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

Percentage Accurate: 85.5% → 99.6%

Time: 31.1s

Alternatives: 9

Speedup: 0.8×

• could not determine a ground truth

  1. x = -6.345523484422262e+49
  2. y = 4.1334891586105655e+249
  3. z = 6.759880953206116e+186

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: 85.5% 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: 99.6% accurate, 0.1× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 5 \cdot 10^{+43}:\\ \;\;\;\;\frac{1}{x \cdot \left(y \cdot \mathsf{fma}\left(z, z, 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x \cdot z}}{z \cdot y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= (* z z) 5e+43)
   (/ 1.0 (* x (* y (fma z z 1.0))))
   (/ (/ 1.0 (* x z)) (* z y))))

C

assert(x < y);
double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 5e+43) {
		tmp = 1.0 / (x * (y * fma(z, z, 1.0)));
	} else {
		tmp = (1.0 / (x * z)) / (z * y);
	}
	return tmp;
}

Julia

x, y = sort([x, y])
function code(x, y, z)
	tmp = 0.0
	if (Float64(z * z) <= 5e+43)
		tmp = Float64(1.0 / Float64(x * Float64(y * fma(z, z, 1.0))));
	else
		tmp = Float64(Float64(1.0 / Float64(x * z)) / Float64(z * y));
	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+43], N[(1.0 / N[(x * N[(y * N[(z * z + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[(x * z), $MachinePrecision]), $MachinePrecision] / N[(z * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 5.0000000000000004e43

   1. Initial program 99.6%

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

      1. associate-/r* 99.7%

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

      2. +-commutative 99.7%

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

      3. fma-def 99.7%

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

   3. Simplified 99.7%

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

   if 5.0000000000000004e43 < (*.f64 z z)

   1. Initial program 62.0%

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

      1. associate-/r* 62.0%

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

      2. +-commutative 62.0%

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

      3. fma-def 62.0%

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

   3. Simplified 62.0%

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

      1. fma-udef 62.0%

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

      2. +-commutative 62.0%

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

      3. associate-/r* 62.0%

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

      4. associate-/r* 45.2%

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

      5. add-sqr-sqrt 45.2%

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

      6. *-un-lft-identity 45.2%

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

      7. times-frac 45.2%

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

      8. hypot-1-def 45.2%

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

      9. hypot-1-def 63.9%

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

   5. Applied egg-rr 63.9%

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

      1. associate-*l/ 64.0%

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

      2. *-lft-identity 64.0%

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

      3. associate-/l/ 92.1%

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

      4. *-rgt-identity 92.1%

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

      5. associate-*r/ 91.9%

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

      6. associate-/l/ 92.0%

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

      7. associate-*l/ 99.4%

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

      8. associate-/l/ 99.4%

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

      9. associate-*r/ 99.5%

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

      10. *-rgt-identity 99.5%

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

      11. associate-/l/ 99.5%

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

      12. *-commutative 99.5%

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

      13. *-commutative 99.5%

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

   7. Simplified 99.5%

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

   8. Taylor expanded in z around inf 50.1%

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

   9. Taylor expanded in z around inf 99.5%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 5 \cdot 10^{+43}:\\ \;\;\;\;\frac{1}{x \cdot \left(y \cdot \mathsf{fma}\left(z, z, 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x \cdot z}}{z \cdot y}\\ \end{array} \]

