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

Percentage Accurate: 90.6% → 97.2%

Time: 8.7s

Alternatives: 5

Speedup: 0.7×

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: 90.6% 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: 97.2% accurate, 0.1× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 1.6 \cdot 10^{+185}:\\ \;\;\;\;\frac{\frac{-1}{y \cdot x}}{-1 - {z}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{1}{x}}{z \cdot y}}{z}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= (* z z) 1.6e+185)
   (/ (/ -1.0 (* y x)) (- -1.0 (pow z 2.0)))
   (/ (/ (/ 1.0 x) (* z y)) z)))

C

assert(x < y);
double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 1.6e+185) {
		tmp = (-1.0 / (y * x)) / (-1.0 - pow(z, 2.0));
	} else {
		tmp = ((1.0 / x) / (z * y)) / z;
	}
	return tmp;
}

Fortran

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) <= 1.6d+185) then
        tmp = ((-1.0d0) / (y * x)) / ((-1.0d0) - (z ** 2.0d0))
    else
        tmp = ((1.0d0 / x) / (z * y)) / z
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 1.6e+185) {
		tmp = (-1.0 / (y * x)) / (-1.0 - Math.pow(z, 2.0));
	} else {
		tmp = ((1.0 / x) / (z * y)) / z;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z):
	tmp = 0
	if (z * z) <= 1.6e+185:
		tmp = (-1.0 / (y * x)) / (-1.0 - math.pow(z, 2.0))
	else:
		tmp = ((1.0 / x) / (z * y)) / z
	return tmp

Julia

x, y = sort([x, y])
function code(x, y, z)
	tmp = 0.0
	if (Float64(z * z) <= 1.6e+185)
		tmp = Float64(Float64(-1.0 / Float64(y * x)) / Float64(-1.0 - (z ^ 2.0)));
	else
		tmp = Float64(Float64(Float64(1.0 / x) / Float64(z * y)) / z);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z * z) <= 1.6e+185)
		tmp = (-1.0 / (y * x)) / (-1.0 - (z ^ 2.0));
	else
		tmp = ((1.0 / x) / (z * y)) / z;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_, z_] := If[LessEqual[N[(z * z), $MachinePrecision], 1.6e+185], N[(N[(-1.0 / N[(y * x), $MachinePrecision]), $MachinePrecision] / N[(-1.0 - N[Power[z, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 / x), $MachinePrecision] / N[(z * y), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 1.60000000000000003e185

   1. Initial program 98.4%

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

      1. frac-2neg 98.4%

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

      2. div-inv 98.2%

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

      3. distribute-neg-frac 98.2%

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

      4. metadata-eval 98.2%

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

      5. distribute-rgt-neg-in 98.2%

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

      6. +-commutative 98.2%

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

      7. fma-udef 98.2%

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

   3. Applied egg-rr 98.2%

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

      1. associate-*r/ 98.4%

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

      2. *-rgt-identity 98.4%

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

      3. associate-/r* 99.6%

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

      4. metadata-eval 99.6%

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

      5. distribute-neg-frac 99.6%

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

      6. distribute-frac-neg 99.6%

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

      7. associate-/r* 99.6%

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

      8. distribute-neg-frac 99.6%

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

      9. metadata-eval 99.6%

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

      10. *-commutative 99.6%

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

      11. neg-sub0 99.6%

          \[\leadsto \frac{\frac{-1}{y \cdot x}}{\color{blue}{0 - \mathsf{fma}\left(z, z, 1\right)}} \]

      12. fma-udef 99.6%

          \[\leadsto \frac{\frac{-1}{y \cdot x}}{0 - \color{blue}{\left(z \cdot z + 1\right)}} \]

      13. unpow2 99.6%

          \[\leadsto \frac{\frac{-1}{y \cdot x}}{0 - \left(\color{blue}{{z}^{2}} + 1\right)} \]

