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

Percentage Accurate: 88.6% → 99.4%

Time: 13.1s

Alternatives: 10

Speedup: 0.7×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 88.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.4% accurate, 0.1× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;z \cdot z \leq 5 \cdot 10^{+296}:\\ \;\;\;\;\frac{\frac{1}{y\_m}}{x\_m \cdot \mathsf{fma}\left(z, z, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{z} \cdot \frac{\frac{1}{y\_m}}{x\_m \cdot z}\\ \end{array}\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (y_s x_s x_m y_m z)
 :precision binary64
 (*
  y_s
  (*
   x_s
   (if (<= (* z z) 5e+296)
     (/ (/ 1.0 y_m) (* x_m (fma z z 1.0)))
     (* (/ 1.0 z) (/ (/ 1.0 y_m) (* x_m z)))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
assert(x_m < y_m && y_m < z);
double code(double y_s, double x_s, double x_m, double y_m, double z) {
	double tmp;
	if ((z * z) <= 5e+296) {
		tmp = (1.0 / y_m) / (x_m * fma(z, z, 1.0));
	} else {
		tmp = (1.0 / z) * ((1.0 / y_m) / (x_m * z));
	}
	return y_s * (x_s * tmp);
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
y\_m = abs(y)
y\_s = copysign(1.0, y)
x_m, y_m, z = sort([x_m, y_m, z])
function code(y_s, x_s, x_m, y_m, z)
	tmp = 0.0
	if (Float64(z * z) <= 5e+296)
		tmp = Float64(Float64(1.0 / y_m) / Float64(x_m * fma(z, z, 1.0)));
	else
		tmp = Float64(Float64(1.0 / z) * Float64(Float64(1.0 / y_m) / Float64(x_m * z)));
	end
	return Float64(y_s * Float64(x_s * tmp))
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
code[y$95$s_, x$95$s_, x$95$m_, y$95$m_, z_] := N[(y$95$s * N[(x$95$s * If[LessEqual[N[(z * z), $MachinePrecision], 5e+296], N[(N[(1.0 / y$95$m), $MachinePrecision] / N[(x$95$m * N[(z * z + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / z), $MachinePrecision] * N[(N[(1.0 / y$95$m), $MachinePrecision] / N[(x$95$m * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 5.0000000000000001e296

   1. Initial program 96.0%

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

      1. associate-/l/ 95.3%

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

      2. associate-*l* 93.7%

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

      3. *-commutative 93.7%

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

      4. sqr-neg 93.7%

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

      5. +-commutative 93.7%

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

      6. sqr-neg 93.7%

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

      7. fma-define 93.7%

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

   3. Simplified 93.7%

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

      1. *-commutative 93.7%

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

      2. associate-*r* 95.3%

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

      3. fma-undefine 95.3%

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

      4. +-commutative 95.3%

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

      5. associate-/l/ 96.0%

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

      6. add-sqr-sqrt 50.1%

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

      7. sqrt-div 22.0%

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

      8. inv-pow 22.0%

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

      9. sqrt-pow1 22.0%

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

      10. metadata-eval 22.0%

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

      11. +-commutative 22.0%

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

      12. fma-undefine 22.0%

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

      13. *-commutative 22.0%

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

      14. sqrt-prod 22.0%

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

      15. fma-undefine 22.0%

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

      16. +-commutative 22.0%

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

      17. hypot-1-def 22.0%

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

      18. sqrt-div 21.9%

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

   6. Applied egg-rr 22.8%

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

      1. unpow2 22.8%

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

   8. Simplified 22.8%

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

      1. unpow2 22.8%

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

      2. frac-times 21.8%

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

      3. pow-prod-up 47.0%

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

      4. metadata-eval 47.0%

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

      5. inv-pow 47.0%

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

      6. *-un-lft-identity 47.0%

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

      7. *-commutative 47.0%

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

      8. *-commutative 47.0%

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

      9. swap-sqr 47.1%

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

      10. add-sqr-sqrt 96.0%

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

      11. hypot-undefine 96.0%

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

      12. hypot-undefine 96.0%

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

      13. rem-square-sqrt 96.0%

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

      14. metadata-eval 96.0%

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

      15. unpow2 96.0%

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

      16. +-commutative 96.0%

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

      17. unpow2 96.0%

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

      18. fma-undefine 96.0%

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

      19. frac-times 95.3%

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

   10. Applied egg-rr 95.4%

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

   if 5.0000000000000001e296 < (*.f64 z z)

