Statistics.Distribution.CauchyLorentz:$cdensity from math-functions-0.1.5.2

Percentage Accurate: 88.9% → 99.3%

Time: 15.0s

Alternatives: 9

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 88.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;z \cdot z \leq 10^{+237}:\\ \;\;\;\;\frac{\frac{1}{y\_m}}{x\_m \cdot \left(1 + z \cdot z\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{y\_m \cdot \left(z \cdot x\_m\right)}}{z}\\ \end{array}\right) \end{array} \]

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 z) 1e+237)
     (/ (/ 1.0 y_m) (* x_m (+ 1.0 (* z z))))
     (/ (/ 1.0 (* y_m (* z x_m))) z)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
assert(x_m < y_m && y_m < z);
double code(double y_s, double x_s, double x_m, double y_m, double z) {
	double tmp;
	if ((z * z) <= 1e+237) {
		tmp = (1.0 / y_m) / (x_m * (1.0 + (z * z)));
	} else {
		tmp = (1.0 / (y_m * (z * x_m))) / z;
	}
	return y_s * (x_s * tmp);
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
y\_m = abs(y)
y\_s = copysign(1.0d0, y)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
real(8) function code(y_s, x_s, x_m, y_m, z)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((z * z) <= 1d+237) then
        tmp = (1.0d0 / y_m) / (x_m * (1.0d0 + (z * z)))
    else
        tmp = (1.0d0 / (y_m * (z * x_m))) / z
    end if
    code = y_s * (x_s * tmp)
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
assert x_m < y_m && y_m < z;
public static double code(double y_s, double x_s, double x_m, double y_m, double z) {
	double tmp;
	if ((z * z) <= 1e+237) {
		tmp = (1.0 / y_m) / (x_m * (1.0 + (z * z)));
	} else {
		tmp = (1.0 / (y_m * (z * x_m))) / z;
	}
	return y_s * (x_s * tmp);
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
[x_m, y_m, z] = sort([x_m, y_m, z])
def code(y_s, x_s, x_m, y_m, z):
	tmp = 0
	if (z * z) <= 1e+237:
		tmp = (1.0 / y_m) / (x_m * (1.0 + (z * z)))
	else:
		tmp = (1.0 / (y_m * (z * x_m))) / z
	return y_s * (x_s * tmp)

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
y\_m = abs(y)
y\_s = copysign(1.0, y)
x_m, y_m, z = sort([x_m, y_m, z])
function code(y_s, x_s, x_m, y_m, z)
	tmp = 0.0
	if (Float64(z * z) <= 1e+237)
		tmp = Float64(Float64(1.0 / y_m) / Float64(x_m * Float64(1.0 + Float64(z * z))));
	else
		tmp = Float64(Float64(1.0 / Float64(y_m * Float64(z * x_m))) / z);
	end
	return Float64(y_s * Float64(x_s * tmp))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
x_m, y_m, z = num2cell(sort([x_m, y_m, z])){:}
function tmp_2 = code(y_s, x_s, x_m, y_m, z)
	tmp = 0.0;
	if ((z * z) <= 1e+237)
		tmp = (1.0 / y_m) / (x_m * (1.0 + (z * z)));
	else
		tmp = (1.0 / (y_m * (z * x_m))) / z;
	end
	tmp_2 = y_s * (x_s * tmp);
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
code[y$95$s_, x$95$s_, x$95$m_, y$95$m_, z_] := N[(y$95$s * N[(x$95$s * If[LessEqual[N[(z * z), $MachinePrecision], 1e+237], N[(N[(1.0 / y$95$m), $MachinePrecision] / N[(x$95$m * N[(1.0 + N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[(y$95$m * N[(z * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 9.9999999999999994e236

   1. Initial program 97.5%

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

      1. associate-/l/ N/A

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

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

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

      3. *-commutative N/A

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

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

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

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

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

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

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

      7. *-lowering-*.f64 97.0%

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

   3. Simplified 97.0%

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

      1. associate-/r* N/A

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

      2. clear-num N/A

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

      3. associate-/l* N/A

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

      4. associate-/r* N/A

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

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

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

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

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

      7. div-inv N/A

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

      8. clear-num N/A

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

      9. /-rgt-identity N/A

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

      10. *-commutative N/A

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

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

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

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

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

      13. *-lowering-*.f64 98.7%

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

   6. Applied egg-rr 98.7%

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

   if 9.9999999999999994e236 < (*.f64 z z)

   1. Initial program 73.6%

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

      1. associate-/l/ N/A

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

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

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

      3. *-commutative N/A

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

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

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

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

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

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

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

      7. *-lowering-*.f64 73.7%

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

   3. Simplified 73.7%

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

   5. Taylor expanded in z around inf

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

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

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(x \cdot \left(y \cdot {z}^{2}\right)\right)}\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \color{blue}{\left(y \cdot {z}^{2}\right)}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \color{blue}{\left({z}^{2}\right)}\right)\right)\right) \]

