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

Percentage Accurate: 91.0% → 98.6%

Time: 9.3s

Alternatives: 11

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: 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: 98.6% accurate, 0.1× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 2 \cdot 10^{-48}:\\ \;\;\;\;\frac{\frac{1}{x}}{\mathsf{hypot}\left(1, z\right)} \cdot \frac{\frac{1}{y}}{\mathsf{hypot}\left(1, z\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{1}{y}}{x \cdot \mathsf{hypot}\left(1, z\right)}}{\mathsf{hypot}\left(1, z\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y 2e-48)
   (* (/ (/ 1.0 x) (hypot 1.0 z)) (/ (/ 1.0 y) (hypot 1.0 z)))
   (/ (/ (/ 1.0 y) (* x (hypot 1.0 z))) (hypot 1.0 z))))

C

assert(x < y);
double code(double x, double y, double z) {
	double tmp;
	if (y <= 2e-48) {
		tmp = ((1.0 / x) / hypot(1.0, z)) * ((1.0 / y) / hypot(1.0, z));
	} else {
		tmp = ((1.0 / y) / (x * hypot(1.0, z))) / hypot(1.0, z);
	}
	return tmp;
}

Java

assert x < y;
public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 2e-48) {
		tmp = ((1.0 / x) / Math.hypot(1.0, z)) * ((1.0 / y) / Math.hypot(1.0, z));
	} else {
		tmp = ((1.0 / y) / (x * Math.hypot(1.0, z))) / Math.hypot(1.0, z);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z):
	tmp = 0
	if y <= 2e-48:
		tmp = ((1.0 / x) / math.hypot(1.0, z)) * ((1.0 / y) / math.hypot(1.0, z))
	else:
		tmp = ((1.0 / y) / (x * math.hypot(1.0, z))) / math.hypot(1.0, z)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y, z)
	tmp = 0.0
	if (y <= 2e-48)
		tmp = Float64(Float64(Float64(1.0 / x) / hypot(1.0, z)) * Float64(Float64(1.0 / y) / hypot(1.0, z)));
	else
		tmp = Float64(Float64(Float64(1.0 / y) / Float64(x * hypot(1.0, z))) / hypot(1.0, z));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 2e-48)
		tmp = ((1.0 / x) / hypot(1.0, z)) * ((1.0 / y) / hypot(1.0, z));
	else
		tmp = ((1.0 / y) / (x * hypot(1.0, z))) / hypot(1.0, 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[y, 2e-48], N[(N[(N[(1.0 / x), $MachinePrecision] / N[Sqrt[1.0 ^ 2 + z ^ 2], $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 / y), $MachinePrecision] / N[Sqrt[1.0 ^ 2 + z ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 / y), $MachinePrecision] / N[(x * N[Sqrt[1.0 ^ 2 + z ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[1.0 ^ 2 + z ^ 2], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.9999999999999999e-48

   1. Initial program 86.1%

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

      1. associate-/r* 86.0%

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

      2. +-commutative 86.0%

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

      3. fma-def 86.0%

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

   3. Simplified 86.0%

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

      1. fma-udef 86.0%

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

      2. +-commutative 86.0%

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

      3. associate-/r* 86.1%

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

      4. associate-/r* 83.9%

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

      5. div-inv 83.9%

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

      6. add-sqr-sqrt 83.9%

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

      7. times-frac 85.6%

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

      8. hypot-1-def 85.6%

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

      9. hypot-1-def 97.1%

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

   5. Applied egg-rr 97.1%

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

   if 1.9999999999999999e-48 < y

   1. Initial program 92.3%

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

      1. associate-/r* 90.8%

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

      2. +-commutative 90.8%

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

      3. fma-def 90.8%

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

   3. Simplified 90.8%

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

      1. fma-udef 90.8%

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

      2. +-commutative 90.8%

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

      3. associate-/r* 92.3%

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

      4. associate-/r* 96.3%

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

      5. add-sqr-sqrt 96.3%

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

      6. *-un-lft-identity 96.3%

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

      7. times-frac 96.2%

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

      8. hypot-1-def 96.2%

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

      9. associate-/l/ 94.7%

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

      10. hypot-1-def 97.2%

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

   5. Applied egg-rr 97.2%

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

      1. associate-*l/ 97.3%

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

      2. *-lft-identity 97.3%

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

      3. associate-/r* 98.8%

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

      4. associate-/l/ 99.8%

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

      5. *-commutative 99.8%

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

   7. Simplified 99.8%

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

4. Final simplification 98.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2 \cdot 10^{-48}:\\ \;\;\;\;\frac{\frac{1}{x}}{\mathsf{hypot}\left(1, z\right)} \cdot \frac{\frac{1}{y}}{\mathsf{hypot}\left(1, z\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{1}{y}}{x \cdot \mathsf{hypot}\left(1, z\right)}}{\mathsf{hypot}\left(1, z\right)}\\ \end{array} \]

