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

Percentage Accurate: 83.8% → 97.8%

Time: 8.3s

Alternatives: 9

Speedup: 1.0×

• 26.2% 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: 83.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 97.8% accurate, 1.0× speedup

Math

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

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

Derivation

1. Initial program 85.4%

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

   1. times-frac N/A

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

   2. *-commutative N/A

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

   3. clear-num N/A

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

   4. associate-/r* N/A

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

   5. frac-times N/A

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

   6. metadata-eval N/A

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

   7. metadata-eval N/A

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

   8. clear-num N/A

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

   9. inv-pow N/A

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

   10. metadata-eval N/A

       \[\leadsto \frac{{1}^{\left(\mathsf{neg}\left(1\right)\right)} \cdot {\left(\frac{z}{x}\right)}^{\left(\mathsf{neg}\left(1\right)\right)}}{\frac{z + 1}{y} \cdot z} \]

   11. unpow-prod-down N/A

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

   12. associate-/l* N/A

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

   13. *-lft-identity N/A

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

   14. metadata-eval N/A

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

   15. inv-pow N/A

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

   16. clear-num N/A

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

   17. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\left(\frac{x}{z}\right), \color{blue}{\left(\frac{z + 1}{y} \cdot z\right)}\right) \]

   18. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, z\right), \left(\color{blue}{\frac{z + 1}{y}} \cdot z\right)\right) \]

   19. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, z\right), \mathsf{*.f64}\left(\left(\frac{z + 1}{y}\right), \color{blue}{z}\right)\right) \]

   20. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, z\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(z + 1\right), y\right), z\right)\right) \]

   21. +-lowering-+.f64 97.7%

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, z\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(z, 1\right), y\right), z\right)\right) \]

4. Applied egg-rr 97.7%

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

5. Final simplification 97.7%

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

Alternative 2: 97.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ \begin{array}{l} t_0 := \left(z + 1\right) \cdot \left(z \cdot z\right)\\ y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -0.2:\\ \;\;\;\;\frac{\frac{x\_m}{z}}{\frac{z}{\frac{y\_m}{z}}}\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{-70}:\\ \;\;\;\;\frac{y\_m}{\frac{z}{\frac{x\_m}{z}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m}{\left(z + 1\right) \cdot \frac{z \cdot z}{y\_m}}\\ \end{array}\right) \end{array} \end{array} \]

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 -0.2)
       (/ (/ x_m z) (/ z (/ y_m z)))
       (if (<= t_0 5e-70)
         (/ y_m (/ z (/ x_m z)))
         (/ x_m (* (+ z 1.0) (/ (* z z) y_m)))))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
assert(x_m < y_m && y_m < z);
double code(double y_s, double x_s, double x_m, double y_m, double z) {
	double t_0 = (z + 1.0) * (z * z);
	double tmp;
	if (t_0 <= -0.2) {
		tmp = (x_m / z) / (z / (y_m / z));
	} else if (t_0 <= 5e-70) {
		tmp = y_m / (z / (x_m / z));
	} else {
		tmp = x_m / ((z + 1.0) * ((z * z) / y_m));
	}
	return y_s * (x_s * tmp);
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
y\_m = abs(y)
y\_s = copysign(1.0d0, y)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
real(8) function code(y_s, x_s, x_m, y_m, z)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (z + 1.0d0) * (z * z)
    if (t_0 <= (-0.2d0)) then
        tmp = (x_m / z) / (z / (y_m / z))
    else if (t_0 <= 5d-70) then
        tmp = y_m / (z / (x_m / z))
    else
        tmp = x_m / ((z + 1.0d0) * ((z * z) / y_m))
    end if
    code = y_s * (x_s * tmp)
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
assert x_m < y_m && y_m < z;
public static double code(double y_s, double x_s, double x_m, double y_m, double z) {
	double t_0 = (z + 1.0) * (z * z);
	double tmp;
	if (t_0 <= -0.2) {
		tmp = (x_m / z) / (z / (y_m / z));
	} else if (t_0 <= 5e-70) {
		tmp = y_m / (z / (x_m / z));
	} else {
		tmp = x_m / ((z + 1.0) * ((z * z) / y_m));
	}
	return y_s * (x_s * tmp);
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
[x_m, y_m, z] = sort([x_m, y_m, z])
def code(y_s, x_s, x_m, y_m, z):
	t_0 = (z + 1.0) * (z * z)
	tmp = 0
	if t_0 <= -0.2:
		tmp = (x_m / z) / (z / (y_m / z))
	elif t_0 <= 5e-70:
		tmp = y_m / (z / (x_m / z))
	else:
		tmp = x_m / ((z + 1.0) * ((z * z) / y_m))
	return y_s * (x_s * tmp)

