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

Percentage Accurate: 91.3% → 97.7%

Time: 9.7s

Alternatives: 10

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 91.3% 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: 97.7% accurate, 0.1× speedup

Math

\[\begin{array}{l} z = |z|\\ [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 4 \cdot 10^{+167}:\\ \;\;\;\;\frac{\frac{1}{y}}{x \cdot \mathsf{fma}\left(z, z, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x \cdot \left(z \cdot y\right)}}{\mathsf{hypot}\left(1, z\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: z should be positive before calling this function
NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= (* z z) 4e+167)
   (/ (/ 1.0 y) (* x (fma z z 1.0)))
   (/ (/ 1.0 (* x (* z y))) (hypot 1.0 z))))

C

z = abs(z);
assert(x < y);
double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 4e+167) {
		tmp = (1.0 / y) / (x * fma(z, z, 1.0));
	} else {
		tmp = (1.0 / (x * (z * y))) / hypot(1.0, z);
	}
	return tmp;
}

Julia

z = abs(z)
x, y = sort([x, y])
function code(x, y, z)
	tmp = 0.0
	if (Float64(z * z) <= 4e+167)
		tmp = Float64(Float64(1.0 / y) / Float64(x * fma(z, z, 1.0)));
	else
		tmp = Float64(Float64(1.0 / Float64(x * Float64(z * y))) / hypot(1.0, z));
	end
	return tmp
end

Wolfram

NOTE: z should be positive before calling this function
NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_, z_] := If[LessEqual[N[(z * z), $MachinePrecision], 4e+167], N[(N[(1.0 / y), $MachinePrecision] / N[(x * N[(z * z + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[(x * N[(z * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[1.0 ^ 2 + z ^ 2], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 4.0000000000000002e167

   1. Initial program 98.0%

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

      1. associate-/r* 97.7%

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

      2. associate-/l/ 98.0%

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

      3. associate-/r* 98.0%

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

      4. associate-/l/ 99.0%

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

      5. sqr-neg 99.0%

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

      6. +-commutative 99.0%

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

      7. sqr-neg 99.0%

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

      8. fma-def 99.0%

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

   3. Simplified 99.0%

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

   if 4.0000000000000002e167 < (*.f64 z z)

   1. Initial program 70.1%

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

   2. Taylor expanded in x around 0 70.1%

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

      1. +-commutative 70.1%

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

      2. unpow2 70.1%

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

      3. fma-udef 70.1%

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

   4. Simplified 70.1%

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

      1. associate-/r* 70.1%

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

      2. inv-pow 70.1%

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

      3. metadata-eval 70.1%

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

      4. pow-prod-up 31.3%

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

      5. frac-times 32.4%

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

      6. *-commutative 32.4%

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

      7. add-sqr-sqrt 32.4%

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

      8. associate-*l* 32.4%

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

      9. sqrt-div 32.4%

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

      10. sqrt-pow1 32.4%

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

      11. metadata-eval 32.4%

          \[\leadsto \frac{{x}^{\color{blue}{-0.25}}}{\sqrt{\mathsf{fma}\left(z, z, 1\right)}} \cdot \left(\sqrt{\frac{{x}^{-0.5}}{\mathsf{fma}\left(z, z, 1\right)}} \cdot \frac{{x}^{-0.5}}{y}\right) \]

      12. sqrt-div 32.4%

          \[\leadsto \frac{{x}^{-0.25}}{\sqrt{\mathsf{fma}\left(z, z, 1\right)}} \cdot \left(\color{blue}{\frac{\sqrt{{x}^{-0.5}}}{\sqrt{\mathsf{fma}\left(z, z, 1\right)}}} \cdot \frac{{x}^{-0.5}}{y}\right) \]

      13. sqrt-pow1 32.4%

          \[\leadsto \frac{{x}^{-0.25}}{\sqrt{\mathsf{fma}\left(z, z, 1\right)}} \cdot \left(\frac{\color{blue}{{x}^{\left(\frac{-0.5}{2}\right)}}}{\sqrt{\mathsf{fma}\left(z, z, 1\right)}} \cdot \frac{{x}^{-0.5}}{y}\right) \]

