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

Percentage Accurate: 83.1% → 97.7%

Time: 13.0s

Alternatives: 8

Speedup: 0.6×

• 26.3% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 83.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 97.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -5.8 \cdot 10^{-29} \lor \neg \left(z \leq 2 \cdot 10^{-39}\right):\\ \;\;\;\;x\_m \cdot \frac{\frac{\frac{y\_m}{z}}{z}}{z + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{y\_m \cdot \frac{x\_m}{z}}{z}\\ \end{array}\right) \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
y_m = (fabs.f64 y)
y_s = (copysign.f64 1 y)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (y_s x_s x_m y_m z)
 :precision binary64
 (*
  y_s
  (*
   x_s
   (if (or (<= z -5.8e-29) (not (<= z 2e-39)))
     (* x_m (/ (/ (/ y_m z) z) (+ z 1.0)))
     (/ (* y_m (/ x_m z)) z)))))

C

x_m = fabs(x);
x_s = copysign(1.0, x);
y_m = fabs(y);
y_s = copysign(1.0, y);
assert(x_m < y_m && y_m < z);
double code(double y_s, double x_s, double x_m, double y_m, double z) {
	double tmp;
	if ((z <= -5.8e-29) || !(z <= 2e-39)) {
		tmp = x_m * (((y_m / z) / z) / (z + 1.0));
	} else {
		tmp = (y_m * (x_m / z)) / z;
	}
	return y_s * (x_s * tmp);
}

Fortran

x_m = abs(x)
x_s = copysign(1.0d0, x)
y_m = abs(y)
y_s = copysign(1.0d0, y)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
real(8) function code(y_s, x_s, x_m, y_m, z)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((z <= (-5.8d-29)) .or. (.not. (z <= 2d-39))) then
        tmp = x_m * (((y_m / z) / z) / (z + 1.0d0))
    else
        tmp = (y_m * (x_m / z)) / z
    end if
    code = y_s * (x_s * tmp)
end function

Java

x_m = Math.abs(x);
x_s = Math.copySign(1.0, x);
y_m = Math.abs(y);
y_s = Math.copySign(1.0, y);
assert x_m < y_m && y_m < z;
public static double code(double y_s, double x_s, double x_m, double y_m, double z) {
	double tmp;
	if ((z <= -5.8e-29) || !(z <= 2e-39)) {
		tmp = x_m * (((y_m / z) / z) / (z + 1.0));
	} else {
		tmp = (y_m * (x_m / z)) / z;
	}
	return y_s * (x_s * tmp);
}

Python

x_m = math.fabs(x)
x_s = math.copysign(1.0, x)
y_m = math.fabs(y)
y_s = math.copysign(1.0, y)
[x_m, y_m, z] = sort([x_m, y_m, z])
def code(y_s, x_s, x_m, y_m, z):
	tmp = 0
	if (z <= -5.8e-29) or not (z <= 2e-39):
		tmp = x_m * (((y_m / z) / z) / (z + 1.0))
	else:
		tmp = (y_m * (x_m / z)) / z
	return y_s * (x_s * tmp)

Julia

x_m = abs(x)
x_s = copysign(1.0, x)
y_m = abs(y)
y_s = copysign(1.0, y)
x_m, y_m, z = sort([x_m, y_m, z])
function code(y_s, x_s, x_m, y_m, z)
	tmp = 0.0
	if ((z <= -5.8e-29) || !(z <= 2e-39))
		tmp = Float64(x_m * Float64(Float64(Float64(y_m / z) / z) / Float64(z + 1.0)));
	else
		tmp = Float64(Float64(y_m * Float64(x_m / z)) / z);
	end
	return Float64(y_s * Float64(x_s * tmp))
end

MATLAB

x_m = abs(x);
x_s = sign(x) * abs(1.0);
y_m = abs(y);
y_s = sign(y) * abs(1.0);
x_m, y_m, z = num2cell(sort([x_m, y_m, z])){:}
function tmp_2 = code(y_s, x_s, x_m, y_m, z)
	tmp = 0.0;
	if ((z <= -5.8e-29) || ~((z <= 2e-39)))
		tmp = x_m * (((y_m / z) / z) / (z + 1.0));
	else
		tmp = (y_m * (x_m / z)) / z;
	end
	tmp_2 = y_s * (x_s * tmp);
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
x_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y_m = N[Abs[y], $MachinePrecision]
y_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
code[y$95$s_, x$95$s_, x$95$m_, y$95$m_, z_] := N[(y$95$s * N[(x$95$s * If[Or[LessEqual[z, -5.8e-29], N[Not[LessEqual[z, 2e-39]], $MachinePrecision]], N[(x$95$m * N[(N[(N[(y$95$m / z), $MachinePrecision] / z), $MachinePrecision] / N[(z + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y$95$m * N[(x$95$m / z), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < -5.80000000000000048e-29 or 1.99999999999999986e-39 < z

   1. Initial program 84.6%

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

      1. associate-/l* 88.3%

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

      2. sqr-neg 88.3%

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

      3. *-commutative 88.3%

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

      4. distribute-rgt1-in 62.8%

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

      5. sqr-neg 62.8%

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

      6. sqr-neg 62.8%

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

      7. cube-unmult 62.9%

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

   3. Simplified 62.9%

      \[\leadsto \color{blue}{x \cdot \frac{y}{z \cdot z + {z}^{3}}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. cube-mult 62.8%

