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

Percentage Accurate: 88.0% → 99.1%

Time: 10.6s

Alternatives: 13

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 88.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.1% accurate, 0.1× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x_m, y_m, z_m] = \mathsf{sort}([x_m, y_m, z_m])\\ \\ y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \cdot z\_m \leq 5 \cdot 10^{+301}:\\ \;\;\;\;\frac{1}{y\_m} \cdot \frac{\frac{1}{x\_m}}{\mathsf{fma}\left(z\_m, z\_m, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\mathsf{hypot}\left(1, z\_m\right) \cdot x\_m}}{z\_m \cdot y\_m}\\ \end{array}\right) \end{array} \]

FPCore

z_m = (fabs.f64 z)
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_m should be sorted in increasing order before calling this function.
(FPCore (y_s x_s x_m y_m z_m)
 :precision binary64
 (*
  y_s
  (*
   x_s
   (if (<= (* z_m z_m) 5e+301)
     (* (/ 1.0 y_m) (/ (/ 1.0 x_m) (fma z_m z_m 1.0)))
     (/ (/ 1.0 (* (hypot 1.0 z_m) x_m)) (* z_m y_m))))))

C

z_m = fabs(z);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
assert(x_m < y_m && y_m < z_m);
double code(double y_s, double x_s, double x_m, double y_m, double z_m) {
	double tmp;
	if ((z_m * z_m) <= 5e+301) {
		tmp = (1.0 / y_m) * ((1.0 / x_m) / fma(z_m, z_m, 1.0));
	} else {
		tmp = (1.0 / (hypot(1.0, z_m) * x_m)) / (z_m * y_m);
	}
	return y_s * (x_s * tmp);
}

Julia

z_m = abs(z)
x\_m = abs(x)
x\_s = copysign(1.0, x)
y\_m = abs(y)
y\_s = copysign(1.0, y)
x_m, y_m, z_m = sort([x_m, y_m, z_m])
function code(y_s, x_s, x_m, y_m, z_m)
	tmp = 0.0
	if (Float64(z_m * z_m) <= 5e+301)
		tmp = Float64(Float64(1.0 / y_m) * Float64(Float64(1.0 / x_m) / fma(z_m, z_m, 1.0)));
	else
		tmp = Float64(Float64(1.0 / Float64(hypot(1.0, z_m) * x_m)) / Float64(z_m * y_m));
	end
	return Float64(y_s * Float64(x_s * tmp))
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, and z_m should be sorted in increasing order before calling this function.
code[y$95$s_, x$95$s_, x$95$m_, y$95$m_, z$95$m_] := N[(y$95$s * N[(x$95$s * If[LessEqual[N[(z$95$m * z$95$m), $MachinePrecision], 5e+301], N[(N[(1.0 / y$95$m), $MachinePrecision] * N[(N[(1.0 / x$95$m), $MachinePrecision] / N[(z$95$m * z$95$m + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[(N[Sqrt[1.0 ^ 2 + z$95$m ^ 2], $MachinePrecision] * x$95$m), $MachinePrecision]), $MachinePrecision] / N[(z$95$m * y$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 5.0000000000000004e301

   1. Initial program 97.2%

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

      1. associate-/l/ 96.8%

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

      2. associate-*l* 96.4%

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

      3. *-commutative 96.4%

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

      4. sqr-neg 96.4%

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

      5. +-commutative 96.4%

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

      6. sqr-neg 96.4%

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

      7. fma-define 96.4%

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

   3. Simplified 96.4%

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

      1. associate-*r* 96.4%

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

      2. *-commutative 96.4%

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

      3. associate-/r* 95.6%

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

      4. *-commutative 95.6%

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

      5. associate-/l/ 95.7%

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

      6. associate-/r* 97.2%

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

      7. *-un-lft-identity 97.2%

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

      8. times-frac 96.6%

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

   6. Applied egg-rr 96.6%

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

   if 5.0000000000000004e301 < (*.f64 z z)

