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

Percentage Accurate: 89.0% → 99.2%

Time: 15.0s

Alternatives: 15

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 89.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.2% accurate, 0.1× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ [x_m, y_m, z_m] = \mathsf{sort}([x_m, y_m, z_m])\\ \\ y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 2 \cdot 10^{+127}:\\ \;\;\;\;\frac{\frac{1}{y\_m}}{x\_m \cdot \left(1 + {z\_m}^{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(1, z\_m\right)} \cdot \frac{1}{y\_m \cdot \left(z\_m \cdot x\_m\right)}\\ \end{array}\right) \end{array} \]

FPCore

z_m = (fabs.f64 z)
x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
y_m = (fabs.f64 y)
y_s = (copysign.f64 1 y)
NOTE: x_m, y_m, and z_m should be sorted in increasing order before calling this function.
(FPCore (y_s x_s x_m y_m z_m)
 :precision binary64
 (*
  y_s
  (*
   x_s
   (if (<= z_m 2e+127)
     (/ (/ 1.0 y_m) (* x_m (+ 1.0 (pow z_m 2.0))))
     (* (/ 1.0 (hypot 1.0 z_m)) (/ 1.0 (* y_m (* z_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) {
	double tmp;
	if (z_m <= 2e+127) {
		tmp = (1.0 / y_m) / (x_m * (1.0 + pow(z_m, 2.0)));
	} else {
		tmp = (1.0 / hypot(1.0, z_m)) * (1.0 / (y_m * (z_m * x_m)));
	}
	return y_s * (x_s * tmp);
}

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 <= 2e+127) {
		tmp = (1.0 / y_m) / (x_m * (1.0 + Math.pow(z_m, 2.0)));
	} else {
		tmp = (1.0 / Math.hypot(1.0, z_m)) * (1.0 / (y_m * (z_m * x_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 <= 2e+127:
		tmp = (1.0 / y_m) / (x_m * (1.0 + math.pow(z_m, 2.0)))
	else:
		tmp = (1.0 / math.hypot(1.0, z_m)) * (1.0 / (y_m * (z_m * x_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 <= 2e+127)
		tmp = Float64(Float64(1.0 / y_m) / Float64(x_m * Float64(1.0 + (z_m ^ 2.0))));
	else
		tmp = Float64(Float64(1.0 / hypot(1.0, z_m)) * Float64(1.0 / Float64(y_m * Float64(z_m * x_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 <= 2e+127)
		tmp = (1.0 / y_m) / (x_m * (1.0 + (z_m ^ 2.0)));
	else
		tmp = (1.0 / hypot(1.0, z_m)) * (1.0 / (y_m * (z_m * x_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, 2e+127], N[(N[(1.0 / y$95$m), $MachinePrecision] / N[(x$95$m * N[(1.0 + N[Power[z$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[Sqrt[1.0 ^ 2 + z$95$m ^ 2], $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(y$95$m * N[(z$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 1.99999999999999991e127

   1. Initial program 94.4%

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

      1. associate-/l/ 94.2%

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

      2. metadata-eval 94.2%

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

      3. associate-*r/ 94.2%

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

      4. associate-/l/ 94.4%

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

      5. associate-*r/ 94.4%

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

      6. associate-/l* 94.1%

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

      7. associate-/r/ 94.2%

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

      8. /-rgt-identity 94.2%

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

      9. associate-*l* 91.6%

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

      10. *-commutative 91.6%

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

      11. sqr-neg 91.6%

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

      12. +-commutative 91.6%

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

      13. sqr-neg 91.6%

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

      14. fma-def 91.6%

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

   3. Simplified 91.6%

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

      1. fma-udef 91.6%

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

      2. +-commutative 91.6%

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

      3. *-commutative 91.6%

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

      4. associate-*l* 94.2%

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

      5. associate-/l/ 94.4%

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

      6. add-sqr-sqrt 48.5%

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

      7. *-un-lft-identity 48.5%

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

      8. times-frac 48.5%

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

      9. *-commutative 48.5%

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

      10. sqrt-prod 48.5%

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

      11. hypot-1-def 48.6%

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

      12. *-commutative 48.6%

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

      13. sqrt-prod 48.9%

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

      14. hypot-1-def 51.9%

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

   6. Applied egg-rr 51.9%

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

      1. frac-times 50.3%

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

      2. *-un-lft-identity 50.3%

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

      3. associate-/l/ 50.1%

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

      4. *-commutative 50.1%

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

      5. *-commutative 50.1%

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

      6. swap-sqr 48.4%

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

      7. add-sqr-sqrt 94.2%

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

      8. pow2 94.2%

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

      9. associate-*r* 91.6%

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

      10. add-sqr-sqrt 50.8%

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

      11. pow2 50.8%

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

      12. pow-prod-down 52.1%

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

      13. associate-/r* 52.2%

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

      14. *-commutative 52.2%

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

      15. unpow-prod-down 50.9%

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

   8. Applied egg-rr 91.8%

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

   if 1.99999999999999991e127 < z

   1. Initial program 65.1%

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

      1. associate-/l/ 65.2%

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

      2. metadata-eval 65.2%

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

      3. associate-*r/ 65.2%

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

      4. associate-/l/ 65.1%

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

      5. associate-*r/ 65.1%

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

      6. associate-/l* 65.2%

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

      7. associate-/r/ 65.2%

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

      8. /-rgt-identity 65.2%

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

      9. associate-*l* 67.6%

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

      10. *-commutative 67.6%

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

      11. sqr-neg 67.6%

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

      12. +-commutative 67.6%

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

      13. sqr-neg 67.6%

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

      14. fma-def 67.6%

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

   3. Simplified 67.6%

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

      1. fma-udef 67.6%

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

      2. +-commutative 67.6%

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

      3. *-commutative 67.6%

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

      4. associate-*l* 65.2%

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

      5. associate-/l/ 65.1%

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

      6. add-sqr-sqrt 29.1%

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

      7. *-un-lft-identity 29.1%

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

      8. times-frac 29.1%

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

      9. *-commutative 29.1%

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

      10. sqrt-prod 29.1%

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

      11. hypot-1-def 29.1%

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

      12. *-commutative 29.1%

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

      13. sqrt-prod 29.1%

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

      14. hypot-1-def 47.4%

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

   6. Applied egg-rr 47.4%

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

      1. frac-times 38.7%

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

      2. *-un-lft-identity 38.7%

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

      3. inv-pow 38.7%

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

      4. metadata-eval 38.7%

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

      5. pow-prod-up 28.6%

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

      6. frac-times 32.4%

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

      7. associate-/r* 32.3%

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

      8. associate-/r* 32.3%

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

      9. frac-times 26.1%

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

      10. add-sqr-sqrt 41.4%

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

   8. Applied egg-rr 41.4%

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

      1. associate-*l/ 41.5%

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

      2. associate-*r/ 41.5%

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

      3. pow-sqr 86.4%

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

      4. metadata-eval 86.4%

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

      5. unpow-1 86.4%

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

      6. *-rgt-identity 86.4%

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

      7. *-commutative 86.4%

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

      8. associate-*l/ 86.3%

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

      9. associate-/l/ 86.5%

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

      10. associate-*r/ 86.5%

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

      11. *-rgt-identity 86.5%

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

   10. Simplified 86.5%

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

       1. associate-/l/ 97.5%

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

       2. div-inv 97.5%

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

   12. Applied egg-rr 97.5%

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

   13. Taylor expanded in z around inf 97.5%

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

4. Final simplification 92.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 2 \cdot 10^{+127}:\\ \;\;\;\;\frac{\frac{1}{y}}{x \cdot \left(1 + {z}^{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(1, z\right)} \cdot \frac{1}{y \cdot \left(z \cdot x\right)}\\ \end{array} \]
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 1 x)
y_m = (fabs.f64 y)
y_s = (copysign.f64 1 y)
NOTE: x_m, y_m, and z_m should be sorted in increasing order before calling this function.
(FPCore (y_s x_s x_m y_m z_m)
 :precision binary64
 (let* ((t_0 (* (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 89.8%