   1. Initial program 70.2%

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

      1. associate-/l/ 70.2%

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

      2. associate-*l* 70.2%

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

      3. *-commutative 70.2%

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

      4. sqr-neg 70.2%

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

      5. +-commutative 70.2%

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

      6. sqr-neg 70.2%

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

      7. fma-define 70.2%

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

   3. Simplified 70.2%

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

      1. *-commutative 70.2%

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

      2. associate-*r* 70.2%

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

      3. fma-undefine 70.2%

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

      4. +-commutative 70.2%

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

      5. associate-/l/ 70.2%

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

      6. add-sqr-sqrt 70.2%

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

      7. sqrt-div 20.1%

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

      8. inv-pow 20.1%

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

      9. sqrt-pow1 20.1%

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

      10. metadata-eval 20.1%

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

      11. +-commutative 20.1%

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

      12. fma-undefine 20.1%

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

      13. *-commutative 20.1%

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

      14. sqrt-prod 20.1%

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

      15. fma-undefine 20.1%

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

      16. +-commutative 20.1%

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

      17. hypot-1-def 20.1%

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

      18. sqrt-div 20.1%

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

   6. Applied egg-rr 27.9%

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

      1. unpow2 27.9%

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

   8. Simplified 27.9%

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

   9. Taylor expanded in z around inf 27.9%

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

       1. *-un-lft-identity 27.9%

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

       2. *-commutative 27.9%

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

       3. times-frac 27.9%

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

       4. unpow-1 27.9%

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

       5. metadata-eval 27.9%

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

       6. sqrt-pow1 20.1%

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

       7. metadata-eval 20.1%

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

       8. sqrt-pow1 20.1%

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

       9. inv-pow 20.1%

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

       10. sqrt-div 41.8%

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

       11. sqrt-prod 70.6%

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

       12. pow2 70.6%

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

       13. add-sqr-sqrt 70.6%

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

       14. sqr-pow 70.6%

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

       15. associate-*l* 83.3%

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

       16. metadata-eval 83.3%

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

       17. unpow-1 83.3%

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

       18. metadata-eval 83.3%

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

       19. unpow-1 83.3%

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

       20. associate-/r* 83.2%

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

   11. Applied egg-rr 83.2%

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

       1. associate-/l/ 83.3%

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

       2. frac-times 96.6%

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

       3. *-un-lft-identity 96.6%

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

   13. Applied egg-rr 96.6%

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

4. Final simplification 95.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 5 \cdot 10^{+296}:\\ \;\;\;\;\frac{\frac{1}{y}}{x \cdot \mathsf{fma}\left(z, z, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{z} \cdot \frac{\frac{1}{y}}{x \cdot z}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.5% accurate, 0.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 87.7%

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

   1. associate-/l/ 87.3%

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

   2. associate-*l* 86.2%

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

   3. *-commutative 86.2%

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

   4. sqr-neg 86.2%

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

   5. +-commutative 86.2%

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

   6. sqr-neg 86.2%

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

   7. fma-define 86.2%

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

3. Simplified 86.2%

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

   1. *-commutative 86.2%

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

   2. associate-*r* 87.3%

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

   3. fma-undefine 87.3%

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

   4. +-commutative 87.3%

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

   5. associate-/l/ 87.7%

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

   6. add-sqr-sqrt 56.5%

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

   7. sqrt-div 21.4%

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

   8. inv-pow 21.4%

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

   9. sqrt-pow1 21.4%

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

   10. metadata-eval 21.4%

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

   11. +-commutative 21.4%

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

   12. fma-undefine 21.4%

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

   13. *-commutative 21.4%

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

   14. sqrt-prod 21.4%

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

   15. fma-undefine 21.4%

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

   16. +-commutative 21.4%

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

   17. hypot-1-def 21.4%

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

   18. sqrt-div 21.3%

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

6. Applied egg-rr 24.4%

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

   1. unpow2 24.4%

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

8. Simplified 24.4%

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

10. Applied egg-rr 54.5%

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

Alternative 3: 99.2% accurate, 0.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 87.7%

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

   1. associate-/l/ 87.3%

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

   2. associate-*l* 86.2%

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

   3. *-commutative 86.2%

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

   4. sqr-neg 86.2%

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

   5. +-commutative 86.2%

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

   6. sqr-neg 86.2%

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

   7. fma-define 86.2%

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

3. Simplified 86.2%

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

   1. *-commutative 86.2%

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

   2. associate-*r* 87.3%

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

   3. fma-undefine 87.3%

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

   4. +-commutative 87.3%

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

   5. associate-/l/ 87.7%

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

   6. add-sqr-sqrt 56.5%

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

   7. sqrt-div 21.4%

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

   8. inv-pow 21.4%

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

   9. sqrt-pow1 21.4%

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

   10. metadata-eval 21.4%

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

   11. +-commutative 21.4%

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

   12. fma-undefine 21.4%

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

   13. *-commutative 21.4%

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

   14. sqrt-prod 21.4%

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

   15. fma-undefine 21.4%

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

   16. +-commutative 21.4%

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

   17. hypot-1-def 21.4%

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

   18. sqrt-div 21.3%

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

6. Applied egg-rr 24.4%

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

   1. unpow2 24.4%

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

8. Simplified 24.4%

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

Alternative 4: 98.5% accurate, 0.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 87.7%

