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

Percentage Accurate: 83.1% → 96.1%

Time: 9.9s

Alternatives: 8

Speedup: 1.0×

• 25.4% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 83.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 96.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ \begin{array}{l} t_0 := \left(z + 1\right) \cdot \left(z \cdot z\right)\\ y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -2 \cdot 10^{+29}:\\ \;\;\;\;\frac{\frac{\frac{x\_m}{z}}{\frac{z}{y\_m}}}{z}\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{-112}:\\ \;\;\;\;\frac{\frac{x\_m}{z} \cdot y\_m}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m}{\frac{z}{y\_m} \cdot \left(z \cdot \left(z + 1\right)\right)}\\ \end{array}\right) \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (y_s x_s x_m y_m z)
 :precision binary64
 (let* ((t_0 (* (+ z 1.0) (* z z))))
   (*
    y_s
    (*
     x_s
     (if (<= t_0 -2e+29)
       (/ (/ (/ x_m z) (/ z y_m)) z)
       (if (<= t_0 5e-112)
         (/ (* (/ x_m z) y_m) z)
         (/ x_m (* (/ z y_m) (* z (+ z 1.0))))))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
assert(x_m < y_m && y_m < z);
double code(double y_s, double x_s, double x_m, double y_m, double z) {
	double t_0 = (z + 1.0) * (z * z);
	double tmp;
	if (t_0 <= -2e+29) {
		tmp = ((x_m / z) / (z / y_m)) / z;
	} else if (t_0 <= 5e-112) {
		tmp = ((x_m / z) * y_m) / z;
	} else {
		tmp = x_m / ((z / y_m) * (z * (z + 1.0)));
	}
	return y_s * (x_s * tmp);
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
y\_m = abs(y)
y\_s = copysign(1.0d0, y)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
real(8) function code(y_s, x_s, x_m, y_m, z)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (z + 1.0d0) * (z * z)
    if (t_0 <= (-2d+29)) then
        tmp = ((x_m / z) / (z / y_m)) / z
    else if (t_0 <= 5d-112) then
        tmp = ((x_m / z) * y_m) / z
    else
        tmp = x_m / ((z / y_m) * (z * (z + 1.0d0)))
    end if
    code = y_s * (x_s * tmp)
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
assert x_m < y_m && y_m < z;
public static double code(double y_s, double x_s, double x_m, double y_m, double z) {
	double t_0 = (z + 1.0) * (z * z);
	double tmp;
	if (t_0 <= -2e+29) {
		tmp = ((x_m / z) / (z / y_m)) / z;
	} else if (t_0 <= 5e-112) {
		tmp = ((x_m / z) * y_m) / z;
	} else {
		tmp = x_m / ((z / y_m) * (z * (z + 1.0)));
	}
	return y_s * (x_s * tmp);
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
[x_m, y_m, z] = sort([x_m, y_m, z])
def code(y_s, x_s, x_m, y_m, z):
	t_0 = (z + 1.0) * (z * z)
	tmp = 0
	if t_0 <= -2e+29:
		tmp = ((x_m / z) / (z / y_m)) / z
	elif t_0 <= 5e-112:
		tmp = ((x_m / z) * y_m) / z
	else:
		tmp = x_m / ((z / y_m) * (z * (z + 1.0)))
	return y_s * (x_s * tmp)

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
y\_m = abs(y)
y\_s = copysign(1.0, y)
x_m, y_m, z = sort([x_m, y_m, z])
function code(y_s, x_s, x_m, y_m, z)
	t_0 = Float64(Float64(z + 1.0) * Float64(z * z))
	tmp = 0.0
	if (t_0 <= -2e+29)
		tmp = Float64(Float64(Float64(x_m / z) / Float64(z / y_m)) / z);
	elseif (t_0 <= 5e-112)
		tmp = Float64(Float64(Float64(x_m / z) * y_m) / z);
	else
		tmp = Float64(x_m / Float64(Float64(z / y_m) * Float64(z * Float64(z + 1.0))));
	end
	return Float64(y_s * Float64(x_s * tmp))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
x_m, y_m, z = num2cell(sort([x_m, y_m, z])){:}
function tmp_2 = code(y_s, x_s, x_m, y_m, z)
	t_0 = (z + 1.0) * (z * z);
	tmp = 0.0;
	if (t_0 <= -2e+29)
		tmp = ((x_m / z) / (z / y_m)) / z;
	elseif (t_0 <= 5e-112)
		tmp = ((x_m / z) * y_m) / z;
	else
		tmp = x_m / ((z / y_m) * (z * (z + 1.0)));
	end
	tmp_2 = y_s * (x_s * tmp);
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
code[y$95$s_, x$95$s_, x$95$m_, y$95$m_, z_] := Block[{t$95$0 = N[(N[(z + 1.0), $MachinePrecision] * N[(z * z), $MachinePrecision]), $MachinePrecision]}, N[(y$95$s * N[(x$95$s * If[LessEqual[t$95$0, -2e+29], N[(N[(N[(x$95$m / z), $MachinePrecision] / N[(z / y$95$m), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], If[LessEqual[t$95$0, 5e-112], N[(N[(N[(x$95$m / z), $MachinePrecision] * y$95$m), $MachinePrecision] / z), $MachinePrecision], N[(x$95$m / N[(N[(z / y$95$m), $MachinePrecision] * N[(z * N[(z + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64))) < -1.99999999999999983e29

