000000000001

   1. Initial program 80.8%

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

      1. associate-*l* 80.8%

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

      2. times-frac 97.4%

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

      3. associate-/r* 97.4%

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

      4. associate-*r/ 97.4%

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

   3. Simplified 97.4%

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

   4. Taylor expanded in z around 0 68.4%

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

      1. +-commutative 68.4%

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

      2. unpow2 68.4%

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

      3. times-frac 84.8%

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

      4. mul-1-neg 84.8%

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

      5. associate-*l/ 82.7%

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

      6. distribute-rgt-neg-in 82.7%

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

      7. distribute-lft-out 94.7%

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

   6. Simplified 94.7%

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

   7. Taylor expanded in x around 0 93.3%

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

4. Final simplification 94.2%

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

Alternative 5: 95.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -1 \cdot 10^{-25}:\\ \;\;\;\;\frac{y}{z \cdot z} \cdot \frac{x}{z + 1}\\ \mathbf{elif}\;z \leq 0.76:\\ \;\;\;\;\frac{x \cdot \left(\frac{y}{z} - y\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{z}}{z \cdot \frac{z}{y}}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= z -1e-25)
   (* (/ y (* z z)) (/ x (+ z 1.0)))
   (if (<= z 0.76) (/ (* x (- (/ y z) y)) z) (/ (/ x z) (* z (/ z y))))))

C

assert(x < y);
double code(double x, double y, double z) {
	double tmp;
	if (z <= -1e-25) {
		tmp = (y / (z * z)) * (x / (z + 1.0));
	} else if (z <= 0.76) {
		tmp = (x * ((y / z) - y)) / z;
	} else {
		tmp = (x / z) / (z * (z / y));
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (z <= (-1d-25)) then
        tmp = (y / (z * z)) * (x / (z + 1.0d0))
    else if (z <= 0.76d0) then
        tmp = (x * ((y / z) - y)) / z
    else
        tmp = (x / z) / (z * (z / y))
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -1e-25) {
		tmp = (y / (z * z)) * (x / (z + 1.0));
	} else if (z <= 0.76) {
		tmp = (x * ((y / z) - y)) / z;
	} else {
		tmp = (x / z) / (z * (z / y));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z):
	tmp = 0
	if z <= -1e-25:
		tmp = (y / (z * z)) * (x / (z + 1.0))
	elif z <= 0.76:
		tmp = (x * ((y / z) - y)) / z
	else:
		tmp = (x / z) / (z * (z / y))
	return tmp

Julia

x, y = sort([x, y])
function code(x, y, z)
	tmp = 0.0
	if (z <= -1e-25)
		tmp = Float64(Float64(y / Float64(z * z)) * Float64(x / Float64(z + 1.0)));
	elseif (z <= 0.76)
		tmp = Float64(Float64(x * Float64(Float64(y / z) - y)) / z);
	else
		tmp = Float64(Float64(x / z) / Float64(z * Float64(z / y)));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -1e-25)
		tmp = (y / (z * z)) * (x / (z + 1.0));
	elseif (z <= 0.76)
		tmp = (x * ((y / z) - y)) / z;
	else
		tmp = (x / z) / (z * (z / y));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_, z_] := If[LessEqual[z, -1e-25], N[(N[(y / N[(z * z), $MachinePrecision]), $MachinePrecision] * N[(x / N[(z + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 0.76], N[(N[(x * N[(N[(y / z), $MachinePrecision] - y), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], N[(N[(x / z), $MachinePrecision] / N[(z * N[(z / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.00000000000000004e-25