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
y\_m = abs(y)
y\_s = copysign(1.0, y)
x_m, y_m, z = sort([x_m, y_m, z])
function code(y_s, x_s, x_m, y_m, z)
	t_0 = Float64(Float64(z + 1.0) * Float64(z * z))
	tmp = 0.0
	if (t_0 <= -0.2)
		tmp = Float64(Float64(x_m / z) / Float64(z / Float64(y_m / z)));
	elseif (t_0 <= 5e-70)
		tmp = Float64(y_m / Float64(z / Float64(x_m / z)));
	else
		tmp = Float64(x_m / Float64(Float64(z + 1.0) * Float64(Float64(z * z) / y_m)));
	end
	return Float64(y_s * Float64(x_s * tmp))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
x_m, y_m, z = num2cell(sort([x_m, y_m, z])){:}
function tmp_2 = code(y_s, x_s, x_m, y_m, z)
	t_0 = (z + 1.0) * (z * z);
	tmp = 0.0;
	if (t_0 <= -0.2)
		tmp = (x_m / z) / (z / (y_m / z));
	elseif (t_0 <= 5e-70)
		tmp = y_m / (z / (x_m / z));
	else
		tmp = x_m / ((z + 1.0) * ((z * z) / y_m));
	end
	tmp_2 = y_s * (x_s * tmp);
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
code[y$95$s_, x$95$s_, x$95$m_, y$95$m_, z_] := Block[{t$95$0 = N[(N[(z + 1.0), $MachinePrecision] * N[(z * z), $MachinePrecision]), $MachinePrecision]}, N[(y$95$s * N[(x$95$s * If[LessEqual[t$95$0, -0.2], N[(N[(x$95$m / z), $MachinePrecision] / N[(z / N[(y$95$m / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 5e-70], N[(y$95$m / N[(z / N[(x$95$m / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x$95$m / N[(N[(z + 1.0), $MachinePrecision] * N[(N[(z * z), $MachinePrecision] / y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64))) < -0.20000000000000001

   1. Initial program 76.7%

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

   3. Taylor expanded in z around inf

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(x \cdot y\right), \color{blue}{\left({z}^{3}\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \left({\color{blue}{z}}^{3}\right)\right) \]

      3. cube-mult N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(z \cdot \color{blue}{\left(z \cdot z\right)}\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(z \cdot {z}^{\color{blue}{2}}\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \color{blue}{\left({z}^{2}\right)}\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \left(z \cdot \color{blue}{z}\right)\right)\right) \]

      7. *-lowering-*.f64 75.5%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(z, \color{blue}{z}\right)\right)\right) \]

   5. Simplified 75.5%

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

      1. times-frac N/A

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{x}{z}\right), \color{blue}{\left(\frac{z \cdot z}{y}\right)}\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, z\right), \left(\frac{\color{blue}{z \cdot z}}{y}\right)\right) \]

      6. clear-num N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, z\right), \left(\frac{1}{\color{blue}{\frac{y}{z \cdot z}}}\right)\right) \]

      7. associate-/r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, z\right), \left(\frac{1}{\frac{\frac{y}{z}}{\color{blue}{z}}}\right)\right) \]

      8. clear-num N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, z\right), \left(\frac{z}{\color{blue}{\frac{y}{z}}}\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, z\right), \mathsf{/.f64}\left(z, \color{blue}{\left(\frac{y}{z}\right)}\right)\right) \]

