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

Percentage Accurate: 83.4% → 96.9%

Time: 10.5s

Alternatives: 13

Speedup: 1.0×

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 83.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 96.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ \begin{array}{l} t_0 := z \cdot \frac{z}{y\_m}\\ y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -2.15 \cdot 10^{+54}:\\ \;\;\;\;\frac{\frac{x\_m}{z}}{t\_0}\\ \mathbf{elif}\;z \leq -6.6 \cdot 10^{-131}:\\ \;\;\;\;y\_m \cdot \frac{\frac{x\_m}{z \cdot z}}{z + 1}\\ \mathbf{elif}\;z \leq 18000000000:\\ \;\;\;\;\frac{\frac{y\_m}{\frac{z}{x\_m}}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m}{\left(z + 1\right) \cdot t\_0}\\ \end{array}\right) \end{array} \end{array} \]

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 (/ z y_m))))
   (*
    y_s
    (*
     x_s
     (if (<= z -2.15e+54)
       (/ (/ x_m z) t_0)
       (if (<= z -6.6e-131)
         (* y_m (/ (/ x_m (* z z)) (+ z 1.0)))
         (if (<= z 18000000000.0)
           (/ (/ y_m (/ z x_m)) z)
           (/ x_m (* (+ z 1.0) t_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 * (z / y_m);
	double tmp;
	if (z <= -2.15e+54) {
		tmp = (x_m / z) / t_0;
	} else if (z <= -6.6e-131) {
		tmp = y_m * ((x_m / (z * z)) / (z + 1.0));
	} else if (z <= 18000000000.0) {
		tmp = (y_m / (z / x_m)) / z;
	} else {
		tmp = x_m / ((z + 1.0) * t_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 * (z / y_m)
    if (z <= (-2.15d+54)) then
        tmp = (x_m / z) / t_0
    else if (z <= (-6.6d-131)) then
        tmp = y_m * ((x_m / (z * z)) / (z + 1.0d0))
    else if (z <= 18000000000.0d0) then
        tmp = (y_m / (z / x_m)) / z
    else
        tmp = x_m / ((z + 1.0d0) * t_0)
    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 * (z / y_m);
	double tmp;
	if (z <= -2.15e+54) {
		tmp = (x_m / z) / t_0;
	} else if (z <= -6.6e-131) {
		tmp = y_m * ((x_m / (z * z)) / (z + 1.0));
	} else if (z <= 18000000000.0) {
		tmp = (y_m / (z / x_m)) / z;
	} else {
		tmp = x_m / ((z + 1.0) * t_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 * (z / y_m)
	tmp = 0
	if z <= -2.15e+54:
		tmp = (x_m / z) / t_0
	elif z <= -6.6e-131:
		tmp = y_m * ((x_m / (z * z)) / (z + 1.0))
	elif z <= 18000000000.0:
		tmp = (y_m / (z / x_m)) / z
	else:
		tmp = x_m / ((z + 1.0) * t_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(z * Float64(z / y_m))
	tmp = 0.0
	if (z <= -2.15e+54)
		tmp = Float64(Float64(x_m / z) / t_0);
	elseif (z <= -6.6e-131)
		tmp = Float64(y_m * Float64(Float64(x_m / Float64(z * z)) / Float64(z + 1.0)));
	elseif (z <= 18000000000.0)
		tmp = Float64(Float64(y_m / Float64(z / x_m)) / z);
	else
		tmp = Float64(x_m / Float64(Float64(z + 1.0) * t_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 * (z / y_m);
	tmp = 0.0;
	if (z <= -2.15e+54)
		tmp = (x_m / z) / t_0;
	elseif (z <= -6.6e-131)
		tmp = y_m * ((x_m / (z * z)) / (z + 1.0));
	elseif (z <= 18000000000.0)
		tmp = (y_m / (z / x_m)) / z;
	else
		tmp = x_m / ((z + 1.0) * t_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[(z * N[(z / y$95$m), $MachinePrecision]), $MachinePrecision]}, N[(y$95$s * N[(x$95$s * If[LessEqual[z, -2.15e+54], N[(N[(x$95$m / z), $MachinePrecision] / t$95$0), $MachinePrecision], If[LessEqual[z, -6.6e-131], N[(y$95$m * N[(N[(x$95$m / N[(z * z), $MachinePrecision]), $MachinePrecision] / N[(z + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 18000000000.0], N[(N[(y$95$m / N[(z / x$95$m), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], N[(x$95$m / N[(N[(z + 1.0), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 4 regimes

