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

Percentage Accurate: 91.0% → 98.1%

Time: 10.9s

Alternatives: 16

Speedup: 1.0×

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: 98.1% accurate, 0.1× speedup

Math

\[\begin{array}{l} z = |z|\\ [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 10^{+112}:\\ \;\;\;\;\frac{\frac{\frac{-1}{x}}{-1 - z \cdot z}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{{z}^{-1}}{x}}{y}}{\mathsf{hypot}\left(1, z\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: z should be positive before calling this function
NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= (* z z) 1e+112)
   (/ (/ (/ -1.0 x) (- -1.0 (* z z))) y)
   (/ (/ (/ (pow z -1.0) x) y) (hypot 1.0 z))))

C

z = abs(z);
assert(x < y);
double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 1e+112) {
		tmp = ((-1.0 / x) / (-1.0 - (z * z))) / y;
	} else {
		tmp = ((pow(z, -1.0) / x) / y) / hypot(1.0, z);
	}
	return tmp;
}

Java

z = Math.abs(z);
assert x < y;
public static double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 1e+112) {
		tmp = ((-1.0 / x) / (-1.0 - (z * z))) / y;
	} else {
		tmp = ((Math.pow(z, -1.0) / x) / y) / Math.hypot(1.0, z);
	}
	return tmp;
}

Python

z = abs(z)
[x, y] = sort([x, y])
def code(x, y, z):
	tmp = 0
	if (z * z) <= 1e+112:
		tmp = ((-1.0 / x) / (-1.0 - (z * z))) / y
	else:
		tmp = ((math.pow(z, -1.0) / x) / y) / math.hypot(1.0, z)
	return tmp

Julia

z = abs(z)
x, y = sort([x, y])
function code(x, y, z)
	tmp = 0.0
	if (Float64(z * z) <= 1e+112)
		tmp = Float64(Float64(Float64(-1.0 / x) / Float64(-1.0 - Float64(z * z))) / y);
	else
		tmp = Float64(Float64(Float64((z ^ -1.0) / x) / y) / hypot(1.0, z));
	end
	return tmp
end

MATLAB

z = abs(z)
x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z * z) <= 1e+112)
		tmp = ((-1.0 / x) / (-1.0 - (z * z))) / y;
	else
		tmp = (((z ^ -1.0) / x) / y) / hypot(1.0, z);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: z should be positive before calling this function
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+112], N[(N[(N[(-1.0 / x), $MachinePrecision] / N[(-1.0 - N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision], N[(N[(N[(N[Power[z, -1.0], $MachinePrecision] / x), $MachinePrecision] / y), $MachinePrecision] / N[Sqrt[1.0 ^ 2 + z ^ 2], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 9.9999999999999993e111

   1. Initial program 97.9%

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

      1. associate-/r* 97.5%

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

      2. +-commutative 97.5%

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

      3. fma-def 97.5%

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

   3. Simplified 97.5%

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

      1. fma-udef 97.5%

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

      2. +-commutative 97.5%

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

      3. associate-/r* 97.9%

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

      4. associate-/r* 99.7%

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

      5. add-sqr-sqrt 99.7%

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

      6. *-un-lft-identity 99.7%

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

      7. times-frac 99.7%

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

      8. hypot-1-def 99.7%

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

      9. associate-/l/ 99.3%

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

      10. hypot-1-def 99.3%

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

   5. Applied egg-rr 99.3%

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

      1. associate-*l/ 99.3%

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

      2. *-lft-identity 99.3%

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

      3. associate-/r* 99.6%

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

      4. associate-/l/ 99.0%

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

      5. *-commutative 99.0%

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

   7. Simplified 99.0%

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

   8. Taylor expanded in y around 0 98.1%

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

      1. *-commutative 98.1%

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

      2. unpow2 98.1%

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

      3. fma-udef 98.1%

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

      4. associate-*l* 98.8%

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

      5. associate-/l/ 99.3%

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

      6. *-rgt-identity 99.3%

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

      7. times-frac 97.7%

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

      8. *-commutative 97.7%

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

      9. associate-*r/ 99.0%

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

      10. associate-*r/ 99.0%

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

      11. *-rgt-identity 99.0%

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

   10. Simplified 99.0%

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

       1. frac-2neg 99.0%

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

       2. div-inv 99.0%

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

       3. distribute-neg-frac 99.0%

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

       4. metadata-eval 99.0%

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

   12. Applied egg-rr 99.0%

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

       1. associate-*r/ 99.0%

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

       2. *-rgt-identity 99.0%

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

       3. neg-sub0 99.0%

          \[\leadsto \frac{\frac{\frac{-1}{x}}{\color{blue}{0 - \mathsf{fma}\left(z, z, 1\right)}}}{y} \]

       4. fma-udef 99.0%

          \[\leadsto \frac{\frac{\frac{-1}{x}}{0 - \color{blue}{\left(z \cdot z + 1\right)}}}{y} \]

       5. unpow2 99.0%

          \[\leadsto \frac{\frac{\frac{-1}{x}}{0 - \left(\color{blue}{{z}^{2}} + 1\right)}}{y} \]

       6. +-commutative 99.0%

          \[\leadsto \frac{\frac{\frac{-1}{x}}{0 - \color{blue}{\left(1 + {z}^{2}\right)}}}{y} \]

       7. associate--r+ 99.0%

          \[\leadsto \frac{\frac{\frac{-1}{x}}{\color{blue}{\left(0 - 1\right) - {z}^{2}}}}{y} \]

       8. metadata-eval 99.0%

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

       9. unpow2 99.0%

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

   14. Simplified 99.0%

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

   if 9.9999999999999993e111 < (*.f64 z z)

   1. Initial program 70.7%

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

      1. associate-/r* 70.7%

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

      2. +-commutative 70.7%

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

      3. fma-def 70.7%

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

   3. Simplified 70.7%

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

      1. associate-/r* 70.7%

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

      2. *-un-lft-identity 70.7%

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

      3. fma-udef 70.7%

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

      4. +-commutative 70.7%

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

      5. *-commutative 70.7%

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

      6. times-frac 72.6%

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

      7. +-commutative 72.6%

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

      8. fma-udef 72.6%

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

      9. associate-/l/ 72.6%

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

   5. Applied egg-rr 72.6%

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

      1. associate-*l/ 72.6%

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

      2. frac-2neg 72.6%

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

      3. metadata-eval 72.6%

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

      4. *-commutative 72.6%

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

      5. distribute-rgt-neg-out 72.6%

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

      6. fma-udef 72.6%

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

      7. +-commutative 72.6%

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

      8. add-sqr-sqrt 72.6%

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

      9. hypot-1-def 72.6%

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

      10. hypot-1-def 72.6%

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

      11. times-frac 86.4%

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

      12. metadata-eval 86.4%

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

      13. distribute-rgt-neg-out 86.4%

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

      14. *-commutative 86.4%

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

      15. frac-2neg 86.4%

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

      16. associate-/r* 86.4%

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

      17. associate-/r* 95.8%

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

      18. times-frac 86.0%

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

      19. *-un-lft-identity 86.0%

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

      20. associate-/l/ 95.8%

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

   7. Applied egg-rr 86.4%

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

   8. Taylor expanded in z around inf 72.4%

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

      1. *-commutative 72.4%

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

      2. *-commutative 72.4%

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

      3. associate-/r* 72.4%

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

      4. *-commutative 72.4%

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

      5. associate-/r* 72.4%

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

      6. unpow-1 72.4%

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

   10. Simplified 72.4%

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

4. Final simplification 87.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 10^{+112}:\\ \;\;\;\;\frac{\frac{\frac{-1}{x}}{-1 - z \cdot z}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{{z}^{-1}}{x}}{y}}{\mathsf{hypot}\left(1, z\right)}\\ \end{array} \]

Alternative 2: 98.0% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

NOTE: z should be positive before calling this function
NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_, z_] := 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. Initial program 85.7%

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

   1. associate-/r* 85.5%

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

   2. +-commutative 85.5%

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

   3. fma-def 85.5%

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

3. Simplified 85.5%

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

   1. fma-udef 85.5%

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

   2. +-commutative 85.5%

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

   3. associate-/r* 85.7%

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

   4. associate-/r* 87.5%

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

   5. add-sqr-sqrt 87.5%

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

   6. *-un-lft-identity 87.5%

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

   7. times-frac 87.5%

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

   8. hypot-1-def 87.5%

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

   9. associate-/l/ 87.3%

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

   10. hypot-1-def 93.5%

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

5. Applied egg-rr 93.5%

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

   1. associate-*l/ 93.5%

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

   2. *-lft-identity 93.5%

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

   3. associate-/r* 93.7%

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

   4. associate-/l/ 97.5%

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

   5. *-commutative 97.5%

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

7. Simplified 97.5%

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

8. Final simplification 97.5%

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

Alternative 3: 97.6% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 85.7%

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

   1. associate-/r* 85.5%

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

   2. +-commutative 85.5%

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

   3. fma-def 85.5%

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

3. Simplified 85.5%

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

   1. associate-/r* 85.7%

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

   2. *-un-lft-identity 85.7%

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

   3. fma-udef 85.7%

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

   4. +-commutative 85.7%

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

   5. *-commutative 85.7%

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

   6. times-frac 87.5%

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

   7. +-commutative 87.5%

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

   8. fma-udef 87.5%

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

   9. associate-/l/ 87.3%

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

5. Applied egg-rr 87.3%

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

   1. associate-*l/ 87.3%

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

   2. frac-2neg 87.3%

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

   3. metadata-eval 87.3%

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

   4. *-commutative 87.3%

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

   5. distribute-rgt-neg-out 87.3%

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

   6. fma-udef 87.3%

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

   7. +-commutative 87.3%

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

   8. add-sqr-sqrt 87.3%

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

   9. hypot-1-def 87.3%

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

   10. hypot-1-def 87.3%

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

   11. times-frac 93.5%

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

   12. metadata-eval 93.5%

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

   13. distribute-rgt-neg-out 93.5%

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

   14. *-commutative 93.5%

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

   15. frac-2neg 93.5%

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

   16. associate-/r* 93.7%

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

   17. associate-/r* 97.5%

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

   18. times-frac 93.2%

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

   19. *-un-lft-identity 93.2%

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

   20. associate-/l/ 97.5%

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

7. Applied egg-rr 97.1%

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

8. Final simplification 97.1%

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

Alternative 4: 98.0% accurate, 0.1× speedup

Math

\[\begin{array}{l} z = |z|\\ [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 10^{+121}:\\ \;\;\;\;\frac{\frac{\frac{-1}{x}}{-1 - z \cdot z}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{z} \cdot \frac{{y}^{-1}}{x \cdot z}\\ \end{array} \end{array} \]

FPCore

NOTE: z should be positive before calling this function
NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= (* z z) 1e+121)
   (/ (/ (/ -1.0 x) (- -1.0 (* z z))) y)
   (* (/ 1.0 z) (/ (pow y -1.0) (* x z)))))

C

z = abs(z);
assert(x < y);
double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 1e+121) {
		tmp = ((-1.0 / x) / (-1.0 - (z * z))) / y;
	} else {
		tmp = (1.0 / z) * (pow(y, -1.0) / (x * z));
	}
	return tmp;
}

