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

Percentage Accurate: 88.0% → 99.2%

Time: 11.6s

Alternatives: 9

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: 88.0% 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: 99.2% accurate, 0.1× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ y_s \cdot \left(x_s \cdot \begin{array}{l} \mathbf{if}\;y_m \cdot \left(1 + z \cdot z\right) \leq 2 \cdot 10^{+307}:\\ \;\;\;\;\frac{\frac{1}{x_m}}{\mathsf{fma}\left(y_m \cdot z, z, y_m\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{z} \cdot \frac{\frac{\frac{1}{z}}{x_m}}{y_m}\\ \end{array}\right) \end{array} \]

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 (<= (* y_m (+ 1.0 (* z z))) 2e+307)
     (/ (/ 1.0 x_m) (fma (* y_m z) z 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 ((y_m * (1.0 + (z * z))) <= 2e+307) {
		tmp = (1.0 / x_m) / fma((y_m * z), z, 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 (Float64(y_m * Float64(1.0 + Float64(z * z))) <= 2e+307)
		tmp = Float64(Float64(1.0 / x_m) / fma(Float64(y_m * z), z, y_m));
	else
		tmp = Float64(Float64(1.0 / z) * Float64(Float64(Float64(1.0 / z) / x_m) / y_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], 2e+307], N[(N[(1.0 / x$95$m), $MachinePrecision] / N[(N[(y$95$m * z), $MachinePrecision] * z + y$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / z), $MachinePrecision] * N[(N[(N[(1.0 / z), $MachinePrecision] / x$95$m), $MachinePrecision] / y$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y (+.f64 1 (*.f64 z z))) < 1.99999999999999997e307

   1. Initial program 95.8%

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

      1. +-commutative 95.8%

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

      2. distribute-lft-in 95.8%

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

      3. associate-*r* 98.4%

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

      4. *-rgt-identity 98.4%

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

      5. fma-def 98.4%

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

   4. Applied egg-rr 98.4%

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

   if 1.99999999999999997e307 < (*.f64 y (+.f64 1 (*.f64 z z)))

   1. Initial program 60.5%

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

      1. associate-/l/ 60.6%

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

      2. metadata-eval 60.6%

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

      3. associate-*r/ 60.6%

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

      4. associate-/l/ 60.5%

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

      5. associate-*r/ 60.5%

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

      6. associate-/l* 60.5%

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

      7. associate-/r/ 60.6%

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

      8. /-rgt-identity 60.6%

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

      9. associate-*l* 72.4%

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

      10. *-commutative 72.4%

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

      11. sqr-neg 72.4%

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

      12. +-commutative 72.4%

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

      13. sqr-neg 72.4%

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

      14. fma-def 72.4%

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

   3. Simplified 72.4%

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

      1. fma-udef 72.4%

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

      2. +-commutative 72.4%

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

      3. *-commutative 72.4%

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

      4. associate-*l* 60.6%

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

      5. associate-/l/ 60.5%

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

      6. add-sqr-sqrt 60.5%

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

      7. sqrt-div 16.5%

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

      8. inv-pow 16.5%

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

      9. sqrt-pow1 16.5%

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

      10. metadata-eval 16.5%

          \[\leadsto \frac{{x}^{\color{blue}{-0.5}}}{\sqrt{y \cdot \left(1 + z \cdot z\right)}} \cdot \sqrt{\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}} \]

      11. *-commutative 16.5%

          \[\leadsto \frac{{x}^{-0.5}}{\sqrt{\color{blue}{\left(1 + z \cdot z\right) \cdot y}}} \cdot \sqrt{\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}} \]

      12. sqrt-prod 16.5%

          \[\leadsto \frac{{x}^{-0.5}}{\color{blue}{\sqrt{1 + z \cdot z} \cdot \sqrt{y}}} \cdot \sqrt{\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}} \]

      13. hypot-1-def 16.5%

          \[\leadsto \frac{{x}^{-0.5}}{\color{blue}{\mathsf{hypot}\left(1, z\right)} \cdot \sqrt{y}} \cdot \sqrt{\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}} \]

      14. sqrt-div 16.5%

          \[\leadsto \frac{{x}^{-0.5}}{\mathsf{hypot}\left(1, z\right) \cdot \sqrt{y}} \cdot \color{blue}{\frac{\sqrt{\frac{1}{x}}}{\sqrt{y \cdot \left(1 + z \cdot z\right)}}} \]

      15. inv-pow 16.5%

          \[\leadsto \frac{{x}^{-0.5}}{\mathsf{hypot}\left(1, z\right) \cdot \sqrt{y}} \cdot \frac{\sqrt{\color{blue}{{x}^{-1}}}}{\sqrt{y \cdot \left(1 + z \cdot z\right)}} \]

      16. sqrt-pow1 16.5%

          \[\leadsto \frac{{x}^{-0.5}}{\mathsf{hypot}\left(1, z\right) \cdot \sqrt{y}} \cdot \frac{\color{blue}{{x}^{\left(\frac{-1}{2}\right)}}}{\sqrt{y \cdot \left(1 + z \cdot z\right)}} \]

      17. metadata-eval 16.5%

          \[\leadsto \frac{{x}^{-0.5}}{\mathsf{hypot}\left(1, z\right) \cdot \sqrt{y}} \cdot \frac{{x}^{\color{blue}{-0.5}}}{\sqrt{y \cdot \left(1 + z \cdot z\right)}} \]

      18. *-commutative 16.5%

          \[\leadsto \frac{{x}^{-0.5}}{\mathsf{hypot}\left(1, z\right) \cdot \sqrt{y}} \cdot \frac{{x}^{-0.5}}{\sqrt{\color{blue}{\left(1 + z \cdot z\right) \cdot y}}} \]

   6. Applied egg-rr 38.1%

      \[\leadsto \color{blue}{\frac{{x}^{-0.5}}{\mathsf{hypot}\left(1, z\right) \cdot \sqrt{y}} \cdot \frac{{x}^{-0.5}}{\mathsf{hypot}\left(1, z\right) \cdot \sqrt{y}}} \]
   7. Step-by-step derivation

