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

Percentage Accurate: 88.9% → 98.6%

Time: 11.8s

Alternatives: 12

Speedup: 0.8×

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: 88.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.6% accurate, 0.1× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ [x_m, y_m, z_m] = \mathsf{sort}([x_m, y_m, z_m])\\ \\ y_s \cdot \left(x_s \cdot \begin{array}{l} \mathbf{if}\;z_m \cdot z_m \leq 10^{+276}:\\ \;\;\;\;\frac{1}{y_m \cdot \left(x_m \cdot \mathsf{fma}\left(z_m, z_m, 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{1}{z_m}}{y_m}}{x_m \cdot \mathsf{hypot}\left(1, z_m\right)}\\ \end{array}\right) \end{array} \]

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 z_m) 1e+276)
     (/ 1.0 (* y_m (* x_m (fma z_m z_m 1.0))))
     (/ (/ (/ 1.0 z_m) y_m) (* x_m (hypot 1.0 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 * z_m) <= 1e+276) {
		tmp = 1.0 / (y_m * (x_m * fma(z_m, z_m, 1.0)));
	} else {
		tmp = ((1.0 / z_m) / y_m) / (x_m * hypot(1.0, 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 (Float64(z_m * z_m) <= 1e+276)
		tmp = Float64(1.0 / Float64(y_m * Float64(x_m * fma(z_m, z_m, 1.0))));
	else
		tmp = Float64(Float64(Float64(1.0 / z_m) / y_m) / Float64(x_m * hypot(1.0, 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[N[(z$95$m * z$95$m), $MachinePrecision], 1e+276], N[(1.0 / N[(y$95$m * N[(x$95$m * N[(z$95$m * z$95$m + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 / z$95$m), $MachinePrecision] / y$95$m), $MachinePrecision] / N[(x$95$m * N[Sqrt[1.0 ^ 2 + z$95$m ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 1.0000000000000001e276

   1. Initial program 97.1%

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

      1. associate-/r* 97.0%

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

      2. sqr-neg 97.0%

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

      3. +-commutative 97.0%

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

      4. sqr-neg 97.0%

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

      5. fma-def 97.0%

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

   3. Simplified 97.0%

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

      1. associate-/r* 97.1%

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

      2. associate-/l/ 95.3%

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

      3. associate-/r* 95.0%

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

      4. add-sqr-sqrt 54.2%

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

      5. *-un-lft-identity 54.2%

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

      6. times-frac 54.3%

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

      7. associate-/r* 53.7%

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

      8. sqrt-div 51.8%

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

      9. inv-pow 51.8%

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

      10. sqrt-pow1 51.9%

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

      11. metadata-eval 51.9%

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

      12. fma-udef 51.9%

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

      13. +-commutative 51.9%

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

      14. hypot-1-def 51.9%

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

   5. Applied egg-rr 54.4%

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

      1. /-rgt-identity 54.4%

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

      2. associate-*r/ 52.0%

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

   7. Applied egg-rr 95.0%

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

   if 1.0000000000000001e276 < (*.f64 z z)

   1. Initial program 75.2%

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

      1. associate-/r* 75.2%

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

      2. sqr-neg 75.2%

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

      3. +-commutative 75.2%

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

      4. sqr-neg 75.2%

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

      5. fma-def 75.2%

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

   3. Simplified 75.2%

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

   4. Taylor expanded in x around 0 75.2%

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

      1. associate-/r* 75.2%

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

      2. *-commutative 75.2%

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

      3. +-commutative 75.2%

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

      4. unpow2 75.2%

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

      5. fma-udef 75.2%

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

      6. associate-/l/ 74.5%

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

      7. associate-/l/ 74.5%

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

      8. associate-/r* 74.5%

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

   6. Simplified 74.5%

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

      1. div-inv 74.5%

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

      2. add-sqr-sqrt 74.5%

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

      3. times-frac 75.2%

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

      4. fma-udef 75.2%

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

      5. +-commutative 75.2%

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

      6. hypot-1-def 75.2%

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

      7. fma-udef 75.2%

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

      8. +-commutative 75.2%

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

      9. hypot-1-def 99.8%

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

   8. Applied egg-rr 99.8%

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

      1. associate-/l/ 99.8%

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

      2. associate-/r* 99.8%

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

   10. Simplified 99.8%

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

   11. Taylor expanded in z around inf 84.1%

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

       1. associate-/l/ 84.1%

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

       2. un-div-inv 84.2%

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

       3. *-commutative 84.2%

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

       4. associate-/r* 84.1%

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

   13. Applied egg-rr 84.1%

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

4. Final simplification 91.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 10^{+276}:\\ \;\;\;\;\frac{1}{y \cdot \left(x \cdot \mathsf{fma}\left(z, z, 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{1}{z}}{y}}{x \cdot \mathsf{hypot}\left(1, z\right)}\\ \end{array} \]

Alternative 2: 99.4% accurate, 0.0× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ [x_m, y_m, z_m] = \mathsf{sort}([x_m, y_m, z_m])\\ \\ \begin{array}{l} t_0 := \frac{{x_m}^{-0.5}}{\mathsf{hypot}\left(1, z_m\right)}\\ y_s \cdot \left(x_s \cdot \left(t_0 \cdot \frac{t_0}{y_m}\right)\right) \end{array} \end{array} \]

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 (/ (pow x_m -0.5) (hypot 1.0 z_m))))
   (* y_s (* x_s (* t_0 (/ t_0 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 t_0 = pow(x_m, -0.5) / hypot(1.0, z_m);
	return y_s * (x_s * (t_0 * (t_0 / 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) {
	double t_0 = Math.pow(x_m, -0.5) / Math.hypot(1.0, z_m);
	return y_s * (x_s * (t_0 * (t_0 / 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):
	t_0 = math.pow(x_m, -0.5) / math.hypot(1.0, z_m)
	return y_s * (x_s * (t_0 * (t_0 / 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)
	t_0 = Float64((x_m ^ -0.5) / hypot(1.0, z_m))
	return Float64(y_s * Float64(x_s * Float64(t_0 * Float64(t_0 / 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)
	t_0 = (x_m ^ -0.5) / hypot(1.0, z_m);
	tmp = y_s * (x_s * (t_0 * (t_0 / 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_] := Block[{t$95$0 = N[(N[Power[x$95$m, -0.5], $MachinePrecision] / N[Sqrt[1.0 ^ 2 + z$95$m ^ 2], $MachinePrecision]), $MachinePrecision]}, N[(y$95$s * N[(x$95$s * N[(t$95$0 * N[(t$95$0 / y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 90.5%

