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

Percentage Accurate: 90.4% → 98.8%

Time: 15.2s

Alternatives: 12

Speedup: 0.8×

Specification

Math

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

FPCore

(FPCore (x y z) :precision binary64 (/ (/ 1.0 x) (* y (+ 1.0 (* z z)))))

C

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

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = (1.0d0 / x) / (y * (1.0d0 + (z * z)))
end function

Java

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

Python

def code(x, y, z):
	return (1.0 / x) / (y * (1.0 + (z * z)))

Julia

function code(x, y, z)
	return Float64(Float64(1.0 / x) / Float64(y * Float64(1.0 + Float64(z * z))))
end

MATLAB

function tmp = code(x, y, z)
	tmp = (1.0 / x) / (y * (1.0 + (z * z)));
end

Wolfram

code[x_, y_, z_] := N[(N[(1.0 / x), $MachinePrecision] / N[(y * N[(1.0 + N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 90.4% 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: 98.8% accurate, 0.1× speedup

Math

\[\begin{array}{l} z = |z|\\ [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \cdot \left(1 + z \cdot z\right) \leq 5 \cdot 10^{+302}:\\ \;\;\;\;\frac{\frac{1}{x}}{\mathsf{fma}\left(y \cdot z, z, y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\mathsf{hypot}\left(1, z\right)}}{y \cdot \left(z \cdot x\right)}\\ \end{array} \end{array} \]

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 (<= (* y (+ 1.0 (* z z))) 5e+302)
   (/ (/ 1.0 x) (fma (* y z) z y))
   (/ (/ 1.0 (hypot 1.0 z)) (* y (* z x)))))

C

z = abs(z);
assert(x < y);
double code(double x, double y, double z) {
	double tmp;
	if ((y * (1.0 + (z * z))) <= 5e+302) {
		tmp = (1.0 / x) / fma((y * z), z, y);
	} else {
		tmp = (1.0 / hypot(1.0, z)) / (y * (z * x));
	}
	return tmp;
}

Julia

z = abs(z)
x, y = sort([x, y])
function code(x, y, z)
	tmp = 0.0
	if (Float64(y * Float64(1.0 + Float64(z * z))) <= 5e+302)
		tmp = Float64(Float64(1.0 / x) / fma(Float64(y * z), z, y));
	else
		tmp = Float64(Float64(1.0 / hypot(1.0, z)) / Float64(y * Float64(z * x)));
	end
	return 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[(y * N[(1.0 + N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 5e+302], N[(N[(1.0 / x), $MachinePrecision] / N[(N[(y * z), $MachinePrecision] * z + y), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[Sqrt[1.0 ^ 2 + z ^ 2], $MachinePrecision]), $MachinePrecision] / N[(y * N[(z * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y (+.f64 1 (*.f64 z z))) < 5e302