      10. /-lowering-/.f64 97.2%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, z\right), \mathsf{/.f64}\left(z, \mathsf{/.f64}\left(y, \color{blue}{z}\right)\right)\right) \]

   7. Applied egg-rr 97.2%

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

   if -0.20000000000000001 < (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64))) < 4.9999999999999998e-70

   1. Initial program 92.0%

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

   3. Taylor expanded in z around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \color{blue}{\left({z}^{2}\right)}\right) \]
   4. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(z \cdot \color{blue}{z}\right)\right) \]

      2. *-lowering-*.f64 92.0%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \color{blue}{z}\right)\right) \]

   5. Simplified 92.0%

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. associate-/l/ N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(y, \color{blue}{\left(\frac{z}{\frac{x}{z}}\right)}\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(y, \mathsf{/.f64}\left(z, \color{blue}{\left(\frac{x}{z}\right)}\right)\right) \]

      8. /-lowering-/.f64 94.3%

         \[\leadsto \mathsf{/.f64}\left(y, \mathsf{/.f64}\left(z, \mathsf{/.f64}\left(x, \color{blue}{z}\right)\right)\right) \]

   7. Applied egg-rr 94.3%

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

   if 4.9999999999999998e-70 < (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64)))

   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. associate-/l* N/A

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \color{blue}{\left(\frac{\left(z \cdot z\right) \cdot \left(z + 1\right)}{y}\right)}\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(x, \left(\frac{\left(z + 1\right) \cdot \left(z \cdot z\right)}{y}\right)\right) \]

      6. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(x, \left(\left(z + 1\right) \cdot \color{blue}{\frac{z \cdot z}{y}}\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(\left(z + 1\right), \color{blue}{\left(\frac{z \cdot z}{y}\right)}\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(\mathsf{+.f64}\left(z, 1\right), \left(\frac{\color{blue}{z \cdot z}}{y}\right)\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(\mathsf{+.f64}\left(z, 1\right), \mathsf{/.f64}\left(\left(z \cdot z\right), \color{blue}{y}\right)\right)\right) \]

      10. *-lowering-*.f64 93.4%

          \[\leadsto \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(\mathsf{+.f64}\left(z, 1\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(z, z\right), y\right)\right)\right) \]

   4. Applied egg-rr 93.4%

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

4. Final simplification 94.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(z + 1\right) \cdot \left(z \cdot z\right) \leq -0.2:\\ \;\;\;\;\frac{\frac{x}{z}}{\frac{z}{\frac{y}{z}}}\\ \mathbf{elif}\;\left(z + 1\right) \cdot \left(z \cdot z\right) \leq 5 \cdot 10^{-70}:\\ \;\;\;\;\frac{y}{\frac{z}{\frac{x}{z}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\left(z + 1\right) \cdot \frac{z \cdot z}{y}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 97.1% accurate, 0.6× speedup

Math

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

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

Derivation

1. Split input into 2 regimes

2. if z < -1 or 1 < z

   1. Initial program 79.6%

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

   3. Taylor expanded in z around inf

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(x \cdot y\right), \color{blue}{\left({z}^{3}\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \left({\color{blue}{z}}^{3}\right)\right) \]

      3. cube-mult N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(z \cdot \color{blue}{\left(z \cdot z\right)}\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(z \cdot {z}^{\color{blue}{2}}\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \color{blue}{\left({z}^{2}\right)}\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \left(z \cdot \color{blue}{z}\right)\right)\right) \]

      7. *-lowering-*.f64 78.3%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(z, \color{blue}{z}\right)\right)\right) \]

   5. Simplified 78.3%

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

      1. times-frac N/A

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{x}{z}\right), \color{blue}{\left(\frac{z \cdot z}{y}\right)}\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, z\right), \left(\frac{\color{blue}{z \cdot z}}{y}\right)\right) \]

      6. clear-num N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, z\right), \left(\frac{1}{\color{blue}{\frac{y}{z \cdot z}}}\right)\right) \]