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

   1. associate-/l/ 87.3%

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

   2. associate-*l* 86.2%

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

   3. *-commutative 86.2%

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

   4. sqr-neg 86.2%

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

   5. +-commutative 86.2%

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

   6. sqr-neg 86.2%

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

   7. fma-define 86.2%

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

3. Simplified 86.2%

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

   1. *-commutative 86.2%

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

   2. associate-*r* 87.3%

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

   3. fma-undefine 87.3%

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

   4. +-commutative 87.3%

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

   5. associate-/l/ 87.7%

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

   6. add-sqr-sqrt 56.5%

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

   7. sqrt-div 21.4%

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

   8. inv-pow 21.4%

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

   9. sqrt-pow1 21.4%

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

   10. metadata-eval 21.4%

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

   11. +-commutative 21.4%

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

   12. fma-undefine 21.4%

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

   13. *-commutative 21.4%

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

   14. sqrt-prod 21.4%

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

   15. fma-undefine 21.4%

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

   16. +-commutative 21.4%

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

   17. hypot-1-def 21.4%

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

   18. sqrt-div 21.3%

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

6. Applied egg-rr 24.4%

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

   1. unpow2 24.4%

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

8. Simplified 24.4%

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

10. Applied egg-rr 54.5%

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

    1. *-commutative 54.5%

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

    2. clear-num 54.4%

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

    3. un-div-inv 54.4%

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

    4. clear-num 54.4%

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

    5. associate-/r/ 54.4%

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

    6. *-commutative 54.4%

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

    7. associate-/r* 54.4%

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

    8. metadata-eval 54.4%

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

    9. sqrt-div 54.4%

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

    10. inv-pow 54.4%

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

    11. sqrt-pow1 54.4%

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

    12. metadata-eval 54.4%

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

    13. frac-times 53.9%

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

    14. *-commutative 53.9%

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

    15. *-un-lft-identity 53.9%

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

12. Applied egg-rr 53.9%

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

    1. associate-/r/ 53.0%

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

    2. associate-*l/ 52.8%

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

    3. associate-*r/ 53.9%

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

    4. associate-/r* 54.4%

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

    5. associate-*l/ 49.0%

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

    6. unpow1 49.0%

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

    7. pow-plus 49.0%

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

    8. metadata-eval 49.0%

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

14. Simplified 49.0%

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

Alternative 5: 99.3% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 91.1%

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

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

   1. Initial program 72.8%

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

      1. associate-/l/ 72.8%

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

      2. associate-*l* 78.8%

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

      3. *-commutative 78.8%

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

      4. sqr-neg 78.8%

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

      5. +-commutative 78.8%

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

      6. sqr-neg 78.8%

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

      7. fma-define 78.8%

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

   3. Simplified 78.8%

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

      1. *-commutative 78.8%

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

      2. associate-*r* 72.8%

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

      3. fma-undefine 72.8%

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

      4. +-commutative 72.8%

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

      5. associate-/l/ 72.8%

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

      6. add-sqr-sqrt 72.8%

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

      7. sqrt-div 38.2%

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

      8. inv-pow 38.2%

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

      9. sqrt-pow1 38.2%

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

      10. metadata-eval 38.2%

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

      11. +-commutative 38.2%

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

      12. fma-undefine 38.2%

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

      13. *-commutative 38.2%

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

      14. sqrt-prod 38.2%

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

      15. fma-undefine 38.2%

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

