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

Percentage Accurate: 91.0% → 97.6%

Time: 12.2s

Alternatives: 8

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: 91.0% 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: 97.6% accurate, 0.1× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 5 \cdot 10^{+206}:\\ \;\;\;\;\frac{\frac{1}{y}}{x \cdot \mathsf{fma}\left(z, z, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{1}{x}}{z \cdot y}}{z}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= (* z z) 5e+206)
   (/ (/ 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) <= 5e+206) {
		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) <= 5e+206)
		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], 5e+206], 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) < 5.0000000000000002e206

   1. Initial program 97.4%

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

      1. associate-/r* 98.0%

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

      2. associate-/r* 98.1%

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

      3. associate-/l/ 98.0%

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

      4. associate-/l/ 98.0%

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

      5. *-commutative 98.0%

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

      6. +-commutative 98.0%

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

      7. fma-def 98.0%

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

   3. Simplified 98.0%

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

   if 5.0000000000000002e206 < (*.f64 z z)

   1. Initial program 76.4%

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

   2. Taylor expanded in z around inf 76.5%

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

      1. associate-*r* 74.0%

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

      2. *-commutative 74.0%

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

   4. Simplified 74.0%

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

      1. associate-/r* 74.2%

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

      2. div-inv 74.2%

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

      3. unpow2 74.2%

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

      4. times-frac 88.8%

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

      5. associate-/r* 88.8%

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

   6. Applied egg-rr 88.8%

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

      1. *-commutative 88.8%

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

      2. div-inv 88.8%

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

      3. associate-/l/ 88.8%

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

      4. associate-/r* 88.9%

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

      5. associate-/l/ 97.4%

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

   8. Applied egg-rr 97.4%

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

4. Final simplification 97.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 5 \cdot 10^{+206}:\\ \;\;\;\;\frac{\frac{1}{y}}{x \cdot \mathsf{fma}\left(z, z, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{1}{x}}{z \cdot y}}{z}\\ \end{array} \]

Alternative 2: 97.6% accurate, 0.1× speedup

Math

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

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)) (* y (* (hypot 1.0 z) x))))

C

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

Java

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

Python

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

Julia

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

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp = code(x, y, z)
	tmp = (1.0 / hypot(1.0, z)) / (y * (hypot(1.0, z) * 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 / N[Sqrt[1.0 ^ 2 + z ^ 2], $MachinePrecision]), $MachinePrecision] / N[(y * N[(N[Sqrt[1.0 ^ 2 + z ^ 2], $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 90.2%

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

   1. associate-/r* 89.8%

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

   2. associate-/r* 89.9%

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

   3. div-inv 89.9%

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

   4. +-commutative 89.9%

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

   5. fma-udef 89.9%

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

   6. add-sqr-sqrt 89.9%

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

   7. times-frac 89.8%

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

   8. fma-udef 89.8%

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

   9. +-commutative 89.8%

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

   10. hypot-1-def 89.8%

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

   11. *-commutative 89.8%

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

   12. fma-udef 89.8%

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

   13. +-commutative 89.8%

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

   14. hypot-1-def 94.8%

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

3. Applied egg-rr 94.8%

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

   1. associate-*r/ 94.8%

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

   2. associate-/l* 94.8%

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

   3. clear-num 94.8%

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

   4. associate-/r* 94.8%

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

   5. associate-/l/ 98.5%

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

   6. associate-/r/ 98.5%

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

   7. clear-num 98.6%

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

   8. /-rgt-identity 98.6%

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

5. Applied egg-rr 98.6%

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

6. Final simplification 98.6%

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

Alternative 3: 97.4% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp = code(x, y, z)
	tmp = (1.0 / (y * (hypot(1.0, z) * x))) / hypot(1.0, z);
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[(y * N[(N[Sqrt[1.0 ^ 2 + z ^ 2], $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[1.0 ^ 2 + z ^ 2], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 90.2%

