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

Percentage Accurate: 88.5% → 99.3%

Time: 11.5s

Alternatives: 11

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 88.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.3% accurate, 0.1× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ [x_m, y_m, z_m] = \mathsf{sort}([x_m, y_m, z_m])\\ \\ y_s \cdot \left(x_s \cdot \begin{array}{l} \mathbf{if}\;z_m \leq 2.22 \cdot 10^{+116}:\\ \;\;\;\;\frac{\frac{1}{y_m}}{x_m \cdot \mathsf{fma}\left(z_m, z_m, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{z_m} \cdot \frac{1}{y_m \cdot \left(x_m \cdot z_m\right)}\\ \end{array}\right) \end{array} \]

FPCore

z_m = (fabs.f64 z)
x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
y_m = (fabs.f64 y)
y_s = (copysign.f64 1 y)
NOTE: x_m, y_m, and z_m should be sorted in increasing order before calling this function.
(FPCore (y_s x_s x_m y_m z_m)
 :precision binary64
 (*
  y_s
  (*
   x_s
   (if (<= z_m 2.22e+116)
     (/ (/ 1.0 y_m) (* x_m (fma z_m z_m 1.0)))
     (* (/ 1.0 z_m) (/ 1.0 (* y_m (* x_m z_m))))))))

C

z_m = fabs(z);
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_m);
double code(double y_s, double x_s, double x_m, double y_m, double z_m) {
	double tmp;
	if (z_m <= 2.22e+116) {
		tmp = (1.0 / y_m) / (x_m * fma(z_m, z_m, 1.0));
	} else {
		tmp = (1.0 / z_m) * (1.0 / (y_m * (x_m * z_m)));
	}
	return y_s * (x_s * tmp);
}

Julia

z_m = abs(z)
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_m = sort([x_m, y_m, z_m])
function code(y_s, x_s, x_m, y_m, z_m)
	tmp = 0.0
	if (z_m <= 2.22e+116)
		tmp = Float64(Float64(1.0 / y_m) / Float64(x_m * fma(z_m, z_m, 1.0)));
	else
		tmp = Float64(Float64(1.0 / z_m) * Float64(1.0 / Float64(y_m * Float64(x_m * z_m))));
	end
	return Float64(y_s * Float64(x_s * tmp))
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
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_m should be sorted in increasing order before calling this function.
code[y$95$s_, x$95$s_, x$95$m_, y$95$m_, z$95$m_] := N[(y$95$s * N[(x$95$s * If[LessEqual[z$95$m, 2.22e+116], N[(N[(1.0 / y$95$m), $MachinePrecision] / N[(x$95$m * N[(z$95$m * z$95$m + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / z$95$m), $MachinePrecision] * N[(1.0 / N[(y$95$m * N[(x$95$m * z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 2.2199999999999999e116

   1. Initial program 93.9%

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

      1. associate-/r* 93.8%

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

      2. sqr-neg 93.8%

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

      3. +-commutative 93.8%

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

      4. sqr-neg 93.8%

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

      5. fma-def 93.8%

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

   3. Simplified 93.8%

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

      1. /-rgt-identity 93.8%

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

      2. *-commutative 93.8%

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

      3. fma-udef 93.8%

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

      4. +-commutative 93.8%

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

      5. associate-/l* 93.7%

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

      6. +-commutative 93.7%

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

      7. fma-udef 93.7%

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

      8. *-commutative 93.7%

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

      9. associate-/l* 92.4%

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

      10. associate-/l/ 92.4%

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

      11. *-commutative 92.4%

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

   5. Applied egg-rr 92.4%

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

   6. Taylor expanded in x around 0 93.8%

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

      1. associate-/r* 93.9%

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

      2. +-commutative 93.9%

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

      3. unpow2 93.9%

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

      4. fma-udef 93.9%

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

      5. associate-/r* 92.6%

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

      6. associate-/r* 92.5%

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

      7. associate-/l/ 92.6%

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

      8. associate-/l/ 92.7%

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

      9. *-commutative 92.7%

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

   8. Simplified 92.7%

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

   if 2.2199999999999999e116 < z

   1. Initial program 70.7%

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

      1. associate-/l/ 70.7%

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

      2. metadata-eval 70.7%

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

      3. associate-/r* 70.7%

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

      4. metadata-eval 70.7%

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

      5. neg-mul-1 70.7%

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

      6. distribute-neg-frac 70.7%

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

      7. distribute-frac-neg 70.7%

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

      8. distribute-frac-neg 70.7%

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

      9. distribute-neg-frac 70.7%

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

      10. metadata-eval 70.7%

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

      11. neg-mul-1 70.7%

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

      12. associate-/r* 70.7%

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

      13. metadata-eval 70.7%

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

      14. associate-/r* 70.8%

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

      15. sqr-neg 70.8%

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

      16. +-commutative 70.8%

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

      17. sqr-neg 70.8%

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

      18. fma-def 70.8%

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

   3. Simplified 70.8%

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

   4. Taylor expanded in z around inf 70.8%

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

      1. div-inv 70.8%

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

      2. add-sqr-sqrt 55.5%

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

      3. associate-*l* 55.4%

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

      4. associate-/r* 55.4%

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

      5. sqrt-div 32.0%

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

      6. inv-pow 32.0%

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

      7. metadata-eval 32.0%

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

      8. pow-prod-up 32.0%

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

      9. sqrt-unprod 32.0%

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

      10. add-sqr-sqrt 32.0%

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

      11. unpow2 32.0%

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

      12. sqrt-prod 32.0%

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

      13. add-sqr-sqrt 32.0%

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

   6. Applied egg-rr 53.4%

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

      1. associate-*r/ 53.5%

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

      2. *-rgt-identity 53.5%

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

      3. associate-*r/ 34.4%

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

      4. unpow2 34.4%

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

   8. Simplified 34.4%

      \[\leadsto \color{blue}{\frac{{\left(\frac{{x}^{-0.5}}{z}\right)}^{2}}{y}} \]
   9. Step-by-step derivation