Alternative 2: 99.6% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 85.0%

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

   1. associate-/r* 85.1%

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

   2. +-commutative 85.1%

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

   3. fma-def 85.1%

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

3. Simplified 85.1%

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

   1. fma-udef 85.1%

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

   2. +-commutative 85.1%

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

   3. associate-/r* 85.0%

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

   4. associate-/r* 78.2%

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

   5. add-sqr-sqrt 78.2%

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

   6. *-un-lft-identity 78.2%

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

   7. times-frac 78.2%

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

   8. hypot-1-def 78.2%

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

   9. hypot-1-def 85.4%

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

5. Applied egg-rr 85.4%

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

   1. associate-*l/ 85.4%

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

   2. *-lft-identity 85.4%

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

   3. associate-/l/ 96.7%

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

   4. *-rgt-identity 96.7%

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

   5. associate-*r/ 96.6%

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

   6. associate-/l/ 96.6%

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

   7. associate-*l/ 99.5%

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

   8. associate-/l/ 99.5%

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

   9. associate-*r/ 99.5%

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

   10. *-rgt-identity 99.5%

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

   11. associate-/l/ 99.5%

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

   12. *-commutative 99.5%

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

   13. *-commutative 99.5%

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

7. Simplified 99.5%

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

   1. expm1-log1p-u 56.6%

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

   2. expm1-udef 41.2%

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

9. Applied egg-rr 41.2%

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

    1. expm1-def 56.6%

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

    2. expm1-log1p 99.5%

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

    3. associate-/r* 99.5%

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

11. Simplified 99.5%

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

12. Final simplification 99.5%

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

Alternative 3: 99.6% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 85.0%

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

   1. associate-/r* 85.1%

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

   2. +-commutative 85.1%

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

   3. fma-def 85.1%

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

3. Simplified 85.1%

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

   1. fma-udef 85.1%

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

   2. +-commutative 85.1%

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

   3. associate-/r* 85.0%

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

   4. associate-/r* 78.2%

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

   5. add-sqr-sqrt 78.2%

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

   6. *-un-lft-identity 78.2%

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

   7. times-frac 78.2%

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

   8. hypot-1-def 78.2%

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

   9. hypot-1-def 85.4%

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

5. Applied egg-rr 85.4%

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

   1. associate-*l/ 85.4%

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

   2. *-lft-identity 85.4%

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

   3. associate-/l/ 96.7%

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

   4. *-rgt-identity 96.7%

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

   5. associate-*r/ 96.6%

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

   6. associate-/l/ 96.6%

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

   7. associate-*l/ 99.5%

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

   8. associate-/l/ 99.5%

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

   9. associate-*r/ 99.5%

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

   10. *-rgt-identity 99.5%

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

   11. associate-/l/ 99.5%

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

   12. *-commutative 99.5%

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

   13. *-commutative 99.5%

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

7. Simplified 99.5%

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

8. Final simplification 99.5%

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

Alternative 4: 99.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 400:\\ \;\;\;\;\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x \cdot z}}{z \cdot y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= (* z z) 400.0)
   (/ (/ 1.0 x) (* y (+ 1.0 (* z z))))
   (/ (/ 1.0 (* x z)) (* z y))))

C

assert(x < y);
double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 400.0) {
		tmp = (1.0 / x) / (y * (1.0 + (z * z)));
	} else {
		tmp = (1.0 / (x * z)) / (z * y);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((z * z) <= 400.0d0) then
        tmp = (1.0d0 / x) / (y * (1.0d0 + (z * z)))
    else
        tmp = (1.0d0 / (x * z)) / (z * y)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 400.0) {
		tmp = (1.0 / x) / (y * (1.0 + (z * z)));
	} else {
		tmp = (1.0 / (x * z)) / (z * y);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z):
	tmp = 0
	if (z * z) <= 400.0:
		tmp = (1.0 / x) / (y * (1.0 + (z * z)))
	else:
		tmp = (1.0 / (x * z)) / (z * y)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y, z)
	tmp = 0.0
	if (Float64(z * z) <= 400.0)
		tmp = Float64(Float64(1.0 / x) / Float64(y * Float64(1.0 + Float64(z * z))));
	else
		tmp = Float64(Float64(1.0 / Float64(x * z)) / Float64(z * y));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z * z) <= 400.0)
		tmp = (1.0 / x) / (y * (1.0 + (z * z)));
	else
		tmp = (1.0 / (x * z)) / (z * y);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_, z_] := If[LessEqual[N[(z * z), $MachinePrecision], 400.0], N[(N[(1.0 / x), $MachinePrecision] / N[(y * N[(1.0 + N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[(x * z), $MachinePrecision]), $MachinePrecision] / N[(z * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 400