      16. +-commutative 38.2%

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

      17. hypot-1-def 38.2%

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

      18. sqrt-div 38.2%

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

   6. Applied egg-rr 55.1%

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

      1. unpow2 55.1%

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

   8. Simplified 55.1%

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

   9. Taylor expanded in z around inf 55.1%

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

       1. *-un-lft-identity 55.1%

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

       2. *-commutative 55.1%

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

       3. times-frac 55.1%

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

       4. unpow-1 55.1%

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

       5. metadata-eval 55.1%

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

       6. sqrt-pow1 41.4%

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

       7. metadata-eval 41.4%

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

       8. sqrt-pow1 41.4%

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

       9. inv-pow 41.4%

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

       10. sqrt-div 45.4%

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

       11. sqrt-prod 75.6%

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

       12. pow2 75.6%

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

       13. add-sqr-sqrt 79.8%

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

       14. sqr-pow 79.8%

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

       15. associate-*l* 91.5%

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

       16. metadata-eval 91.5%

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

       17. unpow-1 91.5%

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

       18. metadata-eval 91.5%

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

       19. unpow-1 91.5%

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

       20. associate-/r* 91.5%

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

   11. Applied egg-rr 91.5%

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

       1. associate-/l/ 91.5%

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

       2. frac-times 94.2%

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

       3. *-un-lft-identity 94.2%

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

   13. Applied egg-rr 94.2%

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

4. Final simplification 91.7%

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

Alternative 6: 97.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{-23}:\\ \;\;\;\;\frac{\frac{1}{x\_m}}{y\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{z} \cdot \frac{\frac{1}{y\_m}}{x\_m \cdot z}\\ \end{array}\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (y_s x_s x_m y_m z)
 :precision binary64
 (*
  y_s
  (*
   x_s
   (if (<= (* z z) 2e-23)
     (/ (/ 1.0 x_m) y_m)
     (* (/ 1.0 z) (/ (/ 1.0 y_m) (* x_m z)))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
assert(x_m < y_m && y_m < z);
double code(double y_s, double x_s, double x_m, double y_m, double z) {
	double tmp;
	if ((z * z) <= 2e-23) {
		tmp = (1.0 / x_m) / y_m;
	} else {
		tmp = (1.0 / z) * ((1.0 / y_m) / (x_m * z));
	}
	return y_s * (x_s * tmp);
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
y\_m = abs(y)
y\_s = copysign(1.0d0, y)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
real(8) function code(y_s, x_s, x_m, y_m, z)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((z * z) <= 2d-23) then
        tmp = (1.0d0 / x_m) / y_m
    else
        tmp = (1.0d0 / z) * ((1.0d0 / y_m) / (x_m * z))
    end if
    code = y_s * (x_s * tmp)
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
assert x_m < y_m && y_m < z;
public static double code(double y_s, double x_s, double x_m, double y_m, double z) {
	double tmp;
	if ((z * z) <= 2e-23) {
		tmp = (1.0 / x_m) / y_m;
	} else {
		tmp = (1.0 / z) * ((1.0 / y_m) / (x_m * z));
	}
	return y_s * (x_s * tmp);
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
[x_m, y_m, z] = sort([x_m, y_m, z])
def code(y_s, x_s, x_m, y_m, z):
	tmp = 0
	if (z * z) <= 2e-23:
		tmp = (1.0 / x_m) / y_m
	else:
		tmp = (1.0 / z) * ((1.0 / y_m) / (x_m * z))
	return y_s * (x_s * tmp)

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
y\_m = abs(y)
y\_s = copysign(1.0, y)
x_m, y_m, z = sort([x_m, y_m, z])
function code(y_s, x_s, x_m, y_m, z)
	tmp = 0.0
	if (Float64(z * z) <= 2e-23)
		tmp = Float64(Float64(1.0 / x_m) / y_m);
	else
		tmp = Float64(Float64(1.0 / z) * Float64(Float64(1.0 / y_m) / Float64(x_m * z)));
	end
	return Float64(y_s * Float64(x_s * tmp))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
x_m, y_m, z = num2cell(sort([x_m, y_m, z])){:}
function tmp_2 = code(y_s, x_s, x_m, y_m, z)
	tmp = 0.0;
	if ((z * z) <= 2e-23)
		tmp = (1.0 / x_m) / y_m;
	else
		tmp = (1.0 / z) * ((1.0 / y_m) / (x_m * z));
	end
	tmp_2 = y_s * (x_s * tmp);
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
code[y$95$s_, x$95$s_, x$95$m_, y$95$m_, z_] := N[(y$95$s * N[(x$95$s * If[LessEqual[N[(z * z), $MachinePrecision], 2e-23], N[(N[(1.0 / x$95$m), $MachinePrecision] / y$95$m), $MachinePrecision], N[(N[(1.0 / z), $MachinePrecision] * N[(N[(1.0 / y$95$m), $MachinePrecision] / N[(x$95$m * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 1.99999999999999992e-23

   1. Initial program 99.7%

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

      1. associate-/l/ 99.2%

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

      2. associate-*l* 99.2%

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

      3. *-commutative 99.2%

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

      4. sqr-neg 99.2%

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

      5. +-commutative 99.2%

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

      6. sqr-neg 99.2%

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

      7. fma-define 99.2%

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

   3. Simplified 99.2%

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

   5. Taylor expanded in z around 0 99.2%

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

      1. associate-/r* 99.7%

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

   7. Simplified 99.7%

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

   if 1.99999999999999992e-23 < (*.f64 z z)