   1. Initial program 83.1%

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

      1. clear-num 83.0%

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

      2. associate-/l* 87.2%

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

      3. pow2 87.2%

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

   4. Applied egg-rr 87.2%

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

      1. associate-*r/ 83.0%

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

      2. clear-num 83.1%

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

      3. *-commutative 83.1%

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

      4. distribute-rgt-in 48.6%

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

      5. *-un-lft-identity 48.6%

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

      6. unpow2 48.6%

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

      7. distribute-lft-in 83.1%

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

      8. unpow2 83.1%

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

      9. fma-undefine 83.0%

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

      10. frac-times 88.5%

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

      11. *-commutative 88.5%

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

      12. frac-times 83.0%

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

      13. associate-/l/ 87.2%

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

      14. clear-num 87.2%

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

      15. *-un-lft-identity 87.2%

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

      16. associate-/l* 92.6%

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

      17. times-frac 85.8%

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

      18. fma-undefine 85.8%

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

      19. *-un-lft-identity 85.8%

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

      20. distribute-rgt-out 85.8%

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

   6. Applied egg-rr 85.8%

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

      1. associate-*l/ 85.8%

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

      2. *-lft-identity 85.8%

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

      3. associate-/r* 92.6%

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

      4. *-commutative 92.6%

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

      5. times-frac 87.2%

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

   8. Simplified 87.2%

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

      1. associate-/r* 88.8%

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

      2. clear-num 88.8%

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

      3. associate-/r/ 88.8%

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

      4. associate-/r* 97.0%

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

   10. Applied egg-rr 97.0%

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

   11. Taylor expanded in z around inf 96.6%

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

   if -1.99999999999999983e29 < (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64))) < 5.00000000000000044e-112

   1. Initial program 83.8%

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

   3. Taylor expanded in z around 0 83.8%

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

      1. associate-/l* 81.7%

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

   5. Simplified 81.7%

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

      1. *-un-lft-identity 81.7%

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

      2. unpow2 81.7%

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

      3. times-frac 92.0%

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

   7. Applied egg-rr 92.0%

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

      1. associate-*l/ 92.0%

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

      2. *-lft-identity 92.0%

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

   9. Simplified 92.0%

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

       1. *-un-lft-identity 92.0%

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

       2. associate-*l/ 92.0%

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

       3. associate-*r* 96.5%

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

       4. div-inv 96.5%

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

       5. frac-2neg 96.5%

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

       6. associate-*r/ 94.0%

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

   11. Applied egg-rr 94.0%

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

   if 5.00000000000000044e-112 < (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64)))