Fortran

NOTE: z should be positive before calling this function
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+121) then
        tmp = (((-1.0d0) / x) / ((-1.0d0) - (z * z))) / y
    else
        tmp = (1.0d0 / z) * ((y ** (-1.0d0)) / (x * z))
    end if
    code = tmp
end function

Java

z = Math.abs(z);
assert x < y;
public static double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 1e+121) {
		tmp = ((-1.0 / x) / (-1.0 - (z * z))) / y;
	} else {
		tmp = (1.0 / z) * (Math.pow(y, -1.0) / (x * z));
	}
	return tmp;
}

Python

z = abs(z)
[x, y] = sort([x, y])
def code(x, y, z):
	tmp = 0
	if (z * z) <= 1e+121:
		tmp = ((-1.0 / x) / (-1.0 - (z * z))) / y
	else:
		tmp = (1.0 / z) * (math.pow(y, -1.0) / (x * z))
	return tmp

Julia

z = abs(z)
x, y = sort([x, y])
function code(x, y, z)
	tmp = 0.0
	if (Float64(z * z) <= 1e+121)
		tmp = Float64(Float64(Float64(-1.0 / x) / Float64(-1.0 - Float64(z * z))) / y);
	else
		tmp = Float64(Float64(1.0 / z) * Float64((y ^ -1.0) / Float64(x * z)));
	end
	return tmp
end

MATLAB

z = abs(z)
x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z * z) <= 1e+121)
		tmp = ((-1.0 / x) / (-1.0 - (z * z))) / y;
	else
		tmp = (1.0 / z) * ((y ^ -1.0) / (x * z));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: z should be positive before calling this function
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+121], N[(N[(N[(-1.0 / x), $MachinePrecision] / N[(-1.0 - N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision], N[(N[(1.0 / z), $MachinePrecision] * N[(N[Power[y, -1.0], $MachinePrecision] / N[(x * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 1.00000000000000004e121

   1. Initial program 97.9%

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

      1. associate-/r* 97.6%

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

      2. +-commutative 97.6%

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

      3. fma-def 97.6%

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

   3. Simplified 97.6%

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

      1. fma-udef 97.6%

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

      2. +-commutative 97.6%

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

      3. associate-/r* 97.9%

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

      4. associate-/r* 99.0%

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

      5. add-sqr-sqrt 99.0%

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

      6. *-un-lft-identity 99.0%

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

      7. times-frac 99.0%

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

      8. hypot-1-def 99.0%

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

      9. associate-/l/ 98.6%

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

      10. hypot-1-def 98.6%

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

   5. Applied egg-rr 98.6%

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

      1. associate-*l/ 98.6%

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

      2. *-lft-identity 98.6%

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

      3. associate-/r* 99.0%

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

      4. associate-/l/ 98.4%

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

      5. *-commutative 98.4%

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

   7. Simplified 98.4%

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

   8. Taylor expanded in y around 0 97.6%

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

      1. *-commutative 97.6%

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

      2. unpow2 97.6%

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

      3. fma-udef 97.6%

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

      4. associate-*l* 98.2%

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

      5. associate-/l/ 98.7%

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

      6. *-rgt-identity 98.7%

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

      7. times-frac 97.8%

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

      8. *-commutative 97.8%

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

      9. associate-*r/ 98.4%

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

      10. associate-*r/ 98.4%

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

      11. *-rgt-identity 98.4%

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

   10. Simplified 98.4%

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

       1. frac-2neg 98.4%

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

       2. div-inv 98.4%

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

       3. distribute-neg-frac 98.4%

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

       4. metadata-eval 98.4%

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

   12. Applied egg-rr 98.4%

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

       1. associate-*r/ 98.4%

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

       2. *-rgt-identity 98.4%

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

       3. neg-sub0 98.4%

          \[\leadsto \frac{\frac{\frac{-1}{x}}{\color{blue}{0 - \mathsf{fma}\left(z, z, 1\right)}}}{y} \]

       4. fma-udef 98.4%

          \[\leadsto \frac{\frac{\frac{-1}{x}}{0 - \color{blue}{\left(z \cdot z + 1\right)}}}{y} \]

       5. unpow2 98.4%

          \[\leadsto \frac{\frac{\frac{-1}{x}}{0 - \left(\color{blue}{{z}^{2}} + 1\right)}}{y} \]

       6. +-commutative 98.4%

          \[\leadsto \frac{\frac{\frac{-1}{x}}{0 - \color{blue}{\left(1 + {z}^{2}\right)}}}{y} \]

       7. associate--r+ 98.4%

          \[\leadsto \frac{\frac{\frac{-1}{x}}{\color{blue}{\left(0 - 1\right) - {z}^{2}}}}{y} \]

       8. metadata-eval 98.4%

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

       9. unpow2 98.4%

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

   14. Simplified 98.4%

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

   if 1.00000000000000004e121 < (*.f64 z z)

   1. Initial program 69.4%

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

      1. associate-/r* 69.4%

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

      2. +-commutative 69.4%

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

      3. fma-def 69.4%

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

   3. Simplified 69.4%

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

      1. fma-udef 69.4%

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

      2. +-commutative 69.4%

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

      3. associate-/r* 69.4%

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

      4. associate-/r* 72.2%

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

      5. add-sqr-sqrt 72.2%

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

      6. *-un-lft-identity 72.2%

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

      7. times-frac 72.2%

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

      8. hypot-1-def 72.2%

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

      9. associate-/l/ 72.2%

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

      10. hypot-1-def 86.6%

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

   5. Applied egg-rr 86.6%

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

      1. associate-*l/ 86.7%

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

      2. *-lft-identity 86.7%

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

      3. associate-/r* 86.6%

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

      4. associate-/l/ 96.4%

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

      5. *-commutative 96.4%

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

   7. Simplified 96.4%

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

   8. Taylor expanded in z around inf 76.0%

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

      1. unpow2 76.0%

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

   10. Simplified 76.0%

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

       1. associate-/r* 76.0%

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

       2. *-un-lft-identity 76.0%

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

       3. associate-*l* 86.3%

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

       4. times-frac 96.4%

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

       5. inv-pow 96.4%

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

       6. *-commutative 96.4%

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

   12. Applied egg-rr 96.4%

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

4. Final simplification 97.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 10^{+121}:\\ \;\;\;\;\frac{\frac{\frac{-1}{x}}{-1 - z \cdot z}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{z} \cdot \frac{{y}^{-1}}{x \cdot z}\\ \end{array} \]

Alternative 5: 97.8% accurate, 0.1× speedup

Math

\[\begin{array}{l} z = |z|\\ [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 5 \cdot 10^{+94}:\\ \;\;\;\;\frac{\frac{\frac{-1}{x}}{-1 - z \cdot z}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(1, z\right) \cdot \left(y \cdot \left(x \cdot z\right)\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: z should be positive before calling this function
NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= (* z z) 5e+94)
   (/ (/ (/ -1.0 x) (- -1.0 (* z z))) y)
   (/ 1.0 (* (hypot 1.0 z) (* y (* x z))))))

C

z = abs(z);
assert(x < y);
double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 5e+94) {
		tmp = ((-1.0 / x) / (-1.0 - (z * z))) / y;
	} else {
		tmp = 1.0 / (hypot(1.0, z) * (y * (x * z)));
	}
	return tmp;
}

Java

z = Math.abs(z);
assert x < y;
public static double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 5e+94) {
		tmp = ((-1.0 / x) / (-1.0 - (z * z))) / y;
	} else {
		tmp = 1.0 / (Math.hypot(1.0, z) * (y * (x * z)));
	}
	return tmp;
}

Python

z = abs(z)
[x, y] = sort([x, y])
def code(x, y, z):
	tmp = 0
	if (z * z) <= 5e+94:
		tmp = ((-1.0 / x) / (-1.0 - (z * z))) / y
	else:
		tmp = 1.0 / (math.hypot(1.0, z) * (y * (x * z)))
	return tmp

Julia

z = abs(z)
x, y = sort([x, y])
function code(x, y, z)
	tmp = 0.0
	if (Float64(z * z) <= 5e+94)
		tmp = Float64(Float64(Float64(-1.0 / x) / Float64(-1.0 - Float64(z * z))) / y);
	else
		tmp = Float64(1.0 / Float64(hypot(1.0, z) * Float64(y * Float64(x * z))));
	end
	return tmp
end

MATLAB

z = abs(z)
x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z * z) <= 5e+94)
		tmp = ((-1.0 / x) / (-1.0 - (z * z))) / y;
	else
		tmp = 1.0 / (hypot(1.0, z) * (y * (x * z)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: z should be positive before calling this function
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+94], N[(N[(N[(-1.0 / x), $MachinePrecision] / N[(-1.0 - N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision], N[(1.0 / N[(N[Sqrt[1.0 ^ 2 + z ^ 2], $MachinePrecision] * N[(y * N[(x * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 5.0000000000000001e94

   1. Initial program 98.5%

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

      1. associate-/r* 98.1%

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

      2. +-commutative 98.1%

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

      3. fma-def 98.1%

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

   3. Simplified 98.1%

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

      1. fma-udef 98.1%

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

      2. +-commutative 98.1%

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

      3. associate-/r* 98.5%

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

      4. associate-/r* 99.7%

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

      5. add-sqr-sqrt 99.7%

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

      6. *-un-lft-identity 99.7%

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

      7. times-frac 99.7%

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

      8. hypot-1-def 99.7%

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

      9. associate-/l/ 99.2%

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

      10. hypot-1-def 99.2%

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

   5. Applied egg-rr 99.2%

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

      1. associate-*l/ 99.2%

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

      2. *-lft-identity 99.2%

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

      3. associate-/r* 99.6%

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

      4. associate-/l/ 99.0%

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

      5. *-commutative 99.0%

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

   7. Simplified 99.0%

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

   8. Taylor expanded in y around 0 98.1%

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

      1. *-commutative 98.1%

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

      2. unpow2 98.1%

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

      3. fma-udef 98.1%

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

      4. associate-*l* 98.8%

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

      5. associate-/l/ 99.3%

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

      6. *-rgt-identity 99.3%

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

      7. times-frac 98.4%

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

      8. *-commutative 98.4%

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

      9. associate-*r/ 99.0%

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

      10. associate-*r/ 99.0%

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

      11. *-rgt-identity 99.0%

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

   10. Simplified 99.0%

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

       1. frac-2neg 99.0%

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

       2. div-inv 99.0%

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

       3. distribute-neg-frac 99.0%

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

       4. metadata-eval 99.0%

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

   12. Applied egg-rr 99.0%

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

       1. associate-*r/ 99.0%

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

       2. *-rgt-identity 99.0%

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

       3. neg-sub0 99.0%

          \[\leadsto \frac{\frac{\frac{-1}{x}}{\color{blue}{0 - \mathsf{fma}\left(z, z, 1\right)}}}{y} \]

       4. fma-udef 99.0%

          \[\leadsto \frac{\frac{\frac{-1}{x}}{0 - \color{blue}{\left(z \cdot z + 1\right)}}}{y} \]

       5. unpow2 99.0%

          \[\leadsto \frac{\frac{\frac{-1}{x}}{0 - \left(\color{blue}{{z}^{2}} + 1\right)}}{y} \]

       6. +-commutative 99.0%

          \[\leadsto \frac{\frac{\frac{-1}{x}}{0 - \color{blue}{\left(1 + {z}^{2}\right)}}}{y} \]

