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

Percentage Accurate: 89.5% → 98.6%

Time: 12.9s

Alternatives: 9

Speedup: 0.7×

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: 89.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.6% accurate, 0.1× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;z \cdot z \leq 5 \cdot 10^{+111}:\\ \;\;\;\;\frac{\frac{\frac{1}{x\_m}}{y\_m}}{\mathsf{fma}\left(z, z, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{y\_m \cdot \left(z \cdot x\_m\right)}}{z}\\ \end{array}\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (y_s x_s x_m y_m z)
 :precision binary64
 (*
  y_s
  (*
   x_s
   (if (<= (* z z) 5e+111)
     (/ (/ (/ 1.0 x_m) y_m) (fma z z 1.0))
     (/ (/ 1.0 (* y_m (* z x_m))) z)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
assert(x_m < y_m && y_m < z);
double code(double y_s, double x_s, double x_m, double y_m, double z) {
	double tmp;
	if ((z * z) <= 5e+111) {
		tmp = ((1.0 / x_m) / y_m) / fma(z, z, 1.0);
	} else {
		tmp = (1.0 / (y_m * (z * x_m))) / z;
	}
	return y_s * (x_s * tmp);
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
y\_m = abs(y)
y\_s = copysign(1.0, y)
x_m, y_m, z = sort([x_m, y_m, z])
function code(y_s, x_s, x_m, y_m, z)
	tmp = 0.0
	if (Float64(z * z) <= 5e+111)
		tmp = Float64(Float64(Float64(1.0 / x_m) / y_m) / fma(z, z, 1.0));
	else
		tmp = Float64(Float64(1.0 / Float64(y_m * Float64(z * x_m))) / z);
	end
	return Float64(y_s * Float64(x_s * tmp))
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
code[y$95$s_, x$95$s_, x$95$m_, y$95$m_, z_] := N[(y$95$s * N[(x$95$s * If[LessEqual[N[(z * z), $MachinePrecision], 5e+111], N[(N[(N[(1.0 / x$95$m), $MachinePrecision] / y$95$m), $MachinePrecision] / N[(z * z + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[(y$95$m * N[(z * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 4.9999999999999997e111

   1. Initial program 98.5%

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

      1. associate-/l/ 98.1%

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

      2. remove-double-neg 98.1%

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

      3. distribute-rgt-neg-out 98.1%

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

      4. distribute-rgt-neg-out 98.1%

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

      5. remove-double-neg 98.1%

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

      6. associate-*l* 98.7%

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

      7. *-commutative 98.7%

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

      8. sqr-neg 98.7%

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

      9. +-commutative 98.7%

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

      10. sqr-neg 98.7%

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

      11. fma-define 98.7%

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

   3. Simplified 98.7%

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

      1. *-commutative 98.7%

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

      2. associate-*r* 98.1%

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

      3. fma-undefine 98.1%

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

      4. +-commutative 98.1%

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

      5. associate-/l/ 98.5%

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

      6. add-sqr-sqrt 53.4%

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

      7. *-un-lft-identity 53.4%

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

      8. times-frac 53.4%

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

      9. +-commutative 53.4%

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

      10. fma-undefine 53.4%

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

      11. *-commutative 53.4%

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

      12. sqrt-prod 53.4%

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

      13. fma-undefine 53.4%

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

      14. +-commutative 53.4%

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

      15. hypot-1-def 53.4%

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

      16. +-commutative 53.4%

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

   6. Applied egg-rr 54.0%

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

      1. frac-times 53.4%

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

      2. *-un-lft-identity 53.4%

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

      3. swap-sqr 53.4%

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

      4. hypot-undefine 53.4%

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

      5. hypot-undefine 53.4%

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

      6. rem-square-sqrt 53.4%

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

      7. metadata-eval 53.4%

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

      8. unpow2 53.4%

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

      9. +-commutative 53.4%

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

      10. unpow2 53.4%

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

      11. fma-undefine 53.4%

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

      12. add-sqr-sqrt 98.5%

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

      13. *-commutative 98.5%

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

      14. associate-/r* 99.7%

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

   8. Applied egg-rr 99.7%

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

   if 4.9999999999999997e111 < (*.f64 z z)

