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

Percentage Accurate: 90.3% → 98.8%

Time: 14.2s

Alternatives: 9

Speedup: 0.5×

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: 90.3% 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: 98.8% accurate, 0.1× speedup

Math

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

FPCore

y_m = (fabs.f64 y)
y_s = (copysign.f64 1 y)
(FPCore (y_s x y_m z)
 :precision binary64
 (*
  y_s
  (if (<= (* y_m (+ 1.0 (* z z))) 1e+301)
    (/ (/ 1.0 x) (fma (* y_m z) z y_m))
    (/ (/ 1.0 z) (* y_m (* z x))))))

C

y_m = fabs(y);
y_s = copysign(1.0, y);
double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if ((y_m * (1.0 + (z * z))) <= 1e+301) {
		tmp = (1.0 / x) / fma((y_m * z), z, y_m);
	} else {
		tmp = (1.0 / z) / (y_m * (z * x));
	}
	return y_s * tmp;
}

Julia

y_m = abs(y)
y_s = copysign(1.0, y)
function code(y_s, x, y_m, z)
	tmp = 0.0
	if (Float64(y_m * Float64(1.0 + Float64(z * z))) <= 1e+301)
		tmp = Float64(Float64(1.0 / x) / fma(Float64(y_m * z), z, y_m));
	else
		tmp = Float64(Float64(1.0 / z) / Float64(y_m * Float64(z * x)));
	end
	return Float64(y_s * tmp)
end

Wolfram

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_, y$95$m_, z_] := N[(y$95$s * If[LessEqual[N[(y$95$m * N[(1.0 + N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1e+301], N[(N[(1.0 / x), $MachinePrecision] / N[(N[(y$95$m * z), $MachinePrecision] * z + y$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / z), $MachinePrecision] / N[(y$95$m * N[(z * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y (+.f64 1 (*.f64 z z))) < 1.00000000000000005e301

   1. Initial program 95.8%

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

      1. +-commutative 95.8%

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

      2. distribute-lft-in 95.8%

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

      3. associate-*r* 97.1%

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

      4. *-rgt-identity 97.1%

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

      5. fma-def 97.1%

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

   4. Applied egg-rr 97.1%

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

   if 1.00000000000000005e301 < (*.f64 y (+.f64 1 (*.f64 z z)))

   1. Initial program 76.9%

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

      1. associate-/l/ 76.9%

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

      2. metadata-eval 76.9%

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

      3. associate-*r/ 76.9%

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

      4. associate-/l/ 76.9%

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

      5. associate-*r/ 76.9%

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

      6. associate-/l* 76.9%

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

      7. associate-/r/ 76.9%

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

      8. /-rgt-identity 76.9%

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

      9. associate-*l* 84.6%

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

      10. *-commutative 84.6%

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

      11. sqr-neg 84.6%

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

      12. +-commutative 84.6%

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

      13. sqr-neg 84.6%

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

      14. fma-def 84.6%

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

   3. Simplified 84.6%

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

   5. Taylor expanded in z around inf 76.9%

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

      1. associate-*r* 84.3%

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

      2. *-commutative 84.3%

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

   7. Simplified 84.3%

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

      1. expm1-log1p-u 84.3%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{{z}^{2} \cdot \left(x \cdot y\right)}\right)\right)} \]

      2. expm1-udef 76.8%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{1}{{z}^{2} \cdot \left(x \cdot y\right)}\right)} - 1} \]

      3. associate-/r* 76.8%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{\frac{1}{{z}^{2}}}{x \cdot y}}\right)} - 1 \]

      4. pow-flip 76.8%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{{z}^{\left(-2\right)}}}{x \cdot y}\right)} - 1 \]

      5. metadata-eval 76.8%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{{z}^{\color{blue}{-2}}}{x \cdot y}\right)} - 1 \]

   9. Applied egg-rr 76.8%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{{z}^{-2}}{x \cdot y}\right)} - 1} \]
   10. Step-by-step derivation

       1. expm1-def 86.4%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{{z}^{-2}}{x \cdot y}\right)\right)} \]

       2. expm1-log1p 86.4%

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

   11. Simplified 86.4%

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

       1. add-sqr-sqrt 86.4%

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

       2. *-un-lft-identity 86.4%

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

       3. times-frac 86.4%

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

       4. sqrt-pow1 79.2%

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

       5. metadata-eval 79.2%

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

       6. unpow-1 79.2%

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

       7. sqrt-pow1 92.4%

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

       8. metadata-eval 92.4%

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

       9. unpow-1 92.4%

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

   13. Applied egg-rr 92.4%

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

       1. /-rgt-identity 92.4%

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

       2. associate-/l/ 92.3%

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

       3. un-div-inv 92.4%

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

       4. *-commutative 92.4%

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

       5. associate-*l* 100.0%

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

   15. Applied egg-rr 100.0%

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

4. Final simplification 97.5%

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

Alternative 2: 98.7% accurate, 0.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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_, y$95$m_, z_] := N[(y$95$s * N[(N[(N[Power[y$95$m, -0.5], $MachinePrecision] / N[Sqrt[1.0 ^ 2 + z ^ 2], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Power[x, -1.0], $MachinePrecision] / N[Sqrt[1.0 ^ 2 + z ^ 2], $MachinePrecision]), $MachinePrecision] / N[Sqrt[y$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 93.2%