       7. associate--r+ 99.0%

          \[\leadsto \frac{\frac{\frac{-1}{x}}{\color{blue}{\left(0 - 1\right) - {z}^{2}}}}{y} \]

       8. metadata-eval 99.0%

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

       9. unpow2 99.0%

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

   14. Simplified 99.0%

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

   if 5.0000000000000001e94 < (*.f64 z z)

   1. Initial program 70.6%

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

      1. associate-/r* 70.6%

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

      2. +-commutative 70.6%

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

      3. fma-def 70.6%

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

   3. Simplified 70.6%

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

      1. associate-/r* 70.6%

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

      2. *-un-lft-identity 70.6%

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

      3. fma-udef 70.6%

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

      4. +-commutative 70.6%

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

      5. *-commutative 70.6%

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

      6. times-frac 73.2%

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

      7. +-commutative 73.2%

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

      8. fma-udef 73.2%

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

      9. associate-/l/ 73.3%

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

   5. Applied egg-rr 73.3%

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

      1. associate-*l/ 73.3%

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

      2. frac-2neg 73.3%

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

      3. metadata-eval 73.3%

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

      4. *-commutative 73.3%

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

      5. distribute-rgt-neg-out 73.3%

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

      6. fma-udef 73.3%

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

      7. +-commutative 73.3%

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

      8. add-sqr-sqrt 73.3%

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

      9. hypot-1-def 73.3%

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

      10. hypot-1-def 73.3%

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

      11. times-frac 86.7%

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

      12. metadata-eval 86.7%

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

      13. distribute-rgt-neg-out 86.7%

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

      14. *-commutative 86.7%

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

      15. frac-2neg 86.7%

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

      16. associate-/r* 86.7%

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

      17. associate-/r* 95.9%

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

      18. times-frac 86.4%

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

      19. *-un-lft-identity 86.4%

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

      20. associate-/l/ 95.9%

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

   7. Applied egg-rr 96.0%

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

   8. Taylor expanded in z around inf 73.1%

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

4. Final simplification 87.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 5 \cdot 10^{+94}:\\ \;\;\;\;\frac{\frac{\frac{-1}{x}}{-1 - z \cdot z}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(1, z\right) \cdot \left(y \cdot \left(x \cdot z\right)\right)}\\ \end{array} \]

Alternative 6: 98.2% accurate, 0.1× speedup

Math

\[\begin{array}{l} z = |z|\\ [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{+151}:\\ \;\;\;\;\frac{\frac{\frac{1}{y}}{x}}{1 + z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{{z}^{-1}}{x}}{y \cdot z}\\ \end{array} \end{array} \]

FPCore

NOTE: z should be positive before calling this function
NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= (* z z) 2e+151)
   (/ (/ (/ 1.0 y) x) (+ 1.0 (* z z)))
   (/ (/ (pow z -1.0) x) (* y z))))

C

z = abs(z);
assert(x < y);
double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 2e+151) {
		tmp = ((1.0 / y) / x) / (1.0 + (z * z));
	} else {
		tmp = (pow(z, -1.0) / x) / (y * z);
	}
	return tmp;
}

Fortran

NOTE: z should be positive before calling this function
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+151) then
        tmp = ((1.0d0 / y) / x) / (1.0d0 + (z * z))
    else
        tmp = ((z ** (-1.0d0)) / x) / (y * z)
    end if
    code = tmp
end function

Java

z = Math.abs(z);
assert x < y;
public static double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 2e+151) {
		tmp = ((1.0 / y) / x) / (1.0 + (z * z));
	} else {
		tmp = (Math.pow(z, -1.0) / x) / (y * z);
	}
	return tmp;
}

Python

z = abs(z)
[x, y] = sort([x, y])
def code(x, y, z):
	tmp = 0
	if (z * z) <= 2e+151:
		tmp = ((1.0 / y) / x) / (1.0 + (z * z))
	else:
		tmp = (math.pow(z, -1.0) / x) / (y * z)
	return tmp

Julia

z = abs(z)
x, y = sort([x, y])
function code(x, y, z)
	tmp = 0.0
	if (Float64(z * z) <= 2e+151)
		tmp = Float64(Float64(Float64(1.0 / y) / x) / Float64(1.0 + Float64(z * z)));
	else
		tmp = Float64(Float64((z ^ -1.0) / x) / Float64(y * z));
	end
	return tmp
end

MATLAB

z = abs(z)
x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z * z) <= 2e+151)
		tmp = ((1.0 / y) / x) / (1.0 + (z * z));
	else
		tmp = ((z ^ -1.0) / x) / (y * z);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: z should be positive before calling this function
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+151], N[(N[(N[(1.0 / y), $MachinePrecision] / x), $MachinePrecision] / N[(1.0 + N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Power[z, -1.0], $MachinePrecision] / x), $MachinePrecision] / N[(y * z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 2.00000000000000003e151

   1. Initial program 96.8%

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

      1. associate-/r* 99.0%

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

   3. Simplified 99.0%

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

      1. *-un-lft-identity 99.0%

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

      2. add-sqr-sqrt 49.3%

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

      3. times-frac 49.3%

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

   5. Applied egg-rr 49.3%

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

      1. frac-times 49.3%

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

      2. *-un-lft-identity 49.3%

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

      3. add-sqr-sqrt 99.0%

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

      4. div-inv 98.8%

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

      5. inv-pow 98.8%

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

      6. inv-pow 98.8%

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

      7. unpow-prod-down 98.7%

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

      8. *-commutative 98.7%

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

      9. inv-pow 98.7%

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

      10. associate-/r* 99.0%

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

   7. Applied egg-rr 99.0%

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

   if 2.00000000000000003e151 < (*.f64 z z)

   1. Initial program 69.1%

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

      1. associate-/r* 69.1%

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

      2. +-commutative 69.1%

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

      3. fma-def 69.1%

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

   3. Simplified 69.1%

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

      1. associate-/r* 69.1%

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

      2. *-un-lft-identity 69.1%

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

      3. fma-udef 69.1%

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

      4. +-commutative 69.1%

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

      5. *-commutative 69.1%

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

      6. times-frac 70.4%

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

      7. +-commutative 70.4%

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

      8. fma-udef 70.4%

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

      9. associate-/l/ 70.3%

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

   5. Applied egg-rr 70.3%

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

      1. associate-*l/ 70.4%

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

      2. frac-2neg 70.4%

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

      3. metadata-eval 70.4%

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

      4. *-commutative 70.4%

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

      5. distribute-rgt-neg-out 70.4%

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

      6. fma-udef 70.4%

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

      7. +-commutative 70.4%

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

      8. add-sqr-sqrt 70.4%

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

      9. hypot-1-def 70.4%

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

      10. hypot-1-def 70.4%

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

      11. times-frac 85.8%

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

      12. metadata-eval 85.8%

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

      13. distribute-rgt-neg-out 85.8%

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

      14. *-commutative 85.8%

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

      15. frac-2neg 85.8%

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

      16. associate-/r* 85.8%

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

      17. associate-/r* 96.2%

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

      18. times-frac 85.3%

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

      19. *-un-lft-identity 85.3%

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

      20. associate-/l/ 96.2%

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

   7. Applied egg-rr 85.8%

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

   8. Taylor expanded in z around inf 74.4%

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

      1. *-commutative 74.4%

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

      2. *-commutative 74.4%

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

      3. unpow2 74.4%

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

      4. associate-*r* 85.4%

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

      5. associate-*r* 95.6%

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

      6. associate-/r* 95.6%

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

      7. *-commutative 95.6%

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

      8. associate-/r* 95.5%

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

      9. unpow-1 95.5%

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

   10. Simplified 95.5%

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

4. Final simplification 97.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{+151}:\\ \;\;\;\;\frac{\frac{\frac{1}{y}}{x}}{1 + z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{{z}^{-1}}{x}}{y \cdot z}\\ \end{array} \]

Alternative 7: 97.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} z = |z|\\ [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_0 := y \cdot \left(1 + z \cdot z\right)\\ \mathbf{if}\;t_0 \leq -\infty:\\ \;\;\;\;\frac{1}{x \cdot \left(z \cdot \left(y \cdot z\right)\right)}\\ \mathbf{elif}\;t_0 \leq 2 \cdot 10^{+302}:\\ \;\;\;\;\frac{\frac{1}{x}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y \cdot \left(z \cdot \left(x \cdot z\right)\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: z should be positive before calling this function
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 (* z (* y z))))
     (if (<= t_0 2e+302) (/ (/ 1.0 x) t_0) (/ 1.0 (* y (* z (* x z))))))))

C

z = abs(z);
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 * (z * (y * z)));
	} else if (t_0 <= 2e+302) {
		tmp = (1.0 / x) / t_0;
	} else {
		tmp = 1.0 / (y * (z * (x * z)));
	}
	return tmp;
}

Java

z = Math.abs(z);
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 * (z * (y * z)));
	} else if (t_0 <= 2e+302) {
		tmp = (1.0 / x) / t_0;
	} else {
		tmp = 1.0 / (y * (z * (x * z)));
	}
	return tmp;
}

Python

z = abs(z)
[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 * (z * (y * z)))
	elif t_0 <= 2e+302:
		tmp = (1.0 / x) / t_0
	else:
		tmp = 1.0 / (y * (z * (x * z)))
	return tmp

Julia

z = abs(z)
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 <= Float64(-Inf))
		tmp = Float64(1.0 / Float64(x * Float64(z * Float64(y * z))));
	elseif (t_0 <= 2e+302)
		tmp = Float64(Float64(1.0 / x) / t_0);
	else
		tmp = Float64(1.0 / Float64(y * Float64(z * Float64(x * z))));
	end
	return tmp
end

MATLAB

z = abs(z)
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 * (z * (y * z)));
	elseif (t_0 <= 2e+302)
		tmp = (1.0 / x) / t_0;
	else
		tmp = 1.0 / (y * (z * (x * z)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: z should be positive before calling this function
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[(1.0 / N[(x * N[(z * N[(y * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 2e+302], N[(N[(1.0 / x), $MachinePrecision] / t$95$0), $MachinePrecision], N[(1.0 / N[(y * N[(z * N[(x * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 58.1%

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

      1. associate-/r* 58.1%

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

      2. +-commutative 58.1%

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

      3. fma-def 58.1%

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

   3. Simplified 58.1%

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

   4. Taylor expanded in z around inf 58.1%

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

      1. unpow2 58.1%

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

      2. associate-*r* 83.6%

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

      3. *-commutative 83.6%

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

   6. Simplified 83.6%

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

   if -inf.0 < (*.f64 y (+.f64 1 (*.f64 z z))) < 2.0000000000000002e302

   1. Initial program 99.6%

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

   if 2.0000000000000002e302 < (*.f64 y (+.f64 1 (*.f64 z z)))

