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

Percentage Accurate: 88.9% → 99.2%

Time: 9.3s

Alternatives: 9

Speedup: 0.6×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 88.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.2% accurate, 0.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 90.1%

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

   1. associate-/r* 90.1%

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

   2. sqr-neg 90.1%

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

   3. +-commutative 90.1%

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

   4. sqr-neg 90.1%

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

   5. fma-def 90.1%

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

3. Simplified 90.1%

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

   1. add-sqr-sqrt 67.0%

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

   2. pow2 67.0%

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

   3. fma-udef 67.0%

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

   4. +-commutative 67.0%

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

   5. associate-/r* 67.0%

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

   6. sqrt-div 27.4%

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

   7. inv-pow 27.4%

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

   8. sqrt-pow1 27.3%

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

   9. metadata-eval 27.3%

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

   10. +-commutative 27.3%

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

   11. fma-udef 27.3%

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

   12. sqrt-prod 28.1%

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

   13. fma-udef 28.1%

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

   14. +-commutative 28.1%

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

   15. hypot-1-def 30.3%

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

5. Applied egg-rr 30.3%

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

6. Final simplification 30.3%

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

Alternative 2: 97.7% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Java

x_m = Math.abs(x);
x_s = Math.copySign(1.0, x);
y_m = Math.abs(y);
y_s = Math.copySign(1.0, y);
assert x_m < y_m && y_m < z;
public static double code(double y_s, double x_s, double x_m, double y_m, double z) {
	return y_s * (x_s * (((1.0 / Math.hypot(1.0, z)) / x_m) / (y_m * Math.hypot(1.0, 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)
[x_m, y_m, z] = sort([x_m, y_m, z])
def code(y_s, x_s, x_m, y_m, z):
	return y_s * (x_s * (((1.0 / math.hypot(1.0, z)) / x_m) / (y_m * math.hypot(1.0, z))))

Julia

x_m = abs(x)
x_s = copysign(1.0, x)
y_m = abs(y)
y_s = copysign(1.0, y)
x_m, y_m, z = sort([x_m, y_m, z])
function code(y_s, x_s, x_m, y_m, z)
	return Float64(y_s * Float64(x_s * Float64(Float64(Float64(1.0 / hypot(1.0, z)) / x_m) / Float64(y_m * hypot(1.0, 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);
x_m, y_m, z = num2cell(sort([x_m, y_m, z])){:}
function tmp = code(y_s, x_s, x_m, y_m, z)
	tmp = y_s * (x_s * (((1.0 / hypot(1.0, z)) / x_m) / (y_m * hypot(1.0, 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]
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
code[y$95$s_, x$95$s_, x$95$m_, y$95$m_, z_] := N[(y$95$s * N[(x$95$s * N[(N[(N[(1.0 / N[Sqrt[1.0 ^ 2 + z ^ 2], $MachinePrecision]), $MachinePrecision] / x$95$m), $MachinePrecision] / N[(y$95$m * N[Sqrt[1.0 ^ 2 + z ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 90.1%

