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

Percentage Accurate: 82.6% → 98.2%

Time: 16.3s

Alternatives: 15

Speedup: 1.0×

• 26.1% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 82.6% 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: 98.2% accurate, 1.0× speedup

Math

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

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 (+ z 1.0)) (/ 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 / (z + 1.0)) * (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 / (z + 1.0d0)) * (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 / (z + 1.0)) * (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 / (z + 1.0)) * (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(Float64(Float64(y_m / Float64(z + 1.0)) * 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 / (z + 1.0)) * (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[(N[(N[(y$95$m / N[(z + 1.0), $MachinePrecision]), $MachinePrecision] * N[(x$95$m / z), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 85.9%

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

   1. associate-*l/ 86.1%

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

   2. *-commutative 86.1%

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

   3. sqr-neg 86.1%

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

   4. associate-/r* 87.2%

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

   5. sqr-neg 87.2%

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

3. Simplified 87.2%

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

   1. associate-*r/ 88.6%

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

   2. associate-*l/ 88.6%

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

   3. associate-/r* 92.7%

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

   4. associate-*r/ 96.1%

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

6. Applied egg-rr 96.1%

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

7. Final simplification 96.1%

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

Alternative 2: 96.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ \begin{array}{l} t_0 := \left(z + 1\right) \cdot \left(z \cdot z\right)\\ y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -2000:\\ \;\;\;\;\frac{\frac{\frac{x\_m}{\frac{z}{y\_m}}}{z}}{z}\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{-56}:\\ \;\;\;\;\frac{y\_m}{z \cdot \frac{z}{x\_m}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y\_m}{z \cdot z} \cdot \frac{x\_m}{z + 1}\\ \end{array}\right) \end{array} \end{array} \]

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
 (let* ((t_0 (* (+ z 1.0) (* z z))))
   (*
    y_s
    (*
     x_s
     (if (<= t_0 -2000.0)
       (/ (/ (/ x_m (/ z y_m)) z) z)
       (if (<= t_0 5e-56)
         (/ y_m (* z (/ z x_m)))
         (* (/ y_m (* z z)) (/ x_m (+ z 1.0)))))))))

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 t_0 = (z + 1.0) * (z * z);
	double tmp;
	if (t_0 <= -2000.0) {
		tmp = ((x_m / (z / y_m)) / z) / z;
	} else if (t_0 <= 5e-56) {
		tmp = y_m / (z * (z / x_m));
	} else {
		tmp = (y_m / (z * z)) * (x_m / (z + 1.0));
	}
	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) :: t_0
    real(8) :: tmp
    t_0 = (z + 1.0d0) * (z * z)
    if (t_0 <= (-2000.0d0)) then
        tmp = ((x_m / (z / y_m)) / z) / z
    else if (t_0 <= 5d-56) then
        tmp = y_m / (z * (z / x_m))
    else
        tmp = (y_m / (z * z)) * (x_m / (z + 1.0d0))
    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 t_0 = (z + 1.0) * (z * z);
	double tmp;
	if (t_0 <= -2000.0) {
		tmp = ((x_m / (z / y_m)) / z) / z;
	} else if (t_0 <= 5e-56) {
		tmp = y_m / (z * (z / x_m));
	} else {
		tmp = (y_m / (z * z)) * (x_m / (z + 1.0));
	}
	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):
	t_0 = (z + 1.0) * (z * z)
	tmp = 0
	if t_0 <= -2000.0:
		tmp = ((x_m / (z / y_m)) / z) / z
	elif t_0 <= 5e-56:
		tmp = y_m / (z * (z / x_m))
	else:
		tmp = (y_m / (z * z)) * (x_m / (z + 1.0))
	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)
	t_0 = Float64(Float64(z + 1.0) * Float64(z * z))
	tmp = 0.0
	if (t_0 <= -2000.0)
		tmp = Float64(Float64(Float64(x_m / Float64(z / y_m)) / z) / z);
	elseif (t_0 <= 5e-56)
		tmp = Float64(y_m / Float64(z * Float64(z / x_m)));
	else
		tmp = Float64(Float64(y_m / Float64(z * z)) * Float64(x_m / Float64(z + 1.0)));
	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)
	t_0 = (z + 1.0) * (z * z);
	tmp = 0.0;
	if (t_0 <= -2000.0)
		tmp = ((x_m / (z / y_m)) / z) / z;
	elseif (t_0 <= 5e-56)
		tmp = y_m / (z * (z / x_m));
	else
		tmp = (y_m / (z * z)) * (x_m / (z + 1.0));
	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_] := Block[{t$95$0 = N[(N[(z + 1.0), $MachinePrecision] * N[(z * z), $MachinePrecision]), $MachinePrecision]}, N[(y$95$s * N[(x$95$s * If[LessEqual[t$95$0, -2000.0], N[(N[(N[(x$95$m / N[(z / y$95$m), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision] / z), $MachinePrecision], If[LessEqual[t$95$0, 5e-56], N[(y$95$m / N[(z * N[(z / x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y$95$m / N[(z * z), $MachinePrecision]), $MachinePrecision] * N[(x$95$m / N[(z + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 z z) (+.f64 z 1)) < -2e3