      1. unpow2 34.4%

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

      2. frac-times 32.0%

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

      3. pow-prod-up 70.7%

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

      4. metadata-eval 70.7%

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

      5. inv-pow 70.7%

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

      6. unpow2 70.7%

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

      7. associate-/r* 70.8%

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

      8. associate-/l/ 70.7%

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

      9. add-sqr-sqrt 70.6%

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

      10. associate-/l* 70.7%

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

      11. sqrt-div 70.8%

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

      12. metadata-eval 70.8%

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

      13. unpow2 70.8%

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

      14. sqrt-prod 70.8%

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

      15. add-sqr-sqrt 70.8%

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

      16. sqrt-div 70.8%

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

      17. metadata-eval 70.8%

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

      18. unpow2 70.8%

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

      19. sqrt-prod 78.0%

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

      20. add-sqr-sqrt 78.1%

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

   10. Applied egg-rr 78.1%

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

       1. associate-/l/ 94.9%

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

       2. div-inv 94.7%

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

       3. associate-/r/ 94.9%

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

       4. /-rgt-identity 94.9%

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

   12. Applied egg-rr 94.9%

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

4. Final simplification 93.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 2.22 \cdot 10^{+116}:\\ \;\;\;\;\frac{\frac{1}{y}}{x \cdot \mathsf{fma}\left(z, z, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{z} \cdot \frac{1}{y \cdot \left(x \cdot z\right)}\\ \end{array} \]

Alternative 2: 99.2% accurate, 0.0× speedup

Math

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

FPCore

z_m = (fabs.f64 z)
x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
y_m = (fabs.f64 y)
y_s = (copysign.f64 1 y)
NOTE: x_m, y_m, and z_m should be sorted in increasing order before calling this function.
(FPCore (y_s x_s x_m y_m z_m)
 :precision binary64
 (* y_s (* x_s (pow (/ (pow x_m -0.5) (* (sqrt y_m) (hypot 1.0 z_m))) 2.0))))

C

z_m = fabs(z);
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_m);
double code(double y_s, double x_s, double x_m, double y_m, double z_m) {
	return y_s * (x_s * pow((pow(x_m, -0.5) / (sqrt(y_m) * hypot(1.0, z_m))), 2.0));
}

Java

z_m = Math.abs(z);
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_m;
public static double code(double y_s, double x_s, double x_m, double y_m, double z_m) {
	return y_s * (x_s * Math.pow((Math.pow(x_m, -0.5) / (Math.sqrt(y_m) * Math.hypot(1.0, z_m))), 2.0));
}

Python

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

Julia

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

MATLAB

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

Wolfram

z_m = N[Abs[z], $MachinePrecision]
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_m should be sorted in increasing order before calling this function.
code[y$95$s_, x$95$s_, x$95$m_, y$95$m_, z$95$m_] := N[(y$95$s * N[(x$95$s * N[Power[N[(N[Power[x$95$m, -0.5], $MachinePrecision] / N[(N[Sqrt[y$95$m], $MachinePrecision] * N[Sqrt[1.0 ^ 2 + z$95$m ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 90.4%

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

   1. associate-/r* 90.3%

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

   2. sqr-neg 90.3%

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

   3. +-commutative 90.3%

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

   4. sqr-neg 90.3%

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

   5. fma-def 90.3%

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

3. Simplified 90.3%

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

   1. fma-udef 90.3%

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

   2. +-commutative 90.3%

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

   3. associate-/r* 90.4%

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

   4. add-sqr-sqrt 57.6%

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

   5. sqrt-div 20.5%

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

   6. inv-pow 20.5%

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

   7. sqrt-pow1 20.5%

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

   8. metadata-eval 20.5%

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

   9. +-commutative 20.5%

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

   10. fma-udef 20.5%

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

   11. sqrt-prod 20.5%

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

   12. fma-udef 20.5%

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

   13. +-commutative 20.5%

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

   14. hypot-1-def 20.5%

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

   15. sqrt-div 20.5%

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

   16. inv-pow 20.5%

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

   17. sqrt-pow1 20.5%

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

   18. metadata-eval 20.5%

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

5. Applied egg-rr 22.5%

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

   1. unpow2 22.5%

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

7. Simplified 22.5%

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

8. Final simplification 22.5%

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

Alternative 3: 98.8% accurate, 0.1× speedup

Math

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

FPCore

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

C

z_m = fabs(z);
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_m);
double code(double y_s, double x_s, double x_m, double y_m, double z_m) {
	return y_s * (x_s * (((1.0 / hypot(1.0, z_m)) * ((1.0 / x_m) / hypot(1.0, z_m))) / y_m));
}

Java

z_m = Math.abs(z);
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_m;
public static double code(double y_s, double x_s, double x_m, double y_m, double z_m) {
	return y_s * (x_s * (((1.0 / Math.hypot(1.0, z_m)) * ((1.0 / x_m) / Math.hypot(1.0, z_m))) / y_m));
}

Python

z_m = math.fabs(z)
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_m] = sort([x_m, y_m, z_m])
def code(y_s, x_s, x_m, y_m, z_m):
	return y_s * (x_s * (((1.0 / math.hypot(1.0, z_m)) * ((1.0 / x_m) / math.hypot(1.0, z_m))) / y_m))

Julia

z_m = abs(z)
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_m = sort([x_m, y_m, z_m])
function code(y_s, x_s, x_m, y_m, z_m)
	return Float64(y_s * Float64(x_s * Float64(Float64(Float64(1.0 / hypot(1.0, z_m)) * Float64(Float64(1.0 / x_m) / hypot(1.0, z_m))) / y_m)))
end