   1. Initial program 93.4%

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

      1. distribute-lft-in 93.4%

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

      2. *-rgt-identity 93.4%

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

      3. +-commutative 93.4%

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

      4. associate-*r* 95.6%

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

      5. fma-def 95.6%

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

   3. Applied egg-rr 95.6%

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

   if 5e302 < (*.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%

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

      3. sqr-neg 64.0%

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

      4. +-commutative 64.0%

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

      5. distribute-lft1-in 64.0%

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

      6. *-commutative 64.0%

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

      7. fma-def 64.0%

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

      8. sqr-neg 64.0%

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

   3. Simplified 64.0%

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

      1. *-commutative 64.0%

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

      2. fma-udef 64.0%

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

      3. *-rgt-identity 64.0%

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

      4. distribute-lft-in 64.0%

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

      5. +-commutative 64.0%

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

      6. /-rgt-identity 64.0%

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

      7. clear-num 64.0%

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

      8. div-inv 64.0%

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

      9. +-commutative 64.0%

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

      10. fma-udef 64.0%

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

      11. *-commutative 64.0%

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

      12. associate-/l* 74.3%

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

      13. add-sqr-sqrt 74.3%

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

      14. associate-/r* 74.3%

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

      15. associate-/l* 74.3%

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

      16. fma-udef 74.3%

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

      17. +-commutative 74.3%

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

      18. hypot-1-def 74.3%

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

      19. *-commutative 74.3%

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

      20. associate-/r* 74.3%

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

      21. fma-udef 74.3%

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

      22. +-commutative 74.3%

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

      23. hypot-1-def 93.0%

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

   5. Applied egg-rr 93.0%

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

      1. associate-/r/ 92.9%

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

      2. associate-/r* 92.9%

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

   7. Simplified 92.9%

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

   8. Taylor expanded in z around inf 78.6%

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

      1. expm1-log1p-u 78.4%

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

      2. expm1-udef 66.5%

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

      3. *-commutative 66.5%

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

      4. associate-/r* 66.5%

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

      5. associate-*r* 63.9%

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

      6. *-commutative 63.9%

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

   10. Applied egg-rr 63.9%

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

       1. expm1-def 74.3%

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

       2. expm1-log1p 74.4%

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

       3. *-commutative 74.4%

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

       4. *-commutative 74.4%

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

       5. associate-*l* 78.8%

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

   12. Simplified 78.8%

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

4. Final simplification 92.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(1 + z \cdot z\right) \leq 5 \cdot 10^{+302}:\\ \;\;\;\;\frac{\frac{1}{x}}{\mathsf{fma}\left(y \cdot z, z, y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\mathsf{hypot}\left(1, z\right)}}{y \cdot \left(z \cdot x\right)}\\ \end{array} \]

Alternative 2: 96.9% accurate, 0.1× speedup

Math

\[\begin{array}{l} z = |z|\\ [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \cdot \left(1 + z \cdot z\right) \leq 5 \cdot 10^{+302}:\\ \;\;\;\;\frac{\frac{1}{x}}{\mathsf{fma}\left(y \cdot z, z, y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y \cdot \left(z \cdot \left(z \cdot x\right)\right)}\\ \end{array} \end{array} \]

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 (<= (* y (+ 1.0 (* z z))) 5e+302)
   (/ (/ 1.0 x) (fma (* y z) z y))
   (/ 1.0 (* y (* z (* z x))))))

C

z = abs(z);
assert(x < y);
double code(double x, double y, double z) {
	double tmp;
	if ((y * (1.0 + (z * z))) <= 5e+302) {
		tmp = (1.0 / x) / fma((y * z), z, y);
	} else {
		tmp = 1.0 / (y * (z * (z * x)));
	}
	return tmp;
}

Julia

z = abs(z)
x, y = sort([x, y])
function code(x, y, z)
	tmp = 0.0
	if (Float64(y * Float64(1.0 + Float64(z * z))) <= 5e+302)
		tmp = Float64(Float64(1.0 / x) / fma(Float64(y * z), z, y));
	else
		tmp = Float64(1.0 / Float64(y * Float64(z * Float64(z * x))));
	end
	return 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[(y * N[(1.0 + N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 5e+302], N[(N[(1.0 / x), $MachinePrecision] / N[(N[(y * z), $MachinePrecision] * z + y), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(y * N[(z * N[(z * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y (+.f64 1 (*.f64 z z))) < 5e302

   1. Initial program 93.4%

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

      1. distribute-lft-in 93.4%

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

      2. *-rgt-identity 93.4%

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

      3. +-commutative 93.4%

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

      4. associate-*r* 95.6%

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

      5. fma-def 95.6%

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

   3. Applied egg-rr 95.6%

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

   if 5e302 < (*.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%

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

      3. sqr-neg 64.0%

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

      4. +-commutative 64.0%

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

      5. distribute-lft1-in 64.0%

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

      6. *-commutative 64.0%

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

      7. fma-def 64.0%

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

      8. sqr-neg 64.0%

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

   3. Simplified 64.0%

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

   4. Taylor expanded in z around inf 64.0%

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

      1. *-commutative 64.0%

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

      2. unpow2 64.0%

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

      3. associate-*l* 74.5%

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

      4. associate-*l* 91.2%

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

   6. Simplified 91.2%

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

4. Final simplification 94.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(1 + z \cdot z\right) \leq 5 \cdot 10^{+302}:\\ \;\;\;\;\frac{\frac{1}{x}}{\mathsf{fma}\left(y \cdot z, z, y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y \cdot \left(z \cdot \left(z \cdot x\right)\right)}\\ \end{array} \]