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

   1. associate-/r* 90.1%

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

   2. sqr-neg 90.1%

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

   3. +-commutative 90.1%

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

   4. sqr-neg 90.1%

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

   5. fma-def 90.1%

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

3. Simplified 90.1%

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

   1. add-sqr-sqrt 67.0%

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

   2. pow2 67.0%

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

   3. fma-udef 67.0%

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

   4. +-commutative 67.0%

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

   5. associate-/r* 67.0%

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

   6. sqrt-div 27.4%

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

   7. inv-pow 27.4%

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

   8. sqrt-pow1 27.3%

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

   9. metadata-eval 27.3%

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

   10. +-commutative 27.3%

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

   11. fma-udef 27.3%

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

   12. sqrt-prod 28.1%

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

   13. fma-udef 28.1%

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

   14. +-commutative 28.1%

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

   15. hypot-1-def 30.3%

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

5. Applied egg-rr 30.3%

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

   1. unpow2 30.3%

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

   2. associate-/r* 30.3%

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

   3. associate-/r* 30.3%

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

   4. frac-times 28.1%

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

   5. times-frac 28.1%

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

   6. pow-prod-up 46.9%

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

   7. metadata-eval 46.9%

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

   8. inv-pow 46.9%

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

   9. add-sqr-sqrt 90.8%

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

   10. associate-/r* 90.1%

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

   11. associate-/l/ 91.0%

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

   12. *-un-lft-identity 91.0%

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

   13. frac-times 95.8%

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

   14. div-inv 95.6%

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

7. Applied egg-rr 97.9%

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

8. Final simplification 97.9%

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

Alternative 3: 98.9% accurate, 0.1× speedup

Math

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

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
y_m = (fabs.f64 y)
y_s = (copysign.f64 1 y)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (y_s x_s x_m y_m z)
 :precision binary64
 (* y_s (* x_s (/ (/ (/ (/ 1.0 (hypot 1.0 z)) x_m) (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);
assert(x_m < y_m && y_m < z);
double code(double y_s, double x_s, double x_m, double y_m, double z) {
	return y_s * (x_s * ((((1.0 / hypot(1.0, z)) / x_m) / 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);
assert x_m < y_m && y_m < z;
public static double code(double y_s, double x_s, double x_m, double y_m, double z) {
	return y_s * (x_s * ((((1.0 / Math.hypot(1.0, z)) / x_m) / 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)
[x_m, y_m, z] = sort([x_m, y_m, z])
def code(y_s, x_s, x_m, y_m, z):
	return y_s * (x_s * ((((1.0 / math.hypot(1.0, z)) / x_m) / 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)
x_m, y_m, z = sort([x_m, y_m, z])
function code(y_s, x_s, x_m, y_m, z)
	return Float64(y_s * Float64(x_s * Float64(Float64(Float64(Float64(1.0 / hypot(1.0, z)) / x_m) / 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);
x_m, y_m, z = num2cell(sort([x_m, y_m, z])){:}
function tmp = code(y_s, x_s, x_m, y_m, z)
	tmp = y_s * (x_s * ((((1.0 / hypot(1.0, z)) / x_m) / 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]
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
code[y$95$s_, x$95$s_, x$95$m_, y$95$m_, z_] := N[(y$95$s * N[(x$95$s * N[(N[(N[(N[(1.0 / N[Sqrt[1.0 ^ 2 + z ^ 2], $MachinePrecision]), $MachinePrecision] / x$95$m), $MachinePrecision] / N[Sqrt[1.0 ^ 2 + z ^ 2], $MachinePrecision]), $MachinePrecision] / y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 90.1%

