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

Percentage Accurate: 90.3% → 97.5%

Time: 8.0s

Alternatives: 7

Speedup: 0.8×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 90.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 97.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 10^{+178}:\\ \;\;\;\;\frac{\frac{1}{x \cdot y}}{1 + z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{z} \cdot \frac{1}{x \cdot \left(z \cdot y\right)}\\ \end{array} \end{array} \]

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+178)
   (/ (/ 1.0 (* x y)) (+ 1.0 (* z z)))
   (* (/ 1.0 z) (/ 1.0 (* x (* z y))))))

C

assert(x < y);
double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 1e+178) {
		tmp = (1.0 / (x * y)) / (1.0 + (z * z));
	} else {
		tmp = (1.0 / z) * (1.0 / (x * (z * y)));
	}
	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+178) then
        tmp = (1.0d0 / (x * y)) / (1.0d0 + (z * z))
    else
        tmp = (1.0d0 / z) * (1.0d0 / (x * (z * y)))
    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+178) {
		tmp = (1.0 / (x * y)) / (1.0 + (z * z));
	} else {
		tmp = (1.0 / z) * (1.0 / (x * (z * y)));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z):
	tmp = 0
	if (z * z) <= 1e+178:
		tmp = (1.0 / (x * y)) / (1.0 + (z * z))
	else:
		tmp = (1.0 / z) * (1.0 / (x * (z * y)))
	return tmp

Julia

x, y = sort([x, y])
function code(x, y, z)
	tmp = 0.0
	if (Float64(z * z) <= 1e+178)
		tmp = Float64(Float64(1.0 / Float64(x * y)) / Float64(1.0 + Float64(z * z)));
	else
		tmp = Float64(Float64(1.0 / z) * Float64(1.0 / Float64(x * Float64(z * y))));
	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+178)
		tmp = (1.0 / (x * y)) / (1.0 + (z * z));
	else
		tmp = (1.0 / z) * (1.0 / (x * (z * y)));
	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+178], N[(N[(1.0 / N[(x * y), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / z), $MachinePrecision] * N[(1.0 / N[(x * N[(z * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 1.0000000000000001e178

   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* 99.0%

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

   3. Simplified 99.0%

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

      1. add-sqr-sqrt 46.7%

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

      2. sqrt-unprod 41.1%

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

      3. frac-times 32.1%

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

      4. inv-pow 32.1%

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

      5. inv-pow 32.1%

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

      6. pow-prod-up 32.1%

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

      7. metadata-eval 32.1%

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

   5. Applied egg-rr 32.1%

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

      1. metadata-eval 32.1%

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

      2. pow-sqr 32.1%

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

      3. unpow-1 32.1%

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

      4. unpow-1 32.1%

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

      5. times-frac 41.1%

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

      6. associate-/r* 41.1%

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

      7. *-commutative 41.1%

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

      8. associate-/r* 41.1%

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

      9. *-commutative 41.1%

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

      10. unpow-1 41.1%

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

      11. unpow-1 41.1%

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

      12. pow-sqr 41.1%

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

      13. metadata-eval 41.1%

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

   7. Simplified 41.1%

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

      1. sqrt-pow1 99.1%

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

      2. metadata-eval 99.1%

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

      3. inv-pow 99.1%

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

      4. *-commutative 99.1%

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

   9. Applied egg-rr 99.1%

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

   if 1.0000000000000001e178 < (*.f64 z z)

   1. Initial program 77.3%

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

      1. associate-/r* 77.3%

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

      2. +-commutative 77.3%

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

      3. fma-def 77.3%

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

   3. Simplified 77.3%

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

   4. Taylor expanded in z around inf 78.3%

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

      1. unpow2 78.3%

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

      2. *-commutative 78.3%

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

      3. associate-*l* 78.9%

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

      4. *-commutative 78.9%

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

      5. associate-*l* 87.8%

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

   6. Simplified 87.8%

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

   7. Taylor expanded in z around 0 97.7%

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

      1. associate-/r* 97.7%

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

      2. div-inv 97.8%

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

      3. *-commutative 97.8%

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

      4. *-commutative 97.8%

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

      5. associate-*l* 96.7%

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

      6. *-commutative 96.7%

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

   9. Applied egg-rr 96.7%

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

4. Final simplification 98.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 10^{+178}:\\ \;\;\;\;\frac{\frac{1}{x \cdot y}}{1 + z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{z} \cdot \frac{1}{x \cdot \left(z \cdot y\right)}\\ \end{array} \]

