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

Percentage Accurate: 90.8% → 99.2%

Time: 13.8s

Alternatives: 12

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: 90.8% 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, 0.0× speedup

Math

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

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
y_m = (fabs.f64 y)
y_s = (copysign.f64 1 y)
(FPCore (y_s x_s x_m y_m z)
 :precision binary64
 (* y_s (* x_s (pow (/ (pow x_m -0.5) (* (hypot 1.0 z) (sqrt y_m))) 2.0))))

C

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

Java

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

Python

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

Julia

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

MATLAB

x_m = abs(x);
x_s = sign(x) * abs(1.0);
y_m = abs(y);
y_s = sign(y) * abs(1.0);
function tmp = code(y_s, x_s, x_m, y_m, z)
	tmp = y_s * (x_s * (((x_m ^ -0.5) / (hypot(1.0, z) * sqrt(y_m))) ^ 2.0));
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
x_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y_m = N[Abs[y], $MachinePrecision]
y_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x$95$s_, x$95$m_, y$95$m_, z_] := N[(y$95$s * N[(x$95$s * N[Power[N[(N[Power[x$95$m, -0.5], $MachinePrecision] / N[(N[Sqrt[1.0 ^ 2 + z ^ 2], $MachinePrecision] * N[Sqrt[y$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 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* 89.3%

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

   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* 90.4%

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

   10. *-commutative 90.4%

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

   11. sqr-neg 90.4%

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

   12. +-commutative 90.4%

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

   13. sqr-neg 90.4%

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

   14. fma-def 90.4%

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

3. Simplified 90.4%

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

   1. fma-udef 90.4%

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

   2. +-commutative 90.4%

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

   3. *-commutative 90.4%

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

   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 58.2%

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

   7. sqrt-div 19.3%

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

   8. inv-pow 19.3%

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

   9. sqrt-pow1 19.3%

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

   10. metadata-eval 19.3%

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

   11. *-commutative 19.3%

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

   12. sqrt-prod 19.3%

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

   13. hypot-1-def 19.3%

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

   14. sqrt-div 19.3%

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

   15. inv-pow 19.3%

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

   16. sqrt-pow1 19.2%

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

   17. metadata-eval 19.2%

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

   18. *-commutative 19.2%

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

6. Applied egg-rr 22.1%

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

   1. unpow2 22.1%

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

8. Simplified 22.1%

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

9. Final simplification 22.1%

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

Alternative 2: 95.0% accurate, 0.1× speedup

Math

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

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
y_m = (fabs.f64 y)
y_s = (copysign.f64 1 y)
(FPCore (y_s x_s x_m y_m z)
 :precision binary64
 (* y_s (* x_s (/ (/ (/ (/ 1.0 x_m) (hypot 1.0 z)) (hypot 1.0 z)) y_m))))

C

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

Java

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

Python

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

Julia

x_m = abs(x)
x_s = copysign(1.0, x)
y_m = abs(y)
y_s = copysign(1.0, y)
function code(y_s, x_s, x_m, y_m, z)
	return Float64(y_s * Float64(x_s * Float64(Float64(Float64(Float64(1.0 / x_m) / hypot(1.0, z)) / hypot(1.0, z)) / y_m)))
end

MATLAB

x_m = abs(x);
x_s = sign(x) * abs(1.0);
y_m = abs(y);
y_s = sign(y) * abs(1.0);
function tmp = code(y_s, x_s, x_m, y_m, z)
	tmp = y_s * (x_s * ((((1.0 / x_m) / hypot(1.0, z)) / hypot(1.0, z)) / y_m));
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
x_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y_m = N[Abs[y], $MachinePrecision]
y_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x$95$s_, x$95$m_, y$95$m_, z_] := N[(y$95$s * N[(x$95$s * N[(N[(N[(N[(1.0 / x$95$m), $MachinePrecision] / N[Sqrt[1.0 ^ 2 + z ^ 2], $MachinePrecision]), $MachinePrecision] / N[Sqrt[1.0 ^ 2 + z ^ 2], $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* 89.3%

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

   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* 90.4%

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

   10. *-commutative 90.4%

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

   11. sqr-neg 90.4%

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

   12. +-commutative 90.4%

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

   13. sqr-neg 90.4%

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

   14. fma-def 90.4%

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

3. Simplified 90.4%

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

   1. fma-udef 90.4%

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

   2. +-commutative 90.4%

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

   3. *-commutative 90.4%

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

   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 58.2%

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

   7. sqrt-div 19.3%

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

   8. inv-pow 19.3%

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

   9. sqrt-pow1 19.3%

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

   10. metadata-eval 19.3%

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

   11. *-commutative 19.3%

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

   12. sqrt-prod 19.3%

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

   13. hypot-1-def 19.3%

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

   14. sqrt-div 19.3%

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

   15. inv-pow 19.3%

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

   16. sqrt-pow1 19.2%

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

   17. metadata-eval 19.2%

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

   18. *-commutative 19.2%

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

6. Applied egg-rr 22.1%

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

   1. unpow2 22.1%

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

8. Simplified 22.1%

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

   1. unpow2 22.1%

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

   2. associate-/r* 22.1%

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

   3. associate-/r* 22.1%

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

   4. frac-times 20.6%

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

   5. add-sqr-sqrt 48.0%

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

10. Applied egg-rr 48.0%

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

    1. associate-*r/ 48.0%

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

    2. associate-*l/ 48.0%

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

    3. pow-sqr 94.5%

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

    4. metadata-eval 94.5%

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

    5. unpow-1 94.5%

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

12. Simplified 94.5%

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

13. Final simplification 94.5%

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

Alternative 3: 95.0% accurate, 0.1× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;z \cdot z \leq 10^{+15}:\\ \;\;\;\;\frac{\frac{1}{x\_m}}{y\_m \cdot \left(1 + z \cdot z\right)}\\ \mathbf{elif}\;z \cdot z \leq 5 \cdot 10^{+305}:\\ \;\;\;\;\frac{1}{y\_m \cdot \left(x\_m \cdot {z}^{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{z} \cdot \frac{\frac{1}{y\_m}}{z}}{x\_m}\\ \end{array}\right) \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
y_m = (fabs.f64 y)
y_s = (copysign.f64 1 y)
(FPCore (y_s x_s x_m y_m z)
 :precision binary64
 (*
  y_s
  (*
   x_s
   (if (<= (* z z) 1e+15)
     (/ (/ 1.0 x_m) (* y_m (+ 1.0 (* z z))))
     (if (<= (* z z) 5e+305)
       (/ 1.0 (* y_m (* x_m (pow z 2.0))))
       (/ (* (/ 1.0 z) (/ (/ 1.0 y_m) z)) x_m))))))

C

x_m = fabs(x);
x_s = copysign(1.0, x);
y_m = fabs(y);
y_s = copysign(1.0, y);
double code(double y_s, double x_s, double x_m, double y_m, double z) {
	double tmp;
	if ((z * z) <= 1e+15) {
		tmp = (1.0 / x_m) / (y_m * (1.0 + (z * z)));
	} else if ((z * z) <= 5e+305) {
		tmp = 1.0 / (y_m * (x_m * pow(z, 2.0)));
	} else {
		tmp = ((1.0 / z) * ((1.0 / y_m) / z)) / x_m;
	}
	return y_s * (x_s * tmp);
}

Fortran

x_m = abs(x)
x_s = copysign(1.0d0, x)
y_m = abs(y)
y_s = copysign(1.0d0, y)
real(8) function code(y_s, x_s, x_m, y_m, z)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((z * z) <= 1d+15) then
        tmp = (1.0d0 / x_m) / (y_m * (1.0d0 + (z * z)))
    else if ((z * z) <= 5d+305) then
        tmp = 1.0d0 / (y_m * (x_m * (z ** 2.0d0)))
    else
        tmp = ((1.0d0 / z) * ((1.0d0 / y_m) / z)) / x_m
    end if
    code = y_s * (x_s * tmp)
end function

Java

x_m = Math.abs(x);
x_s = Math.copySign(1.0, x);
y_m = Math.abs(y);
y_s = Math.copySign(1.0, y);
public static double code(double y_s, double x_s, double x_m, double y_m, double z) {
	double tmp;
	if ((z * z) <= 1e+15) {
		tmp = (1.0 / x_m) / (y_m * (1.0 + (z * z)));
	} else if ((z * z) <= 5e+305) {
		tmp = 1.0 / (y_m * (x_m * Math.pow(z, 2.0)));
	} else {
		tmp = ((1.0 / z) * ((1.0 / y_m) / z)) / x_m;
	}
	return y_s * (x_s * tmp);
}

Python

x_m = math.fabs(x)
x_s = math.copysign(1.0, x)
y_m = math.fabs(y)
y_s = math.copysign(1.0, y)
def code(y_s, x_s, x_m, y_m, z):
	tmp = 0
	if (z * z) <= 1e+15:
		tmp = (1.0 / x_m) / (y_m * (1.0 + (z * z)))
	elif (z * z) <= 5e+305:
		tmp = 1.0 / (y_m * (x_m * math.pow(z, 2.0)))
	else:
		tmp = ((1.0 / z) * ((1.0 / y_m) / z)) / x_m
	return y_s * (x_s * tmp)