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

   1. associate-/l/ 89.7%

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

   2. metadata-eval 89.7%

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

   3. associate-*r/ 89.7%

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

   4. associate-/l/ 89.8%

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

   5. associate-*r/ 89.8%

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

   6. associate-/l* 89.6%

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

   7. associate-/r/ 89.7%

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

   8. /-rgt-identity 89.7%

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

   9. associate-*l* 87.8%

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

   10. *-commutative 87.8%

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

   11. sqr-neg 87.8%

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

   12. +-commutative 87.8%

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

   13. sqr-neg 87.8%

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

   14. fma-def 87.8%

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

3. Simplified 87.8%

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

   1. fma-udef 87.8%

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

   2. +-commutative 87.8%

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

   3. *-commutative 87.8%

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

   4. associate-*l* 89.7%

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

   5. associate-/l/ 89.8%

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

   6. add-sqr-sqrt 45.5%

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

   7. *-un-lft-identity 45.5%

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

   8. times-frac 45.5%

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

   9. *-commutative 45.5%

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

   10. sqrt-prod 45.5%

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

   11. hypot-1-def 45.5%

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

   12. *-commutative 45.5%

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

   13. sqrt-prod 45.8%

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

   14. hypot-1-def 51.2%

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

6. Applied egg-rr 51.2%

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

7. Final simplification 51.2%

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

Alternative 3: 99.2% accurate, 0.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 89.8%

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

   1. associate-/l/ 89.7%

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

   2. metadata-eval 89.7%

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

   3. associate-*r/ 89.7%

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

   4. associate-/l/ 89.8%

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

   5. associate-*r/ 89.8%

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

   6. associate-/l* 89.6%

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

   7. associate-/r/ 89.7%

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

   8. /-rgt-identity 89.7%

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

   9. associate-*l* 87.8%

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

   10. *-commutative 87.8%

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

   11. sqr-neg 87.8%

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

   12. +-commutative 87.8%

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

   13. sqr-neg 87.8%

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

   14. fma-def 87.8%

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

3. Simplified 87.8%

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

   1. fma-udef 87.8%

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

   2. +-commutative 87.8%

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

   3. *-commutative 87.8%

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

   4. associate-*l* 89.7%

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

   5. associate-/l/ 89.8%

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

   6. add-sqr-sqrt 62.5%

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

   7. sqrt-div 28.3%

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

   8. inv-pow 28.3%

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

   9. sqrt-pow1 28.3%

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

   10. metadata-eval 28.3%

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

   11. *-commutative 28.3%

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

   12. sqrt-prod 28.2%

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

   13. hypot-1-def 28.3%

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

   14. sqrt-div 28.2%

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

   15. inv-pow 28.2%

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

   16. sqrt-pow1 28.2%

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

   17. metadata-eval 28.2%

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

   18. *-commutative 28.2%

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

6. Applied egg-rr 31.7%

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

   1. unpow2 31.7%

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

8. Simplified 31.7%

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

9. Final simplification 31.7%

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

Alternative 4: 98.4% accurate, 0.1× speedup

Math

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

FPCore

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

Julia

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

Wolfram

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

Derivation

1. Initial program 89.8%

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

   1. associate-/l/ 89.7%

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

   2. metadata-eval 89.7%

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

   3. associate-*r/ 89.7%

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

   4. associate-/l/ 89.8%

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

   5. associate-*r/ 89.8%

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

   6. associate-/l* 89.6%

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

   7. associate-/r/ 89.7%

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

   8. /-rgt-identity 89.7%

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

   9. associate-*l* 87.8%

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

   10. *-commutative 87.8%

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

   11. sqr-neg 87.8%

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

   12. +-commutative 87.8%

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

   13. sqr-neg 87.8%

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

   14. fma-def 87.8%

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

3. Simplified 87.8%

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

   1. fma-udef 87.8%

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

   2. +-commutative 87.8%

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

   3. *-commutative 87.8%

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

   4. associate-*l* 89.7%

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

   5. associate-/l/ 89.8%

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

   6. add-sqr-sqrt 45.5%

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

   7. *-un-lft-identity 45.5%

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

   8. times-frac 45.5%

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

   9. *-commutative 45.5%

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

   10. sqrt-prod 45.5%

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

   11. hypot-1-def 45.5%

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

   12. *-commutative 45.5%

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

   13. sqrt-prod 45.8%

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

   14. hypot-1-def 51.2%

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

6. Applied egg-rr 51.2%

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

   1. frac-times 48.5%

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

   2. *-un-lft-identity 48.5%

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

   3. inv-pow 48.5%

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

   4. metadata-eval 48.5%

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

   5. pow-prod-up 30.5%

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

   6. frac-times 31.7%

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

   7. associate-/r* 31.7%

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

   8. associate-/r* 31.7%

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

   9. frac-times 28.3%

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

   10. add-sqr-sqrt 50.5%

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

8. Applied egg-rr 50.5%

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

   1. associate-*l/ 50.5%

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

   2. associate-*r/ 50.5%

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

   3. pow-sqr 93.2%

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

   4. metadata-eval 93.2%

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

   5. unpow-1 93.2%

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

   6. *-rgt-identity 93.2%

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

   7. *-commutative 93.2%

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

   8. associate-*l/ 93.2%

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

   9. associate-/l/ 93.2%

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

   10. associate-*r/ 93.2%

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

   11. *-rgt-identity 93.2%

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

10. Simplified 93.2%

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

    1. associate-/l/ 96.5%

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

    2. div-inv 96.4%

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

12. Applied egg-rr 96.4%

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

13. Final simplification 96.4%

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

Alternative 5: 98.9% accurate, 0.1× speedup

Math

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

FPCore

z_m = (fabs.f64 z)
x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
y_m = (fabs.f64 y)
y_s = (copysign.f64 1 y)
NOTE: x_m, y_m, and z_m should be sorted in increasing order before calling this function.
(FPCore (y_s x_s x_m y_m z_m)
 :precision binary64
 (* y_s (* x_s (/ (/ (/ 1.0 (hypot 1.0 z_m)) (* (hypot 1.0 z_m) 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 / hypot(1.0, z_m)) / (hypot(1.0, z_m) * x_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)) / (Math.hypot(1.0, z_m) * 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 / math.hypot(1.0, z_m)) / (math.hypot(1.0, z_m) * 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(Float64(1.0 / hypot(1.0, z_m)) / Float64(hypot(1.0, z_m) * 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 / hypot(1.0, z_m)) / (hypot(1.0, z_m) * 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[(N[(1.0 / N[Sqrt[1.0 ^ 2 + z$95$m ^ 2], $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[1.0 ^ 2 + z$95$m ^ 2], $MachinePrecision] * x$95$m), $MachinePrecision]), $MachinePrecision] / y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 89.8%