Alternative 2: 97.6% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 90.9%

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

   1. associate-/r* 90.5%

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

   2. +-commutative 90.5%

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

   3. fma-def 90.5%

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

3. Simplified 90.5%

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

   1. fma-udef 90.5%

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

   2. +-commutative 90.5%

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

   3. associate-/r* 90.9%

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

   4. associate-/r* 91.8%

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

   5. div-inv 91.7%

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

   6. add-sqr-sqrt 91.7%

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

   7. times-frac 92.3%

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

   8. hypot-1-def 92.3%

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

   9. hypot-1-def 98.1%

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

5. Applied egg-rr 98.1%

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

   1. associate-/l/ 98.2%

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

   2. un-div-inv 98.2%

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

   3. associate-/l/ 98.1%

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

7. Applied egg-rr 98.1%

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

8. Final simplification 98.1%

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

Alternative 3: 97.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 10^{+178}:\\ \;\;\;\;\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{z} \cdot \frac{1}{x \cdot \left(z \cdot y\right)}\\ \end{array} \end{array} \]

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+178)
   (/ (/ 1.0 x) (* y (+ 1.0 (* z z))))
   (* (/ 1.0 z) (/ 1.0 (* x (* z y))))))

C

assert(x < y);
double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 1e+178) {
		tmp = (1.0 / x) / (y * (1.0 + (z * z)));
	} else {
		tmp = (1.0 / z) * (1.0 / (x * (z * y)));
	}
	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+178) then
        tmp = (1.0d0 / x) / (y * (1.0d0 + (z * z)))
    else
        tmp = (1.0d0 / z) * (1.0d0 / (x * (z * y)))
    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+178) {
		tmp = (1.0 / x) / (y * (1.0 + (z * z)));
	} else {
		tmp = (1.0 / z) * (1.0 / (x * (z * y)));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z):
	tmp = 0
	if (z * z) <= 1e+178:
		tmp = (1.0 / x) / (y * (1.0 + (z * z)))
	else:
		tmp = (1.0 / z) * (1.0 / (x * (z * y)))
	return tmp

Julia

x, y = sort([x, y])
function code(x, y, z)
	tmp = 0.0
	if (Float64(z * z) <= 1e+178)
		tmp = Float64(Float64(1.0 / x) / Float64(y * Float64(1.0 + Float64(z * z))));
	else
		tmp = Float64(Float64(1.0 / z) * Float64(1.0 / Float64(x * Float64(z * y))));
	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+178)
		tmp = (1.0 / x) / (y * (1.0 + (z * z)));
	else
		tmp = (1.0 / z) * (1.0 / (x * (z * y)));
	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+178], 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[(1.0 / N[(x * N[(z * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 1.0000000000000001e178

   1. Initial program 98.4%

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

   if 1.0000000000000001e178 < (*.f64 z z)

   1. Initial program 77.3%

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

      1. associate-/r* 77.3%

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

      2. +-commutative 77.3%

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

      3. fma-def 77.3%

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

   3. Simplified 77.3%

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

   4. Taylor expanded in z around inf 78.3%

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

      1. unpow2 78.3%

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

      2. *-commutative 78.3%

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

      3. associate-*l* 78.9%

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

      4. *-commutative 78.9%

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

      5. associate-*l* 87.8%

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

   6. Simplified 87.8%

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

   7. Taylor expanded in z around 0 97.7%

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

      1. associate-/r* 97.7%

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

      2. div-inv 97.8%

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

      3. *-commutative 97.8%

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

      4. *-commutative 97.8%

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

      5. associate-*l* 96.7%

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

      6. *-commutative 96.7%

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

   9. Applied egg-rr 96.7%

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

4. Final simplification 97.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 10^{+178}:\\ \;\;\;\;\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{z} \cdot \frac{1}{x \cdot \left(z \cdot y\right)}\\ \end{array} \]

Alternative 4: 93.3% accurate, 0.8× speedup

Math

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

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (or (<= z -1.0) (not (<= z 1.0)))
   (/ 1.0 (* x (* z (* z y))))
   (/ 1.0 (* x y))))