   1. Initial program 64.0%

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

      1. associate-/r* 64.0%

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

      2. +-commutative 64.0%

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

      3. fma-def 64.0%

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

   3. Simplified 64.0%

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

      1. associate-/r* 64.0%

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

      2. *-un-lft-identity 64.0%

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

      3. fma-udef 64.0%

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

      4. +-commutative 64.0%

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

      5. *-commutative 64.0%

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

      6. times-frac 81.0%

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

      7. +-commutative 81.0%

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

      8. fma-udef 81.0%

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

      9. associate-/l/ 81.0%

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

   5. Applied egg-rr 81.0%

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

      1. associate-*l/ 81.1%

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

      2. frac-2neg 81.1%

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

      3. metadata-eval 81.1%

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

      4. *-commutative 81.1%

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

      5. distribute-rgt-neg-out 81.1%

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

      6. fma-udef 81.1%

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

      7. +-commutative 81.1%

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

      8. add-sqr-sqrt 81.1%

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

      9. hypot-1-def 81.1%

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

      10. hypot-1-def 81.1%

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

      11. times-frac 90.4%

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

      12. metadata-eval 90.4%

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

      13. distribute-rgt-neg-out 90.4%

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

      14. *-commutative 90.4%

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

      15. frac-2neg 90.4%

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

      16. associate-/r* 90.3%

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

      17. associate-/r* 99.6%

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

      18. times-frac 92.9%

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

      19. *-un-lft-identity 92.9%

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

      20. associate-/l/ 99.7%

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

   7. Applied egg-rr 98.3%

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

   8. Taylor expanded in z around inf 79.9%

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

      1. unpow2 79.9%

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

      2. *-commutative 79.9%

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

      3. associate-*r* 91.4%

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

      4. *-commutative 91.4%

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

   10. Simplified 91.4%

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

4. Final simplification 95.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(1 + z \cdot z\right) \leq -\infty:\\ \;\;\;\;\frac{1}{x \cdot \left(z \cdot \left(y \cdot z\right)\right)}\\ \mathbf{elif}\;y \cdot \left(1 + z \cdot z\right) \leq 2 \cdot 10^{+302}:\\ \;\;\;\;\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y \cdot \left(z \cdot \left(x \cdot z\right)\right)}\\ \end{array} \]

Alternative 8: 97.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} z = |z|\\ [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 5 \cdot 10^{+86}:\\ \;\;\;\;\frac{\frac{\frac{-1}{x}}{-1 - z \cdot z}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(x \cdot z\right) \cdot \left(y \cdot z\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: z should be positive before calling this function
NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= (* z z) 5e+86)
   (/ (/ (/ -1.0 x) (- -1.0 (* z z))) y)
   (/ 1.0 (* (* x z) (* y z)))))

C

z = abs(z);
assert(x < y);
double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 5e+86) {
		tmp = ((-1.0 / x) / (-1.0 - (z * z))) / y;
	} else {
		tmp = 1.0 / ((x * z) * (y * z));
	}
	return tmp;
}

Fortran

NOTE: z should be positive before calling this function
NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((z * z) <= 5d+86) then
        tmp = (((-1.0d0) / x) / ((-1.0d0) - (z * z))) / y
    else
        tmp = 1.0d0 / ((x * z) * (y * z))
    end if
    code = tmp
end function

Java

z = Math.abs(z);
assert x < y;
public static double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 5e+86) {
		tmp = ((-1.0 / x) / (-1.0 - (z * z))) / y;
	} else {
		tmp = 1.0 / ((x * z) * (y * z));
	}
	return tmp;
}

Python

z = abs(z)
[x, y] = sort([x, y])
def code(x, y, z):
	tmp = 0
	if (z * z) <= 5e+86:
		tmp = ((-1.0 / x) / (-1.0 - (z * z))) / y
	else:
		tmp = 1.0 / ((x * z) * (y * z))
	return tmp

Julia

z = abs(z)
x, y = sort([x, y])
function code(x, y, z)
	tmp = 0.0
	if (Float64(z * z) <= 5e+86)
		tmp = Float64(Float64(Float64(-1.0 / x) / Float64(-1.0 - Float64(z * z))) / y);
	else
		tmp = Float64(1.0 / Float64(Float64(x * z) * Float64(y * z)));
	end
	return tmp
end

MATLAB

z = abs(z)
x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z * z) <= 5e+86)
		tmp = ((-1.0 / x) / (-1.0 - (z * z))) / y;
	else
		tmp = 1.0 / ((x * z) * (y * z));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: z should be positive before calling this function
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+86], N[(N[(N[(-1.0 / x), $MachinePrecision] / N[(-1.0 - N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision], N[(1.0 / N[(N[(x * z), $MachinePrecision] * N[(y * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 4.9999999999999998e86

   1. Initial program 98.5%

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

      1. associate-/r* 98.1%

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

      2. +-commutative 98.1%

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

      3. fma-def 98.1%

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

   3. Simplified 98.1%

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

      1. fma-udef 98.1%

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

      2. +-commutative 98.1%

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

      3. associate-/r* 98.5%

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

      4. associate-/r* 99.7%

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

      5. add-sqr-sqrt 99.7%

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

      6. *-un-lft-identity 99.7%

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

      7. times-frac 99.7%

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

      8. hypot-1-def 99.7%

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

      9. associate-/l/ 99.3%

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

      10. hypot-1-def 99.3%

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

   5. Applied egg-rr 99.3%

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

      1. associate-*l/ 99.2%

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

      2. *-lft-identity 99.2%

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

      3. associate-/r* 99.6%

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

      4. associate-/l/ 99.0%

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

      5. *-commutative 99.0%

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

   7. Simplified 99.0%

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

   8. Taylor expanded in y around 0 98.1%

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

      1. *-commutative 98.1%

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

      2. unpow2 98.1%

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

      3. fma-udef 98.1%

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

      4. associate-*l* 98.8%

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

      5. associate-/l/ 99.3%

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

      6. *-rgt-identity 99.3%

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

      7. times-frac 98.4%

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

      8. *-commutative 98.4%

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

      9. associate-*r/ 99.0%

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

      10. associate-*r/ 99.0%

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

      11. *-rgt-identity 99.0%

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

   10. Simplified 99.0%

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

       1. frac-2neg 99.0%

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

       2. div-inv 99.0%

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

       3. distribute-neg-frac 99.0%

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

       4. metadata-eval 99.0%

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

   12. Applied egg-rr 99.0%

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

       1. associate-*r/ 99.0%

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

       2. *-rgt-identity 99.0%

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

       3. neg-sub0 99.0%

          \[\leadsto \frac{\frac{\frac{-1}{x}}{\color{blue}{0 - \mathsf{fma}\left(z, z, 1\right)}}}{y} \]

       4. fma-udef 99.0%

          \[\leadsto \frac{\frac{\frac{-1}{x}}{0 - \color{blue}{\left(z \cdot z + 1\right)}}}{y} \]

       5. unpow2 99.0%

          \[\leadsto \frac{\frac{\frac{-1}{x}}{0 - \left(\color{blue}{{z}^{2}} + 1\right)}}{y} \]

       6. +-commutative 99.0%

          \[\leadsto \frac{\frac{\frac{-1}{x}}{0 - \color{blue}{\left(1 + {z}^{2}\right)}}}{y} \]

       7. associate--r+ 99.0%

          \[\leadsto \frac{\frac{\frac{-1}{x}}{\color{blue}{\left(0 - 1\right) - {z}^{2}}}}{y} \]

       8. metadata-eval 99.0%

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

       9. unpow2 99.0%

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

   14. Simplified 99.0%

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

   if 4.9999999999999998e86 < (*.f64 z z)

   1. Initial program 70.9%

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

      1. associate-/r* 70.9%

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

      2. +-commutative 70.9%

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

      3. fma-def 70.9%

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

   3. Simplified 70.9%

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

      1. associate-/r* 70.9%

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

      2. *-un-lft-identity 70.9%

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

      3. fma-udef 70.9%

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

      4. +-commutative 70.9%

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

      5. *-commutative 70.9%

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

      6. times-frac 73.5%

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

      7. +-commutative 73.5%

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

      8. fma-udef 73.5%

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

      9. associate-/l/ 73.5%

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

   5. Applied egg-rr 73.5%

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

      1. associate-*l/ 73.5%

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

      2. frac-2neg 73.5%

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

      3. metadata-eval 73.5%

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

      4. *-commutative 73.5%

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

      5. distribute-rgt-neg-out 73.5%

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

      6. fma-udef 73.5%

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

      7. +-commutative 73.5%

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

      8. add-sqr-sqrt 73.5%

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

      9. hypot-1-def 73.5%

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

      10. hypot-1-def 73.5%

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

      11. times-frac 86.8%

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

      12. metadata-eval 86.8%

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

      13. distribute-rgt-neg-out 86.8%

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

      14. *-commutative 86.8%

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

      15. frac-2neg 86.8%

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

      16. associate-/r* 86.8%

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

      17. associate-/r* 95.9%

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

      18. times-frac 86.5%

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

      19. *-un-lft-identity 86.5%

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

      20. associate-/l/ 95.9%

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

   7. Applied egg-rr 96.0%

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

   8. Taylor expanded in z around inf 77.0%

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

      1. unpow2 77.0%

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

      2. *-commutative 77.0%

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

      3. associate-*r* 86.5%

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

      4. *-commutative 86.5%

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

   10. Simplified 86.5%

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

       1. /-rgt-identity 86.5%

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

       2. *-commutative 86.5%

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

       3. associate-*l* 94.6%

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

       4. *-commutative 94.6%

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

   12. Applied egg-rr 94.6%

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

4. Final simplification 96.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 5 \cdot 10^{+86}:\\ \;\;\;\;\frac{\frac{\frac{-1}{x}}{-1 - z \cdot z}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(x \cdot z\right) \cdot \left(y \cdot z\right)}\\ \end{array} \]

Alternative 9: 97.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} z = |z|\\ [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{+151}:\\ \;\;\;\;\frac{\frac{\frac{1}{x}}{y}}{1 + z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(x \cdot z\right) \cdot \left(y \cdot z\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: z should be positive before calling this function
NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= (* z z) 2e+151)
   (/ (/ (/ 1.0 x) y) (+ 1.0 (* z z)))
   (/ 1.0 (* (* x z) (* y z)))))

C

z = abs(z);
assert(x < y);
double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 2e+151) {
		tmp = ((1.0 / x) / y) / (1.0 + (z * z));
	} else {
		tmp = 1.0 / ((x * z) * (y * z));
	}
	return tmp;
}

Fortran

NOTE: z should be positive before calling this function
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+151) then
        tmp = ((1.0d0 / x) / y) / (1.0d0 + (z * z))
    else
        tmp = 1.0d0 / ((x * z) * (y * z))
    end if
    code = tmp
end function

Java

z = Math.abs(z);
assert x < y;
public static double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 2e+151) {
		tmp = ((1.0 / x) / y) / (1.0 + (z * z));
	} else {
		tmp = 1.0 / ((x * z) * (y * z));
	}
	return tmp;
}

Python

z = abs(z)
[x, y] = sort([x, y])
def code(x, y, z):
	tmp = 0
	if (z * z) <= 2e+151:
		tmp = ((1.0 / x) / y) / (1.0 + (z * z))
	else:
		tmp = 1.0 / ((x * z) * (y * z))
	return tmp

Julia

z = abs(z)
x, y = sort([x, y])
function code(x, y, z)
	tmp = 0.0
	if (Float64(z * z) <= 2e+151)
		tmp = Float64(Float64(Float64(1.0 / x) / y) / Float64(1.0 + Float64(z * z)));
	else
		tmp = Float64(1.0 / Float64(Float64(x * z) * Float64(y * z)));
	end
	return tmp
end