   1. Initial program 86.6%

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

      1. *-commutative 86.6%

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

      2. associate-*l* 86.5%

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

      3. times-frac 96.3%

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

      4. distribute-lft-in 96.3%

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

      5. *-rgt-identity 96.3%

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

      6. fma-define 96.3%

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

   4. Applied egg-rr 96.3%

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

      1. clear-num 96.2%

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

      2. frac-times 89.2%

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

      3. *-un-lft-identity 89.2%

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

      4. fma-undefine 89.2%

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

      5. *-un-lft-identity 89.2%

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

      6. distribute-rgt-out 89.3%

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

   6. Applied egg-rr 89.3%

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

4. Final simplification 93.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(z + 1\right) \cdot \left(z \cdot z\right) \leq -2 \cdot 10^{+29}:\\ \;\;\;\;\frac{\frac{\frac{x}{z}}{\frac{z}{y}}}{z}\\ \mathbf{elif}\;\left(z + 1\right) \cdot \left(z \cdot z\right) \leq 5 \cdot 10^{-112}:\\ \;\;\;\;\frac{\frac{x}{z} \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{y} \cdot \left(z \cdot \left(z + 1\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 96.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;\left(z + 1\right) \cdot \left(z \cdot z\right) \leq -5 \cdot 10^{+61}:\\ \;\;\;\;\frac{\frac{\frac{x\_m}{z}}{\frac{z}{y\_m}}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y\_m \cdot \frac{x\_m}{z \cdot \left(z + 1\right)}}{z}\\ \end{array}\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (y_s x_s x_m y_m z)
 :precision binary64
 (*
  y_s
  (*
   x_s
   (if (<= (* (+ z 1.0) (* z z)) -5e+61)
     (/ (/ (/ x_m z) (/ z y_m)) z)
     (/ (* y_m (/ x_m (* z (+ z 1.0)))) z)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
assert(x_m < y_m && y_m < z);
double code(double y_s, double x_s, double x_m, double y_m, double z) {
	double tmp;
	if (((z + 1.0) * (z * z)) <= -5e+61) {
		tmp = ((x_m / z) / (z / y_m)) / z;
	} else {
		tmp = (y_m * (x_m / (z * (z + 1.0)))) / z;
	}
	return y_s * (x_s * tmp);
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
y\_m = abs(y)
y\_s = copysign(1.0d0, y)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
real(8) function code(y_s, x_s, x_m, y_m, z)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if (((z + 1.0d0) * (z * z)) <= (-5d+61)) then
        tmp = ((x_m / z) / (z / y_m)) / z
    else
        tmp = (y_m * (x_m / (z * (z + 1.0d0)))) / z
    end if
    code = y_s * (x_s * tmp)
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
assert x_m < y_m && y_m < z;
public static double code(double y_s, double x_s, double x_m, double y_m, double z) {
	double tmp;
	if (((z + 1.0) * (z * z)) <= -5e+61) {
		tmp = ((x_m / z) / (z / y_m)) / z;
	} else {
		tmp = (y_m * (x_m / (z * (z + 1.0)))) / z;
	}
	return y_s * (x_s * tmp);
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
[x_m, y_m, z] = sort([x_m, y_m, z])
def code(y_s, x_s, x_m, y_m, z):
	tmp = 0
	if ((z + 1.0) * (z * z)) <= -5e+61:
		tmp = ((x_m / z) / (z / y_m)) / z
	else:
		tmp = (y_m * (x_m / (z * (z + 1.0)))) / z
	return y_s * (x_s * tmp)

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
y\_m = abs(y)
y\_s = copysign(1.0, y)
x_m, y_m, z = sort([x_m, y_m, z])
function code(y_s, x_s, x_m, y_m, z)
	tmp = 0.0
	if (Float64(Float64(z + 1.0) * Float64(z * z)) <= -5e+61)
		tmp = Float64(Float64(Float64(x_m / z) / Float64(z / y_m)) / z);
	else
		tmp = Float64(Float64(y_m * Float64(x_m / Float64(z * Float64(z + 1.0)))) / z);
	end
	return Float64(y_s * Float64(x_s * tmp))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
x_m, y_m, z = num2cell(sort([x_m, y_m, z])){:}
function tmp_2 = code(y_s, x_s, x_m, y_m, z)
	tmp = 0.0;
	if (((z + 1.0) * (z * z)) <= -5e+61)
		tmp = ((x_m / z) / (z / y_m)) / z;
	else
		tmp = (y_m * (x_m / (z * (z + 1.0)))) / z;
	end
	tmp_2 = y_s * (x_s * tmp);
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
code[y$95$s_, x$95$s_, x$95$m_, y$95$m_, z_] := N[(y$95$s * N[(x$95$s * If[LessEqual[N[(N[(z + 1.0), $MachinePrecision] * N[(z * z), $MachinePrecision]), $MachinePrecision], -5e+61], N[(N[(N[(x$95$m / z), $MachinePrecision] / N[(z / y$95$m), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], N[(N[(y$95$m * N[(x$95$m / N[(z * N[(z + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64))) < -5.00000000000000018e61