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

   1. associate-/l/ 89.7%

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

   2. metadata-eval 89.7%

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

   3. associate-*r/ 89.7%

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

   4. associate-/l/ 89.8%

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

   5. associate-*r/ 89.8%

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

   6. associate-/l* 89.6%

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

   7. associate-/r/ 89.7%

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

   8. /-rgt-identity 89.7%

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

   9. associate-*l* 87.8%

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

   10. *-commutative 87.8%

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

   11. sqr-neg 87.8%

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

   12. +-commutative 87.8%

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

   13. sqr-neg 87.8%

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

   14. fma-def 87.8%

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

3. Simplified 87.8%

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

   1. fma-udef 87.8%

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

   2. +-commutative 87.8%

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

   3. *-commutative 87.8%

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

   4. associate-*l* 89.7%

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

   5. associate-/l/ 89.8%

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

   6. add-sqr-sqrt 45.5%

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

   7. *-un-lft-identity 45.5%

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

   8. times-frac 45.5%

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

   9. *-commutative 45.5%

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

   10. sqrt-prod 45.5%

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

   11. hypot-1-def 45.5%

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

   12. *-commutative 45.5%

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

   13. sqrt-prod 45.8%

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

   14. hypot-1-def 51.2%

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

6. Applied egg-rr 51.2%

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

   1. frac-times 48.5%

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

   2. *-un-lft-identity 48.5%

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

   3. inv-pow 48.5%

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

   4. metadata-eval 48.5%

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

   5. pow-prod-up 30.5%

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

   6. frac-times 31.7%

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

   7. associate-/r* 31.7%

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

   8. associate-/r* 31.7%

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

   9. frac-times 28.3%

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

   10. add-sqr-sqrt 50.5%

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

8. Applied egg-rr 50.5%

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

   1. associate-*l/ 50.5%

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

   2. associate-*r/ 50.5%

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

   3. pow-sqr 93.2%

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

   4. metadata-eval 93.2%

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

   5. unpow-1 93.2%

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

   6. *-rgt-identity 93.2%

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

   7. *-commutative 93.2%

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

   8. associate-*l/ 93.2%

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

   9. associate-/l/ 93.2%

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

   10. associate-*r/ 93.2%

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

   11. *-rgt-identity 93.2%

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

10. Simplified 93.2%

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

11. Final simplification 93.2%

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

Alternative 6: 98.5% accurate, 0.1× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ [x_m, y_m, z_m] = \mathsf{sort}([x_m, y_m, z_m])\\ \\ y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \cdot z\_m \leq 2 \cdot 10^{+264}:\\ \;\;\;\;\frac{\frac{1}{y\_m}}{x\_m \cdot \left(1 + {z\_m}^{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(1, z\_m\right)} \cdot \frac{1}{x\_m \cdot \left(z\_m \cdot y\_m\right)}\\ \end{array}\right) \end{array} \]

FPCore

z_m = (fabs.f64 z)
x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
y_m = (fabs.f64 y)
y_s = (copysign.f64 1 y)
NOTE: x_m, y_m, and z_m should be sorted in increasing order before calling this function.
(FPCore (y_s x_s x_m y_m z_m)
 :precision binary64
 (*
  y_s
  (*
   x_s
   (if (<= (* z_m z_m) 2e+264)
     (/ (/ 1.0 y_m) (* x_m (+ 1.0 (pow z_m 2.0))))
     (* (/ 1.0 (hypot 1.0 z_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 * z_m) <= 2e+264) {
		tmp = (1.0 / y_m) / (x_m * (1.0 + pow(z_m, 2.0)));
	} else {
		tmp = (1.0 / hypot(1.0, z_m)) * (1.0 / (x_m * (z_m * y_m)));
	}
	return y_s * (x_s * tmp);
}

Java

z_m = Math.abs(z);
x_m = Math.abs(x);
x_s = Math.copySign(1.0, x);
y_m = Math.abs(y);
y_s = Math.copySign(1.0, y);
assert x_m < y_m && y_m < z_m;
public static double code(double y_s, double x_s, double x_m, double y_m, double z_m) {
	double tmp;
	if ((z_m * z_m) <= 2e+264) {
		tmp = (1.0 / y_m) / (x_m * (1.0 + Math.pow(z_m, 2.0)));
	} else {
		tmp = (1.0 / Math.hypot(1.0, z_m)) * (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 * z_m) <= 2e+264:
		tmp = (1.0 / y_m) / (x_m * (1.0 + math.pow(z_m, 2.0)))
	else:
		tmp = (1.0 / math.hypot(1.0, z_m)) * (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 (Float64(z_m * z_m) <= 2e+264)
		tmp = Float64(Float64(1.0 / y_m) / Float64(x_m * Float64(1.0 + (z_m ^ 2.0))));
	else
		tmp = Float64(Float64(1.0 / hypot(1.0, z_m)) * 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 * z_m) <= 2e+264)
		tmp = (1.0 / y_m) / (x_m * (1.0 + (z_m ^ 2.0)));
	else
		tmp = (1.0 / hypot(1.0, z_m)) * (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[N[(z$95$m * z$95$m), $MachinePrecision], 2e+264], N[(N[(1.0 / y$95$m), $MachinePrecision] / N[(x$95$m * N[(1.0 + N[Power[z$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[Sqrt[1.0 ^ 2 + z$95$m ^ 2], $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(x$95$m * N[(z$95$m * y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 2.00000000000000009e264

   1. Initial program 98.1%

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

      1. associate-/l/ 97.8%

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

      2. metadata-eval 97.8%

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

      3. associate-*r/ 97.8%

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

      4. associate-/l/ 98.1%

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

      5. associate-*r/ 98.1%

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

      6. associate-/l* 97.7%

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

      7. associate-/r/ 97.8%

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

      8. /-rgt-identity 97.8%

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

      9. associate-*l* 95.3%

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

      10. *-commutative 95.3%

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

      11. sqr-neg 95.3%

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

      12. +-commutative 95.3%

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

      13. sqr-neg 95.3%

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

      14. fma-def 95.3%

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

   3. Simplified 95.3%

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

      1. fma-udef 95.3%

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

      2. +-commutative 95.3%

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

      3. *-commutative 95.3%

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

      4. associate-*l* 97.8%

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

      5. associate-/l/ 98.1%

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

      6. add-sqr-sqrt 50.5%

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

      7. *-un-lft-identity 50.5%

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

      8. times-frac 50.5%

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

      9. *-commutative 50.5%

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

      10. sqrt-prod 50.5%

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

      11. hypot-1-def 50.5%

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

      12. *-commutative 50.5%

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

      13. sqrt-prod 51.0%

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

      14. hypot-1-def 51.0%

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

   6. Applied egg-rr 51.0%

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

      1. frac-times 50.5%

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

      2. *-un-lft-identity 50.5%

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

      3. associate-/l/ 50.4%

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

      4. *-commutative 50.4%

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

      5. *-commutative 50.4%

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

      6. swap-sqr 50.3%

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

      7. add-sqr-sqrt 97.8%

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

      8. pow2 97.8%

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

      9. associate-*r* 95.3%

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

      10. add-sqr-sqrt 55.4%

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

      11. pow2 55.4%

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

      12. pow-prod-down 55.4%

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

      13. associate-/r* 55.5%

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

      14. *-commutative 55.5%

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

      15. unpow-prod-down 55.5%

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

   8. Applied egg-rr 95.6%

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

   if 2.00000000000000009e264 < (*.f64 z z)

