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

Percentage Accurate: 82.7% → 98.0%

Time: 16.0s

Alternatives: 10

Speedup: 1.0×

• 26.7% 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: 82.7% 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: 98.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;y\_m \leq 5.2 \cdot 10^{-17}:\\ \;\;\;\;\frac{y\_m}{z} \cdot \frac{\frac{x\_m}{z + 1}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{y\_m}{\frac{z + 1}{x\_m}}}{z}}{z}\\ \end{array}\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) 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 5.2e-17)
     (* (/ y_m z) (/ (/ x_m (+ z 1.0)) z))
     (/ (/ (/ 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) {
	double tmp;
	if (y_m <= 5.2e-17) {
		tmp = (y_m / z) * ((x_m / (z + 1.0)) / z);
	} else {
		tmp = ((y_m / ((z + 1.0) / 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 <= 5.2d-17) then
        tmp = (y_m / z) * ((x_m / (z + 1.0d0)) / z)
    else
        tmp = ((y_m / ((z + 1.0d0) / 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 <= 5.2e-17) {
		tmp = (y_m / z) * ((x_m / (z + 1.0)) / z);
	} else {
		tmp = ((y_m / ((z + 1.0) / 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 <= 5.2e-17:
		tmp = (y_m / z) * ((x_m / (z + 1.0)) / z)
	else:
		tmp = ((y_m / ((z + 1.0) / 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 <= 5.2e-17)
		tmp = Float64(Float64(y_m / z) * Float64(Float64(x_m / Float64(z + 1.0)) / z));
	else
		tmp = Float64(Float64(Float64(y_m / Float64(Float64(z + 1.0) / x_m)) / z) / z);
	end
	return Float64(y_s * Float64(x_s * tmp))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
x_m, y_m, z = num2cell(sort([x_m, y_m, z])){:}
function tmp_2 = code(y_s, x_s, x_m, y_m, z)
	tmp = 0.0;
	if (y_m <= 5.2e-17)
		tmp = (y_m / z) * ((x_m / (z + 1.0)) / z);
	else
		tmp = ((y_m / ((z + 1.0) / 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, 5.2e-17], N[(N[(y$95$m / z), $MachinePrecision] * N[(N[(x$95$m / N[(z + 1.0), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], N[(N[(N[(y$95$m / N[(N[(z + 1.0), $MachinePrecision] / x$95$m), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision] / z), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 5.20000000000000006e-17

   1. Initial program 82.0%

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

      1. *-commutative 82.0%

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

      2. frac-times 86.8%

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

      3. associate-*l/ 85.6%

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

      4. times-frac 96.5%

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

   4. Applied egg-rr 96.5%

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

   if 5.20000000000000006e-17 < y

   1. Initial program 86.9%

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

      1. *-commutative 86.9%

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

      2. associate-/l* 86.7%

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

      3. sqr-neg 86.7%

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

      4. associate-/r* 92.4%

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

      5. sqr-neg 92.4%

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

   3. Simplified 92.4%

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

      1. associate-/l/ 86.7%

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

      2. associate-/r* 92.3%

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

      3. associate-/l* 94.3%

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

      4. associate-/r* 98.4%

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

      5. clear-num 98.3%

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

      6. un-div-inv 98.4%

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

   6. Applied egg-rr 98.4%

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

Alternative 2: 98.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ \begin{array}{l} t_0 := \frac{x\_m}{z + 1}\\ t_1 := \frac{y\_m \cdot x\_m}{\left(z + 1\right) \cdot \left(z \cdot z\right)}\\ y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq 0.001:\\ \;\;\;\;t\_0 \cdot \frac{y\_m}{z \cdot z}\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;\frac{y\_m}{z \cdot \left(z \cdot \frac{z + 1}{x\_m}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{y\_m}{z} \cdot \frac{t\_0}{z}\\ \end{array}\right) \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) 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 (/ x_m (+ z 1.0))) (t_1 (/ (* y_m x_m) (* (+ z 1.0) (* z z)))))
   (*
    y_s
    (*
     x_s
     (if (<= t_1 0.001)
       (* t_0 (/ y_m (* z z)))
       (if (<= t_1 INFINITY)
         (/ y_m (* z (* z (/ (+ z 1.0) x_m))))
         (* (/ y_m z) (/ t_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 t_0 = x_m / (z + 1.0);
	double t_1 = (y_m * x_m) / ((z + 1.0) * (z * z));
	double tmp;
	if (t_1 <= 0.001) {
		tmp = t_0 * (y_m / (z * z));
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = y_m / (z * (z * ((z + 1.0) / x_m)));
	} else {
		tmp = (y_m / z) * (t_0 / z);
	}
	return y_s * (x_s * tmp);
}

