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

Percentage Accurate: 89.0% → 99.4%

Time: 16.2s

Alternatives: 13

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 89.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.4% accurate, 0.1× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ \begin{array}{l} t_0 := z \cdot \sqrt{x\_m}\\ y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{+267}:\\ \;\;\;\;\frac{\frac{\frac{1}{x\_m}}{\mathsf{fma}\left(z, z, 1\right)}}{y\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{t\_0 \cdot \left(y\_m \cdot t\_0\right)}\\ \end{array}\right) \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (y_s x_s x_m y_m z)
 :precision binary64
 (let* ((t_0 (* z (sqrt x_m))))
   (*
    y_s
    (*
     x_s
     (if (<= (* z z) 2e+267)
       (/ (/ (/ 1.0 x_m) (fma z z 1.0)) y_m)
       (/ 1.0 (* t_0 (* y_m t_0))))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
assert(x_m < y_m && y_m < z);
double code(double y_s, double x_s, double x_m, double y_m, double z) {
	double t_0 = z * sqrt(x_m);
	double tmp;
	if ((z * z) <= 2e+267) {
		tmp = ((1.0 / x_m) / fma(z, z, 1.0)) / y_m;
	} else {
		tmp = 1.0 / (t_0 * (y_m * t_0));
	}
	return y_s * (x_s * tmp);
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
y\_m = abs(y)
y\_s = copysign(1.0, y)
x_m, y_m, z = sort([x_m, y_m, z])
function code(y_s, x_s, x_m, y_m, z)
	t_0 = Float64(z * sqrt(x_m))
	tmp = 0.0
	if (Float64(z * z) <= 2e+267)
		tmp = Float64(Float64(Float64(1.0 / x_m) / fma(z, z, 1.0)) / y_m);
	else
		tmp = Float64(1.0 / Float64(t_0 * Float64(y_m * t_0)));
	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_] := Block[{t$95$0 = N[(z * N[Sqrt[x$95$m], $MachinePrecision]), $MachinePrecision]}, N[(y$95$s * N[(x$95$s * If[LessEqual[N[(z * z), $MachinePrecision], 2e+267], N[(N[(N[(1.0 / x$95$m), $MachinePrecision] / N[(z * z + 1.0), $MachinePrecision]), $MachinePrecision] / y$95$m), $MachinePrecision], N[(1.0 / N[(t$95$0 * N[(y$95$m * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 1.9999999999999999e267

   1. Initial program 97.6%

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

      1. *-un-lft-identity 97.6%

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

      2. add-sqr-sqrt 52.2%

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

      3. times-frac 52.2%

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

      4. +-commutative 52.2%

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

      5. fma-undefine 52.2%

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

      6. *-commutative 52.2%

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

      7. sqrt-prod 52.2%

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

      8. fma-undefine 52.2%

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

      9. +-commutative 52.2%

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

      10. hypot-1-def 52.2%

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

      11. +-commutative 52.2%

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

      12. fma-undefine 52.2%

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

      13. *-commutative 52.2%

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

      14. sqrt-prod 53.7%

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

      15. fma-undefine 53.7%

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

      16. +-commutative 53.7%

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

      17. hypot-1-def 53.7%

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

   4. Applied egg-rr 53.7%

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

      1. associate-/r* 53.8%

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

      2. associate-/r* 53.2%

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

      3. frac-times 52.7%

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

      4. associate-/l/ 52.7%

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

      5. add-sqr-sqrt 97.0%

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

   6. Applied egg-rr 97.0%

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

   7. Taylor expanded in x around 0 97.0%

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

      1. associate-/r* 97.0%

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

      2. +-commutative 97.0%

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

      3. unpow2 97.0%

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

      4. fma-undefine 97.0%

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

   9. Simplified 97.0%

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

   if 1.9999999999999999e267 < (*.f64 z z)