      1. unpow2 38.1%

         \[\leadsto \color{blue}{{\left(\frac{{x}^{-0.5}}{\mathsf{hypot}\left(1, z\right) \cdot \sqrt{y}}\right)}^{2}} \]

      2. *-commutative 38.1%

         \[\leadsto {\left(\frac{{x}^{-0.5}}{\color{blue}{\sqrt{y} \cdot \mathsf{hypot}\left(1, z\right)}}\right)}^{2} \]

      3. associate-/r* 38.2%

         \[\leadsto {\color{blue}{\left(\frac{\frac{{x}^{-0.5}}{\sqrt{y}}}{\mathsf{hypot}\left(1, z\right)}\right)}}^{2} \]

   8. Simplified 38.2%

      \[\leadsto \color{blue}{{\left(\frac{\frac{{x}^{-0.5}}{\sqrt{y}}}{\mathsf{hypot}\left(1, z\right)}\right)}^{2}} \]

   9. Taylor expanded in z around inf 60.6%

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

       1. associate-*r* 72.0%

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

       2. associate-/r* 72.1%

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

       3. associate-/l/ 72.1%

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

       4. unpow2 72.1%

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

       5. associate-/r* 87.5%

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

       6. associate-/l/ 87.6%

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

       7. unpow-1 87.6%

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

       8. metadata-eval 87.6%

          \[\leadsto \frac{\frac{{\left(x \cdot y\right)}^{\color{blue}{\left(2 \cdot -0.5\right)}}}{z}}{z} \]

       9. pow-sqr 46.2%

          \[\leadsto \frac{\frac{\color{blue}{{\left(x \cdot y\right)}^{-0.5} \cdot {\left(x \cdot y\right)}^{-0.5}}}{z}}{z} \]

       10. associate-*r/ 46.2%

           \[\leadsto \frac{\color{blue}{{\left(x \cdot y\right)}^{-0.5} \cdot \frac{{\left(x \cdot y\right)}^{-0.5}}{z}}}{z} \]

       11. associate-*l/ 46.3%

           \[\leadsto \color{blue}{\frac{{\left(x \cdot y\right)}^{-0.5}}{z} \cdot \frac{{\left(x \cdot y\right)}^{-0.5}}{z}} \]

       12. *-rgt-identity 46.3%

           \[\leadsto \frac{\color{blue}{{\left(x \cdot y\right)}^{-0.5} \cdot 1}}{z} \cdot \frac{{\left(x \cdot y\right)}^{-0.5}}{z} \]

       13. associate-*r/ 46.2%

           \[\leadsto \color{blue}{\left({\left(x \cdot y\right)}^{-0.5} \cdot \frac{1}{z}\right)} \cdot \frac{{\left(x \cdot y\right)}^{-0.5}}{z} \]

       14. *-rgt-identity 46.2%

           \[\leadsto \left({\left(x \cdot y\right)}^{-0.5} \cdot \frac{1}{z}\right) \cdot \frac{\color{blue}{{\left(x \cdot y\right)}^{-0.5} \cdot 1}}{z} \]

       15. associate-*r/ 46.1%

           \[\leadsto \left({\left(x \cdot y\right)}^{-0.5} \cdot \frac{1}{z}\right) \cdot \color{blue}{\left({\left(x \cdot y\right)}^{-0.5} \cdot \frac{1}{z}\right)} \]

       16. unswap-sqr 38.9%

           \[\leadsto \color{blue}{\left({\left(x \cdot y\right)}^{-0.5} \cdot {\left(x \cdot y\right)}^{-0.5}\right) \cdot \left(\frac{1}{z} \cdot \frac{1}{z}\right)} \]

       17. pow-sqr 73.5%

           \[\leadsto \color{blue}{{\left(x \cdot y\right)}^{\left(2 \cdot -0.5\right)}} \cdot \left(\frac{1}{z} \cdot \frac{1}{z}\right) \]

       18. metadata-eval 73.5%

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

       19. unpow-1 73.5%

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

       20. unpow-1 73.5%

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

       21. unpow-1 73.5%

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

       22. pow-sqr 73.5%

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

       23. metadata-eval 73.5%

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

   11. Simplified 73.5%

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

       1. div-inv 73.5%

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

       2. add-sqr-sqrt 73.5%

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

       3. associate-/r* 73.5%

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

       4. associate-*l* 73.4%

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

       5. sqrt-pow1 69.8%

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

       6. metadata-eval 69.8%

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

       7. unpow-1 69.8%

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

       8. sqrt-pow1 87.4%

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

       9. metadata-eval 87.4%

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

       10. unpow-1 87.4%

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

       11. associate-/l/ 87.5%

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

   13. Applied egg-rr 87.5%

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

       1. un-div-inv 87.4%

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

       2. *-commutative 87.4%

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

       3. associate-/r* 94.8%

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

   15. Applied egg-rr 94.8%

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

4. Final simplification 97.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(1 + z \cdot z\right) \leq 2 \cdot 10^{+307}:\\ \;\;\;\;\frac{\frac{1}{x}}{\mathsf{fma}\left(y \cdot z, z, y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{z} \cdot \frac{\frac{\frac{1}{z}}{x}}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.2% accurate, 0.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ y_s \cdot \left(x_s \cdot {\left(\frac{\frac{{x_m}^{-0.5}}{\sqrt{y_m}}}{\mathsf{hypot}\left(1, z\right)}\right)}^{2}\right) \end{array} \]