Alternative 2: 97.2% accurate, 0.1× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 4 \cdot 10^{+304}:\\ \;\;\;\;\frac{1}{y} \cdot \frac{\frac{1}{x}}{\mathsf{fma}\left(z, z, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x \cdot \left(y \cdot \mathsf{hypot}\left(1, z\right)\right)}}{\mathsf{hypot}\left(1, z\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= (* z z) 4e+304)
   (* (/ 1.0 y) (/ (/ 1.0 x) (fma z z 1.0)))
   (/ (/ 1.0 (* x (* y (hypot 1.0 z)))) (hypot 1.0 z))))

C

assert(x < y);
double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 4e+304) {
		tmp = (1.0 / y) * ((1.0 / x) / fma(z, z, 1.0));
	} else {
		tmp = (1.0 / (x * (y * hypot(1.0, z)))) / hypot(1.0, z);
	}
	return tmp;
}

Julia

x, y = sort([x, y])
function code(x, y, z)
	tmp = 0.0
	if (Float64(z * z) <= 4e+304)
		tmp = Float64(Float64(1.0 / y) * Float64(Float64(1.0 / x) / fma(z, z, 1.0)));
	else
		tmp = Float64(Float64(1.0 / Float64(x * Float64(y * hypot(1.0, z)))) / hypot(1.0, z));
	end
	return tmp
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_, z_] := If[LessEqual[N[(z * z), $MachinePrecision], 4e+304], N[(N[(1.0 / y), $MachinePrecision] * N[(N[(1.0 / x), $MachinePrecision] / N[(z * z + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[(x * N[(y * N[Sqrt[1.0 ^ 2 + z ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[1.0 ^ 2 + z ^ 2], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 3.9999999999999998e304

   1. Initial program 96.6%

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

      1. associate-/r* 95.8%

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

      2. +-commutative 95.8%

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

      3. fma-def 95.8%

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

   3. Simplified 95.8%

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

      1. associate-/r* 96.6%

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

      2. *-un-lft-identity 96.6%

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

      3. fma-udef 96.6%

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

      4. +-commutative 96.6%

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

      5. times-frac 97.9%

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

      6. +-commutative 97.9%

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

      7. fma-udef 97.9%

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

   5. Applied egg-rr 97.9%

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

   if 3.9999999999999998e304 < (*.f64 z z)

   1. Initial program 68.9%

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

      1. associate-/r* 68.9%

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

      2. +-commutative 68.9%

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

      3. fma-def 68.9%

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

   3. Simplified 68.9%

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

      1. fma-udef 68.9%

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

      2. +-commutative 68.9%

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

      3. associate-/r* 68.9%

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

      4. associate-/r* 68.1%

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

      5. div-inv 68.1%

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

      6. add-sqr-sqrt 68.1%

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

      7. times-frac 68.9%

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

      8. hypot-1-def 68.9%

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

      9. hypot-1-def 97.7%

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

   5. Applied egg-rr 97.7%

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

      1. associate-*l/ 98.6%

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

      2. associate-/l/ 98.5%

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

      3. frac-times 98.6%

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

      4. metadata-eval 98.6%

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

   7. Applied egg-rr 98.6%

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

4. Final simplification 98.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 4 \cdot 10^{+304}:\\ \;\;\;\;\frac{1}{y} \cdot \frac{\frac{1}{x}}{\mathsf{fma}\left(z, z, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x \cdot \left(y \cdot \mathsf{hypot}\left(1, z\right)\right)}}{\mathsf{hypot}\left(1, z\right)}\\ \end{array} \]

Alternative 3: 97.1% accurate, 0.1× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 4 \cdot 10^{+304}:\\ \;\;\;\;\frac{1}{y} \cdot \frac{\frac{1}{x}}{\mathsf{fma}\left(z, z, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{z}{\frac{1}{x \cdot \left(y \cdot z\right)}}}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= (* z z) 4e+304)
   (* (/ 1.0 y) (/ (/ 1.0 x) (fma z z 1.0)))
   (/ 1.0 (/ z (/ 1.0 (* x (* y z)))))))

C

assert(x < y);
double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 4e+304) {
		tmp = (1.0 / y) * ((1.0 / x) / fma(z, z, 1.0));
	} else {
		tmp = 1.0 / (z / (1.0 / (x * (y * z))));
	}
	return tmp;
}