   1. Initial program 80.5%

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

      1. associate-/l/ 80.2%

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

      2. associate-*l* 78.4%

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

      3. *-commutative 78.4%

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

      4. sqr-neg 78.4%

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

      5. +-commutative 78.4%

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

      6. sqr-neg 78.4%

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

      7. fma-define 78.4%

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

   3. Simplified 78.4%

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

      1. *-commutative 78.4%

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

      2. associate-*r* 80.2%

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

      3. fma-undefine 80.2%

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

      4. +-commutative 80.2%

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

      5. associate-/l/ 80.5%

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

      6. add-sqr-sqrt 61.3%

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

      7. sqrt-div 22.4%

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

      8. inv-pow 22.4%

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

      9. sqrt-pow1 22.4%

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

      10. metadata-eval 22.4%

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

      11. +-commutative 22.4%

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

      12. fma-undefine 22.4%

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

      13. *-commutative 22.4%

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

      14. sqrt-prod 22.4%

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

      15. fma-undefine 22.4%

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

      16. +-commutative 22.4%

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

      17. hypot-1-def 22.4%

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

      18. sqrt-div 22.4%

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

   6. Applied egg-rr 27.3%

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

      1. unpow2 27.3%

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

   8. Simplified 27.3%

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

   9. Taylor expanded in z around inf 27.3%

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

       1. *-un-lft-identity 27.3%

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

       2. *-commutative 27.3%

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

       3. times-frac 27.3%

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

       4. unpow-1 27.3%

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

       5. metadata-eval 27.3%

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

       6. sqrt-pow1 23.3%

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

       7. metadata-eval 23.3%

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

       8. sqrt-pow1 23.3%

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

       9. inv-pow 23.3%

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

       10. sqrt-div 42.0%

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

       11. sqrt-prod 60.6%

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

       12. pow2 60.6%

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

       13. add-sqr-sqrt 80.3%

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

       14. sqr-pow 80.2%

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

       15. associate-*l* 86.6%

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

       16. metadata-eval 86.6%

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

       17. unpow-1 86.6%

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

       18. metadata-eval 86.6%

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

       19. unpow-1 86.6%

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

       20. associate-/r* 86.6%

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

   11. Applied egg-rr 86.6%

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

       1. associate-/l/ 86.6%

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

       2. frac-times 93.0%

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

       3. *-un-lft-identity 93.0%

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

   13. Applied egg-rr 93.0%

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

4. Final simplification 95.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{-23}:\\ \;\;\;\;\frac{\frac{1}{x}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{z} \cdot \frac{\frac{1}{y}}{x \cdot z}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 95.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{-23}:\\ \;\;\;\;\frac{\frac{1}{x\_m}}{y\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{z} \cdot \frac{1}{x\_m \cdot \left(z \cdot y\_m\right)}\\ \end{array}\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (y_s x_s x_m y_m z)
 :precision binary64
 (*
  y_s
  (*
   x_s
   (if (<= (* z z) 2e-23)
     (/ (/ 1.0 x_m) y_m)
     (* (/ 1.0 z) (/ 1.0 (* x_m (* z y_m))))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
assert(x_m < y_m && y_m < z);
double code(double y_s, double x_s, double x_m, double y_m, double z) {
	double tmp;
	if ((z * z) <= 2e-23) {
		tmp = (1.0 / x_m) / y_m;
	} else {
		tmp = (1.0 / z) * (1.0 / (x_m * (z * y_m)));
	}
	return y_s * (x_s * tmp);
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
y\_m = abs(y)
y\_s = copysign(1.0d0, y)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
real(8) function code(y_s, x_s, x_m, y_m, z)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((z * z) <= 2d-23) then
        tmp = (1.0d0 / x_m) / y_m
    else
        tmp = (1.0d0 / z) * (1.0d0 / (x_m * (z * y_m)))
    end if
    code = y_s * (x_s * tmp)
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
assert x_m < y_m && y_m < z;
public static double code(double y_s, double x_s, double x_m, double y_m, double z) {
	double tmp;
	if ((z * z) <= 2e-23) {
		tmp = (1.0 / x_m) / y_m;
	} else {
		tmp = (1.0 / z) * (1.0 / (x_m * (z * y_m)));
	}
	return y_s * (x_s * tmp);
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
[x_m, y_m, z] = sort([x_m, y_m, z])
def code(y_s, x_s, x_m, y_m, z):
	tmp = 0
	if (z * z) <= 2e-23:
		tmp = (1.0 / x_m) / y_m
	else:
		tmp = (1.0 / z) * (1.0 / (x_m * (z * y_m)))
	return y_s * (x_s * tmp)

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
y\_m = abs(y)
y\_s = copysign(1.0, y)
x_m, y_m, z = sort([x_m, y_m, z])
function code(y_s, x_s, x_m, y_m, z)
	tmp = 0.0
	if (Float64(z * z) <= 2e-23)
		tmp = Float64(Float64(1.0 / x_m) / y_m);
	else
		tmp = Float64(Float64(1.0 / z) * Float64(1.0 / Float64(x_m * Float64(z * y_m))));
	end
	return Float64(y_s * Float64(x_s * tmp))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
x_m, y_m, z = num2cell(sort([x_m, y_m, z])){:}
function tmp_2 = code(y_s, x_s, x_m, y_m, z)
	tmp = 0.0;
	if ((z * z) <= 2e-23)
		tmp = (1.0 / x_m) / y_m;
	else
		tmp = (1.0 / z) * (1.0 / (x_m * (z * y_m)));
	end
	tmp_2 = y_s * (x_s * tmp);
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
code[y$95$s_, x$95$s_, x$95$m_, y$95$m_, z_] := N[(y$95$s * N[(x$95$s * If[LessEqual[N[(z * z), $MachinePrecision], 2e-23], N[(N[(1.0 / x$95$m), $MachinePrecision] / y$95$m), $MachinePrecision], N[(N[(1.0 / z), $MachinePrecision] * N[(1.0 / N[(x$95$m * N[(z * y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 1.99999999999999992e-23

   1. Initial program 99.7%

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

      1. associate-/l/ 99.2%

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

      2. associate-*l* 99.2%

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

      3. *-commutative 99.2%

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

      4. sqr-neg 99.2%

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

      5. +-commutative 99.2%

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

      6. sqr-neg 99.2%

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

      7. fma-define 99.2%

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

   3. Simplified 99.2%

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

   5. Taylor expanded in z around 0 99.2%

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

      1. associate-/r* 99.7%

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

   7. Simplified 99.7%

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

   if 1.99999999999999992e-23 < (*.f64 z z)