      4. unpow2 N/A

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

      5. *-lowering-*.f64 73.7%

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

   7. Simplified 73.7%

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

      1. associate-*r* N/A

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

      2. associate-/r* N/A

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

      3. associate-/r* N/A

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

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

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

      5. clear-num N/A

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

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

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

      7. associate-/r/ N/A

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

      8. /-rgt-identity N/A

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

      9. *-commutative N/A

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

      10. *-commutative N/A

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

      11. associate-*l* N/A

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

      12. *-commutative N/A

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

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

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

      14. *-lowering-*.f64 97.2%

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

   9. Applied egg-rr 97.2%

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

Alternative 2: 99.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ \begin{array}{l} t_0 := y\_m \cdot \left(1 + z \cdot z\right)\\ y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq 5 \cdot 10^{+306}:\\ \;\;\;\;\frac{\frac{1}{x\_m}}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{y\_m \cdot \left(z \cdot x\_m\right)}}{z}\\ \end{array}\right) \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (y_s x_s x_m y_m z)
 :precision binary64
 (let* ((t_0 (* y_m (+ 1.0 (* z z)))))
   (*
    y_s
    (*
     x_s
     (if (<= t_0 5e+306)
       (/ (/ 1.0 x_m) t_0)
       (/ (/ 1.0 (* y_m (* z x_m))) z))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
assert(x_m < y_m && y_m < z);
double code(double y_s, double x_s, double x_m, double y_m, double z) {
	double t_0 = y_m * (1.0 + (z * z));
	double tmp;
	if (t_0 <= 5e+306) {
		tmp = (1.0 / x_m) / t_0;
	} else {
		tmp = (1.0 / (y_m * (z * x_m))) / z;
	}
	return y_s * (x_s * tmp);
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
y\_m = abs(y)
y\_s = copysign(1.0d0, y)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
real(8) function code(y_s, x_s, x_m, y_m, z)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = y_m * (1.0d0 + (z * z))
    if (t_0 <= 5d+306) then
        tmp = (1.0d0 / x_m) / t_0
    else
        tmp = (1.0d0 / (y_m * (z * x_m))) / z
    end if
    code = y_s * (x_s * tmp)
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
assert x_m < y_m && y_m < z;
public static double code(double y_s, double x_s, double x_m, double y_m, double z) {
	double t_0 = y_m * (1.0 + (z * z));
	double tmp;
	if (t_0 <= 5e+306) {
		tmp = (1.0 / x_m) / t_0;
	} else {
		tmp = (1.0 / (y_m * (z * x_m))) / z;
	}
	return y_s * (x_s * tmp);
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
[x_m, y_m, z] = sort([x_m, y_m, z])
def code(y_s, x_s, x_m, y_m, z):
	t_0 = y_m * (1.0 + (z * z))
	tmp = 0
	if t_0 <= 5e+306:
		tmp = (1.0 / x_m) / t_0
	else:
		tmp = (1.0 / (y_m * (z * x_m))) / z
	return y_s * (x_s * tmp)

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
y\_m = abs(y)
y\_s = copysign(1.0, y)
x_m, y_m, z = sort([x_m, y_m, z])
function code(y_s, x_s, x_m, y_m, z)
	t_0 = Float64(y_m * Float64(1.0 + Float64(z * z)))
	tmp = 0.0
	if (t_0 <= 5e+306)
		tmp = Float64(Float64(1.0 / x_m) / t_0);
	else
		tmp = Float64(Float64(1.0 / Float64(y_m * Float64(z * x_m))) / z);
	end
	return Float64(y_s * Float64(x_s * tmp))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
x_m, y_m, z = num2cell(sort([x_m, y_m, z])){:}
function tmp_2 = code(y_s, x_s, x_m, y_m, z)
	t_0 = y_m * (1.0 + (z * z));
	tmp = 0.0;
	if (t_0 <= 5e+306)
		tmp = (1.0 / x_m) / t_0;
	else
		tmp = (1.0 / (y_m * (z * x_m))) / z;
	end
	tmp_2 = y_s * (x_s * tmp);
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
code[y$95$s_, x$95$s_, x$95$m_, y$95$m_, z_] := Block[{t$95$0 = N[(y$95$m * N[(1.0 + N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(y$95$s * N[(x$95$s * If[LessEqual[t$95$0, 5e+306], N[(N[(1.0 / x$95$m), $MachinePrecision] / t$95$0), $MachinePrecision], N[(N[(1.0 / N[(y$95$m * N[(z * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y (+.f64 #s(literal 1 binary64) (*.f64 z z))) < 4.99999999999999993e306

   1. Initial program 93.1%

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

   if 4.99999999999999993e306 < (*.f64 y (+.f64 #s(literal 1 binary64) (*.f64 z z)))