      14. metadata-eval 32.4%

          \[\leadsto \frac{{x}^{-0.25}}{\sqrt{\mathsf{fma}\left(z, z, 1\right)}} \cdot \left(\frac{{x}^{\color{blue}{-0.25}}}{\sqrt{\mathsf{fma}\left(z, z, 1\right)}} \cdot \frac{{x}^{-0.5}}{y}\right) \]

   6. Applied egg-rr 32.4%

      \[\leadsto \color{blue}{\frac{{x}^{-0.25}}{\sqrt{\mathsf{fma}\left(z, z, 1\right)}} \cdot \left(\frac{{x}^{-0.25}}{\sqrt{\mathsf{fma}\left(z, z, 1\right)}} \cdot \frac{{x}^{-0.5}}{y}\right)} \]
   7. Step-by-step derivation

      1. associate-*l/ 32.4%

         \[\leadsto \color{blue}{\frac{{x}^{-0.25} \cdot \left(\frac{{x}^{-0.25}}{\sqrt{\mathsf{fma}\left(z, z, 1\right)}} \cdot \frac{{x}^{-0.5}}{y}\right)}{\sqrt{\mathsf{fma}\left(z, z, 1\right)}}} \]

   8. Simplified 97.5%

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

   9. Taylor expanded in z around inf 75.9%

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

4. Final simplification 91.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 4 \cdot 10^{+167}:\\ \;\;\;\;\frac{\frac{1}{y}}{x \cdot \mathsf{fma}\left(z, z, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x \cdot \left(z \cdot y\right)}}{\mathsf{hypot}\left(1, z\right)}\\ \end{array} \]

Alternative 2: 97.7% accurate, 0.0× speedup

Math

\[\begin{array}{l} z = |z|\\ [x, y] = \mathsf{sort}([x, y])\\ \\ \frac{\frac{{x}^{-1}}{\mathsf{hypot}\left(1, z\right) \cdot y}}{\mathsf{hypot}\left(1, z\right)} \end{array} \]

FPCore

NOTE: z should be positive before calling this function
NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (/ (/ (pow x -1.0) (* (hypot 1.0 z) y)) (hypot 1.0 z)))

C

z = abs(z);
assert(x < y);
double code(double x, double y, double z) {
	return (pow(x, -1.0) / (hypot(1.0, z) * y)) / hypot(1.0, z);
}

Java

z = Math.abs(z);
assert x < y;
public static double code(double x, double y, double z) {
	return (Math.pow(x, -1.0) / (Math.hypot(1.0, z) * y)) / Math.hypot(1.0, z);
}

Python

z = abs(z)
[x, y] = sort([x, y])
def code(x, y, z):
	return (math.pow(x, -1.0) / (math.hypot(1.0, z) * y)) / math.hypot(1.0, z)

Julia

z = abs(z)
x, y = sort([x, y])
function code(x, y, z)
	return Float64(Float64((x ^ -1.0) / Float64(hypot(1.0, z) * y)) / hypot(1.0, z))
end

MATLAB

z = abs(z)
x, y = num2cell(sort([x, y])){:}
function tmp = code(x, y, z)
	tmp = ((x ^ -1.0) / (hypot(1.0, z) * y)) / hypot(1.0, z);
end

Wolfram

NOTE: z should be positive before calling this function
NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_, z_] := N[(N[(N[Power[x, -1.0], $MachinePrecision] / N[(N[Sqrt[1.0 ^ 2 + z ^ 2], $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision] / N[Sqrt[1.0 ^ 2 + z ^ 2], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 88.4%

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

2. Taylor expanded in x around 0 88.2%

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

   1. +-commutative 88.2%

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

   2. unpow2 88.2%

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

   3. fma-udef 88.2%

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

4. Simplified 88.2%

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

   1. associate-/r* 88.4%

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

   2. inv-pow 88.4%

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

   3. metadata-eval 88.4%

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

   4. pow-prod-up 42.6%

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

   5. frac-times 43.8%

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

   6. *-commutative 43.8%

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

   7. add-sqr-sqrt 43.7%

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

   8. associate-*l* 43.7%

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

   9. sqrt-div 43.7%

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

   10. sqrt-pow1 43.7%

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

   11. metadata-eval 43.7%

       \[\leadsto \frac{{x}^{\color{blue}{-0.25}}}{\sqrt{\mathsf{fma}\left(z, z, 1\right)}} \cdot \left(\sqrt{\frac{{x}^{-0.5}}{\mathsf{fma}\left(z, z, 1\right)}} \cdot \frac{{x}^{-0.5}}{y}\right) \]