      7. associate-/r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, z\right), \left(\frac{1}{\frac{\frac{y}{z}}{\color{blue}{z}}}\right)\right) \]

      8. clear-num N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, z\right), \left(\frac{z}{\color{blue}{\frac{y}{z}}}\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, z\right), \mathsf{/.f64}\left(z, \color{blue}{\left(\frac{y}{z}\right)}\right)\right) \]

      10. /-lowering-/.f64 95.8%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, z\right), \mathsf{/.f64}\left(z, \mathsf{/.f64}\left(y, \color{blue}{z}\right)\right)\right) \]

   7. Applied egg-rr 95.8%

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

   if -1 < z < 1

   1. Initial program 92.2%

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

   3. Taylor expanded in z around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \color{blue}{\left({z}^{2}\right)}\right) \]
   4. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(z \cdot \color{blue}{z}\right)\right) \]

      2. *-lowering-*.f64 91.2%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \color{blue}{z}\right)\right) \]

   5. Simplified 91.2%

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. associate-/l/ N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(y, \color{blue}{\left(\frac{z}{\frac{x}{z}}\right)}\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(y, \mathsf{/.f64}\left(z, \color{blue}{\left(\frac{x}{z}\right)}\right)\right) \]

      8. /-lowering-/.f64 94.0%

         \[\leadsto \mathsf{/.f64}\left(y, \mathsf{/.f64}\left(z, \mathsf{/.f64}\left(x, \color{blue}{z}\right)\right)\right) \]

   7. Applied egg-rr 94.0%

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

Alternative 4: 96.9% accurate, 0.6× speedup

Math

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

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

Derivation

1. Split input into 2 regimes

2. if z < -1 or 1 < z

   1. Initial program 79.6%

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

      1. times-frac N/A

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

      2. associate-/r* N/A

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

      3. associate-*l/ N/A

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

      4. clear-num N/A

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

      5. inv-pow N/A

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

      6. metadata-eval N/A

         \[\leadsto \frac{{\left(\frac{z}{x}\right)}^{\left(\mathsf{neg}\left(1\right)\right)} \cdot \frac{y}{z + 1}}{z} \]

      7. clear-num N/A

         \[\leadsto \frac{{\left(\frac{z}{x}\right)}^{\left(\mathsf{neg}\left(1\right)\right)} \cdot \frac{1}{\frac{z + 1}{y}}}{z} \]

      8. inv-pow N/A

         \[\leadsto \frac{{\left(\frac{z}{x}\right)}^{\left(\mathsf{neg}\left(1\right)\right)} \cdot {\left(\frac{z + 1}{y}\right)}^{-1}}{z} \]

      9. metadata-eval N/A

         \[\leadsto \frac{{\left(\frac{z}{x}\right)}^{\left(\mathsf{neg}\left(1\right)\right)} \cdot {\left(\frac{z + 1}{y}\right)}^{\left(\mathsf{neg}\left(1\right)\right)}}{z} \]

      10. unpow-prod-down N/A

          \[\leadsto \frac{{\left(\frac{z}{x} \cdot \frac{z + 1}{y}\right)}^{\left(\mathsf{neg}\left(1\right)\right)}}{z} \]

      11. times-frac N/A

          \[\leadsto \frac{{\left(\frac{z \cdot \left(z + 1\right)}{x \cdot y}\right)}^{\left(\mathsf{neg}\left(1\right)\right)}}{z} \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\left({\left(\frac{z \cdot \left(z + 1\right)}{x \cdot y}\right)}^{\left(\mathsf{neg}\left(1\right)\right)}\right), \color{blue}{z}\right) \]

   4. Applied egg-rr 98.5%

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

   5. Taylor expanded in z around inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\color{blue}{\left(\frac{y}{z}\right)}, \mathsf{/.f64}\left(x, z\right)\right), z\right) \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 97.1%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(y, z\right), \mathsf{/.f64}\left(x, z\right)\right), z\right) \]

   7. Simplified 97.1%

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

   if -1 < z < 1

   1. Initial program 92.2%

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

   3. Taylor expanded in z around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \color{blue}{\left({z}^{2}\right)}\right) \]
   4. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(z \cdot \color{blue}{z}\right)\right) \]