   1. Initial program 68.4%

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

      1. associate-/l/ 68.5%

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

      2. metadata-eval 68.5%

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

      3. associate-*r/ 68.5%

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

      4. associate-/l/ 68.4%

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

      5. associate-*r/ 68.4%

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

      6. associate-/l* 68.5%

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

      7. associate-/r/ 68.5%

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

      8. /-rgt-identity 68.5%

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

      9. associate-*l* 68.5%

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

      10. *-commutative 68.5%

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

      11. sqr-neg 68.5%

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

      12. +-commutative 68.5%

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

      13. sqr-neg 68.5%

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

      14. fma-def 68.5%

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

   3. Simplified 68.5%

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

      1. fma-udef 68.5%

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

      2. +-commutative 68.5%

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

      3. *-commutative 68.5%

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

      4. associate-*l* 68.5%

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

      5. associate-/l/ 68.4%

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

      6. add-sqr-sqrt 32.5%

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

      7. *-un-lft-identity 32.5%

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

      8. times-frac 32.5%

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

      9. *-commutative 32.5%

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

      10. sqrt-prod 32.5%

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

      11. hypot-1-def 32.5%

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

      12. *-commutative 32.5%

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

      13. sqrt-prod 32.5%

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

      14. hypot-1-def 51.9%

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

   6. Applied egg-rr 51.9%

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

      1. frac-times 43.0%

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

      2. *-un-lft-identity 43.0%

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

      3. inv-pow 43.0%

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

      4. metadata-eval 43.0%

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

      5. pow-prod-up 24.7%

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

      6. frac-times 29.4%

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

      7. associate-/r* 29.4%

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

      8. associate-/r* 29.4%

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

      9. frac-times 23.3%

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

      10. add-sqr-sqrt 37.6%

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

   8. Applied egg-rr 37.6%

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

      1. associate-*l/ 37.6%

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

      2. associate-*r/ 37.6%

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

      3. pow-sqr 87.1%

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

      4. metadata-eval 87.1%

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

      5. unpow-1 87.1%

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

      6. *-rgt-identity 87.1%

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

      7. *-commutative 87.1%

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

      8. associate-*l/ 87.0%

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

      9. associate-/l/ 87.1%

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

      10. associate-*r/ 87.1%

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

      11. *-rgt-identity 87.1%

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

   10. Simplified 87.1%

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

       1. associate-/l/ 97.1%

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

       2. div-inv 97.2%

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

   12. Applied egg-rr 97.2%

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

   13. Taylor expanded in z around inf 82.2%

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

4. Final simplification 91.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{+264}:\\ \;\;\;\;\frac{\frac{1}{y}}{x \cdot \left(1 + {z}^{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(1, z\right)} \cdot \frac{1}{x \cdot \left(z \cdot y\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 98.6% accurate, 0.1× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ [x_m, y_m, z_m] = \mathsf{sort}([x_m, y_m, z_m])\\ \\ y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \cdot z\_m \leq 10^{+25}:\\ \;\;\;\;\frac{1}{x\_m \cdot \left(y\_m \cdot \left(1 + z\_m \cdot z\_m\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{1}{\mathsf{hypot}\left(1, z\_m\right)}}{z\_m \cdot x\_m}}{y\_m}\\ \end{array}\right) \end{array} \]

FPCore

z_m = (fabs.f64 z)
x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
y_m = (fabs.f64 y)
y_s = (copysign.f64 1 y)
NOTE: x_m, y_m, and z_m should be sorted in increasing order before calling this function.
(FPCore (y_s x_s x_m y_m z_m)
 :precision binary64
 (*
  y_s
  (*
   x_s
   (if (<= (* z_m z_m) 1e+25)
     (/ 1.0 (* x_m (* y_m (+ 1.0 (* z_m z_m)))))
     (/ (/ (/ 1.0 (hypot 1.0 z_m)) (* z_m 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) {
	double tmp;
	if ((z_m * z_m) <= 1e+25) {
		tmp = 1.0 / (x_m * (y_m * (1.0 + (z_m * z_m))));
	} else {
		tmp = ((1.0 / hypot(1.0, z_m)) / (z_m * x_m)) / y_m;
	}
	return y_s * (x_s * tmp);
}

Java

z_m = Math.abs(z);
x_m = Math.abs(x);
x_s = Math.copySign(1.0, x);
y_m = Math.abs(y);
y_s = Math.copySign(1.0, y);
assert x_m < y_m && y_m < z_m;
public static double code(double y_s, double x_s, double x_m, double y_m, double z_m) {
	double tmp;
	if ((z_m * z_m) <= 1e+25) {
		tmp = 1.0 / (x_m * (y_m * (1.0 + (z_m * z_m))));
	} else {
		tmp = ((1.0 / Math.hypot(1.0, z_m)) / (z_m * x_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 * z_m) <= 1e+25:
		tmp = 1.0 / (x_m * (y_m * (1.0 + (z_m * z_m))))
	else:
		tmp = ((1.0 / math.hypot(1.0, z_m)) / (z_m * x_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) <= 1e+25)
		tmp = Float64(1.0 / Float64(x_m * Float64(y_m * Float64(1.0 + Float64(z_m * z_m)))));
	else
		tmp = Float64(Float64(Float64(1.0 / hypot(1.0, z_m)) / Float64(z_m * x_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 * z_m) <= 1e+25)
		tmp = 1.0 / (x_m * (y_m * (1.0 + (z_m * z_m))));
	else
		tmp = ((1.0 / hypot(1.0, z_m)) / (z_m * x_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[N[(z$95$m * z$95$m), $MachinePrecision], 1e+25], N[(1.0 / N[(x$95$m * N[(y$95$m * N[(1.0 + N[(z$95$m * z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 / N[Sqrt[1.0 ^ 2 + z$95$m ^ 2], $MachinePrecision]), $MachinePrecision] / N[(z$95$m * x$95$m), $MachinePrecision]), $MachinePrecision] / y$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 1.00000000000000009e25

   1. Initial program 99.6%

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

      1. associate-/l/ 99.7%

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

   3. Simplified 99.7%

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

   if 1.00000000000000009e25 < (*.f64 z z)