2. if z < -2.14999999999999988e54

   1. Initial program 75.7%

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

      1. *-commutative 75.7%

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

      2. frac-times 85.5%

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

      3. associate-*l/ 87.1%

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

      4. times-frac 92.7%

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

   4. Applied egg-rr 92.7%

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

      1. clear-num 92.6%

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

      2. frac-times 92.8%

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

      3. *-un-lft-identity 92.8%

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

   6. Applied egg-rr 92.8%

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

   7. Taylor expanded in z around inf 92.8%

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

   if -2.14999999999999988e54 < z < -6.6000000000000004e-131

   1. Initial program 97.0%

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

      1. *-commutative 97.0%

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

      2. associate-/l* 92.1%

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

      3. sqr-neg 92.1%

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

      4. associate-/r* 92.1%

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

      5. sqr-neg 92.1%

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

   3. Simplified 92.1%

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

   if -6.6000000000000004e-131 < z < 1.8e10

   1. Initial program 85.3%

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

      1. *-commutative 85.3%

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

      2. sqr-neg 85.3%

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

      3. times-frac 84.5%

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

      4. sqr-neg 84.5%

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

   3. Simplified 84.5%

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

   5. Taylor expanded in z around 0 83.7%

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

      1. associate-*l/ 84.6%

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

      2. *-commutative 84.6%

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

      3. associate-/r* 92.1%

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

      4. associate-*l/ 97.4%

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

      5. clear-num 97.3%

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

      6. associate-*l/ 97.4%

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

      7. *-un-lft-identity 97.4%

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

   7. Applied egg-rr 97.4%

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

   if 1.8e10 < z

   1. Initial program 79.6%

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

      1. *-commutative 79.6%

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

      2. associate-/l* 88.9%

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

      3. sqr-neg 88.9%

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

      4. associate-/r* 94.3%

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

      5. sqr-neg 94.3%

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

   3. Simplified 94.3%

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

      1. associate-/r* 88.9%

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

      2. associate-*r/ 79.6%

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

      3. frac-times 89.7%

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

      4. clear-num 87.4%

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

      5. frac-times 85.7%

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

      6. *-un-lft-identity 85.7%

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

      7. pow2 85.7%

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

   6. Applied egg-rr 85.7%

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

      1. pow2 85.7%

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

      2. div-inv 85.7%

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

      3. associate-*l* 85.7%

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

      4. div-inv 85.7%

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

   8. Applied egg-rr 85.7%

      \[\leadsto \frac{x}{\color{blue}{\left(z \cdot \frac{z}{y}\right)} \cdot \left(z + 1\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 93.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.15 \cdot 10^{+54}:\\ \;\;\;\;\frac{\frac{x}{z}}{z \cdot \frac{z}{y}}\\ \mathbf{elif}\;z \leq -6.6 \cdot 10^{-131}:\\ \;\;\;\;y \cdot \frac{\frac{x}{z \cdot z}}{z + 1}\\ \mathbf{elif}\;z \leq 18000000000:\\ \;\;\;\;\frac{\frac{y}{\frac{z}{x}}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\left(z + 1\right) \cdot \left(z \cdot \frac{z}{y}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 97.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ \begin{array}{l} t_0 := \frac{x\_m \cdot y\_m}{\left(z + 1\right) \cdot \left(z \cdot z\right)}\\ y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq 0:\\ \;\;\;\;\frac{x\_m}{\left(z + 1\right) \cdot \left(z \cdot \frac{z}{y\_m}\right)}\\ \mathbf{elif}\;t\_0 \leq 4 \cdot 10^{+151}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{y\_m}{z} \cdot \frac{\frac{x\_m}{z + 1}}{z}\\ \end{array}\right) \end{array} \end{array} \]

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 y_m) (* (+ z 1.0) (* z z)))))
   (*
    y_s
    (*
     x_s
     (if (<= t_0 0.0)
       (/ x_m (* (+ z 1.0) (* z (/ z y_m))))
       (if (<= t_0 4e+151) t_0 (* (/ y_m z) (/ (/ x_m (+ z 1.0)) z))))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
assert(x_m < y_m && y_m < z);
double code(double y_s, double x_s, double x_m, double y_m, double z) {
	double t_0 = (x_m * y_m) / ((z + 1.0) * (z * z));
	double tmp;
	if (t_0 <= 0.0) {
		tmp = x_m / ((z + 1.0) * (z * (z / y_m)));
	} else if (t_0 <= 4e+151) {
		tmp = t_0;
	} else {
		tmp = (y_m / z) * ((x_m / (z + 1.0)) / z);
	}
	return y_s * (x_s * tmp);
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
y\_m = abs(y)
y\_s = copysign(1.0d0, y)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
real(8) function code(y_s, x_s, x_m, y_m, z)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (x_m * y_m) / ((z + 1.0d0) * (z * z))
    if (t_0 <= 0.0d0) then
        tmp = x_m / ((z + 1.0d0) * (z * (z / y_m)))
    else if (t_0 <= 4d+151) then
        tmp = t_0
    else
        tmp = (y_m / z) * ((x_m / (z + 1.0d0)) / z)
    end if
    code = y_s * (x_s * tmp)
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
assert x_m < y_m && y_m < z;
public static double code(double y_s, double x_s, double x_m, double y_m, double z) {
	double t_0 = (x_m * y_m) / ((z + 1.0) * (z * z));
	double tmp;
	if (t_0 <= 0.0) {
		tmp = x_m / ((z + 1.0) * (z * (z / y_m)));
	} else if (t_0 <= 4e+151) {
		tmp = t_0;
	} else {
		tmp = (y_m / z) * ((x_m / (z + 1.0)) / z);
	}
	return y_s * (x_s * tmp);
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
[x_m, y_m, z] = sort([x_m, y_m, z])
def code(y_s, x_s, x_m, y_m, z):
	t_0 = (x_m * y_m) / ((z + 1.0) * (z * z))
	tmp = 0
	if t_0 <= 0.0:
		tmp = x_m / ((z + 1.0) * (z * (z / y_m)))
	elif t_0 <= 4e+151:
		tmp = t_0
	else:
		tmp = (y_m / z) * ((x_m / (z + 1.0)) / z)
	return y_s * (x_s * tmp)

