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

Percentage Accurate: 88.7% → 98.7%

Time: 12.1s

Alternatives: 8

Speedup: 0.5×

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: 88.7% 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: 98.7% accurate, 0.1× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;y\_m \cdot \left(1 + z \cdot z\right) \leq 5 \cdot 10^{+301}:\\ \;\;\;\;\frac{\frac{1}{y\_m}}{x\_m \cdot \mathsf{fma}\left(z, z, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{z}}{y\_m \cdot \left(z \cdot x\_m\right)}\\ \end{array}\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 1 x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 1 y)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (y_s x_s x_m y_m z)
 :precision binary64
 (*
  y_s
  (*
   x_s
   (if (<= (* y_m (+ 1.0 (* z z))) 5e+301)
     (/ (/ 1.0 y_m) (* x_m (fma z z 1.0)))
     (/ (/ 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 ((y_m * (1.0 + (z * z))) <= 5e+301) {
		tmp = (1.0 / y_m) / (x_m * fma(z, z, 1.0));
	} 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 (Float64(y_m * Float64(1.0 + Float64(z * z))) <= 5e+301)
		tmp = Float64(Float64(1.0 / y_m) / Float64(x_m * fma(z, z, 1.0)));
	else
		tmp = Float64(Float64(1.0 / z) / Float64(y_m * Float64(z * x_m)));
	end
	return Float64(y_s * Float64(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[(y$95$m * N[(1.0 + N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 5e+301], N[(N[(1.0 / y$95$m), $MachinePrecision] / N[(x$95$m * N[(z * z + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / z), $MachinePrecision] / N[(y$95$m * N[(z * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y (+.f64 1 (*.f64 z z))) < 5.0000000000000004e301

   1. Initial program 94.7%

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

      1. associate-/l/ 94.2%

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

      2. associate-*l* 94.2%

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

      3. *-commutative 94.2%

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

      4. sqr-neg 94.2%

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

      5. +-commutative 94.2%

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

      6. sqr-neg 94.2%

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

      7. fma-define 94.2%

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

   3. Simplified 94.2%

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

      1. associate-*r* 94.2%

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

      2. *-commutative 94.2%

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

      3. associate-/r* 93.8%

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

      4. *-commutative 93.8%

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

      5. associate-/l/ 94.3%

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

      6. fma-undefine 94.3%

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

      7. +-commutative 94.3%

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

      8. associate-/r* 94.7%

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

      9. add-cbrt-cube 61.2%

         \[\leadsto \color{blue}{\sqrt[3]{\left(\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)} \cdot \frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}\right) \cdot \frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}}} \]

      10. pow3 61.2%

          \[\leadsto \sqrt[3]{\color{blue}{{\left(\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}\right)}^{3}}} \]

      11. associate-/l/ 61.2%

          \[\leadsto \sqrt[3]{{\color{blue}{\left(\frac{1}{\left(y \cdot \left(1 + z \cdot z\right)\right) \cdot x}\right)}}^{3}} \]

      12. +-commutative 61.2%

          \[\leadsto \sqrt[3]{{\left(\frac{1}{\left(y \cdot \color{blue}{\left(z \cdot z + 1\right)}\right) \cdot x}\right)}^{3}} \]

      13. fma-undefine 61.2%

          \[\leadsto \sqrt[3]{{\left(\frac{1}{\left(y \cdot \color{blue}{\mathsf{fma}\left(z, z, 1\right)}\right) \cdot x}\right)}^{3}} \]

      14. *-commutative 61.2%

          \[\leadsto \sqrt[3]{{\left(\frac{1}{\color{blue}{\left(\mathsf{fma}\left(z, z, 1\right) \cdot y\right)} \cdot x}\right)}^{3}} \]

      15. associate-*r* 60.8%

          \[\leadsto \sqrt[3]{{\left(\frac{1}{\color{blue}{\mathsf{fma}\left(z, z, 1\right) \cdot \left(y \cdot x\right)}}\right)}^{3}} \]

      16. *-commutative 60.8%

          \[\leadsto \sqrt[3]{{\left(\frac{1}{\mathsf{fma}\left(z, z, 1\right) \cdot \color{blue}{\left(x \cdot y\right)}}\right)}^{3}} \]

      17. associate-/r* 60.8%

          \[\leadsto \sqrt[3]{{\color{blue}{\left(\frac{\frac{1}{\mathsf{fma}\left(z, z, 1\right)}}{x \cdot y}\right)}}^{3}} \]