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 = x_m / (z + 1.0);
	double t_1 = (y_m * x_m) / ((z + 1.0) * (z * z));
	double tmp;
	if (t_1 <= 0.001) {
		tmp = t_0 * (y_m / (z * z));
	} else if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = y_m / (z * (z * ((z + 1.0) / x_m)));
	} else {
		tmp = (y_m / z) * (t_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):
	t_0 = x_m / (z + 1.0)
	t_1 = (y_m * x_m) / ((z + 1.0) * (z * z))
	tmp = 0
	if t_1 <= 0.001:
		tmp = t_0 * (y_m / (z * z))
	elif t_1 <= math.inf:
		tmp = y_m / (z * (z * ((z + 1.0) / x_m)))
	else:
		tmp = (y_m / z) * (t_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)
	t_0 = Float64(x_m / Float64(z + 1.0))
	t_1 = Float64(Float64(y_m * x_m) / Float64(Float64(z + 1.0) * Float64(z * z)))
	tmp = 0.0
	if (t_1 <= 0.001)
		tmp = Float64(t_0 * Float64(y_m / Float64(z * z)));
	elseif (t_1 <= Inf)
		tmp = Float64(y_m / Float64(z * Float64(z * Float64(Float64(z + 1.0) / x_m))));
	else
		tmp = Float64(Float64(y_m / z) * Float64(t_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)
	t_0 = x_m / (z + 1.0);
	t_1 = (y_m * x_m) / ((z + 1.0) * (z * z));
	tmp = 0.0;
	if (t_1 <= 0.001)
		tmp = t_0 * (y_m / (z * z));
	elseif (t_1 <= Inf)
		tmp = y_m / (z * (z * ((z + 1.0) / x_m)));
	else
		tmp = (y_m / z) * (t_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_] := Block[{t$95$0 = N[(x$95$m / N[(z + 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(y$95$m * x$95$m), $MachinePrecision] / N[(N[(z + 1.0), $MachinePrecision] * N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(y$95$s * N[(x$95$s * If[LessEqual[t$95$1, 0.001], N[(t$95$0 * N[(y$95$m / N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[(y$95$m / N[(z * N[(z * N[(N[(z + 1.0), $MachinePrecision] / x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y$95$m / z), $MachinePrecision] * N[(t$95$0 / z), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 x y) (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64)))) < 1e-3

   1. Initial program 89.8%

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

      1. *-commutative 89.8%

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

      2. sqr-neg 89.8%

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

      3. times-frac 93.0%

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

      4. sqr-neg 93.0%

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

   3. Simplified 93.0%

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

   if 1e-3 < (/.f64 (*.f64 x y) (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64)))) < +inf.0

   1. Initial program 86.0%

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

      1. *-commutative 86.0%

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

      2. frac-times 86.0%

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

      3. associate-*l/ 87.5%

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

      4. times-frac 92.1%

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

   4. Applied egg-rr 92.1%

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

      1. *-commutative 92.1%

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

      2. clear-num 92.2%

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

      3. frac-times 90.0%

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

      4. *-un-lft-identity 90.0%

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

      5. div-inv 89.9%

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

      6. clear-num 89.9%

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

   6. Applied egg-rr 89.9%

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

   if +inf.0 < (/.f64 (*.f64 x y) (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64))))