MATLAB

z = abs(z)
x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z * z) <= 2e+151)
		tmp = ((1.0 / x) / y) / (1.0 + (z * z));
	else
		tmp = 1.0 / ((x * z) * (y * z));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: z should be positive before calling this function
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+151], N[(N[(N[(1.0 / x), $MachinePrecision] / y), $MachinePrecision] / N[(1.0 + N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(x * z), $MachinePrecision] * N[(y * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 2.00000000000000003e151

   1. Initial program 96.8%

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

      1. associate-/r* 99.0%

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

   3. Simplified 99.0%

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

   if 2.00000000000000003e151 < (*.f64 z z)

   1. Initial program 69.1%

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

      1. associate-/r* 69.1%

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

      2. +-commutative 69.1%

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

      3. fma-def 69.1%

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

   3. Simplified 69.1%

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

      1. associate-/r* 69.1%

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

      2. *-un-lft-identity 69.1%

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

      3. fma-udef 69.1%

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

      4. +-commutative 69.1%

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

      5. *-commutative 69.1%

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

      6. times-frac 70.4%

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

      7. +-commutative 70.4%

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

      8. fma-udef 70.4%

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

      9. associate-/l/ 70.3%

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

   5. Applied egg-rr 70.3%

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

      1. associate-*l/ 70.4%

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

      2. frac-2neg 70.4%

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

      3. metadata-eval 70.4%

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

      4. *-commutative 70.4%

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

      5. distribute-rgt-neg-out 70.4%

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

      6. fma-udef 70.4%

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

      7. +-commutative 70.4%

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

      8. add-sqr-sqrt 70.4%

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

      9. hypot-1-def 70.4%

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

      10. hypot-1-def 70.4%

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

      11. times-frac 85.8%

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

      12. metadata-eval 85.8%

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

      13. distribute-rgt-neg-out 85.8%

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

      14. *-commutative 85.8%

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

      15. frac-2neg 85.8%

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

      16. associate-/r* 85.8%

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

      17. associate-/r* 96.2%

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

      18. times-frac 85.3%

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

      19. *-un-lft-identity 85.3%

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

      20. associate-/l/ 96.2%

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

   7. Applied egg-rr 96.3%

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

   8. Taylor expanded in z around inf 74.4%

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

      1. unpow2 74.4%

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

      2. *-commutative 74.4%

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

      3. associate-*r* 85.4%

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

      4. *-commutative 85.4%

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

   10. Simplified 85.4%

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

       1. /-rgt-identity 85.4%

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

       2. *-commutative 85.4%

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

       3. associate-*l* 95.6%

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

       4. *-commutative 95.6%

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

   12. Applied egg-rr 95.6%

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

4. Final simplification 97.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{+151}:\\ \;\;\;\;\frac{\frac{\frac{1}{x}}{y}}{1 + z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(x \cdot z\right) \cdot \left(y \cdot z\right)}\\ \end{array} \]

Alternative 10: 97.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} z = |z|\\ [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{+151}:\\ \;\;\;\;\frac{\frac{\frac{1}{y}}{x}}{1 + z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(x \cdot z\right) \cdot \left(y \cdot z\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: z should be positive before calling this function
NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= (* z z) 2e+151)
   (/ (/ (/ 1.0 y) x) (+ 1.0 (* z z)))
   (/ 1.0 (* (* x z) (* y z)))))

C

z = abs(z);
assert(x < y);
double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 2e+151) {
		tmp = ((1.0 / y) / x) / (1.0 + (z * z));
	} else {
		tmp = 1.0 / ((x * z) * (y * z));
	}
	return tmp;
}

Fortran

NOTE: z should be positive before calling this function
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+151) then
        tmp = ((1.0d0 / y) / x) / (1.0d0 + (z * z))
    else
        tmp = 1.0d0 / ((x * z) * (y * z))
    end if
    code = tmp
end function

Java

z = Math.abs(z);
assert x < y;
public static double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 2e+151) {
		tmp = ((1.0 / y) / x) / (1.0 + (z * z));
	} else {
		tmp = 1.0 / ((x * z) * (y * z));
	}
	return tmp;
}

Python

z = abs(z)
[x, y] = sort([x, y])
def code(x, y, z):
	tmp = 0
	if (z * z) <= 2e+151:
		tmp = ((1.0 / y) / x) / (1.0 + (z * z))
	else:
		tmp = 1.0 / ((x * z) * (y * z))
	return tmp

Julia

z = abs(z)
x, y = sort([x, y])
function code(x, y, z)
	tmp = 0.0
	if (Float64(z * z) <= 2e+151)
		tmp = Float64(Float64(Float64(1.0 / y) / x) / Float64(1.0 + Float64(z * z)));
	else
		tmp = Float64(1.0 / Float64(Float64(x * z) * Float64(y * z)));
	end
	return tmp
end

MATLAB

z = abs(z)
x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z * z) <= 2e+151)
		tmp = ((1.0 / y) / x) / (1.0 + (z * z));
	else
		tmp = 1.0 / ((x * z) * (y * z));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: z should be positive before calling this function
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+151], N[(N[(N[(1.0 / y), $MachinePrecision] / x), $MachinePrecision] / N[(1.0 + N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(x * z), $MachinePrecision] * N[(y * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 2.00000000000000003e151

   1. Initial program 96.8%

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

      1. associate-/r* 99.0%

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

   3. Simplified 99.0%

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

      1. *-un-lft-identity 99.0%

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

      2. add-sqr-sqrt 49.3%

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

      3. times-frac 49.3%

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

   5. Applied egg-rr 49.3%

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

      1. frac-times 49.3%

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

      2. *-un-lft-identity 49.3%

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

      3. add-sqr-sqrt 99.0%

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

      4. div-inv 98.8%

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

      5. inv-pow 98.8%

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

      6. inv-pow 98.8%

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

      7. unpow-prod-down 98.7%

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

      8. *-commutative 98.7%

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

      9. inv-pow 98.7%

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

      10. associate-/r* 99.0%

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

   7. Applied egg-rr 99.0%

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

   if 2.00000000000000003e151 < (*.f64 z z)

   1. Initial program 69.1%

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

      1. associate-/r* 69.1%

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

      2. +-commutative 69.1%

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

      3. fma-def 69.1%

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

   3. Simplified 69.1%

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

      1. associate-/r* 69.1%

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

      2. *-un-lft-identity 69.1%

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

      3. fma-udef 69.1%

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

      4. +-commutative 69.1%

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

      5. *-commutative 69.1%

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

      6. times-frac 70.4%

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

      7. +-commutative 70.4%

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

      8. fma-udef 70.4%

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

      9. associate-/l/ 70.3%

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

   5. Applied egg-rr 70.3%

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

      1. associate-*l/ 70.4%

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

      2. frac-2neg 70.4%

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

      3. metadata-eval 70.4%

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

      4. *-commutative 70.4%

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

      5. distribute-rgt-neg-out 70.4%

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

      6. fma-udef 70.4%

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

      7. +-commutative 70.4%

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

      8. add-sqr-sqrt 70.4%

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

      9. hypot-1-def 70.4%

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

      10. hypot-1-def 70.4%

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

      11. times-frac 85.8%

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

      12. metadata-eval 85.8%

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

      13. distribute-rgt-neg-out 85.8%

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

      14. *-commutative 85.8%

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

      15. frac-2neg 85.8%

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

      16. associate-/r* 85.8%

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

      17. associate-/r* 96.2%

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

      18. times-frac 85.3%

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

      19. *-un-lft-identity 85.3%

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

      20. associate-/l/ 96.2%

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

   7. Applied egg-rr 96.3%

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

   8. Taylor expanded in z around inf 74.4%

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

      1. unpow2 74.4%

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

      2. *-commutative 74.4%

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

      3. associate-*r* 85.4%

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

      4. *-commutative 85.4%

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

   10. Simplified 85.4%

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

       1. /-rgt-identity 85.4%

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

       2. *-commutative 85.4%

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

       3. associate-*l* 95.6%

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

       4. *-commutative 95.6%

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

   12. Applied egg-rr 95.6%

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

4. Final simplification 97.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{+151}:\\ \;\;\;\;\frac{\frac{\frac{1}{y}}{x}}{1 + z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(x \cdot z\right) \cdot \left(y \cdot z\right)}\\ \end{array} \]

Alternative 11: 93.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} z = |z|\\ [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{-17}:\\ \;\;\;\;\frac{\frac{1}{x}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y \cdot \left(z \cdot \left(x \cdot z\right)\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: z should be positive before calling this function
NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= (* z z) 2e-17) (/ (/ 1.0 x) y) (/ 1.0 (* y (* z (* x z))))))

C

z = abs(z);
assert(x < y);
double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 2e-17) {
		tmp = (1.0 / x) / y;
	} else {
		tmp = 1.0 / (y * (z * (x * z)));
	}
	return tmp;
}

Fortran

NOTE: z should be positive before calling this function
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-17) then
        tmp = (1.0d0 / x) / y
    else
        tmp = 1.0d0 / (y * (z * (x * z)))
    end if
    code = tmp
end function

Java

z = Math.abs(z);
assert x < y;
public static double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 2e-17) {
		tmp = (1.0 / x) / y;
	} else {
		tmp = 1.0 / (y * (z * (x * z)));
	}
	return tmp;
}

Python

z = abs(z)
[x, y] = sort([x, y])
def code(x, y, z):
	tmp = 0
	if (z * z) <= 2e-17:
		tmp = (1.0 / x) / y
	else:
		tmp = 1.0 / (y * (z * (x * z)))
	return tmp

Julia

z = abs(z)
x, y = sort([x, y])
function code(x, y, z)
	tmp = 0.0
	if (Float64(z * z) <= 2e-17)
		tmp = Float64(Float64(1.0 / x) / y);
	else
		tmp = Float64(1.0 / Float64(y * Float64(z * Float64(x * z))));
	end
	return tmp
end

MATLAB

z = abs(z)
x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z * z) <= 2e-17)
		tmp = (1.0 / x) / y;
	else
		tmp = 1.0 / (y * (z * (x * z)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: z should be positive before calling this function
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-17], N[(N[(1.0 / x), $MachinePrecision] / y), $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) < 2.00000000000000014e-17

   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* 99.2%

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

      2. +-commutative 99.2%

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

      3. fma-def 99.2%

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

   3. Simplified 99.2%

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

   4. Taylor expanded in z around 0 99.2%

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

      1. frac-2neg 99.2%

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

      2. metadata-eval 99.2%

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

      3. *-commutative 99.2%

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

      4. distribute-rgt-neg-out 99.2%

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

      5. add-sqr-sqrt 50.2%

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

      6. pow2 50.2%

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

      7. metadata-eval 50.2%

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

      8. distribute-rgt-neg-out 50.2%

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

      9. *-commutative 50.2%

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

      10. frac-2neg 50.2%

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

      11. inv-pow 50.2%

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

      12. sqrt-pow1 50.2%

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

      13. metadata-eval 50.2%

          \[\leadsto {\left({\left(y \cdot x\right)}^{\color{blue}{-0.5}}\right)}^{2} \]

   6. Applied egg-rr 50.2%

      \[\leadsto \color{blue}{{\left({\left(y \cdot x\right)}^{-0.5}\right)}^{2}} \]
   7. Step-by-step derivation