   1. Initial program 79.0%

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

      1. associate-/l/ 78.5%

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

      2. metadata-eval 78.5%

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

      3. associate-*r/ 78.5%

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

      4. associate-/l/ 79.0%

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

      5. associate-*r/ 79.0%

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

      6. associate-/l* 78.5%

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

      7. associate-/r/ 78.5%

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

      8. /-rgt-identity 78.5%

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

      9. associate-*l* 74.7%

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

      10. *-commutative 74.7%

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

      11. sqr-neg 74.7%

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

      12. +-commutative 74.7%

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

      13. sqr-neg 74.7%

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

      14. fma-def 74.7%

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

   3. Simplified 74.7%

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

      1. fma-udef 74.7%

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

      2. +-commutative 74.7%

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

      3. *-commutative 74.7%

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

      4. associate-*l* 78.5%

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

      5. associate-/l/ 79.0%

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

      6. add-sqr-sqrt 43.0%

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

      7. *-un-lft-identity 43.0%

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

      8. times-frac 42.9%

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

      9. *-commutative 42.9%

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

      10. sqrt-prod 42.9%

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

      11. hypot-1-def 42.9%

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

      12. *-commutative 42.9%

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

      13. sqrt-prod 43.6%

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

      14. hypot-1-def 55.0%

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

   6. Applied egg-rr 55.0%

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

      1. frac-times 49.2%

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

      2. *-un-lft-identity 49.2%

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

      3. inv-pow 49.2%

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

      4. metadata-eval 49.2%

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

      5. pow-prod-up 31.7%

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

      6. frac-times 34.4%

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

      7. associate-/r* 34.4%

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

      8. associate-/r* 34.4%

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

      9. frac-times 27.1%

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

      10. add-sqr-sqrt 43.7%

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

   8. Applied egg-rr 43.7%

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

      1. associate-*l/ 43.7%

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

      2. associate-*r/ 43.8%

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

      3. pow-sqr 86.0%

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

      4. metadata-eval 86.0%

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

      5. unpow-1 86.0%

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

      6. *-rgt-identity 86.0%

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

      7. *-commutative 86.0%

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

      8. associate-*l/ 86.0%

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

      9. associate-/l/ 86.1%

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

      10. associate-*r/ 86.1%

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

      11. *-rgt-identity 86.1%

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

   10. Simplified 86.1%

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

   11. Taylor expanded in z around inf 68.2%

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

4. Final simplification 84.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 10^{+25}:\\ \;\;\;\;\frac{1}{x \cdot \left(y \cdot \left(1 + z \cdot z\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{1}{\mathsf{hypot}\left(1, z\right)}}{z \cdot x}}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 93.7% accurate, 0.1× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ [x_m, y_m, z_m] = \mathsf{sort}([x_m, y_m, z_m])\\ \\ y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \cdot z\_m \leq 2 \cdot 10^{+264}:\\ \;\;\;\;\frac{1}{y\_m \cdot \left(x\_m \cdot \mathsf{fma}\left(z\_m, z\_m, 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x\_m \cdot \mathsf{fma}\left(z\_m \cdot y\_m, z\_m, y\_m\right)}\\ \end{array}\right) \end{array} \]

FPCore

z_m = (fabs.f64 z)
x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
y_m = (fabs.f64 y)
y_s = (copysign.f64 1 y)
NOTE: x_m, y_m, and z_m should be sorted in increasing order before calling this function.
(FPCore (y_s x_s x_m y_m z_m)
 :precision binary64
 (*
  y_s
  (*
   x_s
   (if (<= (* z_m z_m) 2e+264)
     (/ 1.0 (* y_m (* x_m (fma z_m z_m 1.0))))
     (/ 1.0 (* x_m (fma (* z_m y_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) <= 2e+264) {
		tmp = 1.0 / (y_m * (x_m * fma(z_m, z_m, 1.0)));
	} else {
		tmp = 1.0 / (x_m * fma((z_m * y_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) <= 2e+264)
		tmp = Float64(1.0 / Float64(y_m * Float64(x_m * fma(z_m, z_m, 1.0))));
	else
		tmp = Float64(1.0 / Float64(x_m * fma(Float64(z_m * y_m), 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], 2e+264], 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[(1.0 / N[(x$95$m * N[(N[(z$95$m * y$95$m), $MachinePrecision] * z$95$m + y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 2.00000000000000009e264

   1. Initial program 98.1%

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

      1. associate-/l/ 97.8%

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

      2. metadata-eval 97.8%

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

      3. associate-*r/ 97.8%

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

      4. associate-/l/ 98.1%

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

      5. associate-*r/ 98.1%

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

      6. associate-/l* 97.7%

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

      7. associate-/r/ 97.8%

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

      8. /-rgt-identity 97.8%

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

      9. associate-*l* 95.3%

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

      10. *-commutative 95.3%

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

      11. sqr-neg 95.3%

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

      12. +-commutative 95.3%

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

      13. sqr-neg 95.3%

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

      14. fma-def 95.3%

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

   3. Simplified 95.3%

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

   if 2.00000000000000009e264 < (*.f64 z z)

   1. Initial program 68.4%

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

      1. associate-/l/ 68.5%

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

   3. Simplified 68.5%

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

      1. +-commutative 68.5%

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

      2. distribute-lft-in 68.5%

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

      3. associate-*r* 85.7%

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

      4. *-rgt-identity 85.7%

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

      5. fma-def 85.7%

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

   6. Applied egg-rr 85.7%

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

4. Final simplification 92.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{+264}:\\ \;\;\;\;\frac{1}{y \cdot \left(x \cdot \mathsf{fma}\left(z, z, 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x \cdot \mathsf{fma}\left(z \cdot y, z, y\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 94.7% accurate, 0.1× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ [x_m, y_m, z_m] = \mathsf{sort}([x_m, y_m, z_m])\\ \\ y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;y\_m \leq 6.6 \cdot 10^{+39}:\\ \;\;\;\;\frac{1}{x\_m \cdot \mathsf{fma}\left(z\_m \cdot y\_m, z\_m, y\_m\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{y\_m}}{x\_m \cdot \left(1 + {z\_m}^{2}\right)}\\ \end{array}\right) \end{array} \]

FPCore

z_m = (fabs.f64 z)
x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
y_m = (fabs.f64 y)
y_s = (copysign.f64 1 y)
NOTE: x_m, y_m, and z_m should be sorted in increasing order before calling this function.
(FPCore (y_s x_s x_m y_m z_m)
 :precision binary64
 (*
  y_s
  (*
   x_s
   (if (<= y_m 6.6e+39)
     (/ 1.0 (* x_m (fma (* z_m y_m) z_m y_m)))
     (/ (/ 1.0 y_m) (* x_m (+ 1.0 (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 tmp;
	if (y_m <= 6.6e+39) {
		tmp = 1.0 / (x_m * fma((z_m * y_m), z_m, y_m));
	} else {
		tmp = (1.0 / y_m) / (x_m * (1.0 + 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)
	tmp = 0.0
	if (y_m <= 6.6e+39)
		tmp = Float64(1.0 / Float64(x_m * fma(Float64(z_m * y_m), z_m, y_m)));
	else
		tmp = Float64(Float64(1.0 / y_m) / Float64(x_m * Float64(1.0 + (z_m ^ 2.0))));
	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[y$95$m, 6.6e+39], N[(1.0 / N[(x$95$m * N[(N[(z$95$m * y$95$m), $MachinePrecision] * z$95$m + y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / y$95$m), $MachinePrecision] / N[(x$95$m * N[(1.0 + N[Power[z$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 6.60000000000000042e39