   1. Initial program 0.0%

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

      1. *-commutative 0.0%

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

      2. frac-times 40.5%

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

      3. associate-*l/ 38.9%

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

      4. times-frac 93.8%

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

   4. Applied egg-rr 93.8%

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

4. Final simplification 92.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y \cdot x}{\left(z + 1\right) \cdot \left(z \cdot z\right)} \leq 0.001:\\ \;\;\;\;\frac{x}{z + 1} \cdot \frac{y}{z \cdot z}\\ \mathbf{elif}\;\frac{y \cdot x}{\left(z + 1\right) \cdot \left(z \cdot z\right)} \leq \infty:\\ \;\;\;\;\frac{y}{z \cdot \left(z \cdot \frac{z + 1}{x}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z} \cdot \frac{\frac{x}{z + 1}}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 96.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ \begin{array}{l} t_0 := \left(z + 1\right) \cdot \left(z \cdot z\right)\\ y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -1 \cdot 10^{+16} \lor \neg \left(t\_0 \leq 5 \cdot 10^{-10}\right):\\ \;\;\;\;\frac{\frac{\frac{x\_m}{\frac{z}{y\_m}}}{z}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y\_m}{z \cdot \left(z \cdot \frac{z + 1}{x\_m}\right)}\\ \end{array}\right) \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) 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 (or (<= t_0 -1e+16) (not (<= t_0 5e-10)))
       (/ (/ (/ x_m (/ z y_m)) z) z)
       (/ y_m (* z (* z (/ (+ z 1.0) 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 t_0 = (z + 1.0) * (z * z);
	double tmp;
	if ((t_0 <= -1e+16) || !(t_0 <= 5e-10)) {
		tmp = ((x_m / (z / y_m)) / z) / z;
	} else {
		tmp = y_m / (z * (z * ((z + 1.0) / 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) :: t_0
    real(8) :: tmp
    t_0 = (z + 1.0d0) * (z * z)
    if ((t_0 <= (-1d+16)) .or. (.not. (t_0 <= 5d-10))) then
        tmp = ((x_m / (z / y_m)) / z) / z
    else
        tmp = y_m / (z * (z * ((z + 1.0d0) / 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 t_0 = (z + 1.0) * (z * z);
	double tmp;
	if ((t_0 <= -1e+16) || !(t_0 <= 5e-10)) {
		tmp = ((x_m / (z / y_m)) / z) / z;
	} else {
		tmp = y_m / (z * (z * ((z + 1.0) / 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):
	t_0 = (z + 1.0) * (z * z)
	tmp = 0
	if (t_0 <= -1e+16) or not (t_0 <= 5e-10):
		tmp = ((x_m / (z / y_m)) / z) / z
	else:
		tmp = y_m / (z * (z * ((z + 1.0) / 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)
	t_0 = Float64(Float64(z + 1.0) * Float64(z * z))
	tmp = 0.0
	if ((t_0 <= -1e+16) || !(t_0 <= 5e-10))
		tmp = Float64(Float64(Float64(x_m / Float64(z / y_m)) / z) / z);
	else
		tmp = Float64(y_m / Float64(z * Float64(z * Float64(Float64(z + 1.0) / 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)
	t_0 = (z + 1.0) * (z * z);
	tmp = 0.0;
	if ((t_0 <= -1e+16) || ~((t_0 <= 5e-10)))
		tmp = ((x_m / (z / y_m)) / z) / z;
	else
		tmp = y_m / (z * (z * ((z + 1.0) / 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_] := Block[{t$95$0 = N[(N[(z + 1.0), $MachinePrecision] * N[(z * z), $MachinePrecision]), $MachinePrecision]}, N[(y$95$s * N[(x$95$s * If[Or[LessEqual[t$95$0, -1e+16], N[Not[LessEqual[t$95$0, 5e-10]], $MachinePrecision]], N[(N[(N[(x$95$m / N[(z / y$95$m), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision] / z), $MachinePrecision], N[(y$95$m / N[(z * N[(z * N[(N[(z + 1.0), $MachinePrecision] / x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64))) < -1e16 or 5.00000000000000031e-10 < (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64)))

   1. Initial program 83.7%

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

      1. *-commutative 83.7%

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

      2. associate-/l* 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* 91.9%

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

      5. sqr-neg 91.9%

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

   3. Simplified 91.9%

      \[\leadsto \color{blue}{y \cdot \frac{\frac{x}{z \cdot z}}{z + 1}} \]
   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* 91.9%