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

   1. associate-/r* 90.5%

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

   2. sqr-neg 90.5%

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

   3. +-commutative 90.5%

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

   4. sqr-neg 90.5%

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

   5. fma-def 90.5%

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

3. Simplified 90.5%

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

   1. associate-/r* 90.5%

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

   2. associate-/l/ 89.3%

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

   3. associate-/r* 89.1%

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

   4. add-sqr-sqrt 59.3%

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

   5. *-un-lft-identity 59.3%

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

   6. times-frac 59.4%

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

   7. associate-/r* 59.0%

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

   8. sqrt-div 46.9%

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

   9. inv-pow 46.9%

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

   10. sqrt-pow1 46.9%

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

   11. metadata-eval 46.9%

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

   12. fma-udef 46.9%

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

   13. +-commutative 46.9%

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

   14. hypot-1-def 46.9%

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

5. Applied egg-rr 53.2%

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

6. Final simplification 53.2%

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

Alternative 3: 99.2% accurate, 0.0× speedup

Math

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

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) (* (hypot 1.0 z_m) (sqrt y_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) / (hypot(1.0, z_m) * sqrt(y_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.hypot(1.0, z_m) * Math.sqrt(y_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.hypot(1.0, z_m) * math.sqrt(y_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(hypot(1.0, z_m) * sqrt(y_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) / (hypot(1.0, z_m) * sqrt(y_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[1.0 ^ 2 + z$95$m ^ 2], $MachinePrecision] * N[Sqrt[y$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 90.5%

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

   1. associate-/r* 90.5%

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

   2. sqr-neg 90.5%

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

   3. +-commutative 90.5%

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

   4. sqr-neg 90.5%

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

   5. fma-def 90.5%

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

3. Simplified 90.5%

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

   1. fma-udef 90.5%

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

   2. +-commutative 90.5%

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

   3. associate-/r* 90.5%

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

   4. add-sqr-sqrt 61.6%

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

   5. sqrt-div 21.1%

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

   6. inv-pow 21.1%

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

   7. sqrt-pow1 21.1%

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

   8. metadata-eval 21.1%

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

   9. +-commutative 21.1%

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

   10. fma-udef 21.1%

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

   11. sqrt-prod 21.1%

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

   12. fma-udef 21.1%

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

   13. +-commutative 21.1%

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

   14. hypot-1-def 21.1%

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

   15. sqrt-div 21.1%

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

   16. inv-pow 21.1%

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

   17. sqrt-pow1 21.1%

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

   18. metadata-eval 21.1%

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

5. Applied egg-rr 25.2%

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

   1. unpow2 25.2%

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

7. Simplified 25.2%

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

8. Final simplification 25.2%

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

Alternative 4: 98.6% accurate, 0.1× speedup

Math

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

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)) (* y_m (* x_m (hypot 1.0 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) {
	return y_s * (x_s * ((1.0 / hypot(1.0, z_m)) / (y_m * (x_m * hypot(1.0, z_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)) / (y_m * (x_m * Math.hypot(1.0, z_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)) / (y_m * (x_m * math.hypot(1.0, z_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 / hypot(1.0, z_m)) / Float64(y_m * Float64(x_m * hypot(1.0, z_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)) / (y_m * (x_m * hypot(1.0, z_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 / N[Sqrt[1.0 ^ 2 + z$95$m ^ 2], $MachinePrecision]), $MachinePrecision] / N[(y$95$m * N[(x$95$m * N[Sqrt[1.0 ^ 2 + z$95$m ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 90.5%

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

   1. associate-/r* 90.5%

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

   2. sqr-neg 90.5%

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

   3. +-commutative 90.5%

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

   4. sqr-neg 90.5%

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

   5. fma-def 90.5%

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

3. Simplified 90.5%

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

4. Taylor expanded in x around 0 90.5%

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

   1. associate-/r* 90.5%

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

   2. *-commutative 90.5%

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

   3. +-commutative 90.5%

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

   4. unpow2 90.5%

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

   5. fma-udef 90.5%

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

   6. associate-/l/ 91.4%

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

   7. associate-/l/ 91.4%

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

   8. associate-/r* 91.4%

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

6. Simplified 91.4%

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

   1. div-inv 91.3%

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

   2. add-sqr-sqrt 91.3%

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

   3. times-frac 90.3%

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

   4. fma-udef 90.3%

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

   5. +-commutative 90.3%

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

   6. hypot-1-def 90.3%

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

   7. fma-udef 90.3%

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

   8. +-commutative 90.3%

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

   9. hypot-1-def 97.7%

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

8. Applied egg-rr 97.7%

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

   1. associate-/l/ 97.4%

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

   2. associate-/r* 97.7%

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

10. Simplified 97.7%

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

    1. *-commutative 97.7%

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

    2. associate-/l/ 97.7%

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

    3. frac-times 98.4%

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

    4. *-un-lft-identity 98.4%

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

12. Applied egg-rr 98.4%

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

13. Final simplification 98.4%

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

Alternative 5: 98.6% accurate, 0.1× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ [x_m, y_m, z_m] = \mathsf{sort}([x_m, y_m, z_m])\\ \\ y_s \cdot \left(x_s \cdot \begin{array}{l} \mathbf{if}\;z_m \cdot z_m \leq 200000:\\ \;\;\;\;\frac{\frac{1}{x_m}}{y_m \cdot \left(1 + z_m \cdot z_m\right)}\\ \mathbf{elif}\;z_m \cdot z_m \leq 10^{+276}:\\ \;\;\;\;\frac{1}{y_m \cdot \left(x_m \cdot {z_m}^{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{z_m \cdot y_m} \cdot \frac{1}{x_m \cdot z_m}\\ \end{array}\right) \end{array} \]