Julia

x_m = abs(x)
x_s = copysign(1.0, x)
y_m = abs(y)
y_s = copysign(1.0, y)
function code(y_s, x_s, x_m, y_m, z)
	tmp = 0.0
	if (Float64(z * z) <= 1e+15)
		tmp = Float64(Float64(1.0 / x_m) / Float64(y_m * Float64(1.0 + Float64(z * z))));
	elseif (Float64(z * z) <= 5e+305)
		tmp = Float64(1.0 / Float64(y_m * Float64(x_m * (z ^ 2.0))));
	else
		tmp = Float64(Float64(Float64(1.0 / z) * Float64(Float64(1.0 / y_m) / z)) / x_m);
	end
	return Float64(y_s * Float64(x_s * tmp))
end

MATLAB

x_m = abs(x);
x_s = sign(x) * abs(1.0);
y_m = abs(y);
y_s = sign(y) * abs(1.0);
function tmp_2 = code(y_s, x_s, x_m, y_m, z)
	tmp = 0.0;
	if ((z * z) <= 1e+15)
		tmp = (1.0 / x_m) / (y_m * (1.0 + (z * z)));
	elseif ((z * z) <= 5e+305)
		tmp = 1.0 / (y_m * (x_m * (z ^ 2.0)));
	else
		tmp = ((1.0 / z) * ((1.0 / y_m) / z)) / x_m;
	end
	tmp_2 = y_s * (x_s * tmp);
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
x_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y_m = N[Abs[y], $MachinePrecision]
y_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x$95$s_, x$95$m_, y$95$m_, z_] := N[(y$95$s * N[(x$95$s * If[LessEqual[N[(z * z), $MachinePrecision], 1e+15], N[(N[(1.0 / x$95$m), $MachinePrecision] / N[(y$95$m * N[(1.0 + N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(z * z), $MachinePrecision], 5e+305], N[(1.0 / N[(y$95$m * N[(x$95$m * N[Power[z, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 / z), $MachinePrecision] * N[(N[(1.0 / y$95$m), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision] / x$95$m), $MachinePrecision]]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if (*.f64 z z) < 1e15

   1. Initial program 99.0%

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

   if 1e15 < (*.f64 z z) < 5.00000000000000009e305

   1. Initial program 88.1%

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

      1. associate-/l/ 88.1%

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

      2. metadata-eval 88.1%

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

      3. associate-*r/ 88.1%

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

      4. associate-/l/ 88.1%

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

      5. associate-*r/ 88.1%

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

      6. associate-/l* 88.1%

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

      7. associate-/r/ 88.1%

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

      8. /-rgt-identity 88.1%

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

      9. associate-*l* 91.1%

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

      10. *-commutative 91.1%

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

      11. sqr-neg 91.1%

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

      12. +-commutative 91.1%

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

      13. sqr-neg 91.1%

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

      14. fma-def 91.1%

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

   3. Simplified 91.1%

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

   5. Taylor expanded in z around inf 91.1%

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

   if 5.00000000000000009e305 < (*.f64 z z)

   1. Initial program 72.0%

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

      1. distribute-rgt-in 72.0%

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

      2. *-un-lft-identity 72.0%

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

      3. flip-+ 1.6%

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

      4. pow2 1.6%

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

      5. *-commutative 1.6%

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

      6. pow2 1.6%

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

      7. *-commutative 1.6%

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

      8. pow2 1.6%

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

      9. *-commutative 1.6%

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

      10. pow2 1.6%

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

   4. Applied egg-rr 1.6%

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

      1. unpow2 1.6%

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

      2. associate-*l* 1.6%

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

      3. distribute-lft-out-- 1.6%

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

      4. *-commutative 1.6%

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

      5. associate-*r* 1.6%

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

      6. pow-sqr 1.6%

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

      7. metadata-eval 1.6%

         \[\leadsto \frac{\frac{1}{x}}{\frac{y \cdot \left(y - y \cdot {z}^{\color{blue}{4}}\right)}{y - y \cdot {z}^{2}}} \]

      8. *-commutative 1.6%

         \[\leadsto \frac{\frac{1}{x}}{\frac{y \cdot \left(y - y \cdot {z}^{4}\right)}{y - \color{blue}{{z}^{2} \cdot y}}} \]

   6. Simplified 1.6%

      \[\leadsto \frac{\frac{1}{x}}{\color{blue}{\frac{y \cdot \left(y - y \cdot {z}^{4}\right)}{y - {z}^{2} \cdot y}}} \]
   7. Step-by-step derivation

      1. div-inv 1.6%

         \[\leadsto \color{blue}{\frac{1}{x} \cdot \frac{1}{\frac{y \cdot \left(y - y \cdot {z}^{4}\right)}{y - {z}^{2} \cdot y}}} \]

      2. *-commutative 1.6%

         \[\leadsto \frac{1}{x} \cdot \frac{1}{\frac{y \cdot \left(y - y \cdot {z}^{4}\right)}{y - \color{blue}{y \cdot {z}^{2}}}} \]

   8. Applied egg-rr 1.6%

      \[\leadsto \color{blue}{\frac{1}{x} \cdot \frac{1}{\frac{y \cdot \left(y - y \cdot {z}^{4}\right)}{y - y \cdot {z}^{2}}}} \]
   9. Step-by-step derivation

      1. associate-*l/ 1.6%

         \[\leadsto \color{blue}{\frac{1 \cdot \frac{1}{\frac{y \cdot \left(y - y \cdot {z}^{4}\right)}{y - y \cdot {z}^{2}}}}{x}} \]

      2. *-lft-identity 1.6%

         \[\leadsto \frac{\color{blue}{\frac{1}{\frac{y \cdot \left(y - y \cdot {z}^{4}\right)}{y - y \cdot {z}^{2}}}}}{x} \]

      3. associate-/r/ 1.6%

         \[\leadsto \frac{\color{blue}{\frac{1}{y \cdot \left(y - y \cdot {z}^{4}\right)} \cdot \left(y - y \cdot {z}^{2}\right)}}{x} \]

      4. associate-/r* 1.6%

         \[\leadsto \frac{\color{blue}{\frac{\frac{1}{y}}{y - y \cdot {z}^{4}}} \cdot \left(y - y \cdot {z}^{2}\right)}{x} \]

      5. *-commutative 1.6%

         \[\leadsto \frac{\frac{\frac{1}{y}}{y - \color{blue}{{z}^{4} \cdot y}} \cdot \left(y - y \cdot {z}^{2}\right)}{x} \]

      6. cancel-sign-sub-inv 1.6%

         \[\leadsto \frac{\frac{\frac{1}{y}}{\color{blue}{y + \left(-{z}^{4}\right) \cdot y}} \cdot \left(y - y \cdot {z}^{2}\right)}{x} \]

      7. *-lft-identity 1.6%

         \[\leadsto \frac{\frac{\frac{1}{y}}{\color{blue}{1 \cdot y} + \left(-{z}^{4}\right) \cdot y} \cdot \left(y - y \cdot {z}^{2}\right)}{x} \]

      8. mul-1-neg 1.6%

         \[\leadsto \frac{\frac{\frac{1}{y}}{1 \cdot y + \color{blue}{\left(-1 \cdot {z}^{4}\right)} \cdot y} \cdot \left(y - y \cdot {z}^{2}\right)}{x} \]

      9. distribute-rgt-in 1.6%

         \[\leadsto \frac{\frac{\frac{1}{y}}{\color{blue}{y \cdot \left(1 + -1 \cdot {z}^{4}\right)}} \cdot \left(y - y \cdot {z}^{2}\right)}{x} \]

      10. mul-1-neg 1.6%

          \[\leadsto \frac{\frac{\frac{1}{y}}{y \cdot \left(1 + \color{blue}{\left(-{z}^{4}\right)}\right)} \cdot \left(y - y \cdot {z}^{2}\right)}{x} \]

      11. sub-neg 1.6%

          \[\leadsto \frac{\frac{\frac{1}{y}}{y \cdot \color{blue}{\left(1 - {z}^{4}\right)}} \cdot \left(y - y \cdot {z}^{2}\right)}{x} \]

   10. Simplified 1.6%

       \[\leadsto \color{blue}{\frac{\frac{\frac{1}{y}}{y \cdot \left(1 - {z}^{4}\right)} \cdot \left(y - y \cdot {z}^{2}\right)}{x}} \]