MATLAB

z_m = abs(z);
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_m = num2cell(sort([x_m, y_m, z_m])){:}
function tmp = code(y_s, x_s, x_m, y_m, z_m)
	tmp = y_s * (x_s * (((1.0 / hypot(1.0, z_m)) * ((1.0 / x_m) / hypot(1.0, z_m))) / y_m));
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
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_m should be sorted in increasing order before calling this function.
code[y$95$s_, x$95$s_, x$95$m_, y$95$m_, z$95$m_] := N[(y$95$s * N[(x$95$s * N[(N[(N[(1.0 / N[Sqrt[1.0 ^ 2 + z$95$m ^ 2], $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 / x$95$m), $MachinePrecision] / N[Sqrt[1.0 ^ 2 + z$95$m ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 90.4%

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

   1. associate-/l/ 89.4%

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

   2. metadata-eval 89.4%

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

   3. associate-/r* 89.4%

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

   4. metadata-eval 89.4%

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

   5. neg-mul-1 89.4%

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

   6. distribute-neg-frac 89.4%

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

   7. distribute-frac-neg 89.4%

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

   8. distribute-frac-neg 89.4%

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

   9. distribute-neg-frac 89.4%

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

   10. metadata-eval 89.4%

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

   11. neg-mul-1 89.4%

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

   12. associate-/r* 89.4%

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

   13. metadata-eval 89.4%

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

   14. associate-/r* 89.3%

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

   15. sqr-neg 89.3%

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

   16. +-commutative 89.3%

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

   17. sqr-neg 89.3%

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

   18. fma-def 89.3%

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

3. Simplified 89.3%

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

   1. associate-/r* 89.4%

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

   2. *-un-lft-identity 89.4%

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

   3. add-sqr-sqrt 89.3%

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

   4. times-frac 89.3%

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

   5. fma-udef 89.3%

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

   6. +-commutative 89.3%

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

   7. hypot-1-def 89.3%

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

   8. fma-udef 89.3%

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

   9. +-commutative 89.3%

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

   10. hypot-1-def 92.2%

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

5. Applied egg-rr 92.2%

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

6. Final simplification 92.2%

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

Alternative 4: 98.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ [x_m, y_m, z_m] = \mathsf{sort}([x_m, y_m, z_m])\\ \\ \begin{array}{l} t_0 := y_m \cdot \left(1 + z_m \cdot z_m\right)\\ y_s \cdot \left(x_s \cdot \begin{array}{l} \mathbf{if}\;t_0 \leq 10^{+299}:\\ \;\;\;\;\frac{\frac{1}{x_m}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{z_m} \cdot \frac{1}{y_m \cdot \left(x_m \cdot z_m\right)}\\ \end{array}\right) \end{array} \end{array} \]

FPCore

z_m = (fabs.f64 z)
x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
y_m = (fabs.f64 y)
y_s = (copysign.f64 1 y)
NOTE: x_m, y_m, and z_m should be sorted in increasing order before calling this function.
(FPCore (y_s x_s x_m y_m z_m)
 :precision binary64
 (let* ((t_0 (* y_m (+ 1.0 (* z_m z_m)))))
   (*
    y_s
    (*
     x_s
     (if (<= t_0 1e+299)
       (/ (/ 1.0 x_m) t_0)
       (* (/ 1.0 z_m) (/ 1.0 (* y_m (* x_m z_m)))))))))

C

z_m = fabs(z);
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_m);
double code(double y_s, double x_s, double x_m, double y_m, double z_m) {
	double t_0 = y_m * (1.0 + (z_m * z_m));
	double tmp;
	if (t_0 <= 1e+299) {
		tmp = (1.0 / x_m) / t_0;
	} else {
		tmp = (1.0 / z_m) * (1.0 / (y_m * (x_m * z_m)));
	}
	return y_s * (x_s * tmp);
}

Fortran

z_m = abs(z)
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_m should be sorted in increasing order before calling this function.
real(8) function code(y_s, x_s, x_m, y_m, z_m)
    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_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = y_m * (1.0d0 + (z_m * z_m))
    if (t_0 <= 1d+299) then
        tmp = (1.0d0 / x_m) / t_0
    else
        tmp = (1.0d0 / z_m) * (1.0d0 / (y_m * (x_m * z_m)))
    end if
    code = y_s * (x_s * tmp)
end function

Java

z_m = Math.abs(z);
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_m;
public static double code(double y_s, double x_s, double x_m, double y_m, double z_m) {
	double t_0 = y_m * (1.0 + (z_m * z_m));
	double tmp;
	if (t_0 <= 1e+299) {
		tmp = (1.0 / x_m) / t_0;
	} else {
		tmp = (1.0 / z_m) * (1.0 / (y_m * (x_m * z_m)));
	}
	return y_s * (x_s * tmp);
}

Python

z_m = math.fabs(z)
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_m] = sort([x_m, y_m, z_m])
def code(y_s, x_s, x_m, y_m, z_m):
	t_0 = y_m * (1.0 + (z_m * z_m))
	tmp = 0
	if t_0 <= 1e+299:
		tmp = (1.0 / x_m) / t_0
	else:
		tmp = (1.0 / z_m) * (1.0 / (y_m * (x_m * z_m)))
	return y_s * (x_s * tmp)

Julia

z_m = abs(z)
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_m = sort([x_m, y_m, z_m])
function code(y_s, x_s, x_m, y_m, z_m)
	t_0 = Float64(y_m * Float64(1.0 + Float64(z_m * z_m)))
	tmp = 0.0
	if (t_0 <= 1e+299)
		tmp = Float64(Float64(1.0 / x_m) / t_0);
	else
		tmp = Float64(Float64(1.0 / z_m) * Float64(1.0 / Float64(y_m * Float64(x_m * z_m))));
	end
	return Float64(y_s * Float64(x_s * tmp))
end