   12. sqrt-div 43.6%

       \[\leadsto \frac{{x}^{-0.25}}{\sqrt{\mathsf{fma}\left(z, z, 1\right)}} \cdot \left(\color{blue}{\frac{\sqrt{{x}^{-0.5}}}{\sqrt{\mathsf{fma}\left(z, z, 1\right)}}} \cdot \frac{{x}^{-0.5}}{y}\right) \]

   13. sqrt-pow1 43.7%

       \[\leadsto \frac{{x}^{-0.25}}{\sqrt{\mathsf{fma}\left(z, z, 1\right)}} \cdot \left(\frac{\color{blue}{{x}^{\left(\frac{-0.5}{2}\right)}}}{\sqrt{\mathsf{fma}\left(z, z, 1\right)}} \cdot \frac{{x}^{-0.5}}{y}\right) \]

   14. metadata-eval 43.7%

       \[\leadsto \frac{{x}^{-0.25}}{\sqrt{\mathsf{fma}\left(z, z, 1\right)}} \cdot \left(\frac{{x}^{\color{blue}{-0.25}}}{\sqrt{\mathsf{fma}\left(z, z, 1\right)}} \cdot \frac{{x}^{-0.5}}{y}\right) \]

6. Applied egg-rr 43.7%

   \[\leadsto \color{blue}{\frac{{x}^{-0.25}}{\sqrt{\mathsf{fma}\left(z, z, 1\right)}} \cdot \left(\frac{{x}^{-0.25}}{\sqrt{\mathsf{fma}\left(z, z, 1\right)}} \cdot \frac{{x}^{-0.5}}{y}\right)} \]
7. Step-by-step derivation

   1. associate-*l/ 43.7%

      \[\leadsto \color{blue}{\frac{{x}^{-0.25} \cdot \left(\frac{{x}^{-0.25}}{\sqrt{\mathsf{fma}\left(z, z, 1\right)}} \cdot \frac{{x}^{-0.5}}{y}\right)}{\sqrt{\mathsf{fma}\left(z, z, 1\right)}}} \]

8. Simplified 97.8%

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

9. Final simplification 97.8%

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

Alternative 3: 97.8% accurate, 0.1× speedup

Math

\[\begin{array}{l} z = |z|\\ [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 4 \cdot 10^{+167}:\\ \;\;\;\;\frac{\frac{1}{y}}{x \cdot \mathsf{fma}\left(z, z, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-1}{z}}{x \cdot \left(y \cdot \left(-z\right)\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: z should be positive before calling this function
NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= (* z z) 4e+167)
   (/ (/ 1.0 y) (* x (fma z z 1.0)))
   (/ (/ -1.0 z) (* x (* y (- z))))))

C

z = abs(z);
assert(x < y);
double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 4e+167) {
		tmp = (1.0 / y) / (x * fma(z, z, 1.0));
	} else {
		tmp = (-1.0 / z) / (x * (y * -z));
	}
	return tmp;
}

Julia

z = abs(z)
x, y = sort([x, y])
function code(x, y, z)
	tmp = 0.0
	if (Float64(z * z) <= 4e+167)
		tmp = Float64(Float64(1.0 / y) / Float64(x * fma(z, z, 1.0)));
	else
		tmp = Float64(Float64(-1.0 / z) / Float64(x * Float64(y * Float64(-z))));
	end
	return tmp
end

Wolfram

NOTE: z should be positive before calling this function
NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_, z_] := If[LessEqual[N[(z * z), $MachinePrecision], 4e+167], N[(N[(1.0 / y), $MachinePrecision] / N[(x * N[(z * z + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(-1.0 / z), $MachinePrecision] / N[(x * N[(y * (-z)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 4.0000000000000002e167