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

   1. associate-/l/ 92.9%

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

   2. metadata-eval 92.9%

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

   3. associate-*r/ 92.9%

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

   4. associate-/l/ 93.2%

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

   5. associate-*r/ 93.2%

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

   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/ 92.9%

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

   8. /-rgt-identity 92.9%

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

   9. associate-*l* 91.5%

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

   10. *-commutative 91.5%

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

   11. sqr-neg 91.5%

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

   12. +-commutative 91.5%

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

   13. sqr-neg 91.5%

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

   14. fma-def 91.5%

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

3. Simplified 91.5%

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

   1. fma-udef 91.5%

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

   2. +-commutative 91.5%

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

   3. *-commutative 91.5%

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

   4. associate-*l* 92.9%

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

   5. associate-/l/ 93.2%

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

   6. add-sqr-sqrt 62.7%

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

   7. sqrt-div 22.8%

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

   8. inv-pow 22.8%

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

   9. sqrt-pow1 22.7%

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

   10. metadata-eval 22.7%

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

   11. *-commutative 22.7%

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

   12. sqrt-prod 22.8%

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

   13. hypot-1-def 22.8%

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

   14. sqrt-div 22.7%

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

   15. inv-pow 22.7%

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

   16. sqrt-pow1 22.7%

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

   17. metadata-eval 22.7%

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

   18. *-commutative 22.7%

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

6. Applied egg-rr 25.2%

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

   1. unpow2 25.2%

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

8. Simplified 25.2%

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

   1. unpow2 25.2%

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

   2. div-inv 25.2%

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

   3. associate-*l* 24.1%

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

   4. *-commutative 24.1%

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

   5. associate-/r* 24.1%

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

   6. metadata-eval 24.1%

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

   7. sqrt-div 24.1%

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

   8. inv-pow 24.1%

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

   9. sqrt-pow1 24.1%

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

   10. metadata-eval 24.1%

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

10. Applied egg-rr 24.1%

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

    1. associate-*r/ 24.1%

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

    2. *-commutative 24.1%

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

    3. associate-*r/ 25.2%

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

    4. associate-*l/ 25.2%

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

    5. associate-*l* 25.2%

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

    6. *-commutative 25.2%

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

    7. associate-*l/ 25.2%

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

    8. associate-/r* 24.9%

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

    9. pow-sqr 46.7%

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

    10. metadata-eval 46.7%

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

12. Simplified 46.7%

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

13. Final simplification 46.7%

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

Alternative 3: 49.8% accurate, 0.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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_, y$95$m_, z_] := N[(y$95$s * N[Power[N[(N[(N[Power[x, -0.5], $MachinePrecision] / N[Sqrt[1.0 ^ 2 + z ^ 2], $MachinePrecision]), $MachinePrecision] / N[Sqrt[y$95$m], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 93.2%

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

   1. associate-/l/ 92.9%

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

   2. metadata-eval 92.9%

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

   3. associate-*r/ 92.9%

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

   4. associate-/l/ 93.2%

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

   5. associate-*r/ 93.2%

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

   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/ 92.9%

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

   8. /-rgt-identity 92.9%

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

   9. associate-*l* 91.5%

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

   10. *-commutative 91.5%

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

   11. sqr-neg 91.5%

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

   12. +-commutative 91.5%

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

   13. sqr-neg 91.5%

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

   14. fma-def 91.5%

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

3. Simplified 91.5%

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

   1. fma-udef 91.5%

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

   2. +-commutative 91.5%

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

   3. *-commutative 91.5%

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

   4. associate-*l* 92.9%

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

   5. associate-/l/ 93.2%

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

   6. add-sqr-sqrt 62.7%

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

   7. sqrt-div 22.8%

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

   8. inv-pow 22.8%

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

   9. sqrt-pow1 22.7%

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

   10. metadata-eval 22.7%

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

   11. *-commutative 22.7%

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

   12. sqrt-prod 22.8%

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

   13. hypot-1-def 22.8%

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

   14. sqrt-div 22.7%

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

   15. inv-pow 22.7%

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

   16. sqrt-pow1 22.7%

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

   17. metadata-eval 22.7%

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

   18. *-commutative 22.7%

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

6. Applied egg-rr 25.2%

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

   1. unpow2 25.2%

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

8. Simplified 25.2%

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

   1. expm1-log1p-u 24.8%

      \[\leadsto {\color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{{x}^{-0.5}}{\mathsf{hypot}\left(1, z\right) \cdot \sqrt{y}}\right)\right)\right)}}^{2} \]

