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

Percentage Accurate: 90.7% → 97.9%

Time: 9.1s

Alternatives: 10

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: 90.7% 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: 97.9% accurate, 0.0× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_0 := \sqrt{y} \cdot \mathsf{hypot}\left(1, z\right)\\ t_1 := y \cdot \left(1 + z \cdot z\right)\\ \mathbf{if}\;t_1 \leq 4.5 \cdot 10^{+294}:\\ \;\;\;\;\frac{\frac{1}{x}}{t_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{t_0} \cdot \frac{\frac{1}{x}}{t_0}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (sqrt y) (hypot 1.0 z))) (t_1 (* y (+ 1.0 (* z z)))))
   (if (<= t_1 4.5e+294) (/ (/ 1.0 x) t_1) (* (/ 1.0 t_0) (/ (/ 1.0 x) t_0)))))

C

assert(x < y);
double code(double x, double y, double z) {
	double t_0 = sqrt(y) * hypot(1.0, z);
	double t_1 = y * (1.0 + (z * z));
	double tmp;
	if (t_1 <= 4.5e+294) {
		tmp = (1.0 / x) / t_1;
	} else {
		tmp = (1.0 / t_0) * ((1.0 / x) / t_0);
	}
	return tmp;
}

Java

assert x < y;
public static double code(double x, double y, double z) {
	double t_0 = Math.sqrt(y) * Math.hypot(1.0, z);
	double t_1 = y * (1.0 + (z * z));
	double tmp;
	if (t_1 <= 4.5e+294) {
		tmp = (1.0 / x) / t_1;
	} else {
		tmp = (1.0 / t_0) * ((1.0 / x) / t_0);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z):
	t_0 = math.sqrt(y) * math.hypot(1.0, z)
	t_1 = y * (1.0 + (z * z))
	tmp = 0
	if t_1 <= 4.5e+294:
		tmp = (1.0 / x) / t_1
	else:
		tmp = (1.0 / t_0) * ((1.0 / x) / t_0)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y, z)
	t_0 = Float64(sqrt(y) * hypot(1.0, z))
	t_1 = Float64(y * Float64(1.0 + Float64(z * z)))
	tmp = 0.0
	if (t_1 <= 4.5e+294)
		tmp = Float64(Float64(1.0 / x) / t_1);
	else
		tmp = Float64(Float64(1.0 / t_0) * Float64(Float64(1.0 / x) / t_0));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y, z)
	t_0 = sqrt(y) * hypot(1.0, z);
	t_1 = y * (1.0 + (z * z));
	tmp = 0.0;
	if (t_1 <= 4.5e+294)
		tmp = (1.0 / x) / t_1;
	else
		tmp = (1.0 / t_0) * ((1.0 / x) / t_0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_, z_] := Block[{t$95$0 = N[(N[Sqrt[y], $MachinePrecision] * N[Sqrt[1.0 ^ 2 + z ^ 2], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(y * N[(1.0 + N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 4.5e+294], N[(N[(1.0 / x), $MachinePrecision] / t$95$1), $MachinePrecision], N[(N[(1.0 / t$95$0), $MachinePrecision] * N[(N[(1.0 / x), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y (+.f64 1 (*.f64 z z))) < 4.4999999999999998e294

   1. Initial program 95.5%

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

   if 4.4999999999999998e294 < (*.f64 y (+.f64 1 (*.f64 z z)))

   1. Initial program 70.4%

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

      1. *-un-lft-identity 70.4%

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

      2. add-sqr-sqrt 70.4%

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

      3. times-frac 70.4%

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

      4. sqrt-prod 70.4%

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

      5. hypot-1-def 70.4%

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

      6. sqrt-prod 77.0%

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

      7. hypot-1-def 99.6%

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

   3. Applied egg-rr 99.6%

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

4. Final simplification 96.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(1 + z \cdot z\right) \leq 4.5 \cdot 10^{+294}:\\ \;\;\;\;\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{y} \cdot \mathsf{hypot}\left(1, z\right)} \cdot \frac{\frac{1}{x}}{\sqrt{y} \cdot \mathsf{hypot}\left(1, z\right)}\\ \end{array} \]