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

   1. associate-/l/ 91.0%

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

   2. metadata-eval 91.0%

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

   3. associate-/r* 91.0%

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

   4. metadata-eval 91.0%

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

   5. neg-mul-1 91.0%

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

   6. distribute-neg-frac 91.0%

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

   7. distribute-frac-neg 91.0%

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

   8. distribute-frac-neg 91.0%

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

   9. distribute-neg-frac 91.0%

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

   10. metadata-eval 91.0%

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

   11. neg-mul-1 91.0%

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

   12. associate-/r* 91.0%

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

   13. metadata-eval 91.0%

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

   14. associate-/r* 90.9%

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

   15. sqr-neg 90.9%

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

   16. +-commutative 90.9%

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

   17. sqr-neg 90.9%

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

   18. fma-def 90.9%

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

3. Simplified 90.9%

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

   1. associate-/r* 91.0%

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

   2. *-un-lft-identity 91.0%

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

   3. add-sqr-sqrt 91.0%

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

   4. times-frac 91.0%

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

   5. fma-udef 91.0%

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

   6. +-commutative 91.0%

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

   7. hypot-1-def 91.0%

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

   8. fma-udef 91.0%

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

   9. +-commutative 91.0%

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

   10. hypot-1-def 95.8%

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

5. Applied egg-rr 95.8%

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

   1. associate-*r/ 95.7%

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

   2. un-div-inv 95.8%

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

7. Applied egg-rr 95.8%

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

8. Final simplification 95.8%

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

Alternative 4: 95.4% accurate, 0.1× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ y_s \cdot \left(x_s \cdot \begin{array}{l} \mathbf{if}\;z \cdot z \leq 10^{+293}:\\ \;\;\;\;\frac{\frac{1}{x_m \cdot \mathsf{fma}\left(z, z, 1\right)}}{y_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\frac{1}{z}}{x_m}}{\mathsf{hypot}\left(1, z\right)}}{y_m}\\ \end{array}\right) \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
y_m = (fabs.f64 y)
y_s = (copysign.f64 1 y)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (y_s x_s x_m y_m z)
 :precision binary64
 (*
  y_s
  (*
   x_s
   (if (<= (* z z) 1e+293)
     (/ (/ 1.0 (* x_m (fma z z 1.0))) y_m)
     (/ (/ (/ (/ 1.0 z) x_m) (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);
assert(x_m < y_m && y_m < z);
double code(double y_s, double x_s, double x_m, double y_m, double z) {
	double tmp;
	if ((z * z) <= 1e+293) {
		tmp = (1.0 / (x_m * fma(z, z, 1.0))) / y_m;
	} else {
		tmp = (((1.0 / z) / x_m) / 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)
x_m, y_m, z = sort([x_m, y_m, z])
function code(y_s, x_s, x_m, y_m, z)
	tmp = 0.0
	if (Float64(z * z) <= 1e+293)
		tmp = Float64(Float64(1.0 / Float64(x_m * fma(z, z, 1.0))) / y_m);
	else
		tmp = Float64(Float64(Float64(Float64(1.0 / z) / x_m) / 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]
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
code[y$95$s_, x$95$s_, x$95$m_, y$95$m_, z_] := N[(y$95$s * N[(x$95$s * If[LessEqual[N[(z * z), $MachinePrecision], 1e+293], N[(N[(1.0 / N[(x$95$m * N[(z * z + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$95$m), $MachinePrecision], N[(N[(N[(N[(1.0 / z), $MachinePrecision] / x$95$m), $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) < 9.9999999999999992e292

   1. Initial program 97.1%

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

      1. associate-/l/ 97.3%

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

      2. metadata-eval 97.3%

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

      3. associate-/r* 97.3%

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

      4. metadata-eval 97.3%

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

      5. neg-mul-1 97.3%

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

      6. distribute-neg-frac 97.3%

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

      7. distribute-frac-neg 97.3%

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

      8. distribute-frac-neg 97.3%

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

      9. distribute-neg-frac 97.3%

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

      10. metadata-eval 97.3%

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

      11. neg-mul-1 97.3%

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

      12. associate-/r* 97.3%

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

      13. metadata-eval 97.3%

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

      14. associate-/r* 97.1%

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

      15. sqr-neg 97.1%

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

      16. +-commutative 97.1%

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

      17. sqr-neg 97.1%

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

      18. fma-def 97.1%

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

   3. Simplified 97.1%

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

   if 9.9999999999999992e292 < (*.f64 z z)