   1. Initial program 87.9%

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

      1. associate-/l/ 87.8%

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

   3. Simplified 87.8%

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

      1. +-commutative 87.8%

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

      2. distribute-lft-in 87.8%

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

      3. associate-*r* 94.0%

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

      4. *-rgt-identity 94.0%

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

      5. fma-def 94.0%

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

   6. Applied egg-rr 94.0%

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

   if 6.60000000000000042e39 < y

   1. Initial program 96.5%

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

      1. associate-/l/ 96.1%

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

      2. metadata-eval 96.1%

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

      3. associate-*r/ 96.1%

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

      4. associate-/l/ 96.5%

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

      5. associate-*r/ 96.5%

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

      6. associate-/l* 95.9%

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

      7. associate-/r/ 96.1%

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

      8. /-rgt-identity 96.1%

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

      9. associate-*l* 97.6%

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

      10. *-commutative 97.6%

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

      11. sqr-neg 97.6%

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

      12. +-commutative 97.6%

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

      13. sqr-neg 97.6%

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

      14. fma-def 97.6%

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

   3. Simplified 97.6%

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

      1. fma-udef 97.6%

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

      2. +-commutative 97.6%

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

      3. *-commutative 97.6%

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

      4. associate-*l* 96.1%

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

      5. associate-/l/ 96.5%

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

      6. add-sqr-sqrt 96.3%

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

      7. *-un-lft-identity 96.3%

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

      8. times-frac 96.3%

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

      9. *-commutative 96.3%

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

      10. sqrt-prod 96.3%

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

      11. hypot-1-def 96.3%

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

      12. *-commutative 96.3%

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

      13. sqrt-prod 97.8%

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

      14. hypot-1-def 99.5%

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

   6. Applied egg-rr 99.5%

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

      1. frac-times 96.4%

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

      2. *-un-lft-identity 96.4%

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

      3. associate-/l/ 95.8%

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

      4. *-commutative 95.8%

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

      5. *-commutative 95.8%

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

      6. swap-sqr 95.8%

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

      7. add-sqr-sqrt 96.1%

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

      8. pow2 96.1%

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

      9. associate-*r* 97.6%

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

      10. add-sqr-sqrt 64.7%

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

      11. pow2 64.7%

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

      12. pow-prod-down 66.3%

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

      13. associate-/r* 66.4%

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

      14. *-commutative 66.4%

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

      15. unpow-prod-down 64.7%

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

   8. Applied egg-rr 98.2%

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

4. Final simplification 94.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 6.6 \cdot 10^{+39}:\\ \;\;\;\;\frac{1}{x \cdot \mathsf{fma}\left(z \cdot y, z, y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{y}}{x \cdot \left(1 + {z}^{2}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 92.4% 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 1 x)
y_m = (fabs.f64 y)
y_s = (copysign.f64 1 y)
NOTE: x_m, y_m, and z_m should be sorted in increasing order before calling this function.
(FPCore (y_s x_s x_m y_m z_m)
 :precision binary64
 (* y_s (* x_s (/ 1.0 (* y_m (* x_m (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 89.8%

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

   1. associate-/l/ 89.7%

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

   2. metadata-eval 89.7%

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

   3. associate-*r/ 89.7%

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

   4. associate-/l/ 89.8%

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

   5. associate-*r/ 89.8%

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

   6. associate-/l* 89.6%

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

   7. associate-/r/ 89.7%

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

   8. /-rgt-identity 89.7%

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

   9. associate-*l* 87.8%

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

   10. *-commutative 87.8%

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

   11. sqr-neg 87.8%

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

   12. +-commutative 87.8%

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

   13. sqr-neg 87.8%

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

   14. fma-def 87.8%

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

3. Simplified 87.8%

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

5. Final simplification 87.8%

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

Alternative 11: 92.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ [x_m, y_m, z_m] = \mathsf{sort}([x_m, y_m, z_m])\\ \\ \begin{array}{l} t_0 := y\_m \cdot \left(1 + z\_m \cdot z\_m\right)\\ y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq 10^{+307}:\\ \;\;\;\;\frac{\frac{1}{x\_m}}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x\_m}}{y\_m} \cdot \left(\frac{1}{z\_m} \cdot \frac{1}{z\_m}\right)\\ \end{array}\right) \end{array} \end{array} \]

FPCore

z_m = (fabs.f64 z)
x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
y_m = (fabs.f64 y)
y_s = (copysign.f64 1 y)
NOTE: x_m, y_m, and z_m should be sorted in increasing order before calling this function.
(FPCore (y_s x_s x_m y_m z_m)
 :precision binary64
 (let* ((t_0 (* y_m (+ 1.0 (* z_m z_m)))))
   (*
    y_s
    (*
     x_s
     (if (<= t_0 1e+307)
       (/ (/ 1.0 x_m) t_0)
       (* (/ (/ 1.0 x_m) y_m) (* (/ 1.0 z_m) (/ 1.0 z_m))))))))

C

z_m = fabs(z);
x_m = fabs(x);
x_s = copysign(1.0, x);
y_m = fabs(y);
y_s = copysign(1.0, y);
assert(x_m < y_m && y_m < z_m);
double code(double y_s, double x_s, double x_m, double y_m, double z_m) {
	double t_0 = y_m * (1.0 + (z_m * z_m));
	double tmp;
	if (t_0 <= 1e+307) {
		tmp = (1.0 / x_m) / t_0;
	} else {
		tmp = ((1.0 / x_m) / y_m) * ((1.0 / z_m) * (1.0 / z_m));
	}
	return y_s * (x_s * tmp);
}

Fortran

z_m = abs(z)
x_m = abs(x)
x_s = copysign(1.0d0, x)
y_m = abs(y)
y_s = copysign(1.0d0, y)
NOTE: x_m, y_m, and z_m should be sorted in increasing order before calling this function.
real(8) function code(y_s, x_s, x_m, y_m, z_m)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = y_m * (1.0d0 + (z_m * z_m))
    if (t_0 <= 1d+307) then
        tmp = (1.0d0 / x_m) / t_0
    else
        tmp = ((1.0d0 / x_m) / y_m) * ((1.0d0 / z_m) * (1.0d0 / z_m))
    end if
    code = y_s * (x_s * tmp)
end function

Java

z_m = Math.abs(z);
x_m = Math.abs(x);
x_s = Math.copySign(1.0, x);
y_m = Math.abs(y);
y_s = Math.copySign(1.0, y);
assert x_m < y_m && y_m < z_m;
public static double code(double y_s, double x_s, double x_m, double y_m, double z_m) {
	double t_0 = y_m * (1.0 + (z_m * z_m));
	double tmp;
	if (t_0 <= 1e+307) {
		tmp = (1.0 / x_m) / t_0;
	} else {
		tmp = ((1.0 / x_m) / y_m) * ((1.0 / z_m) * (1.0 / z_m));
	}
	return y_s * (x_s * tmp);
}