Alternative 2: 97.3% accurate, 0.1× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 5 \cdot 10^{+285}:\\ \;\;\;\;\frac{\frac{1}{y}}{x \cdot \mathsf{fma}\left(z, z, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{y}}{z} \cdot \frac{\frac{1}{x}}{z}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= (* z z) 5e+285)
   (/ (/ 1.0 y) (* x (fma z z 1.0)))
   (* (/ (/ 1.0 y) z) (/ (/ 1.0 x) z))))

C

assert(x < y);
double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 5e+285) {
		tmp = (1.0 / y) / (x * fma(z, z, 1.0));
	} else {
		tmp = ((1.0 / y) / z) * ((1.0 / x) / z);
	}
	return tmp;
}

Julia

x, y = sort([x, y])
function code(x, y, z)
	tmp = 0.0
	if (Float64(z * z) <= 5e+285)
		tmp = Float64(Float64(1.0 / y) / Float64(x * fma(z, z, 1.0)));
	else
		tmp = Float64(Float64(Float64(1.0 / y) / z) * Float64(Float64(1.0 / x) / 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], 5e+285], N[(N[(1.0 / y), $MachinePrecision] / N[(x * N[(z * z + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 / y), $MachinePrecision] / z), $MachinePrecision] * N[(N[(1.0 / x), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 5.00000000000000016e285

   1. Initial program 97.8%

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

      1. associate-/r* 96.8%

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

      2. associate-/l/ 97.7%

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

      3. associate-/r* 97.7%

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

      4. associate-/l/ 96.7%

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

      5. sqr-neg 96.7%

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

      6. +-commutative 96.7%

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

      7. sqr-neg 96.7%

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

      8. fma-def 96.7%

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

   3. Simplified 96.7%

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

   if 5.00000000000000016e285 < (*.f64 z z)

   1. Initial program 71.2%

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

   2. Taylor expanded in z around inf 71.2%

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

      1. associate-*r* 69.3%

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

      2. associate-/r* 69.3%

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

      3. associate-/l/ 69.3%

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

   4. Simplified 69.3%

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

      1. div-inv 69.3%

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

      2. unpow2 69.3%

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

      3. times-frac 99.1%

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

   6. Applied egg-rr 99.1%

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

4. Final simplification 97.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 5 \cdot 10^{+285}:\\ \;\;\;\;\frac{\frac{1}{y}}{x \cdot \mathsf{fma}\left(z, z, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{y}}{z} \cdot \frac{\frac{1}{x}}{z}\\ \end{array} \]

Alternative 3: 96.9% accurate, 0.1× speedup

Math

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

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= (* z z) 5e+301)
   (/ (/ 1.0 (* y x)) (fma z z 1.0))
   (* (/ (/ 1.0 y) z) (/ (/ 1.0 x) z))))

C

assert(x < y);
double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 5e+301) {
		tmp = (1.0 / (y * x)) / fma(z, z, 1.0);
	} else {
		tmp = ((1.0 / y) / z) * ((1.0 / x) / z);
	}
	return tmp;
}

Julia

x, y = sort([x, y])
function code(x, y, z)
	tmp = 0.0
	if (Float64(z * z) <= 5e+301)
		tmp = Float64(Float64(1.0 / Float64(y * x)) / fma(z, z, 1.0));
	else
		tmp = Float64(Float64(Float64(1.0 / y) / z) * Float64(Float64(1.0 / x) / 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], 5e+301], N[(N[(1.0 / N[(y * x), $MachinePrecision]), $MachinePrecision] / N[(z * z + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 / y), $MachinePrecision] / z), $MachinePrecision] * N[(N[(1.0 / x), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 5.0000000000000004e301

   1. Initial program 97.8%

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

   2. Taylor expanded in x around 0 96.9%

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

      1. associate-/r* 97.8%

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

      2. *-commutative 97.8%

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

      3. +-commutative 97.8%

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

      4. unpow2 97.8%

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

      5. fma-udef 97.8%

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

      6. associate-/l/ 99.2%

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

      7. associate-/r* 98.4%

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

   4. Simplified 98.4%

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

   if 5.0000000000000004e301 < (*.f64 z z)