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 z_m) 200000.0)
     (/ (/ 1.0 x_m) (* y_m (+ 1.0 (* z_m z_m))))
     (if (<= (* z_m z_m) 1e+276)
       (/ 1.0 (* y_m (* x_m (pow z_m 2.0))))
       (* (/ 1.0 (* z_m y_m)) (/ 1.0 (* 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 * z_m) <= 200000.0) {
		tmp = (1.0 / x_m) / (y_m * (1.0 + (z_m * z_m)));
	} else if ((z_m * z_m) <= 1e+276) {
		tmp = 1.0 / (y_m * (x_m * pow(z_m, 2.0)));
	} else {
		tmp = (1.0 / (z_m * y_m)) * (1.0 / (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 * z_m) <= 200000.0d0) then
        tmp = (1.0d0 / x_m) / (y_m * (1.0d0 + (z_m * z_m)))
    else if ((z_m * z_m) <= 1d+276) then
        tmp = 1.0d0 / (y_m * (x_m * (z_m ** 2.0d0)))
    else
        tmp = (1.0d0 / (z_m * y_m)) * (1.0d0 / (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 * z_m) <= 200000.0) {
		tmp = (1.0 / x_m) / (y_m * (1.0 + (z_m * z_m)));
	} else if ((z_m * z_m) <= 1e+276) {
		tmp = 1.0 / (y_m * (x_m * Math.pow(z_m, 2.0)));
	} else {
		tmp = (1.0 / (z_m * y_m)) * (1.0 / (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 * z_m) <= 200000.0:
		tmp = (1.0 / x_m) / (y_m * (1.0 + (z_m * z_m)))
	elif (z_m * z_m) <= 1e+276:
		tmp = 1.0 / (y_m * (x_m * math.pow(z_m, 2.0)))
	else:
		tmp = (1.0 / (z_m * y_m)) * (1.0 / (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 (Float64(z_m * z_m) <= 200000.0)
		tmp = Float64(Float64(1.0 / x_m) / Float64(y_m * Float64(1.0 + Float64(z_m * z_m))));
	elseif (Float64(z_m * z_m) <= 1e+276)
		tmp = Float64(1.0 / Float64(y_m * Float64(x_m * (z_m ^ 2.0))));
	else
		tmp = Float64(Float64(1.0 / Float64(z_m * y_m)) * Float64(1.0 / 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 * z_m) <= 200000.0)
		tmp = (1.0 / x_m) / (y_m * (1.0 + (z_m * z_m)));
	elseif ((z_m * z_m) <= 1e+276)
		tmp = 1.0 / (y_m * (x_m * (z_m ^ 2.0)));
	else
		tmp = (1.0 / (z_m * y_m)) * (1.0 / (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[N[(z$95$m * z$95$m), $MachinePrecision], 200000.0], N[(N[(1.0 / x$95$m), $MachinePrecision] / N[(y$95$m * N[(1.0 + N[(z$95$m * z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(z$95$m * z$95$m), $MachinePrecision], 1e+276], N[(1.0 / N[(y$95$m * N[(x$95$m * N[Power[z$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[(z$95$m * y$95$m), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(x$95$m * z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if (*.f64 z z) < 2e5

   1. Initial program 99.7%

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

   if 2e5 < (*.f64 z z) < 1.0000000000000001e276

   1. Initial program 91.1%

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

      1. associate-/r* 91.1%

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

      2. sqr-neg 91.1%

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

      3. +-commutative 91.1%

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

      4. sqr-neg 91.1%

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

      5. fma-def 91.1%

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

   3. Simplified 91.1%

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

   4. Taylor expanded in x around 0 91.1%

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

      1. associate-/r* 91.1%

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

      2. *-commutative 91.1%

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

      3. +-commutative 91.1%

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

      4. unpow2 91.1%

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

      5. fma-udef 91.1%

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

      6. associate-/l/ 96.2%

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

      7. associate-/l/ 96.1%

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

      8. associate-/r* 96.2%

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

   6. Simplified 96.2%

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

      1. div-inv 96.0%

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

      2. add-sqr-sqrt 96.0%

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

      3. times-frac 90.6%

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

      4. fma-udef 90.6%

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

      5. +-commutative 90.6%

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

      6. hypot-1-def 90.6%

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

      7. fma-udef 90.6%

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

      8. +-commutative 90.6%

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

      9. hypot-1-def 90.6%

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

   8. Applied egg-rr 90.6%

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

      1. associate-/l/ 89.3%

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

      2. associate-/r* 90.5%

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

   10. Simplified 90.5%

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

   11. Taylor expanded in z around inf 91.1%

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

       1. associate-*r* 96.2%

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

       2. *-commutative 96.2%

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

       3. associate-*r* 84.4%

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

   13. Simplified 84.4%

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

   if 1.0000000000000001e276 < (*.f64 z z)