   1. Initial program 81.8%

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

      1. clear-num 81.7%

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

      2. associate-/l* 86.2%

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

      3. pow2 86.2%

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

   4. Applied egg-rr 86.2%

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

      1. associate-*r/ 81.7%

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

      2. clear-num 81.7%

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

      3. *-commutative 81.7%

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

      4. distribute-rgt-in 44.6%

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

      5. *-un-lft-identity 44.6%

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

      6. unpow2 44.6%

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

      7. distribute-lft-in 81.7%

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

      8. unpow2 81.7%

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

      9. fma-undefine 81.7%

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

      10. frac-times 87.6%

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

      11. *-commutative 87.6%

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

      12. frac-times 81.7%

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

      13. associate-/l/ 86.2%

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

      14. clear-num 86.2%

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

      15. *-un-lft-identity 86.2%

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

      16. associate-/l* 92.1%

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

      17. times-frac 84.7%

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

      18. fma-undefine 84.7%

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

      19. *-un-lft-identity 84.7%

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

      20. distribute-rgt-out 84.7%

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

   6. Applied egg-rr 84.7%

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

      1. associate-*l/ 84.7%

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

      2. *-lft-identity 84.7%

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

      3. associate-/r* 92.1%

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

      4. *-commutative 92.1%

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

      5. times-frac 86.2%

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

   8. Simplified 86.2%

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

      1. associate-/r* 87.9%

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

      2. clear-num 88.0%

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

      3. associate-/r/ 87.9%

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

      4. associate-/r* 96.8%

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

   10. Applied egg-rr 96.8%

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

   11. Taylor expanded in z around inf 96.8%

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

   if -5.00000000000000018e61 < (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64)))

   1. Initial program 85.3%

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

      1. *-commutative 85.3%

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

      2. associate-*l* 85.3%

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

      3. times-frac 96.5%

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

      4. distribute-lft-in 96.5%

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

      5. *-rgt-identity 96.5%

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

      6. fma-define 96.5%

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

   4. Applied egg-rr 96.5%

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

      1. associate-*l/ 95.5%

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

      2. fma-undefine 95.5%

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

      3. *-un-lft-identity 95.5%

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

      4. distribute-rgt-out 95.5%

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

   6. Applied egg-rr 95.5%

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

4. Final simplification 95.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(z + 1\right) \cdot \left(z \cdot z\right) \leq -5 \cdot 10^{+61}:\\ \;\;\;\;\frac{\frac{\frac{x}{z}}{\frac{z}{y}}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot \frac{x}{z \cdot \left(z + 1\right)}}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 96.5% accurate, 0.6× speedup

Math

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

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 -5.2) (not (<= z 1.0)))
     (/ (/ (/ x_m z) (/ z y_m)) z)
     (/ (* (/ 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) {
	double tmp;
	if ((z <= -5.2) || !(z <= 1.0)) {
		tmp = ((x_m / z) / (z / y_m)) / z;
	} else {
		tmp = ((x_m / z) * y_m) / z;
	}
	return y_s * (x_s * tmp);
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
y\_m = abs(y)
y\_s = copysign(1.0d0, y)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
real(8) function code(y_s, x_s, x_m, y_m, z)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((z <= (-5.2d0)) .or. (.not. (z <= 1.0d0))) then
        tmp = ((x_m / z) / (z / y_m)) / z
    else
        tmp = ((x_m / z) * y_m) / z
    end if
    code = y_s * (x_s * tmp)
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
assert x_m < y_m && y_m < z;
public static double code(double y_s, double x_s, double x_m, double y_m, double z) {
	double tmp;
	if ((z <= -5.2) || !(z <= 1.0)) {
		tmp = ((x_m / z) / (z / y_m)) / z;
	} else {
		tmp = ((x_m / z) * y_m) / z;
	}
	return y_s * (x_s * tmp);
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
[x_m, y_m, z] = sort([x_m, y_m, z])
def code(y_s, x_s, x_m, y_m, z):
	tmp = 0
	if (z <= -5.2) or not (z <= 1.0):
		tmp = ((x_m / z) / (z / y_m)) / z
	else:
		tmp = ((x_m / z) * y_m) / z
	return y_s * (x_s * tmp)