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

      2. distribute-rgt1-in 88.3%

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

      3. *-commutative 88.3%

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

      4. add-sqr-sqrt 45.3%

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

      5. *-commutative 45.3%

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

      6. times-frac 47.9%

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

      7. pow2 47.9%

         \[\leadsto x \cdot \left(\frac{\sqrt{y}}{z + 1} \cdot \frac{\sqrt{y}}{\color{blue}{{z}^{2}}}\right) \]

   6. Applied egg-rr 47.9%

      \[\leadsto x \cdot \color{blue}{\left(\frac{\sqrt{y}}{z + 1} \cdot \frac{\sqrt{y}}{{z}^{2}}\right)} \]
   7. Step-by-step derivation

      1. associate-*l/ 47.9%

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

      2. associate-*r/ 48.0%

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

      3. rem-square-sqrt 91.6%

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

   8. Simplified 91.6%

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

      1. pow2 91.6%

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

      2. associate-/r* 93.6%

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

      3. div-inv 93.6%

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

   10. Applied egg-rr 93.6%

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

       1. associate-*r/ 93.6%

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

       2. *-rgt-identity 93.6%

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

   12. Simplified 93.6%

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

   if -5.80000000000000048e-29 < z < 1.99999999999999986e-39

   1. Initial program 85.3%

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

      1. *-commutative 85.3%

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

      2. frac-times 81.9%

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

      3. associate-*l/ 85.3%

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

      4. times-frac 96.5%

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

   4. Applied egg-rr 96.5%

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

   5. Taylor expanded in z around 0 96.5%

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

      1. associate-*l/ 98.7%

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

   7. Applied egg-rr 98.7%

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

4. Final simplification 95.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -5.8 \cdot 10^{-29} \lor \neg \left(z \leq 2 \cdot 10^{-39}\right):\\ \;\;\;\;x \cdot \frac{\frac{\frac{y}{z}}{z}}{z + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot \frac{x}{z}}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 97.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{x\_m \cdot y\_m}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \leq 10^{-160}:\\ \;\;\;\;x\_m \cdot \frac{\frac{\frac{y\_m}{z}}{z}}{z + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y\_m}{z} \cdot \frac{x\_m}{z + 1}}{z}\\ \end{array}\right) \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
y_m = (fabs.f64 y)
y_s = (copysign.f64 1 y)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (y_s x_s x_m y_m z)
 :precision binary64
 (*
  y_s
  (*
   x_s
   (if (<= (/ (* x_m y_m) (* (* z z) (+ z 1.0))) 1e-160)
     (* x_m (/ (/ (/ y_m z) z) (+ z 1.0)))
     (/ (* (/ y_m z) (/ x_m (+ z 1.0))) z)))))

C

x_m = fabs(x);
x_s = copysign(1.0, x);
y_m = fabs(y);
y_s = copysign(1.0, y);
assert(x_m < y_m && y_m < z);
double code(double y_s, double x_s, double x_m, double y_m, double z) {
	double tmp;
	if (((x_m * y_m) / ((z * z) * (z + 1.0))) <= 1e-160) {
		tmp = x_m * (((y_m / z) / z) / (z + 1.0));
	} else {
		tmp = ((y_m / z) * (x_m / (z + 1.0))) / z;
	}
	return y_s * (x_s * tmp);
}

Fortran

x_m = abs(x)
x_s = copysign(1.0d0, x)
y_m = abs(y)
y_s = copysign(1.0d0, y)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
real(8) function code(y_s, x_s, x_m, y_m, z)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if (((x_m * y_m) / ((z * z) * (z + 1.0d0))) <= 1d-160) then
        tmp = x_m * (((y_m / z) / z) / (z + 1.0d0))
    else
        tmp = ((y_m / z) * (x_m / (z + 1.0d0))) / z
    end if
    code = y_s * (x_s * tmp)
end function

Java

x_m = Math.abs(x);
x_s = Math.copySign(1.0, x);
y_m = Math.abs(y);
y_s = Math.copySign(1.0, y);
assert x_m < y_m && y_m < z;
public static double code(double y_s, double x_s, double x_m, double y_m, double z) {
	double tmp;
	if (((x_m * y_m) / ((z * z) * (z + 1.0))) <= 1e-160) {
		tmp = x_m * (((y_m / z) / z) / (z + 1.0));
	} else {
		tmp = ((y_m / z) * (x_m / (z + 1.0))) / z;
	}
	return y_s * (x_s * tmp);
}

Python

x_m = math.fabs(x)
x_s = math.copysign(1.0, x)
y_m = math.fabs(y)
y_s = math.copysign(1.0, y)
[x_m, y_m, z] = sort([x_m, y_m, z])
def code(y_s, x_s, x_m, y_m, z):
	tmp = 0
	if ((x_m * y_m) / ((z * z) * (z + 1.0))) <= 1e-160:
		tmp = x_m * (((y_m / z) / z) / (z + 1.0))
	else:
		tmp = ((y_m / z) * (x_m / (z + 1.0))) / z
	return y_s * (x_s * tmp)

Julia

x_m = abs(x)
x_s = copysign(1.0, x)
y_m = abs(y)
y_s = copysign(1.0, y)
x_m, y_m, z = sort([x_m, y_m, z])
function code(y_s, x_s, x_m, y_m, z)
	tmp = 0.0
	if (Float64(Float64(x_m * y_m) / Float64(Float64(z * z) * Float64(z + 1.0))) <= 1e-160)
		tmp = Float64(x_m * Float64(Float64(Float64(y_m / z) / z) / Float64(z + 1.0)));
	else
		tmp = Float64(Float64(Float64(y_m / z) * Float64(x_m / Float64(z + 1.0))) / z);
	end
	return Float64(y_s * Float64(x_s * tmp))
end