   1. Initial program 84.1%

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

      1. associate-*l/ 86.0%

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

      2. *-commutative 86.0%

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

      3. sqr-neg 86.0%

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

      4. associate-/r* 89.1%

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

      5. sqr-neg 89.1%

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

   3. Simplified 89.1%

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

      1. associate-/l/ 86.0%

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

      2. associate-/r* 89.1%

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

      3. associate-/l* 93.5%

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

      4. associate-/r* 99.7%

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

      5. associate-*r/ 91.7%

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

      6. *-commutative 91.7%

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

      7. associate-*r/ 99.7%

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

      8. clear-num 99.6%

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

      9. un-div-inv 99.7%

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

   6. Applied egg-rr 99.7%

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

   7. Taylor expanded in z around inf 96.2%

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

   if -2e3 < (*.f64 (*.f64 z z) (+.f64 z 1)) < 4.99999999999999997e-56

   1. Initial program 82.7%

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

      1. *-commutative 82.7%

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

      2. frac-times 84.5%

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

      3. associate-*l/ 82.7%

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

      4. times-frac 92.3%

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

   4. Applied egg-rr 92.3%

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

   5. Taylor expanded in z around 0 92.3%

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

      1. *-commutative 92.3%

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

      2. clear-num 92.4%

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

      3. frac-times 87.8%

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

      4. *-un-lft-identity 87.8%

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

   7. Applied egg-rr 87.8%

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

   if 4.99999999999999997e-56 < (*.f64 (*.f64 z z) (+.f64 z 1))

   1. Initial program 91.5%

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

      1. sqr-neg 91.5%

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

      2. *-commutative 91.5%

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

      3. times-frac 95.9%

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

      4. sqr-neg 95.9%

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

   3. Simplified 95.9%

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

4. Final simplification 92.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(z + 1\right) \cdot \left(z \cdot z\right) \leq -2000:\\ \;\;\;\;\frac{\frac{\frac{x}{\frac{z}{y}}}{z}}{z}\\ \mathbf{elif}\;\left(z + 1\right) \cdot \left(z \cdot z\right) \leq 5 \cdot 10^{-56}:\\ \;\;\;\;\frac{y}{z \cdot \frac{z}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z \cdot z} \cdot \frac{x}{z + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 96.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{y\_m \cdot x\_m}{\left(z + 1\right) \cdot \left(z \cdot z\right)} \leq 2 \cdot 10^{+78}:\\ \;\;\;\;\frac{y\_m}{z \cdot z} \cdot \frac{x\_m}{z + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{y\_m \cdot \frac{x\_m}{z \cdot \left(z + 1\right)}}{z}\\ \end{array}\right) \end{array} \]

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 (<= (/ (* y_m x_m) (* (+ z 1.0) (* z z))) 2e+78)
     (* (/ y_m (* z z)) (/ x_m (+ z 1.0)))
     (/ (* y_m (/ x_m (* z (+ 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 (((y_m * x_m) / ((z + 1.0) * (z * z))) <= 2e+78) {
		tmp = (y_m / (z * z)) * (x_m / (z + 1.0));
	} else {
		tmp = (y_m * (x_m / (z * (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 (((y_m * x_m) / ((z + 1.0d0) * (z * z))) <= 2d+78) then
        tmp = (y_m / (z * z)) * (x_m / (z + 1.0d0))
    else
        tmp = (y_m * (x_m / (z * (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 (((y_m * x_m) / ((z + 1.0) * (z * z))) <= 2e+78) {
		tmp = (y_m / (z * z)) * (x_m / (z + 1.0));
	} else {
		tmp = (y_m * (x_m / (z * (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 ((y_m * x_m) / ((z + 1.0) * (z * z))) <= 2e+78:
		tmp = (y_m / (z * z)) * (x_m / (z + 1.0))
	else:
		tmp = (y_m * (x_m / (z * (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(y_m * x_m) / Float64(Float64(z + 1.0) * Float64(z * z))) <= 2e+78)
		tmp = Float64(Float64(y_m / Float64(z * z)) * Float64(x_m / Float64(z + 1.0)));
	else
		tmp = Float64(Float64(y_m * Float64(x_m / Float64(z * 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 (((y_m * x_m) / ((z + 1.0) * (z * z))) <= 2e+78)
		tmp = (y_m / (z * z)) * (x_m / (z + 1.0));
	else
		tmp = (y_m * (x_m / (z * (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[(y$95$m * x$95$m), $MachinePrecision] / N[(N[(z + 1.0), $MachinePrecision] * N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2e+78], N[(N[(y$95$m / N[(z * z), $MachinePrecision]), $MachinePrecision] * N[(x$95$m / N[(z + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y$95$m * N[(x$95$m / N[(z * N[(z + 1.0), $MachinePrecision]), $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))) < 2.00000000000000002e78

   1. Initial program 90.0%

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

      1. sqr-neg 90.0%

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

      2. *-commutative 90.0%

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

      3. times-frac 94.2%

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

      4. sqr-neg 94.2%

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

   3. Simplified 94.2%

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

   if 2.00000000000000002e78 < (/.f64 (*.f64 x y) (*.f64 (*.f64 z z) (+.f64 z 1)))