   1. Initial program 69.8%

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

   2. Taylor expanded in z around inf 69.8%

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

      1. associate-*r* 67.8%

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

      2. associate-/r* 67.8%

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

      3. associate-/l/ 67.8%

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

   4. Simplified 67.8%

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

      1. div-inv 67.8%

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

      2. unpow2 67.8%

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

      3. times-frac 99.0%

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

   6. Applied egg-rr 99.0%

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

4. Final simplification 98.5%

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

Alternative 4: 97.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{+204}:\\ \;\;\;\;\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{y}}{z} \cdot \frac{\frac{1}{x}}{z}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= (* z z) 2e+204)
   (/ (/ 1.0 x) (* y (+ 1.0 (* z z))))
   (* (/ (/ 1.0 y) z) (/ (/ 1.0 x) z))))

C

assert(x < y);
double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 2e+204) {
		tmp = (1.0 / x) / (y * (1.0 + (z * z)));
	} else {
		tmp = ((1.0 / y) / z) * ((1.0 / x) / 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) <= 2d+204) then
        tmp = (1.0d0 / x) / (y * (1.0d0 + (z * z)))
    else
        tmp = ((1.0d0 / y) / z) * ((1.0d0 / x) / 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) <= 2e+204) {
		tmp = (1.0 / x) / (y * (1.0 + (z * z)));
	} else {
		tmp = ((1.0 / y) / z) * ((1.0 / x) / z);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z):
	tmp = 0
	if (z * z) <= 2e+204:
		tmp = (1.0 / x) / (y * (1.0 + (z * z)))
	else:
		tmp = ((1.0 / y) / z) * ((1.0 / x) / z)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y, z)
	tmp = 0.0
	if (Float64(z * z) <= 2e+204)
		tmp = Float64(Float64(1.0 / x) / Float64(y * Float64(1.0 + Float64(z * z))));
	else
		tmp = Float64(Float64(Float64(1.0 / y) / z) * Float64(Float64(1.0 / x) / 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) <= 2e+204)
		tmp = (1.0 / x) / (y * (1.0 + (z * z)));
	else
		tmp = ((1.0 / y) / z) * ((1.0 / x) / 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], 2e+204], N[(N[(1.0 / x), $MachinePrecision] / N[(y * N[(1.0 + N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 / y), $MachinePrecision] / z), $MachinePrecision] * N[(N[(1.0 / x), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 1.99999999999999998e204

   1. Initial program 98.7%

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

   if 1.99999999999999998e204 < (*.f64 z z)

   1. Initial program 72.2%

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

   2. Taylor expanded in z around inf 72.2%

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

      1. associate-*r* 73.2%

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

      2. associate-/r* 73.2%

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

      3. associate-/l/ 73.2%

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

   4. Simplified 73.2%

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

      1. div-inv 73.2%

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

      2. unpow2 73.2%

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

      3. times-frac 96.6%

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

   6. Applied egg-rr 96.6%

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

4. Final simplification 98.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{+204}:\\ \;\;\;\;\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{y}}{z} \cdot \frac{\frac{1}{x}}{z}\\ \end{array} \]

Alternative 5: 78.2% accurate, 0.8× speedup

Math

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

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 y) z) (/ (/ 1.0 x) 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 / y) / z) * ((1.0 / x) / 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 / y) / z) * ((1.0d0 / x) / 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 / y) / z) * ((1.0 / x) / 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 / y) / z) * ((1.0 / x) / 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 / y) / z) * Float64(Float64(1.0 / x) / 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 / y) / z) * ((1.0 / x) / 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 / y), $MachinePrecision] / z), $MachinePrecision] * N[(N[(1.0 / x), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 1