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

      3. associate-/l* 94.1%

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

      4. associate-/r* 99.0%

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

      5. clear-num 99.0%

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

      6. un-div-inv 99.0%

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

   6. Applied egg-rr 99.0%

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

   7. Taylor expanded in z around inf 89.0%

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

      1. *-un-lft-identity 89.0%

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

      2. *-commutative 89.0%

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

      3. associate-/l* 98.8%

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

      4. clear-num 98.3%

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

      5. un-div-inv 98.3%

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

   9. Applied egg-rr 98.3%

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

   if -1e16 < (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64))) < 5.00000000000000031e-10

   1. Initial program 83.0%

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

      1. *-commutative 83.0%

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

      2. frac-times 85.2%

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

      3. associate-*l/ 83.0%

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

      4. times-frac 95.2%

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

   4. Applied egg-rr 95.2%

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

      1. *-commutative 95.2%

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

      2. clear-num 95.2%

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

      3. frac-times 90.3%

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

      4. *-un-lft-identity 90.3%

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

      5. div-inv 90.3%

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

      6. clear-num 90.3%

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

   6. Applied egg-rr 90.3%

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

4. Final simplification 93.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(z + 1\right) \cdot \left(z \cdot z\right) \leq -1 \cdot 10^{+16} \lor \neg \left(\left(z + 1\right) \cdot \left(z \cdot z\right) \leq 5 \cdot 10^{-10}\right):\\ \;\;\;\;\frac{\frac{\frac{x}{\frac{z}{y}}}{z}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z \cdot \left(z \cdot \frac{z + 1}{x}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 95.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -1 \lor \neg \left(z \leq 8 \cdot 10^{-20}\right):\\ \;\;\;\;\frac{\frac{\frac{y\_m}{\frac{z}{x\_m}}}{z}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y\_m}{z \cdot \frac{z}{x\_m}}\\ \end{array}\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) 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 8e-20)))
     (/ (/ (/ y_m (/ z x_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 <= 8e-20)) {
		tmp = ((y_m / (z / x_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 <= 8d-20))) then
        tmp = ((y_m / (z / x_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 <= 8e-20)) {
		tmp = ((y_m / (z / x_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 <= 8e-20):
		tmp = ((y_m / (z / x_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 <= 8e-20))
		tmp = Float64(Float64(Float64(y_m / Float64(z / x_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 <= 8e-20)))
		tmp = ((y_m / (z / x_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, 8e-20]], $MachinePrecision]], N[(N[(N[(y$95$m / N[(z / x$95$m), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision] / z), $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 7.99999999999999956e-20 < z