   1. Initial program 72.9%

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

      1. *-commutative 72.9%

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

      2. frac-times 78.0%

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

      3. associate-*l/ 80.8%

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

      4. times-frac 91.5%

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

   4. Applied egg-rr 91.5%

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

      1. associate-*l/ 94.6%

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

      2. associate-*r/ 93.4%

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

      3. associate-*r/ 80.7%

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

      4. associate-/r* 80.5%

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

      5. associate-/l* 89.9%

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

   6. Applied egg-rr 89.9%

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

4. Final simplification 93.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y \cdot x}{\left(z + 1\right) \cdot \left(z \cdot z\right)} \leq 2 \cdot 10^{+78}:\\ \;\;\;\;\frac{y}{z \cdot z} \cdot \frac{x}{z + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot \frac{x}{z \cdot \left(z + 1\right)}}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 97.1% accurate, 0.6× speedup

Math

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

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)))
     (* (/ x_m z) (/ (/ y_m z) z))
     (/ y_m (* z (/ z x_m)))))))

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 = (x_m / z) * ((y_m / z) / z);
	} else {
		tmp = y_m / (z * (z / x_m));
	}
	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 = (x_m / z) * ((y_m / z) / z)
    else
        tmp = y_m / (z * (z / x_m))
    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 = (x_m / z) * ((y_m / z) / z);
	} else {
		tmp = y_m / (z * (z / x_m));
	}
	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 = (x_m / z) * ((y_m / z) / z)
	else:
		tmp = y_m / (z * (z / x_m))
	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(x_m / z) * Float64(Float64(y_m / z) / z));
	else
		tmp = Float64(y_m / Float64(z * Float64(z / x_m)));
	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 = (x_m / z) * ((y_m / z) / z);
	else
		tmp = y_m / (z * (z / x_m));
	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[(x$95$m / z), $MachinePrecision] * N[(N[(y$95$m / z), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], N[(y$95$m / N[(z * N[(z / x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < -1 or 1 < z

   1. Initial program 87.4%

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

      1. sqr-neg 87.4%

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

      2. times-frac 93.9%

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

      3. sqr-neg 93.9%

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

   3. Simplified 93.9%

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

   5. Taylor expanded in z around inf 91.7%

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

      1. *-commutative 91.7%

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

      2. associate-/r* 94.5%

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

      3. associate-*r/ 97.5%

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

   7. Applied egg-rr 97.5%

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

      1. *-commutative 97.5%

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

      2. associate-/l* 96.1%

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

   9. Applied egg-rr 96.1%

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

   if -1 < z < 1

   1. Initial program 84.2%

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

      1. *-commutative 84.2%

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

      2. frac-times 85.8%

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

      3. associate-*l/ 84.2%

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

      4. times-frac 93.0%

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

   4. Applied egg-rr 93.0%

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

   5. Taylor expanded in z around 0 91.5%

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

      1. *-commutative 91.5%

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

      2. clear-num 91.5%

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

      3. frac-times 87.4%

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

      4. *-un-lft-identity 87.4%

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

   7. Applied egg-rr 87.4%

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

4. Final simplification 91.9%

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

Alternative 5: 97.0% accurate, 0.6× speedup

Math

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

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 (* z (/ z x_m))) (* (/ x_m z) (/ (/ y_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 / (z * (z / x_m));
	} else {
		tmp = (x_m / z) * ((y_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 / (z * (z / x_m))
    else
        tmp = (x_m / z) * ((y_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 / (z * (z / x_m));
	} else {
		tmp = (x_m / z) * ((y_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 / (z * (z / x_m))
	else:
		tmp = (x_m / z) * ((y_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(Float64(x_m / z) * Float64(y_m / z)) / z);
	elseif (z <= 1.0)
		tmp = Float64(y_m / Float64(z * Float64(z / x_m)));
	else
		tmp = Float64(Float64(x_m / z) * Float64(Float64(y_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 / (z * (z / x_m));
	else
		tmp = (x_m / z) * ((y_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[(N[(x$95$m / z), $MachinePrecision] * N[(y$95$m / z), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], If[LessEqual[z, 1.0], N[(y$95$m / N[(z * N[(z / x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$95$m / z), $MachinePrecision] * N[(N[(y$95$m / z), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if z < -1