      14. +-commutative 99.6%

          \[\leadsto \frac{\frac{-1}{y \cdot x}}{0 - \color{blue}{\left(1 + {z}^{2}\right)}} \]

      15. associate--r+ 99.6%

          \[\leadsto \frac{\frac{-1}{y \cdot x}}{\color{blue}{\left(0 - 1\right) - {z}^{2}}} \]

      16. metadata-eval 99.6%

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

   5. Simplified 99.6%

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

   if 1.60000000000000003e185 < (*.f64 z z)

   1. Initial program 78.9%

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

      1. associate-/r* 78.8%

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

      2. associate-/l/ 79.0%

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

      3. associate-/r* 79.0%

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

      4. associate-/l/ 76.9%

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

      5. sqr-neg 76.9%

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

      6. +-commutative 76.9%

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

      7. sqr-neg 76.9%

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

      8. fma-def 76.9%

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

   3. Simplified 76.9%

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

   4. Taylor expanded in z around inf 76.9%

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

      1. *-commutative 76.9%

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

   6. Simplified 76.9%

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

      1. *-un-lft-identity 76.9%

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

      2. add-sqr-sqrt 33.8%

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

      3. times-frac 33.8%

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

      4. sqrt-prod 33.8%

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

      5. unpow2 33.8%

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

      6. sqrt-prod 9.8%

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

      7. add-sqr-sqrt 25.5%

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

      8. sqrt-prod 26.4%

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

      9. unpow2 26.4%

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

      10. sqrt-prod 18.3%

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

      11. add-sqr-sqrt 46.4%

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

   8. Applied egg-rr 46.4%

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

      1. associate-*l/ 46.5%

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

      2. *-lft-identity 46.5%

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

      3. associate-/l/ 46.4%

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

      4. associate-*l* 42.2%

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

   10. Simplified 42.2%

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

       1. div-inv 42.2%

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

       2. associate-*r* 46.4%

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

       3. associate-/r* 46.4%

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

       4. *-commutative 46.4%

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

       5. associate-/r* 46.4%

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

       6. metadata-eval 46.4%

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

       7. sqrt-div 46.5%

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

       8. inv-pow 46.5%

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

       9. sqrt-pow1 46.5%

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

       10. metadata-eval 46.5%

           \[\leadsto \frac{\frac{{x}^{\color{blue}{-0.5}}}{z}}{y} \cdot \frac{1}{z \cdot \sqrt{x}} \]

       11. *-commutative 46.5%

           \[\leadsto \frac{\frac{{x}^{-0.5}}{z}}{y} \cdot \frac{1}{\color{blue}{\sqrt{x} \cdot z}} \]

       12. associate-/r* 46.5%

           \[\leadsto \frac{\frac{{x}^{-0.5}}{z}}{y} \cdot \color{blue}{\frac{\frac{1}{\sqrt{x}}}{z}} \]

       13. metadata-eval 46.5%

           \[\leadsto \frac{\frac{{x}^{-0.5}}{z}}{y} \cdot \frac{\frac{\color{blue}{\sqrt{1}}}{\sqrt{x}}}{z} \]

       14. sqrt-div 46.5%

           \[\leadsto \frac{\frac{{x}^{-0.5}}{z}}{y} \cdot \frac{\color{blue}{\sqrt{\frac{1}{x}}}}{z} \]

       15. inv-pow 46.5%

           \[\leadsto \frac{\frac{{x}^{-0.5}}{z}}{y} \cdot \frac{\sqrt{\color{blue}{{x}^{-1}}}}{z} \]

       16. sqrt-pow1 46.5%

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

       17. metadata-eval 46.5%

           \[\leadsto \frac{\frac{{x}^{-0.5}}{z}}{y} \cdot \frac{{x}^{\color{blue}{-0.5}}}{z} \]

   12. Applied egg-rr 46.5%

       \[\leadsto \color{blue}{\frac{\frac{{x}^{-0.5}}{z}}{y} \cdot \frac{{x}^{-0.5}}{z}} \]
   13. Step-by-step derivation