   1. Initial program 75.6%

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

      1. associate-/l/ N/A

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

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

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

      3. *-commutative N/A

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

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

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

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

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

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

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

      7. *-lowering-*.f64 75.6%

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

   3. Simplified 75.6%

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

   5. Taylor expanded in z around inf

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

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

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(x \cdot \left(y \cdot {z}^{2}\right)\right)}\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \color{blue}{\left(y \cdot {z}^{2}\right)}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \color{blue}{\left({z}^{2}\right)}\right)\right)\right) \]

      4. unpow2 N/A

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

      5. *-lowering-*.f64 75.6%

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

   7. Simplified 75.6%

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

      1. associate-*r* N/A

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

      2. associate-/r* N/A

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

      3. associate-/r* N/A

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

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

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

      5. clear-num N/A

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

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

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

      7. associate-/r/ N/A

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

      8. /-rgt-identity N/A

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

      9. *-commutative N/A

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

      10. *-commutative N/A

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

      11. associate-*l* N/A

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

      12. *-commutative N/A

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

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

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

      14. *-lowering-*.f64 97.5%

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

   9. Applied egg-rr 97.5%

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

Alternative 3: 99.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ \begin{array}{l} t_0 := y\_m \cdot \left(1 + z \cdot z\right)\\ y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq 5 \cdot 10^{+306}:\\ \;\;\;\;\frac{1}{x\_m \cdot t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{y\_m \cdot \left(z \cdot x\_m\right)}}{z}\\ \end{array}\right) \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (y_s x_s x_m y_m z)
 :precision binary64
 (let* ((t_0 (* y_m (+ 1.0 (* z z)))))
   (*
    y_s
    (*
     x_s
     (if (<= t_0 5e+306)
       (/ 1.0 (* x_m t_0))
       (/ (/ 1.0 (* y_m (* z x_m))) z))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
assert(x_m < y_m && y_m < z);
double code(double y_s, double x_s, double x_m, double y_m, double z) {
	double t_0 = y_m * (1.0 + (z * z));
	double tmp;
	if (t_0 <= 5e+306) {
		tmp = 1.0 / (x_m * t_0);
	} else {
		tmp = (1.0 / (y_m * (z * x_m))) / z;
	}
	return y_s * (x_s * tmp);
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
y\_m = abs(y)
y\_s = copysign(1.0d0, y)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
real(8) function code(y_s, x_s, x_m, y_m, z)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = y_m * (1.0d0 + (z * z))
    if (t_0 <= 5d+306) then
        tmp = 1.0d0 / (x_m * t_0)
    else
        tmp = (1.0d0 / (y_m * (z * x_m))) / z
    end if
    code = y_s * (x_s * tmp)
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
assert x_m < y_m && y_m < z;
public static double code(double y_s, double x_s, double x_m, double y_m, double z) {
	double t_0 = y_m * (1.0 + (z * z));
	double tmp;
	if (t_0 <= 5e+306) {
		tmp = 1.0 / (x_m * t_0);
	} else {
		tmp = (1.0 / (y_m * (z * x_m))) / z;
	}
	return y_s * (x_s * tmp);
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
[x_m, y_m, z] = sort([x_m, y_m, z])
def code(y_s, x_s, x_m, y_m, z):
	t_0 = y_m * (1.0 + (z * z))
	tmp = 0
	if t_0 <= 5e+306:
		tmp = 1.0 / (x_m * t_0)
	else:
		tmp = (1.0 / (y_m * (z * x_m))) / z
	return y_s * (x_s * tmp)

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
y\_m = abs(y)
y\_s = copysign(1.0, y)
x_m, y_m, z = sort([x_m, y_m, z])
function code(y_s, x_s, x_m, y_m, z)
	t_0 = Float64(y_m * Float64(1.0 + Float64(z * z)))
	tmp = 0.0
	if (t_0 <= 5e+306)
		tmp = Float64(1.0 / Float64(x_m * t_0));
	else
		tmp = Float64(Float64(1.0 / Float64(y_m * Float64(z * x_m))) / z);
	end
	return Float64(y_s * Float64(x_s * tmp))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
x_m, y_m, z = num2cell(sort([x_m, y_m, z])){:}
function tmp_2 = code(y_s, x_s, x_m, y_m, z)
	t_0 = y_m * (1.0 + (z * z));
	tmp = 0.0;
	if (t_0 <= 5e+306)
		tmp = 1.0 / (x_m * t_0);
	else
		tmp = (1.0 / (y_m * (z * x_m))) / z;
	end
	tmp_2 = y_s * (x_s * tmp);
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
code[y$95$s_, x$95$s_, x$95$m_, y$95$m_, z_] := Block[{t$95$0 = N[(y$95$m * N[(1.0 + N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(y$95$s * N[(x$95$s * If[LessEqual[t$95$0, 5e+306], N[(1.0 / N[(x$95$m * t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[(y$95$m * N[(z * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y (+.f64 #s(literal 1 binary64) (*.f64 z z))) < 4.99999999999999993e306