   1. Initial program 80.1%

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

      1. *-commutative 80.1%

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

      2. sqr-neg 80.1%

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

      3. times-frac 92.0%

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

      4. sqr-neg 92.0%

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

   3. Simplified 92.0%

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

   if -1.00000000000000004e-25 < z < 0.76000000000000001

   1. Initial program 80.2%

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

      1. associate-*l* 80.2%

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

      2. times-frac 97.3%

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

      3. associate-/r* 97.3%

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

      4. associate-*r/ 97.3%

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

   3. Simplified 97.3%

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

   4. Taylor expanded in z around 0 68.4%

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

      1. +-commutative 68.4%

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

      2. unpow2 68.4%

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

      3. times-frac 85.4%

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

      4. mul-1-neg 85.4%

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

      5. associate-*l/ 82.8%

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

      6. distribute-rgt-neg-in 82.8%

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

      7. distribute-lft-out 95.8%

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

   6. Simplified 95.8%

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

   7. Taylor expanded in x around 0 95.0%

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

   if 0.76000000000000001 < z

   1. Initial program 85.2%

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

      1. *-commutative 85.2%

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

      2. sqr-neg 85.2%

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

      3. times-frac 92.7%

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

      4. sqr-neg 92.7%

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

   3. Simplified 92.7%

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

      1. frac-times 85.2%

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

      2. *-commutative 85.2%

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

      3. associate-/r* 86.4%

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

      4. frac-times 99.8%

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

      5. associate-/l* 98.4%

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

      6. div-inv 98.4%

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

      7. clear-num 98.4%

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

   5. Applied egg-rr 98.4%

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

   6. Taylor expanded in z around inf 92.0%

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

      1. unpow2 92.0%

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

      2. associate-*r/ 97.7%

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

   8. Simplified 97.7%

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

4. Final simplification 94.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1 \cdot 10^{-25}:\\ \;\;\;\;\frac{y}{z \cdot z} \cdot \frac{x}{z + 1}\\ \mathbf{elif}\;z \leq 0.76:\\ \;\;\;\;\frac{x \cdot \left(\frac{y}{z} - y\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{z}}{z \cdot \frac{z}{y}}\\ \end{array} \]

Alternative 6: 78.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -4 \cdot 10^{+52} \lor \neg \left(z \leq 5 \cdot 10^{+56}\right):\\ \;\;\;\;y \cdot \frac{x}{z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z} \cdot \frac{x}{z}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (or (<= z -4e+52) (not (<= z 5e+56)))
   (* y (/ x (* z z)))
   (* (/ y z) (/ x z))))

C

assert(x < y);
double code(double x, double y, double z) {
	double tmp;
	if ((z <= -4e+52) || !(z <= 5e+56)) {
		tmp = y * (x / (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 ((z <= (-4d+52)) .or. (.not. (z <= 5d+56))) then
        tmp = y * (x / (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 ((z <= -4e+52) || !(z <= 5e+56)) {
		tmp = y * (x / (z * z));
	} else {
		tmp = (y / z) * (x / z);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z):
	tmp = 0
	if (z <= -4e+52) or not (z <= 5e+56):
		tmp = y * (x / (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 ((z <= -4e+52) || !(z <= 5e+56))
		tmp = Float64(y * Float64(x / 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 ((z <= -4e+52) || ~((z <= 5e+56)))
		tmp = y * (x / (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[z, -4e+52], N[Not[LessEqual[z, 5e+56]], $MachinePrecision]], N[(y * N[(x / 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 z < -4e52 or 5.00000000000000024e56 < z

   1. Initial program 79.7%

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

      1. *-commutative 79.7%

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

      2. associate-*r/ 84.0%

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

      3. sqr-neg 84.0%

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

      4. associate-*l* 84.0%

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

      5. associate-*l* 84.0%

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

      6. sqr-neg 84.0%

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

      7. associate-*l* 84.0%

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

      8. distribute-lft-in 84.0%

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

      9. fma-def 84.0%

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

      10. *-rgt-identity 84.0%

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

   3. Simplified 84.0%

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

   4. Taylor expanded in z around 0 73.0%

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

      1. unpow2 73.0%

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

   6. Simplified 73.0%

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

   if -4e52 < z < 5.00000000000000024e56

   1. Initial program 82.6%

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

      1. associate-*l* 82.6%

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

      2. times-frac 97.2%

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

      3. associate-/r* 97.2%

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

      4. associate-*r/ 97.8%

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

   3. Simplified 97.8%

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

   4. Taylor expanded in z around 0 58.7%

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

      1. +-commutative 58.7%

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

      2. unpow2 58.7%

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

      3. times-frac 72.4%

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

      4. mul-1-neg 72.4%

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

      5. associate-*l/ 70.6%

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

      6. distribute-rgt-neg-in 70.6%

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

      7. distribute-lft-out 80.6%

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

   6. Simplified 80.6%

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

   7. Taylor expanded in z around 0 81.4%

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

4. Final simplification 77.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -4 \cdot 10^{+52} \lor \neg \left(z \leq 5 \cdot 10^{+56}\right):\\ \;\;\;\;y \cdot \frac{x}{z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z} \cdot \frac{x}{z}\\ \end{array} \]