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 (pow (/ (/ (pow x_m -0.5) (sqrt y_m)) (hypot 1.0 z)) 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 * pow(((pow(x_m, -0.5) / sqrt(y_m)) / hypot(1.0, z)), 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 * Math.pow(((Math.pow(x_m, -0.5) / Math.sqrt(y_m)) / Math.hypot(1.0, z)), 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 * math.pow(((math.pow(x_m, -0.5) / math.sqrt(y_m)) / math.hypot(1.0, z)), 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(Float64((x_m ^ -0.5) / sqrt(y_m)) / hypot(1.0, z)) ^ 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 * ((((x_m ^ -0.5) / sqrt(y_m)) / hypot(1.0, z)) ^ 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[Power[N[(N[(N[Power[x$95$m, -0.5], $MachinePrecision] / N[Sqrt[y$95$m], $MachinePrecision]), $MachinePrecision] / N[Sqrt[1.0 ^ 2 + z ^ 2], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 90.5%

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

   1. associate-/l/ 90.2%

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

   2. metadata-eval 90.2%

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

   3. associate-*r/ 90.2%

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

   4. associate-/l/ 90.5%

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

   5. associate-*r/ 90.5%

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

   6. associate-/l* 90.1%

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

   7. associate-/r/ 90.2%

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

   8. /-rgt-identity 90.2%

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

   9. associate-*l* 90.1%

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

   10. *-commutative 90.1%

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

   11. sqr-neg 90.1%

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

   12. +-commutative 90.1%

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

   13. sqr-neg 90.1%

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

   14. fma-def 90.1%

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

3. Simplified 90.1%

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

   1. fma-udef 90.1%

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

   2. +-commutative 90.1%

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

   3. *-commutative 90.1%

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

   4. associate-*l* 90.2%

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

   5. associate-/l/ 90.5%

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

   6. add-sqr-sqrt 60.8%

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

   7. sqrt-div 18.8%

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

   8. inv-pow 18.8%

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

   9. sqrt-pow1 18.8%

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

   10. metadata-eval 18.8%

       \[\leadsto \frac{{x}^{\color{blue}{-0.5}}}{\sqrt{y \cdot \left(1 + z \cdot z\right)}} \cdot \sqrt{\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}} \]

   11. *-commutative 18.8%

       \[\leadsto \frac{{x}^{-0.5}}{\sqrt{\color{blue}{\left(1 + z \cdot z\right) \cdot y}}} \cdot \sqrt{\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}} \]

   12. sqrt-prod 18.8%

       \[\leadsto \frac{{x}^{-0.5}}{\color{blue}{\sqrt{1 + z \cdot z} \cdot \sqrt{y}}} \cdot \sqrt{\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}} \]

   13. hypot-1-def 18.8%

       \[\leadsto \frac{{x}^{-0.5}}{\color{blue}{\mathsf{hypot}\left(1, z\right)} \cdot \sqrt{y}} \cdot \sqrt{\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}} \]

   14. sqrt-div 18.8%

       \[\leadsto \frac{{x}^{-0.5}}{\mathsf{hypot}\left(1, z\right) \cdot \sqrt{y}} \cdot \color{blue}{\frac{\sqrt{\frac{1}{x}}}{\sqrt{y \cdot \left(1 + z \cdot z\right)}}} \]

   15. inv-pow 18.8%

       \[\leadsto \frac{{x}^{-0.5}}{\mathsf{hypot}\left(1, z\right) \cdot \sqrt{y}} \cdot \frac{\sqrt{\color{blue}{{x}^{-1}}}}{\sqrt{y \cdot \left(1 + z \cdot z\right)}} \]

   16. sqrt-pow1 18.8%

       \[\leadsto \frac{{x}^{-0.5}}{\mathsf{hypot}\left(1, z\right) \cdot \sqrt{y}} \cdot \frac{\color{blue}{{x}^{\left(\frac{-1}{2}\right)}}}{\sqrt{y \cdot \left(1 + z \cdot z\right)}} \]

   17. metadata-eval 18.8%

       \[\leadsto \frac{{x}^{-0.5}}{\mathsf{hypot}\left(1, z\right) \cdot \sqrt{y}} \cdot \frac{{x}^{\color{blue}{-0.5}}}{\sqrt{y \cdot \left(1 + z \cdot z\right)}} \]

   18. *-commutative 18.8%

       \[\leadsto \frac{{x}^{-0.5}}{\mathsf{hypot}\left(1, z\right) \cdot \sqrt{y}} \cdot \frac{{x}^{-0.5}}{\sqrt{\color{blue}{\left(1 + z \cdot z\right) \cdot y}}} \]

6. Applied egg-rr 22.1%

   \[\leadsto \color{blue}{\frac{{x}^{-0.5}}{\mathsf{hypot}\left(1, z\right) \cdot \sqrt{y}} \cdot \frac{{x}^{-0.5}}{\mathsf{hypot}\left(1, z\right) \cdot \sqrt{y}}} \]
7. Step-by-step derivation

   1. unpow2 22.1%

      \[\leadsto \color{blue}{{\left(\frac{{x}^{-0.5}}{\mathsf{hypot}\left(1, z\right) \cdot \sqrt{y}}\right)}^{2}} \]

   2. *-commutative 22.1%

      \[\leadsto {\left(\frac{{x}^{-0.5}}{\color{blue}{\sqrt{y} \cdot \mathsf{hypot}\left(1, z\right)}}\right)}^{2} \]

   3. associate-/r* 22.1%

      \[\leadsto {\color{blue}{\left(\frac{\frac{{x}^{-0.5}}{\sqrt{y}}}{\mathsf{hypot}\left(1, z\right)}\right)}}^{2} \]

8. Simplified 22.1%

   \[\leadsto \color{blue}{{\left(\frac{\frac{{x}^{-0.5}}{\sqrt{y}}}{\mathsf{hypot}\left(1, z\right)}\right)}^{2}} \]

9. Final simplification 22.1%

   \[\leadsto {\left(\frac{\frac{{x}^{-0.5}}{\sqrt{y}}}{\mathsf{hypot}\left(1, z\right)}\right)}^{2} \]
10. Add Preprocessing