Julia

x, y = sort([x, y])
function code(x, y, z)
	tmp = 0.0
	if (Float64(z * z) <= 4e+304)
		tmp = Float64(Float64(1.0 / y) * Float64(Float64(1.0 / x) / fma(z, z, 1.0)));
	else
		tmp = Float64(1.0 / Float64(z / Float64(1.0 / Float64(x * Float64(y * z)))));
	end
	return tmp
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_, z_] := If[LessEqual[N[(z * z), $MachinePrecision], 4e+304], N[(N[(1.0 / y), $MachinePrecision] * N[(N[(1.0 / x), $MachinePrecision] / N[(z * z + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(z / N[(1.0 / N[(x * N[(y * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 3.9999999999999998e304

   1. Initial program 96.6%

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

      1. associate-/r* 95.8%

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

      2. +-commutative 95.8%

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

      3. fma-def 95.8%

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

   3. Simplified 95.8%

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

      1. associate-/r* 96.6%

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

      2. *-un-lft-identity 96.6%

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

      3. fma-udef 96.6%

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

      4. +-commutative 96.6%

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

      5. times-frac 97.9%

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

      6. +-commutative 97.9%

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

      7. fma-udef 97.9%

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

   5. Applied egg-rr 97.9%

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

   if 3.9999999999999998e304 < (*.f64 z z)

   1. Initial program 68.9%

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

      1. associate-/r* 68.9%

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

      2. +-commutative 68.9%

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

      3. fma-def 68.9%

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

   3. Simplified 68.9%

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

   4. Taylor expanded in z around inf 68.9%

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

      1. unpow2 68.9%

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

      2. associate-*r* 86.3%

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

      3. *-commutative 86.3%

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

   6. Simplified 86.3%

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

      1. *-commutative 86.3%

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

      2. *-commutative 86.3%

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

      3. associate-*r* 97.4%

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

      4. associate-*r* 85.2%

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

      5. /-rgt-identity 85.2%

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

      6. associate-/l* 85.2%

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

      7. associate-*r* 97.4%

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

      8. *-commutative 97.4%

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

   8. Applied egg-rr 97.4%

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

4. Final simplification 97.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 4 \cdot 10^{+304}:\\ \;\;\;\;\frac{1}{y} \cdot \frac{\frac{1}{x}}{\mathsf{fma}\left(z, z, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{z}{\frac{1}{x \cdot \left(y \cdot z\right)}}}\\ \end{array} \]

Alternative 4: 91.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_0 := y \cdot \left(1 + z \cdot z\right)\\ \mathbf{if}\;t_0 \leq \infty:\\ \;\;\;\;\frac{\frac{1}{x}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{y \cdot \left(x \cdot z\right)}}{z}\\ \end{array} \end{array} \]

FPCore

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

C

assert(x < y);
double code(double x, double y, double z) {
	double t_0 = y * (1.0 + (z * z));
	double tmp;
	if (t_0 <= ((double) INFINITY)) {
		tmp = (1.0 / x) / t_0;
	} else {
		tmp = (1.0 / (y * (x * z))) / z;
	}
	return tmp;
}

Java

assert x < y;
public static double code(double x, double y, double z) {
	double t_0 = y * (1.0 + (z * z));
	double tmp;
	if (t_0 <= Double.POSITIVE_INFINITY) {
		tmp = (1.0 / x) / t_0;
	} else {
		tmp = (1.0 / (y * (x * z))) / z;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z):
	t_0 = y * (1.0 + (z * z))
	tmp = 0
	if t_0 <= math.inf:
		tmp = (1.0 / x) / t_0
	else:
		tmp = (1.0 / (y * (x * z))) / z
	return tmp

Julia

x, y = sort([x, y])
function code(x, y, z)
	t_0 = Float64(y * Float64(1.0 + Float64(z * z)))
	tmp = 0.0
	if (t_0 <= Inf)
		tmp = Float64(Float64(1.0 / x) / t_0);
	else
		tmp = Float64(Float64(1.0 / Float64(y * Float64(x * z))) / z);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y, z)
	t_0 = y * (1.0 + (z * z));
	tmp = 0.0;
	if (t_0 <= Inf)
		tmp = (1.0 / x) / t_0;
	else
		tmp = (1.0 / (y * (x * z))) / 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_] := Block[{t$95$0 = N[(y * N[(1.0 + N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, Infinity], N[(N[(1.0 / x), $MachinePrecision] / t$95$0), $MachinePrecision], N[(N[(1.0 / N[(y * N[(x * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y (+.f64 1 (*.f64 z z))) < +inf.0