Alternative 7: 78.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -4 \cdot 10^{+52}:\\ \;\;\;\;y \cdot \frac{x}{z \cdot z}\\ \mathbf{elif}\;z \leq 1.35 \cdot 10^{-40}:\\ \;\;\;\;\frac{y}{z} \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{z \cdot z}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= z -4e+52)
   (* y (/ x (* z z)))
   (if (<= z 1.35e-40) (* (/ y z) (/ x z)) (* x (/ y (* z z))))))

C

assert(x < y);
double code(double x, double y, double z) {
	double tmp;
	if (z <= -4e+52) {
		tmp = y * (x / (z * z));
	} else if (z <= 1.35e-40) {
		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 (z <= (-4d+52)) then
        tmp = y * (x / (z * z))
    else if (z <= 1.35d-40) 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 (z <= -4e+52) {
		tmp = y * (x / (z * z));
	} else if (z <= 1.35e-40) {
		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 z <= -4e+52:
		tmp = y * (x / (z * z))
	elif z <= 1.35e-40:
		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 (z <= -4e+52)
		tmp = Float64(y * Float64(x / Float64(z * z)));
	elseif (z <= 1.35e-40)
		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 (z <= -4e+52)
		tmp = y * (x / (z * z));
	elseif (z <= 1.35e-40)
		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[z, -4e+52], N[(y * N[(x / N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 1.35e-40], 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 z < -4e52

   1. Initial program 76.4%

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

      1. *-commutative 76.4%

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

      2. associate-*r/ 79.0%

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

      3. sqr-neg 79.0%

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

      4. associate-*l* 79.0%

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

      5. associate-*l* 79.0%

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

      6. sqr-neg 79.0%

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

      7. associate-*l* 79.0%

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

      8. distribute-lft-in 79.0%

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

      9. fma-def 79.0%

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

      10. *-rgt-identity 79.0%

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

   3. Simplified 79.0%

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

   4. Taylor expanded in z around 0 67.1%

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

      1. unpow2 67.1%

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

   6. Simplified 67.1%

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

   if -4e52 < z < 1.35e-40

   1. Initial program 80.5%

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

      1. associate-*l* 80.5%

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

      2. times-frac 97.5%

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

      3. associate-/r* 97.5%

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

      4. associate-*r/ 98.2%

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

   3. Simplified 98.2%

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

   4. Taylor expanded in z around 0 60.9%

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

      1. +-commutative 60.9%

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

      2. unpow2 60.9%

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

      3. times-frac 77.0%

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

      4. mul-1-neg 77.0%

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

      5. associate-*l/ 74.9%

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

      6. distribute-rgt-neg-in 74.9%

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

      7. distribute-lft-out 86.5%

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

   6. Simplified 86.5%

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

   7. Taylor expanded in z around 0 86.6%

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

   if 1.35e-40 < z

   1. Initial program 86.3%

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

      1. *-commutative 86.3%

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

      2. sqr-neg 86.3%

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

      3. times-frac 92.3%

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

      4. sqr-neg 92.3%

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

   3. Simplified 92.3%

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

   4. Taylor expanded in z around 0 70.1%

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

4. Final simplification 77.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -4 \cdot 10^{+52}:\\ \;\;\;\;y \cdot \frac{x}{z \cdot z}\\ \mathbf{elif}\;z \leq 1.35 \cdot 10^{-40}:\\ \;\;\;\;\frac{y}{z} \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{z \cdot z}\\ \end{array} \]

Alternative 8: 79.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -4 \cdot 10^{+52}:\\ \;\;\;\;y \cdot \frac{x}{z \cdot z}\\ \mathbf{elif}\;z \leq 7.4 \cdot 10^{-41}:\\ \;\;\;\;\frac{y}{z} \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z \cdot z}{y}}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= z -4e+52)
   (* y (/ x (* z z)))
   (if (<= z 7.4e-41) (* (/ y z) (/ x z)) (/ x (/ (* z z) y)))))