      2. *-lowering-*.f64 91.2%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \color{blue}{z}\right)\right) \]

   5. Simplified 91.2%

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. associate-/l/ N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(y, \color{blue}{\left(\frac{z}{\frac{x}{z}}\right)}\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(y, \mathsf{/.f64}\left(z, \color{blue}{\left(\frac{x}{z}\right)}\right)\right) \]

      8. /-lowering-/.f64 94.0%

         \[\leadsto \mathsf{/.f64}\left(y, \mathsf{/.f64}\left(z, \mathsf{/.f64}\left(x, \color{blue}{z}\right)\right)\right) \]

   7. Applied egg-rr 94.0%

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

4. Final simplification 95.7%

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

Alternative 5: 95.4% accurate, 0.6× speedup

Math

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

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 -1.0) t_0 (if (<= z 1.0) (/ 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 <= -1.0) {
		tmp = t_0;
	} else if (z <= 1.0) {
		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 <= (-1.0d0)) then
        tmp = t_0
    else if (z <= 1.0d0) 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 <= -1.0) {
		tmp = t_0;
	} else if (z <= 1.0) {
		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 <= -1.0:
		tmp = t_0
	elif z <= 1.0:
		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(x_m / Float64(Float64(z * z) * Float64(z / y_m)))
	tmp = 0.0
	if (z <= -1.0)
		tmp = t_0;
	elseif (z <= 1.0)
		tmp = Float64(y_m / Float64(z / Float64(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 <= -1.0)
		tmp = t_0;
	elseif (z <= 1.0)
		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[(x$95$m / N[(N[(z * z), $MachinePrecision] * N[(z / y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(y$95$s * N[(x$95$s * If[LessEqual[z, -1.0], t$95$0, If[LessEqual[z, 1.0], N[(y$95$m / N[(z / N[(x$95$m / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1 or 1 < z

   1. Initial program 79.6%

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

   3. Taylor expanded in z around inf

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(x \cdot y\right), \color{blue}{\left({z}^{3}\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \left({\color{blue}{z}}^{3}\right)\right) \]

      3. cube-mult N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(z \cdot \color{blue}{\left(z \cdot z\right)}\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(z \cdot {z}^{\color{blue}{2}}\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \color{blue}{\left({z}^{2}\right)}\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \left(z \cdot \color{blue}{z}\right)\right)\right) \]

      7. *-lowering-*.f64 78.3%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(z, \color{blue}{z}\right)\right)\right) \]

   5. Simplified 78.3%

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

      1. associate-/l* N/A

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \color{blue}{\left(\frac{z \cdot \left(z \cdot z\right)}{y}\right)}\right) \]

      5. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(x, \left(z \cdot \color{blue}{\frac{z \cdot z}{y}}\right)\right) \]

      6. clear-num N/A

         \[\leadsto \mathsf{/.f64}\left(x, \left(z \cdot \frac{1}{\color{blue}{\frac{y}{z \cdot z}}}\right)\right) \]

      7. un-div-inv N/A

         \[\leadsto \mathsf{/.f64}\left(x, \left(\frac{z}{\color{blue}{\frac{y}{z \cdot z}}}\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{/.f64}\left(z, \color{blue}{\left(\frac{y}{z \cdot z}\right)}\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{/.f64}\left(z, \mathsf{/.f64}\left(y, \color{blue}{\left(z \cdot z\right)}\right)\right)\right) \]

      10. *-lowering-*.f64 89.1%

          \[\leadsto \mathsf{/.f64}\left(x, \mathsf{/.f64}\left(z, \mathsf{/.f64}\left(y, \mathsf{*.f64}\left(z, \color{blue}{z}\right)\right)\right)\right) \]

   7. Applied egg-rr 89.1%

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

      1. associate-/r/ N/A

         \[\leadsto \mathsf{/.f64}\left(x, \left(\frac{z}{y} \cdot \color{blue}{\left(z \cdot z\right)}\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(\left(\frac{z}{y}\right), \color{blue}{\left(z \cdot z\right)}\right)\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(\mathsf{/.f64}\left(z, y\right), \left(\color{blue}{z} \cdot z\right)\right)\right) \]