   11. Taylor expanded in z around inf 72.0%

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

       1. associate-/r* 72.0%

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

       2. *-un-lft-identity 72.0%

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

       3. unpow2 72.0%

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

       4. times-frac 85.3%

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

   13. Applied egg-rr 85.3%

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

4. Final simplification 93.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 10^{+15}:\\ \;\;\;\;\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}\\ \mathbf{elif}\;z \cdot z \leq 5 \cdot 10^{+305}:\\ \;\;\;\;\frac{1}{y \cdot \left(x \cdot {z}^{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{z} \cdot \frac{\frac{1}{y}}{z}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 91.1% accurate, 0.1× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;y\_m \cdot \left(1 + z \cdot z\right) \leq 10^{+305}:\\ \;\;\;\;\frac{\frac{1}{x\_m \cdot \left(1 + {z}^{2}\right)}}{y\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{1}{x\_m \cdot z}}{\mathsf{hypot}\left(1, z\right)}}{y\_m}\\ \end{array}\right) \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
y_m = (fabs.f64 y)
y_s = (copysign.f64 1 y)
(FPCore (y_s x_s x_m y_m z)
 :precision binary64
 (*
  y_s
  (*
   x_s
   (if (<= (* y_m (+ 1.0 (* z z))) 1e+305)
     (/ (/ 1.0 (* x_m (+ 1.0 (pow z 2.0)))) y_m)
     (/ (/ (/ 1.0 (* x_m z)) (hypot 1.0 z)) y_m)))))

C

x_m = fabs(x);
x_s = copysign(1.0, x);
y_m = fabs(y);
y_s = copysign(1.0, y);
double code(double y_s, double x_s, double x_m, double y_m, double z) {
	double tmp;
	if ((y_m * (1.0 + (z * z))) <= 1e+305) {
		tmp = (1.0 / (x_m * (1.0 + pow(z, 2.0)))) / y_m;
	} else {
		tmp = ((1.0 / (x_m * z)) / hypot(1.0, z)) / y_m;
	}
	return y_s * (x_s * tmp);
}

Java

x_m = Math.abs(x);
x_s = Math.copySign(1.0, x);
y_m = Math.abs(y);
y_s = Math.copySign(1.0, y);
public static double code(double y_s, double x_s, double x_m, double y_m, double z) {
	double tmp;
	if ((y_m * (1.0 + (z * z))) <= 1e+305) {
		tmp = (1.0 / (x_m * (1.0 + Math.pow(z, 2.0)))) / y_m;
	} else {
		tmp = ((1.0 / (x_m * z)) / Math.hypot(1.0, z)) / y_m;
	}
	return y_s * (x_s * tmp);
}

Python

x_m = math.fabs(x)
x_s = math.copysign(1.0, x)
y_m = math.fabs(y)
y_s = math.copysign(1.0, y)
def code(y_s, x_s, x_m, y_m, z):
	tmp = 0
	if (y_m * (1.0 + (z * z))) <= 1e+305:
		tmp = (1.0 / (x_m * (1.0 + math.pow(z, 2.0)))) / y_m
	else:
		tmp = ((1.0 / (x_m * z)) / math.hypot(1.0, z)) / y_m
	return y_s * (x_s * tmp)

Julia

x_m = abs(x)
x_s = copysign(1.0, x)
y_m = abs(y)
y_s = copysign(1.0, y)
function code(y_s, x_s, x_m, y_m, z)
	tmp = 0.0
	if (Float64(y_m * Float64(1.0 + Float64(z * z))) <= 1e+305)
		tmp = Float64(Float64(1.0 / Float64(x_m * Float64(1.0 + (z ^ 2.0)))) / y_m);
	else
		tmp = Float64(Float64(Float64(1.0 / Float64(x_m * z)) / hypot(1.0, z)) / y_m);
	end
	return Float64(y_s * Float64(x_s * tmp))
end

MATLAB

x_m = abs(x);
x_s = sign(x) * abs(1.0);
y_m = abs(y);
y_s = sign(y) * abs(1.0);
function tmp_2 = code(y_s, x_s, x_m, y_m, z)
	tmp = 0.0;
	if ((y_m * (1.0 + (z * z))) <= 1e+305)
		tmp = (1.0 / (x_m * (1.0 + (z ^ 2.0)))) / y_m;
	else
		tmp = ((1.0 / (x_m * z)) / hypot(1.0, z)) / y_m;
	end
	tmp_2 = y_s * (x_s * tmp);
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
x_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y_m = N[Abs[y], $MachinePrecision]
y_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x$95$s_, x$95$m_, y$95$m_, z_] := N[(y$95$s * N[(x$95$s * If[LessEqual[N[(y$95$m * N[(1.0 + N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1e+305], N[(N[(1.0 / N[(x$95$m * N[(1.0 + N[Power[z, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$95$m), $MachinePrecision], N[(N[(N[(1.0 / N[(x$95$m * z), $MachinePrecision]), $MachinePrecision] / N[Sqrt[1.0 ^ 2 + z ^ 2], $MachinePrecision]), $MachinePrecision] / y$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y (+.f64 1 (*.f64 z z))) < 9.9999999999999994e304

   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/ 93.4%

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

      2. metadata-eval 93.4%

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

      3. associate-*r/ 93.4%

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

      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* 92.9%

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

      7. associate-/r/ 93.4%

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

      8. /-rgt-identity 93.4%

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

      9. associate-*l* 92.5%

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

      10. *-commutative 92.5%

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

      11. sqr-neg 92.5%

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

      12. +-commutative 92.5%

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

      13. sqr-neg 92.5%

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

      14. fma-def 92.5%

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

   3. Simplified 92.5%

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

      1. fma-udef 92.5%

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

      2. +-commutative 92.5%

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

      3. *-commutative 92.5%

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

      4. associate-*l* 93.4%

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

      5. associate-/l/ 93.6%

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

      6. add-sqr-sqrt 55.9%

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

      7. sqrt-div 15.3%

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

      8. inv-pow 15.3%

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

      9. sqrt-pow1 15.3%

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

      10. metadata-eval 15.3%

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

      11. *-commutative 15.3%

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

      12. sqrt-prod 15.3%

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

      13. hypot-1-def 15.3%

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

      14. sqrt-div 15.3%

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

      15. inv-pow 15.3%

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

      16. sqrt-pow1 15.2%

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

      17. metadata-eval 15.2%

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

      18. *-commutative 15.2%

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

   6. Applied egg-rr 15.2%

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

      1. unpow2 15.2%

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

   8. Simplified 15.2%

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

      1. unpow2 15.2%

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

      2. associate-/r* 15.2%

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

      3. associate-/r* 15.2%

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

      4. frac-times 14.3%

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

      5. add-sqr-sqrt 46.9%

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

   10. Applied egg-rr 46.9%

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

       1. associate-*r/ 46.9%

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

       2. associate-*l/ 46.9%

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

       3. pow-sqr 95.5%

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

       4. metadata-eval 95.5%

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

       5. unpow-1 95.5%

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

   12. Simplified 95.5%

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

   13. Taylor expanded in x around 0 92.7%

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

   if 9.9999999999999994e304 < (*.f64 y (+.f64 1 (*.f64 z z)))

   1. Initial program 70.3%

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

      1. associate-/l/ 70.3%

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

      2. metadata-eval 70.3%

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

      3. associate-*r/ 70.3%

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

      4. associate-/l/ 70.3%

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

      5. associate-*r/ 70.3%

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

      6. associate-/l* 70.3%

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

      7. associate-/r/ 70.3%

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

      8. /-rgt-identity 70.3%

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

      9. associate-*l* 79.2%

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

      10. *-commutative 79.2%

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

      11. sqr-neg 79.2%

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

      12. +-commutative 79.2%

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

      13. sqr-neg 79.2%

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

      14. fma-def 79.2%

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

   3. Simplified 79.2%

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

      1. fma-udef 79.2%

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

      2. +-commutative 79.2%

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

      3. *-commutative 79.2%

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

      4. associate-*l* 70.3%

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

      5. associate-/l/ 70.3%

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

      6. add-sqr-sqrt 70.3%

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

      7. sqrt-div 40.3%

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

      8. inv-pow 40.3%

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

      9. sqrt-pow1 40.3%

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

      10. metadata-eval 40.3%

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

      11. *-commutative 40.3%

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

      12. sqrt-prod 40.3%

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

      13. hypot-1-def 40.3%

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

      14. sqrt-div 40.3%

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

      15. inv-pow 40.3%

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

      16. sqrt-pow1 40.3%

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

      17. metadata-eval 40.3%

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

      18. *-commutative 40.3%

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

   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. Step-by-step derivation

      1. unpow2 58.3%

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

      2. associate-/r* 58.3%

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

      3. associate-/r* 58.3%

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

      4. frac-times 53.7%

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

      5. add-sqr-sqrt 53.7%

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

   10. Applied egg-rr 53.7%

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

       1. associate-*r/ 53.7%

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

       2. associate-*l/ 53.7%

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

       3. pow-sqr 89.3%

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

       4. metadata-eval 89.3%

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

       5. unpow-1 89.3%

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

   12. Simplified 89.3%

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

   13. Taylor expanded in z around inf 79.1%

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

4. Final simplification 90.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(1 + z \cdot z\right) \leq 10^{+305}:\\ \;\;\;\;\frac{\frac{1}{x \cdot \left(1 + {z}^{2}\right)}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{1}{x \cdot z}}{\mathsf{hypot}\left(1, z\right)}}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 85.6% accurate, 0.1× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{+29}:\\ \;\;\;\;\frac{1}{x\_m \cdot y\_m} \cdot \frac{1}{\mathsf{fma}\left(z, z, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{1}{x\_m \cdot z}}{\mathsf{hypot}\left(1, z\right)}}{y\_m}\\ \end{array}\right) \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
y_m = (fabs.f64 y)
y_s = (copysign.f64 1 y)
(FPCore (y_s x_s x_m y_m z)
 :precision binary64
 (*
  y_s
  (*
   x_s
   (if (<= (* z z) 2e+29)
     (* (/ 1.0 (* x_m y_m)) (/ 1.0 (fma z z 1.0)))
     (/ (/ (/ 1.0 (* x_m z)) (hypot 1.0 z)) y_m)))))