   1. Initial program 75.2%

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

      1. associate-/r* 75.2%

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

      2. sqr-neg 75.2%

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

      3. +-commutative 75.2%

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

      4. sqr-neg 75.2%

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

      5. fma-def 75.2%

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

   3. Simplified 75.2%

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

   4. Taylor expanded in x around 0 75.2%

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

      1. associate-/r* 75.2%

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

      2. *-commutative 75.2%

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

      3. +-commutative 75.2%

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

      4. unpow2 75.2%

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

      5. fma-udef 75.2%

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

      6. associate-/l/ 74.5%

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

      7. associate-/l/ 74.5%

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

      8. associate-/r* 74.5%

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

   6. Simplified 74.5%

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

      1. div-inv 74.5%

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

      2. add-sqr-sqrt 74.5%

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

      3. times-frac 75.2%

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

      4. fma-udef 75.2%

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

      5. +-commutative 75.2%

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

      6. hypot-1-def 75.2%

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

      7. fma-udef 75.2%

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

      8. +-commutative 75.2%

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

      9. hypot-1-def 99.8%

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

   8. Applied egg-rr 99.8%

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

      1. associate-/l/ 99.8%

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

      2. associate-/r* 99.8%

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

   10. Simplified 99.8%

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

   11. Taylor expanded in z around inf 84.1%

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

   12. Taylor expanded in z around inf 99.8%

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

4. Final simplification 96.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 200000:\\ \;\;\;\;\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}\\ \mathbf{elif}\;z \cdot z \leq 10^{+276}:\\ \;\;\;\;\frac{1}{y \cdot \left(x \cdot {z}^{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{z \cdot y} \cdot \frac{1}{x \cdot z}\\ \end{array} \]

Alternative 6: 98.3% accurate, 0.1× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ [x_m, y_m, z_m] = \mathsf{sort}([x_m, y_m, z_m])\\ \\ y_s \cdot \left(x_s \cdot \begin{array}{l} \mathbf{if}\;z_m \cdot z_m \leq 200000:\\ \;\;\;\;\frac{1}{x_m \cdot \left(y_m \cdot \mathsf{fma}\left(z_m, z_m, 1\right)\right)}\\ \mathbf{elif}\;z_m \cdot z_m \leq 10^{+276}:\\ \;\;\;\;\frac{1}{y_m \cdot \left(x_m \cdot {z_m}^{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{z_m \cdot y_m} \cdot \frac{1}{x_m \cdot z_m}\\ \end{array}\right) \end{array} \]

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 z_m) 200000.0)
     (/ 1.0 (* x_m (* y_m (fma z_m z_m 1.0))))
     (if (<= (* z_m z_m) 1e+276)
       (/ 1.0 (* y_m (* x_m (pow z_m 2.0))))
       (* (/ 1.0 (* z_m y_m)) (/ 1.0 (* 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 * z_m) <= 200000.0) {
		tmp = 1.0 / (x_m * (y_m * fma(z_m, z_m, 1.0)));
	} else if ((z_m * z_m) <= 1e+276) {
		tmp = 1.0 / (y_m * (x_m * pow(z_m, 2.0)));
	} else {
		tmp = (1.0 / (z_m * y_m)) * (1.0 / (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 (Float64(z_m * z_m) <= 200000.0)
		tmp = Float64(1.0 / Float64(x_m * Float64(y_m * fma(z_m, z_m, 1.0))));
	elseif (Float64(z_m * z_m) <= 1e+276)
		tmp = Float64(1.0 / Float64(y_m * Float64(x_m * (z_m ^ 2.0))));
	else
		tmp = Float64(Float64(1.0 / Float64(z_m * y_m)) * Float64(1.0 / 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[N[(z$95$m * z$95$m), $MachinePrecision], 200000.0], N[(1.0 / N[(x$95$m * N[(y$95$m * N[(z$95$m * z$95$m + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(z$95$m * z$95$m), $MachinePrecision], 1e+276], N[(1.0 / N[(y$95$m * N[(x$95$m * N[Power[z$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[(z$95$m * y$95$m), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(x$95$m * z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if (*.f64 z z) < 2e5

   1. Initial program 99.7%

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

      1. associate-/r* 99.7%

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

      2. sqr-neg 99.7%

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

      3. +-commutative 99.7%

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

      4. sqr-neg 99.7%

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

      5. fma-def 99.7%

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

   3. Simplified 99.7%

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

   if 2e5 < (*.f64 z z) < 1.0000000000000001e276

   1. Initial program 91.1%

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

      1. associate-/r* 91.1%

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

      2. sqr-neg 91.1%

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

      3. +-commutative 91.1%

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

      4. sqr-neg 91.1%

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

      5. fma-def 91.1%

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

   3. Simplified 91.1%

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

   4. Taylor expanded in x around 0 91.1%

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

      1. associate-/r* 91.1%

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

      2. *-commutative 91.1%

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

      3. +-commutative 91.1%

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

      4. unpow2 91.1%

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

      5. fma-udef 91.1%

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

      6. associate-/l/ 96.2%

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

      7. associate-/l/ 96.1%

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

      8. associate-/r* 96.2%

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

   6. Simplified 96.2%

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

      1. div-inv 96.0%

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

      2. add-sqr-sqrt 96.0%

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

      3. times-frac 90.6%

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

      4. fma-udef 90.6%

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

      5. +-commutative 90.6%

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

      6. hypot-1-def 90.6%

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

      7. fma-udef 90.6%

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

      8. +-commutative 90.6%

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

      9. hypot-1-def 90.6%

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

   8. Applied egg-rr 90.6%

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

      1. associate-/l/ 89.3%

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

      2. associate-/r* 90.5%

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

   10. Simplified 90.5%

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

   11. Taylor expanded in z around inf 91.1%

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

       1. associate-*r* 96.2%

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

       2. *-commutative 96.2%

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

       3. associate-*r* 84.4%

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

   13. Simplified 84.4%

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

   if 1.0000000000000001e276 < (*.f64 z z)