MATLAB

x_m = abs(x);
x_s = sign(x) * abs(1.0);
y_m = abs(y);
y_s = sign(y) * abs(1.0);
x_m, y_m, z = num2cell(sort([x_m, y_m, z])){:}
function tmp_2 = code(y_s, x_s, x_m, y_m, z)
	tmp = 0.0;
	if (((x_m * y_m) / ((z * z) * (z + 1.0))) <= 1e-160)
		tmp = x_m * (((y_m / z) / z) / (z + 1.0));
	else
		tmp = ((y_m / z) * (x_m / (z + 1.0))) / z;
	end
	tmp_2 = y_s * (x_s * tmp);
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
x_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y_m = N[Abs[y], $MachinePrecision]
y_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
code[y$95$s_, x$95$s_, x$95$m_, y$95$m_, z_] := N[(y$95$s * N[(x$95$s * If[LessEqual[N[(N[(x$95$m * y$95$m), $MachinePrecision] / N[(N[(z * z), $MachinePrecision] * N[(z + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1e-160], N[(x$95$m * N[(N[(N[(y$95$m / z), $MachinePrecision] / z), $MachinePrecision] / N[(z + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(y$95$m / z), $MachinePrecision] * N[(x$95$m / N[(z + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 x y) (*.f64 (*.f64 z z) (+.f64 z 1))) < 9.9999999999999999e-161

   1. Initial program 91.2%

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

      1. associate-/l* 91.7%

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

      2. sqr-neg 91.7%

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

      3. *-commutative 91.7%

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

      4. distribute-rgt1-in 73.1%

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

      5. sqr-neg 73.1%

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

      6. sqr-neg 73.1%

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

      7. cube-unmult 73.1%

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

   3. Simplified 73.1%

      \[\leadsto \color{blue}{x \cdot \frac{y}{z \cdot z + {z}^{3}}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. cube-mult 73.1%

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

      2. distribute-rgt1-in 91.7%

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

      3. *-commutative 91.7%

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

      4. add-sqr-sqrt 47.7%

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

      5. *-commutative 47.7%

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

      6. times-frac 48.2%

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

      7. pow2 48.2%

         \[\leadsto x \cdot \left(\frac{\sqrt{y}}{z + 1} \cdot \frac{\sqrt{y}}{\color{blue}{{z}^{2}}}\right) \]

   6. Applied egg-rr 48.2%

      \[\leadsto x \cdot \color{blue}{\left(\frac{\sqrt{y}}{z + 1} \cdot \frac{\sqrt{y}}{{z}^{2}}\right)} \]
   7. Step-by-step derivation

      1. associate-*l/ 48.3%

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

      2. associate-*r/ 48.3%

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

      3. rem-square-sqrt 92.8%

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

   8. Simplified 92.8%

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

      1. pow2 92.8%

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

      2. associate-/r* 95.0%

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

      3. div-inv 94.9%

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

   10. Applied egg-rr 94.9%

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

       1. associate-*r/ 95.0%

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

       2. *-rgt-identity 95.0%

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

   12. Simplified 95.0%

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

   if 9.9999999999999999e-161 < (/.f64 (*.f64 x y) (*.f64 (*.f64 z z) (+.f64 z 1)))

   1. Initial program 72.0%

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

      1. associate-/l* 72.5%

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

      2. sqr-neg 72.5%

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

      3. *-commutative 72.5%

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

      4. distribute-rgt1-in 67.9%

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

      5. sqr-neg 67.9%

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

      6. sqr-neg 67.9%

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

      7. cube-unmult 67.9%

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

   3. Simplified 67.9%

      \[\leadsto \color{blue}{x \cdot \frac{y}{z \cdot z + {z}^{3}}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-*r/ 72.0%

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

      2. *-commutative 72.0%

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

      3. cube-mult 72.0%

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

      4. distribute-rgt1-in 72.0%

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

      5. *-commutative 72.0%

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

      6. frac-times 75.9%

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

      7. associate-/r* 88.2%

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

      8. associate-*l/ 96.4%

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

   6. Applied egg-rr 96.4%

      \[\leadsto \color{blue}{\frac{\frac{y}{z} \cdot \frac{x}{z + 1}}{z}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 95.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \leq 10^{-160}:\\ \;\;\;\;x \cdot \frac{\frac{\frac{y}{z}}{z}}{z + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{z} \cdot \frac{x}{z + 1}}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 97.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{x\_m \cdot y\_m}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \leq 2 \cdot 10^{-99}:\\ \;\;\;\;x\_m \cdot \frac{\frac{\frac{y\_m}{z}}{z}}{z + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{y\_m}{z} \cdot \frac{\frac{x\_m}{z + 1}}{z}\\ \end{array}\right) \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
y_m = (fabs.f64 y)
y_s = (copysign.f64 1 y)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (y_s x_s x_m y_m z)
 :precision binary64
 (*
  y_s
  (*
   x_s
   (if (<= (/ (* x_m y_m) (* (* z z) (+ z 1.0))) 2e-99)
     (* x_m (/ (/ (/ y_m z) z) (+ z 1.0)))
     (* (/ y_m z) (/ (/ x_m (+ z 1.0)) z))))))