       1. associate-*r/ 44.5%

          \[\leadsto \color{blue}{\frac{\frac{\frac{{x}^{-0.5}}{z}}{y} \cdot {x}^{-0.5}}{z}} \]

       2. associate-/l/ 44.1%

          \[\leadsto \frac{\color{blue}{\frac{{x}^{-0.5}}{y \cdot z}} \cdot {x}^{-0.5}}{z} \]

       3. associate-*l/ 44.2%

          \[\leadsto \frac{\color{blue}{\frac{{x}^{-0.5} \cdot {x}^{-0.5}}{y \cdot z}}}{z} \]

       4. pow-sqr 95.4%

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

       5. metadata-eval 95.4%

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

       6. unpow-1 95.4%

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

       7. *-commutative 95.4%

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

   14. Simplified 95.4%

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

4. Final simplification 98.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 1.6 \cdot 10^{+185}:\\ \;\;\;\;\frac{\frac{-1}{y \cdot x}}{-1 - {z}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{1}{x}}{z \cdot y}}{z}\\ \end{array} \]

Alternative 2: 97.9% accurate, 0.1× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 4 \cdot 10^{+86}:\\ \;\;\;\;\frac{\frac{1}{y}}{x \cdot \mathsf{fma}\left(z, z, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{1}{x}}{z \cdot y}}{z}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= (* z z) 4e+86)
   (/ (/ 1.0 y) (* x (fma z z 1.0)))
   (/ (/ (/ 1.0 x) (* z y)) z)))

C

assert(x < y);
double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 4e+86) {
		tmp = (1.0 / y) / (x * fma(z, z, 1.0));
	} else {
		tmp = ((1.0 / x) / (z * y)) / z;
	}
	return tmp;
}

Julia

x, y = sort([x, y])
function code(x, y, z)
	tmp = 0.0
	if (Float64(z * z) <= 4e+86)
		tmp = Float64(Float64(1.0 / y) / Float64(x * fma(z, z, 1.0)));
	else
		tmp = Float64(Float64(Float64(1.0 / x) / Float64(z * y)) / z);
	end
	return tmp
end

Wolfram

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+86], N[(N[(1.0 / y), $MachinePrecision] / N[(x * N[(z * z + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 / x), $MachinePrecision] / N[(z * y), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 4.0000000000000001e86

   1. Initial program 98.4%

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

      1. associate-/r* 98.4%

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

      2. associate-/l/ 98.3%

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

      3. associate-/r* 98.4%

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

      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 4.0000000000000001e86 < (*.f64 z z)

   1. Initial program 81.8%

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

      1. associate-/r* 81.7%

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

      2. associate-/l/ 81.8%

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

      3. associate-/r* 82.0%

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

      4. associate-/l/ 75.1%

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

      5. sqr-neg 75.1%

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

      6. +-commutative 75.1%

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

      7. sqr-neg 75.1%

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

      8. fma-def 75.1%

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

   3. Simplified 75.1%

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

   4. Taylor expanded in z around inf 75.1%

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

      1. *-commutative 75.1%

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

   6. Simplified 75.1%

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

      1. *-un-lft-identity 75.1%

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

      2. add-sqr-sqrt 34.5%

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

      3. times-frac 34.5%

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

      4. sqrt-prod 34.5%

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

      5. unpow2 34.5%

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

      6. sqrt-prod 12.2%

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

      7. add-sqr-sqrt 25.6%

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

      8. sqrt-prod 28.1%

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

      9. unpow2 28.1%

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

      10. sqrt-prod 21.1%

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

      11. add-sqr-sqrt 46.9%

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

   8. Applied egg-rr 46.9%

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

      1. associate-*l/ 46.9%

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

      2. *-lft-identity 46.9%

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

      3. associate-/l/ 46.9%

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

      4. associate-*l* 43.3%

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

   10. Simplified 43.3%

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

       1. div-inv 43.3%

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

       2. associate-*r* 46.9%

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

       3. associate-/r* 46.9%

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

       4. *-commutative 46.9%

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

       5. associate-/r* 46.9%

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

       6. metadata-eval 46.9%

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

       7. sqrt-div 47.0%

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

       8. inv-pow 47.0%

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

       9. sqrt-pow1 47.0%

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

       10. metadata-eval 47.0%

           \[\leadsto \frac{\frac{{x}^{\color{blue}{-0.5}}}{z}}{y} \cdot \frac{1}{z \cdot \sqrt{x}} \]