   1. Initial program 80.5%

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

      1. associate-/l/ 80.2%

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

      2. associate-*l* 78.4%

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

      3. *-commutative 78.4%

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

      4. sqr-neg 78.4%

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

      5. +-commutative 78.4%

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

      6. sqr-neg 78.4%

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

      7. fma-define 78.4%

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

   3. Simplified 78.4%

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

      1. *-commutative 78.4%

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

      2. associate-*r* 80.2%

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

      3. fma-undefine 80.2%

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

      4. +-commutative 80.2%

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

      5. associate-/l/ 80.5%

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

      6. add-sqr-sqrt 61.3%

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

      7. sqrt-div 22.4%

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

      8. inv-pow 22.4%

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

      9. sqrt-pow1 22.4%

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

      10. metadata-eval 22.4%

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

      11. +-commutative 22.4%

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

      12. fma-undefine 22.4%

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

      13. *-commutative 22.4%

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

      14. sqrt-prod 22.4%

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

      15. fma-undefine 22.4%

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

      16. +-commutative 22.4%

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

      17. hypot-1-def 22.4%

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

      18. sqrt-div 22.4%

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

   6. Applied egg-rr 27.3%

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

      1. unpow2 27.3%

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

   8. Simplified 27.3%

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

   9. Taylor expanded in z around inf 27.3%

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

       1. *-un-lft-identity 27.3%

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

       2. *-commutative 27.3%

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

       3. times-frac 27.3%

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

       4. unpow-1 27.3%

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

       5. metadata-eval 27.3%

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

       6. sqrt-pow1 23.3%

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

       7. metadata-eval 23.3%

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

       8. sqrt-pow1 23.3%

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

       9. inv-pow 23.3%

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

       10. sqrt-div 42.0%

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

       11. sqrt-prod 60.6%

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

       12. pow2 60.6%

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

       13. add-sqr-sqrt 80.3%

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

       14. sqr-pow 80.2%

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

       15. associate-*l* 86.6%

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

       16. metadata-eval 86.6%

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

       17. unpow-1 86.6%

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

       18. metadata-eval 86.6%

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

       19. unpow-1 86.6%

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

       20. associate-/r* 86.6%

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

   11. Applied egg-rr 86.6%

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

   12. Taylor expanded in z around 0 95.7%

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

4. Final simplification 97.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{-23}:\\ \;\;\;\;\frac{\frac{1}{x}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{z} \cdot \frac{1}{x \cdot \left(z \cdot y\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 93.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{-23}:\\ \;\;\;\;\frac{\frac{1}{x\_m}}{y\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{z \cdot \left(z \cdot \left(x\_m \cdot y\_m\right)\right)}\\ \end{array}\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (y_s x_s x_m y_m z)
 :precision binary64
 (*
  y_s
  (*
   x_s
   (if (<= (* z z) 2e-23)
     (/ (/ 1.0 x_m) y_m)
     (/ 1.0 (* z (* z (* x_m y_m))))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
assert(x_m < y_m && y_m < z);
double code(double y_s, double x_s, double x_m, double y_m, double z) {
	double tmp;
	if ((z * z) <= 2e-23) {
		tmp = (1.0 / x_m) / y_m;
	} else {
		tmp = 1.0 / (z * (z * (x_m * y_m)));
	}
	return y_s * (x_s * tmp);
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
y\_m = abs(y)
y\_s = copysign(1.0d0, y)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
real(8) function code(y_s, x_s, x_m, y_m, z)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((z * z) <= 2d-23) then
        tmp = (1.0d0 / x_m) / y_m
    else
        tmp = 1.0d0 / (z * (z * (x_m * y_m)))
    end if
    code = y_s * (x_s * tmp)
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
assert x_m < y_m && y_m < z;
public static double code(double y_s, double x_s, double x_m, double y_m, double z) {
	double tmp;
	if ((z * z) <= 2e-23) {
		tmp = (1.0 / x_m) / y_m;
	} else {
		tmp = 1.0 / (z * (z * (x_m * y_m)));
	}
	return y_s * (x_s * tmp);
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
[x_m, y_m, z] = sort([x_m, y_m, z])
def code(y_s, x_s, x_m, y_m, z):
	tmp = 0
	if (z * z) <= 2e-23:
		tmp = (1.0 / x_m) / y_m
	else:
		tmp = 1.0 / (z * (z * (x_m * y_m)))
	return y_s * (x_s * tmp)

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
y\_m = abs(y)
y\_s = copysign(1.0, y)
x_m, y_m, z = sort([x_m, y_m, z])
function code(y_s, x_s, x_m, y_m, z)
	tmp = 0.0
	if (Float64(z * z) <= 2e-23)
		tmp = Float64(Float64(1.0 / x_m) / y_m);
	else
		tmp = Float64(1.0 / Float64(z * Float64(z * Float64(x_m * y_m))));
	end
	return Float64(y_s * Float64(x_s * tmp))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
x_m, y_m, z = num2cell(sort([x_m, y_m, z])){:}
function tmp_2 = code(y_s, x_s, x_m, y_m, z)
	tmp = 0.0;
	if ((z * z) <= 2e-23)
		tmp = (1.0 / x_m) / y_m;
	else
		tmp = 1.0 / (z * (z * (x_m * y_m)));
	end
	tmp_2 = y_s * (x_s * tmp);
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
code[y$95$s_, x$95$s_, x$95$m_, y$95$m_, z_] := N[(y$95$s * N[(x$95$s * If[LessEqual[N[(z * z), $MachinePrecision], 2e-23], N[(N[(1.0 / x$95$m), $MachinePrecision] / y$95$m), $MachinePrecision], N[(1.0 / N[(z * N[(z * N[(x$95$m * y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 1.99999999999999992e-23