   1. Initial program 75.2%

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

      1. associate-/r* 75.2%

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

      2. sqr-neg 75.2%

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

      3. +-commutative 75.2%

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

      4. sqr-neg 75.2%

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

      5. fma-def 75.2%

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

   3. Simplified 75.2%

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

   4. Taylor expanded in x around 0 75.2%

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

      1. associate-/r* 75.2%

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

      2. *-commutative 75.2%

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

      3. +-commutative 75.2%

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

      4. unpow2 75.2%

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

      5. fma-udef 75.2%

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

      6. associate-/l/ 74.5%

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

      7. associate-/l/ 74.5%

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

      8. associate-/r* 74.5%

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

   6. Simplified 74.5%

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

      1. div-inv 74.5%

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

      2. add-sqr-sqrt 74.5%

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

      3. times-frac 75.2%

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

      4. fma-udef 75.2%

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

      5. +-commutative 75.2%

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

      6. hypot-1-def 75.2%

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

      7. fma-udef 75.2%

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

      8. +-commutative 75.2%

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

      9. hypot-1-def 99.8%

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

   8. Applied egg-rr 99.8%

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

      1. associate-/l/ 99.8%

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

      2. associate-/r* 99.8%

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

   10. Simplified 99.8%

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

   11. Taylor expanded in z around inf 84.1%

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

   12. Taylor expanded in z around inf 99.8%

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

4. Final simplification 96.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 200000:\\ \;\;\;\;\frac{1}{x \cdot \left(y \cdot \mathsf{fma}\left(z, z, 1\right)\right)}\\ \mathbf{elif}\;z \cdot z \leq 10^{+276}:\\ \;\;\;\;\frac{1}{y \cdot \left(x \cdot {z}^{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{z \cdot y} \cdot \frac{1}{x \cdot z}\\ \end{array} \]

Alternative 7: 98.5% accurate, 0.1× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ [x_m, y_m, z_m] = \mathsf{sort}([x_m, y_m, z_m])\\ \\ y_s \cdot \left(x_s \cdot \begin{array}{l} \mathbf{if}\;z_m \cdot z_m \leq 10^{+276}:\\ \;\;\;\;\frac{1}{y_m \cdot \left(x_m \cdot \mathsf{fma}\left(z_m, z_m, 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{z_m \cdot y_m} \cdot \frac{1}{x_m \cdot z_m}\\ \end{array}\right) \end{array} \]

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 z_m) 1e+276)
     (/ 1.0 (* y_m (* x_m (fma z_m z_m 1.0))))
     (* (/ 1.0 (* z_m y_m)) (/ 1.0 (* 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 * z_m) <= 1e+276) {
		tmp = 1.0 / (y_m * (x_m * fma(z_m, z_m, 1.0)));
	} else {
		tmp = (1.0 / (z_m * y_m)) * (1.0 / (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 (Float64(z_m * z_m) <= 1e+276)
		tmp = Float64(1.0 / Float64(y_m * Float64(x_m * fma(z_m, z_m, 1.0))));
	else
		tmp = Float64(Float64(1.0 / Float64(z_m * y_m)) * Float64(1.0 / 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[N[(z$95$m * z$95$m), $MachinePrecision], 1e+276], N[(1.0 / N[(y$95$m * N[(x$95$m * N[(z$95$m * z$95$m + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[(z$95$m * y$95$m), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(x$95$m * z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 1.0000000000000001e276

   1. Initial program 97.1%

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

      1. associate-/r* 97.0%

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

      2. sqr-neg 97.0%

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

      3. +-commutative 97.0%

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

      4. sqr-neg 97.0%

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

      5. fma-def 97.0%

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

   3. Simplified 97.0%

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

      1. associate-/r* 97.1%

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

      2. associate-/l/ 95.3%

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

      3. associate-/r* 95.0%

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

      4. add-sqr-sqrt 54.2%

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

      5. *-un-lft-identity 54.2%

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

      6. times-frac 54.3%

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

      7. associate-/r* 53.7%

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

      8. sqrt-div 51.8%

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

      9. inv-pow 51.8%

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

      10. sqrt-pow1 51.9%

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

      11. metadata-eval 51.9%

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

      12. fma-udef 51.9%

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

      13. +-commutative 51.9%

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

      14. hypot-1-def 51.9%

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

   5. Applied egg-rr 54.4%

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

      1. /-rgt-identity 54.4%

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

      2. associate-*r/ 52.0%

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

   7. Applied egg-rr 95.0%

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

   if 1.0000000000000001e276 < (*.f64 z z)

   1. Initial program 75.2%

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

      1. associate-/r* 75.2%

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

      2. sqr-neg 75.2%

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

      3. +-commutative 75.2%

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

      4. sqr-neg 75.2%

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

      5. fma-def 75.2%

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

   3. Simplified 75.2%

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

   4. Taylor expanded in x around 0 75.2%

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

      1. associate-/r* 75.2%

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

      2. *-commutative 75.2%

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

      3. +-commutative 75.2%

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

      4. unpow2 75.2%

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

      5. fma-udef 75.2%

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

      6. associate-/l/ 74.5%

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

      7. associate-/l/ 74.5%

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

      8. associate-/r* 74.5%

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

   6. Simplified 74.5%

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

      1. div-inv 74.5%

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

      2. add-sqr-sqrt 74.5%

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

      3. times-frac 75.2%

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

      4. fma-udef 75.2%

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

      5. +-commutative 75.2%

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

      6. hypot-1-def 75.2%

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

      7. fma-udef 75.2%

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

      8. +-commutative 75.2%

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

      9. hypot-1-def 99.8%

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

   8. Applied egg-rr 99.8%

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

      1. associate-/l/ 99.8%

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

      2. associate-/r* 99.8%

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

   10. Simplified 99.8%

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

   11. Taylor expanded in z around inf 84.1%

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

   12. Taylor expanded in z around inf 99.8%

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

4. Final simplification 96.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 10^{+276}:\\ \;\;\;\;\frac{1}{y \cdot \left(x \cdot \mathsf{fma}\left(z, z, 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{z \cdot y} \cdot \frac{1}{x \cdot z}\\ \end{array} \]

Alternative 8: 98.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ [x_m, y_m, z_m] = \mathsf{sort}([x_m, y_m, z_m])\\ \\ y_s \cdot \left(x_s \cdot \begin{array}{l} \mathbf{if}\;z_m \cdot z_m \leq 200000:\\ \;\;\;\;\frac{\frac{1}{x_m}}{y_m \cdot \left(1 + z_m \cdot z_m\right)}\\ \mathbf{elif}\;z_m \cdot z_m \leq 10^{+276}:\\ \;\;\;\;\frac{\frac{1}{z_m} \cdot \frac{\frac{1}{x_m}}{z_m}}{y_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{z_m \cdot y_m} \cdot \frac{1}{x_m \cdot z_m}\\ \end{array}\right) \end{array} \]