   6. Applied egg-rr 60.8%

      \[\leadsto \color{blue}{\sqrt[3]{{\left(\frac{\frac{1}{\mathsf{fma}\left(z, z, 1\right)}}{x \cdot y}\right)}^{3}}} \]

   8. Applied egg-rr 94.6%

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

   if 5.0000000000000004e301 < (*.f64 y (+.f64 1 (*.f64 z z)))

   1. Initial program 69.6%

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

      1. associate-/l/ 69.6%

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

      2. associate-*l* 80.6%

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

      3. *-commutative 80.6%

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

      4. sqr-neg 80.6%

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

      5. +-commutative 80.6%

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

      6. sqr-neg 80.6%

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

      7. fma-define 80.6%

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

   3. Simplified 80.6%

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

      1. add-sqr-sqrt 47.8%

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

      2. pow2 47.8%

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

      3. *-commutative 47.8%

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

      4. sqrt-prod 47.9%

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

      5. fma-undefine 47.9%

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

      6. +-commutative 47.9%

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

      7. hypot-1-def 53.4%

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

   6. Applied egg-rr 53.4%

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

   7. Taylor expanded in z around inf 69.6%

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

      1. associate-*r* 80.2%

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

      2. *-commutative 80.2%

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

      3. *-commutative 80.2%

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

      4. associate-/r* 80.1%

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

      5. unpow-1 80.1%

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

      6. exp-to-pow 29.7%

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

      7. *-commutative 29.7%

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

      8. exp-prod 29.7%

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

      9. *-commutative 29.7%

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

      10. associate-*r* 29.7%

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

      11. metadata-eval 29.7%

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

      12. *-commutative 29.7%

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

      13. exp-to-pow 80.1%

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

      14. *-commutative 80.1%

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

   9. Simplified 80.1%

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

       1. sqr-pow 80.1%

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

       2. *-commutative 80.1%

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

       3. times-frac 99.8%

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

       4. metadata-eval 99.8%

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

       5. unpow-1 99.8%

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

       6. metadata-eval 99.8%

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

       7. unpow-1 99.8%

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

   11. Applied egg-rr 99.8%

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

       1. *-commutative 99.8%

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

       2. associate-/l/ 99.6%

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

       3. frac-times 99.8%

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

       4. *-un-lft-identity 99.8%

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

   13. Applied egg-rr 99.8%

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

4. Final simplification 95.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(1 + z \cdot z\right) \leq 5 \cdot 10^{+301}:\\ \;\;\;\;\frac{\frac{1}{y}}{x \cdot \mathsf{fma}\left(z, z, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{z}}{y \cdot \left(z \cdot x\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 97.9% accurate, 0.0× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ y\_s \cdot \left(x\_s \cdot \frac{1}{y\_m \cdot {\left(\mathsf{hypot}\left(1, z\right) \cdot \sqrt{x\_m}\right)}^{2}}\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 1 x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 1 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 (pow (* (hypot 1.0 z) (sqrt x_m)) 2.0))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
assert(x_m < y_m && y_m < z);
double code(double y_s, double x_s, double x_m, double y_m, double z) {
	return y_s * (x_s * (1.0 / (y_m * pow((hypot(1.0, z) * sqrt(x_m)), 2.0))));
}

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 * Math.pow((Math.hypot(1.0, z) * Math.sqrt(x_m)), 2.0))));
}

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 * math.pow((math.hypot(1.0, z) * math.sqrt(x_m)), 2.0))))

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 * (Float64(hypot(1.0, z) * sqrt(x_m)) ^ 2.0)))))
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 * ((hypot(1.0, z) * sqrt(x_m)) ^ 2.0))));
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 * N[Power[N[(N[Sqrt[1.0 ^ 2 + z ^ 2], $MachinePrecision] * N[Sqrt[x$95$m], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 91.3%

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

   1. associate-/l/ 90.9%

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

   2. associate-*l* 92.4%

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

   3. *-commutative 92.4%

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

   4. sqr-neg 92.4%

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

   5. +-commutative 92.4%

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

   6. sqr-neg 92.4%

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

   7. fma-define 92.4%

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

3. Simplified 92.4%

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

   1. add-sqr-sqrt 47.7%

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

   2. pow2 47.7%

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

   3. *-commutative 47.7%

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

   4. sqrt-prod 47.7%

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

   5. fma-undefine 47.7%

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

   6. +-commutative 47.7%

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

   7. hypot-1-def 49.2%

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

6. Applied egg-rr 49.2%

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

7. Final simplification 49.2%

   \[\leadsto \frac{1}{y \cdot {\left(\mathsf{hypot}\left(1, z\right) \cdot \sqrt{x}\right)}^{2}} \]
8. Add Preprocessing