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

   1. associate-/r* 89.8%

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

   2. associate-/r* 89.9%

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

   3. div-inv 89.9%

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

   4. +-commutative 89.9%

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

   5. fma-udef 89.9%

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

   6. add-sqr-sqrt 89.9%

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

   7. times-frac 89.8%

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

   8. fma-udef 89.8%

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

   9. +-commutative 89.8%

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

   10. hypot-1-def 89.8%

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

   11. *-commutative 89.8%

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

   12. fma-udef 89.8%

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

   13. +-commutative 89.8%

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

   14. hypot-1-def 94.8%

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

3. Applied egg-rr 94.8%

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

   1. associate-*l/ 94.8%

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

   2. metadata-eval 94.8%

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

   3. associate-/r/ 94.8%

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

   4. associate-/r* 94.7%

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

   5. associate-/l/ 98.5%

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

   6. associate-/r/ 98.5%

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

   7. clear-num 98.6%

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

   8. /-rgt-identity 98.6%

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

5. Applied egg-rr 98.6%

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

6. Final simplification 98.6%

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

Alternative 4: 97.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 5 \cdot 10^{+38}:\\ \;\;\;\;\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{z} \cdot \frac{\frac{1}{y}}{z \cdot x}\\ \end{array} \end{array} \]

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+38)
   (/ (/ 1.0 x) (* y (+ 1.0 (* z z))))
   (* (/ 1.0 z) (/ (/ 1.0 y) (* z x)))))

C

assert(x < y);
double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 5e+38) {
		tmp = (1.0 / x) / (y * (1.0 + (z * z)));
	} else {
		tmp = (1.0 / z) * ((1.0 / y) / (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 * z) <= 5d+38) then
        tmp = (1.0d0 / x) / (y * (1.0d0 + (z * z)))
    else
        tmp = (1.0d0 / z) * ((1.0d0 / y) / (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 * z) <= 5e+38) {
		tmp = (1.0 / x) / (y * (1.0 + (z * z)));
	} else {
		tmp = (1.0 / z) * ((1.0 / y) / (z * x));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z):
	tmp = 0
	if (z * z) <= 5e+38:
		tmp = (1.0 / x) / (y * (1.0 + (z * z)))
	else:
		tmp = (1.0 / z) * ((1.0 / y) / (z * x))
	return tmp

Julia

x, y = sort([x, y])
function code(x, y, z)
	tmp = 0.0
	if (Float64(z * z) <= 5e+38)
		tmp = Float64(Float64(1.0 / x) / Float64(y * Float64(1.0 + Float64(z * z))));
	else
		tmp = Float64(Float64(1.0 / z) * Float64(Float64(1.0 / y) / 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 * z) <= 5e+38)
		tmp = (1.0 / x) / (y * (1.0 + (z * z)));
	else
		tmp = (1.0 / z) * ((1.0 / y) / (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[N[(z * z), $MachinePrecision], 5e+38], 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[(N[(1.0 / y), $MachinePrecision] / N[(z * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 4.9999999999999997e38

   1. Initial program 99.7%

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

   if 4.9999999999999997e38 < (*.f64 z z)

   1. Initial program 81.1%

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

   2. Taylor expanded in z around inf 80.7%

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

      1. associate-*r* 79.8%

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

      2. *-commutative 79.8%

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

   4. Simplified 79.8%

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

      1. associate-/r* 80.5%

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

      2. div-inv 80.5%

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

      3. unpow2 80.5%

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

      4. times-frac 90.2%

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

      5. associate-/r* 90.2%

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

   6. Applied egg-rr 90.2%

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

      1. associate-/l/ 97.5%

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

   8. Simplified 97.5%

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

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 5 \cdot 10^{+38}:\\ \;\;\;\;\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{z} \cdot \frac{\frac{1}{y}}{z \cdot x}\\ \end{array} \]