C

assert(x < y);
double code(double x, double y, double z) {
	double tmp;
	if (z <= -4e+52) {
		tmp = y * (x / (z * z));
	} else if (z <= 7.4e-41) {
		tmp = (y / z) * (x / z);
	} else {
		tmp = x / ((z * z) / y);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (z <= (-4d+52)) then
        tmp = y * (x / (z * z))
    else if (z <= 7.4d-41) then
        tmp = (y / z) * (x / z)
    else
        tmp = x / ((z * z) / y)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -4e+52) {
		tmp = y * (x / (z * z));
	} else if (z <= 7.4e-41) {
		tmp = (y / z) * (x / z);
	} else {
		tmp = x / ((z * z) / y);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z):
	tmp = 0
	if z <= -4e+52:
		tmp = y * (x / (z * z))
	elif z <= 7.4e-41:
		tmp = (y / z) * (x / z)
	else:
		tmp = x / ((z * z) / y)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y, z)
	tmp = 0.0
	if (z <= -4e+52)
		tmp = Float64(y * Float64(x / Float64(z * z)));
	elseif (z <= 7.4e-41)
		tmp = Float64(Float64(y / z) * Float64(x / z));
	else
		tmp = Float64(x / Float64(Float64(z * z) / y));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -4e+52)
		tmp = y * (x / (z * z));
	elseif (z <= 7.4e-41)
		tmp = (y / z) * (x / z);
	else
		tmp = x / ((z * z) / y);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_, z_] := If[LessEqual[z, -4e+52], N[(y * N[(x / N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 7.4e-41], N[(N[(y / z), $MachinePrecision] * N[(x / z), $MachinePrecision]), $MachinePrecision], N[(x / N[(N[(z * z), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -4e52

   1. Initial program 76.4%

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

      1. *-commutative 76.4%

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

      2. associate-*r/ 79.0%

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

      3. sqr-neg 79.0%

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

      4. associate-*l* 79.0%

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

      5. associate-*l* 79.0%

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

      6. sqr-neg 79.0%

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

      7. associate-*l* 79.0%

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

      8. distribute-lft-in 79.0%

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

      9. fma-def 79.0%

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

      10. *-rgt-identity 79.0%

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

   3. Simplified 79.0%

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

   4. Taylor expanded in z around 0 67.1%

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

      1. unpow2 67.1%

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

   6. Simplified 67.1%

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

   if -4e52 < z < 7.4000000000000004e-41

   1. Initial program 80.5%

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

      1. associate-*l* 80.5%

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

      2. times-frac 97.5%

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

      3. associate-/r* 97.5%

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

      4. associate-*r/ 98.2%

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

   3. Simplified 98.2%

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

   4. Taylor expanded in z around 0 60.9%

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

      1. +-commutative 60.9%

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

      2. unpow2 60.9%

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

      3. times-frac 77.0%

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

      4. mul-1-neg 77.0%

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

      5. associate-*l/ 74.9%

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

      6. distribute-rgt-neg-in 74.9%

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

      7. distribute-lft-out 86.5%

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

   6. Simplified 86.5%

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

   7. Taylor expanded in z around 0 86.6%

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

   if 7.4000000000000004e-41 < z

   1. Initial program 86.3%

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

      1. *-commutative 86.3%

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

      2. associate-*r/ 91.3%

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

      3. sqr-neg 91.3%

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

      4. associate-*l* 91.3%

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

      5. associate-*l* 91.3%

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

      6. sqr-neg 91.3%

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

      7. associate-*l* 91.3%

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

      8. distribute-lft-in 91.3%

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

      9. fma-def 91.3%

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

      10. *-rgt-identity 91.3%

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

   3. Simplified 91.3%

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

   4. Taylor expanded in z around 0 71.3%

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

      1. unpow2 71.3%

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

   6. Simplified 71.3%

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

      1. associate-*r/ 66.4%

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

      2. *-commutative 66.4%

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

      3. associate-/l* 70.1%

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

   8. Applied egg-rr 70.1%

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

4. Final simplification 77.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -4 \cdot 10^{+52}:\\ \;\;\;\;y \cdot \frac{x}{z \cdot z}\\ \mathbf{elif}\;z \leq 7.4 \cdot 10^{-41}:\\ \;\;\;\;\frac{y}{z} \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z \cdot z}{y}}\\ \end{array} \]

Alternative 9: 96.8% 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 81.4%

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

   1. associate-*l* 81.4%

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

   2. times-frac 94.6%

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

   3. associate-/r* 97.3%

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

   4. associate-*r/ 98.6%

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

3. Simplified 98.6%

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

4. Final simplification 98.6%

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

Alternative 10: 78.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -2.2 \cdot 10^{+62}:\\ \;\;\;\;\frac{x}{z \cdot \frac{z}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z \cdot \frac{z}{x}}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= x -2.2e+62) (/ x (* z (/ z y))) (/ y (* z (/ z x)))))