   1. Initial program 88.4%

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

   if +inf.0 < (*.f64 y (+.f64 1 (*.f64 z z)))

   1. Initial program 88.4%

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

      1. associate-/r* 87.8%

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

      2. +-commutative 87.8%

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

      3. fma-def 87.8%

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

   3. Simplified 87.8%

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

   4. Taylor expanded in z around inf 51.3%

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

      1. unpow2 51.3%

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

      2. associate-*r* 56.5%

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

      3. *-commutative 56.5%

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

   6. Simplified 56.5%

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

      1. *-commutative 56.5%

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

      2. *-commutative 56.5%

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

      3. *-commutative 56.5%

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

      4. associate-*r* 51.3%

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

      5. associate-*l* 52.0%

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

      6. associate-/l/ 52.0%

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

      7. *-un-lft-identity 52.0%

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

      8. associate-*r* 57.2%

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

      9. times-frac 60.2%

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

   8. Applied egg-rr 60.2%

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

      1. associate-*r/ 60.8%

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

      2. frac-times 61.2%

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

      3. metadata-eval 61.2%

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

      4. *-commutative 61.2%

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

   10. Applied egg-rr 61.2%

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

4. Final simplification 88.4%

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

Alternative 5: 91.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 10^{-51}:\\ \;\;\;\;\frac{\frac{1}{x}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x \cdot \left(z \cdot \left(y \cdot z\right)\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= (* z z) 1e-51) (/ (/ 1.0 x) y) (/ 1.0 (* x (* z (* y z))))))

C

assert(x < y);
double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 1e-51) {
		tmp = (1.0 / x) / y;
	} else {
		tmp = 1.0 / (x * (z * (y * z)));
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((z * z) <= 1d-51) then
        tmp = (1.0d0 / x) / y
    else
        tmp = 1.0d0 / (x * (z * (y * z)))
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 1e-51) {
		tmp = (1.0 / x) / y;
	} else {
		tmp = 1.0 / (x * (z * (y * z)));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z):
	tmp = 0
	if (z * z) <= 1e-51:
		tmp = (1.0 / x) / y
	else:
		tmp = 1.0 / (x * (z * (y * z)))
	return tmp

Julia

x, y = sort([x, y])
function code(x, y, z)
	tmp = 0.0
	if (Float64(z * z) <= 1e-51)
		tmp = Float64(Float64(1.0 / x) / y);
	else
		tmp = Float64(1.0 / Float64(x * Float64(z * Float64(y * z))));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z * z) <= 1e-51)
		tmp = (1.0 / x) / y;
	else
		tmp = 1.0 / (x * (z * (y * z)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_, z_] := If[LessEqual[N[(z * z), $MachinePrecision], 1e-51], N[(N[(1.0 / x), $MachinePrecision] / y), $MachinePrecision], N[(1.0 / N[(x * N[(z * N[(y * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 1e-51

   1. Initial program 99.7%

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

      1. associate-/r* 98.4%

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

      2. +-commutative 98.4%

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

      3. fma-def 98.4%

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

   3. Simplified 98.4%

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

      1. fma-udef 98.4%

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

      2. +-commutative 98.4%

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

      3. associate-/r* 99.7%

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

      4. associate-/r* 99.7%

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

      5. div-inv 99.6%

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

      6. add-sqr-sqrt 99.6%

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

      7. times-frac 99.6%

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

      8. hypot-1-def 99.6%

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

      9. hypot-1-def 99.6%

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

   5. Applied egg-rr 99.6%

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

   6. Taylor expanded in z around 0 98.4%

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

      1. associate-/l/ 99.7%

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

   8. Simplified 99.7%

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

   if 1e-51 < (*.f64 z z)

   1. Initial program 79.3%

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

      1. associate-/r* 79.2%

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

      2. +-commutative 79.2%

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

      3. fma-def 79.2%

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

   3. Simplified 79.2%

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

   4. Taylor expanded in z around inf 78.9%

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

      1. unpow2 78.9%

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

      2. associate-*r* 88.2%

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

      3. *-commutative 88.2%

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

   6. Simplified 88.2%

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

4. Final simplification 93.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 10^{-51}:\\ \;\;\;\;\frac{\frac{1}{x}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x \cdot \left(z \cdot \left(y \cdot z\right)\right)}\\ \end{array} \]