Alternative 5: 78.0% accurate, 0.8× speedup

Math

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

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

Derivation

1. Split input into 2 regimes

2. if z < 1

   1. Initial program 93.3%

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

      1. associate-/r* 92.7%

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

      2. associate-/r* 92.8%

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

      3. associate-/l/ 92.7%

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

      4. associate-/l/ 93.8%

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

      5. *-commutative 93.8%

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

      6. +-commutative 93.8%

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

      7. fma-def 93.8%

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

   3. Simplified 93.8%

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

   4. Taylor expanded in z around 0 66.2%

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

   if 1 < z

   1. Initial program 82.0%

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

   2. Taylor expanded in z around inf 81.7%

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

      1. associate-*r* 81.5%

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

      2. *-commutative 81.5%

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

   4. Simplified 81.5%

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

      1. associate-/r* 81.9%

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

      2. div-inv 81.9%

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

      3. unpow2 81.9%

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

      4. times-frac 88.5%

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

      5. associate-/r* 88.5%

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

   6. Applied egg-rr 88.5%

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

      1. associate-/l/ 95.3%

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

   8. Simplified 95.3%

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

4. Final simplification 74.2%

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

Alternative 6: 76.7% 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 93.3%

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

      1. associate-/r* 92.7%

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

      2. associate-/r* 92.8%

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

      3. associate-/l/ 92.7%

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

      4. associate-/l/ 93.8%

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

      5. *-commutative 93.8%

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

      6. +-commutative 93.8%

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

      7. fma-def 93.8%

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

   3. Simplified 93.8%

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

   4. Taylor expanded in z around 0 66.2%

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

   if 1 < z

   1. Initial program 82.0%

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

   2. Taylor expanded in z around inf 81.7%

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

      1. associate-*r* 81.5%

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

      2. *-commutative 81.5%

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

   4. Simplified 81.5%

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

      1. associate-/r* 81.9%

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

      2. div-inv 81.9%

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

      3. unpow2 81.9%

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

      4. times-frac 88.5%

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

      5. associate-/r* 88.5%

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

   6. Applied egg-rr 88.5%

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

      1. *-commutative 88.5%

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

      2. clear-num 88.6%

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

      3. frac-times 88.2%

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

      4. metadata-eval 88.2%

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

      5. associate-/r* 88.2%

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

      6. associate-/r/ 89.2%

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

      7. /-rgt-identity 89.2%

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

   8. Applied egg-rr 89.2%

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

4. Final simplification 72.5%

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

Alternative 7: 78.1% accurate, 1.0× speedup

Math

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

FPCore

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

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) / (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 <= 1.0d0) then
        tmp = (1.0d0 / y) / x
    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 <= 1.0) {
		tmp = (1.0 / y) / x;
	} 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 <= 1.0:
		tmp = (1.0 / y) / x
	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 (z <= 1.0)
		tmp = Float64(Float64(1.0 / y) / x);
	else
		tmp = Float64(Float64(1.0 / 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 <= 1.0)
		tmp = (1.0 / y) / x;
	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[z, 1.0], N[(N[(1.0 / y), $MachinePrecision] / x), $MachinePrecision], N[(N[(1.0 / z), $MachinePrecision] / N[(x * N[(z * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 1

   1. Initial program 93.3%

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

      1. associate-/r* 92.7%

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

      2. associate-/r* 92.8%

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

      3. associate-/l/ 92.7%

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

      4. associate-/l/ 93.8%

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

      5. *-commutative 93.8%

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

      6. +-commutative 93.8%

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

      7. fma-def 93.8%

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

   3. Simplified 93.8%

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

   4. Taylor expanded in z around 0 66.2%

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

   if 1 < z

   1. Initial program 82.0%

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

   2. Taylor expanded in z around inf 81.7%

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

      1. associate-*r* 81.5%

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

      2. *-commutative 81.5%

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

   4. Simplified 81.5%

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

      1. associate-/r* 81.9%

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

      2. div-inv 81.9%

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

      3. unpow2 81.9%

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

      4. times-frac 88.5%

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

      5. associate-/r* 88.5%

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

   6. Applied egg-rr 88.5%

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

      1. associate-*r/ 88.5%

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

      2. associate-/l* 88.6%

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

      3. associate-/r* 88.6%

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

      4. associate-/r/ 89.5%

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

      5. /-rgt-identity 89.5%

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

   8. Applied egg-rr 89.5%

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

   9. Taylor expanded in z around 0 95.2%

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

4. Final simplification 74.1%

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

Alternative 8: 58.2% 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 90.2%

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

2. Taylor expanded in z around 0 52.7%

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

3. Final simplification 52.7%

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

Developer target: 92.8% 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 2023301 
(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)))))