   1. Initial program 76.5%

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

      1. associate-/l/ 76.6%

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

      2. associate-*l* 76.6%

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

      3. *-commutative 76.6%

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

      4. sqr-neg 76.6%

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

      5. +-commutative 76.6%

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

      6. sqr-neg 76.6%

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

      7. fma-define 76.6%

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

   3. Simplified 76.6%

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

      1. associate-*r* 76.1%

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

      2. *-commutative 76.1%

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

      3. *-commutative 76.1%

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

      4. add-sqr-sqrt 76.1%

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

      5. pow2 76.1%

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

      6. sqrt-div 36.8%

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

      7. metadata-eval 36.8%

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

      8. sqrt-prod 36.8%

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

      9. fma-undefine 36.8%

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

      10. +-commutative 36.8%

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

      11. hypot-1-def 40.8%

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

   6. Applied egg-rr 40.8%

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

      1. *-commutative 40.8%

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

      2. sqrt-prod 26.7%

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

   8. Applied egg-rr 26.7%

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

      1. unpow2 26.7%

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

      2. frac-times 26.7%

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

      3. metadata-eval 26.7%

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

      4. *-commutative 26.7%

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

      5. *-commutative 26.7%

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

      6. swap-sqr 21.3%

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

      7. swap-sqr 21.3%

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

      8. add-sqr-sqrt 47.2%

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

      9. add-sqr-sqrt 76.1%

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

      10. *-commutative 76.1%

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

      11. *-commutative 76.1%

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

      12. /-rgt-identity 76.1%

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

      13. clear-num 76.1%

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

      14. div-inv 76.1%

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

      15. associate-/r/ 76.5%

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

      16. associate-/r* 82.0%

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

      17. un-div-inv 82.0%

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

   10. Applied egg-rr 99.8%

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

   11. Taylor expanded in z around inf 83.5%

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

4. Final simplification 92.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 5 \cdot 10^{+301}:\\ \;\;\;\;\frac{1}{y} \cdot \frac{\frac{1}{x}}{\mathsf{fma}\left(z, z, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\mathsf{hypot}\left(1, z\right) \cdot x}}{z \cdot y}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.3% accurate, 0.0× speedup

Math

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

FPCore

z_m = (fabs.f64 z)
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_m should be sorted in increasing order before calling this function.
(FPCore (y_s x_s x_m y_m z_m)
 :precision binary64
 (let* ((t_0 (* (hypot 1.0 z_m) (sqrt y_m))))
   (* y_s (* x_s (* (/ 1.0 t_0) (/ (/ 1.0 x_m) t_0))))))

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 91.4%

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

   1. associate-/l/ 91.2%

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

   2. associate-*l* 90.9%

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

   3. *-commutative 90.9%

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

   4. sqr-neg 90.9%

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

   5. +-commutative 90.9%

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

   6. sqr-neg 90.9%

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

   7. fma-define 90.9%

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

3. Simplified 90.9%

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

   1. associate-*r* 90.8%

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

   2. *-commutative 90.8%

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

   3. associate-/r* 90.2%

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

   4. *-commutative 90.2%

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

   5. associate-/l/ 90.2%

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

   6. fma-undefine 90.2%

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

   7. +-commutative 90.2%

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

   8. associate-/r* 91.4%

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

   9. *-un-lft-identity 91.4%

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

   10. add-sqr-sqrt 41.2%

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

   11. times-frac 41.2%

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

   12. +-commutative 41.2%

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

   13. fma-undefine 41.2%

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

   14. *-commutative 41.2%

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

   15. sqrt-prod 41.2%

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

   16. fma-undefine 41.2%

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

   17. +-commutative 41.2%

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

   18. hypot-1-def 41.2%

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

   19. +-commutative 41.2%

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

6. Applied egg-rr 45.0%

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

Alternative 3: 99.4% accurate, 0.0× speedup

Math

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

FPCore

z_m = (fabs.f64 z)
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_m should be sorted in increasing order before calling this function.
(FPCore (y_s x_s x_m y_m z_m)
 :precision binary64
 (*
  y_s
  (*
   x_s
   (/
    (/ (/ 1.0 (sqrt y_m)) (hypot 1.0 z_m))
    (* (* (hypot 1.0 z_m) (sqrt y_m)) x_m)))))