   1. Initial program 98.0%

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

      1. associate-/r* 97.7%

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

      2. associate-/l/ 98.0%

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

      3. associate-/r* 98.0%

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

      4. associate-/l/ 99.0%

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

      5. sqr-neg 99.0%

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

      6. +-commutative 99.0%

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

      7. sqr-neg 99.0%

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

      8. fma-def 99.0%

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

   3. Simplified 99.0%

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

   if 4.0000000000000002e167 < (*.f64 z z)

   1. Initial program 70.1%

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

   2. Taylor expanded in z around inf 70.1%

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

      1. *-commutative 70.1%

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

      2. *-commutative 70.1%

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

      3. associate-*l* 72.9%

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

   4. Simplified 72.9%

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

      1. clear-num 72.9%

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

      2. associate-/r/ 72.9%

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

      3. associate-/r* 72.9%

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

      4. *-commutative 72.9%

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

      5. associate-/r* 70.1%

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

      6. pow-flip 73.1%

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

      7. metadata-eval 73.1%

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

   6. Applied egg-rr 73.1%

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

      1. associate-/l/ 74.9%

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

      2. add-sqr-sqrt 74.8%

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

      3. times-frac 73.2%

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

      4. sqrt-pow1 66.0%

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

      5. metadata-eval 66.0%

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

      6. unpow-1 66.0%

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

      7. sqrt-pow1 95.6%

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

      8. metadata-eval 95.6%

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

      9. unpow-1 95.6%

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

   8. Applied egg-rr 95.6%

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

      1. associate-/l/ 95.7%

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

      2. frac-2neg 95.7%

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

      3. frac-times 97.6%

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

      4. *-un-lft-identity 97.6%

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

      5. distribute-neg-frac 97.6%

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

      6. metadata-eval 97.6%

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

   10. Applied egg-rr 97.6%

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

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 4 \cdot 10^{+167}:\\ \;\;\;\;\frac{\frac{1}{y}}{x \cdot \mathsf{fma}\left(z, z, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-1}{z}}{x \cdot \left(y \cdot \left(-z\right)\right)}\\ \end{array} \]

Alternative 4: 97.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} z = |z|\\ [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{+126}:\\ \;\;\;\;\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{z}}{y \cdot \left(x \cdot z\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: z should be positive before calling this function
NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= (* z z) 2e+126)
   (/ (/ 1.0 x) (* y (+ 1.0 (* z z))))
   (/ (/ 1.0 z) (* y (* x z)))))

C

z = abs(z);
assert(x < y);
double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 2e+126) {
		tmp = (1.0 / x) / (y * (1.0 + (z * z)));
	} else {
		tmp = (1.0 / z) / (y * (x * z));
	}
	return tmp;
}

Fortran

NOTE: z should be positive before calling this function
NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((z * z) <= 2d+126) then
        tmp = (1.0d0 / x) / (y * (1.0d0 + (z * z)))
    else
        tmp = (1.0d0 / z) / (y * (x * z))
    end if
    code = tmp
end function

Java

z = Math.abs(z);
assert x < y;
public static double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 2e+126) {
		tmp = (1.0 / x) / (y * (1.0 + (z * z)));
	} else {
		tmp = (1.0 / z) / (y * (x * z));
	}
	return tmp;
}

Python

z = abs(z)
[x, y] = sort([x, y])
def code(x, y, z):
	tmp = 0
	if (z * z) <= 2e+126:
		tmp = (1.0 / x) / (y * (1.0 + (z * z)))
	else:
		tmp = (1.0 / z) / (y * (x * z))
	return tmp

Julia

z = abs(z)
x, y = sort([x, y])
function code(x, y, z)
	tmp = 0.0
	if (Float64(z * z) <= 2e+126)
		tmp = Float64(Float64(1.0 / x) / Float64(y * Float64(1.0 + Float64(z * z))));
	else
		tmp = Float64(Float64(1.0 / z) / Float64(y * Float64(x * z)));
	end
	return tmp
end

MATLAB

z = abs(z)
x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z * z) <= 2e+126)
		tmp = (1.0 / x) / (y * (1.0 + (z * z)));
	else
		tmp = (1.0 / z) / (y * (x * z));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: z should be positive before calling this function
NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_, z_] := If[LessEqual[N[(z * z), $MachinePrecision], 2e+126], N[(N[(1.0 / x), $MachinePrecision] / N[(y * N[(1.0 + N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / z), $MachinePrecision] / N[(y * N[(x * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 1.99999999999999985e126