Alternative 6: 91.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 10^{-51}:\\ \;\;\;\;\frac{\frac{1}{x}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{z \cdot \left(z \cdot \left(y \cdot x\right)\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= (* z z) 1e-51) (/ (/ 1.0 x) y) (/ 1.0 (* z (* z (* y x))))))

C

assert(x < y);
double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 1e-51) {
		tmp = (1.0 / x) / y;
	} 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 * z) <= 1d-51) then
        tmp = (1.0d0 / x) / y
    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 * z) <= 1e-51) {
		tmp = (1.0 / x) / y;
	} 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 * z) <= 1e-51:
		tmp = (1.0 / x) / y
	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 (Float64(z * z) <= 1e-51)
		tmp = Float64(Float64(1.0 / x) / y);
	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 * z) <= 1e-51)
		tmp = (1.0 / x) / y;
	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[N[(z * z), $MachinePrecision], 1e-51], N[(N[(1.0 / x), $MachinePrecision] / y), $MachinePrecision], N[(1.0 / N[(z * N[(z * N[(y * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 1e-51

   1. Initial program 99.7%

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

      1. associate-/r* 98.4%

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

      2. +-commutative 98.4%

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

      3. fma-def 98.4%

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

   3. Simplified 98.4%

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

      1. fma-udef 98.4%

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

      2. +-commutative 98.4%

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

      3. associate-/r* 99.7%

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

      4. associate-/r* 99.7%

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

      5. div-inv 99.6%

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

      6. add-sqr-sqrt 99.6%

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

      7. times-frac 99.6%

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

      8. hypot-1-def 99.6%

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

      9. hypot-1-def 99.6%

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

   5. Applied egg-rr 99.6%

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

   6. Taylor expanded in z around 0 98.4%

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

      1. associate-/l/ 99.7%

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

   8. Simplified 99.7%

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

   if 1e-51 < (*.f64 z z)

   1. Initial program 79.3%

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

      1. associate-/r* 79.2%

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

      2. +-commutative 79.2%

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

      3. fma-def 79.2%

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

   3. Simplified 79.2%

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

   4. Taylor expanded in z around inf 80.2%

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

      1. unpow2 80.2%

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

      2. *-commutative 80.2%

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

      3. associate-*l* 79.9%

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

      4. *-commutative 79.9%

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

      5. associate-*l* 88.9%

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

   6. Simplified 88.9%

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

4. Final simplification 93.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 10^{-51}:\\ \;\;\;\;\frac{\frac{1}{x}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{z \cdot \left(z \cdot \left(y \cdot x\right)\right)}\\ \end{array} \]

Alternative 7: 94.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 10^{-51}:\\ \;\;\;\;\frac{\frac{1}{x}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(y \cdot z\right) \cdot \left(x \cdot z\right)}\\ \end{array} \end{array} \]

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-51) (/ (/ 1.0 x) y) (/ 1.0 (* (* y z) (* x z)))))

C

assert(x < y);
double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 1e-51) {
		tmp = (1.0 / x) / y;
	} else {
		tmp = 1.0 / ((y * z) * (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) <= 1d-51) then
        tmp = (1.0d0 / x) / y
    else
        tmp = 1.0d0 / ((y * z) * (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) <= 1e-51) {
		tmp = (1.0 / x) / y;
	} else {
		tmp = 1.0 / ((y * z) * (x * z));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z):
	tmp = 0
	if (z * z) <= 1e-51:
		tmp = (1.0 / x) / y
	else:
		tmp = 1.0 / ((y * z) * (x * z))
	return tmp

Julia

x, y = sort([x, y])
function code(x, y, z)
	tmp = 0.0
	if (Float64(z * z) <= 1e-51)
		tmp = Float64(Float64(1.0 / x) / y);
	else
		tmp = Float64(1.0 / Float64(Float64(y * z) * Float64(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) <= 1e-51)
		tmp = (1.0 / x) / y;
	else
		tmp = 1.0 / ((y * z) * (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], 1e-51], N[(N[(1.0 / x), $MachinePrecision] / y), $MachinePrecision], N[(1.0 / N[(N[(y * z), $MachinePrecision] * N[(x * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 1e-51

   1. Initial program 99.7%

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

      1. associate-/r* 98.4%

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

      2. +-commutative 98.4%

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

      3. fma-def 98.4%

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

   3. Simplified 98.4%

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

      1. fma-udef 98.4%

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

      2. +-commutative 98.4%

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

      3. associate-/r* 99.7%

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

      4. associate-/r* 99.7%

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

      5. div-inv 99.6%

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

      6. add-sqr-sqrt 99.6%

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

      7. times-frac 99.6%

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

      8. hypot-1-def 99.6%

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

      9. hypot-1-def 99.6%

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

   5. Applied egg-rr 99.6%

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

   6. Taylor expanded in z around 0 98.4%

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

      1. associate-/l/ 99.7%

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

   8. Simplified 99.7%

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

   if 1e-51 < (*.f64 z z)