   1. Initial program 79.8%

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

      1. remove-double-neg 79.8%

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

      2. distribute-lft-neg-out 79.8%

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

      3. distribute-rgt-neg-in 79.8%

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

      4. associate-/r* 80.6%

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

      5. associate-/l/ 80.6%

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

      6. associate-/l/ 80.4%

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

      7. distribute-lft-neg-out 80.4%

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

      8. distribute-rgt-neg-in 80.4%

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

      9. distribute-lft-neg-in 80.4%

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

      10. remove-double-neg 80.4%

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

      11. sqr-neg 80.4%

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

      12. +-commutative 80.4%

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

      13. sqr-neg 80.4%

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

      14. fma-define 80.4%

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

      15. *-commutative 80.4%

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

   3. Simplified 80.4%

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

   5. Taylor expanded in z around inf 80.4%

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

      1. *-un-lft-identity 80.4%

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

      2. associate-/r* 80.6%

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

      3. *-commutative 80.6%

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

      4. associate-/r* 80.7%

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

      5. pow-flip 81.2%

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

      6. metadata-eval 81.2%

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

   7. Applied egg-rr 81.2%

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

      1. *-lft-identity 81.2%

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

      2. associate-/l/ 81.0%

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

   9. Simplified 81.0%

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

       1. sqr-pow 81.0%

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

       2. associate-/l* 90.8%

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

       3. metadata-eval 90.8%

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

       4. unpow-1 90.8%

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

       5. metadata-eval 90.8%

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

       6. unpow-1 90.8%

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

   11. Applied egg-rr 90.8%

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

       1. associate-*l/ 90.9%

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

       2. *-un-lft-identity 90.9%

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

       3. clear-num 90.9%

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

       4. *-commutative 90.9%

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

       5. associate-*r/ 98.1%

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

       6. associate-/r/ 98.1%

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

       7. /-rgt-identity 98.1%

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

       8. *-commutative 98.1%

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

   13. Applied egg-rr 98.1%

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

Alternative 2: 98.8% accurate, 0.1× speedup

Math

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

FPCore

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

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 / (Math.hypot(1.0, z) * (Math.hypot(1.0, z) * 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 / (math.hypot(1.0, z) * (math.hypot(1.0, z) * 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 / Float64(hypot(1.0, z) * Float64(hypot(1.0, z) * 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 / (hypot(1.0, z) * (hypot(1.0, z) * 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 / N[(N[Sqrt[1.0 ^ 2 + z ^ 2], $MachinePrecision] * N[(N[Sqrt[1.0 ^ 2 + z ^ 2], $MachinePrecision] * x$95$m), $MachinePrecision]), $MachinePrecision]), $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. Step-by-step derivation