C

assert(x < y);
double code(double x, double y, double z) {
	double tmp;
	if ((z <= -1.0) || !(z <= 1.0)) {
		tmp = 1.0 / (x * (z * (z * y)));
	} else {
		tmp = 1.0 / (x * y);
	}
	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)) .or. (.not. (z <= 1.0d0))) then
        tmp = 1.0d0 / (x * (z * (z * y)))
    else
        tmp = 1.0d0 / (x * y)
    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) || !(z <= 1.0)) {
		tmp = 1.0 / (x * (z * (z * y)));
	} else {
		tmp = 1.0 / (x * y);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z):
	tmp = 0
	if (z <= -1.0) or not (z <= 1.0):
		tmp = 1.0 / (x * (z * (z * y)))
	else:
		tmp = 1.0 / (x * y)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y, z)
	tmp = 0.0
	if ((z <= -1.0) || !(z <= 1.0))
		tmp = Float64(1.0 / Float64(x * Float64(z * Float64(z * y))));
	else
		tmp = Float64(1.0 / Float64(x * y));
	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) || ~((z <= 1.0)))
		tmp = 1.0 / (x * (z * (z * y)));
	else
		tmp = 1.0 / (x * y);
	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[Or[LessEqual[z, -1.0], N[Not[LessEqual[z, 1.0]], $MachinePrecision]], N[(1.0 / N[(x * N[(z * N[(z * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(x * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1 or 1 < z

   1. Initial program 82.4%

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

      1. associate-/r* 81.6%

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

      2. +-commutative 81.6%

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

      3. fma-def 81.6%

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

   3. Simplified 81.6%

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

   4. Taylor expanded in z around inf 81.6%

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

      1. unpow2 81.6%

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

      2. associate-*r* 89.0%

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

      3. *-commutative 89.0%

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

   6. Simplified 89.0%

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

   if -1 < z < 1

   1. Initial program 99.6%

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

      1. associate-/r* 99.7%

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

      2. +-commutative 99.7%

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

      3. fma-def 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in z around 0 99.1%

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

4. Final simplification 94.0%

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

Alternative 5: 96.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 0.1:\\ \;\;\;\;\frac{1}{x \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{z \cdot \left(x \cdot \left(z \cdot y\right)\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= (* z z) 0.1) (/ 1.0 (* x y)) (/ 1.0 (* z (* x (* z y))))))

C

assert(x < y);
double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 0.1) {
		tmp = 1.0 / (x * y);
	} else {
		tmp = 1.0 / (z * (x * (z * y)));
	}
	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) <= 0.1d0) then
        tmp = 1.0d0 / (x * y)
    else
        tmp = 1.0d0 / (z * (x * (z * y)))
    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) <= 0.1) {
		tmp = 1.0 / (x * y);
	} else {
		tmp = 1.0 / (z * (x * (z * y)));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z):
	tmp = 0
	if (z * z) <= 0.1:
		tmp = 1.0 / (x * y)
	else:
		tmp = 1.0 / (z * (x * (z * y)))
	return tmp

Julia

x, y = sort([x, y])
function code(x, y, z)
	tmp = 0.0
	if (Float64(z * z) <= 0.1)
		tmp = Float64(1.0 / Float64(x * y));
	else
		tmp = Float64(1.0 / Float64(z * Float64(x * Float64(z * y))));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z * z) <= 0.1)
		tmp = 1.0 / (x * y);
	else
		tmp = 1.0 / (z * (x * (z * y)));
	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], 0.1], N[(1.0 / N[(x * y), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(z * N[(x * N[(z * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 0.10000000000000001

   1. Initial program 99.6%

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

      1. associate-/r* 99.7%

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

      2. +-commutative 99.7%

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

      3. fma-def 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in z around 0 99.1%

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

   if 0.10000000000000001 < (*.f64 z z)

   1. Initial program 82.4%

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

      1. associate-/r* 81.6%

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

      2. +-commutative 81.6%

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

      3. fma-def 81.6%

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

   3. Simplified 81.6%

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

   4. Taylor expanded in z around inf 83.1%

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

      1. unpow2 83.1%

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

      2. *-commutative 83.1%

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

      3. associate-*l* 83.5%

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

      4. *-commutative 83.5%

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

      5. associate-*l* 89.8%

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

   6. Simplified 89.8%

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

   7. Taylor expanded in z around 0 97.5%

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

      1. expm1-log1p-u 63.4%

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

      2. expm1-udef 40.3%

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

      3. *-commutative 40.3%

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

      4. *-commutative 40.3%

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

      5. associate-*l* 39.0%

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

      6. *-commutative 39.0%

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

   9. Applied egg-rr 39.0%

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

       1. expm1-def 62.1%

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

       2. expm1-log1p 96.1%

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

       3. *-commutative 96.1%

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

   11. Simplified 96.1%

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

4. Final simplification 97.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 0.1:\\ \;\;\;\;\frac{1}{x \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{z \cdot \left(x \cdot \left(z \cdot y\right)\right)}\\ \end{array} \]