       11. *-commutative 47.0%

           \[\leadsto \frac{\frac{{x}^{-0.5}}{z}}{y} \cdot \frac{1}{\color{blue}{\sqrt{x} \cdot z}} \]

       12. associate-/r* 47.0%

           \[\leadsto \frac{\frac{{x}^{-0.5}}{z}}{y} \cdot \color{blue}{\frac{\frac{1}{\sqrt{x}}}{z}} \]

       13. metadata-eval 47.0%

           \[\leadsto \frac{\frac{{x}^{-0.5}}{z}}{y} \cdot \frac{\frac{\color{blue}{\sqrt{1}}}{\sqrt{x}}}{z} \]

       14. sqrt-div 47.0%

           \[\leadsto \frac{\frac{{x}^{-0.5}}{z}}{y} \cdot \frac{\color{blue}{\sqrt{\frac{1}{x}}}}{z} \]

       15. inv-pow 47.0%

           \[\leadsto \frac{\frac{{x}^{-0.5}}{z}}{y} \cdot \frac{\sqrt{\color{blue}{{x}^{-1}}}}{z} \]

       16. sqrt-pow1 47.0%

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

       17. metadata-eval 47.0%

           \[\leadsto \frac{\frac{{x}^{-0.5}}{z}}{y} \cdot \frac{{x}^{\color{blue}{-0.5}}}{z} \]

   12. Applied egg-rr 47.0%

       \[\leadsto \color{blue}{\frac{\frac{{x}^{-0.5}}{z}}{y} \cdot \frac{{x}^{-0.5}}{z}} \]
   13. Step-by-step derivation

       1. associate-*r/ 45.3%

          \[\leadsto \color{blue}{\frac{\frac{\frac{{x}^{-0.5}}{z}}{y} \cdot {x}^{-0.5}}{z}} \]

       2. associate-/l/ 45.0%

          \[\leadsto \frac{\color{blue}{\frac{{x}^{-0.5}}{y \cdot z}} \cdot {x}^{-0.5}}{z} \]

       3. associate-*l/ 45.0%

          \[\leadsto \frac{\color{blue}{\frac{{x}^{-0.5} \cdot {x}^{-0.5}}{y \cdot z}}}{z} \]

       4. pow-sqr 96.0%

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

       5. metadata-eval 96.0%

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

       6. unpow-1 96.0%

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

       7. *-commutative 96.0%

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

   14. Simplified 96.0%

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

4. Final simplification 97.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 4 \cdot 10^{+86}:\\ \;\;\;\;\frac{\frac{1}{y}}{x \cdot \mathsf{fma}\left(z, z, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{1}{x}}{z \cdot y}}{z}\\ \end{array} \]

Alternative 3: 97.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 10^{+50}:\\ \;\;\;\;\frac{\frac{1}{x}}{y \cdot \left(z \cdot z + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{1}{x}}{z \cdot y}}{z}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= (* z z) 1e+50)
   (/ (/ 1.0 x) (* y (+ (* z z) 1.0)))
   (/ (/ (/ 1.0 x) (* z y)) z)))

C

assert(x < y);
double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 1e+50) {
		tmp = (1.0 / x) / (y * ((z * z) + 1.0));
	} else {
		tmp = ((1.0 / x) / (z * y)) / z;
	}
	return tmp;
}