      1. pow-pow 99.2%

         \[\leadsto \color{blue}{{\left(y \cdot x\right)}^{\left(-0.5 \cdot 2\right)}} \]

      2. metadata-eval 99.2%

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

      3. inv-pow 99.2%

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

      4. *-commutative 99.2%

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

      5. associate-/r* 99.7%

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

   8. Applied egg-rr 99.7%

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

   if 2.00000000000000014e-17 < (*.f64 z z)

   1. Initial program 74.1%

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

      1. associate-/r* 74.1%

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

      2. +-commutative 74.1%

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

      3. fma-def 74.1%

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

   3. Simplified 74.1%

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

      1. associate-/r* 74.1%

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

      2. *-un-lft-identity 74.1%

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

      3. fma-udef 74.1%

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

      4. +-commutative 74.1%

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

      5. *-commutative 74.1%

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

      6. times-frac 77.4%

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

      7. +-commutative 77.4%

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

      8. fma-udef 77.4%

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

      9. associate-/l/ 77.5%

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

   5. Applied egg-rr 77.5%

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

      1. associate-*l/ 77.4%

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

      2. frac-2neg 77.4%

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

      3. metadata-eval 77.4%

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

      4. *-commutative 77.4%

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

      5. distribute-rgt-neg-out 77.4%

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

      6. fma-udef 77.4%

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

      7. +-commutative 77.4%

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

      8. add-sqr-sqrt 77.4%

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

      9. hypot-1-def 77.4%

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

      10. hypot-1-def 77.4%

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

      11. times-frac 88.7%

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

      12. metadata-eval 88.7%

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

      13. distribute-rgt-neg-out 88.7%

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

      14. *-commutative 88.7%

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

      15. frac-2neg 88.7%

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

      16. associate-/r* 88.8%

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

      17. associate-/r* 95.8%

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

      18. times-frac 87.8%

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

      19. *-un-lft-identity 87.8%

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

      20. associate-/l/ 95.8%

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

   7. Applied egg-rr 95.4%

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

   8. Taylor expanded in z around inf 79.3%

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

      1. unpow2 79.3%

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

      2. *-commutative 79.3%

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

      3. associate-*r* 87.3%

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

      4. *-commutative 87.3%

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

   10. Simplified 87.3%

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

4. Final simplification 92.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{-17}:\\ \;\;\;\;\frac{\frac{1}{x}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y \cdot \left(z \cdot \left(x \cdot z\right)\right)}\\ \end{array} \]

Alternative 12: 93.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} z = |z|\\ [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{-17}:\\ \;\;\;\;\frac{\frac{1}{x}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{z \cdot \left(z \cdot \left(y \cdot x\right)\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: z should be positive before calling this function
NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= (* z z) 2e-17) (/ (/ 1.0 x) y) (/ 1.0 (* z (* z (* y x))))))

C

z = abs(z);
assert(x < y);
double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 2e-17) {
		tmp = (1.0 / x) / y;
	} else {
		tmp = 1.0 / (z * (z * (y * x)));
	}
	return tmp;
}

Fortran

NOTE: z should be positive before calling this function
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-17) then
        tmp = (1.0d0 / x) / y
    else
        tmp = 1.0d0 / (z * (z * (y * x)))
    end if
    code = tmp
end function

Java

z = Math.abs(z);
assert x < y;
public static double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 2e-17) {
		tmp = (1.0 / x) / y;
	} else {
		tmp = 1.0 / (z * (z * (y * x)));
	}
	return tmp;
}

Python

z = abs(z)
[x, y] = sort([x, y])
def code(x, y, z):
	tmp = 0
	if (z * z) <= 2e-17:
		tmp = (1.0 / x) / y
	else:
		tmp = 1.0 / (z * (z * (y * x)))
	return tmp

Julia

z = abs(z)
x, y = sort([x, y])
function code(x, y, z)
	tmp = 0.0
	if (Float64(z * z) <= 2e-17)
		tmp = Float64(Float64(1.0 / x) / y);
	else
		tmp = Float64(1.0 / Float64(z * Float64(z * Float64(y * x))));
	end
	return tmp
end

MATLAB

z = abs(z)
x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z * z) <= 2e-17)
		tmp = (1.0 / x) / y;
	else
		tmp = 1.0 / (z * (z * (y * x)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: z should be positive before calling this function
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-17], 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) < 2.00000000000000014e-17

   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* 99.2%

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

      2. +-commutative 99.2%

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

      3. fma-def 99.2%

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

   3. Simplified 99.2%

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

   4. Taylor expanded in z around 0 99.2%

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

      1. frac-2neg 99.2%

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

      2. metadata-eval 99.2%

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

      3. *-commutative 99.2%

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

      4. distribute-rgt-neg-out 99.2%

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

      5. add-sqr-sqrt 50.2%

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

      6. pow2 50.2%

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

      7. metadata-eval 50.2%

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

      8. distribute-rgt-neg-out 50.2%

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

      9. *-commutative 50.2%

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

      10. frac-2neg 50.2%

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

      11. inv-pow 50.2%

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

      12. sqrt-pow1 50.2%

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

      13. metadata-eval 50.2%

          \[\leadsto {\left({\left(y \cdot x\right)}^{\color{blue}{-0.5}}\right)}^{2} \]

   6. Applied egg-rr 50.2%

      \[\leadsto \color{blue}{{\left({\left(y \cdot x\right)}^{-0.5}\right)}^{2}} \]
   7. Step-by-step derivation

      1. pow-pow 99.2%

         \[\leadsto \color{blue}{{\left(y \cdot x\right)}^{\left(-0.5 \cdot 2\right)}} \]

      2. metadata-eval 99.2%

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

      3. inv-pow 99.2%

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

      4. *-commutative 99.2%

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

      5. associate-/r* 99.7%

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

   8. Applied egg-rr 99.7%

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

   if 2.00000000000000014e-17 < (*.f64 z z)

   1. Initial program 74.1%

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

      1. associate-/r* 74.1%

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

      2. +-commutative 74.1%

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

      3. fma-def 74.1%

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

   3. Simplified 74.1%

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

   4. Taylor expanded in z around inf 79.3%

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

      1. unpow2 79.3%

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

      2. *-commutative 79.3%

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

      3. associate-*l* 77.4%

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

      4. *-commutative 77.4%

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

      5. associate-*l* 89.4%

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

   6. Simplified 89.4%

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

4. Final simplification 94.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{-17}:\\ \;\;\;\;\frac{\frac{1}{x}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{z \cdot \left(z \cdot \left(y \cdot x\right)\right)}\\ \end{array} \]

Alternative 13: 96.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} z = |z|\\ [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{-17}:\\ \;\;\;\;\frac{\frac{1}{x}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(x \cdot z\right) \cdot \left(y \cdot z\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: z should be positive before calling this function
NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= (* z z) 2e-17) (/ (/ 1.0 x) y) (/ 1.0 (* (* x z) (* y z)))))

C

z = abs(z);
assert(x < y);
double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 2e-17) {
		tmp = (1.0 / x) / y;
	} else {
		tmp = 1.0 / ((x * z) * (y * z));
	}
	return tmp;
}

Fortran

NOTE: z should be positive before calling this function
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-17) then
        tmp = (1.0d0 / x) / y
    else
        tmp = 1.0d0 / ((x * z) * (y * z))
    end if
    code = tmp
end function

Java

z = Math.abs(z);
assert x < y;
public static double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 2e-17) {
		tmp = (1.0 / x) / y;
	} else {
		tmp = 1.0 / ((x * z) * (y * z));
	}
	return tmp;
}

Python

z = abs(z)
[x, y] = sort([x, y])
def code(x, y, z):
	tmp = 0
	if (z * z) <= 2e-17:
		tmp = (1.0 / x) / y
	else:
		tmp = 1.0 / ((x * z) * (y * z))
	return tmp

Julia

z = abs(z)
x, y = sort([x, y])
function code(x, y, z)
	tmp = 0.0
	if (Float64(z * z) <= 2e-17)
		tmp = Float64(Float64(1.0 / x) / y);
	else
		tmp = Float64(1.0 / Float64(Float64(x * z) * Float64(y * z)));
	end
	return tmp
end

MATLAB

z = abs(z)
x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z * z) <= 2e-17)
		tmp = (1.0 / x) / y;
	else
		tmp = 1.0 / ((x * z) * (y * z));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: z should be positive before calling this function
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-17], N[(N[(1.0 / x), $MachinePrecision] / y), $MachinePrecision], N[(1.0 / N[(N[(x * z), $MachinePrecision] * N[(y * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 2.00000000000000014e-17

   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* 99.2%

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

      2. +-commutative 99.2%

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

      3. fma-def 99.2%

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

   3. Simplified 99.2%

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

   4. Taylor expanded in z around 0 99.2%

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

      1. frac-2neg 99.2%

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

      2. metadata-eval 99.2%

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

      3. *-commutative 99.2%

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

      4. distribute-rgt-neg-out 99.2%

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

      5. add-sqr-sqrt 50.2%

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

      6. pow2 50.2%

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

      7. metadata-eval 50.2%

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

      8. distribute-rgt-neg-out 50.2%

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

      9. *-commutative 50.2%

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

      10. frac-2neg 50.2%

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

      11. inv-pow 50.2%

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

      12. sqrt-pow1 50.2%

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

      13. metadata-eval 50.2%

          \[\leadsto {\left({\left(y \cdot x\right)}^{\color{blue}{-0.5}}\right)}^{2} \]

   6. Applied egg-rr 50.2%

      \[\leadsto \color{blue}{{\left({\left(y \cdot x\right)}^{-0.5}\right)}^{2}} \]
   7. Step-by-step derivation

      1. pow-pow 99.2%

         \[\leadsto \color{blue}{{\left(y \cdot x\right)}^{\left(-0.5 \cdot 2\right)}} \]

      2. metadata-eval 99.2%

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

      3. inv-pow 99.2%

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

      4. *-commutative 99.2%

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

      5. associate-/r* 99.7%

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

   8. Applied egg-rr 99.7%

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

   if 2.00000000000000014e-17 < (*.f64 z z)

   1. Initial program 74.1%

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

      1. associate-/r* 74.1%

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

      2. +-commutative 74.1%

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

      3. fma-def 74.1%

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

   3. Simplified 74.1%

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

      1. associate-/r* 74.1%

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

      2. *-un-lft-identity 74.1%

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

      3. fma-udef 74.1%

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

      4. +-commutative 74.1%

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

      5. *-commutative 74.1%

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

      6. times-frac 77.4%

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

      7. +-commutative 77.4%

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

      8. fma-udef 77.4%

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

      9. associate-/l/ 77.5%

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

   5. Applied egg-rr 77.5%

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

      1. associate-*l/ 77.4%

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

      2. frac-2neg 77.4%

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

      3. metadata-eval 77.4%

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

      4. *-commutative 77.4%

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

      5. distribute-rgt-neg-out 77.4%

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

      6. fma-udef 77.4%

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

      7. +-commutative 77.4%

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

      8. add-sqr-sqrt 77.4%

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

      9. hypot-1-def 77.4%

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

      10. hypot-1-def 77.4%

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

      11. times-frac 88.7%

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

      12. metadata-eval 88.7%

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

      13. distribute-rgt-neg-out 88.7%

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

      14. *-commutative 88.7%

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

      15. frac-2neg 88.7%

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

      16. associate-/r* 88.8%

          \[\leadsto \frac{1}{\mathsf{hypot}\left(1, z\right)} \cdot \frac{\color{blue}{\frac{\frac{1}{y}}{x}}}{\mathsf{hypot}\left(1, z\right)} \]