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 91.4%

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

   1. associate-/l/ 91.2%

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

   2. associate-*l* 90.9%

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

   3. *-commutative 90.9%

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

   4. sqr-neg 90.9%

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

   5. +-commutative 90.9%

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

   6. sqr-neg 90.9%

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

   7. fma-define 90.9%

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

3. Simplified 90.9%

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

   1. associate-*r* 90.8%

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

   2. *-commutative 90.8%

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

   3. associate-/r* 90.2%

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

   4. *-commutative 90.2%

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

   5. associate-/l/ 90.2%

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

   6. fma-undefine 90.2%

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

   7. +-commutative 90.2%

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

   8. associate-/r* 91.4%

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

   9. *-un-lft-identity 91.4%

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

   10. add-sqr-sqrt 41.2%

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

   11. times-frac 41.2%

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

   12. +-commutative 41.2%

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

   13. fma-undefine 41.2%

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

   14. *-commutative 41.2%

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

   15. sqrt-prod 41.2%

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

   16. fma-undefine 41.2%

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

   17. +-commutative 41.2%

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

   18. hypot-1-def 41.2%

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

   19. +-commutative 41.2%

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

6. Applied egg-rr 45.0%

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

   1. associate-/l/ 45.0%

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

   2. associate-*r/ 45.1%

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

   3. *-rgt-identity 45.1%

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

   4. *-commutative 45.1%

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

   5. associate-/r* 45.1%

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

   6. *-commutative 45.1%

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

8. Simplified 45.1%

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

9. Final simplification 45.1%

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

Alternative 4: 99.2% accurate, 0.0× speedup

Math

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

FPCore

z_m = (fabs.f64 z)
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_m should be sorted in increasing order before calling this function.
(FPCore (y_s x_s x_m y_m z_m)
 :precision binary64
 (*
  y_s
  (* x_s (pow (/ 1.0 (* (hypot 1.0 z_m) (* (sqrt y_m) (sqrt x_m)))) 2.0))))

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 91.4%

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

   1. associate-/l/ 91.2%

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

   2. associate-*l* 90.9%

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

   3. *-commutative 90.9%

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

   4. sqr-neg 90.9%

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

   5. +-commutative 90.9%

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

   6. sqr-neg 90.9%

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

   7. fma-define 90.9%

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

3. Simplified 90.9%

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

   1. associate-*r* 90.8%

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

   2. *-commutative 90.8%

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

   3. *-commutative 90.8%

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

   4. add-sqr-sqrt 59.9%

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

   5. pow2 59.9%

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

   6. sqrt-div 42.9%

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

   7. metadata-eval 42.9%

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

   8. sqrt-prod 43.0%

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

   9. fma-undefine 43.0%

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

   10. +-commutative 43.0%

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

   11. hypot-1-def 44.1%

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

6. Applied egg-rr 44.1%

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

   1. *-commutative 44.1%

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

   2. sqrt-prod 21.7%

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

8. Applied egg-rr 21.7%

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

Alternative 5: 97.6% accurate, 0.1× speedup

Math

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

FPCore

z_m = (fabs.f64 z)
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_m should be sorted in increasing order before calling this function.
(FPCore (y_s x_s x_m y_m z_m)
 :precision binary64
 (* y_s (* x_s (/ (/ 1.0 (* (hypot 1.0 z_m) x_m)) (* (hypot 1.0 z_m) y_m)))))

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 91.4%

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

   1. associate-/l/ 91.2%

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

   2. associate-*l* 90.9%

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

   3. *-commutative 90.9%

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

   4. sqr-neg 90.9%

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

   5. +-commutative 90.9%

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

   6. sqr-neg 90.9%

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

   7. fma-define 90.9%

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

3. Simplified 90.9%

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

   1. associate-*r* 90.8%

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

   2. *-commutative 90.8%

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

   3. *-commutative 90.8%

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

   4. add-sqr-sqrt 59.9%

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

   5. pow2 59.9%

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

   6. sqrt-div 42.9%

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

   7. metadata-eval 42.9%

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

   8. sqrt-prod 43.0%

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

   9. fma-undefine 43.0%

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

   10. +-commutative 43.0%

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

   11. hypot-1-def 44.1%

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

6. Applied egg-rr 44.1%

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

   1. *-commutative 44.1%

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

   2. sqrt-prod 21.7%

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

8. Applied egg-rr 21.7%

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

   1. unpow2 21.7%

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

   2. frac-times 21.7%

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

   3. metadata-eval 21.7%

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

   4. *-commutative 21.7%

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

   5. *-commutative 21.7%

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

   6. swap-sqr 19.1%

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

   7. swap-sqr 19.1%

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

   8. add-sqr-sqrt 45.1%

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

   9. add-sqr-sqrt 90.8%

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

   10. *-commutative 90.8%

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

   11. *-commutative 90.8%

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

   12. /-rgt-identity 90.8%

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

   13. clear-num 90.7%

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

   14. div-inv 90.7%

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

   15. associate-/r/ 90.9%

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

   16. associate-/r* 92.4%

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

   17. un-div-inv 92.4%

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

10. Applied egg-rr 98.6%

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

Alternative 6: 91.6% accurate, 0.1× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x_m, y_m, z_m] = \mathsf{sort}([x_m, y_m, z_m])\\ \\ \begin{array}{l} t_0 := y\_m \cdot \left(1 + z\_m \cdot z\_m\right)\\ y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq 10^{+304}:\\ \;\;\;\;\frac{\frac{1}{x\_m}}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x\_m \cdot {z\_m}^{2}}}{y\_m}\\ \end{array}\right) \end{array} \end{array} \]