      4. *-lowering-*.f64 89.0%

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(\mathsf{/.f64}\left(z, y\right), \mathsf{*.f64}\left(z, \color{blue}{z}\right)\right)\right) \]

   9. Applied egg-rr 89.0%

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

   if -1 < z < 1

   1. Initial program 92.2%

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

   3. Taylor expanded in z around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \color{blue}{\left({z}^{2}\right)}\right) \]
   4. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(z \cdot \color{blue}{z}\right)\right) \]

      2. *-lowering-*.f64 91.2%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \color{blue}{z}\right)\right) \]

   5. Simplified 91.2%

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. associate-/l/ N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(y, \color{blue}{\left(\frac{z}{\frac{x}{z}}\right)}\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(y, \mathsf{/.f64}\left(z, \color{blue}{\left(\frac{x}{z}\right)}\right)\right) \]

      8. /-lowering-/.f64 94.0%

         \[\leadsto \mathsf{/.f64}\left(y, \mathsf{/.f64}\left(z, \mathsf{/.f64}\left(x, \color{blue}{z}\right)\right)\right) \]

   7. Applied egg-rr 94.0%

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

4. Final simplification 91.3%

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

Alternative 6: 92.5% accurate, 0.6× speedup

Math

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

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

Derivation

1. Split input into 2 regimes

2. if z < -1 or 1 < z

   1. Initial program 79.6%

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

   3. Taylor expanded in z around inf

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(x \cdot y\right), \color{blue}{\left({z}^{3}\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \left({\color{blue}{z}}^{3}\right)\right) \]

      3. cube-mult N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(z \cdot \color{blue}{\left(z \cdot z\right)}\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(z \cdot {z}^{\color{blue}{2}}\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \color{blue}{\left({z}^{2}\right)}\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \left(z \cdot \color{blue}{z}\right)\right)\right) \]

      7. *-lowering-*.f64 78.3%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(z, \color{blue}{z}\right)\right)\right) \]

   5. Simplified 78.3%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{y}{z \cdot \left(z \cdot z\right)}\right), \color{blue}{x}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(y, \left(z \cdot \left(z \cdot z\right)\right)\right), x\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(y, \mathsf{*.f64}\left(z, \left(z \cdot z\right)\right)\right), x\right) \]

      6. *-lowering-*.f64 84.3%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(y, \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(z, z\right)\right)\right), x\right) \]

   7. Applied egg-rr 84.3%

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

   if -1 < z < 1

   1. Initial program 92.2%

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

   3. Taylor expanded in z around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \color{blue}{\left({z}^{2}\right)}\right) \]
   4. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(z \cdot \color{blue}{z}\right)\right) \]

      2. *-lowering-*.f64 91.2%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \color{blue}{z}\right)\right) \]

   5. Simplified 91.2%

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. associate-/l/ N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(y, \color{blue}{\left(\frac{z}{\frac{x}{z}}\right)}\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(y, \mathsf{/.f64}\left(z, \color{blue}{\left(\frac{x}{z}\right)}\right)\right) \]

      8. /-lowering-/.f64 94.0%

         \[\leadsto \mathsf{/.f64}\left(y, \mathsf{/.f64}\left(z, \mathsf{/.f64}\left(x, \color{blue}{z}\right)\right)\right) \]

   7. Applied egg-rr 94.0%

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

4. Final simplification 88.7%

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

Alternative 7: 98.2% accurate, 1.0× speedup

Math

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

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 85.4%

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

   1. times-frac N/A

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

   2. associate-/r* N/A

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

   3. associate-*l/ N/A

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

   4. clear-num N/A

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

   5. inv-pow N/A

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

   6. metadata-eval N/A

      \[\leadsto \frac{{\left(\frac{z}{x}\right)}^{\left(\mathsf{neg}\left(1\right)\right)} \cdot \frac{y}{z + 1}}{z} \]