C

x_m = fabs(x);
x_s = copysign(1.0, x);
y_m = fabs(y);
y_s = copysign(1.0, y);
assert(x_m < y_m && y_m < z);
double code(double y_s, double x_s, double x_m, double y_m, double z) {
	double tmp;
	if (((x_m * y_m) / ((z * z) * (z + 1.0))) <= 2e-99) {
		tmp = x_m * (((y_m / z) / z) / (z + 1.0));
	} else {
		tmp = (y_m / z) * ((x_m / (z + 1.0)) / z);
	}
	return y_s * (x_s * tmp);
}

Fortran

x_m = abs(x)
x_s = copysign(1.0d0, x)
y_m = abs(y)
y_s = copysign(1.0d0, y)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
real(8) function code(y_s, x_s, x_m, y_m, z)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if (((x_m * y_m) / ((z * z) * (z + 1.0d0))) <= 2d-99) then
        tmp = x_m * (((y_m / z) / z) / (z + 1.0d0))
    else
        tmp = (y_m / z) * ((x_m / (z + 1.0d0)) / z)
    end if
    code = y_s * (x_s * tmp)
end function

Java

x_m = Math.abs(x);
x_s = Math.copySign(1.0, x);
y_m = Math.abs(y);
y_s = Math.copySign(1.0, y);
assert x_m < y_m && y_m < z;
public static double code(double y_s, double x_s, double x_m, double y_m, double z) {
	double tmp;
	if (((x_m * y_m) / ((z * z) * (z + 1.0))) <= 2e-99) {
		tmp = x_m * (((y_m / z) / z) / (z + 1.0));
	} else {
		tmp = (y_m / z) * ((x_m / (z + 1.0)) / z);
	}
	return y_s * (x_s * tmp);
}

Python

x_m = math.fabs(x)
x_s = math.copysign(1.0, x)
y_m = math.fabs(y)
y_s = math.copysign(1.0, y)
[x_m, y_m, z] = sort([x_m, y_m, z])
def code(y_s, x_s, x_m, y_m, z):
	tmp = 0
	if ((x_m * y_m) / ((z * z) * (z + 1.0))) <= 2e-99:
		tmp = x_m * (((y_m / z) / z) / (z + 1.0))
	else:
		tmp = (y_m / z) * ((x_m / (z + 1.0)) / z)
	return y_s * (x_s * tmp)

Julia

x_m = abs(x)
x_s = copysign(1.0, x)
y_m = abs(y)
y_s = copysign(1.0, y)
x_m, y_m, z = sort([x_m, y_m, z])
function code(y_s, x_s, x_m, y_m, z)
	tmp = 0.0
	if (Float64(Float64(x_m * y_m) / Float64(Float64(z * z) * Float64(z + 1.0))) <= 2e-99)
		tmp = Float64(x_m * Float64(Float64(Float64(y_m / z) / z) / Float64(z + 1.0)));
	else
		tmp = Float64(Float64(y_m / z) * Float64(Float64(x_m / Float64(z + 1.0)) / z));
	end
	return Float64(y_s * Float64(x_s * tmp))
end

MATLAB

x_m = abs(x);
x_s = sign(x) * abs(1.0);
y_m = abs(y);
y_s = sign(y) * abs(1.0);
x_m, y_m, z = num2cell(sort([x_m, y_m, z])){:}
function tmp_2 = code(y_s, x_s, x_m, y_m, z)
	tmp = 0.0;
	if (((x_m * y_m) / ((z * z) * (z + 1.0))) <= 2e-99)
		tmp = x_m * (((y_m / z) / z) / (z + 1.0));
	else
		tmp = (y_m / z) * ((x_m / (z + 1.0)) / z);
	end
	tmp_2 = y_s * (x_s * tmp);
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
x_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y_m = N[Abs[y], $MachinePrecision]
y_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
code[y$95$s_, x$95$s_, x$95$m_, y$95$m_, z_] := N[(y$95$s * N[(x$95$s * If[LessEqual[N[(N[(x$95$m * y$95$m), $MachinePrecision] / N[(N[(z * z), $MachinePrecision] * N[(z + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2e-99], N[(x$95$m * N[(N[(N[(y$95$m / z), $MachinePrecision] / z), $MachinePrecision] / N[(z + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y$95$m / z), $MachinePrecision] * N[(N[(x$95$m / N[(z + 1.0), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 x y) (*.f64 (*.f64 z z) (+.f64 z 1))) < 2e-99

   1. Initial program 91.3%

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

      1. associate-/l* 91.8%

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

      2. sqr-neg 91.8%

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

      3. *-commutative 91.8%

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

      4. distribute-rgt1-in 73.5%

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

      5. sqr-neg 73.5%

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

      6. sqr-neg 73.5%

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

      7. cube-unmult 73.5%

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

   3. Simplified 73.5%

      \[\leadsto \color{blue}{x \cdot \frac{y}{z \cdot z + {z}^{3}}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. cube-mult 73.5%

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

      2. distribute-rgt1-in 91.8%

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

      3. *-commutative 91.8%

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

      4. add-sqr-sqrt 47.5%

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

      5. *-commutative 47.5%

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

      6. times-frac 48.0%

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

      7. pow2 48.0%

         \[\leadsto x \cdot \left(\frac{\sqrt{y}}{z + 1} \cdot \frac{\sqrt{y}}{\color{blue}{{z}^{2}}}\right) \]

   6. Applied egg-rr 48.0%

      \[\leadsto x \cdot \color{blue}{\left(\frac{\sqrt{y}}{z + 1} \cdot \frac{\sqrt{y}}{{z}^{2}}\right)} \]
   7. Step-by-step derivation