Alternative 3: 97.5% accurate, 0.1× speedup

Math

\[\begin{array}{l} z = |z|\\ [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 10^{+111}:\\ \;\;\;\;\frac{1}{y \cdot \left(x \cdot \mathsf{fma}\left(z, z, 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x}}{y \cdot z} \cdot \frac{1}{z}\\ \end{array} \end{array} \]

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+111)
   (/ 1.0 (* y (* x (fma z z 1.0))))
   (* (/ (/ 1.0 x) (* y z)) (/ 1.0 z))))

C

z = abs(z);
assert(x < y);
double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 1e+111) {
		tmp = 1.0 / (y * (x * fma(z, z, 1.0)));
	} else {
		tmp = ((1.0 / x) / (y * z)) * (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+111)
		tmp = Float64(1.0 / Float64(y * Float64(x * fma(z, z, 1.0))));
	else
		tmp = Float64(Float64(Float64(1.0 / x) / Float64(y * z)) * Float64(1.0 / z));
	end
	return 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+111], N[(1.0 / N[(y * N[(x * N[(z * z + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 / x), $MachinePrecision] / N[(y * z), $MachinePrecision]), $MachinePrecision] * N[(1.0 / z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 9.99999999999999957e110

   1. Initial program 99.0%

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

      1. associate-/r* 98.9%

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

      2. *-commutative 98.9%

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

      3. sqr-neg 98.9%

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

      4. +-commutative 98.9%

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

      5. distribute-lft1-in 98.9%

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

      6. *-commutative 98.9%

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

      7. fma-def 98.9%

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

      8. sqr-neg 98.9%

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

   3. Simplified 98.9%

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

      1. fma-udef 98.9%

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

      2. +-commutative 98.9%

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

      3. *-commutative 98.9%

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

      4. distribute-rgt1-in 98.9%

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

      5. fma-udef 98.9%

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

      6. add-sqr-sqrt 98.9%

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

      7. associate-*l* 99.0%

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

      8. fma-udef 99.0%

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

      9. +-commutative 99.0%

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

      10. hypot-1-def 99.0%

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

      11. fma-udef 99.0%

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

      12. +-commutative 99.0%

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

      13. hypot-1-def 99.0%

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

   5. Applied egg-rr 99.0%

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

   6. Taylor expanded in x around 0 98.9%

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

      1. associate-*r* 99.0%

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

      2. unpow2 99.0%

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

      3. *-commutative 99.0%

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

      4. *-commutative 99.0%

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

      5. associate-*r* 98.9%

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

      6. *-commutative 98.9%

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

      7. associate-*l* 98.9%

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

      8. +-commutative 98.9%

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

      9. fma-def 98.9%

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

   8. Simplified 98.9%

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

   if 9.99999999999999957e110 < (*.f64 z z)

   1. Initial program 75.3%

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

      1. associate-/r* 75.3%

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

      2. *-commutative 75.3%

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

      3. sqr-neg 75.3%

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

      4. +-commutative 75.3%

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

      5. distribute-lft1-in 75.3%

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

      6. *-commutative 75.3%

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

      7. fma-def 75.3%

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

      8. sqr-neg 75.3%

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

   3. Simplified 75.3%

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

      1. fma-udef 75.3%

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

      2. +-commutative 75.3%

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

      3. *-commutative 75.3%

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

      4. distribute-rgt1-in 75.3%

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

      5. fma-udef 75.3%

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

      6. add-sqr-sqrt 75.3%

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

      7. associate-*l* 75.3%

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

      8. fma-udef 75.3%

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

      9. +-commutative 75.3%

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

      10. hypot-1-def 75.3%

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

      11. fma-udef 75.3%

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

      12. +-commutative 75.3%

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

      13. hypot-1-def 85.8%

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

   5. Applied egg-rr 85.8%

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

   6. Taylor expanded in z around inf 75.3%

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

      1. associate-/r* 75.3%

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

      2. unpow2 75.3%

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

      3. associate-*r* 85.8%

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

   8. Simplified 85.8%

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

      1. associate-/r* 94.9%

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

      2. div-inv 94.8%

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

   10. Applied egg-rr 94.8%

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

4. Final simplification 97.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 10^{+111}:\\ \;\;\;\;\frac{1}{y \cdot \left(x \cdot \mathsf{fma}\left(z, z, 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x}}{y \cdot z} \cdot \frac{1}{z}\\ \end{array} \]