C

assert(x < y);
double code(double x, double y, double z) {
	double tmp;
	if (x <= -2.2e+62) {
		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 (x <= (-2.2d+62)) 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 (x <= -2.2e+62) {
		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 x <= -2.2e+62:
		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 (x <= -2.2e+62)
		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 (x <= -2.2e+62)
		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[x, -2.2e+62], 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 x < -2.20000000000000015e62

   1. Initial program 77.9%

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

      1. *-commutative 77.9%

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

      2. associate-*r/ 90.1%

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

      3. sqr-neg 90.1%

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

      4. associate-*l* 90.1%

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

      5. associate-*l* 90.1%

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

      6. sqr-neg 90.1%

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

      7. associate-*l* 90.1%

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

      8. distribute-lft-in 90.1%

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

      9. fma-def 90.1%

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

      10. *-rgt-identity 90.1%

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

   3. Simplified 90.1%

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

   4. Taylor expanded in z around 0 64.9%

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

      1. unpow2 64.9%

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

   6. Simplified 64.9%

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

      1. associate-*r/ 59.6%

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

      2. frac-times 53.0%

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

      3. clear-num 53.0%

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

      4. frac-times 67.1%

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

      5. *-un-lft-identity 67.1%

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

   8. Applied egg-rr 67.1%

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

   if -2.20000000000000015e62 < x

   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/ 81.5%

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

      3. sqr-neg 81.5%

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

      4. associate-*l* 81.5%

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

      5. associate-*l* 81.5%

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

      6. sqr-neg 81.5%

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

      7. associate-*l* 81.5%

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

      8. distribute-lft-in 81.5%

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

      9. fma-def 81.5%

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

      10. *-rgt-identity 81.5%

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

   3. Simplified 81.5%

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

   4. Taylor expanded in z around 0 70.0%

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

      1. unpow2 70.0%

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

   6. Simplified 70.0%

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

      1. associate-*r/ 68.4%

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

      2. frac-times 74.2%

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

      3. clear-num 74.2%

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

      4. div-inv 74.2%

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

      5. associate-/l/ 76.6%

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

   8. Applied egg-rr 76.6%

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

4. Final simplification 75.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.2 \cdot 10^{+62}:\\ \;\;\;\;\frac{x}{z \cdot \frac{z}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z \cdot \frac{z}{x}}\\ \end{array} \]

Alternative 11: 40.0% accurate, 1.4× speedup

Math

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

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= z -2e-310) (* x (- (/ y z))) (* y (/ x z))))

C

assert(x < y);
double code(double x, double y, double z) {
	double tmp;
	if (z <= -2e-310) {
		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 (z <= (-2d-310)) 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 (z <= -2e-310) {
		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 z <= -2e-310:
		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 (z <= -2e-310)
		tmp = Float64(x * Float64(-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 (z <= -2e-310)
		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[z, -2e-310], N[(x * (-N[(y / z), $MachinePrecision])), $MachinePrecision], N[(y * N[(x / z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.999999999999994e-310