Python

z_m = math.fabs(z)
x_m = math.fabs(x)
x_s = math.copysign(1.0, x)
y_m = math.fabs(y)
y_s = math.copysign(1.0, y)
[x_m, y_m, z_m] = sort([x_m, y_m, z_m])
def code(y_s, x_s, x_m, y_m, z_m):
	t_0 = y_m * (1.0 + (z_m * z_m))
	tmp = 0
	if t_0 <= 1e+307:
		tmp = (1.0 / x_m) / t_0
	else:
		tmp = ((1.0 / x_m) / y_m) * ((1.0 / z_m) * (1.0 / z_m))
	return y_s * (x_s * tmp)

Julia

z_m = abs(z)
x_m = abs(x)
x_s = copysign(1.0, x)
y_m = abs(y)
y_s = copysign(1.0, y)
x_m, y_m, z_m = sort([x_m, y_m, z_m])
function code(y_s, x_s, x_m, y_m, z_m)
	t_0 = Float64(y_m * Float64(1.0 + Float64(z_m * z_m)))
	tmp = 0.0
	if (t_0 <= 1e+307)
		tmp = Float64(Float64(1.0 / x_m) / t_0);
	else
		tmp = Float64(Float64(Float64(1.0 / x_m) / y_m) * Float64(Float64(1.0 / z_m) * Float64(1.0 / z_m)));
	end
	return Float64(y_s * Float64(x_s * tmp))
end

MATLAB

z_m = abs(z);
x_m = abs(x);
x_s = sign(x) * abs(1.0);
y_m = abs(y);
y_s = sign(y) * abs(1.0);
x_m, y_m, z_m = num2cell(sort([x_m, y_m, z_m])){:}
function tmp_2 = code(y_s, x_s, x_m, y_m, z_m)
	t_0 = y_m * (1.0 + (z_m * z_m));
	tmp = 0.0;
	if (t_0 <= 1e+307)
		tmp = (1.0 / x_m) / t_0;
	else
		tmp = ((1.0 / x_m) / y_m) * ((1.0 / z_m) * (1.0 / z_m));
	end
	tmp_2 = y_s * (x_s * tmp);
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
x_m = N[Abs[x], $MachinePrecision]
x_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y_m = N[Abs[y], $MachinePrecision]
y_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, and z_m should be sorted in increasing order before calling this function.
code[y$95$s_, x$95$s_, x$95$m_, y$95$m_, z$95$m_] := Block[{t$95$0 = N[(y$95$m * N[(1.0 + N[(z$95$m * z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(y$95$s * N[(x$95$s * If[LessEqual[t$95$0, 1e+307], N[(N[(1.0 / x$95$m), $MachinePrecision] / t$95$0), $MachinePrecision], N[(N[(N[(1.0 / x$95$m), $MachinePrecision] / y$95$m), $MachinePrecision] * N[(N[(1.0 / z$95$m), $MachinePrecision] * N[(1.0 / z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y (+.f64 1 (*.f64 z z))) < 9.99999999999999986e306

   1. Initial program 94.8%

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

   if 9.99999999999999986e306 < (*.f64 y (+.f64 1 (*.f64 z z)))

   1. Initial program 63.8%

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

      1. associate-/l/ 63.8%

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

      2. metadata-eval 63.8%

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

      3. associate-*r/ 63.8%

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

      4. associate-/l/ 63.8%

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

      5. associate-*r/ 63.8%

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

      6. associate-/l* 63.8%

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

      7. associate-/r/ 63.8%

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

      8. /-rgt-identity 63.8%

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

      9. associate-*l* 66.0%

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

      10. *-commutative 66.0%

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

      11. sqr-neg 66.0%

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

      12. +-commutative 66.0%

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

      13. sqr-neg 66.0%

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

      14. fma-def 66.0%

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

   3. Simplified 66.0%

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

      1. fma-udef 66.0%

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

      2. +-commutative 66.0%

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

      3. *-commutative 66.0%

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

      4. associate-*l* 63.8%

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

      5. associate-/l/ 63.8%

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

      6. add-sqr-sqrt 63.8%

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

      7. sqrt-div 36.1%

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

      8. inv-pow 36.1%

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

      9. sqrt-pow1 36.1%

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

      10. metadata-eval 36.1%

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

      11. *-commutative 36.1%

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

      12. sqrt-prod 36.1%

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

      13. hypot-1-def 36.1%

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

      14. sqrt-div 36.1%

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

      15. inv-pow 36.1%

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

      16. sqrt-pow1 36.1%

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

      17. metadata-eval 36.1%

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

      18. *-commutative 36.1%

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

   6. Applied egg-rr 58.3%

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

      1. unpow2 58.3%

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

   8. Simplified 58.3%

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

   9. Taylor expanded in z around inf 58.3%

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

       1. associate-/r* 58.4%

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

       2. div-inv 58.3%

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

       3. metadata-eval 58.3%

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

       4. sqrt-pow1 58.3%

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

       5. inv-pow 58.3%

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

       6. sqrt-div 53.7%

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

       7. associate-/r* 53.7%

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

       8. sqrt-div 48.8%

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

       9. metadata-eval 48.8%

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

   11. Applied egg-rr 48.8%

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

       1. associate-*r/ 48.9%

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

       2. *-rgt-identity 48.9%

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

   13. Simplified 48.9%

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

       1. unpow2 48.9%

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

       2. div-inv 48.8%

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

       3. div-inv 48.8%

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

       4. swap-sqr 37.8%

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

       5. un-div-inv 37.8%

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

       6. associate-/r* 37.8%

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

       7. add-sqr-sqrt 68.8%

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

       8. associate-/r* 68.8%

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

   15. Applied egg-rr 68.8%

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

4. Final simplification 90.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(1 + z \cdot z\right) \leq 10^{+307}:\\ \;\;\;\;\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x}}{y} \cdot \left(\frac{1}{z} \cdot \frac{1}{z}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 67.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ [x_m, y_m, z_m] = \mathsf{sort}([x_m, y_m, z_m])\\ \\ y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 0.00122:\\ \;\;\;\;\frac{1}{y\_m \cdot x\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{z\_m \cdot x\_m}}{y\_m}\\ \end{array}\right) \end{array} \]