      1. associate-*l/ 48.0%

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

      2. associate-*r/ 48.0%

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

      3. rem-square-sqrt 92.9%

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

   8. Simplified 92.9%

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

      1. pow2 92.9%

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

      2. associate-/r* 95.1%

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

      3. div-inv 95.0%

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

   10. Applied egg-rr 95.0%

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

       1. associate-*r/ 95.1%

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

       2. *-rgt-identity 95.1%

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

   12. Simplified 95.1%

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

   if 2e-99 < (/.f64 (*.f64 x y) (*.f64 (*.f64 z z) (+.f64 z 1)))

   1. Initial program 71.0%

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

      1. *-commutative 71.0%

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

      2. frac-times 75.0%

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

      3. associate-*l/ 82.0%

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

      4. times-frac 92.8%

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

   4. Applied egg-rr 92.8%

      \[\leadsto \color{blue}{\frac{y}{z} \cdot \frac{\frac{x}{z + 1}}{z}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 94.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \leq 2 \cdot 10^{-99}:\\ \;\;\;\;x \cdot \frac{\frac{\frac{y}{z}}{z}}{z + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z} \cdot \frac{\frac{x}{z + 1}}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 95.5% accurate, 0.6× speedup

Math

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

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
y_m = (fabs.f64 y)
y_s = (copysign.f64 1 y)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (y_s x_s x_m y_m z)
 :precision binary64
 (*
  y_s
  (*
   x_s
   (if (or (<= z -1.0) (not (<= z 1.0)))
     (* (/ y_m z) (/ (/ x_m z) z))
     (/ (* y_m (/ x_m z)) z)))))

C

x_m = fabs(x);
x_s = copysign(1.0, x);
y_m = fabs(y);
y_s = copysign(1.0, y);
assert(x_m < y_m && y_m < z);
double code(double y_s, double x_s, double x_m, double y_m, double z) {
	double tmp;
	if ((z <= -1.0) || !(z <= 1.0)) {
		tmp = (y_m / z) * ((x_m / z) / z);
	} else {
		tmp = (y_m * (x_m / z)) / z;
	}
	return y_s * (x_s * tmp);
}

Fortran

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

Java

x_m = Math.abs(x);
x_s = Math.copySign(1.0, x);
y_m = Math.abs(y);
y_s = Math.copySign(1.0, y);
assert x_m < y_m && y_m < z;
public static double code(double y_s, double x_s, double x_m, double y_m, double z) {
	double tmp;
	if ((z <= -1.0) || !(z <= 1.0)) {
		tmp = (y_m / z) * ((x_m / z) / z);
	} else {
		tmp = (y_m * (x_m / z)) / z;
	}
	return y_s * (x_s * tmp);
}

Python

x_m = math.fabs(x)
x_s = math.copysign(1.0, x)
y_m = math.fabs(y)
y_s = math.copysign(1.0, y)
[x_m, y_m, z] = sort([x_m, y_m, z])
def code(y_s, x_s, x_m, y_m, z):
	tmp = 0
	if (z <= -1.0) or not (z <= 1.0):
		tmp = (y_m / z) * ((x_m / z) / z)
	else:
		tmp = (y_m * (x_m / z)) / z
	return y_s * (x_s * tmp)

Julia

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

MATLAB

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

Wolfram

x_m = N[Abs[x], $MachinePrecision]
x_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y_m = N[Abs[y], $MachinePrecision]
y_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
code[y$95$s_, x$95$s_, x$95$m_, y$95$m_, z_] := N[(y$95$s * N[(x$95$s * If[Or[LessEqual[z, -1.0], N[Not[LessEqual[z, 1.0]], $MachinePrecision]], N[(N[(y$95$m / z), $MachinePrecision] * N[(N[(x$95$m / z), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], N[(N[(y$95$m * N[(x$95$m / z), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < -1 or 1 < z

   1. Initial program 84.5%

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

      1. *-commutative 84.5%

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

      2. frac-times 90.4%

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

      3. associate-*l/ 92.9%

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

      4. times-frac 95.8%

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

   4. Applied egg-rr 95.8%

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

   5. Taylor expanded in z around inf 94.7%

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

   if -1 < z < 1

   1. Initial program 85.3%

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

      1. *-commutative 85.3%

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

      2. frac-times 83.8%

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

      3. associate-*l/ 85.3%

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

      4. times-frac 96.8%

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

   4. Applied egg-rr 96.8%

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

   5. Taylor expanded in z around 0 95.2%

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

      1. associate-*l/ 96.4%

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

   7. Applied egg-rr 96.4%

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

4. Final simplification 95.6%

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

Alternative 5: 97.1% accurate, 0.6× speedup

Math

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

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
y_m = (fabs.f64 y)
y_s = (copysign.f64 1 y)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (y_s x_s x_m y_m z)
 :precision binary64
 (*
  y_s
  (*
   x_s
   (if (or (<= z -1.0) (not (<= z 1.0)))
     (* (/ (/ y_m z) z) (/ x_m z))
     (/ (* y_m (/ x_m z)) z)))))

C

x_m = fabs(x);
x_s = copysign(1.0, x);
y_m = fabs(y);
y_s = copysign(1.0, y);
assert(x_m < y_m && y_m < z);
double code(double y_s, double x_s, double x_m, double y_m, double z) {
	double tmp;
	if ((z <= -1.0) || !(z <= 1.0)) {
		tmp = ((y_m / z) / z) * (x_m / z);
	} else {
		tmp = (y_m * (x_m / z)) / z;
	}
	return y_s * (x_s * tmp);
}