   1. Initial program 84.3%

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

      1. *-commutative 84.3%

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

      2. associate-/l* 86.6%

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

      3. sqr-neg 86.6%

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

      4. associate-/r* 92.3%

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

      5. sqr-neg 92.3%

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

   3. Simplified 92.3%

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

      1. associate-/l/ 86.6%

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

      2. associate-/r* 92.2%

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

      3. associate-/l* 94.3%

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

      4. associate-/r* 99.1%

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

      5. clear-num 99.0%

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

      6. un-div-inv 99.0%

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

   6. Applied egg-rr 99.0%

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

   7. Taylor expanded in z around inf 97.3%

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

   if -1 < z < 7.99999999999999956e-20

   1. Initial program 82.4%

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

      1. *-commutative 82.4%

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

      2. frac-times 84.7%

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

      3. associate-*l/ 82.4%

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

      4. times-frac 95.1%

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

   4. Applied egg-rr 95.1%

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

      1. *-commutative 95.1%

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

      2. clear-num 95.0%

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

      3. frac-times 90.0%

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

      4. *-un-lft-identity 90.0%

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

      5. div-inv 90.0%

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

      6. clear-num 90.0%

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

   6. Applied egg-rr 90.0%

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

   7. Taylor expanded in z around 0 89.4%

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

4. Final simplification 93.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1 \lor \neg \left(z \leq 8 \cdot 10^{-20}\right):\\ \;\;\;\;\frac{\frac{\frac{y}{\frac{z}{x}}}{z}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z \cdot \frac{z}{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 94.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -1 \lor \neg \left(z \leq 8 \cdot 10^{-20}\right):\\ \;\;\;\;\frac{y\_m}{z} \cdot \frac{\frac{x\_m}{z}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y\_m}{z \cdot \frac{z}{x\_m}}\\ \end{array}\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) 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 8e-20)))
     (* (/ y_m z) (/ (/ x_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 <= 8e-20)) {
		tmp = (y_m / z) * ((x_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 <= 8d-20))) then
        tmp = (y_m / z) * ((x_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 <= 8e-20)) {
		tmp = (y_m / z) * ((x_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 <= 8e-20):
		tmp = (y_m / z) * ((x_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 <= 8e-20))
		tmp = Float64(Float64(y_m / z) * Float64(Float64(x_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 <= 8e-20)))
		tmp = (y_m / z) * ((x_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, 8e-20]], $MachinePrecision]], N[(N[(y$95$m / z), $MachinePrecision] * N[(N[(x$95$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 7.99999999999999956e-20 < z

   1. Initial program 84.3%

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

      1. *-commutative 84.3%

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

      2. frac-times 92.0%

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

      3. associate-*l/ 94.3%

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

      4. times-frac 96.0%

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

   4. Applied egg-rr 96.0%

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

   5. Taylor expanded in z around inf 94.4%

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

   if -1 < z < 7.99999999999999956e-20

   1. Initial program 82.4%

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

      1. *-commutative 82.4%

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

      2. frac-times 84.7%

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

      3. associate-*l/ 82.4%

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

      4. times-frac 95.1%

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

   4. Applied egg-rr 95.1%

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

      1. *-commutative 95.1%

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

      2. clear-num 95.0%

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

      3. frac-times 90.0%

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

      4. *-un-lft-identity 90.0%

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

      5. div-inv 90.0%

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

      6. clear-num 90.0%

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

   6. Applied egg-rr 90.0%

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

   7. Taylor expanded in z around 0 89.4%

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

4. Final simplification 91.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1 \lor \neg \left(z \leq 8 \cdot 10^{-20}\right):\\ \;\;\;\;\frac{y}{z} \cdot \frac{\frac{x}{z}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z \cdot \frac{z}{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 89.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -1 \lor \neg \left(z \leq 8 \cdot 10^{-20}\right):\\ \;\;\;\;y\_m \cdot \frac{\frac{x\_m}{z \cdot z}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y\_m}{z \cdot \frac{z}{x\_m}}\\ \end{array}\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) 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 8e-20)))
     (* y_m (/ (/ x_m (* z 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 <= 8e-20)) {
		tmp = y_m * ((x_m / (z * 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 <= 8d-20))) then
        tmp = y_m * ((x_m / (z * 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 <= 8e-20)) {
		tmp = y_m * ((x_m / (z * 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 <= 8e-20):
		tmp = y_m * ((x_m / (z * 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 <= 8e-20))
		tmp = Float64(y_m * Float64(Float64(x_m / Float64(z * 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 <= 8e-20)))
		tmp = y_m * ((x_m / (z * 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, 8e-20]], $MachinePrecision]], N[(y$95$m * N[(N[(x$95$m / N[(z * z), $MachinePrecision]), $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 7.99999999999999956e-20 < z