   1. Initial program 99.7%

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

      1. associate-/l/ 99.2%

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

      2. associate-*l* 99.2%

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

      3. *-commutative 99.2%

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

      4. sqr-neg 99.2%

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

      5. +-commutative 99.2%

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

      6. sqr-neg 99.2%

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

      7. fma-define 99.2%

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

   3. Simplified 99.2%

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

   5. Taylor expanded in z around 0 99.2%

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

      1. associate-/r* 99.7%

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

   7. Simplified 99.7%

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

   if 1.99999999999999992e-23 < (*.f64 z z)

   1. Initial program 80.5%

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

      1. associate-/l/ 80.2%

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

      2. associate-*l* 78.4%

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

      3. *-commutative 78.4%

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

      4. sqr-neg 78.4%

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

      5. +-commutative 78.4%

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

      6. sqr-neg 78.4%

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

      7. fma-define 78.4%

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

   3. Simplified 78.4%

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

      1. *-commutative 78.4%

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

      2. associate-*r* 80.2%

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

      3. fma-undefine 80.2%

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

      4. +-commutative 80.2%

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

      5. associate-/l/ 80.5%

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

      6. add-sqr-sqrt 61.3%

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

      7. sqrt-div 22.4%

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

      8. inv-pow 22.4%

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

      9. sqrt-pow1 22.4%

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

      10. metadata-eval 22.4%

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

      11. +-commutative 22.4%

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

      12. fma-undefine 22.4%

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

      13. *-commutative 22.4%

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

      14. sqrt-prod 22.4%

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

      15. fma-undefine 22.4%

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

      16. +-commutative 22.4%

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

      17. hypot-1-def 22.4%

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

      18. sqrt-div 22.4%

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

   6. Applied egg-rr 27.3%

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

      1. unpow2 27.3%

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

   8. Simplified 27.3%

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

   9. Taylor expanded in z around inf 27.3%

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

       1. *-un-lft-identity 27.3%

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

       2. *-commutative 27.3%

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

       3. times-frac 27.3%

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

       4. unpow-1 27.3%

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

       5. metadata-eval 27.3%

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

       6. sqrt-pow1 23.3%

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

       7. metadata-eval 23.3%

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

       8. sqrt-pow1 23.3%

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

       9. inv-pow 23.3%

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

       10. sqrt-div 42.0%

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

       11. sqrt-prod 60.6%

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

       12. pow2 60.6%

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

       13. add-sqr-sqrt 80.3%

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

       14. sqr-pow 80.2%

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

       15. associate-*l* 86.6%

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

       16. metadata-eval 86.6%

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

       17. unpow-1 86.6%

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

       18. metadata-eval 86.6%

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

       19. unpow-1 86.6%

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

       20. associate-/r* 86.6%

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

   11. Applied egg-rr 86.6%

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

       1. frac-times 86.7%

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

       2. metadata-eval 86.7%

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

       3. frac-times 84.8%

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

       4. metadata-eval 84.8%

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

   13. Applied egg-rr 84.8%

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

Alternative 9: 76.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq 0.00015:\\ \;\;\;\;\frac{\frac{1}{x\_m}}{y\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{z}}{z \cdot \left(x\_m \cdot y\_m\right)}\\ \end{array}\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (y_s x_s x_m y_m z)
 :precision binary64
 (*
  y_s
  (*
   x_s
   (if (<= z 0.00015) (/ (/ 1.0 x_m) y_m) (/ (/ 1.0 z) (* z (* x_m y_m)))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
assert(x_m < y_m && y_m < z);
double code(double y_s, double x_s, double x_m, double y_m, double z) {
	double tmp;
	if (z <= 0.00015) {
		tmp = (1.0 / x_m) / y_m;
	} else {
		tmp = (1.0 / z) / (z * (x_m * y_m));
	}
	return y_s * (x_s * tmp);
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
y\_m = abs(y)
y\_s = copysign(1.0d0, y)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
real(8) function code(y_s, x_s, x_m, y_m, z)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if (z <= 0.00015d0) then
        tmp = (1.0d0 / x_m) / y_m
    else
        tmp = (1.0d0 / z) / (z * (x_m * y_m))
    end if
    code = y_s * (x_s * tmp)
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
assert x_m < y_m && y_m < z;
public static double code(double y_s, double x_s, double x_m, double y_m, double z) {
	double tmp;
	if (z <= 0.00015) {
		tmp = (1.0 / x_m) / y_m;
	} else {
		tmp = (1.0 / z) / (z * (x_m * y_m));
	}
	return y_s * (x_s * tmp);
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
[x_m, y_m, z] = sort([x_m, y_m, z])
def code(y_s, x_s, x_m, y_m, z):
	tmp = 0
	if z <= 0.00015:
		tmp = (1.0 / x_m) / y_m
	else:
		tmp = (1.0 / z) / (z * (x_m * y_m))
	return y_s * (x_s * tmp)