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
y\_m = abs(y)
y\_s = copysign(1.0, y)
x_m, y_m, z = sort([x_m, y_m, z])
function code(y_s, x_s, x_m, y_m, z)
	t_0 = Float64(Float64(x_m * y_m) / Float64(Float64(z + 1.0) * Float64(z * z)))
	tmp = 0.0
	if (t_0 <= 0.0)
		tmp = Float64(x_m / Float64(Float64(z + 1.0) * Float64(z * Float64(z / y_m))));
	elseif (t_0 <= 4e+151)
		tmp = t_0;
	else
		tmp = Float64(Float64(y_m / z) * Float64(Float64(x_m / Float64(z + 1.0)) / z));
	end
	return Float64(y_s * Float64(x_s * tmp))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
x_m, y_m, z = num2cell(sort([x_m, y_m, z])){:}
function tmp_2 = code(y_s, x_s, x_m, y_m, z)
	t_0 = (x_m * y_m) / ((z + 1.0) * (z * z));
	tmp = 0.0;
	if (t_0 <= 0.0)
		tmp = x_m / ((z + 1.0) * (z * (z / y_m)));
	elseif (t_0 <= 4e+151)
		tmp = t_0;
	else
		tmp = (y_m / z) * ((x_m / (z + 1.0)) / z);
	end
	tmp_2 = y_s * (x_s * tmp);
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
code[y$95$s_, x$95$s_, x$95$m_, y$95$m_, z_] := Block[{t$95$0 = N[(N[(x$95$m * y$95$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$0, 0.0], N[(x$95$m / N[(N[(z + 1.0), $MachinePrecision] * N[(z * N[(z / y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 4e+151], t$95$0, N[(N[(y$95$m / z), $MachinePrecision] * N[(N[(x$95$m / N[(z + 1.0), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 x y) (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64)))) < -0.0

   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. *-commutative 90.0%

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

      2. associate-/l* 88.2%

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

      3. sqr-neg 88.2%

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

      4. associate-/r* 90.0%

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

      5. sqr-neg 90.0%

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

   3. Simplified 90.0%

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

      1. associate-/r* 88.2%

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

      2. associate-*r/ 90.0%

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

      3. frac-times 90.5%

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

      4. clear-num 89.7%

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

      5. frac-times 88.6%

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

      6. *-un-lft-identity 88.6%

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

      7. pow2 88.6%

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

   6. Applied egg-rr 88.6%

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

      1. pow2 88.6%

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

      2. div-inv 88.6%

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

      3. associate-*l* 90.8%

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

      4. div-inv 90.8%

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

   8. Applied egg-rr 90.8%

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

   if -0.0 < (/.f64 (*.f64 x y) (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64)))) < 4.00000000000000007e151

   1. Initial program 98.4%

      \[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \]
   2. Add Preprocessing

   if 4.00000000000000007e151 < (/.f64 (*.f64 x y) (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64))))

   1. Initial program 66.6%

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

      1. *-commutative 66.6%

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

      2. frac-times 80.6%

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

      3. associate-*l/ 79.0%

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

      4. times-frac 95.9%

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

   4. Applied egg-rr 95.9%

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

4. Final simplification 92.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot y}{\left(z + 1\right) \cdot \left(z \cdot z\right)} \leq 0:\\ \;\;\;\;\frac{x}{\left(z + 1\right) \cdot \left(z \cdot \frac{z}{y}\right)}\\ \mathbf{elif}\;\frac{x \cdot y}{\left(z + 1\right) \cdot \left(z \cdot z\right)} \leq 4 \cdot 10^{+151}:\\ \;\;\;\;\frac{x \cdot y}{\left(z + 1\right) \cdot \left(z \cdot z\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z} \cdot \frac{\frac{x}{z + 1}}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 95.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ \begin{array}{l} t_0 := \frac{\frac{x\_m}{z}}{z \cdot \frac{z}{y\_m}}\\ y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -2.15 \cdot 10^{+54}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq -5.4 \cdot 10^{-26}:\\ \;\;\;\;y\_m \cdot \frac{\frac{x\_m}{z \cdot z}}{z + 1}\\ \mathbf{elif}\;z \leq 8.5 \cdot 10^{-40}:\\ \;\;\;\;\frac{\frac{y\_m}{\frac{z}{x\_m}}}{z}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array}\right) \end{array} \end{array} \]

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) (* z (/ z y_m)))))
   (*
    y_s
    (*
     x_s
     (if (<= z -2.15e+54)
       t_0
       (if (<= z -5.4e-26)
         (* y_m (/ (/ x_m (* z z)) (+ z 1.0)))
         (if (<= z 8.5e-40) (/ (/ y_m (/ z x_m)) z) t_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 = (x_m / z) / (z * (z / y_m));
	double tmp;
	if (z <= -2.15e+54) {
		tmp = t_0;
	} else if (z <= -5.4e-26) {
		tmp = y_m * ((x_m / (z * z)) / (z + 1.0));
	} else if (z <= 8.5e-40) {
		tmp = (y_m / (z / x_m)) / z;
	} else {
		tmp = t_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 = (x_m / z) / (z * (z / y_m))
    if (z <= (-2.15d+54)) then
        tmp = t_0
    else if (z <= (-5.4d-26)) then
        tmp = y_m * ((x_m / (z * z)) / (z + 1.0d0))
    else if (z <= 8.5d-40) then
        tmp = (y_m / (z / x_m)) / z
    else
        tmp = t_0
    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 = (x_m / z) / (z * (z / y_m));
	double tmp;
	if (z <= -2.15e+54) {
		tmp = t_0;
	} else if (z <= -5.4e-26) {
		tmp = y_m * ((x_m / (z * z)) / (z + 1.0));
	} else if (z <= 8.5e-40) {
		tmp = (y_m / (z / x_m)) / z;
	} else {
		tmp = t_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 = (x_m / z) / (z * (z / y_m))
	tmp = 0
	if z <= -2.15e+54:
		tmp = t_0
	elif z <= -5.4e-26:
		tmp = y_m * ((x_m / (z * z)) / (z + 1.0))
	elif z <= 8.5e-40:
		tmp = (y_m / (z / x_m)) / z
	else:
		tmp = t_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(x_m / z) / Float64(z * Float64(z / y_m)))
	tmp = 0.0
	if (z <= -2.15e+54)
		tmp = t_0;
	elseif (z <= -5.4e-26)
		tmp = Float64(y_m * Float64(Float64(x_m / Float64(z * z)) / Float64(z + 1.0)));
	elseif (z <= 8.5e-40)
		tmp = Float64(Float64(y_m / Float64(z / x_m)) / z);
	else
		tmp = t_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 = (x_m / z) / (z * (z / y_m));
	tmp = 0.0;
	if (z <= -2.15e+54)
		tmp = t_0;
	elseif (z <= -5.4e-26)
		tmp = y_m * ((x_m / (z * z)) / (z + 1.0));
	elseif (z <= 8.5e-40)
		tmp = (y_m / (z / x_m)) / z;
	else
		tmp = t_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[(x$95$m / z), $MachinePrecision] / N[(z * N[(z / y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(y$95$s * N[(x$95$s * If[LessEqual[z, -2.15e+54], t$95$0, If[LessEqual[z, -5.4e-26], N[(y$95$m * N[(N[(x$95$m / N[(z * z), $MachinePrecision]), $MachinePrecision] / N[(z + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 8.5e-40], N[(N[(y$95$m / N[(z / x$95$m), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], t$95$0]]]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if z < -2.14999999999999988e54 or 8.4999999999999998e-40 < z