MATLAB

z_m = abs(z);
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_m = num2cell(sort([x_m, y_m, z_m])){:}
function tmp_2 = code(y_s, x_s, x_m, y_m, z_m)
	t_0 = y_m * (1.0 + (z_m * z_m));
	tmp = 0.0;
	if (t_0 <= 1e+299)
		tmp = (1.0 / x_m) / t_0;
	else
		tmp = (1.0 / z_m) * (1.0 / (y_m * (x_m * z_m)));
	end
	tmp_2 = y_s * (x_s * tmp);
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
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_m should be sorted in increasing order before calling this function.
code[y$95$s_, x$95$s_, x$95$m_, y$95$m_, z$95$m_] := Block[{t$95$0 = N[(y$95$m * N[(1.0 + N[(z$95$m * z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(y$95$s * N[(x$95$s * If[LessEqual[t$95$0, 1e+299], N[(N[(1.0 / x$95$m), $MachinePrecision] / t$95$0), $MachinePrecision], N[(N[(1.0 / z$95$m), $MachinePrecision] * N[(1.0 / N[(y$95$m * N[(x$95$m * z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y (+.f64 1 (*.f64 z z))) < 1.0000000000000001e299

   1. Initial program 93.7%

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

   if 1.0000000000000001e299 < (*.f64 y (+.f64 1 (*.f64 z z)))

   1. Initial program 71.2%

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

      1. associate-/l/ 73.8%

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

      2. metadata-eval 73.8%

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

      3. associate-/r* 73.8%

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

      4. metadata-eval 73.8%

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

      5. neg-mul-1 73.8%

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

      6. distribute-neg-frac 73.8%

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

      7. distribute-frac-neg 73.8%

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

      8. distribute-frac-neg 73.8%

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

      9. distribute-neg-frac 73.8%

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

      10. metadata-eval 73.8%

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

      11. neg-mul-1 73.8%

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

      12. associate-/r* 73.8%

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

      13. metadata-eval 73.8%

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

      14. associate-/r* 73.8%

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

      15. sqr-neg 73.8%

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

      16. +-commutative 73.8%

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

      17. sqr-neg 73.8%

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

      18. fma-def 73.8%

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

   3. Simplified 73.8%

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

   4. Taylor expanded in z around inf 73.8%

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

      1. div-inv 73.8%

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

      2. add-sqr-sqrt 71.2%

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

      3. associate-*l* 71.2%

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

      4. associate-/r* 71.2%

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

      5. sqrt-div 46.5%

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

      6. inv-pow 46.5%

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

      7. metadata-eval 46.5%

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

      8. pow-prod-up 46.5%

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

      9. sqrt-unprod 46.4%

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

      10. add-sqr-sqrt 46.5%

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

      11. unpow2 46.5%

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

      12. sqrt-prod 18.9%

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

      13. add-sqr-sqrt 43.9%

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

   6. Applied egg-rr 57.7%

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

      1. associate-*r/ 57.8%

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

      2. *-rgt-identity 57.8%

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

      3. associate-*r/ 49.0%

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

      4. unpow2 49.0%

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

   8. Simplified 49.0%

      \[\leadsto \color{blue}{\frac{{\left(\frac{{x}^{-0.5}}{z}\right)}^{2}}{y}} \]
   9. Step-by-step derivation

      1. unpow2 49.0%

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

      2. frac-times 46.4%

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

      3. pow-prod-up 73.8%

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

      4. metadata-eval 73.8%

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

      5. inv-pow 73.8%

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

      6. unpow2 73.8%

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

      7. associate-/r* 73.8%

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

      8. associate-/l/ 73.8%

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

      9. add-sqr-sqrt 73.8%

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

      10. associate-/l* 73.8%

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

      11. sqrt-div 73.8%

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

      12. metadata-eval 73.8%

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

      13. unpow2 73.8%

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

      14. sqrt-prod 35.0%

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

      15. add-sqr-sqrt 71.4%

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

      16. sqrt-div 71.4%

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

      17. metadata-eval 71.4%

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

      18. unpow2 71.4%

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

      19. sqrt-prod 37.5%

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

      20. add-sqr-sqrt 78.9%

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

   10. Applied egg-rr 78.9%

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

       1. associate-/l/ 94.2%

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

       2. div-inv 94.0%

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

       3. associate-/r/ 94.1%

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

       4. /-rgt-identity 94.1%

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

   12. Applied egg-rr 94.1%

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

4. Final simplification 93.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(1 + z \cdot z\right) \leq 10^{+299}:\\ \;\;\;\;\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{z} \cdot \frac{1}{y \cdot \left(x \cdot z\right)}\\ \end{array} \]

Alternative 5: 98.1% accurate, 0.8× speedup

Math

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

FPCore

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

C

z_m = fabs(z);
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_m);
double code(double y_s, double x_s, double x_m, double y_m, double z_m) {
	double tmp;
	if (z_m <= 1.0) {
		tmp = (1.0 / y_m) / x_m;
	} else {
		tmp = (1.0 / z_m) * (1.0 / (y_m * (x_m * z_m)));
	}
	return y_s * (x_s * tmp);
}

Fortran

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

Java

z_m = Math.abs(z);
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_m;
public static double code(double y_s, double x_s, double x_m, double y_m, double z_m) {
	double tmp;
	if (z_m <= 1.0) {
		tmp = (1.0 / y_m) / x_m;
	} else {
		tmp = (1.0 / z_m) * (1.0 / (y_m * (x_m * z_m)));
	}
	return y_s * (x_s * tmp);
}