Fortran

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) <= 1d+50) then
        tmp = (1.0d0 / x) / (y * ((z * z) + 1.0d0))
    else
        tmp = ((1.0d0 / x) / (z * y)) / z
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 1e+50) {
		tmp = (1.0 / x) / (y * ((z * z) + 1.0));
	} else {
		tmp = ((1.0 / x) / (z * y)) / z;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z):
	tmp = 0
	if (z * z) <= 1e+50:
		tmp = (1.0 / x) / (y * ((z * z) + 1.0))
	else:
		tmp = ((1.0 / x) / (z * y)) / z
	return tmp

Julia

x, y = sort([x, y])
function code(x, y, z)
	tmp = 0.0
	if (Float64(z * z) <= 1e+50)
		tmp = Float64(Float64(1.0 / x) / Float64(y * Float64(Float64(z * z) + 1.0)));
	else
		tmp = Float64(Float64(Float64(1.0 / x) / Float64(z * y)) / z);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z * z) <= 1e+50)
		tmp = (1.0 / x) / (y * ((z * z) + 1.0));
	else
		tmp = ((1.0 / x) / (z * y)) / z;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_, z_] := If[LessEqual[N[(z * z), $MachinePrecision], 1e+50], N[(N[(1.0 / x), $MachinePrecision] / N[(y * N[(N[(z * z), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 / x), $MachinePrecision] / N[(z * y), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 1.0000000000000001e50

   1. Initial program 99.7%

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

   if 1.0000000000000001e50 < (*.f64 z z)

   1. Initial program 81.9%

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

      1. associate-/r* 81.8%

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

      2. associate-/l/ 81.9%

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

      3. associate-/r* 82.0%

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

      4. associate-/l/ 76.6%

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

      5. sqr-neg 76.6%

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

      6. +-commutative 76.6%

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

      7. sqr-neg 76.6%

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

      8. fma-def 76.6%

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

   3. Simplified 76.6%

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

   4. Taylor expanded in z around inf 76.6%

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

      1. *-commutative 76.6%

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

   6. Simplified 76.6%

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

      1. *-un-lft-identity 76.6%

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

      2. add-sqr-sqrt 34.6%

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

      3. times-frac 34.7%

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

      4. sqrt-prod 34.7%

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

      5. unpow2 34.7%

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

      6. sqrt-prod 13.6%

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

      7. add-sqr-sqrt 25.7%

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

      8. sqrt-prod 28.8%

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

      9. unpow2 28.8%

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

      10. sqrt-prod 22.5%

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

      11. add-sqr-sqrt 46.8%

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

   8. Applied egg-rr 46.8%

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

      1. associate-*l/ 46.8%

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

      2. *-lft-identity 46.8%

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

      3. associate-/l/ 46.8%

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

      4. associate-*l* 43.5%

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

   10. Simplified 43.5%

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

       1. div-inv 43.5%

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

       2. associate-*r* 46.8%

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

       3. associate-/r* 46.7%

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

       4. *-commutative 46.7%

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

       5. associate-/r* 46.7%

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

       6. metadata-eval 46.7%

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

       7. sqrt-div 46.8%

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

       8. inv-pow 46.8%

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

       9. sqrt-pow1 46.9%

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

       10. metadata-eval 46.9%

           \[\leadsto \frac{\frac{{x}^{\color{blue}{-0.5}}}{z}}{y} \cdot \frac{1}{z \cdot \sqrt{x}} \]

       11. *-commutative 46.9%

           \[\leadsto \frac{\frac{{x}^{-0.5}}{z}}{y} \cdot \frac{1}{\color{blue}{\sqrt{x} \cdot z}} \]

       12. associate-/r* 46.8%

           \[\leadsto \frac{\frac{{x}^{-0.5}}{z}}{y} \cdot \color{blue}{\frac{\frac{1}{\sqrt{x}}}{z}} \]

       13. metadata-eval 46.8%

           \[\leadsto \frac{\frac{{x}^{-0.5}}{z}}{y} \cdot \frac{\frac{\color{blue}{\sqrt{1}}}{\sqrt{x}}}{z} \]