Alternative 3: 97.6% 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 90.5%

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

   1. associate-/l/ 90.2%

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

   2. metadata-eval 90.2%

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

   3. associate-*r/ 90.2%

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

   4. associate-/l/ 90.5%

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

   5. associate-*r/ 90.5%

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

   6. associate-/l* 90.1%

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

   7. associate-/r/ 90.2%

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

   8. /-rgt-identity 90.2%

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

   9. associate-*l* 90.1%

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

   10. *-commutative 90.1%

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

   11. sqr-neg 90.1%

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

   12. +-commutative 90.1%

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

   13. sqr-neg 90.1%

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

   14. fma-def 90.1%

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

3. Simplified 90.1%

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

   1. add-sqr-sqrt 42.0%

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

   2. pow2 42.0%

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

   3. fma-udef 42.0%

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

   4. +-commutative 42.0%

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

   5. *-commutative 42.0%

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

   6. sqrt-prod 42.0%

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

   7. hypot-1-def 43.5%

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

6. Applied egg-rr 43.5%

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

7. Final simplification 43.5%

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

Alternative 4: 98.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ \begin{array}{l} t_0 := y_m \cdot \left(1 + z \cdot z\right)\\ y_s \cdot \left(x_s \cdot \begin{array}{l} \mathbf{if}\;t_0 \leq 2 \cdot 10^{+307}:\\ \;\;\;\;\frac{1}{x_m \cdot t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{z} \cdot \frac{\frac{\frac{1}{z}}{x_m}}{y_m}\\ \end{array}\right) \end{array} \end{array} \]

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 2e+307)
       (/ 1.0 (* x_m t_0))
       (* (/ 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 t_0 = y_m * (1.0 + (z * z));
	double tmp;
	if (t_0 <= 2e+307) {
		tmp = 1.0 / (x_m * t_0);
	} 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) :: t_0
    real(8) :: tmp
    t_0 = y_m * (1.0d0 + (z * z))
    if (t_0 <= 2d+307) then
        tmp = 1.0d0 / (x_m * t_0)
    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 t_0 = y_m * (1.0 + (z * z));
	double tmp;
	if (t_0 <= 2e+307) {
		tmp = 1.0 / (x_m * t_0);
	} 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):
	t_0 = y_m * (1.0 + (z * z))
	tmp = 0
	if t_0 <= 2e+307:
		tmp = 1.0 / (x_m * t_0)
	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)
	t_0 = Float64(y_m * Float64(1.0 + Float64(z * z)))
	tmp = 0.0
	if (t_0 <= 2e+307)
		tmp = Float64(1.0 / Float64(x_m * t_0));
	else
		tmp = Float64(Float64(1.0 / z) * Float64(Float64(Float64(1.0 / 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)
	t_0 = y_m * (1.0 + (z * z));
	tmp = 0.0;
	if (t_0 <= 2e+307)
		tmp = 1.0 / (x_m * t_0);
	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_] := 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, 2e+307], N[(1.0 / N[(x$95$m * t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / z), $MachinePrecision] * N[(N[(N[(1.0 / z), $MachinePrecision] / x$95$m), $MachinePrecision] / y$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y (+.f64 1 (*.f64 z z))) < 1.99999999999999997e307

   1. Initial program 95.8%

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

      1. associate-/l/ 95.5%

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

   3. Simplified 95.5%

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

   if 1.99999999999999997e307 < (*.f64 y (+.f64 1 (*.f64 z z)))

   1. Initial program 60.5%

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

      1. associate-/l/ 60.6%

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

      2. metadata-eval 60.6%

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

      3. associate-*r/ 60.6%

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

      4. associate-/l/ 60.5%

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

      5. associate-*r/ 60.5%

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

      6. associate-/l* 60.5%

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

      7. associate-/r/ 60.6%

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

      8. /-rgt-identity 60.6%

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

      9. associate-*l* 72.4%

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

      10. *-commutative 72.4%

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

      11. sqr-neg 72.4%

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

      12. +-commutative 72.4%

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

      13. sqr-neg 72.4%

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

      14. fma-def 72.4%

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

   3. Simplified 72.4%

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

      1. fma-udef 72.4%

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

      2. +-commutative 72.4%

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

      3. *-commutative 72.4%

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

      4. associate-*l* 60.6%

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

      5. associate-/l/ 60.5%

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

      6. add-sqr-sqrt 60.5%

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

      7. sqrt-div 16.5%

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

      8. inv-pow 16.5%

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

      9. sqrt-pow1 16.5%

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

      10. metadata-eval 16.5%

          \[\leadsto \frac{{x}^{\color{blue}{-0.5}}}{\sqrt{y \cdot \left(1 + z \cdot z\right)}} \cdot \sqrt{\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}} \]

      11. *-commutative 16.5%

          \[\leadsto \frac{{x}^{-0.5}}{\sqrt{\color{blue}{\left(1 + z \cdot z\right) \cdot y}}} \cdot \sqrt{\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}} \]

      12. sqrt-prod 16.5%

          \[\leadsto \frac{{x}^{-0.5}}{\color{blue}{\sqrt{1 + z \cdot z} \cdot \sqrt{y}}} \cdot \sqrt{\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}} \]

      13. hypot-1-def 16.5%

          \[\leadsto \frac{{x}^{-0.5}}{\color{blue}{\mathsf{hypot}\left(1, z\right)} \cdot \sqrt{y}} \cdot \sqrt{\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}} \]

      14. sqrt-div 16.5%

          \[\leadsto \frac{{x}^{-0.5}}{\mathsf{hypot}\left(1, z\right) \cdot \sqrt{y}} \cdot \color{blue}{\frac{\sqrt{\frac{1}{x}}}{\sqrt{y \cdot \left(1 + z \cdot z\right)}}} \]

      15. inv-pow 16.5%

          \[\leadsto \frac{{x}^{-0.5}}{\mathsf{hypot}\left(1, z\right) \cdot \sqrt{y}} \cdot \frac{\sqrt{\color{blue}{{x}^{-1}}}}{\sqrt{y \cdot \left(1 + z \cdot z\right)}} \]

      16. sqrt-pow1 16.5%

          \[\leadsto \frac{{x}^{-0.5}}{\mathsf{hypot}\left(1, z\right) \cdot \sqrt{y}} \cdot \frac{\color{blue}{{x}^{\left(\frac{-1}{2}\right)}}}{\sqrt{y \cdot \left(1 + z \cdot z\right)}} \]