   1. Initial program 78.9%

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

      1. *-commutative 78.9%

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

      2. frac-times 88.3%

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

      3. associate-*l/ 90.5%

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

      4. times-frac 93.4%

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

   4. Applied egg-rr 93.4%

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

      1. clear-num 93.4%

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

      2. frac-times 92.3%

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

      3. *-un-lft-identity 92.3%

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

   6. Applied egg-rr 92.3%

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

   7. Taylor expanded in z around inf 90.4%

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

   if -2.14999999999999988e54 < z < -5.39999999999999963e-26

   1. Initial program 94.9%

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

      1. *-commutative 94.9%

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

      2. associate-/l* 99.5%

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

      3. sqr-neg 99.5%

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

      4. associate-/r* 99.5%

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

      5. sqr-neg 99.5%

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

   3. Simplified 99.5%

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

   if -5.39999999999999963e-26 < z < 8.4999999999999998e-40

   1. Initial program 86.7%

      \[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \]
   2. 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. sqr-neg 86.7%

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

      3. times-frac 85.9%

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

      4. sqr-neg 85.9%

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

   3. Simplified 85.9%

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

   5. Taylor expanded in z around 0 85.9%

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

      1. associate-*l/ 86.7%

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

      2. *-commutative 86.7%

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

      3. associate-/r* 93.5%

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

      4. associate-*l/ 97.5%

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

      5. clear-num 97.5%

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

      6. associate-*l/ 98.1%

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

      7. *-un-lft-identity 98.1%

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

   7. Applied egg-rr 98.1%

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

4. Final simplification 94.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.15 \cdot 10^{+54}:\\ \;\;\;\;\frac{\frac{x}{z}}{z \cdot \frac{z}{y}}\\ \mathbf{elif}\;z \leq -5.4 \cdot 10^{-26}:\\ \;\;\;\;y \cdot \frac{\frac{x}{z \cdot z}}{z + 1}\\ \mathbf{elif}\;z \leq 8.5 \cdot 10^{-40}:\\ \;\;\;\;\frac{\frac{y}{\frac{z}{x}}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{z}}{z \cdot \frac{z}{y}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 96.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -4.1 \cdot 10^{+55}:\\ \;\;\;\;\frac{\frac{x\_m}{z}}{z \cdot \frac{z}{y\_m}}\\ \mathbf{elif}\;z \leq -1 \cdot 10^{-131}:\\ \;\;\;\;y\_m \cdot \frac{\frac{x\_m}{z \cdot z}}{z + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{y\_m}{z} \cdot \frac{\frac{x\_m}{z + 1}}{z}\\ \end{array}\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #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 -4.1e+55)
     (/ (/ x_m z) (* z (/ z y_m)))
     (if (<= z -1e-131)
       (* y_m (/ (/ x_m (* z z)) (+ z 1.0)))
       (* (/ y_m z) (/ (/ x_m (+ z 1.0)) z)))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
assert(x_m < y_m && y_m < z);
double code(double y_s, double x_s, double x_m, double y_m, double z) {
	double tmp;
	if (z <= -4.1e+55) {
		tmp = (x_m / z) / (z * (z / y_m));
	} else if (z <= -1e-131) {
		tmp = y_m * ((x_m / (z * z)) / (z + 1.0));
	} else {
		tmp = (y_m / z) * ((x_m / (z + 1.0)) / z);
	}
	return y_s * (x_s * tmp);
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
y\_m = abs(y)
y\_s = copysign(1.0d0, y)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
real(8) function code(y_s, x_s, x_m, y_m, z)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if (z <= (-4.1d+55)) then
        tmp = (x_m / z) / (z * (z / y_m))
    else if (z <= (-1d-131)) then
        tmp = y_m * ((x_m / (z * z)) / (z + 1.0d0))
    else
        tmp = (y_m / z) * ((x_m / (z + 1.0d0)) / z)
    end if
    code = y_s * (x_s * tmp)
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
assert x_m < y_m && y_m < z;
public static double code(double y_s, double x_s, double x_m, double y_m, double z) {
	double tmp;
	if (z <= -4.1e+55) {
		tmp = (x_m / z) / (z * (z / y_m));
	} else if (z <= -1e-131) {
		tmp = y_m * ((x_m / (z * z)) / (z + 1.0));
	} else {
		tmp = (y_m / z) * ((x_m / (z + 1.0)) / z);
	}
	return y_s * (x_s * tmp);
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
[x_m, y_m, z] = sort([x_m, y_m, z])
def code(y_s, x_s, x_m, y_m, z):
	tmp = 0
	if z <= -4.1e+55:
		tmp = (x_m / z) / (z * (z / y_m))
	elif z <= -1e-131:
		tmp = y_m * ((x_m / (z * z)) / (z + 1.0))
	else:
		tmp = (y_m / z) * ((x_m / (z + 1.0)) / z)
	return y_s * (x_s * tmp)