       14. sqrt-div 46.8%

           \[\leadsto \frac{\frac{{x}^{-0.5}}{z}}{y} \cdot \frac{\color{blue}{\sqrt{\frac{1}{x}}}}{z} \]

       15. inv-pow 46.8%

           \[\leadsto \frac{\frac{{x}^{-0.5}}{z}}{y} \cdot \frac{\sqrt{\color{blue}{{x}^{-1}}}}{z} \]

       16. sqrt-pow1 46.8%

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

       17. metadata-eval 46.8%

           \[\leadsto \frac{\frac{{x}^{-0.5}}{z}}{y} \cdot \frac{{x}^{\color{blue}{-0.5}}}{z} \]

   12. Applied egg-rr 46.8%

       \[\leadsto \color{blue}{\frac{\frac{{x}^{-0.5}}{z}}{y} \cdot \frac{{x}^{-0.5}}{z}} \]
   13. Step-by-step derivation

       1. associate-*r/ 45.3%

          \[\leadsto \color{blue}{\frac{\frac{\frac{{x}^{-0.5}}{z}}{y} \cdot {x}^{-0.5}}{z}} \]

       2. associate-/l/ 45.0%

          \[\leadsto \frac{\color{blue}{\frac{{x}^{-0.5}}{y \cdot z}} \cdot {x}^{-0.5}}{z} \]

       3. associate-*l/ 45.0%

          \[\leadsto \frac{\color{blue}{\frac{{x}^{-0.5} \cdot {x}^{-0.5}}{y \cdot z}}}{z} \]

       4. pow-sqr 96.3%

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

       5. metadata-eval 96.3%

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

       6. unpow-1 96.3%

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

       7. *-commutative 96.3%

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

   14. Simplified 96.3%

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

4. Final simplification 98.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 10^{+50}:\\ \;\;\;\;\frac{\frac{1}{x}}{y \cdot \left(z \cdot z + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{1}{x}}{z \cdot y}}{z}\\ \end{array} \]

Alternative 4: 78.0% accurate, 1.0× speedup

Math

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

FPCore

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)) z)))

C

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)) / z;
	}
	return tmp;
}

Fortran

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)) / z
    end if
    code = tmp
end function

Java

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)) / z;
	}
	return tmp;
}

Python

[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)) / z
	return tmp

Julia

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(Float64(1.0 / x) / Float64(z * y)) / z);
	end
	return tmp
end

MATLAB

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)) / z;
	end
	tmp_2 = tmp;
end

Wolfram

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[(N[(1.0 / x), $MachinePrecision] / N[(z * y), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 1