Alternative 4: 96.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} z = |z|\\ [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_0 := y \cdot \left(1 + z \cdot z\right)\\ \mathbf{if}\;t_0 \leq -\infty:\\ \;\;\;\;\frac{\frac{1}{x}}{z \cdot \left(y \cdot z\right)}\\ \mathbf{elif}\;t_0 \leq 5 \cdot 10^{+302}:\\ \;\;\;\;\frac{\frac{1}{x}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y \cdot \left(z \cdot \left(z \cdot x\right)\right)}\\ \end{array} \end{array} \]

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
 (let* ((t_0 (* y (+ 1.0 (* z z)))))
   (if (<= t_0 (- INFINITY))
     (/ (/ 1.0 x) (* z (* y z)))
     (if (<= t_0 5e+302) (/ (/ 1.0 x) t_0) (/ 1.0 (* y (* z (* z x))))))))

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 <= 5e+302) {
		tmp = (1.0 / x) / t_0;
	} else {
		tmp = 1.0 / (y * (z * (z * x)));
	}
	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 <= 5e+302) {
		tmp = (1.0 / x) / t_0;
	} else {
		tmp = 1.0 / (y * (z * (z * x)));
	}
	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 <= 5e+302:
		tmp = (1.0 / x) / t_0
	else:
		tmp = 1.0 / (y * (z * (z * x)))
	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(Float64(1.0 / x) / Float64(z * Float64(y * z)));
	elseif (t_0 <= 5e+302)
		tmp = Float64(Float64(1.0 / x) / t_0);
	else
		tmp = Float64(1.0 / Float64(y * Float64(z * Float64(z * x))));
	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 <= 5e+302)
		tmp = (1.0 / x) / t_0;
	else
		tmp = 1.0 / (y * (z * (z * 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_] := Block[{t$95$0 = N[(y * N[(1.0 + N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[(1.0 / x), $MachinePrecision] / N[(z * N[(y * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 5e+302], N[(N[(1.0 / x), $MachinePrecision] / t$95$0), $MachinePrecision], N[(1.0 / N[(y * N[(z * N[(z * x), $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 72.3%