   1. Initial program 99.1%

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

   if 1.99999999999999985e126 < (*.f64 z z)

   1. Initial program 71.2%

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

   2. Taylor expanded in z around inf 71.2%

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

      1. *-commutative 71.2%

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

      2. *-commutative 71.2%

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

      3. associate-*l* 73.7%

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

   4. Simplified 73.7%

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

      1. clear-num 73.7%

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

      2. associate-/r/ 73.7%

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

      3. associate-/r* 73.6%

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

      4. *-commutative 73.6%

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

      5. associate-/r* 72.1%

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

      6. pow-flip 74.8%

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

      7. metadata-eval 74.8%

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

   6. Applied egg-rr 74.8%

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

      1. associate-/l/ 75.5%

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

      2. add-sqr-sqrt 75.4%

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

      3. times-frac 74.0%

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

      4. sqrt-pow1 63.5%

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

      5. metadata-eval 63.5%

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

      6. unpow-1 63.5%

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

      7. sqrt-pow1 94.1%

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

      8. metadata-eval 94.1%

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

      9. unpow-1 94.1%

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

   8. Applied egg-rr 94.1%

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

      1. associate-/l/ 94.0%

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

      2. frac-times 96.9%

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

      3. *-rgt-identity 96.9%

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

   10. Applied egg-rr 96.9%

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

4. Final simplification 98.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{+126}:\\ \;\;\;\;\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{z}}{y \cdot \left(x \cdot z\right)}\\ \end{array} \]

Alternative 5: 96.7% accurate, 1.0× speedup

Math

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

FPCore

NOTE: z should be positive before calling this function
NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= z 1.0) (/ (/ 1.0 y) x) (/ 1.0 (* (* z y) (* x z)))))

C

z = abs(z);
assert(x < y);
double code(double x, double y, double z) {
	double tmp;
	if (z <= 1.0) {
		tmp = (1.0 / y) / x;
	} else {
		tmp = 1.0 / ((z * y) * (x * z));
	}
	return tmp;
}

Fortran

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

Java

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

Python

z = abs(z)
[x, y] = sort([x, y])
def code(x, y, z):
	tmp = 0
	if z <= 1.0:
		tmp = (1.0 / y) / x
	else:
		tmp = 1.0 / ((z * y) * (x * z))
	return tmp

Julia

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

MATLAB

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

Wolfram

NOTE: z should be positive before calling this function
NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_, z_] := If[LessEqual[z, 1.0], N[(N[(1.0 / y), $MachinePrecision] / x), $MachinePrecision], N[(1.0 / N[(N[(z * y), $MachinePrecision] * N[(x * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 1

   1. Initial program 92.5%

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

      1. associate-/r* 92.2%

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

      2. associate-/l/ 92.5%

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

      3. associate-/r* 92.5%

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

      4. associate-/l/ 92.0%

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

      5. sqr-neg 92.0%

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

      6. +-commutative 92.0%

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

      7. sqr-neg 92.0%

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

      8. fma-def 92.0%

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

   3. Simplified 92.0%

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

   4. Taylor expanded in z around 0 73.5%

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

   if 1 < z

   1. Initial program 77.9%

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

   2. Taylor expanded in z around inf 75.9%

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

      1. *-commutative 75.9%

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

      2. *-commutative 75.9%

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

      3. associate-*l* 80.9%

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

   4. Simplified 80.9%

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

      1. clear-num 80.9%

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

      2. associate-/r/ 80.9%

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

      3. associate-/r* 80.8%

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

      4. *-commutative 80.8%

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

      5. associate-/r* 79.6%

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

      6. pow-flip 81.4%

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

      7. metadata-eval 81.4%

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

   6. Applied egg-rr 81.4%

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

      1. associate-/l/ 82.6%

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

      2. add-sqr-sqrt 82.5%

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

      3. times-frac 78.7%

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

      4. sqrt-pow1 78.8%

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

      5. metadata-eval 78.8%

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

      6. unpow-1 78.8%

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

      7. sqrt-pow1 91.0%

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

      8. metadata-eval 91.0%

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

      9. unpow-1 91.0%

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

   8. Applied egg-rr 91.0%

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

      1. associate-/l/ 91.1%

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

      2. associate-/l/ 91.1%

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

      3. frac-times 90.2%

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

      4. metadata-eval 90.2%

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

   10. Applied egg-rr 90.2%

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

4. Final simplification 78.2%

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

Alternative 6: 97.0% accurate, 1.0× speedup

Math

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

FPCore

NOTE: z should be positive before calling this function
NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= z 1.0) (/ (/ 1.0 y) x) (/ (/ 1.0 z) (* y (* x z)))))