   1. Initial program 84.3%

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

      1. *-commutative 84.3%

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

      2. associate-/l* 86.6%

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

      3. sqr-neg 86.6%

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

      4. associate-/r* 92.3%

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

      5. sqr-neg 92.3%

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

   3. Simplified 92.3%

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

   5. Taylor expanded in z around inf 90.9%

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

   if -1 < z < 7.99999999999999956e-20

   1. Initial program 82.4%

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

      1. *-commutative 82.4%

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

      2. frac-times 84.7%

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

      3. associate-*l/ 82.4%

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

      4. times-frac 95.1%

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

   4. Applied egg-rr 95.1%

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

      1. *-commutative 95.1%

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

      2. clear-num 95.0%

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

      3. frac-times 90.0%

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

      4. *-un-lft-identity 90.0%

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

      5. div-inv 90.0%

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

      6. clear-num 90.0%

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

   6. Applied egg-rr 90.0%

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

   7. Taylor expanded in z around 0 89.4%

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

4. Final simplification 90.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1 \lor \neg \left(z \leq 8 \cdot 10^{-20}\right):\\ \;\;\;\;y \cdot \frac{\frac{x}{z \cdot z}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z \cdot \frac{z}{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 94.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -1:\\ \;\;\;\;\frac{y\_m}{z \cdot z} \cdot \frac{x\_m}{z}\\ \mathbf{elif}\;z \leq 8 \cdot 10^{-20}:\\ \;\;\;\;\frac{y\_m}{z \cdot \frac{z}{x\_m}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y\_m}{z} \cdot \frac{\frac{x\_m}{z}}{z}\\ \end{array}\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) 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)
     (* (/ y_m (* z z)) (/ x_m z))
     (if (<= z 8e-20)
       (/ y_m (* z (/ z x_m)))
       (* (/ y_m z) (/ (/ x_m z) z)))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
assert(x_m < y_m && y_m < z);
double code(double y_s, double x_s, double x_m, double y_m, double z) {
	double tmp;
	if (z <= -1.0) {
		tmp = (y_m / (z * z)) * (x_m / z);
	} else if (z <= 8e-20) {
		tmp = y_m / (z * (z / x_m));
	} else {
		tmp = (y_m / z) * ((x_m / z) / z);
	}
	return y_s * (x_s * tmp);
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
y\_m = abs(y)
y\_s = copysign(1.0d0, y)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
real(8) function code(y_s, x_s, x_m, y_m, z)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if (z <= (-1.0d0)) then
        tmp = (y_m / (z * z)) * (x_m / z)
    else if (z <= 8d-20) then
        tmp = y_m / (z * (z / x_m))
    else
        tmp = (y_m / z) * ((x_m / z) / z)
    end if
    code = y_s * (x_s * tmp)
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
assert x_m < y_m && y_m < z;
public static double code(double y_s, double x_s, double x_m, double y_m, double z) {
	double tmp;
	if (z <= -1.0) {
		tmp = (y_m / (z * z)) * (x_m / z);
	} else if (z <= 8e-20) {
		tmp = y_m / (z * (z / x_m));
	} else {
		tmp = (y_m / z) * ((x_m / z) / z);
	}
	return y_s * (x_s * tmp);
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
[x_m, y_m, z] = sort([x_m, y_m, z])
def code(y_s, x_s, x_m, y_m, z):
	tmp = 0
	if z <= -1.0:
		tmp = (y_m / (z * z)) * (x_m / z)
	elif z <= 8e-20:
		tmp = y_m / (z * (z / x_m))
	else:
		tmp = (y_m / z) * ((x_m / z) / z)
	return y_s * (x_s * tmp)

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
y\_m = abs(y)
y\_s = copysign(1.0, y)
x_m, y_m, z = sort([x_m, y_m, z])
function code(y_s, x_s, x_m, y_m, z)
	tmp = 0.0
	if (z <= -1.0)
		tmp = Float64(Float64(y_m / Float64(z * z)) * Float64(x_m / z));
	elseif (z <= 8e-20)
		tmp = Float64(y_m / Float64(z * Float64(z / x_m)));
	else
		tmp = Float64(Float64(y_m / z) * Float64(Float64(x_m / z) / z));
	end
	return Float64(y_s * Float64(x_s * tmp))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
x_m, y_m, z = num2cell(sort([x_m, y_m, z])){:}
function tmp_2 = code(y_s, x_s, x_m, y_m, z)
	tmp = 0.0;
	if (z <= -1.0)
		tmp = (y_m / (z * z)) * (x_m / z);
	elseif (z <= 8e-20)
		tmp = y_m / (z * (z / x_m));
	else
		tmp = (y_m / z) * ((x_m / z) / z);
	end
	tmp_2 = y_s * (x_s * tmp);
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
code[y$95$s_, x$95$s_, x$95$m_, y$95$m_, z_] := N[(y$95$s * N[(x$95$s * If[LessEqual[z, -1.0], N[(N[(y$95$m / N[(z * z), $MachinePrecision]), $MachinePrecision] * N[(x$95$m / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 8e-20], N[(y$95$m / N[(z * N[(z / x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y$95$m / z), $MachinePrecision] * N[(N[(x$95$m / z), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if z < -1