   2. expm1-udef 14.4%

      \[\leadsto {\color{blue}{\left(e^{\mathsf{log1p}\left(\frac{{x}^{-0.5}}{\mathsf{hypot}\left(1, z\right) \cdot \sqrt{y}}\right)} - 1\right)}}^{2} \]

10. Applied egg-rr 14.4%

    \[\leadsto {\color{blue}{\left(e^{\mathsf{log1p}\left(\frac{{x}^{-0.5}}{\mathsf{hypot}\left(1, z\right) \cdot \sqrt{y}}\right)} - 1\right)}}^{2} \]
11. Step-by-step derivation

    1. expm1-def 24.8%

       \[\leadsto {\color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{{x}^{-0.5}}{\mathsf{hypot}\left(1, z\right) \cdot \sqrt{y}}\right)\right)\right)}}^{2} \]

    2. expm1-log1p 25.2%

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

    3. associate-/r* 25.2%

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

12. Simplified 25.2%

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

13. Final simplification 25.2%

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

Alternative 4: 47.4% accurate, 0.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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_, y$95$m_, z_] := N[(y$95$s * N[(N[(1.0 / y$95$m), $MachinePrecision] * N[(1.0 / N[Power[N[(N[Sqrt[1.0 ^ 2 + z ^ 2], $MachinePrecision] * N[Sqrt[x], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 93.2%

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

   1. associate-/l/ 92.9%

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

   2. metadata-eval 92.9%

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

   3. associate-*r/ 92.9%

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

   4. associate-/l/ 93.2%

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

   5. associate-*r/ 93.2%

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

   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/ 92.9%

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

   8. /-rgt-identity 92.9%

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

   9. associate-*l* 91.5%

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

   10. *-commutative 91.5%

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

   11. sqr-neg 91.5%

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

   12. +-commutative 91.5%

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

   13. sqr-neg 91.5%

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

   14. fma-def 91.5%

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

3. Simplified 91.5%

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

   1. associate-/r* 91.9%

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

   2. div-inv 91.8%

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

6. Applied egg-rr 91.8%

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

   1. add-sqr-sqrt 51.1%

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

   2. pow2 51.1%

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

   3. fma-udef 51.1%

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

   4. +-commutative 51.1%

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

   5. *-commutative 51.1%

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

   6. sqrt-prod 51.1%

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

   7. hypot-1-def 51.8%

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

8. Applied egg-rr 51.8%

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

9. Final simplification 51.8%

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

Alternative 5: 98.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ \begin{array}{l} t_0 := y\_m \cdot \left(1 + z \cdot z\right)\\ y\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq 10^{+301}:\\ \;\;\;\;\frac{1}{x \cdot t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{z}}{y\_m \cdot \left(z \cdot x\right)}\\ \end{array} \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
y_s = (copysign.f64 1 y)
(FPCore (y_s x y_m z)
 :precision binary64
 (let* ((t_0 (* y_m (+ 1.0 (* z z)))))
   (*
    y_s
    (if (<= t_0 1e+301) (/ 1.0 (* x t_0)) (/ (/ 1.0 z) (* y_m (* z x)))))))

C

y_m = fabs(y);
y_s = copysign(1.0, y);
double code(double y_s, double x, double y_m, double z) {
	double t_0 = y_m * (1.0 + (z * z));
	double tmp;
	if (t_0 <= 1e+301) {
		tmp = 1.0 / (x * t_0);
	} else {
		tmp = (1.0 / z) / (y_m * (z * x));
	}
	return y_s * tmp;
}

Fortran

y_m = abs(y)
y_s = copysign(1.0d0, y)
real(8) function code(y_s, x, y_m, z)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = y_m * (1.0d0 + (z * z))
    if (t_0 <= 1d+301) then
        tmp = 1.0d0 / (x * t_0)
    else
        tmp = (1.0d0 / z) / (y_m * (z * x))
    end if
    code = y_s * tmp
end function

Java

y_m = Math.abs(y);
y_s = Math.copySign(1.0, y);
public static double code(double y_s, double x, double y_m, double z) {
	double t_0 = y_m * (1.0 + (z * z));
	double tmp;
	if (t_0 <= 1e+301) {
		tmp = 1.0 / (x * t_0);
	} else {
		tmp = (1.0 / z) / (y_m * (z * x));
	}
	return y_s * tmp;
}

Python

y_m = math.fabs(y)
y_s = math.copysign(1.0, y)
def code(y_s, x, y_m, z):
	t_0 = y_m * (1.0 + (z * z))
	tmp = 0
	if t_0 <= 1e+301:
		tmp = 1.0 / (x * t_0)
	else:
		tmp = (1.0 / z) / (y_m * (z * x))
	return y_s * tmp