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
y\_m = abs(y)
y\_s = copysign(1.0, y)
x_m, y_m, z = sort([x_m, y_m, z])
function code(y_s, x_s, x_m, y_m, z)
	tmp = 0.0
	if (z <= -4.1e+55)
		tmp = Float64(Float64(x_m / z) / Float64(z * Float64(z / y_m)));
	elseif (z <= -1e-131)
		tmp = Float64(y_m * Float64(Float64(x_m / Float64(z * z)) / Float64(z + 1.0)));
	else
		tmp = Float64(Float64(y_m / z) * Float64(Float64(x_m / Float64(z + 1.0)) / z));
	end
	return Float64(y_s * Float64(x_s * tmp))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
x_m, y_m, z = num2cell(sort([x_m, y_m, z])){:}
function tmp_2 = code(y_s, x_s, x_m, y_m, z)
	tmp = 0.0;
	if (z <= -4.1e+55)
		tmp = (x_m / z) / (z * (z / y_m));
	elseif (z <= -1e-131)
		tmp = y_m * ((x_m / (z * z)) / (z + 1.0));
	else
		tmp = (y_m / z) * ((x_m / (z + 1.0)) / z);
	end
	tmp_2 = y_s * (x_s * tmp);
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
code[y$95$s_, x$95$s_, x$95$m_, y$95$m_, z_] := N[(y$95$s * N[(x$95$s * If[LessEqual[z, -4.1e+55], N[(N[(x$95$m / z), $MachinePrecision] / N[(z * N[(z / y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, -1e-131], N[(y$95$m * N[(N[(x$95$m / N[(z * z), $MachinePrecision]), $MachinePrecision] / N[(z + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y$95$m / z), $MachinePrecision] * N[(N[(x$95$m / N[(z + 1.0), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if z < -4.09999999999999981e55

   1. Initial program 75.7%

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

      1. *-commutative 75.7%

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

      2. frac-times 85.5%

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

      3. associate-*l/ 87.1%

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

      4. times-frac 92.7%

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

   4. Applied egg-rr 92.7%

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

      1. clear-num 92.6%

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

      2. frac-times 92.8%

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

      3. *-un-lft-identity 92.8%

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

   6. Applied egg-rr 92.8%

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

   7. Taylor expanded in z around inf 92.8%

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

   if -4.09999999999999981e55 < z < -9.9999999999999999e-132

   1. Initial program 97.0%

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

      1. *-commutative 97.0%

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

      2. associate-/l* 92.1%

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

      3. sqr-neg 92.1%

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

      4. associate-/r* 92.1%

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

      5. sqr-neg 92.1%

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

   3. Simplified 92.1%

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

   if -9.9999999999999999e-132 < z

   1. Initial program 83.5%

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

      1. *-commutative 83.5%

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

      2. frac-times 86.2%

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

      3. associate-*l/ 87.7%

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

      4. times-frac 97.3%

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

   4. Applied egg-rr 97.3%

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

4. Final simplification 95.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -4.1 \cdot 10^{+55}:\\ \;\;\;\;\frac{\frac{x}{z}}{z \cdot \frac{z}{y}}\\ \mathbf{elif}\;z \leq -1 \cdot 10^{-131}:\\ \;\;\;\;y \cdot \frac{\frac{x}{z \cdot z}}{z + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z} \cdot \frac{\frac{x}{z + 1}}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 93.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -215000000 \lor \neg \left(z \leq 8.4 \cdot 10^{-55}\right):\\ \;\;\;\;\frac{\frac{x\_m}{z}}{z \cdot \frac{z}{y\_m}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y\_m}{\frac{z}{x\_m}}}{z}\\ \end{array}\right) \end{array} \]

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 (or (<= z -215000000.0) (not (<= z 8.4e-55)))
     (/ (/ x_m z) (* z (/ z y_m)))
     (/ (/ y_m (/ z x_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 <= -215000000.0) || !(z <= 8.4e-55)) {
		tmp = (x_m / z) / (z * (z / y_m));
	} else {
		tmp = (y_m / (z / x_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 <= (-215000000.0d0)) .or. (.not. (z <= 8.4d-55))) then
        tmp = (x_m / z) / (z * (z / y_m))
    else
        tmp = (y_m / (z / x_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 <= -215000000.0) || !(z <= 8.4e-55)) {
		tmp = (x_m / z) / (z * (z / y_m));
	} else {
		tmp = (y_m / (z / x_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 <= -215000000.0) or not (z <= 8.4e-55):
		tmp = (x_m / z) / (z * (z / y_m))
	else:
		tmp = (y_m / (z / x_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 <= -215000000.0) || !(z <= 8.4e-55))
		tmp = Float64(Float64(x_m / z) / Float64(z * Float64(z / y_m)));
	else
		tmp = Float64(Float64(y_m / Float64(z / x_m)) / 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 <= -215000000.0) || ~((z <= 8.4e-55)))
		tmp = (x_m / z) / (z * (z / y_m));
	else
		tmp = (y_m / (z / x_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, -215000000.0], N[Not[LessEqual[z, 8.4e-55]], $MachinePrecision]], N[(N[(x$95$m / z), $MachinePrecision] / N[(z * N[(z / y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y$95$m / N[(z / x$95$m), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < -2.15e8 or 8.4000000000000006e-55 < z