   1. Initial program 82.3%

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

      1. remove-double-neg 82.3%

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

      2. distribute-lft-neg-out 82.3%

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

      3. distribute-rgt-neg-in 82.3%

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

      4. associate-/r* 81.8%

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

      5. associate-/l/ 81.8%

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

      6. associate-/l/ 81.9%

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

      7. distribute-lft-neg-out 81.9%

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

      8. distribute-rgt-neg-in 81.9%

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

      9. distribute-lft-neg-in 81.9%

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

      10. remove-double-neg 81.9%

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

      11. sqr-neg 81.9%

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

      12. +-commutative 81.9%

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

      13. sqr-neg 81.9%

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

      14. fma-define 81.9%

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

      15. *-commutative 81.9%

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

   3. Simplified 81.9%

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

   5. Taylor expanded in z around inf 82.3%

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

      1. *-commutative 82.3%

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

   7. Simplified 82.3%

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

      1. add-sqr-sqrt 44.8%

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

      2. pow2 44.8%

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

      3. sqrt-prod 18.9%

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

      4. sqrt-prod 18.9%

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

      5. sqrt-pow1 22.6%

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

      6. metadata-eval 22.6%

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

      7. pow1 22.6%

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

   9. Applied egg-rr 22.6%

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

       1. unpow2 22.6%

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

       2. swap-sqr 20.1%

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

       3. add-sqr-sqrt 40.2%

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

       4. swap-sqr 37.6%

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

       5. unpow2 37.6%

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

       6. add-sqr-sqrt 82.3%

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

       7. associate-*l* 79.8%

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

       8. add-sqr-sqrt 37.6%

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

       9. associate-*l* 37.6%

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

       10. *-commutative 37.6%

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

       11. sqrt-prod 37.6%

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

       12. sqrt-pow1 37.6%

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

       13. metadata-eval 37.6%

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

       14. pow1 37.6%

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

       15. *-commutative 37.6%

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

       16. sqrt-prod 37.6%

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

       17. sqrt-pow1 43.9%

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

       18. metadata-eval 43.9%

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

       19. pow1 43.9%

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

   11. Applied egg-rr 43.9%

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

4. Final simplification 81.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{+267}:\\ \;\;\;\;\frac{\frac{\frac{1}{x}}{\mathsf{fma}\left(z, z, 1\right)}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(z \cdot \sqrt{x}\right) \cdot \left(y \cdot \left(z \cdot \sqrt{x}\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.3% accurate, 0.0× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ \begin{array}{l} t_0 := \mathsf{hypot}\left(1, z\right) \cdot \sqrt{y\_m}\\ y\_s \cdot \left(x\_s \cdot \left(\frac{1}{t\_0} \cdot \frac{\frac{1}{x\_m}}{t\_0}\right)\right) \end{array} \end{array} \]

FPCore

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

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
assert(x_m < y_m && y_m < z);
double code(double y_s, double x_s, double x_m, double y_m, double z) {
	double t_0 = hypot(1.0, z) * sqrt(y_m);
	return y_s * (x_s * ((1.0 / t_0) * ((1.0 / x_m) / t_0)));
}

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
assert x_m < y_m && y_m < z;
public static double code(double y_s, double x_s, double x_m, double y_m, double z) {
	double t_0 = Math.hypot(1.0, z) * Math.sqrt(y_m);
	return y_s * (x_s * ((1.0 / t_0) * ((1.0 / x_m) / t_0)));
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
[x_m, y_m, z] = sort([x_m, y_m, z])
def code(y_s, x_s, x_m, y_m, z):
	t_0 = math.hypot(1.0, z) * math.sqrt(y_m)
	return y_s * (x_s * ((1.0 / t_0) * ((1.0 / x_m) / t_0)))

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
y\_m = abs(y)
y\_s = copysign(1.0, y)
x_m, y_m, z = sort([x_m, y_m, z])
function code(y_s, x_s, x_m, y_m, z)
	t_0 = Float64(hypot(1.0, z) * sqrt(y_m))
	return Float64(y_s * Float64(x_s * Float64(Float64(1.0 / t_0) * Float64(Float64(1.0 / x_m) / t_0))))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
x_m, y_m, z = num2cell(sort([x_m, y_m, z])){:}
function tmp = code(y_s, x_s, x_m, y_m, z)
	t_0 = hypot(1.0, z) * sqrt(y_m);
	tmp = y_s * (x_s * ((1.0 / t_0) * ((1.0 / x_m) / t_0)));
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
code[y$95$s_, x$95$s_, x$95$m_, y$95$m_, z_] := Block[{t$95$0 = N[(N[Sqrt[1.0 ^ 2 + z ^ 2], $MachinePrecision] * N[Sqrt[y$95$m], $MachinePrecision]), $MachinePrecision]}, N[(y$95$s * N[(x$95$s * N[(N[(1.0 / t$95$0), $MachinePrecision] * N[(N[(1.0 / x$95$m), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 93.1%