   1. Initial program 84.1%

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

      1. sqr-neg 84.1%

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

      2. times-frac 91.4%

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

      3. sqr-neg 91.4%

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

   3. Simplified 91.4%

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

   5. Taylor expanded in z around inf 87.8%

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

      1. *-commutative 87.8%

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

      2. associate-/r* 92.4%

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

      3. associate-*r/ 96.1%

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

   7. Applied egg-rr 96.1%

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

   if -1 < z < 1

   1. Initial program 84.2%

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

      1. *-commutative 84.2%

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

      2. frac-times 85.8%

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

      3. associate-*l/ 84.2%

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

      4. times-frac 93.0%

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

   4. Applied egg-rr 93.0%

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

   5. Taylor expanded in z around 0 91.5%

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

      1. *-commutative 91.5%

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

      2. clear-num 91.5%

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

      3. frac-times 87.4%

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

      4. *-un-lft-identity 87.4%

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

   7. Applied egg-rr 87.4%

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

   if 1 < z

   1. Initial program 90.3%

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

      1. sqr-neg 90.3%

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

      2. times-frac 96.0%

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

      3. sqr-neg 96.0%

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

   3. Simplified 96.0%

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

   5. Taylor expanded in z around inf 95.0%

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

      1. *-commutative 95.0%

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

      2. associate-/r* 96.3%

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

      3. associate-*r/ 98.7%

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

   7. Applied egg-rr 98.7%

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

      1. *-commutative 98.7%

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

      2. associate-/l* 96.2%

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

   9. Applied egg-rr 96.2%

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

4. Final simplification 91.9%

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

Alternative 6: 97.1% accurate, 0.6× speedup

Math

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

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) (* z (/ z y_m)))
     (if (<= z 1.0) (/ y_m (* z (/ z x_m))) (* (/ x_m z) (/ (/ y_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) / (z * (z / y_m));
	} else if (z <= 1.0) {
		tmp = y_m / (z * (z / x_m));
	} else {
		tmp = (x_m / z) * ((y_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) / (z * (z / y_m))
    else if (z <= 1.0d0) then
        tmp = y_m / (z * (z / x_m))
    else
        tmp = (x_m / z) * ((y_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) / (z * (z / y_m));
	} else if (z <= 1.0) {
		tmp = y_m / (z * (z / x_m));
	} else {
		tmp = (x_m / z) * ((y_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) / (z * (z / y_m))
	elif z <= 1.0:
		tmp = y_m / (z * (z / x_m))
	else:
		tmp = (x_m / z) * ((y_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(z * Float64(z / y_m)));
	elseif (z <= 1.0)
		tmp = Float64(y_m / Float64(z * Float64(z / x_m)));
	else
		tmp = Float64(Float64(x_m / z) * Float64(Float64(y_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) / (z * (z / y_m));
	elseif (z <= 1.0)
		tmp = y_m / (z * (z / x_m));
	else
		tmp = (x_m / z) * ((y_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[(z * N[(z / y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 1.0], N[(y$95$m / N[(z * N[(z / x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$95$m / z), $MachinePrecision] * N[(N[(y$95$m / z), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if z < -1

   1. Initial program 84.1%

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

      1. sqr-neg 84.1%

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

      2. times-frac 91.4%

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

      3. sqr-neg 91.4%

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

   3. Simplified 91.4%

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

   5. Taylor expanded in z around inf 87.8%

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

      1. *-commutative 87.8%

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

      2. clear-num 87.8%

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

      3. associate-/r* 92.4%

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

      4. frac-times 96.2%

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

      5. *-un-lft-identity 96.2%

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

   7. Applied egg-rr 96.2%

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

   if -1 < z < 1

   1. Initial program 84.2%

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

      1. *-commutative 84.2%

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

      2. frac-times 85.8%

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

      3. associate-*l/ 84.2%

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

      4. times-frac 93.0%

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

   4. Applied egg-rr 93.0%

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

   5. Taylor expanded in z around 0 91.5%

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

      1. *-commutative 91.5%

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

      2. clear-num 91.5%

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

      3. frac-times 87.4%

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

      4. *-un-lft-identity 87.4%

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

   7. Applied egg-rr 87.4%

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

   if 1 < z

   1. Initial program 90.3%

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

      1. sqr-neg 90.3%

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

      2. times-frac 96.0%

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

      3. sqr-neg 96.0%

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

   3. Simplified 96.0%

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

   5. Taylor expanded in z around inf 95.0%

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

      1. *-commutative 95.0%

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

      2. associate-/r* 96.3%

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

      3. associate-*r/ 98.7%

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

   7. Applied egg-rr 98.7%

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

      1. *-commutative 98.7%

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

      2. associate-/l* 96.2%

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

   9. Applied egg-rr 96.2%

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

4. Final simplification 92.0%

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

Alternative 7: 97.1% accurate, 0.6× speedup

Math

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

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) (/ z y_m)) z)
     (if (<= z 1.0) (/ y_m (* z (/ z x_m))) (* (/ x_m z) (/ (/ y_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) / (z / y_m)) / z;
	} else if (z <= 1.0) {
		tmp = y_m / (z * (z / x_m));
	} else {
		tmp = (x_m / z) * ((y_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) / (z / y_m)) / z
    else if (z <= 1.0d0) then
        tmp = y_m / (z * (z / x_m))
    else
        tmp = (x_m / z) * ((y_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) / (z / y_m)) / z;
	} else if (z <= 1.0) {
		tmp = y_m / (z * (z / x_m));
	} else {
		tmp = (x_m / z) * ((y_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) / (z / y_m)) / z
	elif z <= 1.0:
		tmp = y_m / (z * (z / x_m))
	else:
		tmp = (x_m / z) * ((y_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(Float64(x_m / z) / Float64(z / y_m)) / z);
	elseif (z <= 1.0)
		tmp = Float64(y_m / Float64(z * Float64(z / x_m)));
	else
		tmp = Float64(Float64(x_m / z) * Float64(Float64(y_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) / (z / y_m)) / z;
	elseif (z <= 1.0)
		tmp = y_m / (z * (z / x_m));
	else
		tmp = (x_m / z) * ((y_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[(N[(x$95$m / z), $MachinePrecision] / N[(z / y$95$m), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], If[LessEqual[z, 1.0], N[(y$95$m / N[(z * N[(z / x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$95$m / z), $MachinePrecision] * N[(N[(y$95$m / z), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if z < -1