FPCore

z_m = (fabs.f64 z)
x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
y_m = (fabs.f64 y)
y_s = (copysign.f64 1 y)
NOTE: x_m, y_m, and z_m should be sorted in increasing order before calling this function.
(FPCore (y_s x_s x_m y_m z_m)
 :precision binary64
 (*
  y_s
  (*
   x_s
   (if (<= z_m 0.00122) (/ 1.0 (* y_m x_m)) (/ (/ 1.0 (* z_m 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) {
	double tmp;
	if (z_m <= 0.00122) {
		tmp = 1.0 / (y_m * x_m);
	} else {
		tmp = (1.0 / (z_m * x_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 <= 0.00122d0) then
        tmp = 1.0d0 / (y_m * x_m)
    else
        tmp = (1.0d0 / (z_m * x_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 <= 0.00122) {
		tmp = 1.0 / (y_m * x_m);
	} else {
		tmp = (1.0 / (z_m * x_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 <= 0.00122:
		tmp = 1.0 / (y_m * x_m)
	else:
		tmp = (1.0 / (z_m * x_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 <= 0.00122)
		tmp = Float64(1.0 / Float64(y_m * x_m));
	else
		tmp = Float64(Float64(1.0 / Float64(z_m * x_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 <= 0.00122)
		tmp = 1.0 / (y_m * x_m);
	else
		tmp = (1.0 / (z_m * x_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, 0.00122], N[(1.0 / N[(y$95$m * x$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[(z$95$m * x$95$m), $MachinePrecision]), $MachinePrecision] / y$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 0.00121999999999999995

   1. Initial program 93.6%

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

      1. associate-/l/ 93.6%

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

      2. metadata-eval 93.6%

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

      3. associate-*r/ 93.6%

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

      4. associate-/l/ 93.6%

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

      5. associate-*r/ 93.6%

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

      6. associate-/l* 93.5%

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

      7. associate-/r/ 93.6%

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

      8. /-rgt-identity 93.6%

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

      9. associate-*l* 93.1%

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

      10. *-commutative 93.1%

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

      11. sqr-neg 93.1%

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

      12. +-commutative 93.1%

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

      13. sqr-neg 93.1%

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

      14. fma-def 93.1%

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

   3. Simplified 93.1%

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

   5. Taylor expanded in z around 0 74.0%

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

   if 0.00121999999999999995 < z

   1. Initial program 79.4%

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

      1. associate-/l/ 78.8%

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

      2. metadata-eval 78.8%

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

      3. associate-*r/ 78.8%

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

      4. associate-/l/ 79.4%

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

      5. associate-*r/ 79.4%

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

      6. associate-/l* 78.8%

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

      7. associate-/r/ 78.8%

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

      8. /-rgt-identity 78.8%

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

      9. associate-*l* 73.4%

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

      10. *-commutative 73.4%

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

      11. sqr-neg 73.4%

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

      12. +-commutative 73.4%

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

      13. sqr-neg 73.4%

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

      14. fma-def 73.4%

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

   3. Simplified 73.4%

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

      1. fma-udef 73.4%

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

      2. +-commutative 73.4%

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

      3. *-commutative 73.4%

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

      4. associate-*l* 78.8%

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

      5. associate-/l/ 79.4%

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

      6. add-sqr-sqrt 40.5%

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

      7. *-un-lft-identity 40.5%

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

      8. times-frac 40.4%

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

      9. *-commutative 40.4%

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

      10. sqrt-prod 40.4%

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

      11. hypot-1-def 40.4%

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

      12. *-commutative 40.4%

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

      13. sqrt-prod 40.4%

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

      14. hypot-1-def 51.2%

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

   6. Applied egg-rr 51.2%

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

      1. frac-times 46.2%

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

      2. *-un-lft-identity 46.2%

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

      3. inv-pow 46.2%

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

      4. metadata-eval 46.2%

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

      5. pow-prod-up 37.3%

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

      6. frac-times 39.4%

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

      7. associate-/r* 39.5%

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

      8. associate-/r* 39.4%

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

      9. frac-times 30.3%

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

      10. add-sqr-sqrt 48.0%

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

   8. Applied egg-rr 48.0%

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

      1. associate-*l/ 48.1%

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

      2. associate-*r/ 48.1%

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

      3. pow-sqr 84.9%

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

      4. metadata-eval 84.9%

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

      5. unpow-1 84.9%

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

      6. *-rgt-identity 84.9%

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

      7. *-commutative 84.9%

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

      8. associate-*l/ 84.8%

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

      9. associate-/l/ 84.9%

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

      10. associate-*r/ 84.9%

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

      11. *-rgt-identity 84.9%

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

   10. Simplified 84.9%

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

   11. Taylor expanded in z around 0 43.8%

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

   12. Taylor expanded in z around inf 43.8%

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

4. Final simplification 66.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 0.00122:\\ \;\;\;\;\frac{1}{y \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{z \cdot x}}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 88.7% accurate, 1.0× speedup

Math

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

FPCore

z_m = (fabs.f64 z)
x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
y_m = (fabs.f64 y)
y_s = (copysign.f64 1 y)
NOTE: x_m, y_m, and z_m should be sorted in increasing order before calling this function.
(FPCore (y_s x_s x_m y_m z_m)
 :precision binary64
 (* y_s (* x_s (/ 1.0 (* 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(1.0 / Float64(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[(1.0 / N[(x$95$m * N[(y$95$m * N[(1.0 + N[(z$95$m * z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 89.8%

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

   1. associate-/l/ 89.7%

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

3. Simplified 89.7%

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

5. Final simplification 89.7%

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

Alternative 14: 89.0% 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 1 x)
y_m = (fabs.f64 y)
y_s = (copysign.f64 1 y)
NOTE: x_m, y_m, and z_m should be sorted in increasing order before calling this function.
(FPCore (y_s x_s x_m y_m z_m)
 :precision binary64
 (* y_s (* x_s (/ (/ 1.0 x_m) (* y_m (+ 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 89.8%

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

3. Final simplification 89.8%

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

Alternative 15: 60.0% 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 1 x)
y_m = (fabs.f64 y)
y_s = (copysign.f64 1 y)
NOTE: x_m, y_m, and z_m should be sorted in increasing order before calling this function.
(FPCore (y_s x_s x_m y_m z_m)
 :precision binary64
 (* y_s (* x_s (/ 1.0 (* y_m x_m)))))

C

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

Fortran

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

Java

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

Python

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

Julia

z_m = abs(z)
x_m = abs(x)
x_s = copysign(1.0, x)
y_m = abs(y)
y_s = copysign(1.0, y)
x_m, y_m, z_m = sort([x_m, y_m, z_m])
function code(y_s, x_s, x_m, y_m, z_m)
	return Float64(y_s * Float64(x_s * Float64(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 89.8%

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

   1. associate-/l/ 89.7%

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

   2. metadata-eval 89.7%

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

   3. associate-*r/ 89.7%

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

   4. associate-/l/ 89.8%

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

   5. associate-*r/ 89.8%

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

   6. associate-/l* 89.6%

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

   7. associate-/r/ 89.7%

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

   8. /-rgt-identity 89.7%

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

   9. associate-*l* 87.8%

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

   10. *-commutative 87.8%

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

   11. sqr-neg 87.8%

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

   12. +-commutative 87.8%

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

   13. sqr-neg 87.8%

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

   14. fma-def 87.8%

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

3. Simplified 87.8%

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

5. Taylor expanded in z around 0 59.7%

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

6. Final simplification 59.7%

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

Developer target: 92.7% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024041 
(FPCore (x y z)
  :name "Statistics.Distribution.CauchyLorentz:$cdensity from math-functions-0.1.5.2"
  :precision binary64

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

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