C

x_m = fabs(x);
x_s = copysign(1.0, x);
y_m = fabs(y);
y_s = copysign(1.0, y);
double code(double y_s, double x_s, double x_m, double y_m, double z) {
	double tmp;
	if ((z * z) <= 2e+29) {
		tmp = (1.0 / (x_m * y_m)) * (1.0 / fma(z, z, 1.0));
	} else {
		tmp = ((1.0 / (x_m * z)) / hypot(1.0, z)) / y_m;
	}
	return y_s * (x_s * tmp);
}

Julia

x_m = abs(x)
x_s = copysign(1.0, x)
y_m = abs(y)
y_s = copysign(1.0, y)
function code(y_s, x_s, x_m, y_m, z)
	tmp = 0.0
	if (Float64(z * z) <= 2e+29)
		tmp = Float64(Float64(1.0 / Float64(x_m * y_m)) * Float64(1.0 / fma(z, z, 1.0)));
	else
		tmp = Float64(Float64(Float64(1.0 / Float64(x_m * z)) / hypot(1.0, z)) / y_m);
	end
	return Float64(y_s * Float64(x_s * tmp))
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
x_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y_m = N[Abs[y], $MachinePrecision]
y_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x$95$s_, x$95$m_, y$95$m_, z_] := N[(y$95$s * N[(x$95$s * If[LessEqual[N[(z * z), $MachinePrecision], 2e+29], N[(N[(1.0 / N[(x$95$m * y$95$m), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(z * z + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 / N[(x$95$m * z), $MachinePrecision]), $MachinePrecision] / N[Sqrt[1.0 ^ 2 + z ^ 2], $MachinePrecision]), $MachinePrecision] / y$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 1.99999999999999983e29

   1. Initial program 99.0%

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

      1. associate-/l/ 98.7%

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

      2. metadata-eval 98.7%

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

      3. associate-*r/ 98.7%

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

      4. associate-/l/ 99.0%

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

      5. associate-*r/ 99.0%

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

      6. associate-/l* 97.9%

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

      7. associate-/r/ 98.7%

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

      8. /-rgt-identity 98.7%

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

      9. associate-*l* 98.7%

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

      10. *-commutative 98.7%

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

      11. sqr-neg 98.7%

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

      12. +-commutative 98.7%

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

      13. sqr-neg 98.7%

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

      14. fma-def 98.7%

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

   3. Simplified 98.7%

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

      1. fma-udef 98.7%

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

      2. +-commutative 98.7%

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

      3. *-commutative 98.7%

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

      4. associate-*l* 98.7%

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

      5. associate-/l/ 99.0%

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

      6. *-commutative 99.0%

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

      7. *-un-lft-identity 99.0%

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

      8. times-frac 99.0%

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

      9. +-commutative 99.0%

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

      10. fma-udef 99.0%

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

      11. associate-/l/ 99.2%

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

      12. *-commutative 99.2%

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

   6. Applied egg-rr 99.2%

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

   if 1.99999999999999983e29 < (*.f64 z z)

   1. Initial program 79.2%

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

      1. associate-/l/ 79.2%

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

      2. metadata-eval 79.2%

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

      3. associate-*r/ 79.2%

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

      4. associate-/l/ 79.2%

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

      5. associate-*r/ 79.2%

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

      6. associate-/l* 79.1%

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

      7. associate-/r/ 79.2%

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

      8. /-rgt-identity 79.2%

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

      9. associate-*l* 80.6%

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

      10. *-commutative 80.6%

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

      11. sqr-neg 80.6%

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

      12. +-commutative 80.6%

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

      13. sqr-neg 80.6%

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

      14. fma-def 80.6%

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

   3. Simplified 80.6%

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

      1. fma-udef 80.6%

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

      2. +-commutative 80.6%

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

      3. *-commutative 80.6%

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

      4. associate-*l* 79.2%

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

      5. associate-/l/ 79.2%

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

      6. add-sqr-sqrt 64.7%

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

      7. sqrt-div 22.5%

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

      8. inv-pow 22.5%

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

      9. sqrt-pow1 22.4%

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

      10. metadata-eval 22.4%

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

      11. *-commutative 22.4%

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

      12. sqrt-prod 22.4%

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

      13. hypot-1-def 22.4%

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

      14. sqrt-div 22.4%

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

      15. inv-pow 22.4%

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

      16. sqrt-pow1 22.4%

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

      17. metadata-eval 22.4%

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

      18. *-commutative 22.4%

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

   6. Applied egg-rr 28.7%

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

      1. unpow2 28.7%

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

   8. Simplified 28.7%

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

      1. unpow2 28.7%

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

      2. associate-/r* 28.7%

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

      3. associate-/r* 28.7%

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

      4. frac-times 25.5%

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

      5. add-sqr-sqrt 47.7%

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

   10. Applied egg-rr 47.7%

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

       1. associate-*r/ 47.7%

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

       2. associate-*l/ 47.6%

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

       3. pow-sqr 89.3%

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

       4. metadata-eval 89.3%

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

       5. unpow-1 89.3%

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

   12. Simplified 89.3%

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

   13. Taylor expanded in z around inf 70.5%

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

4. Final simplification 86.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{+29}:\\ \;\;\;\;\frac{1}{x \cdot y} \cdot \frac{1}{\mathsf{fma}\left(z, z, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{1}{x \cdot z}}{\mathsf{hypot}\left(1, z\right)}}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 95.1% accurate, 0.1× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;z \cdot z \leq 5 \cdot 10^{+305}:\\ \;\;\;\;\frac{\frac{1}{x\_m \cdot \left(1 + {z}^{2}\right)}}{y\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{z} \cdot \frac{\frac{1}{y\_m}}{z}}{x\_m}\\ \end{array}\right) \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
y_m = (fabs.f64 y)
y_s = (copysign.f64 1 y)
(FPCore (y_s x_s x_m y_m z)
 :precision binary64
 (*
  y_s
  (*
   x_s
   (if (<= (* z z) 5e+305)
     (/ (/ 1.0 (* x_m (+ 1.0 (pow z 2.0)))) y_m)
     (/ (* (/ 1.0 z) (/ (/ 1.0 y_m) z)) x_m)))))

C

x_m = fabs(x);
x_s = copysign(1.0, x);
y_m = fabs(y);
y_s = copysign(1.0, y);
double code(double y_s, double x_s, double x_m, double y_m, double z) {
	double tmp;
	if ((z * z) <= 5e+305) {
		tmp = (1.0 / (x_m * (1.0 + pow(z, 2.0)))) / y_m;
	} else {
		tmp = ((1.0 / z) * ((1.0 / y_m) / z)) / x_m;
	}
	return y_s * (x_s * tmp);
}

Fortran

x_m = abs(x)
x_s = copysign(1.0d0, x)
y_m = abs(y)
y_s = copysign(1.0d0, y)
real(8) function code(y_s, x_s, x_m, y_m, z)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((z * z) <= 5d+305) then
        tmp = (1.0d0 / (x_m * (1.0d0 + (z ** 2.0d0)))) / y_m
    else
        tmp = ((1.0d0 / z) * ((1.0d0 / y_m) / z)) / x_m
    end if
    code = y_s * (x_s * tmp)
end function

Java

x_m = Math.abs(x);
x_s = Math.copySign(1.0, x);
y_m = Math.abs(y);
y_s = Math.copySign(1.0, y);
public static double code(double y_s, double x_s, double x_m, double y_m, double z) {
	double tmp;
	if ((z * z) <= 5e+305) {
		tmp = (1.0 / (x_m * (1.0 + Math.pow(z, 2.0)))) / y_m;
	} else {
		tmp = ((1.0 / z) * ((1.0 / y_m) / z)) / x_m;
	}
	return y_s * (x_s * tmp);
}

Python

x_m = math.fabs(x)
x_s = math.copysign(1.0, x)
y_m = math.fabs(y)
y_s = math.copysign(1.0, y)
def code(y_s, x_s, x_m, y_m, z):
	tmp = 0
	if (z * z) <= 5e+305:
		tmp = (1.0 / (x_m * (1.0 + math.pow(z, 2.0)))) / y_m
	else:
		tmp = ((1.0 / z) * ((1.0 / y_m) / z)) / x_m
	return y_s * (x_s * tmp)

Julia

x_m = abs(x)
x_s = copysign(1.0, x)
y_m = abs(y)
y_s = copysign(1.0, y)
function code(y_s, x_s, x_m, y_m, z)
	tmp = 0.0
	if (Float64(z * z) <= 5e+305)
		tmp = Float64(Float64(1.0 / Float64(x_m * Float64(1.0 + (z ^ 2.0)))) / y_m);
	else
		tmp = Float64(Float64(Float64(1.0 / z) * Float64(Float64(1.0 / y_m) / z)) / x_m);
	end
	return Float64(y_s * Float64(x_s * tmp))
end