   1. Initial program 80.5%

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

      1. *-commutative 80.5%

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

      2. frac-times 88.8%

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

      3. associate-*l/ 91.5%

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

      4. times-frac 93.4%

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

   4. Applied egg-rr 93.4%

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

      1. clear-num 93.4%

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

      2. frac-times 92.4%

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

      3. *-un-lft-identity 92.4%

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

   6. Applied egg-rr 92.4%

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

   7. Taylor expanded in z around inf 88.5%

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

   if -2.15e8 < z < 8.4000000000000006e-55

   1. Initial program 87.2%

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

      1. *-commutative 87.2%

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

      2. sqr-neg 87.2%

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

      3. times-frac 86.5%

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

      4. sqr-neg 86.5%

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

   3. Simplified 86.5%

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

   5. Taylor expanded in z around 0 83.3%

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

      1. associate-*l/ 84.0%

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

      2. *-commutative 84.0%

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

      3. associate-/r* 90.5%

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

      4. associate-*l/ 94.4%

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

      5. clear-num 94.3%

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

      6. associate-*l/ 94.9%

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

      7. *-un-lft-identity 94.9%

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

   7. Applied egg-rr 94.9%

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

4. Final simplification 91.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -215000000 \lor \neg \left(z \leq 8.4 \cdot 10^{-55}\right):\\ \;\;\;\;\frac{\frac{x}{z}}{z \cdot \frac{z}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{\frac{z}{x}}}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 93.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -215000000 \lor \neg \left(z \leq 8.4 \cdot 10^{-55}\right):\\ \;\;\;\;\frac{\frac{x\_m}{z} \cdot \frac{y\_m}{z}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y\_m}{\frac{z}{x\_m}}}{z}\\ \end{array}\right) \end{array} \]

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 (or (<= z -215000000.0) (not (<= z 8.4e-55)))
     (/ (* (/ x_m z) (/ y_m z)) z)
     (/ (/ y_m (/ z x_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 <= -215000000.0) || !(z <= 8.4e-55)) {
		tmp = ((x_m / z) * (y_m / z)) / z;
	} else {
		tmp = (y_m / (z / x_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 <= (-215000000.0d0)) .or. (.not. (z <= 8.4d-55))) then
        tmp = ((x_m / z) * (y_m / z)) / z
    else
        tmp = (y_m / (z / x_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 <= -215000000.0) || !(z <= 8.4e-55)) {
		tmp = ((x_m / z) * (y_m / z)) / z;
	} else {
		tmp = (y_m / (z / x_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 <= -215000000.0) or not (z <= 8.4e-55):
		tmp = ((x_m / z) * (y_m / z)) / z
	else:
		tmp = (y_m / (z / x_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 <= -215000000.0) || !(z <= 8.4e-55))
		tmp = Float64(Float64(Float64(x_m / z) * Float64(y_m / z)) / z);
	else
		tmp = Float64(Float64(y_m / Float64(z / x_m)) / 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 <= -215000000.0) || ~((z <= 8.4e-55)))
		tmp = ((x_m / z) * (y_m / z)) / z;
	else
		tmp = (y_m / (z / x_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, -215000000.0], N[Not[LessEqual[z, 8.4e-55]], $MachinePrecision]], N[(N[(N[(x$95$m / z), $MachinePrecision] * N[(y$95$m / z), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], N[(N[(y$95$m / N[(z / x$95$m), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < -2.15e8 or 8.4000000000000006e-55 < z

   1. Initial program 80.5%

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

      1. *-commutative 80.5%

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

      2. associate-/l* 82.4%

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

      3. sqr-neg 82.4%

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

      4. associate-/r* 87.8%

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

      5. sqr-neg 87.8%

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

   3. Simplified 87.8%

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

      1. associate-*r/ 89.7%

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

      2. *-commutative 89.7%

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

      3. associate-*r/ 87.4%

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

      4. associate-/r* 93.4%

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

      5. associate-*l/ 96.7%

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

   6. Applied egg-rr 96.7%

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

   7. Taylor expanded in z around inf 93.6%

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

   if -2.15e8 < z < 8.4000000000000006e-55

   1. Initial program 87.2%

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

      1. *-commutative 87.2%

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

      2. sqr-neg 87.2%

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

      3. times-frac 86.5%

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

      4. sqr-neg 86.5%

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

   3. Simplified 86.5%

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

   5. Taylor expanded in z around 0 83.3%

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

      1. associate-*l/ 84.0%

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

      2. *-commutative 84.0%

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

      3. associate-/r* 90.5%

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

      4. associate-*l/ 94.4%

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

      5. clear-num 94.3%

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

      6. associate-*l/ 94.9%

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

      7. *-un-lft-identity 94.9%

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

   7. Applied egg-rr 94.9%

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

4. Final simplification 94.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -215000000 \lor \neg \left(z \leq 8.4 \cdot 10^{-55}\right):\\ \;\;\;\;\frac{\frac{x}{z} \cdot \frac{y}{z}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{\frac{z}{x}}}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 92.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -1 \lor \neg \left(z \leq 3.5 \cdot 10^{-35}\right):\\ \;\;\;\;\frac{x\_m}{z} \cdot \frac{y\_m}{z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y\_m}{\frac{z}{x\_m}}}{z}\\ \end{array}\right) \end{array} \]