   1. Initial program 84.1%

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

      1. sqr-neg 84.1%

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

      2. times-frac 91.4%

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

      3. sqr-neg 91.4%

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

   3. Simplified 91.4%

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

   5. Taylor expanded in z around inf 87.8%

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

      1. *-commutative 87.8%

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

      2. associate-/r* 92.4%

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

      3. associate-*r/ 96.1%

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

   7. Applied egg-rr 96.1%

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

      1. *-commutative 96.1%

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

      2. clear-num 96.2%

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

      3. un-div-inv 96.2%

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

   9. Applied egg-rr 96.2%

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

   if -1 < z < 1

   1. Initial program 84.2%

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

      1. *-commutative 84.2%

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

      2. frac-times 85.8%

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

      3. associate-*l/ 84.2%

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

      4. times-frac 93.0%

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

   4. Applied egg-rr 93.0%

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

   5. Taylor expanded in z around 0 91.5%

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

      1. *-commutative 91.5%

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

      2. clear-num 91.5%

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

      3. frac-times 87.4%

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

      4. *-un-lft-identity 87.4%

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

   7. Applied egg-rr 87.4%

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

   if 1 < z

   1. Initial program 90.3%

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

      1. sqr-neg 90.3%

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

      2. times-frac 96.0%

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

      3. sqr-neg 96.0%

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

   3. Simplified 96.0%

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

   5. Taylor expanded in z around inf 95.0%

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

      1. *-commutative 95.0%

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

      2. associate-/r* 96.3%

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

      3. associate-*r/ 98.7%

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

   7. Applied egg-rr 98.7%

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

      1. *-commutative 98.7%

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

      2. associate-/l* 96.2%

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

   9. Applied egg-rr 96.2%

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

4. Final simplification 92.0%

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

Alternative 8: 96.8% accurate, 0.6× speedup

Math

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

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 (* z (/ z x_m))) (* (/ x_m z) (/ (/ y_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 / (z * (z / x_m));
	} else {
		tmp = (x_m / z) * ((y_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 / (z * (z / x_m))
    else
        tmp = (x_m / z) * ((y_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 / (z * (z / x_m));
	} else {
		tmp = (x_m / z) * ((y_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 / (z * (z / x_m))
	else:
		tmp = (x_m / z) * ((y_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(Float64(x_m / Float64(z / y_m)) / z) / z);
	elseif (z <= 1.0)
		tmp = Float64(y_m / Float64(z * Float64(z / x_m)));
	else
		tmp = Float64(Float64(x_m / z) * Float64(Float64(y_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 / (z * (z / x_m));
	else
		tmp = (x_m / z) * ((y_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[(N[(x$95$m / N[(z / y$95$m), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision] / z), $MachinePrecision], If[LessEqual[z, 1.0], N[(y$95$m / N[(z * N[(z / x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$95$m / z), $MachinePrecision] * N[(N[(y$95$m / z), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if z < -1

   1. Initial program 84.1%

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

      1. associate-*l/ 86.0%

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

      2. *-commutative 86.0%

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

      3. sqr-neg 86.0%

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

      4. associate-/r* 89.1%

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

      5. sqr-neg 89.1%

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

   3. Simplified 89.1%

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

      1. associate-/l/ 86.0%

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

      2. associate-/r* 89.1%

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

      3. associate-/l* 93.5%

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

      4. associate-/r* 99.7%

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

      5. associate-*r/ 91.7%

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

      6. *-commutative 91.7%

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

      7. associate-*r/ 99.7%

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

      8. clear-num 99.6%

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

      9. un-div-inv 99.7%

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

   6. Applied egg-rr 99.7%

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

   7. Taylor expanded in z around inf 96.2%

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

   if -1 < z < 1

   1. Initial program 84.2%

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

      1. *-commutative 84.2%

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

      2. frac-times 85.8%

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

      3. associate-*l/ 84.2%

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

      4. times-frac 93.0%

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

   4. Applied egg-rr 93.0%

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

   5. Taylor expanded in z around 0 91.5%

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

      1. *-commutative 91.5%

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

      2. clear-num 91.5%

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

      3. frac-times 87.4%

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

      4. *-un-lft-identity 87.4%

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

   7. Applied egg-rr 87.4%

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

   if 1 < z

   1. Initial program 90.3%

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

      1. sqr-neg 90.3%

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

      2. times-frac 96.0%

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

      3. sqr-neg 96.0%

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

   3. Simplified 96.0%

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

   5. Taylor expanded in z around inf 95.0%

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

      1. *-commutative 95.0%

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

      2. associate-/r* 96.3%

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

      3. associate-*r/ 98.7%

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

   7. Applied egg-rr 98.7%

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

      1. *-commutative 98.7%

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

      2. associate-/l* 96.2%

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

   9. Applied egg-rr 96.2%

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

4. Final simplification 92.0%

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

Alternative 9: 96.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -1.15 \cdot 10^{-15}:\\ \;\;\;\;\frac{y\_m}{z} \cdot \frac{\frac{x\_m}{z + 1}}{z}\\ \mathbf{elif}\;z \leq 1:\\ \;\;\;\;\frac{y\_m}{z \cdot \frac{z}{x\_m}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m}{z} \cdot \frac{\frac{y\_m}{z}}{z}\\ \end{array}\right) \end{array} \]