   1. Initial program 80.1%

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

      1. associate-*l* 80.1%

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

      2. times-frac 94.6%

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

      3. associate-/r* 96.8%

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

      4. associate-*r/ 98.9%

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

   3. Simplified 98.9%

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

   4. Taylor expanded in z around 0 51.5%

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

      1. +-commutative 51.5%

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

      2. unpow2 51.5%

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

      3. times-frac 59.6%

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

      4. mul-1-neg 59.6%

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

      5. associate-*l/ 62.8%

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

      6. distribute-rgt-neg-in 62.8%

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

      7. distribute-lft-out 62.8%

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

   6. Simplified 62.8%

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

   7. Taylor expanded in z around inf 31.3%

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

      1. mul-1-neg 31.3%

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

      2. associate-*l/ 36.5%

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

      3. associate-*l/ 31.3%

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

      4. associate-*r/ 36.5%

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

      5. distribute-rgt-neg-in 36.5%

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

   9. Simplified 36.5%

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

   if -1.999999999999994e-310 < z

   1. Initial program 82.8%

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

      1. associate-*l* 82.8%

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

      2. times-frac 94.6%

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

      3. associate-/r* 97.7%

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

      4. associate-*r/ 98.3%

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

   3. Simplified 98.3%

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

   4. Taylor expanded in z around 0 44.0%

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

      1. +-commutative 44.0%

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

      2. unpow2 44.0%

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

      3. times-frac 52.0%

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

      4. mul-1-neg 52.0%

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

      5. associate-*l/ 54.4%

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

      6. distribute-rgt-neg-in 54.4%

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

      7. distribute-lft-out 66.2%

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

   6. Simplified 66.2%

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

   7. Taylor expanded in z around inf 18.0%

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

      1. mul-1-neg 18.0%

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

      2. associate-*l/ 22.8%

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

      3. associate-*l/ 18.0%

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

      4. associate-*r/ 18.7%

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

      5. distribute-rgt-neg-in 18.7%

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

   9. Simplified 18.7%

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

       1. expm1-log1p-u 18.6%

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

       2. expm1-udef 33.3%

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

       3. add-sqr-sqrt 20.4%

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

       4. sqrt-unprod 41.0%

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

       5. sqr-neg 41.0%

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

       6. sqrt-unprod 24.7%

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

       7. add-sqr-sqrt 43.2%

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

   11. Applied egg-rr 43.2%

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

       1. expm1-def 29.3%

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

       2. expm1-log1p 33.6%

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

       3. associate-*r/ 30.6%

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

       4. associate-*l/ 37.6%

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

       5. *-commutative 37.6%

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

   13. Simplified 37.6%

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

4. Final simplification 37.1%

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

Alternative 12: 40.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -2 \cdot 10^{-310}:\\ \;\;\;\;y \cdot \frac{-x}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= z -2e-310) (* y (/ (- x) z)) (* y (/ x z))))

C

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

Python

[x, y] = sort([x, y])
def code(x, y, z):
	tmp = 0
	if z <= -2e-310:
		tmp = y * (-x / z)
	else:
		tmp = y * (x / z)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y, z)
	tmp = 0.0
	if (z <= -2e-310)
		tmp = Float64(y * Float64(Float64(-x) / 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 (z <= -2e-310)
		tmp = y * (-x / 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[z, -2e-310], N[(y * N[((-x) / z), $MachinePrecision]), $MachinePrecision], N[(y * N[(x / z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.999999999999994e-310

   1. Initial program 80.1%

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

      1. associate-*l* 80.1%

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

      2. times-frac 94.6%

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

      3. associate-/r* 96.8%

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

      4. associate-*r/ 98.9%

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

   3. Simplified 98.9%

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

   4. Taylor expanded in z around 0 51.5%

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

      1. +-commutative 51.5%

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

      2. unpow2 51.5%

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

      3. times-frac 59.6%

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

      4. mul-1-neg 59.6%

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

      5. associate-*l/ 62.8%

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

      6. distribute-rgt-neg-in 62.8%

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

      7. distribute-lft-out 62.8%

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

   6. Simplified 62.8%

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

   7. Taylor expanded in z around inf 36.5%

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

      1. mul-1-neg 36.5%

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

   9. Simplified 36.5%

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

   if -1.999999999999994e-310 < z

   1. Initial program 82.8%

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

      1. associate-*l* 82.8%

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

      2. times-frac 94.6%

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

      3. associate-/r* 97.7%

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

      4. associate-*r/ 98.3%

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

   3. Simplified 98.3%

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

   4. Taylor expanded in z around 0 44.0%

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

      1. +-commutative 44.0%

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

      2. unpow2 44.0%

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

      3. times-frac 52.0%

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

      4. mul-1-neg 52.0%

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

      5. associate-*l/ 54.4%

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

      6. distribute-rgt-neg-in 54.4%

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

      7. distribute-lft-out 66.2%

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

   6. Simplified 66.2%

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

   7. Taylor expanded in z around inf 18.0%

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

      1. mul-1-neg 18.0%

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

      2. associate-*l/ 22.8%

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

      3. associate-*l/ 18.0%

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

      4. associate-*r/ 18.7%

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

      5. distribute-rgt-neg-in 18.7%

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

   9. Simplified 18.7%

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

       1. expm1-log1p-u 18.6%

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

       2. expm1-udef 33.3%

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

       3. add-sqr-sqrt 20.4%

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

       4. sqrt-unprod 41.0%

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

       5. sqr-neg 41.0%

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

       6. sqrt-unprod 24.7%

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

       7. add-sqr-sqrt 43.2%

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

   11. Applied egg-rr 43.2%

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

       1. expm1-def 29.3%

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

       2. expm1-log1p 33.6%

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

       3. associate-*r/ 30.6%

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

       4. associate-*l/ 37.6%

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

       5. *-commutative 37.6%

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

   13. Simplified 37.6%

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

4. Final simplification 37.1%

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

Alternative 13: 40.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\frac{-x}{\frac{z}{y}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= z -2e-310) (/ (- x) (/ z y)) (* y (/ x z))))