Alternative 6: 69.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -1 \lor \neg \left(z \leq 1.3 \cdot 10^{+61}\right):\\ \;\;\;\;\frac{-1}{x \cdot \left(z \cdot y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x \cdot y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (or (<= z -1.0) (not (<= z 1.3e+61)))
   (/ -1.0 (* x (* z y)))
   (/ 1.0 (* x y))))

C

assert(x < y);
double code(double x, double y, double z) {
	double tmp;
	if ((z <= -1.0) || !(z <= 1.3e+61)) {
		tmp = -1.0 / (x * (z * y));
	} else {
		tmp = 1.0 / (x * y);
	}
	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)) .or. (.not. (z <= 1.3d+61))) then
        tmp = (-1.0d0) / (x * (z * y))
    else
        tmp = 1.0d0 / (x * y)
    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) || !(z <= 1.3e+61)) {
		tmp = -1.0 / (x * (z * y));
	} else {
		tmp = 1.0 / (x * y);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z):
	tmp = 0
	if (z <= -1.0) or not (z <= 1.3e+61):
		tmp = -1.0 / (x * (z * y))
	else:
		tmp = 1.0 / (x * y)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y, z)
	tmp = 0.0
	if ((z <= -1.0) || !(z <= 1.3e+61))
		tmp = Float64(-1.0 / Float64(x * Float64(z * y)));
	else
		tmp = Float64(1.0 / Float64(x * y));
	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) || ~((z <= 1.3e+61)))
		tmp = -1.0 / (x * (z * y));
	else
		tmp = 1.0 / (x * y);
	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[Or[LessEqual[z, -1.0], N[Not[LessEqual[z, 1.3e+61]], $MachinePrecision]], N[(-1.0 / N[(x * N[(z * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(x * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1 or 1.29999999999999986e61 < z

   1. Initial program 80.8%

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

      1. associate-/r* 80.6%

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

      2. +-commutative 80.6%

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

      3. fma-def 80.6%

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

   3. Simplified 80.6%

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

      1. fma-udef 80.6%

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

      2. +-commutative 80.6%

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

      3. associate-/r* 80.8%

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

      4. associate-/r* 82.8%

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

      5. div-inv 82.8%

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

      6. add-sqr-sqrt 82.8%

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

      7. times-frac 84.0%

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

      8. hypot-1-def 84.0%

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

      9. hypot-1-def 96.6%

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

   5. Applied egg-rr 96.6%

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

      1. associate-/l/ 96.7%

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

      2. un-div-inv 96.7%

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

      3. associate-/l/ 96.5%

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

   7. Applied egg-rr 96.5%

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

   8. Taylor expanded in z around -inf 78.8%

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

   9. Taylor expanded in z around 0 45.5%

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

       1. associate-*r* 42.6%

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

       2. *-commutative 42.6%

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

       3. *-commutative 42.6%

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

   11. Simplified 42.6%

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

   if -1 < z < 1.29999999999999986e61

   1. Initial program 99.6%

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

      1. associate-/r* 99.1%

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

      2. +-commutative 99.1%

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

      3. fma-def 99.1%

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

   3. Simplified 99.1%

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

   4. Taylor expanded in z around 0 93.7%

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

4. Final simplification 69.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1 \lor \neg \left(z \leq 1.3 \cdot 10^{+61}\right):\\ \;\;\;\;\frac{-1}{x \cdot \left(z \cdot y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x \cdot y}\\ \end{array} \]

Alternative 7: 58.4% accurate, 2.2× speedup

Math

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

FPCore

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

C

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

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 / (x * y)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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 90.9%

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

   1. associate-/r* 90.5%

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

   2. +-commutative 90.5%

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

   3. fma-def 90.5%

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

3. Simplified 90.5%

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

4. Taylor expanded in z around 0 59.7%

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

5. Final simplification 59.7%

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

Developer target: 92.3% 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 2023196 
(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)))))