   1. Initial program 79.3%

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

      1. associate-/r* 79.2%

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

      2. +-commutative 79.2%

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

      3. fma-def 79.2%

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

   3. Simplified 79.2%

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

   4. Taylor expanded in z around inf 78.9%

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

      1. unpow2 78.9%

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

      2. associate-*r* 88.2%

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

      3. *-commutative 88.2%

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

   6. Simplified 88.2%

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

      1. *-commutative 88.2%

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

      2. *-commutative 88.2%

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

      3. *-commutative 88.2%

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

      4. associate-*r* 78.9%

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

      5. associate-*l* 80.2%

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

      6. associate-/l/ 80.2%

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

      7. *-un-lft-identity 80.2%

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

      8. associate-*r* 89.6%

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

      9. times-frac 94.9%

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

   8. Applied egg-rr 94.9%

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

      1. *-commutative 94.9%

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

      2. associate-/l/ 94.0%

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

      3. frac-times 93.6%

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

      4. metadata-eval 93.6%

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

      5. *-commutative 93.6%

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

   10. Applied egg-rr 93.6%

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

4. Final simplification 96.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 10^{-51}:\\ \;\;\;\;\frac{\frac{1}{x}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(y \cdot z\right) \cdot \left(x \cdot z\right)}\\ \end{array} \]

Alternative 8: 94.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 10^{-51}:\\ \;\;\;\;\frac{\frac{1}{x}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{y \cdot z}}{x \cdot z}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= (* z z) 1e-51) (/ (/ 1.0 x) y) (/ (/ 1.0 (* y z)) (* x z))))

C

assert(x < y);
double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 1e-51) {
		tmp = (1.0 / x) / y;
	} else {
		tmp = (1.0 / (y * z)) / (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) <= 1d-51) then
        tmp = (1.0d0 / x) / y
    else
        tmp = (1.0d0 / (y * z)) / (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) <= 1e-51) {
		tmp = (1.0 / x) / y;
	} else {
		tmp = (1.0 / (y * z)) / (x * z);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z):
	tmp = 0
	if (z * z) <= 1e-51:
		tmp = (1.0 / x) / y
	else:
		tmp = (1.0 / (y * z)) / (x * z)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y, z)
	tmp = 0.0
	if (Float64(z * z) <= 1e-51)
		tmp = Float64(Float64(1.0 / x) / y);
	else
		tmp = Float64(Float64(1.0 / Float64(y * z)) / Float64(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) <= 1e-51)
		tmp = (1.0 / x) / y;
	else
		tmp = (1.0 / (y * z)) / (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], 1e-51], N[(N[(1.0 / x), $MachinePrecision] / y), $MachinePrecision], N[(N[(1.0 / N[(y * z), $MachinePrecision]), $MachinePrecision] / N[(x * z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 1e-51

   1. Initial program 99.7%

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

      1. associate-/r* 98.4%

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

      2. +-commutative 98.4%

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

      3. fma-def 98.4%

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

   3. Simplified 98.4%

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

      1. fma-udef 98.4%

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

      2. +-commutative 98.4%

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

      3. associate-/r* 99.7%

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

      4. associate-/r* 99.7%

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

      5. div-inv 99.6%

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

      6. add-sqr-sqrt 99.6%

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

      7. times-frac 99.6%

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

      8. hypot-1-def 99.6%

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

      9. hypot-1-def 99.6%

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

   5. Applied egg-rr 99.6%

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

   6. Taylor expanded in z around 0 98.4%

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

      1. associate-/l/ 99.7%

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

   8. Simplified 99.7%

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

   if 1e-51 < (*.f64 z z)

   1. Initial program 79.3%

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

      1. associate-/r* 79.2%

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

      2. +-commutative 79.2%

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

      3. fma-def 79.2%

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

   3. Simplified 79.2%

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

   4. Taylor expanded in z around inf 78.9%

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

      1. unpow2 78.9%

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

      2. associate-*r* 88.2%

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

      3. *-commutative 88.2%

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

   6. Simplified 88.2%

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

      1. *-commutative 88.2%

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

      2. *-commutative 88.2%

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

      3. *-commutative 88.2%

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

      4. associate-*r* 78.9%

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

      5. associate-*l* 80.2%

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

      6. associate-/l/ 80.2%

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

      7. *-un-lft-identity 80.2%

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

      8. associate-*r* 89.6%

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

      9. times-frac 94.9%

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

   8. Applied egg-rr 94.9%

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

      1. associate-*l/ 94.9%

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

      2. *-un-lft-identity 94.9%

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

      3. div-inv 94.8%

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

      4. frac-times 94.0%

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

      5. metadata-eval 94.0%

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

      6. *-commutative 94.0%

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

   10. Applied egg-rr 94.0%

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

4. Final simplification 96.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 10^{-51}:\\ \;\;\;\;\frac{\frac{1}{x}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{y \cdot z}}{x \cdot z}\\ \end{array} \]