   1. Initial program 85.0%

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

      1. *-commutative 85.0%

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

      2. sqr-neg 85.0%

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

      3. times-frac 91.8%

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

      4. sqr-neg 91.8%

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

   3. Simplified 91.8%

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

   5. Taylor expanded in z around inf 88.3%

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

   if -1 < z < 7.99999999999999956e-20

   1. Initial program 82.4%

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

      1. *-commutative 82.4%

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

      2. frac-times 84.7%

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

      3. associate-*l/ 82.4%

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

      4. times-frac 95.1%

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

   4. Applied egg-rr 95.1%

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

      1. *-commutative 95.1%

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

      2. clear-num 95.0%

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

      3. frac-times 90.0%

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

      4. *-un-lft-identity 90.0%

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

      5. div-inv 90.0%

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

      6. clear-num 90.0%

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

   6. Applied egg-rr 90.0%

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

   7. Taylor expanded in z around 0 89.4%

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

   if 7.99999999999999956e-20 < z

   1. Initial program 83.7%

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

      1. *-commutative 83.7%

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

      2. frac-times 92.1%

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

      3. associate-*l/ 93.7%

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

      4. times-frac 97.3%

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

   4. Applied egg-rr 97.3%

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

   5. Taylor expanded in z around inf 97.3%

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

4. Final simplification 91.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1:\\ \;\;\;\;\frac{y}{z \cdot z} \cdot \frac{x}{z}\\ \mathbf{elif}\;z \leq 8 \cdot 10^{-20}:\\ \;\;\;\;\frac{y}{z \cdot \frac{z}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z} \cdot \frac{\frac{x}{z}}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 95.6% accurate, 1.0× speedup

Math

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

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) 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 (+ 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) {
	return y_s * (x_s * ((y_m / z) * ((x_m / (z + 1.0)) / 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) * ((x_m / (z + 1.0d0)) / 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) * ((x_m / (z + 1.0)) / 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) * ((x_m / (z + 1.0)) / 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(y_m / z) * Float64(Float64(x_m / Float64(z + 1.0)) / 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) * ((x_m / (z + 1.0)) / 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[(y$95$m / z), $MachinePrecision] * N[(N[(x$95$m / N[(z + 1.0), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 83.3%

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

   1. *-commutative 83.3%

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

   2. frac-times 88.1%

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

   3. associate-*l/ 87.9%

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

   4. times-frac 95.5%

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

4. Applied egg-rr 95.5%

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

Alternative 9: 80.3% 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 #s(literal 1 binary64) x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) 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 83.3%

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

   1. *-commutative 83.3%

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

   2. frac-times 88.1%

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

   3. associate-*l/ 87.9%

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

   4. times-frac 95.5%

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

4. Applied egg-rr 95.5%

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

   1. *-commutative 95.5%

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

   2. clear-num 95.2%

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

   3. frac-times 90.8%

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

   4. *-un-lft-identity 90.8%

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

   5. div-inv 90.8%

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

   6. clear-num 90.8%

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

6. Applied egg-rr 90.8%

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

7. Taylor expanded in z around 0 75.8%

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

8. Final simplification 75.8%

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

Alternative 10: 79.8% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 83.3%

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

   1. *-commutative 83.3%

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

   2. sqr-neg 83.3%

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

   3. times-frac 88.1%

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

   4. sqr-neg 88.1%

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

3. Simplified 88.1%

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

5. Taylor expanded in z around 0 73.7%

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

   1. *-commutative 73.7%

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

   2. associate-/l* 71.5%

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

   3. associate-/r* 71.7%

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

   4. *-commutative 71.7%

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

7. Applied egg-rr 71.7%

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

   1. associate-/l* 74.7%

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

   2. associate-/l* 74.8%

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

9. Applied egg-rr 74.8%

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

Developer Target 1: 97.1% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024132 
(FPCore (x y z)
  :name "Statistics.Distribution.Beta:$cvariance from math-functions-0.1.5.2"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< z 2496182814532307/10000000000000) (/ (* y (/ x z)) (+ z (* z z))) (/ (* (/ (/ y z) (+ 1 z)) x) z)))

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