C

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

Python

[x, y] = sort([x, y])
def code(x, y, z):
	tmp = 0
	if z <= -2e-310:
		tmp = -x / (z / y)
	else:
		tmp = y * (x / z)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y, z)
	tmp = 0.0
	if (z <= -2e-310)
		tmp = Float64(Float64(-x) / Float64(z / y));
	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 (z <= -2e-310)
		tmp = -x / (z / y);
	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[z, -2e-310], N[((-x) / N[(z / y), $MachinePrecision]), $MachinePrecision], N[(y * N[(x / z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.999999999999994e-310

   1. Initial program 80.1%

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

      1. associate-*l* 80.1%

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

      2. times-frac 94.6%

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

      3. associate-/r* 96.8%

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

      4. associate-*r/ 98.9%

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

   3. Simplified 98.9%

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

   4. Taylor expanded in z around 0 51.5%

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

      1. +-commutative 51.5%

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

      2. unpow2 51.5%

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

      3. times-frac 59.6%

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

      4. mul-1-neg 59.6%

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

      5. associate-*l/ 62.8%

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

      6. distribute-rgt-neg-in 62.8%

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

      7. distribute-lft-out 62.8%

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

   6. Simplified 62.8%

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

   7. Taylor expanded in z around inf 31.3%

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

      1. mul-1-neg 31.3%

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

      2. associate-*l/ 36.5%

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

      3. associate-*l/ 31.3%

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

      4. associate-*r/ 36.5%

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

      5. distribute-rgt-neg-in 36.5%

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

   9. Simplified 36.5%

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

       1. add-sqr-sqrt 24.6%

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

       2. sqrt-unprod 32.0%

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

       3. sqr-neg 32.0%

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

       4. sqrt-unprod 13.2%

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

       5. add-sqr-sqrt 18.8%

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

       6. clear-num 19.6%

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

       7. div-inv 19.6%

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

       8. frac-2neg 19.6%

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

       9. frac-2neg 19.6%

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

       10. add-sqr-sqrt 19.6%

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

       11. sqrt-unprod 29.2%

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

       12. sqr-neg 29.2%

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

       13. sqrt-unprod 0.0%

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

       14. add-sqr-sqrt 36.5%

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

       15. distribute-frac-neg 36.5%

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

       16. frac-2neg 36.5%

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

   11. Applied egg-rr 36.5%

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

   if -1.999999999999994e-310 < z

   1. Initial program 82.8%

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

      1. associate-*l* 82.8%

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

      2. times-frac 94.6%

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

      3. associate-/r* 97.7%

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

      4. associate-*r/ 98.3%

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

   3. Simplified 98.3%

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

   4. Taylor expanded in z around 0 44.0%

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

      1. +-commutative 44.0%

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

      2. unpow2 44.0%

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

      3. times-frac 52.0%

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

      4. mul-1-neg 52.0%

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

      5. associate-*l/ 54.4%

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

      6. distribute-rgt-neg-in 54.4%

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

      7. distribute-lft-out 66.2%

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

   6. Simplified 66.2%

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

   7. Taylor expanded in z around inf 18.0%

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

      1. mul-1-neg 18.0%

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

      2. associate-*l/ 22.8%

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

      3. associate-*l/ 18.0%

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

      4. associate-*r/ 18.7%

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

      5. distribute-rgt-neg-in 18.7%

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

   9. Simplified 18.7%

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

       1. expm1-log1p-u 18.6%

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

       2. expm1-udef 33.3%

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

       3. add-sqr-sqrt 20.4%

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

       4. sqrt-unprod 41.0%

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

       5. sqr-neg 41.0%

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

       6. sqrt-unprod 24.7%

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

       7. add-sqr-sqrt 43.2%

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

   11. Applied egg-rr 43.2%

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

       1. expm1-def 29.3%

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

       2. expm1-log1p 33.6%

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

       3. associate-*r/ 30.6%

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

       4. associate-*l/ 37.6%

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

       5. *-commutative 37.6%

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

   13. Simplified 37.6%

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

4. Final simplification 37.1%

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

Alternative 14: 72.2% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 81.4%

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

   1. *-commutative 81.4%

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

   2. associate-*r/ 82.8%

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

   3. sqr-neg 82.8%

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

   4. associate-*l* 82.8%