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

      1. associate-/r* 72.3%

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

      2. *-commutative 72.3%

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

      3. sqr-neg 72.3%

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

      4. +-commutative 72.3%

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

      5. distribute-lft1-in 72.3%

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

      6. *-commutative 72.3%

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

      7. fma-def 72.3%

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

      8. sqr-neg 72.3%

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

   3. Simplified 72.3%

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

      1. fma-udef 72.3%

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

      2. +-commutative 72.3%

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

      3. *-commutative 72.3%

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

      4. distribute-rgt1-in 72.3%

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

      5. fma-udef 72.3%

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

      6. add-sqr-sqrt 72.3%

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

      7. associate-*l* 72.3%

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

      8. fma-udef 72.3%

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

      9. +-commutative 72.3%

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

      10. hypot-1-def 72.3%

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

      11. fma-udef 72.3%

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

      12. +-commutative 72.3%

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

      13. hypot-1-def 82.2%

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

   5. Applied egg-rr 82.2%

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

   6. Taylor expanded in z around inf 72.3%

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

      1. associate-/r* 72.3%

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

      2. unpow2 72.3%

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

      3. associate-*r* 82.2%

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

   8. Simplified 82.2%

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

   if -inf.0 < (*.f64 y (+.f64 1 (*.f64 z z))) < 5e302

   1. Initial program 99.6%

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

   if 5e302 < (*.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%

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

      3. sqr-neg 64.0%

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

      4. +-commutative 64.0%

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

      5. distribute-lft1-in 64.0%

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

      6. *-commutative 64.0%

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

      7. fma-def 64.0%

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

      8. sqr-neg 64.0%

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

   3. Simplified 64.0%

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

   4. Taylor expanded in z around inf 64.0%

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

      1. *-commutative 64.0%

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

      2. unpow2 64.0%

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

      3. associate-*l* 74.5%

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

      4. associate-*l* 91.2%

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

   6. Simplified 91.2%

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

4. Final simplification 94.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(1 + z \cdot z\right) \leq -\infty:\\ \;\;\;\;\frac{\frac{1}{x}}{z \cdot \left(y \cdot z\right)}\\ \mathbf{elif}\;y \cdot \left(1 + z \cdot z\right) \leq 5 \cdot 10^{+302}:\\ \;\;\;\;\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y \cdot \left(z \cdot \left(z \cdot x\right)\right)}\\ \end{array} \]

Alternative 5: 97.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} z = |z|\\ [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 10^{+111}:\\ \;\;\;\;\frac{1}{1 + z \cdot z} \cdot \frac{\frac{1}{x}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x}}{y \cdot z} \cdot \frac{1}{z}\\ \end{array} \end{array} \]

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+111)
   (* (/ 1.0 (+ 1.0 (* z z))) (/ (/ 1.0 x) y))
   (* (/ (/ 1.0 x) (* y z)) (/ 1.0 z))))

C

z = abs(z);
assert(x < y);
double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 1e+111) {
		tmp = (1.0 / (1.0 + (z * z))) * ((1.0 / x) / y);
	} else {
		tmp = ((1.0 / x) / (y * z)) * (1.0 / 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+111) then
        tmp = (1.0d0 / (1.0d0 + (z * z))) * ((1.0d0 / x) / y)
    else
        tmp = ((1.0d0 / x) / (y * z)) * (1.0d0 / 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+111) {
		tmp = (1.0 / (1.0 + (z * z))) * ((1.0 / x) / y);
	} else {
		tmp = ((1.0 / x) / (y * z)) * (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+111:
		tmp = (1.0 / (1.0 + (z * z))) * ((1.0 / x) / y)
	else:
		tmp = ((1.0 / x) / (y * z)) * (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+111)
		tmp = Float64(Float64(1.0 / Float64(1.0 + Float64(z * z))) * Float64(Float64(1.0 / x) / y));
	else
		tmp = Float64(Float64(Float64(1.0 / x) / Float64(y * z)) * Float64(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+111)
		tmp = (1.0 / (1.0 + (z * z))) * ((1.0 / x) / y);
	else
		tmp = ((1.0 / x) / (y * z)) * (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+111], N[(N[(1.0 / N[(1.0 + N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 / x), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 / x), $MachinePrecision] / N[(y * z), $MachinePrecision]), $MachinePrecision] * N[(1.0 / z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 9.99999999999999957e110

   1. Initial program 99.0%

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

      1. associate-/r* 98.9%

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

      2. *-commutative 98.9%

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

      3. sqr-neg 98.9%

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

      4. +-commutative 98.9%

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

      5. distribute-lft1-in 98.9%

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

      6. *-commutative 98.9%

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

      7. fma-def 98.9%

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

      8. sqr-neg 98.9%

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

   3. Simplified 98.9%

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

      1. fma-udef 98.9%

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

      2. +-commutative 98.9%

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

      3. *-commutative 98.9%

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

      4. distribute-rgt1-in 98.9%

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

      5. fma-udef 98.9%

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

      6. add-sqr-sqrt 98.9%

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

      7. associate-*l* 99.0%

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

      8. fma-udef 99.0%

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

      9. +-commutative 99.0%

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

      10. hypot-1-def 99.0%

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

      11. fma-udef 99.0%

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

      12. +-commutative 99.0%

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

      13. hypot-1-def 99.0%

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

   5. Applied egg-rr 99.0%

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

      1. associate-/r* 99.0%

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

      2. *-un-lft-identity 99.0%

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

      3. associate-*r* 99.0%

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

      4. times-frac 99.0%

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

      5. hypot-udef 99.0%

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

      6. hypot-udef 99.0%

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

      7. add-sqr-sqrt 99.0%

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

      8. metadata-eval 99.0%

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

   7. Applied egg-rr 99.0%

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

   if 9.99999999999999957e110 < (*.f64 z z)