MATLAB

x_m = abs(x);
x_s = sign(x) * abs(1.0);
y_m = abs(y);
y_s = sign(y) * abs(1.0);
function tmp_2 = code(y_s, x_s, x_m, y_m, z)
	tmp = 0.0;
	if ((z * z) <= 5e+305)
		tmp = (1.0 / (x_m * (1.0 + (z ^ 2.0)))) / y_m;
	else
		tmp = ((1.0 / z) * ((1.0 / y_m) / z)) / x_m;
	end
	tmp_2 = y_s * (x_s * tmp);
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
x_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y_m = N[Abs[y], $MachinePrecision]
y_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x$95$s_, x$95$m_, y$95$m_, z_] := N[(y$95$s * N[(x$95$s * If[LessEqual[N[(z * z), $MachinePrecision], 5e+305], N[(N[(1.0 / N[(x$95$m * N[(1.0 + N[Power[z, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$95$m), $MachinePrecision], N[(N[(N[(1.0 / z), $MachinePrecision] * N[(N[(1.0 / y$95$m), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision] / x$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 5.00000000000000009e305

   1. Initial program 95.8%

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

      1. associate-/l/ 95.6%

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

      2. metadata-eval 95.6%

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

      3. associate-*r/ 95.6%

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

      4. associate-/l/ 95.8%

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

      5. associate-*r/ 95.8%

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

      6. associate-/l* 95.0%

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

      7. associate-/r/ 95.6%

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

      8. /-rgt-identity 95.6%

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

      9. associate-*l* 96.5%

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

      10. *-commutative 96.5%

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

      11. sqr-neg 96.5%

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

      12. +-commutative 96.5%

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

      13. sqr-neg 96.5%

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

      14. fma-def 96.5%

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

   3. Simplified 96.5%

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

      1. fma-udef 96.5%

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

      2. +-commutative 96.5%

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

      3. *-commutative 96.5%

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

      4. associate-*l* 95.6%

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

      5. associate-/l/ 95.8%

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

      6. add-sqr-sqrt 53.6%

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

      7. sqrt-div 19.8%

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

      8. inv-pow 19.8%

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

      9. sqrt-pow1 19.8%

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

      10. metadata-eval 19.8%

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

      11. *-commutative 19.8%

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

      12. sqrt-prod 19.8%

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

      13. hypot-1-def 19.8%

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

      14. sqrt-div 19.8%

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

      15. inv-pow 19.8%

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

      16. sqrt-pow1 19.8%

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

      17. metadata-eval 19.8%

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

      18. *-commutative 19.8%

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

   6. Applied egg-rr 20.7%

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

      1. unpow2 20.7%

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

   8. Simplified 20.7%

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

      1. unpow2 20.7%

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

      2. associate-/r* 20.7%

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

      3. associate-/r* 20.7%

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

      4. frac-times 19.7%

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

      5. add-sqr-sqrt 47.7%

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

   10. Applied egg-rr 47.7%

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

       1. associate-*r/ 47.7%

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

       2. associate-*l/ 47.7%

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

       3. pow-sqr 96.9%

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

       4. metadata-eval 96.9%

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

       5. unpow-1 96.9%

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

   12. Simplified 96.9%

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

   13. Taylor expanded in x around 0 96.7%

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

   if 5.00000000000000009e305 < (*.f64 z z)

   1. Initial program 72.0%

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

      1. distribute-rgt-in 72.0%

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

      2. *-un-lft-identity 72.0%

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

      3. flip-+ 1.6%

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

      4. pow2 1.6%

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

      5. *-commutative 1.6%

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

      6. pow2 1.6%

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

      7. *-commutative 1.6%

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

      8. pow2 1.6%

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

      9. *-commutative 1.6%

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

      10. pow2 1.6%

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

   4. Applied egg-rr 1.6%

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

      1. unpow2 1.6%

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

      2. associate-*l* 1.6%

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

      3. distribute-lft-out-- 1.6%

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

      4. *-commutative 1.6%

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

      5. associate-*r* 1.6%

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

      6. pow-sqr 1.6%

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

      7. metadata-eval 1.6%

         \[\leadsto \frac{\frac{1}{x}}{\frac{y \cdot \left(y - y \cdot {z}^{\color{blue}{4}}\right)}{y - y \cdot {z}^{2}}} \]

      8. *-commutative 1.6%

         \[\leadsto \frac{\frac{1}{x}}{\frac{y \cdot \left(y - y \cdot {z}^{4}\right)}{y - \color{blue}{{z}^{2} \cdot y}}} \]

   6. Simplified 1.6%

      \[\leadsto \frac{\frac{1}{x}}{\color{blue}{\frac{y \cdot \left(y - y \cdot {z}^{4}\right)}{y - {z}^{2} \cdot y}}} \]
   7. Step-by-step derivation

      1. div-inv 1.6%

         \[\leadsto \color{blue}{\frac{1}{x} \cdot \frac{1}{\frac{y \cdot \left(y - y \cdot {z}^{4}\right)}{y - {z}^{2} \cdot y}}} \]

      2. *-commutative 1.6%

         \[\leadsto \frac{1}{x} \cdot \frac{1}{\frac{y \cdot \left(y - y \cdot {z}^{4}\right)}{y - \color{blue}{y \cdot {z}^{2}}}} \]

   8. Applied egg-rr 1.6%

      \[\leadsto \color{blue}{\frac{1}{x} \cdot \frac{1}{\frac{y \cdot \left(y - y \cdot {z}^{4}\right)}{y - y \cdot {z}^{2}}}} \]
   9. Step-by-step derivation

      1. associate-*l/ 1.6%

         \[\leadsto \color{blue}{\frac{1 \cdot \frac{1}{\frac{y \cdot \left(y - y \cdot {z}^{4}\right)}{y - y \cdot {z}^{2}}}}{x}} \]

      2. *-lft-identity 1.6%

         \[\leadsto \frac{\color{blue}{\frac{1}{\frac{y \cdot \left(y - y \cdot {z}^{4}\right)}{y - y \cdot {z}^{2}}}}}{x} \]

      3. associate-/r/ 1.6%

         \[\leadsto \frac{\color{blue}{\frac{1}{y \cdot \left(y - y \cdot {z}^{4}\right)} \cdot \left(y - y \cdot {z}^{2}\right)}}{x} \]

      4. associate-/r* 1.6%

         \[\leadsto \frac{\color{blue}{\frac{\frac{1}{y}}{y - y \cdot {z}^{4}}} \cdot \left(y - y \cdot {z}^{2}\right)}{x} \]

      5. *-commutative 1.6%

         \[\leadsto \frac{\frac{\frac{1}{y}}{y - \color{blue}{{z}^{4} \cdot y}} \cdot \left(y - y \cdot {z}^{2}\right)}{x} \]

      6. cancel-sign-sub-inv 1.6%

         \[\leadsto \frac{\frac{\frac{1}{y}}{\color{blue}{y + \left(-{z}^{4}\right) \cdot y}} \cdot \left(y - y \cdot {z}^{2}\right)}{x} \]

      7. *-lft-identity 1.6%

         \[\leadsto \frac{\frac{\frac{1}{y}}{\color{blue}{1 \cdot y} + \left(-{z}^{4}\right) \cdot y} \cdot \left(y - y \cdot {z}^{2}\right)}{x} \]

      8. mul-1-neg 1.6%

         \[\leadsto \frac{\frac{\frac{1}{y}}{1 \cdot y + \color{blue}{\left(-1 \cdot {z}^{4}\right)} \cdot y} \cdot \left(y - y \cdot {z}^{2}\right)}{x} \]

      9. distribute-rgt-in 1.6%

         \[\leadsto \frac{\frac{\frac{1}{y}}{\color{blue}{y \cdot \left(1 + -1 \cdot {z}^{4}\right)}} \cdot \left(y - y \cdot {z}^{2}\right)}{x} \]

      10. mul-1-neg 1.6%

          \[\leadsto \frac{\frac{\frac{1}{y}}{y \cdot \left(1 + \color{blue}{\left(-{z}^{4}\right)}\right)} \cdot \left(y - y \cdot {z}^{2}\right)}{x} \]

      11. sub-neg 1.6%

          \[\leadsto \frac{\frac{\frac{1}{y}}{y \cdot \color{blue}{\left(1 - {z}^{4}\right)}} \cdot \left(y - y \cdot {z}^{2}\right)}{x} \]

   10. Simplified 1.6%

       \[\leadsto \color{blue}{\frac{\frac{\frac{1}{y}}{y \cdot \left(1 - {z}^{4}\right)} \cdot \left(y - y \cdot {z}^{2}\right)}{x}} \]