   1. Initial program 72.9%

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

      1. associate-/l/ 75.5%

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

      2. metadata-eval 75.5%

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

      3. associate-/r* 75.5%

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

      4. metadata-eval 75.5%

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

      5. neg-mul-1 75.5%

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

      6. distribute-neg-frac 75.5%

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

      7. distribute-frac-neg 75.5%

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

      8. distribute-frac-neg 75.5%

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

      9. distribute-neg-frac 75.5%

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

      10. metadata-eval 75.5%

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

      11. neg-mul-1 75.5%

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

      12. associate-/r* 75.5%

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

      13. metadata-eval 75.5%

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

      14. associate-/r* 75.5%

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

      15. sqr-neg 75.5%

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

      16. +-commutative 75.5%

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

      17. sqr-neg 75.5%

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

      18. fma-def 75.5%

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

   3. Simplified 75.5%

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

      1. associate-/r* 75.5%

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

      2. *-un-lft-identity 75.5%

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

      3. add-sqr-sqrt 75.5%

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

      4. times-frac 75.4%

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

      5. fma-udef 75.4%

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

      6. +-commutative 75.4%

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

      7. hypot-1-def 75.4%

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

      8. fma-udef 75.4%

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

      9. +-commutative 75.4%

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

      10. hypot-1-def 91.9%

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

   5. Applied egg-rr 91.9%

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

      1. associate-*r/ 92.0%

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

      2. un-div-inv 92.1%

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

   7. Applied egg-rr 92.1%

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

   8. Taylor expanded in z around inf 84.6%

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

4. Final simplification 93.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 10^{+293}:\\ \;\;\;\;\frac{\frac{1}{x \cdot \mathsf{fma}\left(z, z, 1\right)}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\frac{1}{z}}{x}}{\mathsf{hypot}\left(1, z\right)}}{y}\\ \end{array} \]

Alternative 5: 98.9% accurate, 0.1× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ y_s \cdot \left(x_s \cdot \begin{array}{l} \mathbf{if}\;z \cdot z \leq 5 \cdot 10^{+247}:\\ \;\;\;\;\frac{\frac{1}{x_m \cdot \mathsf{fma}\left(z, z, 1\right)}}{y_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{z} \cdot \frac{1}{x_m \cdot z}}{y_m}\\ \end{array}\right) \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
y_m = (fabs.f64 y)
y_s = (copysign.f64 1 y)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (y_s x_s x_m y_m z)
 :precision binary64
 (*
  y_s
  (*
   x_s
   (if (<= (* z z) 5e+247)
     (/ (/ 1.0 (* x_m (fma z z 1.0))) y_m)
     (/ (* (/ 1.0 z) (/ 1.0 (* x_m z))) y_m)))))

C

x_m = fabs(x);
x_s = copysign(1.0, x);
y_m = fabs(y);
y_s = copysign(1.0, y);
assert(x_m < y_m && y_m < z);
double code(double y_s, double x_s, double x_m, double y_m, double z) {
	double tmp;
	if ((z * z) <= 5e+247) {
		tmp = (1.0 / (x_m * fma(z, z, 1.0))) / y_m;
	} else {
		tmp = ((1.0 / z) * (1.0 / (x_m * z))) / y_m;
	}
	return y_s * (x_s * tmp);
}

Julia

x_m = abs(x)
x_s = copysign(1.0, x)
y_m = abs(y)
y_s = copysign(1.0, y)
x_m, y_m, z = sort([x_m, y_m, z])
function code(y_s, x_s, x_m, y_m, z)
	tmp = 0.0
	if (Float64(z * z) <= 5e+247)
		tmp = Float64(Float64(1.0 / Float64(x_m * fma(z, z, 1.0))) / y_m);
	else
		tmp = Float64(Float64(Float64(1.0 / z) * Float64(1.0 / Float64(x_m * 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]
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
code[y$95$s_, x$95$s_, x$95$m_, y$95$m_, z_] := N[(y$95$s * N[(x$95$s * If[LessEqual[N[(z * z), $MachinePrecision], 5e+247], N[(N[(1.0 / N[(x$95$m * N[(z * z + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$95$m), $MachinePrecision], N[(N[(N[(1.0 / z), $MachinePrecision] * N[(1.0 / N[(x$95$m * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 5.00000000000000023e247

   1. Initial program 98.4%

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

      1. associate-/l/ 97.7%

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

      2. metadata-eval 97.7%

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

      3. associate-/r* 97.7%

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

      4. metadata-eval 97.7%

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

      5. neg-mul-1 97.7%

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

      6. distribute-neg-frac 97.7%

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

      7. distribute-frac-neg 97.7%

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

      8. distribute-frac-neg 97.7%

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

      9. distribute-neg-frac 97.7%

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

      10. metadata-eval 97.7%

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

      11. neg-mul-1 97.7%

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

      12. associate-/r* 97.7%

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

      13. metadata-eval 97.7%

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

      14. associate-/r* 97.5%

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

      15. sqr-neg 97.5%

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

      16. +-commutative 97.5%

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

      17. sqr-neg 97.5%

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

      18. fma-def 97.5%

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

   3. Simplified 97.5%

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

   if 5.00000000000000023e247 < (*.f64 z z)