FPCore

z_m = (fabs.f64 z)
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_m should be sorted in increasing order before calling this function.
(FPCore (y_s x_s x_m y_m z_m)
 :precision binary64
 (let* ((t_0 (* y_m (+ 1.0 (* z_m z_m)))))
   (*
    y_s
    (*
     x_s
     (if (<= t_0 1e+304)
       (/ (/ 1.0 x_m) t_0)
       (/ (/ 1.0 (* x_m (pow z_m 2.0))) y_m))))))

C

z_m = fabs(z);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
assert(x_m < y_m && y_m < z_m);
double code(double y_s, double x_s, double x_m, double y_m, double z_m) {
	double t_0 = y_m * (1.0 + (z_m * z_m));
	double tmp;
	if (t_0 <= 1e+304) {
		tmp = (1.0 / x_m) / t_0;
	} else {
		tmp = (1.0 / (x_m * pow(z_m, 2.0))) / y_m;
	}
	return y_s * (x_s * tmp);
}

Fortran

z_m = abs(z)
x\_m = abs(x)
x\_s = copysign(1.0d0, x)
y\_m = abs(y)
y\_s = copysign(1.0d0, y)
NOTE: x_m, y_m, and z_m should be sorted in increasing order before calling this function.
real(8) function code(y_s, x_s, x_m, y_m, z_m)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = y_m * (1.0d0 + (z_m * z_m))
    if (t_0 <= 1d+304) then
        tmp = (1.0d0 / x_m) / t_0
    else
        tmp = (1.0d0 / (x_m * (z_m ** 2.0d0))) / y_m
    end if
    code = y_s * (x_s * tmp)
end function

Java

z_m = Math.abs(z);
x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
assert x_m < y_m && y_m < z_m;
public static double code(double y_s, double x_s, double x_m, double y_m, double z_m) {
	double t_0 = y_m * (1.0 + (z_m * z_m));
	double tmp;
	if (t_0 <= 1e+304) {
		tmp = (1.0 / x_m) / t_0;
	} else {
		tmp = (1.0 / (x_m * Math.pow(z_m, 2.0))) / y_m;
	}
	return y_s * (x_s * tmp);
}

Python

z_m = math.fabs(z)
x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
[x_m, y_m, z_m] = sort([x_m, y_m, z_m])
def code(y_s, x_s, x_m, y_m, z_m):
	t_0 = y_m * (1.0 + (z_m * z_m))
	tmp = 0
	if t_0 <= 1e+304:
		tmp = (1.0 / x_m) / t_0
	else:
		tmp = (1.0 / (x_m * math.pow(z_m, 2.0))) / y_m
	return y_s * (x_s * tmp)

Julia

z_m = abs(z)
x\_m = abs(x)
x\_s = copysign(1.0, x)
y\_m = abs(y)
y\_s = copysign(1.0, y)
x_m, y_m, z_m = sort([x_m, y_m, z_m])
function code(y_s, x_s, x_m, y_m, z_m)
	t_0 = Float64(y_m * Float64(1.0 + Float64(z_m * z_m)))
	tmp = 0.0
	if (t_0 <= 1e+304)
		tmp = Float64(Float64(1.0 / x_m) / t_0);
	else
		tmp = Float64(Float64(1.0 / Float64(x_m * (z_m ^ 2.0))) / y_m);
	end
	return Float64(y_s * Float64(x_s * tmp))
end