   11. Taylor expanded in z around inf 72.0%

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

       1. associate-/r* 72.0%

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

       2. *-un-lft-identity 72.0%

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

       3. unpow2 72.0%

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

       4. times-frac 85.3%

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

   13. Applied egg-rr 85.3%

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

4. Final simplification 93.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 5 \cdot 10^{+305}:\\ \;\;\;\;\frac{\frac{1}{x \cdot \left(1 + {z}^{2}\right)}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{z} \cdot \frac{\frac{1}{y}}{z}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 85.6% accurate, 0.1× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{+29}:\\ \;\;\;\;\frac{\frac{1}{x\_m \cdot y\_m}}{\mathsf{fma}\left(z, z, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{1}{x\_m \cdot z}}{\mathsf{hypot}\left(1, z\right)}}{y\_m}\\ \end{array}\right) \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
y_m = (fabs.f64 y)
y_s = (copysign.f64 1 y)
(FPCore (y_s x_s x_m y_m z)
 :precision binary64
 (*
  y_s
  (*
   x_s
   (if (<= (* z z) 2e+29)
     (/ (/ 1.0 (* x_m y_m)) (fma z z 1.0))
     (/ (/ (/ 1.0 (* x_m z)) (hypot 1.0 z)) y_m)))))

C

x_m = fabs(x);
x_s = copysign(1.0, x);
y_m = fabs(y);
y_s = copysign(1.0, y);
double code(double y_s, double x_s, double x_m, double y_m, double z) {
	double tmp;
	if ((z * z) <= 2e+29) {
		tmp = (1.0 / (x_m * y_m)) / fma(z, z, 1.0);
	} else {
		tmp = ((1.0 / (x_m * z)) / hypot(1.0, z)) / y_m;
	}
	return y_s * (x_s * tmp);
}

Julia

x_m = abs(x)
x_s = copysign(1.0, x)
y_m = abs(y)
y_s = copysign(1.0, y)
function code(y_s, x_s, x_m, y_m, z)
	tmp = 0.0
	if (Float64(z * z) <= 2e+29)
		tmp = Float64(Float64(1.0 / Float64(x_m * y_m)) / fma(z, z, 1.0));
	else
		tmp = Float64(Float64(Float64(1.0 / Float64(x_m * z)) / hypot(1.0, z)) / y_m);
	end
	return Float64(y_s * Float64(x_s * tmp))
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
x_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y_m = N[Abs[y], $MachinePrecision]
y_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x$95$s_, x$95$m_, y$95$m_, z_] := N[(y$95$s * N[(x$95$s * If[LessEqual[N[(z * z), $MachinePrecision], 2e+29], N[(N[(1.0 / N[(x$95$m * y$95$m), $MachinePrecision]), $MachinePrecision] / N[(z * z + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 / N[(x$95$m * z), $MachinePrecision]), $MachinePrecision] / N[Sqrt[1.0 ^ 2 + z ^ 2], $MachinePrecision]), $MachinePrecision] / y$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 1.99999999999999983e29

   1. Initial program 99.0%

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

      1. associate-/l/ 98.7%

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

   3. Simplified 98.7%

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

      1. +-commutative 98.7%

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

      2. distribute-lft-in 98.7%

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

      3. associate-*r* 98.7%

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

      4. *-rgt-identity 98.7%

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

      5. fma-def 98.7%

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

   6. Applied egg-rr 98.7%

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

   7. Taylor expanded in y around 0 98.7%

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

      1. associate-*r* 98.7%

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

      2. +-commutative 98.7%

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

      3. unpow2 98.7%

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

      4. fma-udef 98.7%

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

      5. associate-/r* 99.2%

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

      6. associate-/r* 99.0%

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

   9. Simplified 99.0%

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

       1. *-un-lft-identity 99.0%

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

       2. associate-/l/ 99.2%

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

       3. *-commutative 99.2%

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

   11. Applied egg-rr 99.2%

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

   if 1.99999999999999983e29 < (*.f64 z z)

   1. Initial program 79.2%

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

      1. associate-/l/ 79.2%

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

      2. metadata-eval 79.2%

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

      3. associate-*r/ 79.2%

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

      4. associate-/l/ 79.2%

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

      5. associate-*r/ 79.2%

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

      6. associate-/l* 79.1%

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

      7. associate-/r/ 79.2%

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

      8. /-rgt-identity 79.2%

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

      9. associate-*l* 80.6%

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

      10. *-commutative 80.6%

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

      11. sqr-neg 80.6%

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

      12. +-commutative 80.6%

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

      13. sqr-neg 80.6%

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

      14. fma-def 80.6%

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

   3. Simplified 80.6%

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

      1. fma-udef 80.6%

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

      2. +-commutative 80.6%

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

      3. *-commutative 80.6%

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

      4. associate-*l* 79.2%

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

      5. associate-/l/ 79.2%

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

      6. add-sqr-sqrt 64.7%

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

      7. sqrt-div 22.5%

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

      8. inv-pow 22.5%

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

      9. sqrt-pow1 22.4%

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

      10. metadata-eval 22.4%

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

      11. *-commutative 22.4%

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

      12. sqrt-prod 22.4%

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

      13. hypot-1-def 22.4%

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

      14. sqrt-div 22.4%

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

      15. inv-pow 22.4%

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

      16. sqrt-pow1 22.4%

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

      17. metadata-eval 22.4%

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

      18. *-commutative 22.4%

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

   6. Applied egg-rr 28.7%

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

      1. unpow2 28.7%

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

   8. Simplified 28.7%

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

      1. unpow2 28.7%

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

      2. associate-/r* 28.7%

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

      3. associate-/r* 28.7%

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

      4. frac-times 25.5%

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

      5. add-sqr-sqrt 47.7%

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

   10. Applied egg-rr 47.7%

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

       1. associate-*r/ 47.7%

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

       2. associate-*l/ 47.6%

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

       3. pow-sqr 89.3%

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

       4. metadata-eval 89.3%

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

       5. unpow-1 89.3%

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

   12. Simplified 89.3%

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

   13. Taylor expanded in z around inf 70.5%

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

4. Final simplification 86.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{+29}:\\ \;\;\;\;\frac{\frac{1}{x \cdot y}}{\mathsf{fma}\left(z, z, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{1}{x \cdot z}}{\mathsf{hypot}\left(1, z\right)}}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 94.8% accurate, 0.1× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;z \cdot z \leq 5 \cdot 10^{+305}:\\ \;\;\;\;\frac{1}{y\_m \cdot \left(x\_m \cdot \mathsf{fma}\left(z, z, 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{z} \cdot \frac{\frac{1}{y\_m}}{z}}{x\_m}\\ \end{array}\right) \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
y_m = (fabs.f64 y)
y_s = (copysign.f64 1 y)
(FPCore (y_s x_s x_m y_m z)
 :precision binary64
 (*
  y_s
  (*
   x_s
   (if (<= (* z z) 5e+305)
     (/ 1.0 (* y_m (* x_m (fma z z 1.0))))
     (/ (* (/ 1.0 z) (/ (/ 1.0 y_m) z)) x_m)))))

C

x_m = fabs(x);
x_s = copysign(1.0, x);
y_m = fabs(y);
y_s = copysign(1.0, y);
double code(double y_s, double x_s, double x_m, double y_m, double z) {
	double tmp;
	if ((z * z) <= 5e+305) {
		tmp = 1.0 / (y_m * (x_m * fma(z, z, 1.0)));
	} else {
		tmp = ((1.0 / z) * ((1.0 / y_m) / z)) / x_m;
	}
	return y_s * (x_s * tmp);
}

Julia

x_m = abs(x)
x_s = copysign(1.0, x)
y_m = abs(y)
y_s = copysign(1.0, y)
function code(y_s, x_s, x_m, y_m, z)
	tmp = 0.0
	if (Float64(z * z) <= 5e+305)
		tmp = Float64(1.0 / Float64(y_m * Float64(x_m * fma(z, z, 1.0))));
	else
		tmp = Float64(Float64(Float64(1.0 / z) * Float64(Float64(1.0 / y_m) / z)) / x_m);
	end
	return Float64(y_s * Float64(x_s * tmp))
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
x_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y_m = N[Abs[y], $MachinePrecision]
y_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x$95$s_, x$95$m_, y$95$m_, z_] := N[(y$95$s * N[(x$95$s * If[LessEqual[N[(z * z), $MachinePrecision], 5e+305], N[(1.0 / N[(y$95$m * N[(x$95$m * N[(z * z + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 / z), $MachinePrecision] * N[(N[(1.0 / y$95$m), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision] / x$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 5.00000000000000009e305

   1. Initial program 95.8%

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

      1. associate-/l/ 95.6%

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

      2. metadata-eval 95.6%

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

      3. associate-*r/ 95.6%

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

      4. associate-/l/ 95.8%

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

      5. associate-*r/ 95.8%

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

      6. associate-/l* 95.0%

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

      7. associate-/r/ 95.6%

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

      8. /-rgt-identity 95.6%

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

      9. associate-*l* 96.5%

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

      10. *-commutative 96.5%

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

      11. sqr-neg 96.5%

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

      12. +-commutative 96.5%

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

      13. sqr-neg 96.5%

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

      14. fma-def 96.5%

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

   3. Simplified 96.5%

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

   if 5.00000000000000009e305 < (*.f64 z z)