   7. clear-num N/A

      \[\leadsto \frac{{\left(\frac{z}{x}\right)}^{\left(\mathsf{neg}\left(1\right)\right)} \cdot \frac{1}{\frac{z + 1}{y}}}{z} \]

   8. inv-pow N/A

      \[\leadsto \frac{{\left(\frac{z}{x}\right)}^{\left(\mathsf{neg}\left(1\right)\right)} \cdot {\left(\frac{z + 1}{y}\right)}^{-1}}{z} \]

   9. metadata-eval N/A

      \[\leadsto \frac{{\left(\frac{z}{x}\right)}^{\left(\mathsf{neg}\left(1\right)\right)} \cdot {\left(\frac{z + 1}{y}\right)}^{\left(\mathsf{neg}\left(1\right)\right)}}{z} \]

   10. unpow-prod-down N/A

       \[\leadsto \frac{{\left(\frac{z}{x} \cdot \frac{z + 1}{y}\right)}^{\left(\mathsf{neg}\left(1\right)\right)}}{z} \]

   11. times-frac N/A

       \[\leadsto \frac{{\left(\frac{z \cdot \left(z + 1\right)}{x \cdot y}\right)}^{\left(\mathsf{neg}\left(1\right)\right)}}{z} \]

   12. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\left({\left(\frac{z \cdot \left(z + 1\right)}{x \cdot y}\right)}^{\left(\mathsf{neg}\left(1\right)\right)}\right), \color{blue}{z}\right) \]

4. Applied egg-rr 98.5%

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

5. Final simplification 98.5%

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

Alternative 8: 81.1% 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 85.4%

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

3. Taylor expanded in z around 0

   \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \color{blue}{\left({z}^{2}\right)}\right) \]
4. Step-by-step derivation

   1. unpow2 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(z \cdot \color{blue}{z}\right)\right) \]

   2. *-lowering-*.f64 74.0%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \color{blue}{z}\right)\right) \]

5. Simplified 74.0%

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

   1. *-commutative N/A

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

   2. associate-/l* N/A

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

   3. associate-/l/ N/A

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

   4. clear-num N/A

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

   5. un-div-inv N/A

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

   6. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(y, \color{blue}{\left(\frac{z}{\frac{x}{z}}\right)}\right) \]

   7. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(y, \mathsf{/.f64}\left(z, \color{blue}{\left(\frac{x}{z}\right)}\right)\right) \]

   8. /-lowering-/.f64 75.9%

      \[\leadsto \mathsf{/.f64}\left(y, \mathsf{/.f64}\left(z, \mathsf{/.f64}\left(x, \color{blue}{z}\right)\right)\right) \]

7. Applied egg-rr 75.9%

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

Alternative 9: 70.0% accurate, 1.6× speedup

Math

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

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(y_m / Float64(z * z)))))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
x_m, y_m, z = num2cell(sort([x_m, y_m, z])){:}
function tmp = code(y_s, x_s, x_m, y_m, z)
	tmp = y_s * (x_s * (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[(y$95$m / N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 85.4%

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

3. Taylor expanded in z around 0

   \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \color{blue}{\left({z}^{2}\right)}\right) \]
4. Step-by-step derivation

   1. unpow2 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(z \cdot \color{blue}{z}\right)\right) \]

   2. *-lowering-*.f64 74.0%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \color{blue}{z}\right)\right) \]

5. Simplified 74.0%

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

   1. associate-/l* N/A

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

   2. *-commutative N/A

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

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\frac{y}{z \cdot z}\right), \color{blue}{x}\right) \]

   4. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(y, \left(z \cdot z\right)\right), x\right) \]

   5. *-lowering-*.f64 77.1%

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(y, \mathsf{*.f64}\left(z, z\right)\right), x\right) \]

7. Applied egg-rr 77.1%

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

8. Final simplification 77.1%

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

Developer Target 1: 96.9% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024150 
(FPCore (x y z)
  :name "Statistics.Distribution.Beta:$cvariance from math-functions-0.1.5.2"
  :precision binary64

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

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