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 (or (<= z -1.0) (not (<= z 3.5e-35)))
     (* (/ x_m z) (/ y_m (* z z)))
     (/ (/ y_m (/ z x_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 <= 3.5e-35)) {
		tmp = (x_m / z) * (y_m / (z * z));
	} else {
		tmp = (y_m / (z / x_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 <= 3.5d-35))) then
        tmp = (x_m / z) * (y_m / (z * z))
    else
        tmp = (y_m / (z / x_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 <= 3.5e-35)) {
		tmp = (x_m / z) * (y_m / (z * z));
	} else {
		tmp = (y_m / (z / x_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 <= 3.5e-35):
		tmp = (x_m / z) * (y_m / (z * z))
	else:
		tmp = (y_m / (z / x_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 <= 3.5e-35))
		tmp = Float64(Float64(x_m / z) * Float64(y_m / Float64(z * z)));
	else
		tmp = Float64(Float64(y_m / Float64(z / x_m)) / 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 <= 3.5e-35)))
		tmp = (x_m / z) * (y_m / (z * z));
	else
		tmp = (y_m / (z / x_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, 3.5e-35]], $MachinePrecision]], N[(N[(x$95$m / z), $MachinePrecision] * N[(y$95$m / N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y$95$m / N[(z / x$95$m), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < -1 or 3.49999999999999996e-35 < z

   1. Initial program 79.7%

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

      1. *-commutative 79.7%

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

      2. sqr-neg 79.7%

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

      3. times-frac 88.4%

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

      4. sqr-neg 88.4%

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

   3. Simplified 88.4%

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

   5. Taylor expanded in z around inf 87.0%

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

   if -1 < z < 3.49999999999999996e-35

   1. Initial program 87.7%

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

      1. *-commutative 87.7%

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

      2. sqr-neg 87.7%

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

      3. times-frac 87.0%

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

      4. sqr-neg 87.0%

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

   3. Simplified 87.0%

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

   5. Taylor expanded in z around 0 84.6%

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

      1. associate-*l/ 85.3%

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

      2. *-commutative 85.3%

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

      3. associate-/r* 91.6%

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

      4. associate-*l/ 94.6%

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

      5. clear-num 94.5%

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

      6. associate-*l/ 95.1%

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

      7. *-un-lft-identity 95.1%

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

   7. Applied egg-rr 95.1%

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

4. Final simplification 91.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1 \lor \neg \left(z \leq 3.5 \cdot 10^{-35}\right):\\ \;\;\;\;\frac{x}{z} \cdot \frac{y}{z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{\frac{z}{x}}}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 93.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -1 \lor \neg \left(z \leq 9.8 \cdot 10^{-26}\right):\\ \;\;\;\;\frac{y\_m}{z} \cdot \frac{\frac{x\_m}{z}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y\_m}{\frac{z}{x\_m}}}{z}\\ \end{array}\right) \end{array} \]

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 (or (<= z -1.0) (not (<= z 9.8e-26)))
     (* (/ y_m z) (/ (/ x_m z) z))
     (/ (/ y_m (/ z x_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 <= 9.8e-26)) {
		tmp = (y_m / z) * ((x_m / z) / z);
	} else {
		tmp = (y_m / (z / x_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 <= 9.8d-26))) then
        tmp = (y_m / z) * ((x_m / z) / z)
    else
        tmp = (y_m / (z / x_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 <= 9.8e-26)) {
		tmp = (y_m / z) * ((x_m / z) / z);
	} else {
		tmp = (y_m / (z / x_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 <= 9.8e-26):
		tmp = (y_m / z) * ((x_m / z) / z)
	else:
		tmp = (y_m / (z / x_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 <= 9.8e-26))
		tmp = Float64(Float64(y_m / z) * Float64(Float64(x_m / z) / z));
	else
		tmp = Float64(Float64(y_m / Float64(z / x_m)) / 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 <= 9.8e-26)))
		tmp = (y_m / z) * ((x_m / z) / z);
	else
		tmp = (y_m / (z / x_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, 9.8e-26]], $MachinePrecision]], N[(N[(y$95$m / z), $MachinePrecision] * N[(N[(x$95$m / z), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], N[(N[(y$95$m / N[(z / x$95$m), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < -1 or 9.7999999999999998e-26 < z

   1. Initial program 79.5%

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

      1. *-commutative 79.5%

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

      2. frac-times 88.3%

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

      3. associate-*l/ 91.1%

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

      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 inf 93.3%

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

   if -1 < z < 9.7999999999999998e-26

   1. Initial program 87.8%

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

      1. *-commutative 87.8%

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

      2. sqr-neg 87.8%

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

      3. times-frac 87.1%

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

      4. sqr-neg 87.1%

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

   3. Simplified 87.1%

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

   5. Taylor expanded in z around 0 84.7%

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

      1. associate-*l/ 85.4%

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

      2. *-commutative 85.4%

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

      3. associate-/r* 91.6%

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

      4. associate-*l/ 94.6%

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

      5. clear-num 94.6%

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

      6. associate-*l/ 95.1%

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

      7. *-un-lft-identity 95.1%

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

   7. Applied egg-rr 95.1%

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

4. Final simplification 94.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1 \lor \neg \left(z \leq 9.8 \cdot 10^{-26}\right):\\ \;\;\;\;\frac{y}{z} \cdot \frac{\frac{x}{z}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{\frac{z}{x}}}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 77.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 2.6 \cdot 10^{-90}:\\ \;\;\;\;\frac{x\_m}{z} \cdot \frac{y\_m}{z}\\ \mathbf{else}:\\ \;\;\;\;x\_m \cdot \frac{y\_m}{z \cdot z}\\ \end{array}\right) \end{array} \]