C

z = abs(z);
assert(x < y);
double code(double x, double y, double z) {
	double tmp;
	if (z <= 1.0) {
		tmp = (1.0 / y) / x;
	} else {
		tmp = (1.0 / z) / (y * (x * z));
	}
	return tmp;
}

Fortran

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

Java

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

Python

z = abs(z)
[x, y] = sort([x, y])
def code(x, y, z):
	tmp = 0
	if z <= 1.0:
		tmp = (1.0 / y) / x
	else:
		tmp = (1.0 / z) / (y * (x * z))
	return tmp

Julia

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

MATLAB

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

Wolfram

NOTE: z should be positive before calling this function
NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_, z_] := If[LessEqual[z, 1.0], N[(N[(1.0 / y), $MachinePrecision] / x), $MachinePrecision], N[(N[(1.0 / z), $MachinePrecision] / N[(y * N[(x * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 1

   1. Initial program 92.5%

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

      1. associate-/r* 92.2%

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

      2. associate-/l/ 92.5%

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

      3. associate-/r* 92.5%

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

      4. associate-/l/ 92.0%

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

      5. sqr-neg 92.0%

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

      6. +-commutative 92.0%

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

      7. sqr-neg 92.0%

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

      8. fma-def 92.0%

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

   3. Simplified 92.0%

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

   4. Taylor expanded in z around 0 73.5%

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

   if 1 < z

   1. Initial program 77.9%

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

   2. Taylor expanded in z around inf 75.9%

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

      1. *-commutative 75.9%

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

      2. *-commutative 75.9%

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

      3. associate-*l* 80.9%

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

   4. Simplified 80.9%

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

      1. clear-num 80.9%

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

      2. associate-/r/ 80.9%

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

      3. associate-/r* 80.8%

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

      4. *-commutative 80.8%

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

      5. associate-/r* 79.6%

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

      6. pow-flip 81.4%

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

      7. metadata-eval 81.4%

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

   6. Applied egg-rr 81.4%

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

      1. associate-/l/ 82.6%

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

      2. add-sqr-sqrt 82.5%

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

      3. times-frac 78.7%

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

      4. sqrt-pow1 78.8%

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

      5. metadata-eval 78.8%

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

      6. unpow-1 78.8%

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

      7. sqrt-pow1 91.0%

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

      8. metadata-eval 91.0%

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

      9. unpow-1 91.0%

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

   8. Applied egg-rr 91.0%

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

      1. associate-/l/ 91.0%

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

      2. frac-times 96.4%

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

      3. *-rgt-identity 96.4%

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

   10. Applied egg-rr 96.4%

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

4. Final simplification 79.9%

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

Alternative 7: 70.2% accurate, 1.2× speedup

Math

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

FPCore

NOTE: z should be positive before calling this function
NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= z 1.0) (/ (/ 1.0 y) x) (/ 1.0 (* x (* z y)))))

C

z = abs(z);
assert(x < y);
double code(double x, double y, double z) {
	double tmp;
	if (z <= 1.0) {
		tmp = (1.0 / y) / x;
	} else {
		tmp = 1.0 / (x * (z * y));
	}
	return tmp;
}

Fortran

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

Java

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

Python

z = abs(z)
[x, y] = sort([x, y])
def code(x, y, z):
	tmp = 0
	if z <= 1.0:
		tmp = (1.0 / y) / x
	else:
		tmp = 1.0 / (x * (z * y))
	return tmp

Julia

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

MATLAB

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

Wolfram

NOTE: z should be positive before calling this function
NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_, z_] := If[LessEqual[z, 1.0], N[(N[(1.0 / y), $MachinePrecision] / x), $MachinePrecision], N[(1.0 / N[(x * N[(z * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 1