   1. Initial program 93.1%

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

      1. associate-/l/ N/A

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

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

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

      3. *-commutative N/A

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

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

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

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

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

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

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

      7. *-lowering-*.f64 92.7%

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

   3. Simplified 92.7%

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

   if 4.99999999999999993e306 < (*.f64 y (+.f64 #s(literal 1 binary64) (*.f64 z z)))

   1. Initial program 75.6%

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

      1. associate-/l/ N/A

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

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

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

      3. *-commutative N/A

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

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

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

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

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

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

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

      7. *-lowering-*.f64 75.6%

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

   3. Simplified 75.6%

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

   5. Taylor expanded in z around inf

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

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

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(x \cdot \left(y \cdot {z}^{2}\right)\right)}\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \color{blue}{\left(y \cdot {z}^{2}\right)}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \color{blue}{\left({z}^{2}\right)}\right)\right)\right) \]

      4. unpow2 N/A

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

      5. *-lowering-*.f64 75.6%

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

   7. Simplified 75.6%

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

      1. associate-*r* N/A

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

      2. associate-/r* N/A

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

      3. associate-/r* N/A

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

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

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

      5. clear-num N/A

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

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

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

      7. associate-/r/ N/A

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

      8. /-rgt-identity N/A

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

      9. *-commutative N/A

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

      10. *-commutative N/A

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

      11. associate-*l* N/A

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

      12. *-commutative N/A

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

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

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

      14. *-lowering-*.f64 97.5%

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

   9. Applied egg-rr 97.5%

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

Alternative 4: 98.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;z \cdot z \leq 0.01:\\ \;\;\;\;\frac{1 - z \cdot z}{y\_m \cdot x\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{y\_m \cdot \left(z \cdot x\_m\right)}}{z}\\ \end{array}\right) \end{array} \]

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 z) 0.01)
     (/ (- 1.0 (* z z)) (* y_m x_m))
     (/ (/ 1.0 (* y_m (* z x_m))) z)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
assert(x_m < y_m && y_m < z);
double code(double y_s, double x_s, double x_m, double y_m, double z) {
	double tmp;
	if ((z * z) <= 0.01) {
		tmp = (1.0 - (z * z)) / (y_m * x_m);
	} else {
		tmp = (1.0 / (y_m * (z * x_m))) / z;
	}
	return y_s * (x_s * tmp);
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
y\_m = abs(y)
y\_s = copysign(1.0d0, y)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
real(8) function code(y_s, x_s, x_m, y_m, z)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((z * z) <= 0.01d0) then
        tmp = (1.0d0 - (z * z)) / (y_m * x_m)
    else
        tmp = (1.0d0 / (y_m * (z * x_m))) / z
    end if
    code = y_s * (x_s * tmp)
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
assert x_m < y_m && y_m < z;
public static double code(double y_s, double x_s, double x_m, double y_m, double z) {
	double tmp;
	if ((z * z) <= 0.01) {
		tmp = (1.0 - (z * z)) / (y_m * x_m);
	} else {
		tmp = (1.0 / (y_m * (z * x_m))) / z;
	}
	return y_s * (x_s * tmp);
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
[x_m, y_m, z] = sort([x_m, y_m, z])
def code(y_s, x_s, x_m, y_m, z):
	tmp = 0
	if (z * z) <= 0.01:
		tmp = (1.0 - (z * z)) / (y_m * x_m)
	else:
		tmp = (1.0 / (y_m * (z * x_m))) / z
	return y_s * (x_s * tmp)

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
y\_m = abs(y)
y\_s = copysign(1.0, y)
x_m, y_m, z = sort([x_m, y_m, z])
function code(y_s, x_s, x_m, y_m, z)
	tmp = 0.0
	if (Float64(z * z) <= 0.01)
		tmp = Float64(Float64(1.0 - Float64(z * z)) / Float64(y_m * x_m));
	else
		tmp = Float64(Float64(1.0 / Float64(y_m * Float64(z * x_m))) / z);
	end
	return Float64(y_s * Float64(x_s * tmp))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
x_m, y_m, z = num2cell(sort([x_m, y_m, z])){:}
function tmp_2 = code(y_s, x_s, x_m, y_m, z)
	tmp = 0.0;
	if ((z * z) <= 0.01)
		tmp = (1.0 - (z * z)) / (y_m * x_m);
	else
		tmp = (1.0 / (y_m * (z * x_m))) / z;
	end
	tmp_2 = y_s * (x_s * tmp);
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
code[y$95$s_, x$95$s_, x$95$m_, y$95$m_, z_] := N[(y$95$s * N[(x$95$s * If[LessEqual[N[(z * z), $MachinePrecision], 0.01], N[(N[(1.0 - N[(z * z), $MachinePrecision]), $MachinePrecision] / N[(y$95$m * x$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[(y$95$m * N[(z * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 0.0100000000000000002