Fortran

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

Java

x_m = Math.abs(x);
x_s = Math.copySign(1.0, x);
y_m = Math.abs(y);
y_s = Math.copySign(1.0, y);
assert x_m < y_m && y_m < z;
public static double code(double y_s, double x_s, double x_m, double y_m, double z) {
	double tmp;
	if ((z <= -1.0) || !(z <= 1.0)) {
		tmp = ((y_m / z) / z) * (x_m / z);
	} else {
		tmp = (y_m * (x_m / z)) / z;
	}
	return y_s * (x_s * tmp);
}

Python

x_m = math.fabs(x)
x_s = math.copysign(1.0, x)
y_m = math.fabs(y)
y_s = math.copysign(1.0, y)
[x_m, y_m, z] = sort([x_m, y_m, z])
def code(y_s, x_s, x_m, y_m, z):
	tmp = 0
	if (z <= -1.0) or not (z <= 1.0):
		tmp = ((y_m / z) / z) * (x_m / z)
	else:
		tmp = (y_m * (x_m / z)) / z
	return y_s * (x_s * tmp)

Julia

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

MATLAB

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

Wolfram

x_m = N[Abs[x], $MachinePrecision]
x_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y_m = N[Abs[y], $MachinePrecision]
y_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
code[y$95$s_, x$95$s_, x$95$m_, y$95$m_, z_] := N[(y$95$s * N[(x$95$s * If[Or[LessEqual[z, -1.0], N[Not[LessEqual[z, 1.0]], $MachinePrecision]], N[(N[(N[(y$95$m / z), $MachinePrecision] / z), $MachinePrecision] * N[(x$95$m / z), $MachinePrecision]), $MachinePrecision], N[(N[(y$95$m * N[(x$95$m / z), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < -1 or 1 < z

   1. Initial program 84.5%

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

      1. *-commutative 84.5%

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

      2. sqr-neg 84.5%

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

      3. times-frac 90.4%

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

      4. sqr-neg 90.4%

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

   3. Simplified 90.4%

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

      1. associate-/r* 94.8%

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

      2. div-inv 94.8%

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

   6. Applied egg-rr 94.8%

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

      1. associate-*r/ 94.8%

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

      2. *-rgt-identity 94.8%

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

   8. Simplified 94.8%

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

   9. Taylor expanded in z around inf 93.7%

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

   if -1 < z < 1

   1. Initial program 85.3%

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

      1. *-commutative 85.3%

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

      2. frac-times 83.8%

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

      3. associate-*l/ 85.3%

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

      4. times-frac 96.8%

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

   4. Applied egg-rr 96.8%

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

   5. Taylor expanded in z around 0 95.2%

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

      1. associate-*l/ 96.4%

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

   7. Applied egg-rr 96.4%

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

4. Final simplification 95.1%

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

Alternative 6: 96.8% accurate, 0.6× speedup

Math

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

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
y_m = (fabs.f64 y)
y_s = (copysign.f64 1 y)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (y_s x_s x_m y_m z)
 :precision binary64
 (*
  y_s
  (*
   x_s
   (if (or (<= z -1.0) (not (<= z 1.0)))
     (/ (* (/ y_m z) (/ x_m z)) z)
     (/ (* y_m (/ x_m z)) z)))))

C

x_m = fabs(x);
x_s = copysign(1.0, x);
y_m = fabs(y);
y_s = copysign(1.0, y);
assert(x_m < y_m && y_m < z);
double code(double y_s, double x_s, double x_m, double y_m, double z) {
	double tmp;
	if ((z <= -1.0) || !(z <= 1.0)) {
		tmp = ((y_m / z) * (x_m / z)) / z;
	} else {
		tmp = (y_m * (x_m / z)) / z;
	}
	return y_s * (x_s * tmp);
}

Fortran

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

Java

x_m = Math.abs(x);
x_s = Math.copySign(1.0, x);
y_m = Math.abs(y);
y_s = Math.copySign(1.0, y);
assert x_m < y_m && y_m < z;
public static double code(double y_s, double x_s, double x_m, double y_m, double z) {
	double tmp;
	if ((z <= -1.0) || !(z <= 1.0)) {
		tmp = ((y_m / z) * (x_m / z)) / z;
	} else {
		tmp = (y_m * (x_m / z)) / z;
	}
	return y_s * (x_s * tmp);
}

Python

x_m = math.fabs(x)
x_s = math.copysign(1.0, x)
y_m = math.fabs(y)
y_s = math.copysign(1.0, y)
[x_m, y_m, z] = sort([x_m, y_m, z])
def code(y_s, x_s, x_m, y_m, z):
	tmp = 0
	if (z <= -1.0) or not (z <= 1.0):
		tmp = ((y_m / z) * (x_m / z)) / z
	else:
		tmp = (y_m * (x_m / z)) / z
	return y_s * (x_s * tmp)

Julia

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

MATLAB

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

Wolfram

x_m = N[Abs[x], $MachinePrecision]
x_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y_m = N[Abs[y], $MachinePrecision]
y_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
code[y$95$s_, x$95$s_, x$95$m_, y$95$m_, z_] := N[(y$95$s * N[(x$95$s * If[Or[LessEqual[z, -1.0], N[Not[LessEqual[z, 1.0]], $MachinePrecision]], N[(N[(N[(y$95$m / z), $MachinePrecision] * N[(x$95$m / z), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], N[(N[(y$95$m * N[(x$95$m / z), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < -1 or 1 < z