Julia

y_m = abs(y)
y_s = copysign(1.0, y)
function code(y_s, x, y_m, z)
	t_0 = Float64(y_m * Float64(1.0 + Float64(z * z)))
	tmp = 0.0
	if (t_0 <= 1e+301)
		tmp = Float64(1.0 / Float64(x * t_0));
	else
		tmp = Float64(Float64(1.0 / z) / Float64(y_m * Float64(z * x)));
	end
	return Float64(y_s * tmp)
end

MATLAB

y_m = abs(y);
y_s = sign(y) * abs(1.0);
function tmp_2 = code(y_s, x, y_m, z)
	t_0 = y_m * (1.0 + (z * z));
	tmp = 0.0;
	if (t_0 <= 1e+301)
		tmp = 1.0 / (x * t_0);
	else
		tmp = (1.0 / z) / (y_m * (z * x));
	end
	tmp_2 = y_s * tmp;
end

Wolfram

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_, 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 * If[LessEqual[t$95$0, 1e+301], N[(1.0 / N[(x * t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / z), $MachinePrecision] / N[(y$95$m * N[(z * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y (+.f64 1 (*.f64 z z))) < 1.00000000000000005e301

   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/ 95.5%

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

   3. Simplified 95.5%

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

   if 1.00000000000000005e301 < (*.f64 y (+.f64 1 (*.f64 z z)))

   1. Initial program 76.9%

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

      1. associate-/l/ 76.9%

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

      2. metadata-eval 76.9%

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

      3. associate-*r/ 76.9%

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

      4. associate-/l/ 76.9%

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

      5. associate-*r/ 76.9%

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

      6. associate-/l* 76.9%

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

      7. associate-/r/ 76.9%

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

      8. /-rgt-identity 76.9%

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

      9. associate-*l* 84.6%

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

      10. *-commutative 84.6%

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

      11. sqr-neg 84.6%

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

      12. +-commutative 84.6%

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

      13. sqr-neg 84.6%

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

      14. fma-def 84.6%

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

   3. Simplified 84.6%

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

   5. Taylor expanded in z around inf 76.9%

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

      1. associate-*r* 84.3%

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

      2. *-commutative 84.3%

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

   7. Simplified 84.3%

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

      1. expm1-log1p-u 84.3%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{{z}^{2} \cdot \left(x \cdot y\right)}\right)\right)} \]

      2. expm1-udef 76.8%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{1}{{z}^{2} \cdot \left(x \cdot y\right)}\right)} - 1} \]

      3. associate-/r* 76.8%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{\frac{1}{{z}^{2}}}{x \cdot y}}\right)} - 1 \]

      4. pow-flip 76.8%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{{z}^{\left(-2\right)}}}{x \cdot y}\right)} - 1 \]

      5. metadata-eval 76.8%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{{z}^{\color{blue}{-2}}}{x \cdot y}\right)} - 1 \]

   9. Applied egg-rr 76.8%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{{z}^{-2}}{x \cdot y}\right)} - 1} \]
   10. Step-by-step derivation

       1. expm1-def 86.4%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{{z}^{-2}}{x \cdot y}\right)\right)} \]

       2. expm1-log1p 86.4%

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

   11. Simplified 86.4%

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

       1. add-sqr-sqrt 86.4%

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

       2. *-un-lft-identity 86.4%

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

       3. times-frac 86.4%

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

       4. sqrt-pow1 79.2%

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

       5. metadata-eval 79.2%

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

       6. unpow-1 79.2%

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

       7. sqrt-pow1 92.4%

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

       8. metadata-eval 92.4%

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

       9. unpow-1 92.4%

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

   13. Applied egg-rr 92.4%

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

       1. /-rgt-identity 92.4%

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

       2. associate-/l/ 92.3%

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

       3. un-div-inv 92.4%

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

       4. *-commutative 92.4%

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

       5. associate-*l* 100.0%

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

   15. Applied egg-rr 100.0%

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

4. Final simplification 96.2%

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

Alternative 6: 98.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ \begin{array}{l} t_0 := y\_m \cdot \left(1 + z \cdot z\right)\\ y\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq 10^{+301}:\\ \;\;\;\;\frac{\frac{1}{x}}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{z}}{y\_m \cdot \left(z \cdot x\right)}\\ \end{array} \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
y_s = (copysign.f64 1 y)
(FPCore (y_s x y_m z)
 :precision binary64
 (let* ((t_0 (* y_m (+ 1.0 (* z z)))))
   (*
    y_s
    (if (<= t_0 1e+301) (/ (/ 1.0 x) t_0) (/ (/ 1.0 z) (* y_m (* z x)))))))

C

y_m = fabs(y);
y_s = copysign(1.0, y);
double code(double y_s, double x, double y_m, double z) {
	double t_0 = y_m * (1.0 + (z * z));
	double tmp;
	if (t_0 <= 1e+301) {
		tmp = (1.0 / x) / t_0;
	} else {
		tmp = (1.0 / z) / (y_m * (z * x));
	}
	return y_s * tmp;
}