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 z_m) 200000.0)
     (/ (/ 1.0 x_m) (* y_m (+ 1.0 (* z_m z_m))))
     (if (<= (* z_m z_m) 1e+276)
       (/ (* (/ 1.0 z_m) (/ (/ 1.0 x_m) z_m)) y_m)
       (* (/ 1.0 (* z_m y_m)) (/ 1.0 (* 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 * z_m) <= 200000.0) {
		tmp = (1.0 / x_m) / (y_m * (1.0 + (z_m * z_m)));
	} else if ((z_m * z_m) <= 1e+276) {
		tmp = ((1.0 / z_m) * ((1.0 / x_m) / z_m)) / y_m;
	} else {
		tmp = (1.0 / (z_m * y_m)) * (1.0 / (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 * z_m) <= 200000.0d0) then
        tmp = (1.0d0 / x_m) / (y_m * (1.0d0 + (z_m * z_m)))
    else if ((z_m * z_m) <= 1d+276) then
        tmp = ((1.0d0 / z_m) * ((1.0d0 / x_m) / z_m)) / y_m
    else
        tmp = (1.0d0 / (z_m * y_m)) * (1.0d0 / (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 * z_m) <= 200000.0) {
		tmp = (1.0 / x_m) / (y_m * (1.0 + (z_m * z_m)));
	} else if ((z_m * z_m) <= 1e+276) {
		tmp = ((1.0 / z_m) * ((1.0 / x_m) / z_m)) / y_m;
	} else {
		tmp = (1.0 / (z_m * y_m)) * (1.0 / (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 * z_m) <= 200000.0:
		tmp = (1.0 / x_m) / (y_m * (1.0 + (z_m * z_m)))
	elif (z_m * z_m) <= 1e+276:
		tmp = ((1.0 / z_m) * ((1.0 / x_m) / z_m)) / y_m
	else:
		tmp = (1.0 / (z_m * y_m)) * (1.0 / (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 (Float64(z_m * z_m) <= 200000.0)
		tmp = Float64(Float64(1.0 / x_m) / Float64(y_m * Float64(1.0 + Float64(z_m * z_m))));
	elseif (Float64(z_m * z_m) <= 1e+276)
		tmp = Float64(Float64(Float64(1.0 / z_m) * Float64(Float64(1.0 / x_m) / z_m)) / y_m);
	else
		tmp = Float64(Float64(1.0 / Float64(z_m * y_m)) * Float64(1.0 / 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 * z_m) <= 200000.0)
		tmp = (1.0 / x_m) / (y_m * (1.0 + (z_m * z_m)));
	elseif ((z_m * z_m) <= 1e+276)
		tmp = ((1.0 / z_m) * ((1.0 / x_m) / z_m)) / y_m;
	else
		tmp = (1.0 / (z_m * y_m)) * (1.0 / (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[N[(z$95$m * z$95$m), $MachinePrecision], 200000.0], N[(N[(1.0 / x$95$m), $MachinePrecision] / N[(y$95$m * N[(1.0 + N[(z$95$m * z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(z$95$m * z$95$m), $MachinePrecision], 1e+276], N[(N[(N[(1.0 / z$95$m), $MachinePrecision] * N[(N[(1.0 / x$95$m), $MachinePrecision] / z$95$m), $MachinePrecision]), $MachinePrecision] / y$95$m), $MachinePrecision], N[(N[(1.0 / N[(z$95$m * y$95$m), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(x$95$m * z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if (*.f64 z z) < 2e5

   1. Initial program 99.7%

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

   if 2e5 < (*.f64 z z) < 1.0000000000000001e276

   1. Initial program 91.1%

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

      1. associate-/l/ 85.4%

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

      2. metadata-eval 85.4%

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

      3. associate-/r* 85.4%

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

      4. metadata-eval 85.4%

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

      5. neg-mul-1 85.4%

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

      6. distribute-neg-frac 85.4%

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

      7. distribute-frac-neg 85.4%

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

      8. distribute-frac-neg 85.4%

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

      9. distribute-neg-frac 85.4%

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

      10. metadata-eval 85.4%

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

      11. neg-mul-1 85.4%

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

      12. associate-/r* 85.4%

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

      13. metadata-eval 85.4%

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

      14. associate-/r* 84.5%

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

      15. sqr-neg 84.5%

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

      16. +-commutative 84.5%

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

      17. sqr-neg 84.5%

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

      18. fma-def 84.5%

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

   3. Simplified 84.5%

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

   4. Taylor expanded in z around inf 84.5%

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

      1. associate-/r* 85.4%

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

      2. *-un-lft-identity 85.4%

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

      3. unpow2 85.4%

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

      4. times-frac 85.4%

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

   6. Applied egg-rr 85.4%

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

   if 1.0000000000000001e276 < (*.f64 z z)

   1. Initial program 75.2%

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

      1. associate-/r* 75.2%

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

      2. sqr-neg 75.2%

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

      3. +-commutative 75.2%

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

      4. sqr-neg 75.2%

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

      5. fma-def 75.2%

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

   3. Simplified 75.2%

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

   4. Taylor expanded in x around 0 75.2%

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

      1. associate-/r* 75.2%

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

      2. *-commutative 75.2%

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

      3. +-commutative 75.2%

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

      4. unpow2 75.2%

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

      5. fma-udef 75.2%

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

      6. associate-/l/ 74.5%

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

      7. associate-/l/ 74.5%

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

      8. associate-/r* 74.5%

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

   6. Simplified 74.5%

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

      1. div-inv 74.5%

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

      2. add-sqr-sqrt 74.5%

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

      3. times-frac 75.2%

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

      4. fma-udef 75.2%

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

      5. +-commutative 75.2%

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

      6. hypot-1-def 75.2%

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

      7. fma-udef 75.2%

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

      8. +-commutative 75.2%

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

      9. hypot-1-def 99.8%

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

   8. Applied egg-rr 99.8%

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

      1. associate-/l/ 99.8%

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

      2. associate-/r* 99.8%

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

   10. Simplified 99.8%

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

   11. Taylor expanded in z around inf 84.1%

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

   12. Taylor expanded in z around inf 99.8%

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

4. Final simplification 96.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 200000:\\ \;\;\;\;\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}\\ \mathbf{elif}\;z \cdot z \leq 10^{+276}:\\ \;\;\;\;\frac{\frac{1}{z} \cdot \frac{\frac{1}{x}}{z}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{z \cdot y} \cdot \frac{1}{x \cdot z}\\ \end{array} \]