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
y\_m = abs(y)
y\_s = copysign(1.0, y)
x_m, y_m, z = sort([x_m, y_m, z])
function code(y_s, x_s, x_m, y_m, z)
	tmp = 0.0
	if ((z <= -5.2) || !(z <= 1.0))
		tmp = Float64(Float64(Float64(x_m / z) / Float64(z / y_m)) / z);
	else
		tmp = Float64(Float64(Float64(x_m / z) * y_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 <= -5.2) || ~((z <= 1.0)))
		tmp = ((x_m / z) / (z / y_m)) / z;
	else
		tmp = ((x_m / z) * y_m) / z;
	end
	tmp_2 = y_s * (x_s * tmp);
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
code[y$95$s_, x$95$s_, x$95$m_, y$95$m_, z_] := N[(y$95$s * N[(x$95$s * If[Or[LessEqual[z, -5.2], N[Not[LessEqual[z, 1.0]], $MachinePrecision]], N[(N[(N[(x$95$m / z), $MachinePrecision] / N[(z / y$95$m), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], N[(N[(N[(x$95$m / z), $MachinePrecision] * y$95$m), $MachinePrecision] / z), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < -5.20000000000000018 or 1 < z

   1. Initial program 82.9%

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

      1. clear-num 82.9%

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

      2. associate-/l* 88.0%

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

      3. pow2 88.0%

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

   4. Applied egg-rr 88.0%

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

      1. associate-*r/ 82.9%

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

      2. clear-num 82.9%

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

      3. *-commutative 82.9%

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

      4. distribute-rgt-in 64.7%

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

      5. *-un-lft-identity 64.7%

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

      6. unpow2 64.7%

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

      7. distribute-lft-in 82.9%

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

      8. unpow2 82.9%

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

      9. fma-undefine 82.9%

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

      10. frac-times 91.7%

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

      11. *-commutative 91.7%

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

      12. frac-times 82.9%

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

      13. associate-/l/ 88.1%

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

      14. clear-num 88.0%

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

      15. *-un-lft-identity 88.0%

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

      16. associate-/l* 93.9%

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

      17. times-frac 86.0%

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

      18. fma-undefine 86.0%

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

      19. *-un-lft-identity 86.0%

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

      20. distribute-rgt-out 86.0%

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

   6. Applied egg-rr 86.0%

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

      1. associate-*l/ 86.0%

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

      2. *-lft-identity 86.0%

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

      3. associate-/r* 93.9%

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

      4. *-commutative 93.9%

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

      5. times-frac 90.4%

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

   8. Simplified 90.4%

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

      1. associate-/r* 91.3%

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

      2. clear-num 91.3%

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

      3. associate-/r/ 91.3%

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

      4. associate-/r* 96.9%

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

   10. Applied egg-rr 96.9%

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

   11. Taylor expanded in z around inf 95.1%

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

   if -5.20000000000000018 < z < 1

   1. Initial program 86.0%

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

   3. Taylor expanded in z around 0 84.6%

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

      1. associate-/l* 82.8%

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

   5. Simplified 82.8%

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

      1. *-un-lft-identity 82.8%

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

      2. unpow2 82.8%

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

      3. times-frac 91.6%

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

   7. Applied egg-rr 91.6%

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

      1. associate-*l/ 91.6%

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

      2. *-lft-identity 91.6%

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

   9. Simplified 91.6%

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

       1. *-un-lft-identity 91.6%

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

       2. associate-*l/ 91.6%

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

       3. associate-*r* 95.5%

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

       4. div-inv 95.5%

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

       5. frac-2neg 95.5%

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

       6. associate-*r/ 93.3%

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

   11. Applied egg-rr 93.3%

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

4. Final simplification 94.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -5.2 \lor \neg \left(z \leq 1\right):\\ \;\;\;\;\frac{\frac{\frac{x}{z}}{\frac{z}{y}}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{z} \cdot y}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.0% accurate, 1.0× speedup