Python

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

Julia

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

MATLAB

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

Wolfram

z_m = N[Abs[z], $MachinePrecision]
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_m should be sorted in increasing order before calling this function.
code[y$95$s_, x$95$s_, x$95$m_, y$95$m_, z$95$m_] := N[(y$95$s * N[(x$95$s * If[LessEqual[z$95$m, 1.0], N[(N[(1.0 / y$95$m), $MachinePrecision] / x$95$m), $MachinePrecision], N[(N[(1.0 / z$95$m), $MachinePrecision] * N[(1.0 / N[(y$95$m * N[(x$95$m * z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 1

   1. Initial program 93.2%

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

      1. associate-/r* 93.1%

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

      2. sqr-neg 93.1%

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

      3. +-commutative 93.1%

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

      4. sqr-neg 93.1%

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

      5. fma-def 93.1%

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

   3. Simplified 93.1%

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

   4. Taylor expanded in z around 0 75.9%

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

      1. *-commutative 75.9%

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

      2. associate-/r* 75.9%

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

   6. Simplified 75.9%

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

   if 1 < z

   1. Initial program 81.3%

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

      1. associate-/l/ 80.0%

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

      2. metadata-eval 80.0%

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

      3. associate-/r* 80.0%

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

      4. metadata-eval 80.0%

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

      5. neg-mul-1 80.0%

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

      6. distribute-neg-frac 80.0%

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

      7. distribute-frac-neg 80.0%

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

      8. distribute-frac-neg 80.0%

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

      9. distribute-neg-frac 80.0%

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

      10. metadata-eval 80.0%

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

      11. neg-mul-1 80.0%

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

      12. associate-/r* 80.0%

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

      13. metadata-eval 80.0%

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

      14. associate-/r* 80.0%

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

      15. sqr-neg 80.0%

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

      16. +-commutative 80.0%

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

      17. sqr-neg 80.0%

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

      18. fma-def 80.0%

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

   3. Simplified 80.0%

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

   4. Taylor expanded in z around inf 78.5%

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

      1. div-inv 78.4%

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

      2. add-sqr-sqrt 49.0%

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

      3. associate-*l* 49.0%

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

      4. associate-/r* 49.0%

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

      5. sqrt-div 31.0%

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

      6. inv-pow 31.0%

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

      7. metadata-eval 31.0%

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

      8. pow-prod-up 30.9%

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

      9. sqrt-unprod 30.9%

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

      10. add-sqr-sqrt 30.9%

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

      11. unpow2 30.9%

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

      12. sqrt-prod 30.9%

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

      13. add-sqr-sqrt 30.9%

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

   6. Applied egg-rr 44.4%

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

      1. associate-*r/ 44.4%

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

      2. *-rgt-identity 44.4%

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

      3. associate-*r/ 32.4%

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

      4. unpow2 32.4%

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

   8. Simplified 32.4%

      \[\leadsto \color{blue}{\frac{{\left(\frac{{x}^{-0.5}}{z}\right)}^{2}}{y}} \]
   9. Step-by-step derivation

      1. unpow2 32.4%

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

      2. frac-times 30.9%

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

      3. pow-prod-up 78.4%

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

      4. metadata-eval 78.4%

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

      5. inv-pow 78.4%

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

      6. unpow2 78.4%

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

      7. associate-/r* 78.5%

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

      8. associate-/l/ 78.4%

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

      9. add-sqr-sqrt 78.3%

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

      10. associate-/l* 78.4%

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

      11. sqrt-div 78.4%

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

      12. metadata-eval 78.4%

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

      13. unpow2 78.4%

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

      14. sqrt-prod 78.3%

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

      15. add-sqr-sqrt 78.4%

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

      16. sqrt-div 78.4%

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

      17. metadata-eval 78.4%

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

      18. unpow2 78.4%

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

      19. sqrt-prod 82.9%

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

      20. add-sqr-sqrt 83.0%

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

   10. Applied egg-rr 83.0%

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

       1. associate-/l/ 94.0%

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

       2. div-inv 93.4%

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

       3. associate-/r/ 93.6%

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

       4. /-rgt-identity 93.6%

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

   12. Applied egg-rr 93.6%

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

4. Final simplification 80.2%

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

Alternative 6: 97.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

z_m = fabs(z);
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_m);
double code(double y_s, double x_s, double x_m, double y_m, double z_m) {
	double tmp;
	if (z_m <= 1.0) {
		tmp = (1.0 / y_m) / x_m;
	} else {
		tmp = 1.0 / (y_m * (z_m * (x_m * z_m)));
	}
	return y_s * (x_s * tmp);
}

Fortran

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

Java

z_m = Math.abs(z);
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_m;
public static double code(double y_s, double x_s, double x_m, double y_m, double z_m) {
	double tmp;
	if (z_m <= 1.0) {
		tmp = (1.0 / y_m) / x_m;
	} else {
		tmp = 1.0 / (y_m * (z_m * (x_m * z_m)));
	}
	return y_s * (x_s * tmp);
}

Python

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

Julia

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

MATLAB

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

Wolfram

z_m = N[Abs[z], $MachinePrecision]
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_m should be sorted in increasing order before calling this function.
code[y$95$s_, x$95$s_, x$95$m_, y$95$m_, z$95$m_] := N[(y$95$s * N[(x$95$s * If[LessEqual[z$95$m, 1.0], N[(N[(1.0 / y$95$m), $MachinePrecision] / x$95$m), $MachinePrecision], N[(1.0 / N[(y$95$m * N[(z$95$m * N[(x$95$m * z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 1