Alternative 9: 98.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ [x_m, y_m, z_m] = \mathsf{sort}([x_m, y_m, z_m])\\ \\ y_s \cdot \left(x_s \cdot \begin{array}{l} \mathbf{if}\;z_m \leq 1:\\ \;\;\;\;\frac{\frac{1}{y_m}}{x_m}\\ \mathbf{elif}\;z_m \leq 1.95 \cdot 10^{+139}:\\ \;\;\;\;\frac{\frac{1}{z_m} \cdot \frac{\frac{1}{x_m}}{z_m}}{y_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{z_m \cdot y_m} \cdot \frac{1}{x_m \cdot z_m}\\ \end{array}\right) \end{array} \]

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)
     (if (<= z_m 1.95e+139)
       (/ (* (/ 1.0 z_m) (/ (/ 1.0 x_m) z_m)) y_m)
       (* (/ 1.0 (* z_m y_m)) (/ 1.0 (* 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 if (z_m <= 1.95e+139) {
		tmp = ((1.0 / z_m) * ((1.0 / x_m) / z_m)) / y_m;
	} else {
		tmp = (1.0 / (z_m * y_m)) * (1.0 / (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 if (z_m <= 1.95d+139) then
        tmp = ((1.0d0 / z_m) * ((1.0d0 / x_m) / z_m)) / y_m
    else
        tmp = (1.0d0 / (z_m * y_m)) * (1.0d0 / (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 if (z_m <= 1.95e+139) {
		tmp = ((1.0 / z_m) * ((1.0 / x_m) / z_m)) / y_m;
	} else {
		tmp = (1.0 / (z_m * y_m)) * (1.0 / (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
	elif z_m <= 1.95e+139:
		tmp = ((1.0 / z_m) * ((1.0 / x_m) / z_m)) / y_m
	else:
		tmp = (1.0 / (z_m * y_m)) * (1.0 / (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);
	elseif (z_m <= 1.95e+139)
		tmp = Float64(Float64(Float64(1.0 / z_m) * Float64(Float64(1.0 / x_m) / z_m)) / y_m);
	else
		tmp = Float64(Float64(1.0 / Float64(z_m * y_m)) * Float64(1.0 / 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;
	elseif (z_m <= 1.95e+139)
		tmp = ((1.0 / z_m) * ((1.0 / x_m) / z_m)) / y_m;
	else
		tmp = (1.0 / (z_m * y_m)) * (1.0 / (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], If[LessEqual[z$95$m, 1.95e+139], N[(N[(N[(1.0 / z$95$m), $MachinePrecision] * N[(N[(1.0 / x$95$m), $MachinePrecision] / z$95$m), $MachinePrecision]), $MachinePrecision] / y$95$m), $MachinePrecision], N[(N[(1.0 / N[(z$95$m * y$95$m), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(x$95$m * z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if z < 1