   1. Initial program 73.1%

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

      1. associate-/l/ 77.2%

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

      2. metadata-eval 77.2%

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

      3. associate-/r* 77.2%

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

      4. metadata-eval 77.2%

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

      5. neg-mul-1 77.2%

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

      6. distribute-neg-frac 77.2%

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

      7. distribute-frac-neg 77.2%

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

      8. distribute-frac-neg 77.2%

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

      9. distribute-neg-frac 77.2%

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

      10. metadata-eval 77.2%

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

      11. neg-mul-1 77.2%

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

      12. associate-/r* 77.2%

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

      13. metadata-eval 77.2%

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

      14. associate-/r* 77.2%

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

      15. sqr-neg 77.2%

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

      16. +-commutative 77.2%

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

      17. sqr-neg 77.2%

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

      18. fma-def 77.2%

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

   3. Simplified 77.2%

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

   4. Taylor expanded in z around inf 77.2%

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

      1. associate-/r* 77.2%

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

      2. *-un-lft-identity 77.2%

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

      3. unpow2 77.2%

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

      4. times-frac 91.8%

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

      5. associate-/r* 91.8%

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

      6. *-commutative 91.8%

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

   6. Applied egg-rr 91.8%

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

4. Final simplification 95.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 5 \cdot 10^{+247}:\\ \;\;\;\;\frac{\frac{1}{x \cdot \mathsf{fma}\left(z, z, 1\right)}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{z} \cdot \frac{1}{x \cdot z}}{y}\\ \end{array} \]

Alternative 6: 98.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ \begin{array}{l} t_0 := y_m \cdot \left(1 + z \cdot z\right)\\ y_s \cdot \left(x_s \cdot \begin{array}{l} \mathbf{if}\;t_0 \leq 2 \cdot 10^{+307}:\\ \;\;\;\;\frac{\frac{1}{x_m}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{z} \cdot \frac{1}{x_m \cdot z}}{y_m}\\ \end{array}\right) \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
y_m = (fabs.f64 y)
y_s = (copysign.f64 1 y)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (y_s x_s x_m y_m z)
 :precision binary64
 (let* ((t_0 (* y_m (+ 1.0 (* z z)))))
   (*
    y_s
    (*
     x_s
     (if (<= t_0 2e+307)
       (/ (/ 1.0 x_m) t_0)
       (/ (* (/ 1.0 z) (/ 1.0 (* x_m z))) y_m))))))

C

x_m = fabs(x);
x_s = copysign(1.0, x);
y_m = fabs(y);
y_s = copysign(1.0, y);
assert(x_m < y_m && y_m < z);
double code(double y_s, double x_s, double x_m, double y_m, double z) {
	double t_0 = y_m * (1.0 + (z * z));
	double tmp;
	if (t_0 <= 2e+307) {
		tmp = (1.0 / x_m) / t_0;
	} else {
		tmp = ((1.0 / z) * (1.0 / (x_m * z))) / y_m;
	}
	return y_s * (x_s * tmp);
}

Fortran

x_m = abs(x)
x_s = copysign(1.0d0, x)
y_m = abs(y)
y_s = copysign(1.0d0, y)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
real(8) function code(y_s, x_s, x_m, y_m, z)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = y_m * (1.0d0 + (z * z))
    if (t_0 <= 2d+307) then
        tmp = (1.0d0 / x_m) / t_0
    else
        tmp = ((1.0d0 / z) * (1.0d0 / (x_m * z))) / y_m
    end if
    code = y_s * (x_s * tmp)
end function

Java

x_m = Math.abs(x);
x_s = Math.copySign(1.0, x);
y_m = Math.abs(y);
y_s = Math.copySign(1.0, y);
assert x_m < y_m && y_m < z;
public static double code(double y_s, double x_s, double x_m, double y_m, double z) {
	double t_0 = y_m * (1.0 + (z * z));
	double tmp;
	if (t_0 <= 2e+307) {
		tmp = (1.0 / x_m) / t_0;
	} else {
		tmp = ((1.0 / z) * (1.0 / (x_m * z))) / y_m;
	}
	return y_s * (x_s * tmp);
}