   1. Initial program 99.6%

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

      1. associate-/l/ N/A

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

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

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

      3. *-commutative N/A

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

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

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

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

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

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

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

      7. *-lowering-*.f64 99.4%

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

   3. Simplified 99.4%

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

   5. Taylor expanded in z around 0

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

         \[\leadsto \left(0 - \frac{{z}^{2}}{x \cdot y}\right) + \frac{\color{blue}{1}}{x \cdot y} \]

      3. associate-+l- N/A

         \[\leadsto 0 - \color{blue}{\left(\frac{{z}^{2}}{x \cdot y} - \frac{1}{x \cdot y}\right)} \]

      4. sub0-neg N/A

         \[\leadsto \mathsf{neg}\left(\left(\frac{{z}^{2}}{x \cdot y} - \frac{1}{x \cdot y}\right)\right) \]

      5. sub-neg N/A

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

      6. *-rgt-identity N/A

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

      7. associate-*r/ N/A

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

      8. neg-mul-1 N/A

         \[\leadsto \mathsf{neg}\left(\left({z}^{2} \cdot \frac{1}{x \cdot y} + -1 \cdot \frac{1}{x \cdot y}\right)\right) \]

      9. distribute-rgt-out N/A

         \[\leadsto \mathsf{neg}\left(\frac{1}{x \cdot y} \cdot \left({z}^{2} + -1\right)\right) \]

      10. associate-*l/ N/A

          \[\leadsto \mathsf{neg}\left(\frac{1 \cdot \left({z}^{2} + -1\right)}{x \cdot y}\right) \]

      11. *-lft-identity N/A

          \[\leadsto \mathsf{neg}\left(\frac{{z}^{2} + -1}{x \cdot y}\right) \]

      12. distribute-neg-frac N/A

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

      13. sub0-neg N/A

          \[\leadsto \frac{0 - \left({z}^{2} + -1\right)}{\color{blue}{x} \cdot y} \]

      14. metadata-eval N/A

          \[\leadsto \frac{0 - \left({z}^{2} + \left(\mathsf{neg}\left(1\right)\right)\right)}{x \cdot y} \]

      15. sub-neg N/A

          \[\leadsto \frac{0 - \left({z}^{2} - 1\right)}{x \cdot y} \]

      16. associate-+l- N/A

          \[\leadsto \frac{\left(0 - {z}^{2}\right) + 1}{\color{blue}{x} \cdot y} \]

      17. neg-sub0 N/A

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

      18. mul-1-neg N/A

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

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

          \[\leadsto \mathsf{/.f64}\left(\left(-1 \cdot {z}^{2} + 1\right), \color{blue}{\left(x \cdot y\right)}\right) \]

   7. Simplified 99.0%

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

   if 0.0100000000000000002 < (*.f64 z z)

   1. Initial program 82.4%

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

      1. associate-/l/ N/A

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

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

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

      3. *-commutative N/A

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

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

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

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

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

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

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

      7. *-lowering-*.f64 81.9%

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

   3. Simplified 81.9%

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

   5. Taylor expanded in z around inf

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

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

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(x \cdot \left(y \cdot {z}^{2}\right)\right)}\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \color{blue}{\left(y \cdot {z}^{2}\right)}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \color{blue}{\left({z}^{2}\right)}\right)\right)\right) \]

      4. unpow2 N/A

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

      5. *-lowering-*.f64 81.2%

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

   7. Simplified 81.2%

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

      1. associate-*r* N/A

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

      2. associate-/r* N/A

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

      3. associate-/r* N/A

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

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

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

      5. clear-num N/A

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

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

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

      7. associate-/r/ N/A

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

      8. /-rgt-identity N/A

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

      9. *-commutative N/A

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

      10. *-commutative N/A

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

      11. associate-*l* N/A

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

      12. *-commutative N/A

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

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

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

      14. *-lowering-*.f64 94.4%

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

   9. Applied egg-rr 94.4%

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

4. Final simplification 96.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 0.01:\\ \;\;\;\;\frac{1 - z \cdot z}{y \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{y \cdot \left(z \cdot x\right)}}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 88.1% accurate, 0.7× speedup

Math

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

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 z) 1.0) (/ (/ 1.0 y_m) x_m) (/ 1.0 (* x_m (* y_m (* z z))))))))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 1

   1. Initial program 99.6%

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

      1. associate-/l/ N/A

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

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

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

      3. *-commutative N/A

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

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

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

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

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

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

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

      7. *-lowering-*.f64 99.4%

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

   3. Simplified 99.4%

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

   5. Taylor expanded in z around 0

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

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

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

      2. *-lowering-*.f64 98.5%

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

   7. Simplified 98.5%

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

      1. *-commutative N/A

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

      2. associate-/r* N/A

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

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

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

      4. /-lowering-/.f64 98.6%

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

   9. Applied egg-rr 98.6%

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

   if 1 < (*.f64 z z)