Fortran

y_m = abs(y)
y_s = copysign(1.0d0, y)
real(8) function code(y_s, x, y_m, z)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = y_m * (1.0d0 + (z * z))
    if (t_0 <= 1d+301) then
        tmp = (1.0d0 / x) / t_0
    else
        tmp = (1.0d0 / z) / (y_m * (z * x))
    end if
    code = y_s * tmp
end function

Java

y_m = Math.abs(y);
y_s = Math.copySign(1.0, y);
public static double code(double y_s, double x, double y_m, double z) {
	double t_0 = y_m * (1.0 + (z * z));
	double tmp;
	if (t_0 <= 1e+301) {
		tmp = (1.0 / x) / t_0;
	} else {
		tmp = (1.0 / z) / (y_m * (z * x));
	}
	return y_s * tmp;
}

Python

y_m = math.fabs(y)
y_s = math.copysign(1.0, y)
def code(y_s, x, y_m, z):
	t_0 = y_m * (1.0 + (z * z))
	tmp = 0
	if t_0 <= 1e+301:
		tmp = (1.0 / x) / t_0
	else:
		tmp = (1.0 / z) / (y_m * (z * x))
	return y_s * tmp

Julia

y_m = abs(y)
y_s = copysign(1.0, y)
function code(y_s, x, y_m, z)
	t_0 = Float64(y_m * Float64(1.0 + Float64(z * z)))
	tmp = 0.0
	if (t_0 <= 1e+301)
		tmp = Float64(Float64(1.0 / x) / t_0);
	else
		tmp = Float64(Float64(1.0 / z) / Float64(y_m * Float64(z * x)));
	end
	return Float64(y_s * tmp)
end

MATLAB

y_m = abs(y);
y_s = sign(y) * abs(1.0);
function tmp_2 = code(y_s, x, y_m, z)
	t_0 = y_m * (1.0 + (z * z));
	tmp = 0.0;
	if (t_0 <= 1e+301)
		tmp = (1.0 / x) / t_0;
	else
		tmp = (1.0 / z) / (y_m * (z * x));
	end
	tmp_2 = y_s * tmp;
end

Wolfram

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_, 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 * If[LessEqual[t$95$0, 1e+301], N[(N[(1.0 / x), $MachinePrecision] / t$95$0), $MachinePrecision], N[(N[(1.0 / z), $MachinePrecision] / N[(y$95$m * N[(z * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y (+.f64 1 (*.f64 z z))) < 1.00000000000000005e301

   1. Initial program 95.8%

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

   if 1.00000000000000005e301 < (*.f64 y (+.f64 1 (*.f64 z z)))

   1. Initial program 76.9%

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

      1. associate-/l/ 76.9%

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

      2. metadata-eval 76.9%

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

      3. associate-*r/ 76.9%

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

      4. associate-/l/ 76.9%

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

      5. associate-*r/ 76.9%

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

      6. associate-/l* 76.9%

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

      7. associate-/r/ 76.9%

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

      8. /-rgt-identity 76.9%

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

      9. associate-*l* 84.6%

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

      10. *-commutative 84.6%

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

      11. sqr-neg 84.6%

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

      12. +-commutative 84.6%

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

      13. sqr-neg 84.6%

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

      14. fma-def 84.6%

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

   3. Simplified 84.6%

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

   5. Taylor expanded in z around inf 76.9%

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

      1. associate-*r* 84.3%

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

      2. *-commutative 84.3%

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

   7. Simplified 84.3%

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

      1. expm1-log1p-u 84.3%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{{z}^{2} \cdot \left(x \cdot y\right)}\right)\right)} \]

      2. expm1-udef 76.8%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{1}{{z}^{2} \cdot \left(x \cdot y\right)}\right)} - 1} \]

      3. associate-/r* 76.8%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{\frac{1}{{z}^{2}}}{x \cdot y}}\right)} - 1 \]

      4. pow-flip 76.8%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{{z}^{\left(-2\right)}}}{x \cdot y}\right)} - 1 \]

      5. metadata-eval 76.8%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{{z}^{\color{blue}{-2}}}{x \cdot y}\right)} - 1 \]

   9. Applied egg-rr 76.8%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{{z}^{-2}}{x \cdot y}\right)} - 1} \]
   10. Step-by-step derivation

       1. expm1-def 86.4%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{{z}^{-2}}{x \cdot y}\right)\right)} \]