   1. associate-/l/ 90.3%

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

   2. remove-double-neg 90.3%

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

   3. distribute-rgt-neg-out 90.3%

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

   4. distribute-rgt-neg-out 90.3%

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

   5. remove-double-neg 90.3%

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

   6. associate-*l* 89.6%

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

   7. *-commutative 89.6%

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

   8. sqr-neg 89.6%

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

   9. +-commutative 89.6%

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

   10. sqr-neg 89.6%

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

   11. fma-define 89.6%

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

3. Simplified 89.6%

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

   1. *-commutative 89.6%

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

   2. associate-*r* 90.3%

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

   3. fma-undefine 90.3%

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

   4. +-commutative 90.3%

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

   5. associate-/l/ 90.5%

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

   6. add-sqr-sqrt 45.8%

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

   7. *-un-lft-identity 45.8%

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

   8. times-frac 45.8%

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

   9. +-commutative 45.8%

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

   10. fma-undefine 45.8%

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

   11. *-commutative 45.8%

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

   12. sqrt-prod 45.8%

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

   13. fma-undefine 45.8%

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

   14. +-commutative 45.8%

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

   15. hypot-1-def 45.8%

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

   16. +-commutative 45.8%

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

6. Applied egg-rr 51.2%

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

   1. *-commutative 51.2%

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

   2. associate-/r* 51.2%

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

   3. associate-/r* 51.2%

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

   4. frac-times 49.4%

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

   5. add-sqr-sqrt 95.5%

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

8. Applied egg-rr 95.5%

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

   1. un-div-inv 95.6%

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

   2. clear-num 95.5%

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

   3. associate-/l/ 95.0%

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

   4. associate-/r/ 95.0%

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

   5. /-rgt-identity 95.0%

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

   6. *-commutative 95.0%

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

10. Applied egg-rr 95.0%

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

11. Final simplification 95.0%

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

Alternative 3: 98.6% accurate, 0.1× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;z \cdot z \leq 10^{+23}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(z, z, 1\right) \cdot \left(x\_m \cdot y\_m\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{y\_m \cdot \left(z \cdot x\_m\right)}}{z}\\ \end{array}\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (y_s x_s x_m y_m z)
 :precision binary64
 (*
  y_s
  (*
   x_s
   (if (<= (* z z) 1e+23)
     (/ 1.0 (* (fma z z 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 * z) <= 1e+23) {
		tmp = 1.0 / (fma(z, z, 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 (Float64(z * z) <= 1e+23)
		tmp = Float64(1.0 / Float64(fma(z, z, 1.0) * Float64(x_m * y_m)));
	else
		tmp = Float64(Float64(1.0 / Float64(y_m * Float64(z * x_m))) / z);
	end
	return Float64(y_s * Float64(x_s * tmp))
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
code[y$95$s_, x$95$s_, x$95$m_, y$95$m_, z_] := N[(y$95$s * N[(x$95$s * If[LessEqual[N[(z * z), $MachinePrecision], 1e+23], N[(1.0 / N[(N[(z * z + 1.0), $MachinePrecision] * N[(x$95$m * y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[(y$95$m * N[(z * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 9.9999999999999992e22

   1. Initial program 99.1%

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

      1. remove-double-neg 99.1%

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

      2. distribute-lft-neg-out 99.1%

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

      3. distribute-rgt-neg-in 99.1%

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

      4. associate-/r* 99.7%

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

      5. associate-/l/ 99.5%

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

      6. associate-/l/ 99.5%

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

      7. distribute-lft-neg-out 99.5%

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

      8. distribute-rgt-neg-in 99.5%

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

      9. distribute-lft-neg-in 99.5%

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

      10. remove-double-neg 99.5%

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

      11. sqr-neg 99.5%

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

      12. +-commutative 99.5%

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

      13. sqr-neg 99.5%

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

      14. fma-define 99.5%

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

      15. *-commutative 99.5%

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

   3. Simplified 99.5%

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

   if 9.9999999999999992e22 < (*.f64 z z)