MATLAB

z_m = abs(z);
x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
x_m, y_m, z_m = num2cell(sort([x_m, y_m, z_m])){:}
function tmp_2 = code(y_s, x_s, x_m, y_m, z_m)
	t_0 = y_m * (1.0 + (z_m * z_m));
	tmp = 0.0;
	if (t_0 <= 1e+304)
		tmp = (1.0 / x_m) / t_0;
	else
		tmp = (1.0 / (x_m * (z_m ^ 2.0))) / y_m;
	end
	tmp_2 = y_s * (x_s * tmp);
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, and z_m should be sorted in increasing order before calling this function.
code[y$95$s_, x$95$s_, x$95$m_, y$95$m_, z$95$m_] := Block[{t$95$0 = N[(y$95$m * N[(1.0 + N[(z$95$m * z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(y$95$s * N[(x$95$s * If[LessEqual[t$95$0, 1e+304], N[(N[(1.0 / x$95$m), $MachinePrecision] / t$95$0), $MachinePrecision], N[(N[(1.0 / N[(x$95$m * N[Power[z$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y (+.f64 #s(literal 1 binary64) (*.f64 z z))) < 9.9999999999999994e303

   1. Initial program 94.5%

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

   if 9.9999999999999994e303 < (*.f64 y (+.f64 #s(literal 1 binary64) (*.f64 z z)))

   1. Initial program 73.7%

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

      1. associate-/l/ 73.7%

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

      2. associate-*l* 78.6%

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

      3. *-commutative 78.6%

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

      4. sqr-neg 78.6%

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

      5. +-commutative 78.6%

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

      6. sqr-neg 78.6%

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

      7. fma-define 78.6%

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

   3. Simplified 78.6%

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

      1. associate-*r* 78.1%

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

      2. *-commutative 78.1%

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

      3. associate-/r* 78.0%

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

      4. *-commutative 78.0%

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

      5. associate-/l/ 78.0%

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

      6. fma-undefine 78.0%

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

      7. +-commutative 78.0%

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

      8. associate-/r* 73.7%

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

      9. *-un-lft-identity 73.7%

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

      10. add-sqr-sqrt 73.7%

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

      11. times-frac 73.7%

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

      12. +-commutative 73.7%

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

      13. fma-undefine 73.7%

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

      14. *-commutative 73.7%

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

      15. sqrt-prod 73.7%

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

      16. fma-undefine 73.7%

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

      17. +-commutative 73.7%

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

      18. hypot-1-def 73.7%

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

      19. +-commutative 73.7%

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

   6. Applied egg-rr 99.6%

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

      1. *-commutative 99.6%

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

      2. associate-/r* 99.8%

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

      3. associate-/r* 99.7%

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

      4. frac-times 83.6%

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

      5. add-sqr-sqrt 83.6%

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

   8. Applied egg-rr 83.6%

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

   9. Taylor expanded in z around inf 78.5%

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

Alternative 7: 91.8% accurate, 0.1× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x_m, y_m, z_m] = \mathsf{sort}([x_m, y_m, z_m])\\ \\ \begin{array}{l} t_0 := y\_m \cdot \left(1 + z\_m \cdot z\_m\right)\\ y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq 4 \cdot 10^{+306}:\\ \;\;\;\;\frac{\frac{1}{x\_m}}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y\_m \cdot \left(x\_m \cdot {z\_m}^{2}\right)}\\ \end{array}\right) \end{array} \end{array} \]

FPCore

z_m = (fabs.f64 z)
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_m should be sorted in increasing order before calling this function.
(FPCore (y_s x_s x_m y_m z_m)
 :precision binary64
 (let* ((t_0 (* y_m (+ 1.0 (* z_m z_m)))))
   (*
    y_s
    (*
     x_s
     (if (<= t_0 4e+306)
       (/ (/ 1.0 x_m) t_0)
       (/ 1.0 (* y_m (* x_m (pow z_m 2.0)))))))))

C

z_m = fabs(z);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
assert(x_m < y_m && y_m < z_m);
double code(double y_s, double x_s, double x_m, double y_m, double z_m) {
	double t_0 = y_m * (1.0 + (z_m * z_m));
	double tmp;
	if (t_0 <= 4e+306) {
		tmp = (1.0 / x_m) / t_0;
	} else {
		tmp = 1.0 / (y_m * (x_m * pow(z_m, 2.0)));
	}
	return y_s * (x_s * tmp);
}

Fortran

z_m = abs(z)
x\_m = abs(x)
x\_s = copysign(1.0d0, x)
y\_m = abs(y)
y\_s = copysign(1.0d0, y)
NOTE: x_m, y_m, and z_m should be sorted in increasing order before calling this function.
real(8) function code(y_s, x_s, x_m, y_m, z_m)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = y_m * (1.0d0 + (z_m * z_m))
    if (t_0 <= 4d+306) then
        tmp = (1.0d0 / x_m) / t_0
    else
        tmp = 1.0d0 / (y_m * (x_m * (z_m ** 2.0d0)))
    end if
    code = y_s * (x_s * tmp)
end function