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 (<= x_m 2.6e-90) (* (/ x_m z) (/ y_m z)) (* x_m (/ 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 (x_m <= 2.6e-90) {
		tmp = (x_m / z) * (y_m / z);
	} else {
		tmp = x_m * (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 (x_m <= 2.6d-90) then
        tmp = (x_m / z) * (y_m / z)
    else
        tmp = x_m * (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 (x_m <= 2.6e-90) {
		tmp = (x_m / z) * (y_m / z);
	} else {
		tmp = x_m * (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 x_m <= 2.6e-90:
		tmp = (x_m / z) * (y_m / z)
	else:
		tmp = x_m * (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 (x_m <= 2.6e-90)
		tmp = Float64(Float64(x_m / z) * Float64(y_m / z));
	else
		tmp = Float64(x_m * Float64(y_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 (x_m <= 2.6e-90)
		tmp = (x_m / z) * (y_m / z);
	else
		tmp = x_m * (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[x$95$m, 2.6e-90], N[(N[(x$95$m / z), $MachinePrecision] * N[(y$95$m / z), $MachinePrecision]), $MachinePrecision], N[(x$95$m * N[(y$95$m / N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 2.6e-90

   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 88.1%

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

      3. associate-*l/ 88.5%

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

      4. times-frac 96.2%

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

   4. Applied egg-rr 96.2%

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

   5. Taylor expanded in z around 0 78.9%

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

   if 2.6e-90 < x

   1. Initial program 82.9%

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

      1. *-commutative 82.9%

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

      2. sqr-neg 82.9%

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

      3. times-frac 86.8%

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

      4. sqr-neg 86.8%

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

   3. Simplified 86.8%

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

   5. Taylor expanded in z around 0 70.9%

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

4. Final simplification 76.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.6 \cdot 10^{-90}:\\ \;\;\;\;\frac{x}{z} \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{z \cdot z}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 97.9% accurate, 1.0× speedup

Math

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

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

Derivation

1. Initial program 83.9%

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

   1. *-commutative 83.9%

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

   2. associate-/l* 82.8%

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

   3. sqr-neg 82.8%

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

   4. associate-/r* 85.5%

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

   5. sqr-neg 85.5%

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

3. Simplified 85.5%

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

   1. associate-*r/ 86.4%

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

   2. *-commutative 86.4%

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

   3. associate-*r/ 85.3%

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

   4. associate-/r* 91.2%

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

   5. associate-*l/ 97.2%

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

6. Applied egg-rr 97.2%

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

Alternative 11: 81.0% accurate, 1.6× speedup

Math

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

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))))))

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))));
}

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))))
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))));
}

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))))

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

Derivation

1. Initial program 83.9%

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

   1. *-commutative 83.9%

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

   2. frac-times 87.6%

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

   3. associate-*l/ 89.3%

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

   4. times-frac 95.6%

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

4. Applied egg-rr 95.6%

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

   1. associate-*r/ 98.7%

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

   2. frac-times 89.1%

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

   3. *-commutative 89.1%

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

   4. frac-times 97.2%

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

   5. associate-/l* 95.2%

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

   6. clear-num 95.1%

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

   7. frac-times 91.2%

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

   8. *-un-lft-identity 91.2%

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

6. Applied egg-rr 91.2%

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

7. Taylor expanded in z around 0 73.4%

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

   1. *-commutative 73.4%

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

   2. clear-num 73.4%

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

   3. un-div-inv 73.4%

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

9. Applied egg-rr 73.4%

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

Alternative 12: 81.0% 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 83.9%

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

   1. *-commutative 83.9%

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

   2. frac-times 87.6%

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

   3. associate-*l/ 89.3%

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

   4. times-frac 95.6%

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

4. Applied egg-rr 95.6%

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

   1. associate-*r/ 98.7%

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

   2. frac-times 89.1%

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

   3. *-commutative 89.1%

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

   4. frac-times 97.2%

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

   5. associate-/l* 95.2%

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

   6. clear-num 95.1%

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

   7. frac-times 91.2%

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

   8. *-un-lft-identity 91.2%

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

6. Applied egg-rr 91.2%

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

7. Taylor expanded in z around 0 73.4%

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

8. Final simplification 73.4%

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

Alternative 13: 74.3% accurate, 1.6× speedup

Math

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

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 (* (/ x_m z) (/ 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) {
	return y_s * (x_s * ((x_m / z) * (y_m / 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 * ((x_m / z) * (y_m / 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 * ((x_m / z) * (y_m / 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 * ((x_m / z) * (y_m / 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(x_m / z) * Float64(y_m / 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 * ((x_m / z) * (y_m / 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[(x$95$m / z), $MachinePrecision] * N[(y$95$m / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 83.9%

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

   1. *-commutative 83.9%

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

   2. frac-times 87.6%

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

   3. associate-*l/ 89.3%

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

   4. times-frac 95.6%

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

4. Applied egg-rr 95.6%

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

5. Taylor expanded in z around 0 75.4%

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

6. Final simplification 75.4%

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

Developer target: 96.7% 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 2024107 
(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))))