   1. Initial program 82.4%

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

      1. associate-/l/ N/A

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

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

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

      3. *-commutative N/A

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

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

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

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

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

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

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

      7. *-lowering-*.f64 81.9%

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

   3. Simplified 81.9%

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

   5. Taylor expanded in z around inf

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

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

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(x \cdot \left(y \cdot {z}^{2}\right)\right)}\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \color{blue}{\left(y \cdot {z}^{2}\right)}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \color{blue}{\left({z}^{2}\right)}\right)\right)\right) \]

      4. unpow2 N/A

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

      5. *-lowering-*.f64 81.2%

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

   7. Simplified 81.2%

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

Alternative 6: 75.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq 0.86:\\ \;\;\;\;\frac{1 - z \cdot z}{y\_m \cdot x\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{z \cdot \left(y\_m \cdot \left(z \cdot x\_m\right)\right)}\\ \end{array}\right) \end{array} \]

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 0.86)
     (/ (- 1.0 (* z z)) (* y_m x_m))
     (/ 1.0 (* z (* y_m (* z x_m))))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
assert(x_m < y_m && y_m < z);
double code(double y_s, double x_s, double x_m, double y_m, double z) {
	double tmp;
	if (z <= 0.86) {
		tmp = (1.0 - (z * z)) / (y_m * x_m);
	} else {
		tmp = 1.0 / (z * (y_m * (z * x_m)));
	}
	return y_s * (x_s * tmp);
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
y\_m = abs(y)
y\_s = copysign(1.0d0, y)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
real(8) function code(y_s, x_s, x_m, y_m, z)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if (z <= 0.86d0) then
        tmp = (1.0d0 - (z * z)) / (y_m * x_m)
    else
        tmp = 1.0d0 / (z * (y_m * (z * x_m)))
    end if
    code = y_s * (x_s * tmp)
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
assert x_m < y_m && y_m < z;
public static double code(double y_s, double x_s, double x_m, double y_m, double z) {
	double tmp;
	if (z <= 0.86) {
		tmp = (1.0 - (z * z)) / (y_m * x_m);
	} else {
		tmp = 1.0 / (z * (y_m * (z * x_m)));
	}
	return y_s * (x_s * tmp);
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
[x_m, y_m, z] = sort([x_m, y_m, z])
def code(y_s, x_s, x_m, y_m, z):
	tmp = 0
	if z <= 0.86:
		tmp = (1.0 - (z * z)) / (y_m * x_m)
	else:
		tmp = 1.0 / (z * (y_m * (z * x_m)))
	return y_s * (x_s * tmp)

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
y\_m = abs(y)
y\_s = copysign(1.0, y)
x_m, y_m, z = sort([x_m, y_m, z])
function code(y_s, x_s, x_m, y_m, z)
	tmp = 0.0
	if (z <= 0.86)
		tmp = Float64(Float64(1.0 - Float64(z * z)) / Float64(y_m * x_m));
	else
		tmp = Float64(1.0 / Float64(z * Float64(y_m * Float64(z * x_m))));
	end
	return Float64(y_s * Float64(x_s * tmp))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
x_m, y_m, z = num2cell(sort([x_m, y_m, z])){:}
function tmp_2 = code(y_s, x_s, x_m, y_m, z)
	tmp = 0.0;
	if (z <= 0.86)
		tmp = (1.0 - (z * z)) / (y_m * x_m);
	else
		tmp = 1.0 / (z * (y_m * (z * x_m)));
	end
	tmp_2 = y_s * (x_s * tmp);
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
code[y$95$s_, x$95$s_, x$95$m_, y$95$m_, z_] := N[(y$95$s * N[(x$95$s * If[LessEqual[z, 0.86], N[(N[(1.0 - N[(z * z), $MachinePrecision]), $MachinePrecision] / N[(y$95$m * x$95$m), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(z * N[(y$95$m * N[(z * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 0.859999999999999987

   1. Initial program 93.0%

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

      1. associate-/l/ N/A

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

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

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

      3. *-commutative N/A

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

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

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

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

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

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

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

      7. *-lowering-*.f64 92.5%

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

   3. Simplified 92.5%

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

   5. Taylor expanded in z around 0

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

         \[\leadsto \left(0 - \frac{{z}^{2}}{x \cdot y}\right) + \frac{\color{blue}{1}}{x \cdot y} \]

      3. associate-+l- N/A

         \[\leadsto 0 - \color{blue}{\left(\frac{{z}^{2}}{x \cdot y} - \frac{1}{x \cdot y}\right)} \]

      4. sub0-neg N/A

         \[\leadsto \mathsf{neg}\left(\left(\frac{{z}^{2}}{x \cdot y} - \frac{1}{x \cdot y}\right)\right) \]

      5. sub-neg N/A

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

      6. *-rgt-identity N/A

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

      7. associate-*r/ N/A

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

      8. neg-mul-1 N/A

         \[\leadsto \mathsf{neg}\left(\left({z}^{2} \cdot \frac{1}{x \cdot y} + -1 \cdot \frac{1}{x \cdot y}\right)\right) \]