Java

z_m = Math.abs(z);
x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
assert x_m < y_m && y_m < z_m;
public static double code(double y_s, double x_s, double x_m, double y_m, double z_m) {
	double t_0 = y_m * (1.0 + (z_m * z_m));
	double tmp;
	if (t_0 <= 4e+306) {
		tmp = (1.0 / x_m) / t_0;
	} else {
		tmp = 1.0 / (y_m * (x_m * Math.pow(z_m, 2.0)));
	}
	return y_s * (x_s * tmp);
}

Python

z_m = math.fabs(z)
x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
[x_m, y_m, z_m] = sort([x_m, y_m, z_m])
def code(y_s, x_s, x_m, y_m, z_m):
	t_0 = y_m * (1.0 + (z_m * z_m))
	tmp = 0
	if t_0 <= 4e+306:
		tmp = (1.0 / x_m) / t_0
	else:
		tmp = 1.0 / (y_m * (x_m * math.pow(z_m, 2.0)))
	return y_s * (x_s * tmp)

Julia

z_m = abs(z)
x\_m = abs(x)
x\_s = copysign(1.0, x)
y\_m = abs(y)
y\_s = copysign(1.0, y)
x_m, y_m, z_m = sort([x_m, y_m, z_m])
function code(y_s, x_s, x_m, y_m, z_m)
	t_0 = Float64(y_m * Float64(1.0 + Float64(z_m * z_m)))
	tmp = 0.0
	if (t_0 <= 4e+306)
		tmp = Float64(Float64(1.0 / x_m) / t_0);
	else
		tmp = Float64(1.0 / Float64(y_m * Float64(x_m * (z_m ^ 2.0))));
	end
	return Float64(y_s * Float64(x_s * tmp))
end

MATLAB

z_m = abs(z);
x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
x_m, y_m, z_m = num2cell(sort([x_m, y_m, z_m])){:}
function tmp_2 = code(y_s, x_s, x_m, y_m, z_m)
	t_0 = y_m * (1.0 + (z_m * z_m));
	tmp = 0.0;
	if (t_0 <= 4e+306)
		tmp = (1.0 / x_m) / t_0;
	else
		tmp = 1.0 / (y_m * (x_m * (z_m ^ 2.0)));
	end
	tmp_2 = y_s * (x_s * tmp);
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, and z_m should be sorted in increasing order before calling this function.
code[y$95$s_, x$95$s_, x$95$m_, y$95$m_, z$95$m_] := Block[{t$95$0 = N[(y$95$m * N[(1.0 + N[(z$95$m * z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(y$95$s * N[(x$95$s * If[LessEqual[t$95$0, 4e+306], N[(N[(1.0 / x$95$m), $MachinePrecision] / t$95$0), $MachinePrecision], N[(1.0 / N[(y$95$m * N[(x$95$m * N[Power[z$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y (+.f64 #s(literal 1 binary64) (*.f64 z z))) < 4.00000000000000007e306

   1. Initial program 94.5%

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

   if 4.00000000000000007e306 < (*.f64 y (+.f64 #s(literal 1 binary64) (*.f64 z z)))

   1. Initial program 73.7%

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

      1. associate-/l/ 73.7%

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

      2. associate-*l* 78.6%

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

      3. *-commutative 78.6%

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

      4. sqr-neg 78.6%

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

      5. +-commutative 78.6%

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

      6. sqr-neg 78.6%

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

      7. fma-define 78.6%

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

   3. Simplified 78.6%

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

   5. Taylor expanded in z around inf 78.6%

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

Alternative 8: 98.6% accurate, 0.1× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x_m, y_m, z_m] = \mathsf{sort}([x_m, y_m, z_m])\\ \\ y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \cdot z\_m \leq 5 \cdot 10^{+274}:\\ \;\;\;\;\frac{1}{y\_m \cdot \left(x\_m \cdot \mathsf{fma}\left(z\_m, z\_m, 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\mathsf{hypot}\left(1, z\_m\right) \cdot x\_m}}{z\_m \cdot y\_m}\\ \end{array}\right) \end{array} \]