      17. associate-/r* 95.8%

          \[\leadsto \frac{1}{\mathsf{hypot}\left(1, z\right)} \cdot \color{blue}{\frac{\frac{1}{y}}{x \cdot \mathsf{hypot}\left(1, z\right)}} \]

      18. times-frac 87.8%

          \[\leadsto \color{blue}{\frac{1 \cdot \frac{1}{y}}{\mathsf{hypot}\left(1, z\right) \cdot \left(x \cdot \mathsf{hypot}\left(1, z\right)\right)}} \]

      19. *-un-lft-identity 87.8%

          \[\leadsto \frac{\color{blue}{\frac{1}{y}}}{\mathsf{hypot}\left(1, z\right) \cdot \left(x \cdot \mathsf{hypot}\left(1, z\right)\right)} \]

      20. associate-/l/ 95.8%

          \[\leadsto \color{blue}{\frac{\frac{\frac{1}{y}}{x \cdot \mathsf{hypot}\left(1, z\right)}}{\mathsf{hypot}\left(1, z\right)}} \]

   7. Applied egg-rr 95.4%

      \[\leadsto \color{blue}{\frac{1}{\left(\left(x \cdot \mathsf{hypot}\left(1, z\right)\right) \cdot y\right) \cdot \mathsf{hypot}\left(1, z\right)}} \]

   8. Taylor expanded in z around inf 79.3%

      \[\leadsto \color{blue}{\frac{1}{y \cdot \left({z}^{2} \cdot x\right)}} \]
   9. Step-by-step derivation

      1. unpow2 79.3%

         \[\leadsto \frac{1}{y \cdot \left(\color{blue}{\left(z \cdot z\right)} \cdot x\right)} \]

      2. *-commutative 79.3%

         \[\leadsto \frac{1}{y \cdot \color{blue}{\left(x \cdot \left(z \cdot z\right)\right)}} \]

      3. associate-*r* 87.3%

         \[\leadsto \frac{1}{y \cdot \color{blue}{\left(\left(x \cdot z\right) \cdot z\right)}} \]

      4. *-commutative 87.3%

         \[\leadsto \frac{1}{y \cdot \left(\color{blue}{\left(z \cdot x\right)} \cdot z\right)} \]

   10. Simplified 87.3%

       \[\leadsto \color{blue}{\frac{1}{y \cdot \left(\left(z \cdot x\right) \cdot z\right)}} \]
   11. Step-by-step derivation

       1. /-rgt-identity 87.3%

          \[\leadsto \frac{1}{\color{blue}{\frac{y \cdot \left(\left(z \cdot x\right) \cdot z\right)}{1}}} \]

       2. *-commutative 87.3%

          \[\leadsto \frac{1}{\frac{\color{blue}{\left(\left(z \cdot x\right) \cdot z\right) \cdot y}}{1}} \]

       3. associate-*l* 93.5%

          \[\leadsto \frac{1}{\frac{\color{blue}{\left(z \cdot x\right) \cdot \left(z \cdot y\right)}}{1}} \]

       4. *-commutative 93.5%

          \[\leadsto \frac{1}{\frac{\color{blue}{\left(x \cdot z\right)} \cdot \left(z \cdot y\right)}{1}} \]

   12. Applied egg-rr 93.5%

       \[\leadsto \frac{1}{\color{blue}{\frac{\left(x \cdot z\right) \cdot \left(z \cdot y\right)}{1}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 96.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{-17}:\\ \;\;\;\;\frac{\frac{1}{x}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(x \cdot z\right) \cdot \left(y \cdot z\right)}\\ \end{array} \]

Alternative 14: 93.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} z = |z|\\ [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq 2.5 \cdot 10^{-5}:\\ \;\;\;\;\frac{\frac{1}{x}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x \cdot \left(z \cdot \left(y \cdot z\right)\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: z should be positive before calling this function
NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= z 2.5e-5) (/ (/ 1.0 x) y) (/ 1.0 (* x (* z (* y z))))))

C

z = abs(z);
assert(x < y);
double code(double x, double y, double z) {
	double tmp;
	if (z <= 2.5e-5) {
		tmp = (1.0 / x) / y;
	} else {
		tmp = 1.0 / (x * (z * (y * z)));
	}
	return tmp;
}

Fortran

NOTE: z should be positive before calling this function
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 <= 2.5d-5) then
        tmp = (1.0d0 / x) / y
    else
        tmp = 1.0d0 / (x * (z * (y * z)))
    end if
    code = tmp
end function

Java

z = Math.abs(z);
assert x < y;
public static double code(double x, double y, double z) {
	double tmp;
	if (z <= 2.5e-5) {
		tmp = (1.0 / x) / y;
	} else {
		tmp = 1.0 / (x * (z * (y * z)));
	}
	return tmp;
}

Python

z = abs(z)
[x, y] = sort([x, y])
def code(x, y, z):
	tmp = 0
	if z <= 2.5e-5:
		tmp = (1.0 / x) / y
	else:
		tmp = 1.0 / (x * (z * (y * z)))
	return tmp

Julia

z = abs(z)
x, y = sort([x, y])
function code(x, y, z)
	tmp = 0.0
	if (z <= 2.5e-5)
		tmp = Float64(Float64(1.0 / x) / y);
	else
		tmp = Float64(1.0 / Float64(x * Float64(z * Float64(y * z))));
	end
	return tmp
end

MATLAB

z = abs(z)
x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= 2.5e-5)
		tmp = (1.0 / x) / y;
	else
		tmp = 1.0 / (x * (z * (y * z)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: z should be positive before calling this function
NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_, z_] := If[LessEqual[z, 2.5e-5], 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 z < 2.50000000000000012e-5

   1. Initial program 89.9%

      \[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)} \]
   2. Step-by-step derivation

      1. associate-/r* 89.6%

         \[\leadsto \color{blue}{\frac{1}{x \cdot \left(y \cdot \left(1 + z \cdot z\right)\right)}} \]

      2. +-commutative 89.6%

         \[\leadsto \frac{1}{x \cdot \left(y \cdot \color{blue}{\left(z \cdot z + 1\right)}\right)} \]

      3. fma-def 89.6%

         \[\leadsto \frac{1}{x \cdot \left(y \cdot \color{blue}{\mathsf{fma}\left(z, z, 1\right)}\right)} \]

   3. Simplified 89.6%

      \[\leadsto \color{blue}{\frac{1}{x \cdot \left(y \cdot \mathsf{fma}\left(z, z, 1\right)\right)}} \]

   4. Taylor expanded in z around 0 70.1%

      \[\leadsto \frac{1}{\color{blue}{y \cdot x}} \]
   5. Step-by-step derivation

      1. frac-2neg 70.1%

         \[\leadsto \color{blue}{\frac{-1}{-y \cdot x}} \]

      2. metadata-eval 70.1%

         \[\leadsto \frac{\color{blue}{-1}}{-y \cdot x} \]

      3. *-commutative 70.1%

         \[\leadsto \frac{-1}{-\color{blue}{x \cdot y}} \]

      4. distribute-rgt-neg-out 70.1%

         \[\leadsto \frac{-1}{\color{blue}{x \cdot \left(-y\right)}} \]

      5. add-sqr-sqrt 37.6%

         \[\leadsto \color{blue}{\sqrt{\frac{-1}{x \cdot \left(-y\right)}} \cdot \sqrt{\frac{-1}{x \cdot \left(-y\right)}}} \]

      6. pow2 37.6%

         \[\leadsto \color{blue}{{\left(\sqrt{\frac{-1}{x \cdot \left(-y\right)}}\right)}^{2}} \]

      7. metadata-eval 37.6%

         \[\leadsto {\left(\sqrt{\frac{\color{blue}{-1}}{x \cdot \left(-y\right)}}\right)}^{2} \]

      8. distribute-rgt-neg-out 37.6%

         \[\leadsto {\left(\sqrt{\frac{-1}{\color{blue}{-x \cdot y}}}\right)}^{2} \]

      9. *-commutative 37.6%

         \[\leadsto {\left(\sqrt{\frac{-1}{-\color{blue}{y \cdot x}}}\right)}^{2} \]

      10. frac-2neg 37.6%

          \[\leadsto {\left(\sqrt{\color{blue}{\frac{1}{y \cdot x}}}\right)}^{2} \]

      11. inv-pow 37.6%

          \[\leadsto {\left(\sqrt{\color{blue}{{\left(y \cdot x\right)}^{-1}}}\right)}^{2} \]

      12. sqrt-pow1 37.6%

          \[\leadsto {\color{blue}{\left({\left(y \cdot x\right)}^{\left(\frac{-1}{2}\right)}\right)}}^{2} \]

      13. metadata-eval 37.6%

          \[\leadsto {\left({\left(y \cdot x\right)}^{\color{blue}{-0.5}}\right)}^{2} \]

   6. Applied egg-rr 37.6%

      \[\leadsto \color{blue}{{\left({\left(y \cdot x\right)}^{-0.5}\right)}^{2}} \]
   7. Step-by-step derivation

      1. pow-pow 70.1%

         \[\leadsto \color{blue}{{\left(y \cdot x\right)}^{\left(-0.5 \cdot 2\right)}} \]

      2. metadata-eval 70.1%

         \[\leadsto {\left(y \cdot x\right)}^{\color{blue}{-1}} \]

      3. inv-pow 70.1%

         \[\leadsto \color{blue}{\frac{1}{y \cdot x}} \]

      4. *-commutative 70.1%

         \[\leadsto \frac{1}{\color{blue}{x \cdot y}} \]

      5. associate-/r* 70.4%

         \[\leadsto \color{blue}{\frac{\frac{1}{x}}{y}} \]

   8. Applied egg-rr 70.4%

      \[\leadsto \color{blue}{\frac{\frac{1}{x}}{y}} \]

   if 2.50000000000000012e-5 < z

   1. Initial program 75.2%

      \[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)} \]
   2. Step-by-step derivation

      1. associate-/r* 75.2%

         \[\leadsto \color{blue}{\frac{1}{x \cdot \left(y \cdot \left(1 + z \cdot z\right)\right)}} \]

      2. +-commutative 75.2%

         \[\leadsto \frac{1}{x \cdot \left(y \cdot \color{blue}{\left(z \cdot z + 1\right)}\right)} \]

      3. fma-def 75.2%

         \[\leadsto \frac{1}{x \cdot \left(y \cdot \color{blue}{\mathsf{fma}\left(z, z, 1\right)}\right)} \]

   3. Simplified 75.2%

      \[\leadsto \color{blue}{\frac{1}{x \cdot \left(y \cdot \mathsf{fma}\left(z, z, 1\right)\right)}} \]

   4. Taylor expanded in z around inf 75.2%

      \[\leadsto \frac{1}{x \cdot \color{blue}{\left(y \cdot {z}^{2}\right)}} \]
   5. Step-by-step derivation