Alternative 3: 98.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ \begin{array}{l} t_0 := y\_m \cdot \left(1 + z \cdot z\right)\\ y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq 5 \cdot 10^{+301}:\\ \;\;\;\;\frac{\frac{1}{x\_m}}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{z}}{y\_m \cdot \left(z \cdot x\_m\right)}\\ \end{array}\right) \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 1 x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 1 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+301)
       (/ (/ 1.0 x_m) t_0)
       (/ (/ 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 t_0 = y_m * (1.0 + (z * z));
	double tmp;
	if (t_0 <= 5e+301) {
		tmp = (1.0 / x_m) / t_0;
	} 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) :: t_0
    real(8) :: tmp
    t_0 = y_m * (1.0d0 + (z * z))
    if (t_0 <= 5d+301) then
        tmp = (1.0d0 / x_m) / t_0
    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 t_0 = y_m * (1.0 + (z * z));
	double tmp;
	if (t_0 <= 5e+301) {
		tmp = (1.0 / x_m) / t_0;
	} 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):
	t_0 = y_m * (1.0 + (z * z))
	tmp = 0
	if t_0 <= 5e+301:
		tmp = (1.0 / x_m) / t_0
	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)
	t_0 = Float64(y_m * Float64(1.0 + Float64(z * z)))
	tmp = 0.0
	if (t_0 <= 5e+301)
		tmp = Float64(Float64(1.0 / x_m) / t_0);
	else
		tmp = Float64(Float64(1.0 / 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)
	t_0 = y_m * (1.0 + (z * z));
	tmp = 0.0;
	if (t_0 <= 5e+301)
		tmp = (1.0 / x_m) / t_0;
	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_] := 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+301], N[(N[(1.0 / x$95$m), $MachinePrecision] / t$95$0), $MachinePrecision], N[(N[(1.0 / z), $MachinePrecision] / N[(y$95$m * N[(z * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y (+.f64 1 (*.f64 z z))) < 5.0000000000000004e301

   1. Initial program 94.7%

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

   if 5.0000000000000004e301 < (*.f64 y (+.f64 1 (*.f64 z z)))

   1. Initial program 69.6%

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

      1. associate-/l/ 69.6%

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

      2. associate-*l* 80.6%

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

      3. *-commutative 80.6%

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

      4. sqr-neg 80.6%

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

      5. +-commutative 80.6%

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

      6. sqr-neg 80.6%

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

      7. fma-define 80.6%

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

   3. Simplified 80.6%

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

      1. add-sqr-sqrt 47.8%

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

      2. pow2 47.8%

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

      3. *-commutative 47.8%

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

      4. sqrt-prod 47.9%

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

      5. fma-undefine 47.9%

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

      6. +-commutative 47.9%

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

      7. hypot-1-def 53.4%

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

   6. Applied egg-rr 53.4%

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

   7. Taylor expanded in z around inf 69.6%

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

      1. associate-*r* 80.2%

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

      2. *-commutative 80.2%

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

      3. *-commutative 80.2%

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

      4. associate-/r* 80.1%

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

      5. unpow-1 80.1%

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

      6. exp-to-pow 29.7%

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

      7. *-commutative 29.7%

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

      8. exp-prod 29.7%

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

      9. *-commutative 29.7%

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

      10. associate-*r* 29.7%

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

      11. metadata-eval 29.7%

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

      12. *-commutative 29.7%

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

      13. exp-to-pow 80.1%

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

      14. *-commutative 80.1%

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

   9. Simplified 80.1%

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

       1. sqr-pow 80.1%

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

       2. *-commutative 80.1%

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

       3. times-frac 99.8%

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

       4. metadata-eval 99.8%

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

       5. unpow-1 99.8%

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

       6. metadata-eval 99.8%

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

       7. unpow-1 99.8%

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

   11. Applied egg-rr 99.8%

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

       1. *-commutative 99.8%

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

       2. associate-/l/ 99.6%

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

       3. frac-times 99.8%

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

       4. *-un-lft-identity 99.8%

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

   13. Applied egg-rr 99.8%

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

4. Final simplification 95.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(1 + z \cdot z\right) \leq 5 \cdot 10^{+301}:\\ \;\;\;\;\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{z}}{y \cdot \left(z \cdot x\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 78.1% accurate, 0.7× speedup