   1. Initial program 72.0%

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

      1. distribute-rgt-in 72.0%

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

      2. *-un-lft-identity 72.0%

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

      3. flip-+ 1.6%

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

      4. pow2 1.6%

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

      5. *-commutative 1.6%

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

      6. pow2 1.6%

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

      7. *-commutative 1.6%

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

      8. pow2 1.6%

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

      9. *-commutative 1.6%

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

      10. pow2 1.6%

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

   4. Applied egg-rr 1.6%

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

      1. unpow2 1.6%

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

      2. associate-*l* 1.6%

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

      3. distribute-lft-out-- 1.6%

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

      4. *-commutative 1.6%

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

      5. associate-*r* 1.6%

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

      6. pow-sqr 1.6%

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

      7. metadata-eval 1.6%

         \[\leadsto \frac{\frac{1}{x}}{\frac{y \cdot \left(y - y \cdot {z}^{\color{blue}{4}}\right)}{y - y \cdot {z}^{2}}} \]

      8. *-commutative 1.6%

         \[\leadsto \frac{\frac{1}{x}}{\frac{y \cdot \left(y - y \cdot {z}^{4}\right)}{y - \color{blue}{{z}^{2} \cdot y}}} \]

   6. Simplified 1.6%

      \[\leadsto \frac{\frac{1}{x}}{\color{blue}{\frac{y \cdot \left(y - y \cdot {z}^{4}\right)}{y - {z}^{2} \cdot y}}} \]
   7. Step-by-step derivation

      1. div-inv 1.6%

         \[\leadsto \color{blue}{\frac{1}{x} \cdot \frac{1}{\frac{y \cdot \left(y - y \cdot {z}^{4}\right)}{y - {z}^{2} \cdot y}}} \]

      2. *-commutative 1.6%

         \[\leadsto \frac{1}{x} \cdot \frac{1}{\frac{y \cdot \left(y - y \cdot {z}^{4}\right)}{y - \color{blue}{y \cdot {z}^{2}}}} \]

   8. Applied egg-rr 1.6%

      \[\leadsto \color{blue}{\frac{1}{x} \cdot \frac{1}{\frac{y \cdot \left(y - y \cdot {z}^{4}\right)}{y - y \cdot {z}^{2}}}} \]
   9. Step-by-step derivation

      1. associate-*l/ 1.6%

         \[\leadsto \color{blue}{\frac{1 \cdot \frac{1}{\frac{y \cdot \left(y - y \cdot {z}^{4}\right)}{y - y \cdot {z}^{2}}}}{x}} \]

      2. *-lft-identity 1.6%

         \[\leadsto \frac{\color{blue}{\frac{1}{\frac{y \cdot \left(y - y \cdot {z}^{4}\right)}{y - y \cdot {z}^{2}}}}}{x} \]

      3. associate-/r/ 1.6%

         \[\leadsto \frac{\color{blue}{\frac{1}{y \cdot \left(y - y \cdot {z}^{4}\right)} \cdot \left(y - y \cdot {z}^{2}\right)}}{x} \]

      4. associate-/r* 1.6%

         \[\leadsto \frac{\color{blue}{\frac{\frac{1}{y}}{y - y \cdot {z}^{4}}} \cdot \left(y - y \cdot {z}^{2}\right)}{x} \]

      5. *-commutative 1.6%

         \[\leadsto \frac{\frac{\frac{1}{y}}{y - \color{blue}{{z}^{4} \cdot y}} \cdot \left(y - y \cdot {z}^{2}\right)}{x} \]

      6. cancel-sign-sub-inv 1.6%

         \[\leadsto \frac{\frac{\frac{1}{y}}{\color{blue}{y + \left(-{z}^{4}\right) \cdot y}} \cdot \left(y - y \cdot {z}^{2}\right)}{x} \]

      7. *-lft-identity 1.6%

         \[\leadsto \frac{\frac{\frac{1}{y}}{\color{blue}{1 \cdot y} + \left(-{z}^{4}\right) \cdot y} \cdot \left(y - y \cdot {z}^{2}\right)}{x} \]

      8. mul-1-neg 1.6%

         \[\leadsto \frac{\frac{\frac{1}{y}}{1 \cdot y + \color{blue}{\left(-1 \cdot {z}^{4}\right)} \cdot y} \cdot \left(y - y \cdot {z}^{2}\right)}{x} \]

      9. distribute-rgt-in 1.6%

         \[\leadsto \frac{\frac{\frac{1}{y}}{\color{blue}{y \cdot \left(1 + -1 \cdot {z}^{4}\right)}} \cdot \left(y - y \cdot {z}^{2}\right)}{x} \]

      10. mul-1-neg 1.6%

          \[\leadsto \frac{\frac{\frac{1}{y}}{y \cdot \left(1 + \color{blue}{\left(-{z}^{4}\right)}\right)} \cdot \left(y - y \cdot {z}^{2}\right)}{x} \]

      11. sub-neg 1.6%

          \[\leadsto \frac{\frac{\frac{1}{y}}{y \cdot \color{blue}{\left(1 - {z}^{4}\right)}} \cdot \left(y - y \cdot {z}^{2}\right)}{x} \]

   10. Simplified 1.6%

       \[\leadsto \color{blue}{\frac{\frac{\frac{1}{y}}{y \cdot \left(1 - {z}^{4}\right)} \cdot \left(y - y \cdot {z}^{2}\right)}{x}} \]

   11. Taylor expanded in z around inf 72.0%

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

       1. associate-/r* 72.0%

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

       2. *-un-lft-identity 72.0%

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

       3. unpow2 72.0%

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

       4. times-frac 85.3%

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

   13. Applied egg-rr 85.3%

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

4. Final simplification 93.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 5 \cdot 10^{+305}:\\ \;\;\;\;\frac{1}{y \cdot \left(x \cdot \mathsf{fma}\left(z, z, 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{z} \cdot \frac{\frac{1}{y}}{z}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 90.6% accurate, 1.0× speedup

Math

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

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
y_m = (fabs.f64 y)
y_s = (copysign.f64 1 y)
(FPCore (y_s x_s x_m y_m z)
 :precision binary64
 (* y_s (* x_s (/ 1.0 (* x_m (* y_m (+ 1.0 (* z z))))))))

C

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

Fortran

x_m = abs(x)
x_s = copysign(1.0d0, x)
y_m = abs(y)
y_s = copysign(1.0d0, y)
real(8) function code(y_s, x_s, x_m, y_m, z)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    code = y_s * (x_s * (1.0d0 / (x_m * (y_m * (1.0d0 + (z * z))))))
end function

Java

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

Python

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

Julia

x_m = abs(x)
x_s = copysign(1.0, x)
y_m = abs(y)
y_s = copysign(1.0, y)
function code(y_s, x_s, x_m, y_m, z)
	return Float64(y_s * Float64(x_s * Float64(1.0 / Float64(x_m * Float64(y_m * Float64(1.0 + Float64(z * z)))))))
end

MATLAB

x_m = abs(x);
x_s = sign(x) * abs(1.0);
y_m = abs(y);
y_s = sign(y) * abs(1.0);
function tmp = code(y_s, x_s, x_m, y_m, z)
	tmp = y_s * (x_s * (1.0 / (x_m * (y_m * (1.0 + (z * z))))));
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
x_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y_m = N[Abs[y], $MachinePrecision]
y_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x$95$s_, x$95$m_, y$95$m_, z_] := N[(y$95$s * N[(x$95$s * N[(1.0 / N[(x$95$m * N[(y$95$m * N[(1.0 + N[(z * z), $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 10: 94.6% accurate, 1.0× speedup

Math

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

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
y_m = (fabs.f64 y)
y_s = (copysign.f64 1 y)
(FPCore (y_s x_s x_m y_m z)
 :precision binary64
 (* y_s (* x_s (/ 1.0 (* x_m (+ y_m (* z (* z y_m))))))))

C

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

Fortran

x_m = abs(x)
x_s = copysign(1.0d0, x)
y_m = abs(y)
y_s = copysign(1.0d0, y)
real(8) function code(y_s, x_s, x_m, y_m, z)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    code = y_s * (x_s * (1.0d0 / (x_m * (y_m + (z * (z * y_m))))))
end function

Java

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

Python

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

Julia

x_m = abs(x)
x_s = copysign(1.0, x)
y_m = abs(y)
y_s = copysign(1.0, y)
function code(y_s, x_s, x_m, y_m, z)
	return Float64(y_s * Float64(x_s * Float64(1.0 / Float64(x_m * Float64(y_m + Float64(z * Float64(z * y_m)))))))
end

MATLAB

x_m = abs(x);
x_s = sign(x) * abs(1.0);
y_m = abs(y);
y_s = sign(y) * abs(1.0);
function tmp = code(y_s, x_s, x_m, y_m, z)
	tmp = y_s * (x_s * (1.0 / (x_m * (y_m + (z * (z * y_m))))));
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
x_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y_m = N[Abs[y], $MachinePrecision]
y_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x$95$s_, x$95$m_, y$95$m_, z_] := N[(y$95$s * N[(x$95$s * N[(1.0 / N[(x$95$m * N[(y$95$m + N[(z * N[(z * y$95$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. Step-by-step derivation