       2. expm1-log1p 86.4%

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

   11. Simplified 86.4%

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

       1. add-sqr-sqrt 86.4%

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

       2. *-un-lft-identity 86.4%

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

       3. times-frac 86.4%

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

       4. sqrt-pow1 79.2%

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

       5. metadata-eval 79.2%

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

       6. unpow-1 79.2%

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

       7. sqrt-pow1 92.4%

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

       8. metadata-eval 92.4%

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

       9. unpow-1 92.4%

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

   13. Applied egg-rr 92.4%

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

       1. /-rgt-identity 92.4%

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

       2. associate-/l/ 92.3%

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

       3. un-div-inv 92.4%

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

       4. *-commutative 92.4%

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

       5. associate-*l* 100.0%

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

   15. Applied egg-rr 100.0%

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

4. Final simplification 96.4%

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

Alternative 7: 77.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq 0.019:\\ \;\;\;\;\frac{\frac{-1}{y\_m}}{-x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{z \cdot \left(y\_m \cdot \left(z \cdot x\right)\right)}\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
y_s = (copysign.f64 1 y)
(FPCore (y_s x y_m z)
 :precision binary64
 (*
  y_s
  (if (<= z 0.019) (/ (/ -1.0 y_m) (- x)) (/ 1.0 (* z (* y_m (* z x)))))))

C

y_m = fabs(y);
y_s = copysign(1.0, y);
double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if (z <= 0.019) {
		tmp = (-1.0 / y_m) / -x;
	} else {
		tmp = 1.0 / (z * (y_m * (z * x)));
	}
	return y_s * tmp;
}

Fortran

y_m = abs(y)
y_s = copysign(1.0d0, y)
real(8) function code(y_s, x, y_m, z)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if (z <= 0.019d0) then
        tmp = ((-1.0d0) / y_m) / -x
    else
        tmp = 1.0d0 / (z * (y_m * (z * x)))
    end if
    code = y_s * tmp
end function

Java

y_m = Math.abs(y);
y_s = Math.copySign(1.0, y);
public static double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if (z <= 0.019) {
		tmp = (-1.0 / y_m) / -x;
	} else {
		tmp = 1.0 / (z * (y_m * (z * x)));
	}
	return y_s * tmp;
}

Python

y_m = math.fabs(y)
y_s = math.copysign(1.0, y)
def code(y_s, x, y_m, z):
	tmp = 0
	if z <= 0.019:
		tmp = (-1.0 / y_m) / -x
	else:
		tmp = 1.0 / (z * (y_m * (z * x)))
	return y_s * tmp

Julia

y_m = abs(y)
y_s = copysign(1.0, y)
function code(y_s, x, y_m, z)
	tmp = 0.0
	if (z <= 0.019)
		tmp = Float64(Float64(-1.0 / y_m) / Float64(-x));
	else
		tmp = Float64(1.0 / Float64(z * Float64(y_m * Float64(z * x))));
	end
	return Float64(y_s * tmp)
end

MATLAB

y_m = abs(y);
y_s = sign(y) * abs(1.0);
function tmp_2 = code(y_s, x, y_m, z)
	tmp = 0.0;
	if (z <= 0.019)
		tmp = (-1.0 / y_m) / -x;
	else
		tmp = 1.0 / (z * (y_m * (z * x)));
	end
	tmp_2 = y_s * tmp;
end

Wolfram

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_, y$95$m_, z_] := N[(y$95$s * If[LessEqual[z, 0.019], N[(N[(-1.0 / y$95$m), $MachinePrecision] / (-x)), $MachinePrecision], N[(1.0 / N[(z * N[(y$95$m * N[(z * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 0.0189999999999999995

   1. Initial program 94.0%

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

      1. associate-/l/ 93.7%

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

      2. metadata-eval 93.7%

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

      3. associate-*r/ 93.7%

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

      4. associate-/l/ 94.0%

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

      5. associate-*r/ 94.0%

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

      6. associate-/l* 93.7%

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

      7. associate-/r/ 93.7%

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

      8. /-rgt-identity 93.7%

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

      9. associate-*l* 93.3%

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

      10. *-commutative 93.3%

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

      11. sqr-neg 93.3%

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

      12. +-commutative 93.3%

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

      13. sqr-neg 93.3%

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

      14. fma-def 93.3%

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

   3. Simplified 93.3%

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

      1. associate-/r* 93.8%

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

      2. div-inv 93.8%

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

   6. Applied egg-rr 93.8%

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

   7. Taylor expanded in z around 0 67.4%

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

      1. un-div-inv 67.5%

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

      2. frac-2neg 67.5%

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

      3. neg-mul-1 67.5%

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

      4. un-div-inv 67.5%

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

   9. Applied egg-rr 67.5%

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

   if 0.0189999999999999995 < z

   1. Initial program 90.4%

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

      1. associate-/l/ 90.3%

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

      2. metadata-eval 90.3%

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

      3. associate-*r/ 90.3%

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

      4. associate-/l/ 90.4%

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

      5. associate-*r/ 90.4%

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

      6. associate-/l* 90.3%

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

      7. associate-/r/ 90.3%

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

      8. /-rgt-identity 90.3%

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

      9. associate-*l* 85.7%

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

      10. *-commutative 85.7%

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

      11. sqr-neg 85.7%

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

      12. +-commutative 85.7%

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

      13. sqr-neg 85.7%

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

      14. fma-def 85.7%

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

   3. Simplified 85.7%

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

   5. Taylor expanded in z around inf 88.6%

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

      1. associate-*r* 88.3%

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

      2. *-commutative 88.3%

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

   7. Simplified 88.3%

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

      1. expm1-log1p-u 81.9%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{{z}^{2} \cdot \left(x \cdot y\right)}\right)\right)} \]