      17. metadata-eval 16.5%

          \[\leadsto \frac{{x}^{-0.5}}{\mathsf{hypot}\left(1, z\right) \cdot \sqrt{y}} \cdot \frac{{x}^{\color{blue}{-0.5}}}{\sqrt{y \cdot \left(1 + z \cdot z\right)}} \]

      18. *-commutative 16.5%

          \[\leadsto \frac{{x}^{-0.5}}{\mathsf{hypot}\left(1, z\right) \cdot \sqrt{y}} \cdot \frac{{x}^{-0.5}}{\sqrt{\color{blue}{\left(1 + z \cdot z\right) \cdot y}}} \]

   6. Applied egg-rr 38.1%

      \[\leadsto \color{blue}{\frac{{x}^{-0.5}}{\mathsf{hypot}\left(1, z\right) \cdot \sqrt{y}} \cdot \frac{{x}^{-0.5}}{\mathsf{hypot}\left(1, z\right) \cdot \sqrt{y}}} \]
   7. Step-by-step derivation

      1. unpow2 38.1%

         \[\leadsto \color{blue}{{\left(\frac{{x}^{-0.5}}{\mathsf{hypot}\left(1, z\right) \cdot \sqrt{y}}\right)}^{2}} \]

      2. *-commutative 38.1%

         \[\leadsto {\left(\frac{{x}^{-0.5}}{\color{blue}{\sqrt{y} \cdot \mathsf{hypot}\left(1, z\right)}}\right)}^{2} \]

      3. associate-/r* 38.2%

         \[\leadsto {\color{blue}{\left(\frac{\frac{{x}^{-0.5}}{\sqrt{y}}}{\mathsf{hypot}\left(1, z\right)}\right)}}^{2} \]

   8. Simplified 38.2%

      \[\leadsto \color{blue}{{\left(\frac{\frac{{x}^{-0.5}}{\sqrt{y}}}{\mathsf{hypot}\left(1, z\right)}\right)}^{2}} \]

   9. Taylor expanded in z around inf 60.6%

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

       1. associate-*r* 72.0%

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

       2. associate-/r* 72.1%

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

       3. associate-/l/ 72.1%

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

       4. unpow2 72.1%

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

       5. associate-/r* 87.5%

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

       6. associate-/l/ 87.6%

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

       7. unpow-1 87.6%

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

       8. metadata-eval 87.6%

          \[\leadsto \frac{\frac{{\left(x \cdot y\right)}^{\color{blue}{\left(2 \cdot -0.5\right)}}}{z}}{z} \]

       9. pow-sqr 46.2%

          \[\leadsto \frac{\frac{\color{blue}{{\left(x \cdot y\right)}^{-0.5} \cdot {\left(x \cdot y\right)}^{-0.5}}}{z}}{z} \]

       10. associate-*r/ 46.2%

           \[\leadsto \frac{\color{blue}{{\left(x \cdot y\right)}^{-0.5} \cdot \frac{{\left(x \cdot y\right)}^{-0.5}}{z}}}{z} \]

       11. associate-*l/ 46.3%

           \[\leadsto \color{blue}{\frac{{\left(x \cdot y\right)}^{-0.5}}{z} \cdot \frac{{\left(x \cdot y\right)}^{-0.5}}{z}} \]

       12. *-rgt-identity 46.3%

           \[\leadsto \frac{\color{blue}{{\left(x \cdot y\right)}^{-0.5} \cdot 1}}{z} \cdot \frac{{\left(x \cdot y\right)}^{-0.5}}{z} \]

       13. associate-*r/ 46.2%

           \[\leadsto \color{blue}{\left({\left(x \cdot y\right)}^{-0.5} \cdot \frac{1}{z}\right)} \cdot \frac{{\left(x \cdot y\right)}^{-0.5}}{z} \]

       14. *-rgt-identity 46.2%

           \[\leadsto \left({\left(x \cdot y\right)}^{-0.5} \cdot \frac{1}{z}\right) \cdot \frac{\color{blue}{{\left(x \cdot y\right)}^{-0.5} \cdot 1}}{z} \]

       15. associate-*r/ 46.1%

           \[\leadsto \left({\left(x \cdot y\right)}^{-0.5} \cdot \frac{1}{z}\right) \cdot \color{blue}{\left({\left(x \cdot y\right)}^{-0.5} \cdot \frac{1}{z}\right)} \]

       16. unswap-sqr 38.9%

           \[\leadsto \color{blue}{\left({\left(x \cdot y\right)}^{-0.5} \cdot {\left(x \cdot y\right)}^{-0.5}\right) \cdot \left(\frac{1}{z} \cdot \frac{1}{z}\right)} \]

       17. pow-sqr 73.5%

           \[\leadsto \color{blue}{{\left(x \cdot y\right)}^{\left(2 \cdot -0.5\right)}} \cdot \left(\frac{1}{z} \cdot \frac{1}{z}\right) \]

       18. metadata-eval 73.5%

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

       19. unpow-1 73.5%

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

       20. unpow-1 73.5%

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

       21. unpow-1 73.5%

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

       22. pow-sqr 73.5%

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

       23. metadata-eval 73.5%

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

   11. Simplified 73.5%

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

       1. div-inv 73.5%

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

       2. add-sqr-sqrt 73.5%

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

       3. associate-/r* 73.5%

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

       4. associate-*l* 73.4%

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

       5. sqrt-pow1 69.8%

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

       6. metadata-eval 69.8%

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

       7. unpow-1 69.8%

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

       8. sqrt-pow1 87.4%

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

       9. metadata-eval 87.4%

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

       10. unpow-1 87.4%

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

       11. associate-/l/ 87.5%

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

   13. Applied egg-rr 87.5%

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

       1. un-div-inv 87.4%

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

       2. *-commutative 87.4%

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

       3. associate-/r* 94.8%

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

   15. Applied egg-rr 94.8%

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

4. Final simplification 95.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(1 + z \cdot z\right) \leq 2 \cdot 10^{+307}:\\ \;\;\;\;\frac{1}{x \cdot \left(y \cdot \left(1 + z \cdot z\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{z} \cdot \frac{\frac{\frac{1}{z}}{x}}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ \begin{array}{l} t_0 := y_m \cdot \left(1 + z \cdot z\right)\\ y_s \cdot \left(x_s \cdot \begin{array}{l} \mathbf{if}\;t_0 \leq 2 \cdot 10^{+307}:\\ \;\;\;\;\frac{\frac{1}{x_m}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{z} \cdot \frac{\frac{\frac{1}{z}}{x_m}}{y_m}\\ \end{array}\right) \end{array} \end{array} \]