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 -1.15e-15)
     (* (/ y_m z) (/ (/ x_m (+ z 1.0)) z))
     (if (<= z 1.0) (/ y_m (* z (/ z x_m))) (* (/ x_m z) (/ (/ y_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 <= -1.15e-15) {
		tmp = (y_m / z) * ((x_m / (z + 1.0)) / z);
	} else if (z <= 1.0) {
		tmp = y_m / (z * (z / x_m));
	} else {
		tmp = (x_m / z) * ((y_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 <= (-1.15d-15)) then
        tmp = (y_m / z) * ((x_m / (z + 1.0d0)) / z)
    else if (z <= 1.0d0) then
        tmp = y_m / (z * (z / x_m))
    else
        tmp = (x_m / z) * ((y_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 <= -1.15e-15) {
		tmp = (y_m / z) * ((x_m / (z + 1.0)) / z);
	} else if (z <= 1.0) {
		tmp = y_m / (z * (z / x_m));
	} else {
		tmp = (x_m / z) * ((y_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 <= -1.15e-15:
		tmp = (y_m / z) * ((x_m / (z + 1.0)) / z)
	elif z <= 1.0:
		tmp = y_m / (z * (z / x_m))
	else:
		tmp = (x_m / z) * ((y_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 <= -1.15e-15)
		tmp = Float64(Float64(y_m / z) * Float64(Float64(x_m / Float64(z + 1.0)) / z));
	elseif (z <= 1.0)
		tmp = Float64(y_m / Float64(z * Float64(z / x_m)));
	else
		tmp = Float64(Float64(x_m / z) * Float64(Float64(y_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 <= -1.15e-15)
		tmp = (y_m / z) * ((x_m / (z + 1.0)) / z);
	elseif (z <= 1.0)
		tmp = y_m / (z * (z / x_m));
	else
		tmp = (x_m / z) * ((y_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, -1.15e-15], N[(N[(y$95$m / z), $MachinePrecision] * N[(N[(x$95$m / N[(z + 1.0), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 1.0], N[(y$95$m / N[(z * N[(z / x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$95$m / z), $MachinePrecision] * N[(N[(y$95$m / z), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if z < -1.14999999999999995e-15

   1. Initial program 84.6%

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

      1. *-commutative 84.6%

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

      2. frac-times 93.7%

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

      3. associate-*l/ 93.8%

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

      4. times-frac 96.0%

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

   4. Applied egg-rr 96.0%

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

   if -1.14999999999999995e-15 < z < 1

   1. Initial program 84.0%

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

      1. *-commutative 84.0%

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

      2. frac-times 85.6%

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

      3. associate-*l/ 84.0%

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

      4. times-frac 92.9%

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

   4. Applied egg-rr 92.9%

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

   5. Taylor expanded in z around 0 92.0%

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

      1. *-commutative 92.0%

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

      2. clear-num 92.0%

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

      3. frac-times 87.8%

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

      4. *-un-lft-identity 87.8%

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

   7. Applied egg-rr 87.8%

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

   if 1 < z

   1. Initial program 90.3%

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

      1. sqr-neg 90.3%

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

      2. times-frac 96.0%

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

      3. sqr-neg 96.0%

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

   3. Simplified 96.0%

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

   5. Taylor expanded in z around inf 95.0%

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

      1. *-commutative 95.0%

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

      2. associate-/r* 96.3%

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

      3. associate-*r/ 98.7%

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

   7. Applied egg-rr 98.7%

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

      1. *-commutative 98.7%

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

      2. associate-/l* 96.2%

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

   9. Applied egg-rr 96.2%

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

4. Final simplification 92.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.15 \cdot 10^{-15}:\\ \;\;\;\;\frac{y}{z} \cdot \frac{\frac{x}{z + 1}}{z}\\ \mathbf{elif}\;z \leq 1:\\ \;\;\;\;\frac{y}{z \cdot \frac{z}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot \frac{\frac{y}{z}}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 44.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -1 \lor \neg \left(z \leq -5 \cdot 10^{-310}\right):\\ \;\;\;\;\frac{y\_m}{\frac{z}{x\_m}}\\ \mathbf{else}:\\ \;\;\;\;x\_m \cdot \frac{y\_m}{-z}\\ \end{array}\right) \end{array} \]

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 -5e-310)))
     (/ y_m (/ z x_m))
     (* x_m (/ y_m (- 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 <= -5e-310)) {
		tmp = y_m / (z / x_m);
	} else {
		tmp = x_m * (y_m / -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 <= (-5d-310)))) then
        tmp = y_m / (z / x_m)
    else
        tmp = x_m * (y_m / -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 <= -5e-310)) {
		tmp = y_m / (z / x_m);
	} else {
		tmp = x_m * (y_m / -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 <= -5e-310):
		tmp = y_m / (z / x_m)
	else:
		tmp = x_m * (y_m / -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 <= -5e-310))
		tmp = Float64(y_m / Float64(z / x_m));
	else
		tmp = Float64(x_m * Float64(y_m / Float64(-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 <= -5e-310)))
		tmp = y_m / (z / x_m);
	else
		tmp = x_m * (y_m / -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, -5e-310]], $MachinePrecision]], N[(y$95$m / N[(z / x$95$m), $MachinePrecision]), $MachinePrecision], N[(x$95$m * N[(y$95$m / (-z)), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < -1 or -4.999999999999985e-310 < z