FPCore

z_m = (fabs.f64 z)
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_m should be sorted in increasing order before calling this function.
(FPCore (y_s x_s x_m y_m z_m)
 :precision binary64
 (*
  y_s
  (*
   x_s
   (if (<= (* z_m z_m) 5e+274)
     (/ 1.0 (* y_m (* x_m (fma z_m z_m 1.0))))
     (/ (/ 1.0 (* (hypot 1.0 z_m) x_m)) (* z_m y_m))))))

C

z_m = fabs(z);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
assert(x_m < y_m && y_m < z_m);
double code(double y_s, double x_s, double x_m, double y_m, double z_m) {
	double tmp;
	if ((z_m * z_m) <= 5e+274) {
		tmp = 1.0 / (y_m * (x_m * fma(z_m, z_m, 1.0)));
	} else {
		tmp = (1.0 / (hypot(1.0, z_m) * x_m)) / (z_m * y_m);
	}
	return y_s * (x_s * tmp);
}

Julia

z_m = abs(z)
x\_m = abs(x)
x\_s = copysign(1.0, x)
y\_m = abs(y)
y\_s = copysign(1.0, y)
x_m, y_m, z_m = sort([x_m, y_m, z_m])
function code(y_s, x_s, x_m, y_m, z_m)
	tmp = 0.0
	if (Float64(z_m * z_m) <= 5e+274)
		tmp = Float64(1.0 / Float64(y_m * Float64(x_m * fma(z_m, z_m, 1.0))));
	else
		tmp = Float64(Float64(1.0 / Float64(hypot(1.0, z_m) * x_m)) / Float64(z_m * y_m));
	end
	return Float64(y_s * Float64(x_s * tmp))
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, and z_m should be sorted in increasing order before calling this function.
code[y$95$s_, x$95$s_, x$95$m_, y$95$m_, z$95$m_] := N[(y$95$s * N[(x$95$s * If[LessEqual[N[(z$95$m * z$95$m), $MachinePrecision], 5e+274], N[(1.0 / N[(y$95$m * N[(x$95$m * N[(z$95$m * z$95$m + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[(N[Sqrt[1.0 ^ 2 + z$95$m ^ 2], $MachinePrecision] * x$95$m), $MachinePrecision]), $MachinePrecision] / N[(z$95$m * y$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 4.9999999999999998e274

   1. Initial program 97.6%

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

      1. associate-/l/ 97.3%

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

      2. associate-*l* 96.8%

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

      3. *-commutative 96.8%

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

      4. sqr-neg 96.8%

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

      5. +-commutative 96.8%

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

      6. sqr-neg 96.8%

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

      7. fma-define 96.8%

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

   3. Simplified 96.8%

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

   if 4.9999999999999998e274 < (*.f64 z z)

   1. Initial program 77.4%

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

      1. associate-/l/ 77.4%

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

      2. associate-*l* 77.4%

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

      3. *-commutative 77.4%

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

      4. sqr-neg 77.4%

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

      5. +-commutative 77.4%

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

      6. sqr-neg 77.4%

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

      7. fma-define 77.4%

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

   3. Simplified 77.4%

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

      1. associate-*r* 75.8%

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

      2. *-commutative 75.8%

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

      3. *-commutative 75.8%

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

      4. add-sqr-sqrt 75.8%

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

      5. pow2 75.8%

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

      6. sqrt-div 38.7%

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

      7. metadata-eval 38.7%

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

      8. sqrt-prod 38.7%

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

      9. fma-undefine 38.7%

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

      10. +-commutative 38.7%

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

      11. hypot-1-def 42.4%

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

   6. Applied egg-rr 42.4%

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

      1. *-commutative 42.4%

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

      2. sqrt-prod 29.4%

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

   8. Applied egg-rr 29.4%

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

      1. unpow2 29.4%

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

      2. frac-times 29.4%

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

      3. metadata-eval 29.4%

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

      4. *-commutative 29.4%

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

      5. *-commutative 29.4%

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

      6. swap-sqr 23.3%

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

      7. swap-sqr 23.3%

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

      8. add-sqr-sqrt 46.8%

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

      9. add-sqr-sqrt 75.8%

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

      10. *-commutative 75.8%

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

      11. *-commutative 75.8%

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

      12. /-rgt-identity 75.8%

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

      13. clear-num 75.8%

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

      14. div-inv 75.8%

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

      15. associate-/r/ 77.4%

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

      16. associate-/r* 82.3%

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

      17. un-div-inv 82.3%

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

   10. Applied egg-rr 99.8%

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

   11. Taylor expanded in z around inf 83.7%

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

4. Final simplification 92.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 5 \cdot 10^{+274}:\\ \;\;\;\;\frac{1}{y \cdot \left(x \cdot \mathsf{fma}\left(z, z, 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\mathsf{hypot}\left(1, z\right) \cdot x}}{z \cdot y}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 91.8% accurate, 0.1× speedup