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

   1. *-un-lft-identity 93.1%

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

   2. add-sqr-sqrt 47.9%

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

   3. times-frac 47.9%

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

   4. +-commutative 47.9%

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

   5. fma-undefine 47.9%

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

   6. *-commutative 47.9%

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

   7. sqrt-prod 47.9%

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

   8. fma-undefine 47.9%

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

   9. +-commutative 47.9%

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

   10. hypot-1-def 47.9%

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

   11. +-commutative 47.9%

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

   12. fma-undefine 47.9%

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

   13. *-commutative 47.9%

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

   14. sqrt-prod 49.0%

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

   15. fma-undefine 49.0%

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

   16. +-commutative 49.0%

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

   17. hypot-1-def 50.5%

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

4. Applied egg-rr 50.5%

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

Alternative 3: 99.2% accurate, 0.0× speedup

Math

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

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 (pow (/ (pow x_m -0.5) (* (hypot 1.0 z) (sqrt y_m))) 2.0))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
assert(x_m < y_m && y_m < z);
double code(double y_s, double x_s, double x_m, double y_m, double z) {
	return y_s * (x_s * pow((pow(x_m, -0.5) / (hypot(1.0, z) * sqrt(y_m))), 2.0));
}

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
assert x_m < y_m && y_m < z;
public static double code(double y_s, double x_s, double x_m, double y_m, double z) {
	return y_s * (x_s * Math.pow((Math.pow(x_m, -0.5) / (Math.hypot(1.0, z) * Math.sqrt(y_m))), 2.0));
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
[x_m, y_m, z] = sort([x_m, y_m, z])
def code(y_s, x_s, x_m, y_m, z):
	return y_s * (x_s * math.pow((math.pow(x_m, -0.5) / (math.hypot(1.0, z) * math.sqrt(y_m))), 2.0))

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
y\_m = abs(y)
y\_s = copysign(1.0, y)
x_m, y_m, z = sort([x_m, y_m, z])
function code(y_s, x_s, x_m, y_m, z)
	return Float64(y_s * Float64(x_s * (Float64((x_m ^ -0.5) / Float64(hypot(1.0, z) * sqrt(y_m))) ^ 2.0)))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
x_m, y_m, z = num2cell(sort([x_m, y_m, z])){:}
function tmp = code(y_s, x_s, x_m, y_m, z)
	tmp = y_s * (x_s * (((x_m ^ -0.5) / (hypot(1.0, z) * sqrt(y_m))) ^ 2.0));
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
code[y$95$s_, x$95$s_, x$95$m_, y$95$m_, z_] := N[(y$95$s * N[(x$95$s * N[Power[N[(N[Power[x$95$m, -0.5], $MachinePrecision] / N[(N[Sqrt[1.0 ^ 2 + z ^ 2], $MachinePrecision] * N[Sqrt[y$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 93.1%

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

   1. add-sqr-sqrt 49.0%

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

   2. add-sqr-sqrt 27.1%

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

   3. times-frac 27.2%

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

   4. inv-pow 27.2%

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

   5. sqrt-pow1 27.2%

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

   6. metadata-eval 27.2%

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

   7. +-commutative 27.2%

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

   8. fma-undefine 27.2%

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

   9. *-commutative 27.2%

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

   10. sqrt-prod 27.2%

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

   11. fma-undefine 27.2%

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

   12. +-commutative 27.2%

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

   13. hypot-1-def 27.2%

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

   14. inv-pow 27.2%

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

   15. sqrt-pow1 27.2%

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

   16. metadata-eval 27.2%

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

   17. +-commutative 27.2%

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

   18. fma-undefine 27.2%

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

4. Applied egg-rr 28.3%

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

   1. unpow2 28.3%

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

6. Simplified 28.3%

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

Alternative 4: 98.7% accurate, 0.1× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{+251}:\\ \;\;\;\;\frac{\frac{\frac{1}{x\_m}}{\mathsf{fma}\left(z, z, 1\right)}}{y\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(z \cdot \sqrt{x\_m}\right)}^{-2}}{y\_m}\\ \end{array}\right) \end{array} \]

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) 2e+251)
     (/ (/ (/ 1.0 x_m) (fma z z 1.0)) y_m)
     (/ (pow (* z (sqrt x_m)) -2.0) y_m)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
assert(x_m < y_m && y_m < z);
double code(double y_s, double x_s, double x_m, double y_m, double z) {
	double tmp;
	if ((z * z) <= 2e+251) {
		tmp = ((1.0 / x_m) / fma(z, z, 1.0)) / y_m;
	} else {
		tmp = pow((z * sqrt(x_m)), -2.0) / y_m;
	}
	return y_s * (x_s * tmp);
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
y\_m = abs(y)
y\_s = copysign(1.0, y)
x_m, y_m, z = sort([x_m, y_m, z])
function code(y_s, x_s, x_m, y_m, z)
	tmp = 0.0
	if (Float64(z * z) <= 2e+251)
		tmp = Float64(Float64(Float64(1.0 / x_m) / fma(z, z, 1.0)) / y_m);
	else
		tmp = Float64((Float64(z * sqrt(x_m)) ^ -2.0) / y_m);
	end
	return Float64(y_s * Float64(x_s * tmp))
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
code[y$95$s_, x$95$s_, x$95$m_, y$95$m_, z_] := N[(y$95$s * N[(x$95$s * If[LessEqual[N[(z * z), $MachinePrecision], 2e+251], N[(N[(N[(1.0 / x$95$m), $MachinePrecision] / N[(z * z + 1.0), $MachinePrecision]), $MachinePrecision] / y$95$m), $MachinePrecision], N[(N[Power[N[(z * N[Sqrt[x$95$m], $MachinePrecision]), $MachinePrecision], -2.0], $MachinePrecision] / y$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 2.0000000000000001e251