   1. Initial program 84.5%

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

      1. associate-/l* 87.0%

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

      2. sqr-neg 87.0%

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

      3. *-commutative 87.0%

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

      4. distribute-rgt1-in 58.8%

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

      5. sqr-neg 58.8%

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

      6. sqr-neg 58.8%

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

      7. cube-unmult 58.8%

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

   3. Simplified 58.8%

      \[\leadsto \color{blue}{x \cdot \frac{y}{z \cdot z + {z}^{3}}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-*r/ 59.3%

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

      2. *-commutative 59.3%

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

      3. cube-mult 59.2%

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

      4. distribute-rgt1-in 84.5%

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

      5. *-commutative 84.5%

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

      6. frac-times 90.4%

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

      7. associate-/r* 94.8%

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

      8. associate-*l/ 97.6%

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

   6. Applied egg-rr 97.6%

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

   7. Taylor expanded in z around inf 96.6%

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

   if -1 < z < 1

   1. Initial program 85.3%

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

      1. *-commutative 85.3%

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

      2. frac-times 83.8%

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

      3. associate-*l/ 85.3%

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

      4. times-frac 96.8%

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

   4. Applied egg-rr 96.8%

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

   5. Taylor expanded in z around 0 95.2%

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

      1. associate-*l/ 96.4%

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

   7. Applied egg-rr 96.4%

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

4. Final simplification 96.5%

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

Alternative 7: 95.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -1:\\ \;\;\;\;\frac{x\_m}{z} \cdot \frac{y\_m}{z \cdot z}\\ \mathbf{elif}\;z \leq 1:\\ \;\;\;\;\frac{y\_m \cdot \frac{x\_m}{z}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y\_m}{z} \cdot \frac{\frac{x\_m}{z}}{z}\\ \end{array}\right) \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
y_m = (fabs.f64 y)
y_s = (copysign.f64 1 y)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (y_s x_s x_m y_m z)
 :precision binary64
 (*
  y_s
  (*
   x_s
   (if (<= z -1.0)
     (* (/ x_m z) (/ y_m (* z z)))
     (if (<= z 1.0) (/ (* y_m (/ x_m z)) z) (* (/ y_m z) (/ (/ x_m z) z)))))))

C

x_m = fabs(x);
x_s = copysign(1.0, x);
y_m = fabs(y);
y_s = copysign(1.0, y);
assert(x_m < y_m && y_m < z);
double code(double y_s, double x_s, double x_m, double y_m, double z) {
	double tmp;
	if (z <= -1.0) {
		tmp = (x_m / z) * (y_m / (z * z));
	} else if (z <= 1.0) {
		tmp = (y_m * (x_m / z)) / z;
	} else {
		tmp = (y_m / z) * ((x_m / z) / z);
	}
	return y_s * (x_s * tmp);
}

Fortran

x_m = abs(x)
x_s = copysign(1.0d0, x)
y_m = abs(y)
y_s = copysign(1.0d0, y)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
real(8) function code(y_s, x_s, x_m, y_m, z)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if (z <= (-1.0d0)) then
        tmp = (x_m / z) * (y_m / (z * z))
    else if (z <= 1.0d0) then
        tmp = (y_m * (x_m / z)) / z
    else
        tmp = (y_m / z) * ((x_m / z) / z)
    end if
    code = y_s * (x_s * tmp)
end function

Java

x_m = Math.abs(x);
x_s = Math.copySign(1.0, x);
y_m = Math.abs(y);
y_s = Math.copySign(1.0, y);
assert x_m < y_m && y_m < z;
public static double code(double y_s, double x_s, double x_m, double y_m, double z) {
	double tmp;
	if (z <= -1.0) {
		tmp = (x_m / z) * (y_m / (z * z));
	} else if (z <= 1.0) {
		tmp = (y_m * (x_m / z)) / z;
	} else {
		tmp = (y_m / z) * ((x_m / z) / z);
	}
	return y_s * (x_s * tmp);
}

Python

x_m = math.fabs(x)
x_s = math.copysign(1.0, x)
y_m = math.fabs(y)
y_s = math.copysign(1.0, y)
[x_m, y_m, z] = sort([x_m, y_m, z])
def code(y_s, x_s, x_m, y_m, z):
	tmp = 0
	if z <= -1.0:
		tmp = (x_m / z) * (y_m / (z * z))
	elif z <= 1.0:
		tmp = (y_m * (x_m / z)) / z
	else:
		tmp = (y_m / z) * ((x_m / z) / z)
	return y_s * (x_s * tmp)

Julia

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

MATLAB

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

Wolfram

x_m = N[Abs[x], $MachinePrecision]
x_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y_m = N[Abs[y], $MachinePrecision]
y_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
code[y$95$s_, x$95$s_, x$95$m_, y$95$m_, z_] := N[(y$95$s * N[(x$95$s * If[LessEqual[z, -1.0], N[(N[(x$95$m / z), $MachinePrecision] * N[(y$95$m / N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 1.0], N[(N[(y$95$m * N[(x$95$m / z), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], N[(N[(y$95$m / z), $MachinePrecision] * N[(N[(x$95$m / z), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if z < -1

   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. sqr-neg 79.7%

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

      3. times-frac 91.6%

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

      4. sqr-neg 91.6%

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

   3. Simplified 91.6%

      \[\leadsto \color{blue}{\frac{y}{z \cdot z} \cdot \frac{x}{z + 1}} \]
   4. Add Preprocessing