Math

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

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

Derivation

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. clear-num 84.4%

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

   2. associate-/l* 86.7%

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

   3. pow2 86.7%

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

4. Applied egg-rr 86.7%

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

   1. associate-*r/ 84.4%

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

   2. clear-num 84.4%

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

   3. *-commutative 84.4%

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

   4. distribute-rgt-in 75.3%

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

   5. *-un-lft-identity 75.3%

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

   6. unpow2 75.3%

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

   7. distribute-lft-in 84.4%

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

   8. unpow2 84.4%

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

   9. fma-undefine 84.4%

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

   10. frac-times 94.3%

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

   11. *-commutative 94.3%

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

   12. frac-times 84.4%

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

   13. associate-/l/ 91.1%

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

   14. clear-num 91.1%

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

   15. *-un-lft-identity 91.1%

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

   16. associate-/l* 95.7%

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

   17. times-frac 89.7%

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

   18. fma-undefine 89.8%

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

   19. *-un-lft-identity 89.8%

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

   20. distribute-rgt-out 89.7%

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

6. Applied egg-rr 89.7%

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

   1. associate-*l/ 89.8%

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

   2. *-lft-identity 89.8%

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

   3. associate-/r* 95.7%

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

   4. *-commutative 95.7%

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

   5. times-frac 93.8%

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

8. Simplified 93.8%

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

   1. associate-/r* 94.1%

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

   2. clear-num 94.1%

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

   3. associate-/r/ 94.1%

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

   4. associate-/r* 95.8%

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

10. Applied egg-rr 95.8%

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

Alternative 5: 96.0% accurate, 1.0× speedup

Math

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

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 1.0)) (/ 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) {
	return y_s * (x_s * (((x_m / (z + 1.0)) * (y_m / z)) / z));
}

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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. clear-num 84.4%

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

   2. associate-/l* 86.7%

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

   3. pow2 86.7%

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

4. Applied egg-rr 86.7%

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

   1. associate-*r/ 84.4%

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

   2. clear-num 84.4%

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

   3. *-commutative 84.4%

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

   4. distribute-rgt-in 75.3%

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

   5. *-un-lft-identity 75.3%

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

   6. unpow2 75.3%

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

   7. distribute-lft-in 84.4%

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

   8. unpow2 84.4%

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

   9. fma-undefine 84.4%

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

   10. frac-times 94.3%

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

   11. *-commutative 94.3%

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

   12. frac-times 84.4%

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

   13. associate-/l/ 91.1%

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

   14. clear-num 91.1%

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

   15. *-un-lft-identity 91.1%

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

   16. associate-/l* 95.7%

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

   17. times-frac 89.7%

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

   18. fma-undefine 89.8%

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

   19. *-un-lft-identity 89.8%

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

   20. distribute-rgt-out 89.7%

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

6. Applied egg-rr 89.7%

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

   1. associate-*l/ 89.8%

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

   2. *-lft-identity 89.8%

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

   3. associate-/r* 95.7%

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

   4. *-commutative 95.7%

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

   5. times-frac 93.8%

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

8. Simplified 93.8%

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

   1. associate-/r* 94.1%

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

   2. clear-num 94.1%

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

   3. associate-/r/ 94.1%

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

   4. associate-/r* 95.8%

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

10. Applied egg-rr 95.8%

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

    1. div-inv 95.8%

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

    2. div-inv 95.8%

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

    3. times-frac 97.2%

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

    4. associate-/r* 97.1%

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

    5. div-inv 97.2%

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

    6. clear-num 97.2%

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

12. Applied egg-rr 97.2%

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

Alternative 6: 80.6% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 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. associate-*l* 84.4%

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

   3. times-frac 94.4%

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

   4. distribute-lft-in 94.4%

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

   5. *-rgt-identity 94.4%

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

   6. fma-define 94.4%

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

4. Applied egg-rr 94.4%

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

5. Taylor expanded in z around 0 75.3%

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

   1. clear-num 75.3%

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

   2. associate-*l/ 75.3%

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

   3. *-un-lft-identity 75.3%

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

   4. associate-/r/ 72.6%

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

7. Applied egg-rr 72.6%

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

8. Final simplification 72.6%

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

Alternative 7: 74.0% accurate, 1.6× speedup

Math

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

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 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. associate-*l* 84.4%

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

   3. times-frac 94.4%

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

   4. distribute-lft-in 94.4%

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

   5. *-rgt-identity 94.4%

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

   6. fma-define 94.4%

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

4. Applied egg-rr 94.4%

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

5. Taylor expanded in z around 0 75.3%

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

6. Final simplification 75.3%

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

Alternative 8: 66.9% accurate, 1.6× speedup

Math

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

FPCore

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 84.5%

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

3. Taylor expanded in z around 0 71.4%

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

   1. associate-/l* 72.5%

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

5. Simplified 72.5%

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

   1. *-un-lft-identity 72.5%

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

   2. unpow2 72.5%

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

   3. times-frac 74.6%

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

7. Applied egg-rr 74.6%

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

   1. associate-*l/ 74.6%

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

   2. *-lft-identity 74.6%

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

9. Simplified 74.6%

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

Developer target: 96.8% 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 2024096 
(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))))