   1. Initial program 99.6%

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

   if 400 < (*.f64 z z)

   1. Initial program 63.8%

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

      1. associate-/r* 63.8%

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

      2. +-commutative 63.8%

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

      3. fma-def 63.8%

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

   3. Simplified 63.8%

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

      1. fma-udef 63.8%

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

      2. +-commutative 63.8%

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

      3. associate-/r* 63.8%

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

      4. associate-/r* 47.0%

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

      5. add-sqr-sqrt 47.0%

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

      6. *-un-lft-identity 47.0%

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

      7. times-frac 46.9%

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

      8. hypot-1-def 46.9%

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

      9. hypot-1-def 64.8%

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

   5. Applied egg-rr 64.8%

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

      1. associate-*l/ 64.8%

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

      2. *-lft-identity 64.8%

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

      3. associate-/l/ 92.4%

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

      4. *-rgt-identity 92.4%

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

      5. associate-*r/ 92.3%

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

      6. associate-/l/ 92.4%

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

      7. associate-*l/ 99.5%

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

      8. associate-/l/ 99.4%

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

      9. associate-*r/ 99.5%

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

      10. *-rgt-identity 99.5%

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

      11. associate-/l/ 99.5%

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

      12. *-commutative 99.5%

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

      13. *-commutative 99.5%

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

   7. Simplified 99.5%

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

   8. Taylor expanded in z around inf 50.6%

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

   9. Taylor expanded in z around inf 99.5%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 400:\\ \;\;\;\;\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x \cdot z}}{z \cdot y}\\ \end{array} \]

Alternative 5: 84.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{-11}:\\ \;\;\;\;\frac{1}{x \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y \cdot \left(x \cdot \left(z \cdot z\right)\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= (* z z) 2e-11) (/ 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) <= 2e-11) {
		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) <= 2d-11) 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) <= 2e-11) {
		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) <= 2e-11:
		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) <= 2e-11)
		tmp = Float64(1.0 / Float64(x * y));
	else
		tmp = Float64(1.0 / Float64(y * Float64(x * Float64(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) <= 2e-11)
		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], 2e-11], N[(1.0 / N[(x * y), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(y * N[(x * N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 1.99999999999999988e-11

   1. Initial program 99.6%

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

      1. associate-/r* 99.7%

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

      2. +-commutative 99.7%

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

      3. fma-def 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in z around 0 99.5%

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

   if 1.99999999999999988e-11 < (*.f64 z z)

   1. Initial program 64.1%

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

      1. fma-udef 64.2%

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

      2. +-commutative 64.2%

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

      3. associate-/r* 64.1%

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

      4. associate-/r* 47.5%

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

      5. add-sqr-sqrt 47.5%

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

      6. *-un-lft-identity 47.5%

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

      7. times-frac 47.4%

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

      8. hypot-1-def 47.4%

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

      9. hypot-1-def 65.1%

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

   5. Applied egg-rr 65.1%

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

      1. associate-*l/ 65.1%

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

      2. *-lft-identity 65.1%

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

      3. associate-/l/ 92.5%

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

      4. *-rgt-identity 92.5%

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

      5. associate-*r/ 92.3%

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

      6. associate-/l/ 92.4%

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

      7. associate-*l/ 99.5%

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

      8. associate-/l/ 99.4%

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

      9. associate-*r/ 99.5%

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

      10. *-rgt-identity 99.5%

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

      11. associate-/l/ 99.5%

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

      12. *-commutative 99.5%

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

      13. *-commutative 99.5%

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

   7. Simplified 99.5%

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

   8. Taylor expanded in z around inf 63.5%

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

      1. unpow2 63.5%

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

      2. *-commutative 63.5%

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

   10. Simplified 63.5%

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

4. Final simplification 84.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{-11}:\\ \;\;\;\;\frac{1}{x \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y \cdot \left(x \cdot \left(z \cdot z\right)\right)}\\ \end{array} \]