   1. Initial program 92.5%

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

      1. associate-/r* 92.2%

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

      2. associate-/l/ 92.5%

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

      3. associate-/r* 92.5%

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

      4. associate-/l/ 92.0%

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

      5. sqr-neg 92.0%

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

      6. +-commutative 92.0%

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

      7. sqr-neg 92.0%

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

      8. fma-def 92.0%

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

   3. Simplified 92.0%

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

   4. Taylor expanded in z around 0 73.5%

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

   if 1 < z

   1. Initial program 77.9%

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

   2. Taylor expanded in x around 0 77.9%

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

      1. +-commutative 77.9%

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

      2. unpow2 77.9%

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

      3. fma-udef 77.9%

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

   4. Simplified 77.9%

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

      1. associate-/r* 77.9%

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

      2. inv-pow 77.9%

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

      3. metadata-eval 77.9%

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

      4. pow-prod-up 34.0%

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

      5. frac-times 36.6%

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

      6. *-commutative 36.6%

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

      7. add-sqr-sqrt 36.7%

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

      8. associate-*l* 36.7%

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

      9. sqrt-div 36.7%

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

      10. sqrt-pow1 36.7%

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

      11. metadata-eval 36.7%

          \[\leadsto \frac{{x}^{\color{blue}{-0.25}}}{\sqrt{\mathsf{fma}\left(z, z, 1\right)}} \cdot \left(\sqrt{\frac{{x}^{-0.5}}{\mathsf{fma}\left(z, z, 1\right)}} \cdot \frac{{x}^{-0.5}}{y}\right) \]

      12. sqrt-div 36.7%

          \[\leadsto \frac{{x}^{-0.25}}{\sqrt{\mathsf{fma}\left(z, z, 1\right)}} \cdot \left(\color{blue}{\frac{\sqrt{{x}^{-0.5}}}{\sqrt{\mathsf{fma}\left(z, z, 1\right)}}} \cdot \frac{{x}^{-0.5}}{y}\right) \]

      13. sqrt-pow1 36.7%

          \[\leadsto \frac{{x}^{-0.25}}{\sqrt{\mathsf{fma}\left(z, z, 1\right)}} \cdot \left(\frac{\color{blue}{{x}^{\left(\frac{-0.5}{2}\right)}}}{\sqrt{\mathsf{fma}\left(z, z, 1\right)}} \cdot \frac{{x}^{-0.5}}{y}\right) \]

      14. metadata-eval 36.7%

          \[\leadsto \frac{{x}^{-0.25}}{\sqrt{\mathsf{fma}\left(z, z, 1\right)}} \cdot \left(\frac{{x}^{\color{blue}{-0.25}}}{\sqrt{\mathsf{fma}\left(z, z, 1\right)}} \cdot \frac{{x}^{-0.5}}{y}\right) \]

   6. Applied egg-rr 36.7%

      \[\leadsto \color{blue}{\frac{{x}^{-0.25}}{\sqrt{\mathsf{fma}\left(z, z, 1\right)}} \cdot \left(\frac{{x}^{-0.25}}{\sqrt{\mathsf{fma}\left(z, z, 1\right)}} \cdot \frac{{x}^{-0.5}}{y}\right)} \]
   7. Step-by-step derivation

      1. associate-*l/ 36.7%

         \[\leadsto \color{blue}{\frac{{x}^{-0.25} \cdot \left(\frac{{x}^{-0.25}}{\sqrt{\mathsf{fma}\left(z, z, 1\right)}} \cdot \frac{{x}^{-0.5}}{y}\right)}{\sqrt{\mathsf{fma}\left(z, z, 1\right)}}} \]

   8. Simplified 94.4%

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

   9. Taylor expanded in z around inf 92.6%

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

   10. Taylor expanded in z around 0 37.8%

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

4. Final simplification 63.4%

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

Alternative 8: 58.7% accurate, 2.2× speedup

Math

\[\begin{array}{l} z = |z|\\ [x, y] = \mathsf{sort}([x, y])\\ \\ \frac{1}{x \cdot y} \end{array} \]

FPCore

NOTE: z should be positive before calling this function
NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z) :precision binary64 (/ 1.0 (* x y)))