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 2e+307)
       (/ (/ 1.0 x_m) t_0)
       (* (/ 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 t_0 = y_m * (1.0 + (z * z));
	double tmp;
	if (t_0 <= 2e+307) {
		tmp = (1.0 / x_m) / t_0;
	} 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) :: t_0
    real(8) :: tmp
    t_0 = y_m * (1.0d0 + (z * z))
    if (t_0 <= 2d+307) then
        tmp = (1.0d0 / x_m) / t_0
    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 t_0 = y_m * (1.0 + (z * z));
	double tmp;
	if (t_0 <= 2e+307) {
		tmp = (1.0 / x_m) / t_0;
	} 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):
	t_0 = y_m * (1.0 + (z * z))
	tmp = 0
	if t_0 <= 2e+307:
		tmp = (1.0 / x_m) / t_0
	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)
	t_0 = Float64(y_m * Float64(1.0 + Float64(z * z)))
	tmp = 0.0
	if (t_0 <= 2e+307)
		tmp = Float64(Float64(1.0 / x_m) / t_0);
	else
		tmp = Float64(Float64(1.0 / z) * Float64(Float64(Float64(1.0 / 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)
	t_0 = y_m * (1.0 + (z * z));
	tmp = 0.0;
	if (t_0 <= 2e+307)
		tmp = (1.0 / x_m) / t_0;
	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_] := 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, 2e+307], N[(N[(1.0 / x$95$m), $MachinePrecision] / t$95$0), $MachinePrecision], N[(N[(1.0 / z), $MachinePrecision] * N[(N[(N[(1.0 / z), $MachinePrecision] / x$95$m), $MachinePrecision] / y$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y (+.f64 1 (*.f64 z z))) < 1.99999999999999997e307

   1. Initial program 95.8%

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

   if 1.99999999999999997e307 < (*.f64 y (+.f64 1 (*.f64 z z)))

   1. Initial program 60.5%

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

      1. associate-/l/ 60.6%

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

      2. metadata-eval 60.6%

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

      3. associate-*r/ 60.6%

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

      4. associate-/l/ 60.5%

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

      5. associate-*r/ 60.5%

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

      6. associate-/l* 60.5%

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

      7. associate-/r/ 60.6%

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

      8. /-rgt-identity 60.6%

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

      9. associate-*l* 72.4%

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

      10. *-commutative 72.4%

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

      11. sqr-neg 72.4%

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

      12. +-commutative 72.4%

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

      13. sqr-neg 72.4%

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

      14. fma-def 72.4%

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

   3. Simplified 72.4%

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

      1. fma-udef 72.4%

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

      2. +-commutative 72.4%

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

      3. *-commutative 72.4%

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

      4. associate-*l* 60.6%

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

      5. associate-/l/ 60.5%

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

      6. add-sqr-sqrt 60.5%

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

      7. sqrt-div 16.5%

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

      8. inv-pow 16.5%

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

      9. sqrt-pow1 16.5%

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

      10. metadata-eval 16.5%

          \[\leadsto \frac{{x}^{\color{blue}{-0.5}}}{\sqrt{y \cdot \left(1 + z \cdot z\right)}} \cdot \sqrt{\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}} \]

      11. *-commutative 16.5%

          \[\leadsto \frac{{x}^{-0.5}}{\sqrt{\color{blue}{\left(1 + z \cdot z\right) \cdot y}}} \cdot \sqrt{\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}} \]

      12. sqrt-prod 16.5%

          \[\leadsto \frac{{x}^{-0.5}}{\color{blue}{\sqrt{1 + z \cdot z} \cdot \sqrt{y}}} \cdot \sqrt{\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}} \]

      13. hypot-1-def 16.5%

          \[\leadsto \frac{{x}^{-0.5}}{\color{blue}{\mathsf{hypot}\left(1, z\right)} \cdot \sqrt{y}} \cdot \sqrt{\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}} \]

      14. sqrt-div 16.5%

          \[\leadsto \frac{{x}^{-0.5}}{\mathsf{hypot}\left(1, z\right) \cdot \sqrt{y}} \cdot \color{blue}{\frac{\sqrt{\frac{1}{x}}}{\sqrt{y \cdot \left(1 + z \cdot z\right)}}} \]

      15. inv-pow 16.5%

          \[\leadsto \frac{{x}^{-0.5}}{\mathsf{hypot}\left(1, z\right) \cdot \sqrt{y}} \cdot \frac{\sqrt{\color{blue}{{x}^{-1}}}}{\sqrt{y \cdot \left(1 + z \cdot z\right)}} \]

      16. sqrt-pow1 16.5%

          \[\leadsto \frac{{x}^{-0.5}}{\mathsf{hypot}\left(1, z\right) \cdot \sqrt{y}} \cdot \frac{\color{blue}{{x}^{\left(\frac{-1}{2}\right)}}}{\sqrt{y \cdot \left(1 + z \cdot z\right)}} \]

      17. metadata-eval 16.5%

          \[\leadsto \frac{{x}^{-0.5}}{\mathsf{hypot}\left(1, z\right) \cdot \sqrt{y}} \cdot \frac{{x}^{\color{blue}{-0.5}}}{\sqrt{y \cdot \left(1 + z \cdot z\right)}} \]