Alternative 6: 93.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{-11}:\\ \;\;\;\;\frac{1}{x \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y \cdot \left(z \cdot \left(x \cdot z\right)\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= (* z z) 2e-11) (/ 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) <= 2e-11) {
		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) <= 2d-11) 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) <= 2e-11) {
		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) <= 2e-11:
		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) <= 2e-11)
		tmp = Float64(1.0 / Float64(x * y));
	else
		tmp = Float64(1.0 / Float64(y * Float64(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) <= 2e-11)
		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], 2e-11], N[(1.0 / N[(x * y), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(y * N[(z * N[(x * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 1.99999999999999988e-11

   1. Initial program 99.6%

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

      1. associate-/r* 99.7%

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

      2. +-commutative 99.7%

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

      3. fma-def 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in z around 0 99.5%

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

   if 1.99999999999999988e-11 < (*.f64 z z)

   1. Initial program 64.1%

      \[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)} \]
   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 63.5%

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

      1. unpow2 63.5%

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

      2. *-commutative 63.5%

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

      3. associate-*r* 86.2%

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

   6. Simplified 86.2%

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

4. Final simplification 94.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{-11}:\\ \;\;\;\;\frac{1}{x \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y \cdot \left(z \cdot \left(x \cdot z\right)\right)}\\ \end{array} \]

Alternative 7: 93.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{-11}:\\ \;\;\;\;\frac{1}{x \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{y}}{z \cdot \left(x \cdot z\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= (* z z) 2e-11) (/ 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) <= 2e-11) {
		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) <= 2d-11) 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) <= 2e-11) {
		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) <= 2e-11:
		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) <= 2e-11)
		tmp = Float64(1.0 / Float64(x * y));
	else
		tmp = Float64(Float64(1.0 / y) / Float64(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) <= 2e-11)
		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], 2e-11], N[(1.0 / N[(x * y), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / y), $MachinePrecision] / N[(z * N[(x * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 1.99999999999999988e-11

   1. Initial program 99.6%

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

      1. associate-/r* 99.7%

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

      2. +-commutative 99.7%

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

      3. fma-def 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in z around 0 99.5%

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

   if 1.99999999999999988e-11 < (*.f64 z z)

   1. Initial program 64.1%

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

      1. fma-udef 64.2%

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

      2. +-commutative 64.2%

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

      3. associate-/r* 64.1%

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

      4. associate-/r* 47.5%

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

      5. add-sqr-sqrt 47.5%

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

      6. *-un-lft-identity 47.5%

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

      7. times-frac 47.4%

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

      8. hypot-1-def 47.4%

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

      9. hypot-1-def 65.1%

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

   5. Applied egg-rr 65.1%

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

      1. associate-*l/ 65.1%

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

      2. *-lft-identity 65.1%

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

      3. associate-/l/ 92.5%

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

      4. *-rgt-identity 92.5%

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

      5. associate-*r/ 92.3%

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

      6. associate-/l/ 92.4%

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

      7. associate-*l/ 99.5%

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

      8. associate-/l/ 99.4%

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

      9. associate-*r/ 99.5%

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

      10. *-rgt-identity 99.5%

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

      11. associate-/l/ 99.5%

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

      12. *-commutative 99.5%

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

      13. *-commutative 99.5%

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

   7. Simplified 99.5%

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

      1. expm1-log1p-u 77.7%

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

      2. expm1-udef 43.1%

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

   9. Applied egg-rr 43.1%

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

       1. expm1-def 77.7%

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

       2. expm1-log1p 99.5%

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

       3. associate-/r* 99.5%

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

   11. Simplified 99.5%

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

   12. Taylor expanded in z around inf 63.5%

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

       1. associate-/r* 63.5%

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

       2. unpow2 63.5%

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

       3. associate-*l* 86.4%

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

   14. Simplified 86.4%

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

4. Final simplification 94.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{-11}:\\ \;\;\;\;\frac{1}{x \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{y}}{z \cdot \left(x \cdot z\right)}\\ \end{array} \]