   1. Initial program 92.9%

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

      1. associate-/r* 92.9%

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

      2. sqr-neg 92.9%

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

      3. +-commutative 92.9%

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

      4. sqr-neg 92.9%

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

      5. fma-def 92.9%

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

   3. Simplified 92.9%

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

   4. Taylor expanded in z around 0 70.3%

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

      1. *-commutative 70.3%

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

      2. associate-/r* 70.3%

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

   6. Simplified 70.3%

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

   if 1 < z < 1.95000000000000003e139

   1. Initial program 87.8%

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

      1. associate-/l/ 86.9%

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

      2. metadata-eval 86.9%

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

      3. associate-/r* 86.9%

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

      4. metadata-eval 86.9%

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

      5. neg-mul-1 86.9%

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

      6. distribute-neg-frac 86.9%

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

      7. distribute-frac-neg 86.9%

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

      8. distribute-frac-neg 86.9%

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

      9. distribute-neg-frac 86.9%

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

      10. metadata-eval 86.9%

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

      11. neg-mul-1 86.9%

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

      12. associate-/r* 86.9%

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

      13. metadata-eval 86.9%

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

      14. associate-/r* 85.4%

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

      15. sqr-neg 85.4%

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

      16. +-commutative 85.4%

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

      17. sqr-neg 85.4%

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

      18. fma-def 85.4%

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

   3. Simplified 85.4%

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

   4. Taylor expanded in z around inf 81.2%

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

      1. associate-/r* 82.8%

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

      2. *-un-lft-identity 82.8%

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

      3. unpow2 82.8%

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

      4. times-frac 82.8%

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

   6. Applied egg-rr 82.8%

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

   if 1.95000000000000003e139 < z

   1. Initial program 80.1%

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

      1. associate-/r* 80.1%

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

      2. sqr-neg 80.1%

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

      3. +-commutative 80.1%

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

      4. sqr-neg 80.1%

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

      5. fma-def 80.1%

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

   3. Simplified 80.1%

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

   4. Taylor expanded in x around 0 80.1%

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

      1. associate-/r* 80.1%

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

      2. *-commutative 80.1%

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

      3. +-commutative 80.1%

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

      4. unpow2 80.1%

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

      5. fma-udef 80.1%

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

      6. associate-/l/ 79.4%

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

      7. associate-/l/ 79.4%

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

      8. associate-/r* 79.4%

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

   6. Simplified 79.4%

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

      1. div-inv 79.4%

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

      2. add-sqr-sqrt 79.4%

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

      3. times-frac 80.1%

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

      4. fma-udef 80.1%

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

      5. +-commutative 80.1%

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

      6. hypot-1-def 80.1%

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

      7. fma-udef 80.1%

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

      8. +-commutative 80.1%

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

      9. hypot-1-def 99.7%

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

   8. Applied egg-rr 99.7%

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

      1. associate-/l/ 99.8%

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

      2. associate-/r* 99.8%

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

   10. Simplified 99.8%

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

   11. Taylor expanded in z around inf 99.8%

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

   12. Taylor expanded in z around inf 99.8%

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

4. Final simplification 76.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 1:\\ \;\;\;\;\frac{\frac{1}{y}}{x}\\ \mathbf{elif}\;z \leq 1.95 \cdot 10^{+139}:\\ \;\;\;\;\frac{\frac{1}{z} \cdot \frac{\frac{1}{x}}{z}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{z \cdot y} \cdot \frac{1}{x \cdot z}\\ \end{array} \]

Alternative 10: 96.9% accurate, 0.8× speedup

Math

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

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 y_m)) (/ 1.0 (* 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 * y_m)) * (1.0 / (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 * y_m)) * (1.0d0 / (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 * y_m)) * (1.0 / (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 * y_m)) * (1.0 / (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 / Float64(z_m * y_m)) * Float64(1.0 / 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 * y_m)) * (1.0 / (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 / N[(z$95$m * y$95$m), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(x$95$m * z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 1

   1. Initial program 92.9%

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

      1. associate-/r* 92.9%

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

      2. sqr-neg 92.9%

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

      3. +-commutative 92.9%

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

      4. sqr-neg 92.9%

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

      5. fma-def 92.9%

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

   3. Simplified 92.9%

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

   4. Taylor expanded in z around 0 70.3%

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

      1. *-commutative 70.3%

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

      2. associate-/r* 70.3%

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

   6. Simplified 70.3%

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

   if 1 < z

   1. Initial program 83.7%

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

      1. associate-/r* 83.7%

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

      2. sqr-neg 83.7%

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

      3. +-commutative 83.7%

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

      4. sqr-neg 83.7%

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

      5. fma-def 83.7%

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

   3. Simplified 83.7%

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

   4. Taylor expanded in x around 0 83.7%

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

      1. associate-/r* 83.7%

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

      2. *-commutative 83.7%

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

      3. +-commutative 83.7%

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

      4. unpow2 83.7%

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

      5. fma-udef 83.7%

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

      6. associate-/l/ 88.9%

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

      7. associate-/l/ 88.8%

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

      8. associate-/r* 88.9%

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

   6. Simplified 88.9%

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

      1. div-inv 88.8%

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

      2. add-sqr-sqrt 88.8%

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

      3. times-frac 86.1%

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

      4. fma-udef 86.1%

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

      5. +-commutative 86.1%

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

      6. hypot-1-def 86.1%

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

      7. fma-udef 86.1%

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

      8. +-commutative 86.1%

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

      9. hypot-1-def 96.5%

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

   8. Applied egg-rr 96.5%

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

      1. associate-/l/ 95.5%

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

      2. associate-/r* 96.5%

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

   10. Simplified 96.5%

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

   11. Taylor expanded in z around inf 93.6%

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

   12. Taylor expanded in z around inf 93.5%

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

4. Final simplification 76.5%

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

Alternative 11: 73.3% accurate, 1.2× speedup

Math

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

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

Derivation

1. Split input into 2 regimes

2. if z < 1

   1. Initial program 92.9%

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

      1. associate-/r* 92.9%

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

      2. sqr-neg 92.9%

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

      3. +-commutative 92.9%

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

      4. sqr-neg 92.9%

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

      5. fma-def 92.9%

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

   3. Simplified 92.9%

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

   4. Taylor expanded in z around 0 70.3%

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

      1. *-commutative 70.3%

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

      2. associate-/r* 70.3%

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

   6. Simplified 70.3%

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

   if 1 < z

   1. Initial program 83.7%

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

      1. associate-/r* 83.7%

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

      2. sqr-neg 83.7%

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

      3. +-commutative 83.7%

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

      4. sqr-neg 83.7%

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

      5. fma-def 83.7%

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

   3. Simplified 83.7%

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

   4. Taylor expanded in x around 0 83.7%

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

      1. associate-/r* 83.7%

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

      2. *-commutative 83.7%

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

      3. +-commutative 83.7%

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

      4. unpow2 83.7%

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

      5. fma-udef 83.7%

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

      6. associate-/l/ 88.9%

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

      7. associate-/l/ 88.8%

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

      8. associate-/r* 88.9%

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

   6. Simplified 88.9%

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

      1. div-inv 88.8%

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

      2. add-sqr-sqrt 88.8%

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

      3. times-frac 86.1%

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

      4. fma-udef 86.1%

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

      5. +-commutative 86.1%

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

      6. hypot-1-def 86.1%

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

      7. fma-udef 86.1%

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

      8. +-commutative 86.1%

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

      9. hypot-1-def 96.5%

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

   8. Applied egg-rr 96.5%

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

      1. associate-/l/ 95.5%

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

      2. associate-/r* 96.5%

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

   10. Simplified 96.5%

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

   11. Taylor expanded in z around inf 93.6%

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

   12. Taylor expanded in z around 0 47.8%

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

4. Final simplification 64.3%

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

Alternative 12: 59.2% 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 90.5%

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

   1. associate-/r* 90.5%

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

   2. sqr-neg 90.5%

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

   3. +-commutative 90.5%

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

   4. sqr-neg 90.5%

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

   5. fma-def 90.5%

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

3. Simplified 90.5%

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

4. Taylor expanded in z around 0 57.5%

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

5. Final simplification 57.5%

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

Developer target: 92.5% 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 2023334 
(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)))))