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
y\_m = abs(y)
y\_s = copysign(1.0, y)
x_m, y_m, z = sort([x_m, y_m, z])
function code(y_s, x_s, x_m, y_m, z)
	tmp = 0.0
	if (z <= 0.00015)
		tmp = Float64(Float64(1.0 / x_m) / y_m);
	else
		tmp = Float64(Float64(1.0 / z) / Float64(z * Float64(x_m * y_m)));
	end
	return Float64(y_s * Float64(x_s * tmp))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
x_m, y_m, z = num2cell(sort([x_m, y_m, z])){:}
function tmp_2 = code(y_s, x_s, x_m, y_m, z)
	tmp = 0.0;
	if (z <= 0.00015)
		tmp = (1.0 / x_m) / y_m;
	else
		tmp = (1.0 / z) / (z * (x_m * y_m));
	end
	tmp_2 = y_s * (x_s * tmp);
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
code[y$95$s_, x$95$s_, x$95$m_, y$95$m_, z_] := N[(y$95$s * N[(x$95$s * If[LessEqual[z, 0.00015], N[(N[(1.0 / x$95$m), $MachinePrecision] / y$95$m), $MachinePrecision], N[(N[(1.0 / z), $MachinePrecision] / N[(z * N[(x$95$m * y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 1.49999999999999987e-4

   1. Initial program 88.4%

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

      1. associate-/l/ 87.7%

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

      2. associate-*l* 86.7%

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

      3. *-commutative 86.7%

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

      4. sqr-neg 86.7%

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

      5. +-commutative 86.7%

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

      6. sqr-neg 86.7%

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

      7. fma-define 86.7%

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

   3. Simplified 86.7%

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

   5. Taylor expanded in z around 0 59.5%

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

      1. associate-/r* 59.6%

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

   7. Simplified 59.6%

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

   if 1.49999999999999987e-4 < z

   1. Initial program 86.2%

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

      1. associate-/l/ 86.2%

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

      2. associate-*l* 84.9%

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

      3. *-commutative 84.9%

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

      4. sqr-neg 84.9%

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

      5. +-commutative 84.9%

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

      6. sqr-neg 84.9%

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

      7. fma-define 84.9%

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

   3. Simplified 84.9%

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

      1. *-commutative 84.9%

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

      2. associate-*r* 86.2%

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

      3. fma-undefine 86.2%

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

      4. +-commutative 86.2%

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

      5. associate-/l/ 86.2%

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

      6. add-sqr-sqrt 72.3%

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

      7. sqrt-div 28.6%

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

      8. inv-pow 28.6%

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

      9. sqrt-pow1 28.5%

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

      10. metadata-eval 28.5%

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

      11. +-commutative 28.5%

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

      12. fma-undefine 28.5%

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

      13. *-commutative 28.5%

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

      14. sqrt-prod 28.5%

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

      15. fma-undefine 28.5%

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

      16. +-commutative 28.5%

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

      17. hypot-1-def 28.5%

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

      18. sqrt-div 28.5%

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

   6. Applied egg-rr 33.1%

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

      1. unpow2 33.1%

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

   8. Simplified 33.1%

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

   9. Taylor expanded in z around inf 33.1%

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

       1. *-un-lft-identity 33.1%

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

       2. *-commutative 33.1%

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

       3. times-frac 33.1%

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

       4. unpow-1 33.1%

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

       5. metadata-eval 33.1%

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

       6. sqrt-pow1 29.3%

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

       7. metadata-eval 29.3%

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

       8. sqrt-pow1 29.3%

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

       9. inv-pow 29.3%

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

       10. sqrt-div 46.0%

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

       11. sqrt-prod 69.7%

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

       12. pow2 69.7%

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

       13. add-sqr-sqrt 84.0%

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

       14. sqr-pow 84.0%

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

       15. associate-*l* 86.6%

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

       16. metadata-eval 86.6%

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

       17. unpow-1 86.6%

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

       18. metadata-eval 86.6%

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

       19. unpow-1 86.6%

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

       20. associate-/r* 86.5%

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

   11. Applied egg-rr 86.5%

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

       1. un-div-inv 86.6%

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

       2. frac-times 86.7%

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

       3. *-un-lft-identity 86.7%

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

   13. Applied egg-rr 86.7%

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

Alternative 10: 58.4% accurate, 2.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 87.7%

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

   1. associate-/l/ 87.3%

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

   2. associate-*l* 86.2%

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

   3. *-commutative 86.2%

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

   4. sqr-neg 86.2%

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

   5. +-commutative 86.2%

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

   6. sqr-neg 86.2%

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

   7. fma-define 86.2%

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

3. Simplified 86.2%

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

5. Taylor expanded in z around 0 47.5%

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

6. Final simplification 47.5%

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

Developer Target 1: 92.7% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024150 
(FPCore (x y z)
  :name "Statistics.Distribution.CauchyLorentz:$cdensity from math-functions-0.1.5.2"
  :precision binary64

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

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