   1. Initial program 93.2%

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

      1. associate-/r* 93.1%

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

      2. sqr-neg 93.1%

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

      3. +-commutative 93.1%

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

      4. sqr-neg 93.1%

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

      5. fma-def 93.1%

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

   3. Simplified 93.1%

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

   4. Taylor expanded in z around 0 75.9%

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

      1. *-commutative 75.9%

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

      2. associate-/r* 75.9%

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

   6. Simplified 75.9%

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

   if 1 < z

   1. Initial program 81.3%

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

      1. associate-/r* 81.4%

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

      2. sqr-neg 81.4%

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

      3. +-commutative 81.4%

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

      4. sqr-neg 81.4%

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

      5. fma-def 81.4%

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

   3. Simplified 81.4%

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

      1. fma-udef 81.4%

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

      2. +-commutative 81.4%

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

      3. associate-/r* 81.3%

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

      4. add-sqr-sqrt 60.4%

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

      5. sqrt-div 16.4%

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

      6. inv-pow 16.4%

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

      7. sqrt-pow1 16.4%

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

      8. metadata-eval 16.4%

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

      9. +-commutative 16.4%

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

      10. fma-udef 16.4%

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

      11. sqrt-prod 16.4%

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

      12. fma-udef 16.4%

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

      13. +-commutative 16.4%

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

      14. hypot-1-def 16.4%

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

      15. sqrt-div 16.3%

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

      16. inv-pow 16.3%

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

      17. sqrt-pow1 16.3%

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

      18. metadata-eval 16.3%

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

   5. Applied egg-rr 20.8%

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

      1. unpow2 20.8%

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

   7. Simplified 20.8%

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

   8. Taylor expanded in z around inf 79.8%

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

      1. associate-*r* 76.5%

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

      2. *-commutative 76.5%

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

      3. associate-*r* 78.3%

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

   10. Simplified 78.3%

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

       1. add-sqr-sqrt 31.0%

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

       2. pow2 31.0%

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

       3. *-commutative 31.0%

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

       4. sqrt-prod 31.0%

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

       5. unpow2 31.0%

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

       6. sqrt-prod 32.4%

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

       7. add-sqr-sqrt 32.5%

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

   12. Applied egg-rr 32.5%

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

       1. unpow2 32.5%

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

       2. *-commutative 32.5%

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

       3. *-commutative 32.5%

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

       4. swap-sqr 30.9%

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

       5. add-sqr-sqrt 78.3%

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

       6. associate-*l* 83.0%

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

   14. Applied egg-rr 83.0%

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

4. Final simplification 77.6%

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

Alternative 7: 97.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

z_m = fabs(z);
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_m);
double code(double y_s, double x_s, double x_m, double y_m, double z_m) {
	double tmp;
	if (z_m <= 1.0) {
		tmp = (1.0 / y_m) / x_m;
	} else {
		tmp = (1.0 / (z_m * (x_m * z_m))) / y_m;
	}
	return y_s * (x_s * tmp);
}

Fortran

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

Java

z_m = Math.abs(z);
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_m;
public static double code(double y_s, double x_s, double x_m, double y_m, double z_m) {
	double tmp;
	if (z_m <= 1.0) {
		tmp = (1.0 / y_m) / x_m;
	} else {
		tmp = (1.0 / (z_m * (x_m * z_m))) / y_m;
	}
	return y_s * (x_s * tmp);
}

Python

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

Julia

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

MATLAB

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

Wolfram

z_m = N[Abs[z], $MachinePrecision]
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_m should be sorted in increasing order before calling this function.
code[y$95$s_, x$95$s_, x$95$m_, y$95$m_, z$95$m_] := N[(y$95$s * N[(x$95$s * If[LessEqual[z$95$m, 1.0], N[(N[(1.0 / y$95$m), $MachinePrecision] / x$95$m), $MachinePrecision], N[(N[(1.0 / N[(z$95$m * N[(x$95$m * z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 1

   1. Initial program 93.2%

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

      1. associate-/r* 93.1%

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

      2. sqr-neg 93.1%

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

      3. +-commutative 93.1%

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

      4. sqr-neg 93.1%

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

      5. fma-def 93.1%

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

   3. Simplified 93.1%

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

   4. Taylor expanded in z around 0 75.9%

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

      1. *-commutative 75.9%

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

      2. associate-/r* 75.9%

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

   6. Simplified 75.9%

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

   if 1 < z

   1. Initial program 81.3%

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

      1. associate-/l/ 80.0%

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

      2. metadata-eval 80.0%

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

      3. associate-/r* 80.0%

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

      4. metadata-eval 80.0%

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

      5. neg-mul-1 80.0%

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

      6. distribute-neg-frac 80.0%

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

      7. distribute-frac-neg 80.0%

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

      8. distribute-frac-neg 80.0%

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

      9. distribute-neg-frac 80.0%

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

      10. metadata-eval 80.0%

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

      11. neg-mul-1 80.0%

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

      12. associate-/r* 80.0%

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

      13. metadata-eval 80.0%

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

      14. associate-/r* 80.0%

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

      15. sqr-neg 80.0%

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

      16. +-commutative 80.0%

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

      17. sqr-neg 80.0%

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

      18. fma-def 80.0%

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

   3. Simplified 80.0%

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

   4. Taylor expanded in z around inf 78.5%

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

      1. /-rgt-identity 78.5%

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

      2. associate-/l* 78.4%

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

   6. Applied egg-rr 78.4%

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

      1. associate-/r/ 78.5%

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

      2. /-rgt-identity 78.5%

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

      3. unpow2 78.5%

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

      4. associate-*r* 83.1%

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

   8. Applied egg-rr 83.1%

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

4. Final simplification 77.7%

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

Alternative 8: 98.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

z_m = fabs(z);
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_m);
double code(double y_s, double x_s, double x_m, double y_m, double z_m) {
	double tmp;
	if (z_m <= 1.0) {
		tmp = (1.0 / y_m) / x_m;
	} else {
		tmp = ((1.0 / z_m) / (x_m * z_m)) / y_m;
	}
	return y_s * (x_s * tmp);
}