   1. Initial program 86.2%

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

      1. *-commutative 86.2%

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

      2. frac-times 91.9%

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

      3. associate-*l/ 91.4%

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

      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 0 53.7%

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

      1. neg-mul-1 53.7%

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

      2. +-commutative 53.7%

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

      3. unsub-neg 53.7%

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

   7. Simplified 53.7%

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

   8. Taylor expanded in z around inf 18.6%

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

      1. associate-*r/ 24.0%

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

      2. associate-*r* 24.0%

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

      3. neg-mul-1 24.0%

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

   10. Simplified 24.0%

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

       1. *-commutative 24.0%

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

       2. add-sqr-sqrt 11.4%

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

       3. sqrt-unprod 30.3%

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

       4. sqr-neg 30.3%

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

       5. sqrt-unprod 18.4%

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

       6. add-sqr-sqrt 38.5%

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

       7. associate-/r/ 41.2%

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

   12. Applied egg-rr 41.2%

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

   if -1 < z < -4.999999999999985e-310

   1. Initial program 84.9%

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

      1. *-commutative 84.9%

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

      2. frac-times 84.9%

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

      3. associate-*l/ 85.0%

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

      4. times-frac 91.6%

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

   4. Applied egg-rr 91.6%

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

   5. Taylor expanded in z around 0 90.7%

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

      1. neg-mul-1 90.7%

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

      2. +-commutative 90.7%

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

      3. unsub-neg 90.7%

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

   7. Simplified 90.7%

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

   8. Taylor expanded in z around inf 26.2%

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

      1. associate-*r/ 34.7%

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

      2. associate-*r* 34.7%

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

      3. neg-mul-1 34.7%

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

   10. Simplified 34.7%

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

4. Final simplification 39.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1 \lor \neg \left(z \leq -5 \cdot 10^{-310}\right):\\ \;\;\;\;\frac{y}{\frac{z}{x}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{-z}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 80.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;y\_m \leq 5 \cdot 10^{+53}:\\ \;\;\;\;\frac{x\_m}{z} \cdot \frac{y\_m}{z}\\ \mathbf{else}:\\ \;\;\;\;y\_m \cdot \frac{x\_m}{z \cdot z}\\ \end{array}\right) \end{array} \]

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 (<= y_m 5e+53) (* (/ x_m z) (/ y_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 (y_m <= 5e+53) {
		tmp = (x_m / z) * (y_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 (y_m <= 5d+53) then
        tmp = (x_m / z) * (y_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 (y_m <= 5e+53) {
		tmp = (x_m / z) * (y_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 y_m <= 5e+53:
		tmp = (x_m / z) * (y_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 (y_m <= 5e+53)
		tmp = Float64(Float64(x_m / z) * Float64(y_m / z));
	else
		tmp = Float64(y_m * Float64(x_m / Float64(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 (y_m <= 5e+53)
		tmp = (x_m / z) * (y_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[LessEqual[y$95$m, 5e+53], N[(N[(x$95$m / z), $MachinePrecision] * N[(y$95$m / z), $MachinePrecision]), $MachinePrecision], N[(y$95$m * N[(x$95$m / N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 5.0000000000000004e53

   1. Initial program 86.7%

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

      1. *-commutative 86.7%

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

      2. frac-times 89.1%

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

      3. associate-*l/ 88.2%

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

      4. times-frac 94.8%

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

   4. Applied egg-rr 94.8%

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

   5. Taylor expanded in z around 0 72.2%

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

   if 5.0000000000000004e53 < y

   1. Initial program 83.3%

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

      1. sqr-neg 83.3%

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

      2. times-frac 95.4%

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

      3. sqr-neg 95.4%

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

   3. Simplified 95.4%

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

   5. Taylor expanded in z around 0 76.8%

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

4. Final simplification 73.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 5 \cdot 10^{+53}:\\ \;\;\;\;\frac{x}{z} \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{z \cdot z}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 40.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -5 \cdot 10^{-310}:\\ \;\;\;\;y\_m \cdot \frac{x\_m}{-z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y\_m}{\frac{z}{x\_m}}\\ \end{array}\right) \end{array} \]

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 -5e-310) (* y_m (/ x_m (- z))) (/ y_m (/ z x_m))))))

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 <= -5e-310) {
		tmp = y_m * (x_m / -z);
	} else {
		tmp = y_m / (z / x_m);
	}
	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 <= (-5d-310)) then
        tmp = y_m * (x_m / -z)
    else
        tmp = y_m / (z / x_m)
    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 <= -5e-310) {
		tmp = y_m * (x_m / -z);
	} else {
		tmp = y_m / (z / x_m);
	}
	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 <= -5e-310:
		tmp = y_m * (x_m / -z)
	else:
		tmp = y_m / (z / x_m)
	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 <= -5e-310)
		tmp = Float64(y_m * Float64(x_m / Float64(-z)));
	else
		tmp = Float64(y_m / Float64(z / x_m));
	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 <= -5e-310)
		tmp = y_m * (x_m / -z);
	else
		tmp = y_m / (z / x_m);
	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, -5e-310], N[(y$95$m * N[(x$95$m / (-z)), $MachinePrecision]), $MachinePrecision], N[(y$95$m / N[(z / x$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < -4.999999999999985e-310