   1. Initial program 80.4%

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

      1. remove-double-neg 80.4%

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

      2. distribute-lft-neg-out 80.4%

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

      3. distribute-rgt-neg-in 80.4%

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

      4. associate-/r* 81.9%

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

      5. associate-/l/ 81.9%

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

      6. associate-/l/ 81.5%

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

      7. distribute-lft-neg-out 81.5%

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

      8. distribute-rgt-neg-in 81.5%

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

      9. distribute-lft-neg-in 81.5%

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

      10. remove-double-neg 81.5%

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

      11. sqr-neg 81.5%

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

      12. +-commutative 81.5%

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

      13. sqr-neg 81.5%

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

      14. fma-define 81.5%

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

      15. *-commutative 81.5%

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

   3. Simplified 81.5%

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

   5. Taylor expanded in z around inf 81.5%

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

      1. *-un-lft-identity 81.5%

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

      2. associate-/r* 81.9%

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

      3. *-commutative 81.9%

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

      4. associate-/r* 81.2%

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

      5. pow-flip 81.6%

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

      6. metadata-eval 81.6%

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

   7. Applied egg-rr 81.6%

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

      1. *-lft-identity 81.6%

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

      2. associate-/l/ 82.3%

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

   9. Simplified 82.3%

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

       1. sqr-pow 82.3%

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

       2. associate-/l* 91.4%

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

       3. metadata-eval 91.4%

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

       4. unpow-1 91.4%

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

       5. metadata-eval 91.4%

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

       6. unpow-1 91.4%

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

   11. Applied egg-rr 91.4%

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

       1. associate-*l/ 91.5%

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

       2. *-un-lft-identity 91.5%

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

       3. clear-num 91.5%

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

       4. *-commutative 91.5%

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

       5. associate-*r/ 98.2%

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

       6. associate-/r/ 98.2%

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

       7. /-rgt-identity 98.2%

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

       8. *-commutative 98.2%

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

   13. Applied egg-rr 98.2%

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

Alternative 4: 98.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;z \cdot z \leq 40000000:\\ \;\;\;\;\frac{\frac{1}{x\_m}}{y\_m \cdot \left(1 + z \cdot z\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{y\_m \cdot \left(z \cdot x\_m\right)}}{z}\\ \end{array}\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (y_s x_s x_m y_m z)
 :precision binary64
 (*
  y_s
  (*
   x_s
   (if (<= (* z z) 40000000.0)
     (/ (/ 1.0 x_m) (* y_m (+ 1.0 (* z z))))
     (/ (/ 1.0 (* y_m (* z x_m))) z)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
assert(x_m < y_m && y_m < z);
double code(double y_s, double x_s, double x_m, double y_m, double z) {
	double tmp;
	if ((z * z) <= 40000000.0) {
		tmp = (1.0 / x_m) / (y_m * (1.0 + (z * z)));
	} else {
		tmp = (1.0 / (y_m * (z * x_m))) / z;
	}
	return y_s * (x_s * tmp);
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
y\_m = abs(y)
y\_s = copysign(1.0d0, y)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
real(8) function code(y_s, x_s, x_m, y_m, z)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((z * z) <= 40000000.0d0) then
        tmp = (1.0d0 / x_m) / (y_m * (1.0d0 + (z * z)))
    else
        tmp = (1.0d0 / (y_m * (z * x_m))) / z
    end if
    code = y_s * (x_s * tmp)
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
assert x_m < y_m && y_m < z;
public static double code(double y_s, double x_s, double x_m, double y_m, double z) {
	double tmp;
	if ((z * z) <= 40000000.0) {
		tmp = (1.0 / x_m) / (y_m * (1.0 + (z * z)));
	} else {
		tmp = (1.0 / (y_m * (z * x_m))) / z;
	}
	return y_s * (x_s * tmp);
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
[x_m, y_m, z] = sort([x_m, y_m, z])
def code(y_s, x_s, x_m, y_m, z):
	tmp = 0
	if (z * z) <= 40000000.0:
		tmp = (1.0 / x_m) / (y_m * (1.0 + (z * z)))
	else:
		tmp = (1.0 / (y_m * (z * x_m))) / z
	return y_s * (x_s * tmp)

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
y\_m = abs(y)
y\_s = copysign(1.0, y)
x_m, y_m, z = sort([x_m, y_m, z])
function code(y_s, x_s, x_m, y_m, z)
	tmp = 0.0
	if (Float64(z * z) <= 40000000.0)
		tmp = Float64(Float64(1.0 / x_m) / Float64(y_m * Float64(1.0 + Float64(z * z))));
	else
		tmp = Float64(Float64(1.0 / Float64(y_m * Float64(z * x_m))) / z);
	end
	return Float64(y_s * Float64(x_s * tmp))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
x_m, y_m, z = num2cell(sort([x_m, y_m, z])){:}
function tmp_2 = code(y_s, x_s, x_m, y_m, z)
	tmp = 0.0;
	if ((z * z) <= 40000000.0)
		tmp = (1.0 / x_m) / (y_m * (1.0 + (z * z)));
	else
		tmp = (1.0 / (y_m * (z * x_m))) / z;
	end
	tmp_2 = y_s * (x_s * tmp);
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
code[y$95$s_, x$95$s_, x$95$m_, y$95$m_, z_] := N[(y$95$s * N[(x$95$s * If[LessEqual[N[(z * z), $MachinePrecision], 40000000.0], N[(N[(1.0 / x$95$m), $MachinePrecision] / N[(y$95$m * N[(1.0 + N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[(y$95$m * N[(z * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 4e7