Math

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

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 1 x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 1 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 x_m) y_m)
     (/ (* (/ 1.0 z) (/ 1.0 (* z x_m))) y_m)))))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if z < 1

   1. Initial program 94.7%

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

   3. Taylor expanded in z around 0 74.4%

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

   if 1 < z

   1. Initial program 78.2%

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

      1. associate-/l/ 78.2%

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

      2. associate-*l* 85.4%

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

      3. *-commutative 85.4%

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

      4. sqr-neg 85.4%

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

      5. +-commutative 85.4%

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

      6. sqr-neg 85.4%

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

      7. fma-define 85.4%

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

   3. Simplified 85.4%

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

      1. add-sqr-sqrt 46.8%

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

      2. pow2 46.8%

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

      3. *-commutative 46.8%

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

      4. sqrt-prod 46.8%

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

      5. fma-undefine 46.8%

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

      6. +-commutative 46.8%

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

      7. hypot-1-def 52.2%

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

   6. Applied egg-rr 52.2%

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

   7. Taylor expanded in z around inf 78.2%

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

      1. associate-*r* 85.0%

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

      2. *-commutative 85.0%

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

      3. *-commutative 85.0%

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

      4. associate-/r* 85.1%

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

      5. unpow-1 85.1%

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

      6. exp-to-pow 82.6%

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

      7. *-commutative 82.6%

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

      8. exp-prod 82.6%

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

      9. *-commutative 82.6%

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

      10. associate-*r* 82.6%

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

      11. metadata-eval 82.6%

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

      12. *-commutative 82.6%

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

      13. exp-to-pow 85.1%

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

      14. *-commutative 85.1%

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

   9. Simplified 85.1%

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

       1. sqr-pow 85.0%

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

       2. *-commutative 85.0%

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

       3. times-frac 94.7%

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

       4. metadata-eval 94.7%

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

       5. unpow-1 94.7%

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

       6. metadata-eval 94.7%

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

       7. unpow-1 94.7%

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

   11. Applied egg-rr 94.7%

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

       1. *-commutative 94.7%

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

       2. associate-*r/ 92.6%

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

       3. associate-/l/ 92.7%

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

   13. Applied egg-rr 92.7%

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

4. Final simplification 78.1%

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

Alternative 5: 77.4% accurate, 0.8× speedup

Math

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

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 1 x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 1 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 x_m) y_m) (/ 1.0 (* (* z x_m) (* y_m z)))))))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if z < 1