   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 89.4%

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

      3. associate-*l/ 89.4%

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

      4. times-frac 93.9%

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

   4. Applied egg-rr 93.9%

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

   5. Taylor expanded in z around 0 59.6%

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

      1. neg-mul-1 59.6%

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

      2. +-commutative 59.6%

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

      3. unsub-neg 59.6%

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

   7. Simplified 59.6%

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

   8. Taylor expanded in z around inf 26.0%

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

      1. associate-*r/ 32.8%

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

      2. associate-*r* 32.8%

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

      3. neg-mul-1 32.8%

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

   10. Simplified 32.8%

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

       1. distribute-lft-neg-out 32.8%

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

       2. associate-/l* 26.0%

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

       3. div-inv 26.0%

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

       4. *-commutative 26.0%

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

       5. associate-*r* 31.8%

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

       6. div-inv 31.8%

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

   12. Applied egg-rr 31.8%

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

   if -4.999999999999985e-310 < z

   1. Initial program 87.1%

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

      1. *-commutative 87.1%

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

      2. frac-times 91.2%

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

      3. associate-*l/ 90.4%

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

      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 0 63.6%

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

      1. neg-mul-1 63.6%

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

      2. +-commutative 63.6%

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

      3. unsub-neg 63.6%

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

   7. Simplified 63.6%

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

   8. Taylor expanded in z around inf 15.4%

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

      1. associate-*r/ 20.9%

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

      2. associate-*r* 20.9%

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

      3. neg-mul-1 20.9%

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

   10. Simplified 20.9%

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

       1. *-commutative 20.9%

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

       2. add-sqr-sqrt 10.8%

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

       3. sqrt-unprod 29.8%

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

       4. sqr-neg 29.8%

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

       5. sqrt-unprod 18.0%

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

       6. add-sqr-sqrt 40.9%

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

       7. associate-/r/ 44.4%

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

   12. Applied egg-rr 44.4%

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

4. Final simplification 38.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -5 \cdot 10^{-310}:\\ \;\;\;\;y \cdot \frac{x}{-z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{z}{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 74.8% accurate, 1.6× speedup

Math

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

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(x_m / Float64(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[(x$95$m / N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 85.9%

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

   1. sqr-neg 85.9%

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

   2. times-frac 88.6%

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

   3. sqr-neg 88.6%

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

3. Simplified 88.6%

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

5. Taylor expanded in z around 0 69.8%

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

6. Final simplification 69.8%

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

Alternative 14: 80.7% accurate, 1.6× speedup

Math

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

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 (* z (/ z x_m))))))

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 / (z * (z / x_m))));
}

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 / (z * (z / x_m))))
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 / (z * (z / x_m))));
}

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 / (z * (z / x_m))))

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(z * Float64(z / x_m)))))
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 / (z * (z / x_m))));
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[(z * N[(z / x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 85.9%

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

   1. *-commutative 85.9%

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

   2. frac-times 90.3%

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

   3. associate-*l/ 90.0%

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

   4. times-frac 94.9%

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

4. Applied egg-rr 94.9%

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

5. Taylor expanded in z around 0 71.2%

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

   1. *-commutative 71.2%

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

   2. clear-num 71.2%

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

   3. frac-times 72.3%

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

   4. *-un-lft-identity 72.3%

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

7. Applied egg-rr 72.3%

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

8. Final simplification 72.3%

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

Alternative 15: 33.1% accurate, 2.2× speedup

Math

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

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 (/ z x_m)))))

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 / (z / x_m)));
}

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 / (z / x_m)))
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 / (z / x_m)));
}

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 / (z / x_m)))

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(z / x_m))))
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 / (z / x_m)));
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[(z / x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 85.9%

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

   1. *-commutative 85.9%

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

   2. frac-times 90.3%

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

   3. associate-*l/ 90.0%

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

   4. times-frac 94.9%

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

4. Applied egg-rr 94.9%

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

5. Taylor expanded in z around 0 61.8%

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

   1. neg-mul-1 61.8%

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

   2. +-commutative 61.8%

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

   3. unsub-neg 61.8%

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

7. Simplified 61.8%

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

8. Taylor expanded in z around inf 20.3%

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

   1. associate-*r/ 26.3%

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

   2. associate-*r* 26.3%

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

   3. neg-mul-1 26.3%

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

10. Simplified 26.3%

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

    1. *-commutative 26.3%

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

    2. add-sqr-sqrt 12.4%

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

    3. sqrt-unprod 26.2%

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

    4. sqr-neg 26.2%

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

    5. sqrt-unprod 14.5%

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

    6. add-sqr-sqrt 30.3%

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

    7. associate-/r/ 32.4%

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

12. Applied egg-rr 32.4%

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

13. Final simplification 32.4%

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

Developer target: 96.9% 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 2024048 
(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))))