   1. Initial program 98.0%

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

      1. *-un-lft-identity 98.0%

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

      2. add-sqr-sqrt 52.0%

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

      3. times-frac 52.0%

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

      4. +-commutative 52.0%

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

      5. fma-undefine 52.0%

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

      6. *-commutative 52.0%

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

      7. sqrt-prod 52.0%

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

      8. fma-undefine 52.0%

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

      9. +-commutative 52.0%

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

      10. hypot-1-def 52.0%

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

      11. +-commutative 52.0%

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

      12. fma-undefine 52.0%

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

      13. *-commutative 52.0%

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

      14. sqrt-prod 53.6%

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

      15. fma-undefine 53.6%

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

      16. +-commutative 53.6%

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

      17. hypot-1-def 53.6%

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

   4. Applied egg-rr 53.6%

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

      1. associate-/r* 53.6%

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

      2. associate-/r* 53.0%

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

      3. frac-times 52.5%

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

      4. associate-/l/ 52.5%

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

      5. add-sqr-sqrt 96.9%

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

   6. Applied egg-rr 96.9%

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

   7. Taylor expanded in x around 0 96.9%

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

      1. associate-/r* 96.9%

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

      2. +-commutative 96.9%

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

      3. unpow2 96.9%

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

      4. fma-undefine 96.9%

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

   9. Simplified 96.9%

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

   if 2.0000000000000001e251 < (*.f64 z z)

   1. Initial program 82.3%

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

   3. Taylor expanded in x around 0 82.3%

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

      1. associate-/r* 82.3%

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

      2. *-commutative 82.3%

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

      3. +-commutative 82.3%

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

      4. unpow2 82.3%

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

      5. fma-undefine 82.3%

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

      6. associate-/r* 82.3%

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

      7. associate-*r* 81.1%

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

      8. associate-/l/ 81.1%

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

   5. Simplified 81.1%

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

   6. Taylor expanded in z around inf 81.1%

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

      1. add-sqr-sqrt 39.0%

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

      2. pow2 39.0%

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

      3. *-commutative 39.0%

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

      4. sqrt-prod 39.0%

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

      5. sqrt-pow1 43.7%

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

      6. metadata-eval 43.7%

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

      7. pow1 43.7%

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

   8. Applied egg-rr 43.7%

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

      1. *-un-lft-identity 43.7%

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

      2. associate-/l/ 43.7%

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

      3. associate-/r* 43.7%

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

      4. pow-flip 44.0%

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

      5. metadata-eval 44.0%

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

   10. Applied egg-rr 44.0%

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

       1. *-lft-identity 44.0%

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

   12. Simplified 44.0%

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

Alternative 5: 98.7% accurate, 0.1× speedup

Math

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

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)) x_m) (hypot 1.0 z)) y_m))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
assert(x_m < y_m && y_m < z);
double code(double y_s, double x_s, double x_m, double y_m, double z) {
	return y_s * (x_s * ((((1.0 / hypot(1.0, z)) / x_m) / hypot(1.0, z)) / 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)) / x_m) / Math.hypot(1.0, z)) / 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)) / x_m) / math.hypot(1.0, z)) / 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(Float64(Float64(1.0 / hypot(1.0, z)) / x_m) / hypot(1.0, z)) / 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)) / x_m) / hypot(1.0, z)) / 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[(N[(N[(1.0 / N[Sqrt[1.0 ^ 2 + z ^ 2], $MachinePrecision]), $MachinePrecision] / x$95$m), $MachinePrecision] / N[Sqrt[1.0 ^ 2 + z ^ 2], $MachinePrecision]), $MachinePrecision] / y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 93.1%