   1. Initial program 94.7%

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

   3. Taylor expanded in z around 0 74.4%

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

   if 1 < z

   1. Initial program 78.2%

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

      1. associate-/l/ 78.2%

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

      2. associate-*l* 85.4%

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

      3. *-commutative 85.4%

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

      4. sqr-neg 85.4%

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

      5. +-commutative 85.4%

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

      6. sqr-neg 85.4%

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

      7. fma-define 85.4%

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

   3. Simplified 85.4%

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

      1. add-sqr-sqrt 46.8%

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

      2. pow2 46.8%

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

      3. *-commutative 46.8%

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

      4. sqrt-prod 46.8%

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

      5. fma-undefine 46.8%

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

      6. +-commutative 46.8%

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

      7. hypot-1-def 52.2%

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

   6. Applied egg-rr 52.2%

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

   7. Taylor expanded in z around inf 78.2%

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

      1. associate-*r* 85.0%

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

      2. *-commutative 85.0%

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

      3. *-commutative 85.0%

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

      4. associate-/r* 85.1%

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

      5. unpow-1 85.1%

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

      6. exp-to-pow 82.6%

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

      7. *-commutative 82.6%

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

      8. exp-prod 82.6%

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

      9. *-commutative 82.6%

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

      10. associate-*r* 82.6%

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

      11. metadata-eval 82.6%

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

      12. *-commutative 82.6%

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

      13. exp-to-pow 85.1%

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

      14. *-commutative 85.1%

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

   9. Simplified 85.1%

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

       1. sqr-pow 85.0%

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

       2. *-commutative 85.0%

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

       3. times-frac 94.7%

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

       4. metadata-eval 94.7%

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

       5. unpow-1 94.7%

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

       6. metadata-eval 94.7%

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

       7. unpow-1 94.7%

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

   11. Applied egg-rr 94.7%

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

       1. associate-/l/ 94.3%

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

       2. associate-/l/ 94.3%

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

       3. frac-times 94.4%

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

       4. metadata-eval 94.4%

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

   13. Applied egg-rr 94.4%

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

4. Final simplification 78.5%

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

Alternative 6: 78.2% accurate, 0.8× speedup

Math

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

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 1 x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 1 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 x_m) y_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 / x_m) / y_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 / x_m) / y_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 / x_m) / y_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 / x_m) / y_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 / x_m) / y_m);
	else
		tmp = Float64(Float64(1.0 / 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 / x_m) / y_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 / x$95$m), $MachinePrecision] / y$95$m), $MachinePrecision], N[(N[(1.0 / z), $MachinePrecision] / N[(y$95$m * N[(z * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 1

   1. Initial program 94.7%

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

   3. Taylor expanded in z around 0 74.4%

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

   if 1 < z

   1. Initial program 78.2%

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

      1. associate-/l/ 78.2%

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

      2. associate-*l* 85.4%

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

      3. *-commutative 85.4%

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

      4. sqr-neg 85.4%

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

      5. +-commutative 85.4%

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

      6. sqr-neg 85.4%

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

      7. fma-define 85.4%

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

   3. Simplified 85.4%

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

      1. add-sqr-sqrt 46.8%

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

      2. pow2 46.8%

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

      3. *-commutative 46.8%

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

      4. sqrt-prod 46.8%

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

      5. fma-undefine 46.8%

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

      6. +-commutative 46.8%

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

      7. hypot-1-def 52.2%

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

   6. Applied egg-rr 52.2%

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

   7. Taylor expanded in z around inf 78.2%

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

      1. associate-*r* 85.0%

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

      2. *-commutative 85.0%

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

      3. *-commutative 85.0%

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

      4. associate-/r* 85.1%

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

      5. unpow-1 85.1%

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

      6. exp-to-pow 82.6%

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

      7. *-commutative 82.6%

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

      8. exp-prod 82.6%

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

      9. *-commutative 82.6%

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

      10. associate-*r* 82.6%

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

      11. metadata-eval 82.6%

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

      12. *-commutative 82.6%

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

      13. exp-to-pow 85.1%

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

      14. *-commutative 85.1%

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

   9. Simplified 85.1%

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

       1. sqr-pow 85.0%

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

       2. *-commutative 85.0%

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

       3. times-frac 94.7%

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

       4. metadata-eval 94.7%

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

       5. unpow-1 94.7%

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

       6. metadata-eval 94.7%

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

       7. unpow-1 94.7%

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

   11. Applied egg-rr 94.7%

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

       1. *-commutative 94.7%

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

       2. associate-/l/ 94.3%

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

       3. frac-times 97.9%

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

       4. *-un-lft-identity 97.9%

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

   13. Applied egg-rr 97.9%

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

4. Final simplification 79.2%

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

Alternative 7: 58.4% 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 1 x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 1 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 91.3%

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

   1. associate-/l/ 90.9%

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

   2. associate-*l* 92.4%

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

   3. *-commutative 92.4%

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

   4. sqr-neg 92.4%

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

   5. +-commutative 92.4%

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

   6. sqr-neg 92.4%

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

   7. fma-define 92.4%

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

3. Simplified 92.4%

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

5. Taylor expanded in z around 0 63.1%

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

6. Final simplification 63.1%

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

Alternative 8: 58.4% accurate, 2.2× speedup

Math

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

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 1 x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 1 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 x_m) y_m))))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 91.3%

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

3. Taylor expanded in z around 0 63.2%

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

4. Final simplification 63.2%

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

Developer target: 93.1% 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 2024054 
(FPCore (x y z)
  :name "Statistics.Distribution.CauchyLorentz:$cdensity from math-functions-0.1.5.2"
  :precision binary64

  :alt
  (if (< (* y (+ 1.0 (* z z))) (- INFINITY)) (/ (/ 1.0 y) (* (+ 1.0 (* z z)) x)) (if (< (* y (+ 1.0 (* z z))) 8.680743250567252e+305) (/ (/ 1.0 x) (* (+ 1.0 (* z z)) y)) (/ (/ 1.0 y) (* (+ 1.0 (* z z)) x))))

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