Fortran

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

Java

z_m = Math.abs(z);
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_m;
public static double code(double y_s, double x_s, double x_m, double y_m, double z_m) {
	double tmp;
	if (z_m <= 1.0) {
		tmp = (1.0 / y_m) / x_m;
	} else {
		tmp = ((1.0 / z_m) / (x_m * z_m)) / y_m;
	}
	return y_s * (x_s * tmp);
}

Python

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

Julia

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

MATLAB

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

Wolfram

z_m = N[Abs[z], $MachinePrecision]
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_m should be sorted in increasing order before calling this function.
code[y$95$s_, x$95$s_, x$95$m_, y$95$m_, z$95$m_] := N[(y$95$s * N[(x$95$s * If[LessEqual[z$95$m, 1.0], N[(N[(1.0 / y$95$m), $MachinePrecision] / x$95$m), $MachinePrecision], N[(N[(N[(1.0 / z$95$m), $MachinePrecision] / N[(x$95$m * z$95$m), $MachinePrecision]), $MachinePrecision] / y$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 1

   1. Initial program 93.2%

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

      1. associate-/r* 93.1%

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

      2. sqr-neg 93.1%

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

      3. +-commutative 93.1%

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

      4. sqr-neg 93.1%

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

      5. fma-def 93.1%

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

   3. Simplified 93.1%

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

   4. Taylor expanded in z around 0 75.9%

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

      1. *-commutative 75.9%

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

      2. associate-/r* 75.9%

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

   6. Simplified 75.9%

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

   if 1 < z

   1. Initial program 81.3%

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

      1. associate-/l/ 80.0%

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

      2. metadata-eval 80.0%

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

      3. associate-/r* 80.0%

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

      4. metadata-eval 80.0%

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

      5. neg-mul-1 80.0%

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

      6. distribute-neg-frac 80.0%

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

      7. distribute-frac-neg 80.0%

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

      8. distribute-frac-neg 80.0%

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

      9. distribute-neg-frac 80.0%

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

      10. metadata-eval 80.0%

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

      11. neg-mul-1 80.0%

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

      12. associate-/r* 80.0%

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

      13. metadata-eval 80.0%

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

      14. associate-/r* 80.0%

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

      15. sqr-neg 80.0%

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

      16. +-commutative 80.0%

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

      17. sqr-neg 80.0%

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

      18. fma-def 80.0%

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

   3. Simplified 80.0%

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

   4. Taylor expanded in z around inf 78.5%

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

      1. div-inv 78.4%

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

      2. add-sqr-sqrt 49.0%

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

      3. associate-*l* 49.0%

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

      4. associate-/r* 49.0%

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

      5. sqrt-div 31.0%

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

      6. inv-pow 31.0%

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

      7. metadata-eval 31.0%

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

      8. pow-prod-up 30.9%

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

      9. sqrt-unprod 30.9%

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

      10. add-sqr-sqrt 30.9%

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

      11. unpow2 30.9%

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

      12. sqrt-prod 30.9%

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

      13. add-sqr-sqrt 30.9%

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

   6. Applied egg-rr 44.4%

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

      1. associate-*r/ 44.4%

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

      2. *-rgt-identity 44.4%

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

      3. associate-*r/ 32.4%

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

      4. unpow2 32.4%

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

   8. Simplified 32.4%

      \[\leadsto \color{blue}{\frac{{\left(\frac{{x}^{-0.5}}{z}\right)}^{2}}{y}} \]
   9. Step-by-step derivation

      1. unpow2 32.4%

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

      2. frac-times 30.9%

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

      3. pow-prod-up 78.4%

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

      4. metadata-eval 78.4%

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

      5. inv-pow 78.4%

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

      6. unpow2 78.4%

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

      7. associate-/r* 78.5%

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

      8. associate-/l/ 78.4%

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

      9. add-sqr-sqrt 78.3%

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

      10. associate-/l* 78.4%

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

      11. sqrt-div 78.4%

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

      12. metadata-eval 78.4%

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

      13. unpow2 78.4%

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

      14. sqrt-prod 78.3%

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

      15. add-sqr-sqrt 78.4%

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

      16. sqrt-div 78.4%

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

      17. metadata-eval 78.4%

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

      18. unpow2 78.4%

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

      19. sqrt-prod 82.9%

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

      20. add-sqr-sqrt 83.0%

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

   10. Applied egg-rr 83.0%

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

   11. Taylor expanded in x around 0 83.1%

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

4. Final simplification 77.7%

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

Alternative 9: 57.7% accurate, 2.2× speedup

Math

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

FPCore

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

C

z_m = fabs(z);
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_m);
double code(double y_s, double x_s, double x_m, double y_m, double z_m) {
	return y_s * (x_s * (1.0 / (x_m * y_m)));
}

Fortran

z_m = abs(z)
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_m should be sorted in increasing order before calling this function.
real(8) function code(y_s, x_s, x_m, y_m, z_m)
    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_m
    code = y_s * (x_s * (1.0d0 / (x_m * y_m)))
end function

Java

z_m = Math.abs(z);
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_m;
public static double code(double y_s, double x_s, double x_m, double y_m, double z_m) {
	return y_s * (x_s * (1.0 / (x_m * y_m)));
}

Python

z_m = math.fabs(z)
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_m] = sort([x_m, y_m, z_m])
def code(y_s, x_s, x_m, y_m, z_m):
	return y_s * (x_s * (1.0 / (x_m * y_m)))