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

   1. *-un-lft-identity 93.1%

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

   2. add-sqr-sqrt 47.9%

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

   3. times-frac 47.9%

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

   4. +-commutative 47.9%

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

   5. fma-undefine 47.9%

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

   6. *-commutative 47.9%

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

   7. sqrt-prod 47.9%

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

   8. fma-undefine 47.9%

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

   9. +-commutative 47.9%

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

   10. hypot-1-def 47.9%

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

   11. +-commutative 47.9%

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

   12. fma-undefine 47.9%

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

   13. *-commutative 47.9%

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

   14. sqrt-prod 49.0%

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

   15. fma-undefine 49.0%

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

   16. +-commutative 49.0%

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

   17. hypot-1-def 50.5%

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

4. Applied egg-rr 50.5%

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

   1. associate-/r* 50.5%

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

   2. associate-/r* 50.1%

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

   3. frac-times 49.1%

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

   4. associate-/l/ 49.1%

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

   5. add-sqr-sqrt 94.3%

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

6. Applied egg-rr 94.3%

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

   1. un-div-inv 94.3%

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

   2. associate-/l/ 94.3%

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

8. Applied egg-rr 94.3%

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

Alternative 6: 92.8% accurate, 0.1× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ \begin{array}{l} t_0 := y\_m \cdot \left(1 + z \cdot z\right)\\ y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq 10^{+308}:\\ \;\;\;\;\frac{\frac{1}{x\_m}}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{{z}^{2} \cdot \left(y\_m \cdot x\_m\right)}\\ \end{array}\right) \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (y_s x_s x_m y_m z)
 :precision binary64
 (let* ((t_0 (* y_m (+ 1.0 (* z z)))))
   (*
    y_s
    (*
     x_s
     (if (<= t_0 1e+308)
       (/ (/ 1.0 x_m) t_0)
       (/ 1.0 (* (pow z 2.0) (* y_m x_m))))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
assert(x_m < y_m && y_m < z);
double code(double y_s, double x_s, double x_m, double y_m, double z) {
	double t_0 = y_m * (1.0 + (z * z));
	double tmp;
	if (t_0 <= 1e+308) {
		tmp = (1.0 / x_m) / t_0;
	} else {
		tmp = 1.0 / (pow(z, 2.0) * (y_m * x_m));
	}
	return y_s * (x_s * tmp);
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
y\_m = abs(y)
y\_s = copysign(1.0d0, y)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
real(8) function code(y_s, x_s, x_m, y_m, z)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = y_m * (1.0d0 + (z * z))
    if (t_0 <= 1d+308) then
        tmp = (1.0d0 / x_m) / t_0
    else
        tmp = 1.0d0 / ((z ** 2.0d0) * (y_m * x_m))
    end if
    code = y_s * (x_s * tmp)
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
assert x_m < y_m && y_m < z;
public static double code(double y_s, double x_s, double x_m, double y_m, double z) {
	double t_0 = y_m * (1.0 + (z * z));
	double tmp;
	if (t_0 <= 1e+308) {
		tmp = (1.0 / x_m) / t_0;
	} else {
		tmp = 1.0 / (Math.pow(z, 2.0) * (y_m * x_m));
	}
	return y_s * (x_s * tmp);
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
[x_m, y_m, z] = sort([x_m, y_m, z])
def code(y_s, x_s, x_m, y_m, z):
	t_0 = y_m * (1.0 + (z * z))
	tmp = 0
	if t_0 <= 1e+308:
		tmp = (1.0 / x_m) / t_0
	else:
		tmp = 1.0 / (math.pow(z, 2.0) * (y_m * x_m))
	return y_s * (x_s * tmp)

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
y\_m = abs(y)
y\_s = copysign(1.0, y)
x_m, y_m, z = sort([x_m, y_m, z])
function code(y_s, x_s, x_m, y_m, z)
	t_0 = Float64(y_m * Float64(1.0 + Float64(z * z)))
	tmp = 0.0
	if (t_0 <= 1e+308)
		tmp = Float64(Float64(1.0 / x_m) / t_0);
	else
		tmp = Float64(1.0 / Float64((z ^ 2.0) * Float64(y_m * x_m)));
	end
	return Float64(y_s * Float64(x_s * tmp))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
x_m, y_m, z = num2cell(sort([x_m, y_m, z])){:}
function tmp_2 = code(y_s, x_s, x_m, y_m, z)
	t_0 = y_m * (1.0 + (z * z));
	tmp = 0.0;
	if (t_0 <= 1e+308)
		tmp = (1.0 / x_m) / t_0;
	else
		tmp = 1.0 / ((z ^ 2.0) * (y_m * x_m));
	end
	tmp_2 = y_s * (x_s * tmp);
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
code[y$95$s_, x$95$s_, x$95$m_, y$95$m_, z_] := Block[{t$95$0 = N[(y$95$m * N[(1.0 + N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(y$95$s * N[(x$95$s * If[LessEqual[t$95$0, 1e+308], N[(N[(1.0 / x$95$m), $MachinePrecision] / t$95$0), $MachinePrecision], N[(1.0 / N[(N[Power[z, 2.0], $MachinePrecision] * N[(y$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y (+.f64 #s(literal 1 binary64) (*.f64 z z))) < 1e308