Alternative 8: 98.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{-11}:\\ \;\;\;\;\frac{1}{x \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x \cdot z}}{z \cdot y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= (* z z) 2e-11) (/ 1.0 (* x y)) (/ (/ 1.0 (* x z)) (* z y))))

C

assert(x < y);
double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 2e-11) {
		tmp = 1.0 / (x * y);
	} else {
		tmp = (1.0 / (x * z)) / (z * y);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((z * z) <= 2d-11) then
        tmp = 1.0d0 / (x * y)
    else
        tmp = (1.0d0 / (x * z)) / (z * y)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 2e-11) {
		tmp = 1.0 / (x * y);
	} else {
		tmp = (1.0 / (x * z)) / (z * y);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z):
	tmp = 0
	if (z * z) <= 2e-11:
		tmp = 1.0 / (x * y)
	else:
		tmp = (1.0 / (x * z)) / (z * y)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y, z)
	tmp = 0.0
	if (Float64(z * z) <= 2e-11)
		tmp = Float64(1.0 / Float64(x * y));
	else
		tmp = Float64(Float64(1.0 / Float64(x * z)) / Float64(z * y));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z * z) <= 2e-11)
		tmp = 1.0 / (x * y);
	else
		tmp = (1.0 / (x * z)) / (z * y);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_, z_] := If[LessEqual[N[(z * z), $MachinePrecision], 2e-11], N[(1.0 / N[(x * y), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[(x * z), $MachinePrecision]), $MachinePrecision] / N[(z * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 1.99999999999999988e-11

   1. Initial program 99.6%

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

      1. associate-/r* 99.7%

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

      2. +-commutative 99.7%

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

      3. fma-def 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in z around 0 99.5%

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

   if 1.99999999999999988e-11 < (*.f64 z z)

   1. Initial program 64.1%

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

      1. fma-udef 64.2%

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

      2. +-commutative 64.2%

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

      3. associate-/r* 64.1%

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

      4. associate-/r* 47.5%

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

      5. add-sqr-sqrt 47.5%

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

      6. *-un-lft-identity 47.5%

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

      7. times-frac 47.4%

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

      8. hypot-1-def 47.4%

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

      9. hypot-1-def 65.1%

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

   5. Applied egg-rr 65.1%

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

      1. associate-*l/ 65.1%

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

      2. *-lft-identity 65.1%

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

      3. associate-/l/ 92.5%

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

      4. *-rgt-identity 92.5%

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

      5. associate-*r/ 92.3%

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

      6. associate-/l/ 92.4%

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

      7. associate-*l/ 99.5%

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

      8. associate-/l/ 99.4%

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

      9. associate-*r/ 99.5%

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

      10. *-rgt-identity 99.5%

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

      11. associate-/l/ 99.5%

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

      12. *-commutative 99.5%

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

      13. *-commutative 99.5%

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

   7. Simplified 99.5%

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

   8. Taylor expanded in z around inf 50.4%

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

   9. Taylor expanded in z around inf 98.9%

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

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{-11}:\\ \;\;\;\;\frac{1}{x \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x \cdot z}}{z \cdot y}\\ \end{array} \]

Alternative 9: 66.8% accurate, 2.2× speedup

Math

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

FPCore

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

C

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

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = 1.0d0 / (x * y)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 85.0%

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

   1. associate-/r* 85.1%

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

   2. +-commutative 85.1%

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

   3. fma-def 85.1%

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

3. Simplified 85.1%

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

4. Taylor expanded in z around 0 63.3%

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

5. Final simplification 63.3%

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

Developer target: 85.5% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2023278 
(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)))))