   5. Taylor expanded in z around inf 90.4%

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

   if -1 < z < 1

   1. Initial program 85.3%

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

      1. *-commutative 85.3%

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

      2. frac-times 83.8%

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

      3. associate-*l/ 85.3%

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

      4. times-frac 96.8%

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

   4. Applied egg-rr 96.8%

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

   5. Taylor expanded in z around 0 95.2%

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

      1. associate-*l/ 96.4%

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

   7. Applied egg-rr 96.4%

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

   if 1 < z

   1. Initial program 90.8%

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

      1. *-commutative 90.8%

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

      2. frac-times 88.8%

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

      3. associate-*l/ 92.9%

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

      4. times-frac 95.4%

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

   4. Applied egg-rr 95.4%

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

   5. Taylor expanded in z around inf 94.5%

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

4. Final simplification 94.3%

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

Alternative 8: 80.5% accurate, 1.6× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ y\_s \cdot \left(x\_s \cdot \left(y\_m \cdot \frac{\frac{x\_m}{z}}{z}\right)\right) \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
y_m = (fabs.f64 y)
y_s = (copysign.f64 1 y)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (y_s x_s x_m y_m z)
 :precision binary64
 (* y_s (* x_s (* y_m (/ (/ x_m z) z)))))

C

x_m = fabs(x);
x_s = copysign(1.0, x);
y_m = fabs(y);
y_s = copysign(1.0, y);
assert(x_m < y_m && y_m < z);
double code(double y_s, double x_s, double x_m, double y_m, double z) {
	return y_s * (x_s * (y_m * ((x_m / z) / z)));
}

Fortran

x_m = abs(x)
x_s = copysign(1.0d0, x)
y_m = abs(y)
y_s = copysign(1.0d0, y)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
real(8) function code(y_s, x_s, x_m, y_m, z)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    code = y_s * (x_s * (y_m * ((x_m / z) / z)))
end function

Java

x_m = Math.abs(x);
x_s = Math.copySign(1.0, x);
y_m = Math.abs(y);
y_s = Math.copySign(1.0, y);
assert x_m < y_m && y_m < z;
public static double code(double y_s, double x_s, double x_m, double y_m, double z) {
	return y_s * (x_s * (y_m * ((x_m / z) / z)));
}

Python

x_m = math.fabs(x)
x_s = math.copysign(1.0, x)
y_m = math.fabs(y)
y_s = math.copysign(1.0, y)
[x_m, y_m, z] = sort([x_m, y_m, z])
def code(y_s, x_s, x_m, y_m, z):
	return y_s * (x_s * (y_m * ((x_m / z) / z)))

Julia

x_m = abs(x)
x_s = copysign(1.0, x)
y_m = abs(y)
y_s = copysign(1.0, y)
x_m, y_m, z = sort([x_m, y_m, z])
function code(y_s, x_s, x_m, y_m, z)
	return Float64(y_s * Float64(x_s * Float64(y_m * Float64(Float64(x_m / z) / z))))
end

MATLAB

x_m = abs(x);
x_s = sign(x) * abs(1.0);
y_m = abs(y);
y_s = sign(y) * abs(1.0);
x_m, y_m, z = num2cell(sort([x_m, y_m, z])){:}
function tmp = code(y_s, x_s, x_m, y_m, z)
	tmp = y_s * (x_s * (y_m * ((x_m / z) / z)));
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
x_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y_m = N[Abs[y], $MachinePrecision]
y_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
code[y$95$s_, x$95$s_, x$95$m_, y$95$m_, z_] := N[(y$95$s * N[(x$95$s * N[(y$95$m * N[(N[(x$95$m / z), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 84.9%

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

   1. associate-/l* 85.4%

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

   2. sqr-neg 85.4%

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

   3. *-commutative 85.4%

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

   4. distribute-rgt1-in 71.4%

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

   5. sqr-neg 71.4%

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

   6. sqr-neg 71.4%

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

   7. cube-unmult 71.4%

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

3. Simplified 71.4%

   \[\leadsto \color{blue}{x \cdot \frac{y}{z \cdot z + {z}^{3}}} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. associate-*r/ 72.4%

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

   2. *-commutative 72.4%

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

   3. cube-mult 72.4%

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

   4. distribute-rgt1-in 84.9%

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

   5. *-commutative 84.9%

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

   6. frac-times 87.1%

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

   7. associate-/r* 92.6%

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

   8. associate-*l/ 97.1%

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

6. Applied egg-rr 97.1%

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

   1. associate-*r/ 96.3%

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

   2. associate-*l/ 97.3%

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

   3. associate-/l* 89.9%

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

   4. associate-/l/ 89.9%

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

   5. associate-/r* 89.9%

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

8. Applied egg-rr 89.9%

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

9. Taylor expanded in z around 0 75.5%

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

10. Final simplification 75.5%

    \[\leadsto y \cdot \frac{\frac{x}{z}}{z} \]
11. Add Preprocessing

Developer target: 97.1% accurate, 0.7× 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 2024046 
(FPCore (x y z)
  :name "Statistics.Distribution.Beta:$cvariance from math-functions-0.1.5.2"
  :precision binary64

  :alt
  (if (< z 249.6182814532307) (/ (* y (/ x z)) (+ z (* z z))) (/ (* (/ (/ y z) (+ 1.0 z)) x) z))

  (/ (* x y) (* (* z z) (+ z 1.0))))