Python

x_m = math.fabs(x)
x_s = math.copysign(1.0, x)
y_m = math.fabs(y)
y_s = math.copysign(1.0, y)
[x_m, y_m, z] = sort([x_m, y_m, z])
def code(y_s, x_s, x_m, y_m, z):
	t_0 = y_m * (1.0 + (z * z))
	tmp = 0
	if t_0 <= 2e+307:
		tmp = (1.0 / x_m) / t_0
	else:
		tmp = ((1.0 / z) * (1.0 / (x_m * z))) / y_m
	return y_s * (x_s * tmp)

Julia

x_m = abs(x)
x_s = copysign(1.0, x)
y_m = abs(y)
y_s = copysign(1.0, y)
x_m, y_m, z = sort([x_m, y_m, z])
function code(y_s, x_s, x_m, y_m, z)
	t_0 = Float64(y_m * Float64(1.0 + Float64(z * z)))
	tmp = 0.0
	if (t_0 <= 2e+307)
		tmp = Float64(Float64(1.0 / x_m) / t_0);
	else
		tmp = Float64(Float64(Float64(1.0 / z) * Float64(1.0 / Float64(x_m * z))) / y_m);
	end
	return Float64(y_s * Float64(x_s * tmp))
end

MATLAB

x_m = abs(x);
x_s = sign(x) * abs(1.0);
y_m = abs(y);
y_s = sign(y) * abs(1.0);
x_m, y_m, z = num2cell(sort([x_m, y_m, z])){:}
function tmp_2 = code(y_s, x_s, x_m, y_m, z)
	t_0 = y_m * (1.0 + (z * z));
	tmp = 0.0;
	if (t_0 <= 2e+307)
		tmp = (1.0 / x_m) / t_0;
	else
		tmp = ((1.0 / z) * (1.0 / (x_m * z))) / y_m;
	end
	tmp_2 = y_s * (x_s * tmp);
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
x_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y_m = N[Abs[y], $MachinePrecision]
y_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
code[y$95$s_, x$95$s_, x$95$m_, y$95$m_, z_] := Block[{t$95$0 = N[(y$95$m * N[(1.0 + N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(y$95$s * N[(x$95$s * If[LessEqual[t$95$0, 2e+307], N[(N[(1.0 / x$95$m), $MachinePrecision] / t$95$0), $MachinePrecision], N[(N[(N[(1.0 / z), $MachinePrecision] * N[(1.0 / N[(x$95$m * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y (+.f64 1 (*.f64 z z))) < 1.99999999999999997e307

   1. Initial program 93.1%

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

   if 1.99999999999999997e307 < (*.f64 y (+.f64 1 (*.f64 z z)))

   1. Initial program 75.0%

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

      1. associate-/l/ 79.8%

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

      2. metadata-eval 79.8%

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

      3. associate-/r* 79.8%

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

      4. metadata-eval 79.8%

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

      5. neg-mul-1 79.8%

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

      6. distribute-neg-frac 79.8%

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

      7. distribute-frac-neg 79.8%

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

      8. distribute-frac-neg 79.8%

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

      9. distribute-neg-frac 79.8%

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

      10. metadata-eval 79.8%

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

      11. neg-mul-1 79.8%

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

      12. associate-/r* 79.8%

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

      13. metadata-eval 79.8%

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

      14. associate-/r* 79.8%

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

      15. sqr-neg 79.8%

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

      16. +-commutative 79.8%

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

      17. sqr-neg 79.8%

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

      18. fma-def 79.8%

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

   3. Simplified 79.8%

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

   4. Taylor expanded in z around inf 79.8%

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

      1. associate-/r* 79.8%

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

      2. *-un-lft-identity 79.8%

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

      3. unpow2 79.8%

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

      4. times-frac 91.0%

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

      5. associate-/r* 91.1%

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

      6. *-commutative 91.1%

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

   6. Applied egg-rr 91.1%

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

4. Final simplification 92.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(1 + z \cdot z\right) \leq 2 \cdot 10^{+307}:\\ \;\;\;\;\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{z} \cdot \frac{1}{x \cdot z}}{y}\\ \end{array} \]