   1. +-commutative 89.7%

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

   2. distribute-lft-in 89.7%

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

   3. associate-*r* 93.0%

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

   4. *-rgt-identity 93.0%

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

   5. fma-def 93.0%

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

6. Applied egg-rr 93.0%

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

   1. fma-udef 93.0%

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

   2. *-commutative 93.0%

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

8. Applied egg-rr 93.0%

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

9. Final simplification 93.0%

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

Alternative 11: 58.6% accurate, 2.2× speedup

Math

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

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
y_m = (fabs.f64 y)
y_s = (copysign.f64 1 y)
(FPCore (y_s x_s x_m y_m z)
 :precision binary64
 (* y_s (* x_s (/ 1.0 (* x_m y_m)))))

C

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

Fortran

x_m = abs(x)
x_s = copysign(1.0d0, x)
y_m = abs(y)
y_s = copysign(1.0d0, y)
real(8) function code(y_s, x_s, x_m, y_m, z)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    code = y_s * (x_s * (1.0d0 / (x_m * y_m)))
end function

Java

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

Python

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

Julia

x_m = abs(x)
x_s = copysign(1.0, x)
y_m = abs(y)
y_s = copysign(1.0, y)
function code(y_s, x_s, x_m, y_m, z)
	return Float64(y_s * Float64(x_s * Float64(1.0 / Float64(x_m * y_m))))
end

MATLAB

x_m = abs(x);
x_s = sign(x) * abs(1.0);
y_m = abs(y);
y_s = sign(y) * abs(1.0);
function tmp = code(y_s, x_s, x_m, y_m, z)
	tmp = y_s * (x_s * (1.0 / (x_m * y_m)));
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
x_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y_m = N[Abs[y], $MachinePrecision]
y_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x$95$s_, x$95$m_, y$95$m_, z_] := N[(y$95$s * N[(x$95$s * N[(1.0 / N[(x$95$m * y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 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* 89.3%

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

   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* 90.4%

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

   10. *-commutative 90.4%

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

   11. sqr-neg 90.4%

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

   12. +-commutative 90.4%

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

   13. sqr-neg 90.4%

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

   14. fma-def 90.4%

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

3. Simplified 90.4%

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

5. Taylor expanded in z around 0 57.8%

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

6. Final simplification 57.8%

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

Alternative 12: 58.6% accurate, 2.2× speedup

Math

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

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
y_m = (fabs.f64 y)
y_s = (copysign.f64 1 y)
(FPCore (y_s x_s x_m y_m z)
 :precision binary64
 (* y_s (* x_s (/ (/ 1.0 y_m) x_m))))

C

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

Fortran

x_m = abs(x)
x_s = copysign(1.0d0, x)
y_m = abs(y)
y_s = copysign(1.0d0, y)
real(8) function code(y_s, x_s, x_m, y_m, z)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    code = y_s * (x_s * ((1.0d0 / y_m) / x_m))
end function

Java

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

Python

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

Julia

x_m = abs(x)
x_s = copysign(1.0, x)
y_m = abs(y)
y_s = copysign(1.0, y)
function code(y_s, x_s, x_m, y_m, z)
	return Float64(y_s * Float64(x_s * Float64(Float64(1.0 / y_m) / x_m)))
end

MATLAB

x_m = abs(x);
x_s = sign(x) * abs(1.0);
y_m = abs(y);
y_s = sign(y) * abs(1.0);
function tmp = code(y_s, x_s, x_m, y_m, z)
	tmp = y_s * (x_s * ((1.0 / y_m) / x_m));
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
x_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y_m = N[Abs[y], $MachinePrecision]
y_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x$95$s_, x$95$m_, y$95$m_, z_] := N[(y$95$s * N[(x$95$s * N[(N[(1.0 / y$95$m), $MachinePrecision] / x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 89.8%

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

   1. distribute-rgt-in 89.8%

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

   2. *-un-lft-identity 89.8%

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

   3. flip-+ 49.2%

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

   4. pow2 49.2%

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

   5. *-commutative 49.2%

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

   6. pow2 49.2%

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

   7. *-commutative 49.2%

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

   8. pow2 49.2%

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

   9. *-commutative 49.2%

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

   10. pow2 49.2%

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

4. Applied egg-rr 49.2%

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

   1. unpow2 49.2%

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

   2. associate-*l* 48.4%

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

   3. distribute-lft-out-- 50.3%

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

   4. *-commutative 50.3%

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

   5. associate-*r* 46.9%

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

   6. pow-sqr 46.9%

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

   7. metadata-eval 46.9%

      \[\leadsto \frac{\frac{1}{x}}{\frac{y \cdot \left(y - y \cdot {z}^{\color{blue}{4}}\right)}{y - y \cdot {z}^{2}}} \]

   8. *-commutative 46.9%

      \[\leadsto \frac{\frac{1}{x}}{\frac{y \cdot \left(y - y \cdot {z}^{4}\right)}{y - \color{blue}{{z}^{2} \cdot y}}} \]

6. Simplified 46.9%

   \[\leadsto \frac{\frac{1}{x}}{\color{blue}{\frac{y \cdot \left(y - y \cdot {z}^{4}\right)}{y - {z}^{2} \cdot y}}} \]
7. Step-by-step derivation

   1. div-inv 46.9%

      \[\leadsto \color{blue}{\frac{1}{x} \cdot \frac{1}{\frac{y \cdot \left(y - y \cdot {z}^{4}\right)}{y - {z}^{2} \cdot y}}} \]

   2. *-commutative 46.9%

      \[\leadsto \frac{1}{x} \cdot \frac{1}{\frac{y \cdot \left(y - y \cdot {z}^{4}\right)}{y - \color{blue}{y \cdot {z}^{2}}}} \]

8. Applied egg-rr 46.9%

   \[\leadsto \color{blue}{\frac{1}{x} \cdot \frac{1}{\frac{y \cdot \left(y - y \cdot {z}^{4}\right)}{y - y \cdot {z}^{2}}}} \]
9. Step-by-step derivation

   1. associate-*l/ 46.9%

      \[\leadsto \color{blue}{\frac{1 \cdot \frac{1}{\frac{y \cdot \left(y - y \cdot {z}^{4}\right)}{y - y \cdot {z}^{2}}}}{x}} \]

   2. *-lft-identity 46.9%

      \[\leadsto \frac{\color{blue}{\frac{1}{\frac{y \cdot \left(y - y \cdot {z}^{4}\right)}{y - y \cdot {z}^{2}}}}}{x} \]

   3. associate-/r/ 46.7%

      \[\leadsto \frac{\color{blue}{\frac{1}{y \cdot \left(y - y \cdot {z}^{4}\right)} \cdot \left(y - y \cdot {z}^{2}\right)}}{x} \]

   4. associate-/r* 46.9%

      \[\leadsto \frac{\color{blue}{\frac{\frac{1}{y}}{y - y \cdot {z}^{4}}} \cdot \left(y - y \cdot {z}^{2}\right)}{x} \]

   5. *-commutative 46.9%

      \[\leadsto \frac{\frac{\frac{1}{y}}{y - \color{blue}{{z}^{4} \cdot y}} \cdot \left(y - y \cdot {z}^{2}\right)}{x} \]

   6. cancel-sign-sub-inv 46.9%

      \[\leadsto \frac{\frac{\frac{1}{y}}{\color{blue}{y + \left(-{z}^{4}\right) \cdot y}} \cdot \left(y - y \cdot {z}^{2}\right)}{x} \]

   7. *-lft-identity 46.9%

      \[\leadsto \frac{\frac{\frac{1}{y}}{\color{blue}{1 \cdot y} + \left(-{z}^{4}\right) \cdot y} \cdot \left(y - y \cdot {z}^{2}\right)}{x} \]

   8. mul-1-neg 46.9%

      \[\leadsto \frac{\frac{\frac{1}{y}}{1 \cdot y + \color{blue}{\left(-1 \cdot {z}^{4}\right)} \cdot y} \cdot \left(y - y \cdot {z}^{2}\right)}{x} \]

   9. distribute-rgt-in 46.9%

      \[\leadsto \frac{\frac{\frac{1}{y}}{\color{blue}{y \cdot \left(1 + -1 \cdot {z}^{4}\right)}} \cdot \left(y - y \cdot {z}^{2}\right)}{x} \]

   10. mul-1-neg 46.9%

       \[\leadsto \frac{\frac{\frac{1}{y}}{y \cdot \left(1 + \color{blue}{\left(-{z}^{4}\right)}\right)} \cdot \left(y - y \cdot {z}^{2}\right)}{x} \]

   11. sub-neg 46.9%

       \[\leadsto \frac{\frac{\frac{1}{y}}{y \cdot \color{blue}{\left(1 - {z}^{4}\right)}} \cdot \left(y - y \cdot {z}^{2}\right)}{x} \]

10. Simplified 46.9%

    \[\leadsto \color{blue}{\frac{\frac{\frac{1}{y}}{y \cdot \left(1 - {z}^{4}\right)} \cdot \left(y - y \cdot {z}^{2}\right)}{x}} \]

11. Taylor expanded in z around 0 57.8%

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

12. Final simplification 57.8%

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

Developer target: 92.8% 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 2024027 
(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)))))