   1. Initial program 94.9%

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

   if 1e308 < (*.f64 y (+.f64 #s(literal 1 binary64) (*.f64 z z)))

   1. Initial program 83.8%

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

      1. remove-double-neg 83.8%

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

      2. distribute-lft-neg-out 83.8%

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

      3. distribute-rgt-neg-in 83.8%

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

      4. associate-/r* 90.5%

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

      5. associate-/l/ 90.5%

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

      6. associate-/l/ 90.7%

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

      7. distribute-lft-neg-out 90.7%

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

      8. distribute-rgt-neg-in 90.7%

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

      9. distribute-lft-neg-in 90.7%

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

      10. remove-double-neg 90.7%

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

      11. sqr-neg 90.7%

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

      12. +-commutative 90.7%

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

      13. sqr-neg 90.7%

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

      14. fma-define 90.7%

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

      15. *-commutative 90.7%

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

   3. Simplified 90.7%

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

   5. Taylor expanded in z around inf 90.7%

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

4. Final simplification 94.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(1 + z \cdot z\right) \leq 10^{+308}:\\ \;\;\;\;\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{{z}^{2} \cdot \left(y \cdot x\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 93.0% accurate, 0.1× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;z \cdot z \leq 5 \cdot 10^{+15}:\\ \;\;\;\;\frac{\frac{1}{x\_m}}{y\_m \cdot \left(1 + z \cdot z\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{1}{x\_m}}{{z}^{2}}}{y\_m}\\ \end{array}\right) \end{array} \]

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+15)
     (/ (/ 1.0 x_m) (* y_m (+ 1.0 (* z z))))
     (/ (/ (/ 1.0 x_m) (pow z 2.0)) y_m)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
assert(x_m < y_m && y_m < z);
double code(double y_s, double x_s, double x_m, double y_m, double z) {
	double tmp;
	if ((z * z) <= 5e+15) {
		tmp = (1.0 / x_m) / (y_m * (1.0 + (z * z)));
	} else {
		tmp = ((1.0 / x_m) / pow(z, 2.0)) / y_m;
	}
	return y_s * (x_s * tmp);
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
y\_m = abs(y)
y\_s = copysign(1.0d0, y)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
real(8) function code(y_s, x_s, x_m, y_m, z)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((z * z) <= 5d+15) then
        tmp = (1.0d0 / x_m) / (y_m * (1.0d0 + (z * z)))
    else
        tmp = ((1.0d0 / x_m) / (z ** 2.0d0)) / y_m
    end if
    code = y_s * (x_s * tmp)
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
assert x_m < y_m && y_m < z;
public static double code(double y_s, double x_s, double x_m, double y_m, double z) {
	double tmp;
	if ((z * z) <= 5e+15) {
		tmp = (1.0 / x_m) / (y_m * (1.0 + (z * z)));
	} else {
		tmp = ((1.0 / x_m) / Math.pow(z, 2.0)) / y_m;
	}
	return y_s * (x_s * tmp);
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
[x_m, y_m, z] = sort([x_m, y_m, z])
def code(y_s, x_s, x_m, y_m, z):
	tmp = 0
	if (z * z) <= 5e+15:
		tmp = (1.0 / x_m) / (y_m * (1.0 + (z * z)))
	else:
		tmp = ((1.0 / x_m) / math.pow(z, 2.0)) / y_m
	return y_s * (x_s * tmp)