   1. Initial program 99.7%

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

   if 4e7 < (*.f64 z z)

   1. Initial program 80.3%

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

      1. remove-double-neg 80.3%

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

      2. distribute-lft-neg-out 80.3%

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

      3. distribute-rgt-neg-in 80.3%

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

      4. associate-/r* 82.4%

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

      5. associate-/l/ 82.5%

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

      6. associate-/l/ 82.1%

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

      7. distribute-lft-neg-out 82.1%

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

      8. distribute-rgt-neg-in 82.1%

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

      9. distribute-lft-neg-in 82.1%

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

      10. remove-double-neg 82.1%

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

      11. sqr-neg 82.1%

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

      12. +-commutative 82.1%

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

      13. sqr-neg 82.1%

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

      14. fma-define 82.1%

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

      15. *-commutative 82.1%

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

   3. Simplified 82.1%

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

   5. Taylor expanded in z around inf 81.9%

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

      1. *-un-lft-identity 81.9%

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

      2. associate-/r* 82.2%

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

      3. *-commutative 82.2%

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

      4. associate-/r* 81.6%

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

      5. pow-flip 82.0%

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

      6. metadata-eval 82.0%

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

   7. Applied egg-rr 82.0%

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

      1. *-lft-identity 82.0%

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

      2. associate-/l/ 82.7%

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

   9. Simplified 82.7%

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

       1. sqr-pow 82.6%

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

       2. associate-/l* 91.4%

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

       3. metadata-eval 91.4%

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

       4. unpow-1 91.4%

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

       5. metadata-eval 91.4%

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

       6. unpow-1 91.4%

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

   11. Applied egg-rr 91.4%

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

       1. associate-*l/ 91.5%

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

       2. *-un-lft-identity 91.5%

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

       3. clear-num 91.5%

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

       4. *-commutative 91.5%

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

       5. associate-*r/ 98.0%

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

       6. associate-/r/ 98.0%

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

       7. /-rgt-identity 98.0%

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

       8. *-commutative 98.0%

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

   13. Applied egg-rr 98.0%

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

Alternative 5: 98.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;z \cdot z \leq 0.2:\\ \;\;\;\;\frac{\frac{1}{x\_m}}{y\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{y\_m \cdot \left(z \cdot x\_m\right)}}{z}\\ \end{array}\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (y_s x_s x_m y_m z)
 :precision binary64
 (*
  y_s
  (*
   x_s
   (if (<= (* z z) 0.2) (/ (/ 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 * z) <= 0.2) {
		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 * z) <= 0.2d0) 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 * z) <= 0.2) {
		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 * z) <= 0.2:
		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 (Float64(z * z) <= 0.2)
		tmp = Float64(Float64(1.0 / x_m) / y_m);
	else
		tmp = Float64(Float64(1.0 / Float64(y_m * Float64(z * x_m))) / z);
	end
	return Float64(y_s * Float64(x_s * tmp))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
x_m, y_m, z = num2cell(sort([x_m, y_m, z])){:}
function tmp_2 = code(y_s, x_s, x_m, y_m, z)
	tmp = 0.0;
	if ((z * z) <= 0.2)
		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[N[(z * z), $MachinePrecision], 0.2], N[(N[(1.0 / x$95$m), $MachinePrecision] / y$95$m), $MachinePrecision], N[(N[(1.0 / N[(y$95$m * N[(z * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 0.20000000000000001