      9. distribute-rgt-out N/A

         \[\leadsto \mathsf{neg}\left(\frac{1}{x \cdot y} \cdot \left({z}^{2} + -1\right)\right) \]

      10. associate-*l/ N/A

          \[\leadsto \mathsf{neg}\left(\frac{1 \cdot \left({z}^{2} + -1\right)}{x \cdot y}\right) \]

      11. *-lft-identity N/A

          \[\leadsto \mathsf{neg}\left(\frac{{z}^{2} + -1}{x \cdot y}\right) \]

      12. distribute-neg-frac N/A

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

      13. sub0-neg N/A

          \[\leadsto \frac{0 - \left({z}^{2} + -1\right)}{\color{blue}{x} \cdot y} \]

      14. metadata-eval N/A

          \[\leadsto \frac{0 - \left({z}^{2} + \left(\mathsf{neg}\left(1\right)\right)\right)}{x \cdot y} \]

      15. sub-neg N/A

          \[\leadsto \frac{0 - \left({z}^{2} - 1\right)}{x \cdot y} \]

      16. associate-+l- N/A

          \[\leadsto \frac{\left(0 - {z}^{2}\right) + 1}{\color{blue}{x} \cdot y} \]

      17. neg-sub0 N/A

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

      18. mul-1-neg N/A

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

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

          \[\leadsto \mathsf{/.f64}\left(\left(-1 \cdot {z}^{2} + 1\right), \color{blue}{\left(x \cdot y\right)}\right) \]

   7. Simplified 65.8%

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

   if 0.859999999999999987 < z

   1. Initial program 82.9%

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

      1. associate-/l/ N/A

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

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

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

      3. *-commutative N/A

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

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

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

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

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

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

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

      7. *-lowering-*.f64 82.8%

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

   3. Simplified 82.8%

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

   5. Taylor expanded in z around inf

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

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

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(x \cdot \left(y \cdot {z}^{2}\right)\right)}\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \color{blue}{\left(y \cdot {z}^{2}\right)}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \color{blue}{\left({z}^{2}\right)}\right)\right)\right) \]

      4. unpow2 N/A

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

      5. *-lowering-*.f64 82.4%

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

   7. Simplified 82.4%

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. /-rgt-identity N/A

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

      5. associate-/r/ N/A

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

      6. *-commutative N/A

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

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

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

      8. associate-/r/ N/A

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

      9. /-rgt-identity N/A

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

      10. *-commutative N/A

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

      11. *-commutative N/A

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

      12. associate-*l* N/A

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

      13. *-commutative N/A

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

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

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

      15. *-lowering-*.f64 94.6%

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

   9. Applied egg-rr 94.6%

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

4. Final simplification 73.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 0.86:\\ \;\;\;\;\frac{1 - z \cdot z}{y \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{z \cdot \left(y \cdot \left(z \cdot x\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 78.2% accurate, 0.8× speedup

Math

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

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) (/ (/ 1.0 y_m) x_m) (/ 1.0 (* z (* y_m (* z x_m))))))))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if z < 1

   1. Initial program 93.0%

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

      1. associate-/l/ N/A

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

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

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

      3. *-commutative N/A

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

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

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

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

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

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

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

      7. *-lowering-*.f64 92.5%

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

   3. Simplified 92.5%

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

   5. Taylor expanded in z around 0

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

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

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

      2. *-lowering-*.f64 68.7%

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

   7. Simplified 68.7%

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

      1. *-commutative N/A

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

      2. associate-/r* N/A

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

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

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

      4. /-lowering-/.f64 68.0%

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

   9. Applied egg-rr 68.0%

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

   if 1 < z

   1. Initial program 82.9%

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

      1. associate-/l/ N/A

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

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

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

      3. *-commutative N/A

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

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

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

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

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

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

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

      7. *-lowering-*.f64 82.8%

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

   3. Simplified 82.8%

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

   5. Taylor expanded in z around inf

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

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

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(x \cdot \left(y \cdot {z}^{2}\right)\right)}\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \color{blue}{\left(y \cdot {z}^{2}\right)}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \color{blue}{\left({z}^{2}\right)}\right)\right)\right) \]

      4. unpow2 N/A

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

      5. *-lowering-*.f64 82.4%

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

   7. Simplified 82.4%

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. /-rgt-identity N/A

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

      5. associate-/r/ N/A

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

      6. *-commutative N/A

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

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

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

      8. associate-/r/ N/A

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

      9. /-rgt-identity N/A

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

      10. *-commutative N/A

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

      11. *-commutative N/A

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

      12. associate-*l* N/A

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

      13. *-commutative N/A

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

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

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

      15. *-lowering-*.f64 94.6%

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

   9. Applied egg-rr 94.6%

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

4. Final simplification 74.7%

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

Alternative 8: 98.6% accurate, 1.0× speedup

Math

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

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 (/ (/ 1.0 y_m) (+ x_m (* z (* z x_m)))))))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 90.4%