Alternative 9: 94.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 10^{-51}:\\ \;\;\;\;\frac{\frac{1}{x}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{y \cdot \left(x \cdot z\right)}}{z}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= (* z z) 1e-51) (/ (/ 1.0 x) y) (/ (/ 1.0 (* y (* x z))) z)))

C

assert(x < y);
double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 1e-51) {
		tmp = (1.0 / x) / y;
	} else {
		tmp = (1.0 / (y * (x * z))) / z;
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((z * z) <= 1d-51) then
        tmp = (1.0d0 / x) / y
    else
        tmp = (1.0d0 / (y * (x * z))) / z
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 1e-51) {
		tmp = (1.0 / x) / y;
	} else {
		tmp = (1.0 / (y * (x * z))) / z;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z):
	tmp = 0
	if (z * z) <= 1e-51:
		tmp = (1.0 / x) / y
	else:
		tmp = (1.0 / (y * (x * z))) / z
	return tmp

Julia

x, y = sort([x, y])
function code(x, y, z)
	tmp = 0.0
	if (Float64(z * z) <= 1e-51)
		tmp = Float64(Float64(1.0 / x) / y);
	else
		tmp = Float64(Float64(1.0 / Float64(y * Float64(x * z))) / z);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z * z) <= 1e-51)
		tmp = (1.0 / x) / y;
	else
		tmp = (1.0 / (y * (x * z))) / z;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_, z_] := If[LessEqual[N[(z * z), $MachinePrecision], 1e-51], N[(N[(1.0 / x), $MachinePrecision] / y), $MachinePrecision], N[(N[(1.0 / N[(y * N[(x * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 1e-51

   1. Initial program 99.7%

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

      1. associate-/r* 98.4%

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

      2. +-commutative 98.4%

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

      3. fma-def 98.4%

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

   3. Simplified 98.4%

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

      1. fma-udef 98.4%

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

      2. +-commutative 98.4%

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

      3. associate-/r* 99.7%

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

      4. associate-/r* 99.7%

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

      5. div-inv 99.6%

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

      6. add-sqr-sqrt 99.6%

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

      7. times-frac 99.6%

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

      8. hypot-1-def 99.6%

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

      9. hypot-1-def 99.6%

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

   5. Applied egg-rr 99.6%

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

   6. Taylor expanded in z around 0 98.4%

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

      1. associate-/l/ 99.7%

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

   8. Simplified 99.7%

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

   if 1e-51 < (*.f64 z z)

   1. Initial program 79.3%

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

      1. associate-/r* 79.2%

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

      2. +-commutative 79.2%

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

      3. fma-def 79.2%

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

   3. Simplified 79.2%

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

   4. Taylor expanded in z around inf 78.9%

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

      1. unpow2 78.9%

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

      2. associate-*r* 88.2%

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

      3. *-commutative 88.2%

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

   6. Simplified 88.2%

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

      1. *-commutative 88.2%

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

      2. *-commutative 88.2%

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

      3. *-commutative 88.2%

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

      4. associate-*r* 78.9%

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

      5. associate-*l* 80.2%

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

      6. associate-/l/ 80.2%

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

      7. *-un-lft-identity 80.2%

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

      8. associate-*r* 89.6%

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

      9. times-frac 94.9%

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

   8. Applied egg-rr 94.9%

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

      1. associate-*r/ 94.1%

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

      2. frac-times 94.1%

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

      3. metadata-eval 94.1%

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

      4. *-commutative 94.1%

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

   10. Applied egg-rr 94.1%

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

4. Final simplification 96.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 10^{-51}:\\ \;\;\;\;\frac{\frac{1}{x}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{y \cdot \left(x \cdot z\right)}}{z}\\ \end{array} \]