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
y\_m = abs(y)
y\_s = copysign(1.0, y)
x_m, y_m, z = sort([x_m, y_m, z])
function code(y_s, x_s, x_m, y_m, z)
	tmp = 0.0
	if (Float64(z * z) <= 5e+15)
		tmp = Float64(Float64(1.0 / x_m) / Float64(y_m * Float64(1.0 + Float64(z * z))));
	else
		tmp = Float64(Float64(Float64(1.0 / x_m) / (z ^ 2.0)) / y_m);
	end
	return Float64(y_s * Float64(x_s * tmp))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
x_m, y_m, z = num2cell(sort([x_m, y_m, z])){:}
function tmp_2 = code(y_s, x_s, x_m, y_m, z)
	tmp = 0.0;
	if ((z * z) <= 5e+15)
		tmp = (1.0 / x_m) / (y_m * (1.0 + (z * z)));
	else
		tmp = ((1.0 / x_m) / (z ^ 2.0)) / y_m;
	end
	tmp_2 = y_s * (x_s * tmp);
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
code[y$95$s_, x$95$s_, x$95$m_, y$95$m_, z_] := N[(y$95$s * N[(x$95$s * If[LessEqual[N[(z * z), $MachinePrecision], 5e+15], N[(N[(1.0 / x$95$m), $MachinePrecision] / N[(y$95$m * N[(1.0 + N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 / x$95$m), $MachinePrecision] / N[Power[z, 2.0], $MachinePrecision]), $MachinePrecision] / y$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 5e15

   1. Initial program 99.7%

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

   if 5e15 < (*.f64 z z)

   1. Initial program 85.8%

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

      1. *-un-lft-identity 85.8%

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

      2. add-sqr-sqrt 40.6%

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

      3. times-frac 40.7%

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

      4. +-commutative 40.7%

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

      5. fma-undefine 40.7%

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

      6. *-commutative 40.7%

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

      7. sqrt-prod 40.7%

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

      8. fma-undefine 40.7%

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

      9. +-commutative 40.7%

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

      10. hypot-1-def 40.7%

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

      11. +-commutative 40.7%

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

      12. fma-undefine 40.7%

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

      13. *-commutative 40.7%

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

      14. sqrt-prod 43.0%

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

      15. fma-undefine 43.0%

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

      16. +-commutative 43.0%

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

      17. hypot-1-def 46.1%

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

   4. Applied egg-rr 46.1%

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

      1. associate-/r* 46.2%

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

      2. associate-/r* 45.2%

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

      3. frac-times 43.1%

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

      4. associate-/l/ 43.1%

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

      5. add-sqr-sqrt 88.2%

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

   6. Applied egg-rr 88.2%

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

   7. Taylor expanded in z around inf 83.3%

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

      1. associate-/r* 83.4%

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

   9. Simplified 83.4%

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

Alternative 8: 93.0% accurate, 0.1× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;z \cdot z \leq 5 \cdot 10^{+15}:\\ \;\;\;\;\frac{\frac{1}{x\_m}}{y\_m \cdot \left(1 + z \cdot z\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{y\_m}}{x\_m \cdot {z}^{2}}\\ \end{array}\right) \end{array} \]

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+15)
     (/ (/ 1.0 x_m) (* y_m (+ 1.0 (* z z))))
     (/ (/ 1.0 y_m) (* x_m (pow z 2.0)))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
assert(x_m < y_m && y_m < z);
double code(double y_s, double x_s, double x_m, double y_m, double z) {
	double tmp;
	if ((z * z) <= 5e+15) {
		tmp = (1.0 / x_m) / (y_m * (1.0 + (z * z)));
	} else {
		tmp = (1.0 / y_m) / (x_m * pow(z, 2.0));
	}
	return y_s * (x_s * tmp);
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
y\_m = abs(y)
y\_s = copysign(1.0d0, y)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
real(8) function code(y_s, x_s, x_m, y_m, z)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((z * z) <= 5d+15) then
        tmp = (1.0d0 / x_m) / (y_m * (1.0d0 + (z * z)))
    else
        tmp = (1.0d0 / y_m) / (x_m * (z ** 2.0d0))
    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) <= 5e+15) {
		tmp = (1.0 / x_m) / (y_m * (1.0 + (z * z)));
	} else {
		tmp = (1.0 / y_m) / (x_m * Math.pow(z, 2.0));
	}
	return y_s * (x_s * tmp);
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
[x_m, y_m, z] = sort([x_m, y_m, z])
def code(y_s, x_s, x_m, y_m, z):
	tmp = 0
	if (z * z) <= 5e+15:
		tmp = (1.0 / x_m) / (y_m * (1.0 + (z * z)))
	else:
		tmp = (1.0 / y_m) / (x_m * math.pow(z, 2.0))
	return y_s * (x_s * tmp)