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

   1. associate-/l/ N/A

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

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

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

   3. *-commutative N/A

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

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

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

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

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

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

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

   7. *-lowering-*.f64 90.1%

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

3. Simplified 90.1%

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

   1. associate-/r* N/A

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

   2. clear-num N/A

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

   3. associate-/l* N/A

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

   4. associate-/r* N/A

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

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

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

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

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

   7. div-inv N/A

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

   8. clear-num N/A

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

   9. /-rgt-identity N/A

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

   10. *-commutative N/A

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

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

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

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

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

   13. *-lowering-*.f64 91.6%

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

6. Applied egg-rr 91.6%

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

   1. +-commutative N/A

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

   2. distribute-rgt-in N/A

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

   3. *-lft-identity N/A

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

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

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

   5. associate-*l* N/A

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

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

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

   7. *-lowering-*.f64 94.3%

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

8. Applied egg-rr 94.3%

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

9. Final simplification 94.3%

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

Alternative 9: 58.7% accurate, 2.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 90.4%

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

   1. associate-/l/ N/A

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

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

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

   3. *-commutative N/A

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

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

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

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

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

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

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

   7. *-lowering-*.f64 90.1%

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

3. Simplified 90.1%

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

5. Taylor expanded in z around 0

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

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

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

   2. *-lowering-*.f64 57.6%

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

7. Simplified 57.6%

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

8. Final simplification 57.6%

   \[\leadsto \frac{1}{y \cdot x} \]
9. Add Preprocessing

Developer Target 1: 92.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + z \cdot z\\ t_1 := y \cdot t\_0\\ t_2 := \frac{\frac{1}{y}}{t\_0 \cdot x}\\ \mathbf{if}\;t\_1 < -\infty:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 < 8.680743250567252 \cdot 10^{+305}:\\ \;\;\;\;\frac{\frac{1}{x}}{t\_0 \cdot y}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ 1.0 (* z z))) (t_1 (* y t_0)) (t_2 (/ (/ 1.0 y) (* t_0 x))))
   (if (< t_1 (- INFINITY))
     t_2
     (if (< t_1 8.680743250567252e+305) (/ (/ 1.0 x) (* t_0 y)) t_2))))

C

double code(double x, double y, double z) {
	double t_0 = 1.0 + (z * z);
	double t_1 = y * t_0;
	double t_2 = (1.0 / y) / (t_0 * x);
	double tmp;
	if (t_1 < -((double) INFINITY)) {
		tmp = t_2;
	} else if (t_1 < 8.680743250567252e+305) {
		tmp = (1.0 / x) / (t_0 * y);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Java

public static double code(double x, double y, double z) {
	double t_0 = 1.0 + (z * z);
	double t_1 = y * t_0;
	double t_2 = (1.0 / y) / (t_0 * x);
	double tmp;
	if (t_1 < -Double.POSITIVE_INFINITY) {
		tmp = t_2;
	} else if (t_1 < 8.680743250567252e+305) {
		tmp = (1.0 / x) / (t_0 * y);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = 1.0 + (z * z)
	t_1 = y * t_0
	t_2 = (1.0 / y) / (t_0 * x)
	tmp = 0
	if t_1 < -math.inf:
		tmp = t_2
	elif t_1 < 8.680743250567252e+305:
		tmp = (1.0 / x) / (t_0 * y)
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(1.0 + Float64(z * z))
	t_1 = Float64(y * t_0)
	t_2 = Float64(Float64(1.0 / y) / Float64(t_0 * x))
	tmp = 0.0
	if (t_1 < Float64(-Inf))
		tmp = t_2;
	elseif (t_1 < 8.680743250567252e+305)
		tmp = Float64(Float64(1.0 / x) / Float64(t_0 * y));
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = 1.0 + (z * z);
	t_1 = y * t_0;
	t_2 = (1.0 / y) / (t_0 * x);
	tmp = 0.0;
	if (t_1 < -Inf)
		tmp = t_2;
	elseif (t_1 < 8.680743250567252e+305)
		tmp = (1.0 / x) / (t_0 * y);
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(1.0 + N[(z * z), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(y * t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(1.0 / y), $MachinePrecision] / N[(t$95$0 * x), $MachinePrecision]), $MachinePrecision]}, If[Less[t$95$1, (-Infinity)], t$95$2, If[Less[t$95$1, 8.680743250567252e+305], N[(N[(1.0 / x), $MachinePrecision] / N[(t$95$0 * y), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]

Reproduce

herbie shell --seed 2024129 
(FPCore (x y z)
  :name "Statistics.Distribution.CauchyLorentz:$cdensity from math-functions-0.1.5.2"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< (* y (+ 1 (* z z))) -inf.0) (/ (/ 1 y) (* (+ 1 (* z z)) x)) (if (< (* y (+ 1 (* z z))) 868074325056725200000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000) (/ (/ 1 x) (* (+ 1 (* z z)) y)) (/ (/ 1 y) (* (+ 1 (* z z)) x)))))

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