      2. expm1-udef 55.0%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{1}{{z}^{2} \cdot \left(x \cdot y\right)}\right)} - 1} \]

      3. associate-/r* 55.0%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{\frac{1}{{z}^{2}}}{x \cdot y}}\right)} - 1 \]

      4. pow-flip 55.0%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{{z}^{\left(-2\right)}}}{x \cdot y}\right)} - 1 \]

      5. metadata-eval 55.0%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{{z}^{\color{blue}{-2}}}{x \cdot y}\right)} - 1 \]

   9. Applied egg-rr 55.0%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{{z}^{-2}}{x \cdot y}\right)} - 1} \]
   10. Step-by-step derivation

       1. expm1-def 81.9%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{{z}^{-2}}{x \cdot y}\right)\right)} \]

       2. expm1-log1p 88.3%

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

   11. Simplified 88.3%

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

       1. add-sqr-sqrt 88.0%

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

       2. *-un-lft-identity 88.0%

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

       3. times-frac 88.2%

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

       4. sqrt-pow1 88.2%

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

       5. metadata-eval 88.2%

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

       6. unpow-1 88.2%

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

       7. sqrt-pow1 88.4%

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

       8. metadata-eval 88.4%

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

       9. unpow-1 88.4%

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

   13. Applied egg-rr 88.4%

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

       1. /-rgt-identity 88.4%

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

       2. *-commutative 88.4%

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

       3. clear-num 88.3%

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

       4. frac-times 88.4%

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

       5. metadata-eval 88.4%

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

       6. div-inv 88.4%

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

       7. remove-double-div 88.5%

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

       8. *-commutative 88.5%

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

       9. associate-*l* 95.4%

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

   15. Applied egg-rr 95.4%

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

4. Final simplification 74.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 0.019:\\ \;\;\;\;\frac{\frac{-1}{y}}{-x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{z \cdot \left(y \cdot \left(z \cdot x\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 77.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq 0.019:\\ \;\;\;\;\frac{\frac{-1}{y\_m}}{-x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{z}}{y\_m \cdot \left(z \cdot x\right)}\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
y_s = (copysign.f64 1 y)
(FPCore (y_s x y_m z)
 :precision binary64
 (*
  y_s
  (if (<= z 0.019) (/ (/ -1.0 y_m) (- x)) (/ (/ 1.0 z) (* y_m (* z x))))))

C

y_m = fabs(y);
y_s = copysign(1.0, y);
double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if (z <= 0.019) {
		tmp = (-1.0 / y_m) / -x;
	} else {
		tmp = (1.0 / z) / (y_m * (z * x));
	}
	return y_s * tmp;
}

Fortran

y_m = abs(y)
y_s = copysign(1.0d0, y)
real(8) function code(y_s, x, y_m, z)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if (z <= 0.019d0) then
        tmp = ((-1.0d0) / y_m) / -x
    else
        tmp = (1.0d0 / z) / (y_m * (z * x))
    end if
    code = y_s * tmp
end function

Java

y_m = Math.abs(y);
y_s = Math.copySign(1.0, y);
public static double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if (z <= 0.019) {
		tmp = (-1.0 / y_m) / -x;
	} else {
		tmp = (1.0 / z) / (y_m * (z * x));
	}
	return y_s * tmp;
}

Python

y_m = math.fabs(y)
y_s = math.copysign(1.0, y)
def code(y_s, x, y_m, z):
	tmp = 0
	if z <= 0.019:
		tmp = (-1.0 / y_m) / -x
	else:
		tmp = (1.0 / z) / (y_m * (z * x))
	return y_s * tmp

Julia

y_m = abs(y)
y_s = copysign(1.0, y)
function code(y_s, x, y_m, z)
	tmp = 0.0
	if (z <= 0.019)
		tmp = Float64(Float64(-1.0 / y_m) / Float64(-x));
	else
		tmp = Float64(Float64(1.0 / z) / Float64(y_m * Float64(z * x)));
	end
	return Float64(y_s * tmp)
end

MATLAB

y_m = abs(y);
y_s = sign(y) * abs(1.0);
function tmp_2 = code(y_s, x, y_m, z)
	tmp = 0.0;
	if (z <= 0.019)
		tmp = (-1.0 / y_m) / -x;
	else
		tmp = (1.0 / z) / (y_m * (z * x));
	end
	tmp_2 = y_s * tmp;
end

Wolfram

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_, y$95$m_, z_] := N[(y$95$s * If[LessEqual[z, 0.019], N[(N[(-1.0 / y$95$m), $MachinePrecision] / (-x)), $MachinePrecision], N[(N[(1.0 / z), $MachinePrecision] / N[(y$95$m * N[(z * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 0.0189999999999999995