Alternative 10: 93.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq 5.4 \cdot 10^{+152}:\\ \;\;\;\;\frac{\frac{1}{y}}{x \cdot \left(1 + z \cdot z\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{z}{\frac{1}{x \cdot \left(y \cdot z\right)}}}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= z 5.4e+152)
   (/ (/ 1.0 y) (* x (+ 1.0 (* z z))))
   (/ 1.0 (/ z (/ 1.0 (* x (* y z)))))))

C

assert(x < y);
double code(double x, double y, double z) {
	double tmp;
	if (z <= 5.4e+152) {
		tmp = (1.0 / y) / (x * (1.0 + (z * z)));
	} else {
		tmp = 1.0 / (z / (1.0 / (x * (y * z))));
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (z <= 5.4d+152) then
        tmp = (1.0d0 / y) / (x * (1.0d0 + (z * z)))
    else
        tmp = 1.0d0 / (z / (1.0d0 / (x * (y * z))))
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y, double z) {
	double tmp;
	if (z <= 5.4e+152) {
		tmp = (1.0 / y) / (x * (1.0 + (z * z)));
	} else {
		tmp = 1.0 / (z / (1.0 / (x * (y * z))));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z):
	tmp = 0
	if z <= 5.4e+152:
		tmp = (1.0 / y) / (x * (1.0 + (z * z)))
	else:
		tmp = 1.0 / (z / (1.0 / (x * (y * z))))
	return tmp

Julia

x, y = sort([x, y])
function code(x, y, z)
	tmp = 0.0
	if (z <= 5.4e+152)
		tmp = Float64(Float64(1.0 / y) / Float64(x * Float64(1.0 + Float64(z * z))));
	else
		tmp = Float64(1.0 / Float64(z / Float64(1.0 / Float64(x * Float64(y * z)))));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= 5.4e+152)
		tmp = (1.0 / y) / (x * (1.0 + (z * z)));
	else
		tmp = 1.0 / (z / (1.0 / (x * (y * z))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_, z_] := If[LessEqual[z, 5.4e+152], N[(N[(1.0 / y), $MachinePrecision] / N[(x * N[(1.0 + N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(z / N[(1.0 / N[(x * N[(y * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 5.4000000000000003e152

   1. Initial program 93.4%

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

      1. associate-/r* 92.7%

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

      2. +-commutative 92.7%

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

      3. fma-def 92.7%

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

   3. Simplified 92.7%

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

      1. fma-udef 92.7%

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

      2. +-commutative 92.7%

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

      3. associate-/r* 93.4%

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

      4. associate-/r* 93.7%

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

      5. div-inv 93.6%

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

      6. add-sqr-sqrt 93.6%

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

      7. times-frac 93.9%

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

      8. hypot-1-def 93.9%

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

      9. hypot-1-def 97.1%

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

   5. Applied egg-rr 97.1%

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

      1. associate-*l/ 97.2%

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

      2. associate-/l/ 96.6%

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

      3. frac-times 96.0%

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

      4. metadata-eval 96.0%

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

   7. Applied egg-rr 96.0%

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

   8. Taylor expanded in x around 0 93.5%

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

      1. associate-/r* 94.3%

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

      2. unpow2 94.3%

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

      3. +-commutative 94.3%

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

   10. Simplified 94.3%

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

   if 5.4000000000000003e152 < z

   1. Initial program 64.2%

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

      1. associate-/r* 64.2%

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

      2. +-commutative 64.2%

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

      3. fma-def 64.2%

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

   3. Simplified 64.2%

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

   4. Taylor expanded in z around inf 64.2%

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

      1. unpow2 64.2%

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

      2. associate-*r* 83.7%

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

      3. *-commutative 83.7%

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

   6. Simplified 83.7%

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

      1. *-commutative 83.7%

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

      2. *-commutative 83.7%

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

      3. associate-*r* 96.6%

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

      4. associate-*r* 79.9%

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

      5. /-rgt-identity 79.9%

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

      6. associate-/l* 79.9%

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

      7. associate-*r* 96.5%

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

      8. *-commutative 96.5%

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

   8. Applied egg-rr 96.5%

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

4. Final simplification 94.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 5.4 \cdot 10^{+152}:\\ \;\;\;\;\frac{\frac{1}{y}}{x \cdot \left(1 + z \cdot z\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{z}{\frac{1}{x \cdot \left(y \cdot z\right)}}}\\ \end{array} \]

Alternative 11: 58.0% 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 88.4%

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

   1. associate-/r* 87.8%

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

   2. +-commutative 87.8%

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

   3. fma-def 87.8%

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

3. Simplified 87.8%

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

4. Taylor expanded in z around 0 53.7%

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

5. Final simplification 53.7%

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

Developer target: 93.0% 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 2023200 
(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)))))