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
y\_m = abs(y)
y\_s = copysign(1.0, y)
x_m, y_m, z = sort([x_m, y_m, z])
function code(y_s, x_s, x_m, y_m, z)
	tmp = 0.0
	if (Float64(z * z) <= 5e+15)
		tmp = Float64(Float64(1.0 / x_m) / Float64(y_m * Float64(1.0 + Float64(z * z))));
	else
		tmp = Float64(Float64(1.0 / y_m) / Float64(x_m * (z ^ 2.0)));
	end
	return Float64(y_s * Float64(x_s * tmp))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
x_m, y_m, z = num2cell(sort([x_m, y_m, z])){:}
function tmp_2 = code(y_s, x_s, x_m, y_m, z)
	tmp = 0.0;
	if ((z * z) <= 5e+15)
		tmp = (1.0 / x_m) / (y_m * (1.0 + (z * z)));
	else
		tmp = (1.0 / y_m) / (x_m * (z ^ 2.0));
	end
	tmp_2 = y_s * (x_s * tmp);
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
code[y$95$s_, x$95$s_, x$95$m_, y$95$m_, z_] := N[(y$95$s * N[(x$95$s * If[LessEqual[N[(z * z), $MachinePrecision], 5e+15], 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 / y$95$m), $MachinePrecision] / N[(x$95$m * N[Power[z, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 5e15

   1. Initial program 99.7%

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

   if 5e15 < (*.f64 z z)

   1. Initial program 85.8%

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

   3. Taylor expanded in x around 0 85.7%

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

      1. associate-/r* 85.8%

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

      2. *-commutative 85.8%

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

      3. +-commutative 85.8%

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

      4. unpow2 85.8%

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

      5. fma-undefine 85.8%

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

      6. associate-/r* 85.7%

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

      7. associate-*r* 83.4%

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

      8. associate-/l/ 83.3%

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

   5. Simplified 83.3%

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

   6. Taylor expanded in z around inf 83.3%

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

Alternative 9: 93.0% accurate, 0.1× speedup

Math

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

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 x_m) (fma z z 1.0)) y_m))))

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 93.1%

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

   1. *-un-lft-identity 93.1%

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

   2. add-sqr-sqrt 47.9%

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

   3. times-frac 47.9%

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

   4. +-commutative 47.9%

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

   5. fma-undefine 47.9%

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

   6. *-commutative 47.9%

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

   7. sqrt-prod 47.9%

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

   8. fma-undefine 47.9%

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

   9. +-commutative 47.9%

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

   10. hypot-1-def 47.9%

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

   11. +-commutative 47.9%

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

   12. fma-undefine 47.9%

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

   13. *-commutative 47.9%

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

   14. sqrt-prod 49.0%

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

   15. fma-undefine 49.0%

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

   16. +-commutative 49.0%

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

   17. hypot-1-def 50.5%

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

4. Applied egg-rr 50.5%

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

   1. associate-/r* 50.5%

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

   2. associate-/r* 50.1%

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

   3. frac-times 49.1%

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

   4. associate-/l/ 49.1%

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

   5. add-sqr-sqrt 94.3%

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

6. Applied egg-rr 94.3%

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

7. Taylor expanded in x around 0 92.0%

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

   1. associate-/r* 92.0%

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

   2. +-commutative 92.0%

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

   3. unpow2 92.0%

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

   4. fma-undefine 92.0%

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

9. Simplified 92.0%

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

Alternative 10: 93.0% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 93.1%

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

3. Taylor expanded in x around 0 93.0%

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

   1. associate-/r* 93.1%

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

   2. *-commutative 93.1%

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

   3. +-commutative 93.1%

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

   4. unpow2 93.1%

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

   5. fma-undefine 93.1%

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

   6. associate-/r* 93.0%

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

   7. associate-*r* 91.9%

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

   8. associate-/l/ 91.9%

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

5. Simplified 91.9%

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

Alternative 11: 92.6% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 93.1%

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

   1. associate-/l/ 93.0%

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

   2. associate-*l* 91.9%

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

   3. *-commutative 91.9%

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

   4. sqr-neg 91.9%

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

   5. +-commutative 91.9%

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

   6. sqr-neg 91.9%

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

   7. fma-define 91.9%

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

3. Simplified 91.9%

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

Alternative 12: 89.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 93.1%

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

Alternative 13: 59.1% accurate, 2.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 93.1%

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

   1. associate-/l/ 93.0%

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

   2. associate-*l* 91.9%

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

   3. *-commutative 91.9%

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

   4. sqr-neg 91.9%

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

   5. +-commutative 91.9%

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

   6. sqr-neg 91.9%

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

   7. fma-define 91.9%

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

3. Simplified 91.9%

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

5. Taylor expanded in z around 0 61.3%

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

Developer target: 93.0% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024111 
(FPCore (x y z)
  :name "Statistics.Distribution.CauchyLorentz:$cdensity from math-functions-0.1.5.2"
  :precision binary64

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

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