C

z = abs(z);
assert(x < y);
double code(double x, double y, double z) {
	return 1.0 / (x * y);
}

Fortran

NOTE: z should be positive before calling this function
NOTE: x and y should be sorted in increasing order before calling this function.
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)
end function

Java

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

Python

z = abs(z)
[x, y] = sort([x, y])
def code(x, y, z):
	return 1.0 / (x * y)

Julia

z = abs(z)
x, y = sort([x, y])
function code(x, y, z)
	return Float64(1.0 / Float64(x * y))
end

MATLAB

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

Wolfram

NOTE: z should be positive before calling this function
NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_, z_] := N[(1.0 / N[(x * y), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 88.4%

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

2. Taylor expanded in z around 0 58.1%

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

   1. *-commutative 58.1%

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

4. Simplified 58.1%

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

5. Final simplification 58.1%

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

Alternative 9: 58.7% accurate, 2.2× speedup

Math

\[\begin{array}{l} z = |z|\\ [x, y] = \mathsf{sort}([x, y])\\ \\ \frac{\frac{1}{x}}{y} \end{array} \]

FPCore

NOTE: z should be positive before calling this function
NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z) :precision binary64 (/ (/ 1.0 x) y))

C

z = abs(z);
assert(x < y);
double code(double x, double y, double z) {
	return (1.0 / x) / y;
}

Fortran

NOTE: z should be positive before calling this function
NOTE: x and y should be sorted in increasing order before calling this function.
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
end function

Java

z = Math.abs(z);
assert x < y;
public static double code(double x, double y, double z) {
	return (1.0 / x) / y;
}

Python

z = abs(z)
[x, y] = sort([x, y])
def code(x, y, z):
	return (1.0 / x) / y

Julia

z = abs(z)
x, y = sort([x, y])
function code(x, y, z)
	return Float64(Float64(1.0 / x) / y)
end

MATLAB

z = abs(z)
x, y = num2cell(sort([x, y])){:}
function tmp = code(x, y, z)
	tmp = (1.0 / x) / y;
end

Wolfram

NOTE: z should be positive before calling this function
NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_, z_] := N[(N[(1.0 / x), $MachinePrecision] / y), $MachinePrecision]

Derivation

1. Initial program 88.4%

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

2. Taylor expanded in z around 0 58.2%

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

3. Final simplification 58.2%

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

Alternative 10: 58.7% accurate, 2.2× speedup

Math

\[\begin{array}{l} z = |z|\\ [x, y] = \mathsf{sort}([x, y])\\ \\ \frac{\frac{1}{y}}{x} \end{array} \]

FPCore

NOTE: z should be positive before calling this function
NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z) :precision binary64 (/ (/ 1.0 y) x))

C

z = abs(z);
assert(x < y);
double code(double x, double y, double z) {
	return (1.0 / y) / x;
}

Fortran

NOTE: z should be positive before calling this function
NOTE: x and y should be sorted in increasing order before calling this function.
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 / y) / x
end function

Java

z = Math.abs(z);
assert x < y;
public static double code(double x, double y, double z) {
	return (1.0 / y) / x;
}

Python

z = abs(z)
[x, y] = sort([x, y])
def code(x, y, z):
	return (1.0 / y) / x

Julia

z = abs(z)
x, y = sort([x, y])
function code(x, y, z)
	return Float64(Float64(1.0 / y) / x)
end

MATLAB

z = abs(z)
x, y = num2cell(sort([x, y])){:}
function tmp = code(x, y, z)
	tmp = (1.0 / y) / x;
end

Wolfram

NOTE: z should be positive before calling this function
NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_, z_] := N[(N[(1.0 / y), $MachinePrecision] / x), $MachinePrecision]

Derivation

1. Initial program 88.4%

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

   1. associate-/r* 88.2%

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

   2. associate-/l/ 88.4%

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

   3. associate-/r* 88.4%

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

   4. associate-/l/ 89.1%

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

   5. sqr-neg 89.1%

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

   6. +-commutative 89.1%

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

   7. sqr-neg 89.1%

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

   8. fma-def 89.1%

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

3. Simplified 89.1%

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

4. Taylor expanded in z around 0 58.2%

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

5. Final simplification 58.2%

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

Developer target: 93.2% 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 2023310 
(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)))))