      1. unpow2 75.2%

         \[\leadsto \frac{1}{x \cdot \left(y \cdot \color{blue}{\left(z \cdot z\right)}\right)} \]

      2. associate-*r* 89.1%

         \[\leadsto \frac{1}{x \cdot \color{blue}{\left(\left(y \cdot z\right) \cdot z\right)}} \]

      3. *-commutative 89.1%

         \[\leadsto \frac{1}{x \cdot \left(\color{blue}{\left(z \cdot y\right)} \cdot z\right)} \]

   6. Simplified 89.1%

      \[\leadsto \frac{1}{x \cdot \color{blue}{\left(\left(z \cdot y\right) \cdot z\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 75.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 2.5 \cdot 10^{-5}:\\ \;\;\;\;\frac{\frac{1}{x}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x \cdot \left(z \cdot \left(y \cdot z\right)\right)}\\ \end{array} \]

Alternative 15: 58.1% accurate, 2.2× speedup

Math

\[\begin{array}{l} z = |z|\\ [x, y] = \mathsf{sort}([x, y])\\ \\ \frac{1}{y \cdot x} \end{array} \]

FPCore

NOTE: z should be positive before calling this function
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

z = abs(z);
assert(x < y);
double code(double x, double y, double z) {
	return 1.0 / (y * x);
}

Fortran

NOTE: z should be positive before calling this function
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

z = Math.abs(z);
assert x < y;
public static double code(double x, double y, double z) {
	return 1.0 / (y * x);
}

Python

z = abs(z)
[x, y] = sort([x, y])
def code(x, y, z):
	return 1.0 / (y * x)

Julia

z = abs(z)
x, y = sort([x, y])
function code(x, y, z)
	return Float64(1.0 / Float64(y * x))
end

MATLAB

z = abs(z)
x, y = num2cell(sort([x, y])){:}
function tmp = code(x, y, z)
	tmp = 1.0 / (y * x);
end

Wolfram

NOTE: z should be positive before calling this function
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 85.7%

   \[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)} \]
2. Step-by-step derivation

   1. associate-/r* 85.5%

      \[\leadsto \color{blue}{\frac{1}{x \cdot \left(y \cdot \left(1 + z \cdot z\right)\right)}} \]

   2. +-commutative 85.5%

      \[\leadsto \frac{1}{x \cdot \left(y \cdot \color{blue}{\left(z \cdot z + 1\right)}\right)} \]

   3. fma-def 85.5%

      \[\leadsto \frac{1}{x \cdot \left(y \cdot \color{blue}{\mathsf{fma}\left(z, z, 1\right)}\right)} \]

3. Simplified 85.5%

   \[\leadsto \color{blue}{\frac{1}{x \cdot \left(y \cdot \mathsf{fma}\left(z, z, 1\right)\right)}} \]

4. Taylor expanded in z around 0 54.6%

   \[\leadsto \frac{1}{\color{blue}{y \cdot x}} \]

5. Final simplification 54.6%

   \[\leadsto \frac{1}{y \cdot x} \]

Alternative 16: 58.1% accurate, 2.2× speedup

Math

\[\begin{array}{l} z = |z|\\ [x, y] = \mathsf{sort}([x, y])\\ \\ \frac{\frac{1}{x}}{y} \end{array} \]

FPCore

NOTE: z should be positive before calling this function
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

z = abs(z);
assert(x < y);
double code(double x, double y, double z) {
	return (1.0 / x) / y;
}

Fortran

NOTE: z should be positive before calling this function
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

z = Math.abs(z);
assert x < y;
public static double code(double x, double y, double z) {
	return (1.0 / x) / y;
}

Python

z = abs(z)
[x, y] = sort([x, y])
def code(x, y, z):
	return (1.0 / x) / y

Julia

z = abs(z)
x, y = sort([x, y])
function code(x, y, z)
	return Float64(Float64(1.0 / x) / y)
end

MATLAB

z = abs(z)
x, y = num2cell(sort([x, y])){:}
function tmp = code(x, y, z)
	tmp = (1.0 / x) / y;
end

Wolfram

NOTE: z should be positive before calling this function
NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_, z_] := N[(N[(1.0 / x), $MachinePrecision] / y), $MachinePrecision]

Derivation

1. Initial program 85.7%

   \[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)} \]
2. Step-by-step derivation

   1. associate-/r* 85.5%

      \[\leadsto \color{blue}{\frac{1}{x \cdot \left(y \cdot \left(1 + z \cdot z\right)\right)}} \]

   2. +-commutative 85.5%

      \[\leadsto \frac{1}{x \cdot \left(y \cdot \color{blue}{\left(z \cdot z + 1\right)}\right)} \]

   3. fma-def 85.5%

      \[\leadsto \frac{1}{x \cdot \left(y \cdot \color{blue}{\mathsf{fma}\left(z, z, 1\right)}\right)} \]

3. Simplified 85.5%

   \[\leadsto \color{blue}{\frac{1}{x \cdot \left(y \cdot \mathsf{fma}\left(z, z, 1\right)\right)}} \]

4. Taylor expanded in z around 0 54.6%

   \[\leadsto \frac{1}{\color{blue}{y \cdot x}} \]
5. Step-by-step derivation

   1. frac-2neg 54.6%

      \[\leadsto \color{blue}{\frac{-1}{-y \cdot x}} \]

   2. metadata-eval 54.6%

      \[\leadsto \frac{\color{blue}{-1}}{-y \cdot x} \]

   3. *-commutative 54.6%

      \[\leadsto \frac{-1}{-\color{blue}{x \cdot y}} \]

   4. distribute-rgt-neg-out 54.6%

      \[\leadsto \frac{-1}{\color{blue}{x \cdot \left(-y\right)}} \]

   5. add-sqr-sqrt 30.8%

      \[\leadsto \color{blue}{\sqrt{\frac{-1}{x \cdot \left(-y\right)}} \cdot \sqrt{\frac{-1}{x \cdot \left(-y\right)}}} \]

   6. pow2 30.8%

      \[\leadsto \color{blue}{{\left(\sqrt{\frac{-1}{x \cdot \left(-y\right)}}\right)}^{2}} \]

   7. metadata-eval 30.8%

      \[\leadsto {\left(\sqrt{\frac{\color{blue}{-1}}{x \cdot \left(-y\right)}}\right)}^{2} \]

   8. distribute-rgt-neg-out 30.8%

      \[\leadsto {\left(\sqrt{\frac{-1}{\color{blue}{-x \cdot y}}}\right)}^{2} \]

   9. *-commutative 30.8%

      \[\leadsto {\left(\sqrt{\frac{-1}{-\color{blue}{y \cdot x}}}\right)}^{2} \]

   10. frac-2neg 30.8%

       \[\leadsto {\left(\sqrt{\color{blue}{\frac{1}{y \cdot x}}}\right)}^{2} \]

   11. inv-pow 30.8%

       \[\leadsto {\left(\sqrt{\color{blue}{{\left(y \cdot x\right)}^{-1}}}\right)}^{2} \]

   12. sqrt-pow1 30.8%

       \[\leadsto {\color{blue}{\left({\left(y \cdot x\right)}^{\left(\frac{-1}{2}\right)}\right)}}^{2} \]

   13. metadata-eval 30.8%

       \[\leadsto {\left({\left(y \cdot x\right)}^{\color{blue}{-0.5}}\right)}^{2} \]

6. Applied egg-rr 30.8%

   \[\leadsto \color{blue}{{\left({\left(y \cdot x\right)}^{-0.5}\right)}^{2}} \]
7. Step-by-step derivation

   1. pow-pow 54.6%

      \[\leadsto \color{blue}{{\left(y \cdot x\right)}^{\left(-0.5 \cdot 2\right)}} \]

   2. metadata-eval 54.6%

      \[\leadsto {\left(y \cdot x\right)}^{\color{blue}{-1}} \]

   3. inv-pow 54.6%

      \[\leadsto \color{blue}{\frac{1}{y \cdot x}} \]

   4. *-commutative 54.6%

      \[\leadsto \frac{1}{\color{blue}{x \cdot y}} \]

   5. associate-/r* 54.8%

      \[\leadsto \color{blue}{\frac{\frac{1}{x}}{y}} \]

8. Applied egg-rr 54.8%

   \[\leadsto \color{blue}{\frac{\frac{1}{x}}{y}} \]

9. Final simplification 54.8%

   \[\leadsto \frac{\frac{1}{x}}{y} \]

Developer target: 92.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + z \cdot z\\ t_1 := y \cdot t_0\\ t_2 := \frac{\frac{1}{y}}{t_0 \cdot x}\\ \mathbf{if}\;t_1 < -\infty:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t_1 < 8.680743250567252 \cdot 10^{+305}:\\ \;\;\;\;\frac{\frac{1}{x}}{t_0 \cdot y}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ 1.0 (* z z))) (t_1 (* y t_0)) (t_2 (/ (/ 1.0 y) (* t_0 x))))
   (if (< t_1 (- INFINITY))
     t_2
     (if (< t_1 8.680743250567252e+305) (/ (/ 1.0 x) (* t_0 y)) t_2))))

C

double code(double x, double y, double z) {
	double t_0 = 1.0 + (z * z);
	double t_1 = y * t_0;
	double t_2 = (1.0 / y) / (t_0 * x);
	double tmp;
	if (t_1 < -((double) INFINITY)) {
		tmp = t_2;
	} else if (t_1 < 8.680743250567252e+305) {
		tmp = (1.0 / x) / (t_0 * y);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Java

public static double code(double x, double y, double z) {
	double t_0 = 1.0 + (z * z);
	double t_1 = y * t_0;
	double t_2 = (1.0 / y) / (t_0 * x);
	double tmp;
	if (t_1 < -Double.POSITIVE_INFINITY) {
		tmp = t_2;
	} else if (t_1 < 8.680743250567252e+305) {
		tmp = (1.0 / x) / (t_0 * y);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = 1.0 + (z * z)
	t_1 = y * t_0
	t_2 = (1.0 / y) / (t_0 * x)
	tmp = 0
	if t_1 < -math.inf:
		tmp = t_2
	elif t_1 < 8.680743250567252e+305:
		tmp = (1.0 / x) / (t_0 * y)
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(1.0 + Float64(z * z))
	t_1 = Float64(y * t_0)
	t_2 = Float64(Float64(1.0 / y) / Float64(t_0 * x))
	tmp = 0.0
	if (t_1 < Float64(-Inf))
		tmp = t_2;
	elseif (t_1 < 8.680743250567252e+305)
		tmp = Float64(Float64(1.0 / x) / Float64(t_0 * y));
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = 1.0 + (z * z);
	t_1 = y * t_0;
	t_2 = (1.0 / y) / (t_0 * x);
	tmp = 0.0;
	if (t_1 < -Inf)
		tmp = t_2;
	elseif (t_1 < 8.680743250567252e+305)
		tmp = (1.0 / x) / (t_0 * y);
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(1.0 + N[(z * z), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(y * t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(1.0 / y), $MachinePrecision] / N[(t$95$0 * x), $MachinePrecision]), $MachinePrecision]}, If[Less[t$95$1, (-Infinity)], t$95$2, If[Less[t$95$1, 8.680743250567252e+305], N[(N[(1.0 / x), $MachinePrecision] / N[(t$95$0 * y), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]

Reproduce

herbie shell --seed 2023230 
(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)))))