   1. Initial program 99.7%

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

      1. associate-/l/ 99.5%

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

      2. remove-double-neg 99.5%

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

      3. distribute-rgt-neg-out 99.5%

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

      4. distribute-rgt-neg-out 99.5%

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

      5. remove-double-neg 99.5%

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

      6. associate-*l* 99.5%

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

      7. *-commutative 99.5%

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

      8. sqr-neg 99.5%

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

      9. +-commutative 99.5%

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

      10. sqr-neg 99.5%

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

      11. fma-define 99.5%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in z around 0 98.8%

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

      1. associate-/r* 99.1%

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

   7. Simplified 99.1%

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

   if 0.20000000000000001 < (*.f64 z z)

   1. Initial program 80.6%

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

      1. remove-double-neg 80.6%

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

      2. distribute-lft-neg-out 80.6%

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

      3. distribute-rgt-neg-in 80.6%

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

      4. associate-/r* 82.7%

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

      5. associate-/l/ 82.8%

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

      6. associate-/l/ 82.4%

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

      7. distribute-lft-neg-out 82.4%

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

      8. distribute-rgt-neg-in 82.4%

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

      9. distribute-lft-neg-in 82.4%

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

      10. remove-double-neg 82.4%

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

      11. sqr-neg 82.4%

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

      12. +-commutative 82.4%

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

      13. sqr-neg 82.4%

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

      14. fma-define 82.4%

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

      15. *-commutative 82.4%

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

   3. Simplified 82.4%

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

   5. Taylor expanded in z around inf 81.8%

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

      1. *-un-lft-identity 81.8%

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

      2. associate-/r* 82.2%

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

      3. *-commutative 82.2%

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

      4. associate-/r* 81.5%

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

      5. pow-flip 81.9%

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

      6. metadata-eval 81.9%

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

   7. Applied egg-rr 81.9%

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

      1. *-lft-identity 81.9%

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

      2. associate-/l/ 82.6%

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

   9. Simplified 82.6%

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

       1. sqr-pow 82.5%

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

       2. associate-/l* 91.2%

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

       3. metadata-eval 91.2%

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

       4. unpow-1 91.2%

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

       5. metadata-eval 91.2%

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

       6. unpow-1 91.2%

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

   11. Applied egg-rr 91.2%

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

       1. associate-*l/ 91.3%

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

       2. *-un-lft-identity 91.3%

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

       3. clear-num 91.3%

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

       4. *-commutative 91.3%

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

       5. associate-*r/ 97.7%

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

       6. associate-/r/ 97.7%

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

       7. /-rgt-identity 97.7%

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

       8. *-commutative 97.7%

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

   13. Applied egg-rr 97.7%

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

Alternative 6: 98.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;z \cdot z \leq 0.2:\\ \;\;\;\;\frac{\frac{1}{x\_m}}{y\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{z}}{y\_m \cdot \left(z \cdot x\_m\right)}\\ \end{array}\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (y_s x_s x_m y_m z)
 :precision binary64
 (*
  y_s
  (*
   x_s
   (if (<= (* z z) 0.2) (/ (/ 1.0 x_m) y_m) (/ (/ 1.0 z) (* y_m (* z x_m)))))))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 0.20000000000000001

   1. Initial program 99.7%

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

      1. associate-/l/ 99.5%

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

      2. remove-double-neg 99.5%

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

      3. distribute-rgt-neg-out 99.5%

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

      4. distribute-rgt-neg-out 99.5%

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

      5. remove-double-neg 99.5%

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

      6. associate-*l* 99.5%

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

      7. *-commutative 99.5%

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

      8. sqr-neg 99.5%

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

      9. +-commutative 99.5%

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

      10. sqr-neg 99.5%

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

      11. fma-define 99.5%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in z around 0 98.8%

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

      1. associate-/r* 99.1%

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

   7. Simplified 99.1%

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

   if 0.20000000000000001 < (*.f64 z z)