Julia

z_m = abs(z)
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_m = sort([x_m, y_m, z_m])
function code(y_s, x_s, x_m, y_m, z_m)
	return Float64(y_s * Float64(x_s * Float64(1.0 / Float64(x_m * y_m))))
end

MATLAB

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

Wolfram

z_m = N[Abs[z], $MachinePrecision]
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_m should be sorted in increasing order before calling this function.
code[y$95$s_, x$95$s_, x$95$m_, y$95$m_, z$95$m_] := N[(y$95$s * N[(x$95$s * N[(1.0 / N[(x$95$m * y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 90.4%

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

   1. associate-/r* 90.3%

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

   2. sqr-neg 90.3%

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

   3. +-commutative 90.3%

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

   4. sqr-neg 90.3%

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

   5. fma-def 90.3%

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

3. Simplified 90.3%

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

4. Taylor expanded in z around 0 61.2%

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

5. Final simplification 61.2%

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

Alternative 10: 57.7% accurate, 2.2× speedup

Math

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

FPCore

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

C

z_m = fabs(z);
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_m);
double code(double y_s, double x_s, double x_m, double y_m, double z_m) {
	return y_s * (x_s * ((1.0 / x_m) / y_m));
}

Fortran

z_m = abs(z)
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_m should be sorted in increasing order before calling this function.
real(8) function code(y_s, x_s, x_m, y_m, z_m)
    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_m
    code = y_s * (x_s * ((1.0d0 / x_m) / y_m))
end function

Java

z_m = Math.abs(z);
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_m;
public static double code(double y_s, double x_s, double x_m, double y_m, double z_m) {
	return y_s * (x_s * ((1.0 / x_m) / y_m));
}

Python

z_m = math.fabs(z)
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_m] = sort([x_m, y_m, z_m])
def code(y_s, x_s, x_m, y_m, z_m):
	return y_s * (x_s * ((1.0 / x_m) / y_m))

Julia

z_m = abs(z)
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_m = sort([x_m, y_m, z_m])
function code(y_s, x_s, x_m, y_m, z_m)
	return Float64(y_s * Float64(x_s * Float64(Float64(1.0 / x_m) / y_m)))
end

MATLAB

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

Wolfram

z_m = N[Abs[z], $MachinePrecision]
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_m should be sorted in increasing order before calling this function.
code[y$95$s_, x$95$s_, x$95$m_, y$95$m_, z$95$m_] := N[(y$95$s * N[(x$95$s * N[(N[(1.0 / x$95$m), $MachinePrecision] / y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 90.4%

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

   1. associate-/l/ 89.4%

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

   2. metadata-eval 89.4%

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

   3. associate-/r* 89.4%

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

   4. metadata-eval 89.4%

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

   5. neg-mul-1 89.4%

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

   6. distribute-neg-frac 89.4%

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

   7. distribute-frac-neg 89.4%

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

   8. distribute-frac-neg 89.4%

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

   9. distribute-neg-frac 89.4%

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

   10. metadata-eval 89.4%

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

   11. neg-mul-1 89.4%

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

   12. associate-/r* 89.4%

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

   13. metadata-eval 89.4%

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

   14. associate-/r* 89.3%

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

   15. sqr-neg 89.3%

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

   16. +-commutative 89.3%

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

   17. sqr-neg 89.3%

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

   18. fma-def 89.3%

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

3. Simplified 89.3%

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

4. Taylor expanded in z around 0 61.3%

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

5. Final simplification 61.3%

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

Alternative 11: 57.7% accurate, 2.2× speedup

Math

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

FPCore

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

C

z_m = fabs(z);
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_m);
double code(double y_s, double x_s, double x_m, double y_m, double z_m) {
	return y_s * (x_s * ((1.0 / y_m) / x_m));
}

Fortran

z_m = abs(z)
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_m should be sorted in increasing order before calling this function.
real(8) function code(y_s, x_s, x_m, y_m, z_m)
    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_m
    code = y_s * (x_s * ((1.0d0 / y_m) / x_m))
end function

Java

z_m = Math.abs(z);
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_m;
public static double code(double y_s, double x_s, double x_m, double y_m, double z_m) {
	return y_s * (x_s * ((1.0 / y_m) / x_m));
}

Python

z_m = math.fabs(z)
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_m] = sort([x_m, y_m, z_m])
def code(y_s, x_s, x_m, y_m, z_m):
	return y_s * (x_s * ((1.0 / y_m) / x_m))

Julia

z_m = abs(z)
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_m = sort([x_m, y_m, z_m])
function code(y_s, x_s, x_m, y_m, z_m)
	return Float64(y_s * Float64(x_s * Float64(Float64(1.0 / y_m) / x_m)))
end

MATLAB

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

Wolfram

z_m = N[Abs[z], $MachinePrecision]
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_m should be sorted in increasing order before calling this function.
code[y$95$s_, x$95$s_, x$95$m_, y$95$m_, z$95$m_] := N[(y$95$s * N[(x$95$s * N[(N[(1.0 / y$95$m), $MachinePrecision] / x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 90.4%

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

   1. associate-/r* 90.3%

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

   2. sqr-neg 90.3%

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

   3. +-commutative 90.3%

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

   4. sqr-neg 90.3%

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

   5. fma-def 90.3%

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

3. Simplified 90.3%

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

4. Taylor expanded in z around 0 61.2%

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

   1. *-commutative 61.2%

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

   2. associate-/r* 61.3%

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

6. Simplified 61.3%

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

7. Final simplification 61.3%

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

Developer target: 92.4% accurate, 0.4× 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 2023333 
(FPCore (x y z)
  :name "Statistics.Distribution.CauchyLorentz:$cdensity from math-functions-0.1.5.2"
  :precision binary64

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

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