      18. *-commutative 16.5%

          \[\leadsto \frac{{x}^{-0.5}}{\mathsf{hypot}\left(1, z\right) \cdot \sqrt{y}} \cdot \frac{{x}^{-0.5}}{\sqrt{\color{blue}{\left(1 + z \cdot z\right) \cdot y}}} \]

   6. Applied egg-rr 38.1%

      \[\leadsto \color{blue}{\frac{{x}^{-0.5}}{\mathsf{hypot}\left(1, z\right) \cdot \sqrt{y}} \cdot \frac{{x}^{-0.5}}{\mathsf{hypot}\left(1, z\right) \cdot \sqrt{y}}} \]
   7. Step-by-step derivation

      1. unpow2 38.1%

         \[\leadsto \color{blue}{{\left(\frac{{x}^{-0.5}}{\mathsf{hypot}\left(1, z\right) \cdot \sqrt{y}}\right)}^{2}} \]

      2. *-commutative 38.1%

         \[\leadsto {\left(\frac{{x}^{-0.5}}{\color{blue}{\sqrt{y} \cdot \mathsf{hypot}\left(1, z\right)}}\right)}^{2} \]

      3. associate-/r* 38.2%

         \[\leadsto {\color{blue}{\left(\frac{\frac{{x}^{-0.5}}{\sqrt{y}}}{\mathsf{hypot}\left(1, z\right)}\right)}}^{2} \]

   8. Simplified 38.2%

      \[\leadsto \color{blue}{{\left(\frac{\frac{{x}^{-0.5}}{\sqrt{y}}}{\mathsf{hypot}\left(1, z\right)}\right)}^{2}} \]

   9. Taylor expanded in z around inf 60.6%

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

       1. associate-*r* 72.0%

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

       2. associate-/r* 72.1%

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

       3. associate-/l/ 72.1%

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

       4. unpow2 72.1%

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

       5. associate-/r* 87.5%

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

       6. associate-/l/ 87.6%

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

       7. unpow-1 87.6%

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

       8. metadata-eval 87.6%

          \[\leadsto \frac{\frac{{\left(x \cdot y\right)}^{\color{blue}{\left(2 \cdot -0.5\right)}}}{z}}{z} \]

       9. pow-sqr 46.2%

          \[\leadsto \frac{\frac{\color{blue}{{\left(x \cdot y\right)}^{-0.5} \cdot {\left(x \cdot y\right)}^{-0.5}}}{z}}{z} \]

       10. associate-*r/ 46.2%

           \[\leadsto \frac{\color{blue}{{\left(x \cdot y\right)}^{-0.5} \cdot \frac{{\left(x \cdot y\right)}^{-0.5}}{z}}}{z} \]

       11. associate-*l/ 46.3%

           \[\leadsto \color{blue}{\frac{{\left(x \cdot y\right)}^{-0.5}}{z} \cdot \frac{{\left(x \cdot y\right)}^{-0.5}}{z}} \]

       12. *-rgt-identity 46.3%

           \[\leadsto \frac{\color{blue}{{\left(x \cdot y\right)}^{-0.5} \cdot 1}}{z} \cdot \frac{{\left(x \cdot y\right)}^{-0.5}}{z} \]

       13. associate-*r/ 46.2%

           \[\leadsto \color{blue}{\left({\left(x \cdot y\right)}^{-0.5} \cdot \frac{1}{z}\right)} \cdot \frac{{\left(x \cdot y\right)}^{-0.5}}{z} \]

       14. *-rgt-identity 46.2%

           \[\leadsto \left({\left(x \cdot y\right)}^{-0.5} \cdot \frac{1}{z}\right) \cdot \frac{\color{blue}{{\left(x \cdot y\right)}^{-0.5} \cdot 1}}{z} \]

       15. associate-*r/ 46.1%

           \[\leadsto \left({\left(x \cdot y\right)}^{-0.5} \cdot \frac{1}{z}\right) \cdot \color{blue}{\left({\left(x \cdot y\right)}^{-0.5} \cdot \frac{1}{z}\right)} \]

       16. unswap-sqr 38.9%

           \[\leadsto \color{blue}{\left({\left(x \cdot y\right)}^{-0.5} \cdot {\left(x \cdot y\right)}^{-0.5}\right) \cdot \left(\frac{1}{z} \cdot \frac{1}{z}\right)} \]

       17. pow-sqr 73.5%

           \[\leadsto \color{blue}{{\left(x \cdot y\right)}^{\left(2 \cdot -0.5\right)}} \cdot \left(\frac{1}{z} \cdot \frac{1}{z}\right) \]

       18. metadata-eval 73.5%

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

       19. unpow-1 73.5%

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

       20. unpow-1 73.5%

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

       21. unpow-1 73.5%

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

       22. pow-sqr 73.5%

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

       23. metadata-eval 73.5%

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

   11. Simplified 73.5%

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

       1. div-inv 73.5%

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

       2. add-sqr-sqrt 73.5%

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

       3. associate-/r* 73.5%

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

       4. associate-*l* 73.4%

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

       5. sqrt-pow1 69.8%

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

       6. metadata-eval 69.8%

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

       7. unpow-1 69.8%

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

       8. sqrt-pow1 87.4%

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

       9. metadata-eval 87.4%

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

       10. unpow-1 87.4%

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

       11. associate-/l/ 87.5%

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

   13. Applied egg-rr 87.5%

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

       1. un-div-inv 87.4%

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

       2. *-commutative 87.4%

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

       3. associate-/r* 94.8%

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

   15. Applied egg-rr 94.8%

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

4. Final simplification 95.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(1 + z \cdot z\right) \leq 2 \cdot 10^{+307}:\\ \;\;\;\;\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{z} \cdot \frac{\frac{\frac{1}{z}}{x}}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 75.9% accurate, 0.8× speedup