   1. Initial program 95.6%

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

   2. Taylor expanded in z around 0 74.2%

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

      1. associate-/l/ 74.8%

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

   4. Simplified 74.8%

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

   if 1 < z

   1. Initial program 74.8%

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

   2. Taylor expanded in z around inf 74.8%

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

      1. associate-*r* 76.5%

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

      2. associate-/r* 76.6%

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

      3. associate-/l/ 76.6%

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

   4. Simplified 76.6%

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

      1. div-inv 76.6%

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

      2. unpow2 76.6%

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

      3. times-frac 97.0%

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

   6. Applied egg-rr 97.0%

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

4. Final simplification 79.3%

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

Alternative 6: 76.5% accurate, 1.0× speedup

Math

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

FPCore

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

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 / (z * (z * (y * x)));
	}
	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 / (z * (z * (y * x)))
    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 / (z * (z * (y * x)));
	}
	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 / (z * (z * (y * x)))
	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(1.0 / Float64(z * Float64(z * Float64(y * x))));
	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 / (z * (z * (y * x)));
	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[(1.0 / N[(z * N[(z * N[(y * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 1

   1. Initial program 95.6%

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

   2. Taylor expanded in z around 0 74.2%

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

      1. associate-/l/ 74.8%

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

   4. Simplified 74.8%

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

   if 1 < z

   1. Initial program 74.8%

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

   2. Taylor expanded in z around inf 74.8%

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

      1. associate-*r* 76.5%

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

      2. associate-/r* 76.6%

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

      3. associate-/l/ 76.6%

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

   4. Simplified 76.6%

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

      1. *-un-lft-identity 76.6%

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

      2. unpow2 76.6%

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

      3. times-frac 88.7%

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

   6. Applied egg-rr 88.7%

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

      1. *-commutative 88.7%

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

      2. clear-num 88.8%

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

      3. frac-times 87.4%

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

      4. metadata-eval 87.4%

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

      5. div-inv 87.4%

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

      6. clear-num 87.4%

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

      7. associate-/r/ 87.4%

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

      8. /-rgt-identity 87.4%

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

   8. Applied egg-rr 87.4%

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

4. Final simplification 77.4%

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

Alternative 7: 76.6% accurate, 1.0× speedup

Math

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

FPCore

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

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 / y) / (z * (z * x));
	}
	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 / y) / (z * (z * x))
    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 / y) / (z * (z * x));
	}
	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 / y) / (z * (z * x))
	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(1.0 / y) / Float64(z * Float64(z * x)));
	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 / y) / (z * (z * x));
	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[(1.0 / y), $MachinePrecision] / N[(z * N[(z * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 1

   1. Initial program 95.6%

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

   2. Taylor expanded in z around 0 74.2%

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

      1. associate-/l/ 74.8%

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

   4. Simplified 74.8%

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

   if 1 < z

   1. Initial program 74.8%

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

   2. Taylor expanded in z around inf 74.8%

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

      1. associate-*r* 76.5%

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

      2. associate-/r* 76.6%

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

      3. associate-/l/ 76.6%

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

   4. Simplified 76.6%

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

      1. *-un-lft-identity 76.6%

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

      2. unpow2 76.6%

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

      3. times-frac 88.7%

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

   6. Applied egg-rr 88.7%

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

      1. *-commutative 88.7%

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

      2. associate-/l/ 93.3%

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

      3. frac-times 86.0%

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

      4. associate-/r/ 86.0%

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

      5. clear-num 86.0%

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

      6. *-commutative 86.0%

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

   8. Applied egg-rr 86.0%

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

4. Final simplification 77.1%

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

Alternative 8: 59.0% 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 64.5%

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

3. Final simplification 64.5%

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

Alternative 9: 59.0% accurate, 2.2× speedup

Math

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

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(Float64(1.0 / 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[(N[(1.0 / x), $MachinePrecision] / y), $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 64.8%

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

3. Final simplification 64.8%

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

Alternative 10: 59.0% accurate, 2.2× speedup

Math

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

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(Float64(1.0 / 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[(N[(1.0 / y), $MachinePrecision] / x), $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 64.5%

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

   1. associate-/l/ 64.8%

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

4. Simplified 64.8%

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

5. Final simplification 64.8%

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

Developer target: 92.5% 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 2023320 
(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)))))