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

   5. associate-*l* 82.8%

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

   6. sqr-neg 82.8%

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

   7. associate-*l* 82.8%

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

   8. distribute-lft-in 82.8%

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

   9. fma-def 82.8%

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

   10. *-rgt-identity 82.8%

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

3. Simplified 82.8%

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

4. Taylor expanded in z around 0 69.2%

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

   1. unpow2 69.2%

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

6. Simplified 69.2%

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

7. Final simplification 69.2%

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

Alternative 15: 30.9% accurate, 2.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 81.4%

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

   1. associate-*l* 81.4%

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

   2. times-frac 94.6%

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

   3. associate-/r* 97.3%

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

   4. associate-*r/ 98.6%

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

3. Simplified 98.6%

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

4. Taylor expanded in z around 0 47.7%

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

   1. +-commutative 47.7%

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

   2. unpow2 47.7%

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

   3. times-frac 55.8%

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

   4. mul-1-neg 55.8%

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

   5. associate-*l/ 58.6%

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

   6. distribute-rgt-neg-in 58.6%

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

   7. distribute-lft-out 64.5%

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

6. Simplified 64.5%

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

7. Taylor expanded in z around inf 24.7%

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

   1. mul-1-neg 24.7%

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

   2. associate-*l/ 29.7%

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

   3. associate-*l/ 24.7%

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

   4. associate-*r/ 27.6%

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

   5. distribute-rgt-neg-in 27.6%

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

9. Simplified 27.6%

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

    1. expm1-log1p-u 23.0%

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

    2. expm1-udef 35.8%

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

    3. add-sqr-sqrt 23.1%

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

    4. sqrt-unprod 38.5%

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

    5. sqr-neg 38.5%

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

    6. sqrt-unprod 22.2%

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

    7. add-sqr-sqrt 36.4%

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

11. Applied egg-rr 36.4%

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

    1. expm1-def 23.8%

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

    2. expm1-log1p 26.2%

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

    3. associate-*r/ 23.6%

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

    4. associate-*l/ 28.6%

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

    5. *-commutative 28.6%

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

13. Simplified 28.6%

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

14. Taylor expanded in y around 0 23.6%

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

    1. associate-*r/ 26.2%

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

16. Simplified 26.2%

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

17. Final simplification 26.2%

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

Alternative 16: 31.1% 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 81.4%

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

   1. associate-*l* 81.4%

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

   2. times-frac 94.6%

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

   3. associate-/r* 97.3%

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

   4. associate-*r/ 98.6%

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

3. Simplified 98.6%

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

4. Taylor expanded in z around 0 47.7%

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

   1. +-commutative 47.7%

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

   2. unpow2 47.7%

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

   3. times-frac 55.8%

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

   4. mul-1-neg 55.8%

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

   5. associate-*l/ 58.6%

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

   6. distribute-rgt-neg-in 58.6%

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

   7. distribute-lft-out 64.5%

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

6. Simplified 64.5%

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

7. Taylor expanded in z around inf 24.7%

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

   1. mul-1-neg 24.7%

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

   2. associate-*l/ 29.7%

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

   3. associate-*l/ 24.7%

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

   4. associate-*r/ 27.6%

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

   5. distribute-rgt-neg-in 27.6%

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

9. Simplified 27.6%

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

    1. expm1-log1p-u 23.0%

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

    2. expm1-udef 35.8%

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

    3. add-sqr-sqrt 23.1%

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

    4. sqrt-unprod 38.5%

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

    5. sqr-neg 38.5%

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

    6. sqrt-unprod 22.2%

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

    7. add-sqr-sqrt 36.4%

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

11. Applied egg-rr 36.4%

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

    1. expm1-def 23.8%

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

    2. expm1-log1p 26.2%

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

    3. associate-*r/ 23.6%

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

    4. associate-*l/ 28.6%

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

    5. *-commutative 28.6%

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

13. Simplified 28.6%

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

14. Final simplification 28.6%

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

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 2023271 
(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))))