Alternative 7: 78.2% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if z < 1

   1. Initial program 93.6%

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

      1. associate-/r* 93.6%

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

      2. sqr-neg 93.6%

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

      3. +-commutative 93.6%

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

      4. sqr-neg 93.6%

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

      5. fma-def 93.6%

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

   3. Simplified 93.6%

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

   4. Taylor expanded in z around 0 71.1%

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

   if 1 < z

   1. Initial program 80.9%

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

      1. associate-/l/ 78.7%

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

      2. metadata-eval 78.7%

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

      3. associate-/r* 78.7%

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

      4. metadata-eval 78.7%

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

      5. neg-mul-1 78.7%

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

      6. distribute-neg-frac 78.7%

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

      7. distribute-frac-neg 78.7%

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

      8. distribute-frac-neg 78.7%

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

      9. distribute-neg-frac 78.7%

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

      10. metadata-eval 78.7%

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

      11. neg-mul-1 78.7%

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

      12. associate-/r* 78.7%

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

      13. metadata-eval 78.7%

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

      14. associate-/r* 78.2%

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

      15. sqr-neg 78.2%

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

      16. +-commutative 78.2%

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

      17. sqr-neg 78.2%

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

      18. fma-def 78.2%

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

   3. Simplified 78.2%

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

   4. Taylor expanded in z around inf 77.5%

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

      1. associate-/r* 77.9%

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

      2. *-un-lft-identity 77.9%

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

      3. unpow2 77.9%

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

      4. times-frac 90.2%

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

      5. associate-/r* 90.1%

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

      6. *-commutative 90.1%

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

   6. Applied egg-rr 90.1%

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

4. Final simplification 76.4%

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

Alternative 8: 65.5% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if z < 1

   1. Initial program 93.6%

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

      1. associate-/r* 93.6%

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

      2. sqr-neg 93.6%

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

      3. +-commutative 93.6%

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

      4. sqr-neg 93.6%

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

      5. fma-def 93.6%

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

   3. Simplified 93.6%

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

   4. Taylor expanded in z around 0 71.1%

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

   if 1 < z

   1. Initial program 80.9%

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

      1. associate-/l/ 78.7%

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

      2. metadata-eval 78.7%

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

      3. associate-/r* 78.7%

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

      4. metadata-eval 78.7%

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

      5. neg-mul-1 78.7%

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

      6. distribute-neg-frac 78.7%

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

      7. distribute-frac-neg 78.7%

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

      8. distribute-frac-neg 78.7%

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

      9. distribute-neg-frac 78.7%

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

      10. metadata-eval 78.7%

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

      11. neg-mul-1 78.7%

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

      12. associate-/r* 78.7%

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

      13. metadata-eval 78.7%

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

      14. associate-/r* 78.2%

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

      15. sqr-neg 78.2%

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

      16. +-commutative 78.2%

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

      17. sqr-neg 78.2%

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

      18. fma-def 78.2%

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

   3. Simplified 78.2%

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

      1. associate-/r* 78.7%

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

      2. *-un-lft-identity 78.7%

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

      3. add-sqr-sqrt 78.6%

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

      4. times-frac 78.7%

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

      5. fma-udef 78.7%

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

      6. +-commutative 78.7%

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

      7. hypot-1-def 78.7%

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

      8. fma-udef 78.7%

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

      9. +-commutative 78.7%

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

      10. hypot-1-def 90.9%

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

   5. Applied egg-rr 90.9%

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

   6. Taylor expanded in z around inf 90.2%

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

   7. Taylor expanded in z around 0 45.8%

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

4. Final simplification 64.1%

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

Alternative 9: 58.6% accurate, 2.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 90.1%

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

   1. associate-/r* 90.1%

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

   2. sqr-neg 90.1%

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

   3. +-commutative 90.1%

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

   4. sqr-neg 90.1%

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

   5. fma-def 90.1%

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

3. Simplified 90.1%

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

4. Taylor expanded in z around 0 57.9%

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

5. Final simplification 57.9%

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

Developer target: 92.7% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2023319 
(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)))))