   1. Initial program 80.6%

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

      1. remove-double-neg 80.6%

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

      2. distribute-lft-neg-out 80.6%

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

      3. distribute-rgt-neg-in 80.6%

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

      4. associate-/r* 82.7%

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

      5. associate-/l/ 82.8%

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

      6. associate-/l/ 82.4%

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

      7. distribute-lft-neg-out 82.4%

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

      8. distribute-rgt-neg-in 82.4%

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

      9. distribute-lft-neg-in 82.4%

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

      10. remove-double-neg 82.4%

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

      11. sqr-neg 82.4%

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

      12. +-commutative 82.4%

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

      13. sqr-neg 82.4%

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

      14. fma-define 82.4%

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

      15. *-commutative 82.4%

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

   3. Simplified 82.4%

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

   5. Taylor expanded in z around inf 81.8%

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

      1. *-un-lft-identity 81.8%

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

      2. associate-/r* 82.2%

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

      3. *-commutative 82.2%

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

      4. associate-/r* 81.5%

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

      5. pow-flip 81.9%

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

      6. metadata-eval 81.9%

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

   7. Applied egg-rr 81.9%

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

      1. *-lft-identity 81.9%

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

      2. associate-/l/ 82.6%

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

   9. Simplified 82.6%

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

       1. sqr-pow 82.5%

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

       2. associate-/l* 91.2%

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

       3. metadata-eval 91.2%

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

       4. unpow-1 91.2%

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

       5. metadata-eval 91.2%

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

       6. unpow-1 91.2%

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

   11. Applied egg-rr 91.2%

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

       1. associate-/l/ 91.3%

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

       2. un-div-inv 91.2%

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

       3. remove-double-div 91.2%

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

       4. div-inv 91.2%

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

       5. *-commutative 91.2%

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

       6. associate-*l/ 98.1%

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

       7. associate-/r/ 98.2%

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

       8. /-rgt-identity 98.2%

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

       9. associate-*l* 97.6%

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

   13. Applied egg-rr 97.6%

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

Alternative 7: 97.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;z \cdot z \leq 0.2:\\ \;\;\;\;\frac{\frac{1}{x\_m}}{y\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y\_m \cdot \left(z \cdot \left(z \cdot x\_m\right)\right)}\\ \end{array}\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (y_s x_s x_m y_m z)
 :precision binary64
 (*
  y_s
  (*
   x_s
   (if (<= (* z z) 0.2) (/ (/ 1.0 x_m) y_m) (/ 1.0 (* y_m (* z (* z x_m))))))))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 0.20000000000000001

   1. Initial program 99.7%

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

      1. associate-/l/ 99.5%

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

      2. remove-double-neg 99.5%

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

      3. distribute-rgt-neg-out 99.5%

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

      4. distribute-rgt-neg-out 99.5%

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

      5. remove-double-neg 99.5%

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

      6. associate-*l* 99.5%

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

      7. *-commutative 99.5%

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

      8. sqr-neg 99.5%

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

      9. +-commutative 99.5%

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

      10. sqr-neg 99.5%

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

      11. fma-define 99.5%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in z around 0 98.8%

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

      1. associate-/r* 99.1%

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

   7. Simplified 99.1%

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

   if 0.20000000000000001 < (*.f64 z z)