   1. Initial program 94.0%

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

      1. associate-/l/ 93.7%

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

      2. metadata-eval 93.7%

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

      3. associate-*r/ 93.7%

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

      4. associate-/l/ 94.0%

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

      5. associate-*r/ 94.0%

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

      6. associate-/l* 93.7%

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

      7. associate-/r/ 93.7%

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

      8. /-rgt-identity 93.7%

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

      9. associate-*l* 93.3%

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

      10. *-commutative 93.3%

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

      11. sqr-neg 93.3%

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

      12. +-commutative 93.3%

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

      13. sqr-neg 93.3%

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

      14. fma-def 93.3%

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

   3. Simplified 93.3%

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

      1. associate-/r* 93.8%

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

      2. div-inv 93.8%

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

   6. Applied egg-rr 93.8%

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

   7. Taylor expanded in z around 0 67.4%

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

      1. un-div-inv 67.5%

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

      2. frac-2neg 67.5%

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

      3. neg-mul-1 67.5%

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

      4. un-div-inv 67.5%

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

   9. Applied egg-rr 67.5%

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

   if 0.0189999999999999995 < z

   1. Initial program 90.4%

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

      1. associate-/l/ 90.3%

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

      2. metadata-eval 90.3%

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

      3. associate-*r/ 90.3%

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

      4. associate-/l/ 90.4%

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

      5. associate-*r/ 90.4%

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

      6. associate-/l* 90.3%

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

      7. associate-/r/ 90.3%

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

      8. /-rgt-identity 90.3%

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

      9. associate-*l* 85.7%

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

      10. *-commutative 85.7%

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

      11. sqr-neg 85.7%

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

      12. +-commutative 85.7%

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

      13. sqr-neg 85.7%

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

      14. fma-def 85.7%

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

   3. Simplified 85.7%

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

   5. Taylor expanded in z around inf 88.6%

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

      1. associate-*r* 88.3%

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

      2. *-commutative 88.3%

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

   7. Simplified 88.3%

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

      1. expm1-log1p-u 81.9%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{{z}^{2} \cdot \left(x \cdot y\right)}\right)\right)} \]

      2. expm1-udef 55.0%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{1}{{z}^{2} \cdot \left(x \cdot y\right)}\right)} - 1} \]

      3. associate-/r* 55.0%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{\frac{1}{{z}^{2}}}{x \cdot y}}\right)} - 1 \]

      4. pow-flip 55.0%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{{z}^{\left(-2\right)}}}{x \cdot y}\right)} - 1 \]

      5. metadata-eval 55.0%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{{z}^{\color{blue}{-2}}}{x \cdot y}\right)} - 1 \]

   9. Applied egg-rr 55.0%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{{z}^{-2}}{x \cdot y}\right)} - 1} \]
   10. Step-by-step derivation

       1. expm1-def 81.9%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{{z}^{-2}}{x \cdot y}\right)\right)} \]

       2. expm1-log1p 88.3%

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

   11. Simplified 88.3%

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

       1. add-sqr-sqrt 88.0%

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

       2. *-un-lft-identity 88.0%

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

       3. times-frac 88.2%

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

       4. sqrt-pow1 88.2%

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

       5. metadata-eval 88.2%

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

       6. unpow-1 88.2%

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

       7. sqrt-pow1 88.4%

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

       8. metadata-eval 88.4%

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

       9. unpow-1 88.4%

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

   13. Applied egg-rr 88.4%

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

       1. /-rgt-identity 88.4%

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

       2. associate-/l/ 88.4%

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

       3. un-div-inv 88.5%

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

       4. *-commutative 88.5%

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

       5. associate-*l* 95.5%

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

   15. Applied egg-rr 95.5%

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

4. Final simplification 74.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 0.019:\\ \;\;\;\;\frac{\frac{-1}{y}}{-x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{z}}{y \cdot \left(z \cdot x\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 58.2% accurate, 2.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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_, y$95$m_, z_] := N[(y$95$s * N[(1.0 / N[(y$95$m * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 93.2%

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

   1. associate-/l/ 92.9%

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

   2. metadata-eval 92.9%

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

   3. associate-*r/ 92.9%

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

   4. associate-/l/ 93.2%

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

   5. associate-*r/ 93.2%

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

   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/ 92.9%

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

   8. /-rgt-identity 92.9%

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

   9. associate-*l* 91.5%

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

   10. *-commutative 91.5%

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

   11. sqr-neg 91.5%

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

   12. +-commutative 91.5%

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

   13. sqr-neg 91.5%

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

   14. fma-def 91.5%

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

3. Simplified 91.5%

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

5. Taylor expanded in z around 0 55.1%

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

6. Final simplification 55.1%

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

Developer target: 92.3% 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 2024026 
(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)))))