Math

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

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

Derivation

1. Split input into 2 regimes

2. if z < 1

   1. Initial program 96.2%

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

   3. Taylor expanded in z around 0 74.1%

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

   if 1 < z

   1. Initial program 75.2%

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

      1. associate-/l/ 75.2%

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

      2. metadata-eval 75.2%

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

      3. associate-*r/ 75.2%

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

      4. associate-/l/ 75.2%

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

      5. associate-*r/ 75.2%

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

      6. associate-/l* 75.2%

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

      7. associate-/r/ 75.2%

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

      8. /-rgt-identity 75.2%

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

      9. associate-*l* 81.9%

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

      10. *-commutative 81.9%

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

      11. sqr-neg 81.9%

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

      12. +-commutative 81.9%

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

      13. sqr-neg 81.9%

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

      14. fma-def 81.9%

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

   3. Simplified 81.9%

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

   5. Taylor expanded in z around inf 74.9%

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

      1. associate-/r* 74.9%

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

      2. associate-/r* 78.7%

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

      3. associate-/l/ 78.7%

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

      4. associate-/r* 78.6%

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

   7. Simplified 78.6%

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

      1. *-un-lft-identity 78.6%

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

      2. unpow2 78.6%

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

      3. times-frac 88.4%

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

   9. Applied egg-rr 88.4%

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

       1. *-commutative 88.4%

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

       2. associate-/r* 88.5%

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

       3. *-un-lft-identity 88.5%

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

       4. associate-*l/ 88.4%

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

       5. frac-times 88.4%

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

       6. metadata-eval 88.4%

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

       7. frac-times 88.1%

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

       8. metadata-eval 88.1%

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

       9. *-commutative 88.1%

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

   11. Applied egg-rr 88.1%

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

4. Final simplification 77.9%

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

Alternative 7: 77.9% accurate, 0.8× speedup

Math

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

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 y_m) (* z (* x_m z)))))))

C

x_m = fabs(x);
x_s = copysign(1.0, x);
y_m = fabs(y);
y_s = copysign(1.0, y);
assert(x_m < y_m && y_m < z);
double code(double y_s, double x_s, double x_m, double y_m, double z) {
	double tmp;
	if (z <= 1.0) {
		tmp = (1.0 / x_m) / y_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 <= 1.0d0) then
        tmp = (1.0d0 / x_m) / y_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 <= 1.0) {
		tmp = (1.0 / x_m) / y_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 <= 1.0:
		tmp = (1.0 / x_m) / y_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 (z <= 1.0)
		tmp = Float64(Float64(1.0 / x_m) / y_m);
	else
		tmp = Float64(Float64(1.0 / y_m) / Float64(z * Float64(x_m * z)));
	end
	return Float64(y_s * Float64(x_s * tmp))
end

MATLAB

x_m = abs(x);
x_s = sign(x) * abs(1.0);
y_m = abs(y);
y_s = sign(y) * abs(1.0);
x_m, y_m, z = num2cell(sort([x_m, y_m, z])){:}
function tmp_2 = code(y_s, x_s, x_m, y_m, z)
	tmp = 0.0;
	if (z <= 1.0)
		tmp = (1.0 / x_m) / y_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[z, 1.0], N[(N[(1.0 / x$95$m), $MachinePrecision] / y$95$m), $MachinePrecision], N[(N[(1.0 / y$95$m), $MachinePrecision] / N[(z * N[(x$95$m * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 1

   1. Initial program 96.2%

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

   3. Taylor expanded in z around 0 74.1%

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

   if 1 < z

   1. Initial program 75.2%

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

      1. associate-/l/ 75.2%

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

      2. metadata-eval 75.2%

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

      3. associate-*r/ 75.2%

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

      4. associate-/l/ 75.2%

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

      5. associate-*r/ 75.2%

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

      6. associate-/l* 75.2%

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

      7. associate-/r/ 75.2%

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

      8. /-rgt-identity 75.2%

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

      9. associate-*l* 81.9%

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

      10. *-commutative 81.9%

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

      11. sqr-neg 81.9%

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

      12. +-commutative 81.9%

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

      13. sqr-neg 81.9%

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

      14. fma-def 81.9%

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

   3. Simplified 81.9%

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

   5. Taylor expanded in z around inf 74.9%

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

      1. associate-/r* 74.9%

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

      2. associate-/r* 78.7%

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

      3. associate-/l/ 78.7%

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

      4. associate-/r* 78.6%

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

   7. Simplified 78.6%

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

      1. *-un-lft-identity 78.6%

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

      2. unpow2 78.6%

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

      3. times-frac 88.4%

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

   9. Applied egg-rr 88.4%

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

       1. *-commutative 88.4%

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

       2. associate-/l/ 92.8%

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

       3. frac-times 91.1%

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

       4. associate-/r/ 91.1%

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

       5. clear-num 91.1%

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

   11. Applied egg-rr 91.1%

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

4. Final simplification 78.7%

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

Alternative 8: 57.6% accurate, 2.2× speedup

Math

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

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

Derivation

1. Initial program 90.5%

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

   1. associate-/l/ 90.2%

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

   2. metadata-eval 90.2%

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

   3. associate-*r/ 90.2%

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

   4. associate-/l/ 90.5%

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

   5. associate-*r/ 90.5%

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

   6. associate-/l* 90.1%

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

   7. associate-/r/ 90.2%

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

   8. /-rgt-identity 90.2%

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

   9. associate-*l* 90.1%

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

   10. *-commutative 90.1%

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

   11. sqr-neg 90.1%

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

   12. +-commutative 90.1%

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

   13. sqr-neg 90.1%

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

   14. fma-def 90.1%

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

3. Simplified 90.1%

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

5. Taylor expanded in z around 0 58.8%

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

6. Final simplification 58.8%

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

Alternative 9: 57.7% accurate, 2.2× speedup

Math

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

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 90.5%

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

3. Taylor expanded in z around 0 59.1%

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

4. Final simplification 59.1%

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

Developer target: 92.0% 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 2024019 
(FPCore (x y z)
  :name "Statistics.Distribution.CauchyLorentz:$cdensity from math-functions-0.1.5.2"
  :precision binary64

  :herbie-target
  (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)))))