   1. Initial program 94.9%

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

      1. associate-/r* 94.9%

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

      2. associate-/l/ 94.9%

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

      3. associate-/r* 94.9%

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

      4. associate-/l/ 94.8%

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

      5. sqr-neg 94.8%

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

      6. +-commutative 94.8%

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

      7. sqr-neg 94.8%

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

      8. fma-def 94.8%

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

   3. Simplified 94.8%

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

   4. Taylor expanded in z around 0 68.8%

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

   if 1 < z

   1. Initial program 80.2%

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

      1. associate-/r* 80.1%

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

      2. associate-/l/ 80.2%

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

      3. associate-/r* 80.6%

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

      4. associate-/l/ 70.1%

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

      5. sqr-neg 70.1%

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

      6. +-commutative 70.1%

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

      7. sqr-neg 70.1%

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

      8. fma-def 70.1%

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

   3. Simplified 70.1%

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

   4. Taylor expanded in z around inf 69.0%

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

      1. *-commutative 69.0%

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

   6. Simplified 69.0%

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

      1. *-un-lft-identity 69.0%

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

      2. add-sqr-sqrt 29.8%

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

      3. times-frac 29.8%

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

      4. sqrt-prod 29.8%

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

      5. unpow2 29.8%

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

      6. sqrt-prod 29.7%

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

      7. add-sqr-sqrt 29.8%

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

      8. sqrt-prod 35.8%

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

      9. unpow2 35.8%

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

      10. sqrt-prod 47.2%

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

      11. add-sqr-sqrt 47.3%

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

   8. Applied egg-rr 47.3%

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

      1. associate-*l/ 47.3%

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

      2. *-lft-identity 47.3%

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

      3. associate-/l/ 47.3%

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

      4. associate-*l* 42.3%

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

   10. Simplified 42.3%

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

       1. div-inv 42.3%

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

       2. associate-*r* 47.3%

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

       3. associate-/r* 47.3%

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

       4. *-commutative 47.3%

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

       5. associate-/r* 47.2%

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

       6. metadata-eval 47.2%

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

       7. sqrt-div 47.4%

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

       8. inv-pow 47.4%

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

       9. sqrt-pow1 47.3%

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

       10. metadata-eval 47.3%

           \[\leadsto \frac{\frac{{x}^{\color{blue}{-0.5}}}{z}}{y} \cdot \frac{1}{z \cdot \sqrt{x}} \]

       11. *-commutative 47.3%

           \[\leadsto \frac{\frac{{x}^{-0.5}}{z}}{y} \cdot \frac{1}{\color{blue}{\sqrt{x} \cdot z}} \]

       12. associate-/r* 47.3%

           \[\leadsto \frac{\frac{{x}^{-0.5}}{z}}{y} \cdot \color{blue}{\frac{\frac{1}{\sqrt{x}}}{z}} \]

       13. metadata-eval 47.3%

           \[\leadsto \frac{\frac{{x}^{-0.5}}{z}}{y} \cdot \frac{\frac{\color{blue}{\sqrt{1}}}{\sqrt{x}}}{z} \]

       14. sqrt-div 47.3%

           \[\leadsto \frac{\frac{{x}^{-0.5}}{z}}{y} \cdot \frac{\color{blue}{\sqrt{\frac{1}{x}}}}{z} \]

       15. inv-pow 47.3%

           \[\leadsto \frac{\frac{{x}^{-0.5}}{z}}{y} \cdot \frac{\sqrt{\color{blue}{{x}^{-1}}}}{z} \]

       16. sqrt-pow1 47.2%

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

       17. metadata-eval 47.2%

           \[\leadsto \frac{\frac{{x}^{-0.5}}{z}}{y} \cdot \frac{{x}^{\color{blue}{-0.5}}}{z} \]

   12. Applied egg-rr 47.2%

       \[\leadsto \color{blue}{\frac{\frac{{x}^{-0.5}}{z}}{y} \cdot \frac{{x}^{-0.5}}{z}} \]
   13. Step-by-step derivation

       1. associate-*r/ 45.7%

          \[\leadsto \color{blue}{\frac{\frac{\frac{{x}^{-0.5}}{z}}{y} \cdot {x}^{-0.5}}{z}} \]

       2. associate-/l/ 45.2%

          \[\leadsto \frac{\color{blue}{\frac{{x}^{-0.5}}{y \cdot z}} \cdot {x}^{-0.5}}{z} \]

       3. associate-*l/ 45.1%

          \[\leadsto \frac{\color{blue}{\frac{{x}^{-0.5} \cdot {x}^{-0.5}}{y \cdot z}}}{z} \]

       4. pow-sqr 96.4%

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

       5. metadata-eval 96.4%

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

       6. unpow-1 96.4%

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

       7. *-commutative 96.4%

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

   14. Simplified 96.4%

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

4. Final simplification 75.4%

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

Alternative 5: 57.8% accurate, 2.2× speedup

Math

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

FPCore

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

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

Fortran

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

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 91.4%

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

2. Taylor expanded in z around 0 55.7%

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

   1. *-commutative 55.7%

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

4. Simplified 55.7%

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

5. Final simplification 55.7%

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

Developer target: 92.7% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2023308 
(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)))))