   1. Initial program 80.6%

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

      1. remove-double-neg 80.6%

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

      2. distribute-lft-neg-out 80.6%

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

      3. distribute-rgt-neg-in 80.6%

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

      4. associate-/r* 82.7%

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

      5. associate-/l/ 82.8%

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

      6. associate-/l/ 82.4%

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

      7. distribute-lft-neg-out 82.4%

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

      8. distribute-rgt-neg-in 82.4%

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

      9. distribute-lft-neg-in 82.4%

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

      10. remove-double-neg 82.4%

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

      11. sqr-neg 82.4%

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

      12. +-commutative 82.4%

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

      13. sqr-neg 82.4%

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

      14. fma-define 82.4%

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

      15. *-commutative 82.4%

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

   3. Simplified 82.4%

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

   5. Taylor expanded in z around inf 81.8%

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

      1. add-sqr-sqrt 38.4%

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

      2. pow2 38.4%

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

      3. *-commutative 38.4%

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

      4. sqrt-prod 38.4%

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

      5. sqrt-pow1 43.6%

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

      6. metadata-eval 43.6%

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

      7. pow1 43.6%

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

   7. Applied egg-rr 43.6%

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

      1. /-rgt-identity 43.6%

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

      2. clear-num 43.6%

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

      3. inv-pow 43.6%

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

      4. unpow2 43.6%

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

      5. swap-sqr 38.3%

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

      6. pow2 38.3%

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

      7. add-sqr-sqrt 81.8%

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

      8. *-commutative 81.8%

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

      9. unpow-prod-down 81.8%

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

      10. pow2 81.8%

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

      11. pow-prod-down 82.1%

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

      12. pow-prod-up 82.2%

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

      13. metadata-eval 82.2%

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

      14. inv-pow 82.2%

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

      15. div-inv 82.2%

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

      16. clear-num 82.3%

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

      17. *-commutative 82.3%

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

      18. sqr-pow 82.2%

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

      19. times-frac 96.5%

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

      20. metadata-eval 96.5%

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

      21. unpow-1 96.5%

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

   9. Applied egg-rr 96.5%

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

       1. clear-num 96.4%

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

       2. frac-times 87.4%

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

       3. *-un-lft-identity 87.4%

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

       4. associate-/l/ 87.4%

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

       5. *-commutative 87.4%

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

   11. Applied egg-rr 87.4%

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

       1. associate-/r* 98.0%

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

       2. associate-/r/ 98.1%

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

       3. /-rgt-identity 98.1%

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

       4. *-commutative 98.1%

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

       5. *-commutative 98.1%

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

       6. associate-*r* 97.3%

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

       7. associate-/l* 88.9%

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

       8. associate-/r/ 88.9%

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

       9. /-rgt-identity 88.9%

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

       10. *-commutative 88.9%

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

   13. Simplified 88.9%

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

4. Final simplification 94.2%

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

Alternative 8: 58.8% 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 #s(literal 1 binary64) x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (y_s x_s x_m y_m z)
 :precision binary64
 (* y_s (* x_s (/ (/ 1.0 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. Step-by-step derivation

   1. associate-/l/ 90.3%

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

   2. remove-double-neg 90.3%

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

   3. distribute-rgt-neg-out 90.3%

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

   4. distribute-rgt-neg-out 90.3%

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

   5. remove-double-neg 90.3%

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

   6. associate-*l* 89.6%

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

   7. *-commutative 89.6%

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

   8. sqr-neg 89.6%

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

   9. +-commutative 89.6%

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

   10. sqr-neg 89.6%

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

   11. fma-define 89.6%

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

3. Simplified 89.6%

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

5. Taylor expanded in z around 0 60.2%

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

   1. associate-/r* 60.3%

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

7. Simplified 60.3%

   \[\leadsto \color{blue}{\frac{\frac{1}{x}}{y}} \]
8. Add Preprocessing

Alternative 9: 58.9% 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 #s(literal 1 binary64) x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (y_s x_s x_m y_m z)
 :precision binary64
 (* y_s (* x_s (/ 1.0 (* 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/ 90.3%

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

   2. remove-double-neg 90.3%

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

   3. distribute-rgt-neg-out 90.3%

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

   4. distribute-rgt-neg-out 90.3%

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

   5. remove-double-neg 90.3%

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

   6. associate-*l* 89.6%

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

   7. *-commutative 89.6%

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

   8. sqr-neg 89.6%

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

   9. +-commutative 89.6%

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

   10. sqr-neg 89.6%

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

   11. fma-define 89.6%

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

3. Simplified 89.6%

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

5. Taylor expanded in z around 0 60.2%

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

6. Final simplification 60.2%

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

Developer Target 1: 93.3% 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 2024185 
(FPCore (x y z)
  :name "Statistics.Distribution.CauchyLorentz:$cdensity from math-functions-0.1.5.2"
  :precision binary64

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

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