Math

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

FPCore

z_m = (fabs.f64 z)
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_m should be sorted in increasing order before calling this function.
(FPCore (y_s x_s x_m y_m z_m)
 :precision binary64
 (* y_s (* x_s (/ 1.0 (* y_m (* x_m (fma z_m z_m 1.0)))))))

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 91.4%

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

   1. associate-/l/ 91.2%

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

   2. associate-*l* 90.9%

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

   3. *-commutative 90.9%

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

   4. sqr-neg 90.9%

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

   5. +-commutative 90.9%

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

   6. sqr-neg 90.9%

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

   7. fma-define 90.9%

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

3. Simplified 90.9%

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

Alternative 10: 72.6% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if z < 1

   1. Initial program 93.0%

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

   3. Taylor expanded in z around 0 71.2%

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

   if 1 < z

   1. Initial program 87.5%

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

      1. associate-/l/ 86.9%

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

      2. associate-*l* 86.9%

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

      3. *-commutative 86.9%

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

      4. sqr-neg 86.9%

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

      5. +-commutative 86.9%

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

      6. sqr-neg 86.9%

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

      7. fma-define 86.9%

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

   3. Simplified 86.9%

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

      1. associate-*r* 88.1%

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

      2. *-commutative 88.1%

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

      3. associate-/r* 86.1%

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

      4. *-commutative 86.1%

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

      5. associate-/l/ 86.1%

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

      6. fma-undefine 86.1%

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

      7. +-commutative 86.1%

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

      8. associate-/r* 87.5%

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

      9. *-un-lft-identity 87.5%

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

      10. add-sqr-sqrt 36.3%

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

      11. times-frac 36.3%

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

      12. +-commutative 36.3%

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

      13. fma-undefine 36.3%

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

      14. *-commutative 36.3%

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

      15. sqrt-prod 36.4%

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

      16. fma-undefine 36.4%

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

      17. +-commutative 36.4%

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

      18. hypot-1-def 36.4%

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

      19. +-commutative 36.4%

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

   6. Applied egg-rr 40.8%

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

      1. *-commutative 40.8%

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

      2. associate-/r* 39.6%

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

      3. associate-/r* 39.6%

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

      4. frac-times 36.4%

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

      5. add-sqr-sqrt 88.8%

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

   8. Applied egg-rr 88.8%

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

   9. Taylor expanded in z around 0 48.6%

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

   10. Taylor expanded in z around inf 46.1%

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

4. Final simplification 64.0%

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

Alternative 11: 88.0% accurate, 1.0× speedup

Math

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

FPCore

z_m = (fabs.f64 z)
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_m should be sorted in increasing order before calling this function.
(FPCore (y_s x_s x_m y_m z_m)
 :precision binary64
 (* y_s (* x_s (/ (/ 1.0 x_m) (* y_m (+ 1.0 (* z_m z_m)))))))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 91.4%

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

Alternative 12: 58.3% accurate, 2.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 91.4%

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

3. Taylor expanded in z around 0 56.7%

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

Alternative 13: 58.4% accurate, 2.2× speedup

Math

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

FPCore

z_m = (fabs.f64 z)
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_m should be sorted in increasing order before calling this function.
(FPCore (y_s x_s x_m y_m z_m)
 :precision binary64
 (* y_s (* x_s (/ 1.0 (* y_m x_m)))))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 91.4%

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

   1. associate-/l/ 91.2%

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

   2. associate-*l* 90.9%

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

   3. *-commutative 90.9%

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

   4. sqr-neg 90.9%

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

   5. +-commutative 90.9%

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

   6. sqr-neg 90.9%

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

   7. fma-define 90.9%

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

3. Simplified 90.9%

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

5. Taylor expanded in z around 0 56.7%

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

Developer target: 92.2% 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 2024106 
(FPCore (x y z)
  :name "Statistics.Distribution.CauchyLorentz:$cdensity from math-functions-0.1.5.2"
  :precision binary64

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

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