   1. Initial program 75.3%

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

      1. associate-/r* 75.3%

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

      2. *-commutative 75.3%

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

      3. sqr-neg 75.3%

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

      4. +-commutative 75.3%

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

      5. distribute-lft1-in 75.3%

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

      6. *-commutative 75.3%

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

      7. fma-def 75.3%

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

      8. sqr-neg 75.3%

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

   3. Simplified 75.3%

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

      1. fma-udef 75.3%

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

      2. +-commutative 75.3%

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

      3. *-commutative 75.3%

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

      4. distribute-rgt1-in 75.3%

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

      5. fma-udef 75.3%

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

      6. add-sqr-sqrt 75.3%

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

      7. associate-*l* 75.3%

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

      8. fma-udef 75.3%

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

      9. +-commutative 75.3%

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

      10. hypot-1-def 75.3%

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

      11. fma-udef 75.3%

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

      12. +-commutative 75.3%

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

      13. hypot-1-def 85.8%

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

   5. Applied egg-rr 85.8%

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

   6. Taylor expanded in z around inf 75.3%

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

      1. associate-/r* 75.3%

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

      2. unpow2 75.3%

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

      3. associate-*r* 85.8%

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

   8. Simplified 85.8%

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

      1. associate-/r* 94.9%

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

      2. div-inv 94.8%

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

   10. Applied egg-rr 94.8%

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

4. Final simplification 97.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 10^{+111}:\\ \;\;\;\;\frac{1}{1 + z \cdot z} \cdot \frac{\frac{1}{x}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x}}{y \cdot z} \cdot \frac{1}{z}\\ \end{array} \]

Alternative 6: 96.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} z = |z|\\ [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{-13}:\\ \;\;\;\;\frac{1 - z \cdot z}{y \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x}}{y \cdot z} \cdot \frac{1}{z}\\ \end{array} \end{array} \]

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-13)
   (/ (- 1.0 (* z z)) (* y x))
   (* (/ (/ 1.0 x) (* y z)) (/ 1.0 z))))

C

z = abs(z);
assert(x < y);
double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 2e-13) {
		tmp = (1.0 - (z * z)) / (y * x);
	} else {
		tmp = ((1.0 / x) / (y * z)) * (1.0 / 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-13) then
        tmp = (1.0d0 - (z * z)) / (y * x)
    else
        tmp = ((1.0d0 / x) / (y * z)) * (1.0d0 / 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-13) {
		tmp = (1.0 - (z * z)) / (y * x);
	} else {
		tmp = ((1.0 / x) / (y * z)) * (1.0 / z);
	}
	return tmp;
}

Python

z = abs(z)
[x, y] = sort([x, y])
def code(x, y, z):
	tmp = 0
	if (z * z) <= 2e-13:
		tmp = (1.0 - (z * z)) / (y * x)
	else:
		tmp = ((1.0 / x) / (y * z)) * (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) <= 2e-13)
		tmp = Float64(Float64(1.0 - Float64(z * z)) / Float64(y * x));
	else
		tmp = Float64(Float64(Float64(1.0 / x) / Float64(y * z)) * Float64(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) <= 2e-13)
		tmp = (1.0 - (z * z)) / (y * x);
	else
		tmp = ((1.0 / x) / (y * z)) * (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], 2e-13], N[(N[(1.0 - N[(z * z), $MachinePrecision]), $MachinePrecision] / N[(y * x), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 / x), $MachinePrecision] / N[(y * z), $MachinePrecision]), $MachinePrecision] * N[(1.0 / z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 2.0000000000000001e-13

   1. Initial program 99.6%

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

      1. associate-/r* 99.6%

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

      2. *-commutative 99.6%

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

      3. sqr-neg 99.6%

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

      4. +-commutative 99.6%

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

      5. distribute-lft1-in 99.6%

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

      6. *-commutative 99.6%

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

      7. fma-def 99.6%

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

      8. sqr-neg 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in z around 0 89.8%

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

      1. associate-/l/ 89.8%

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

      2. +-commutative 89.8%

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

      3. mul-1-neg 89.8%

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

      4. unsub-neg 89.8%

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

      5. associate-/r* 89.8%

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

      6. unpow2 89.8%

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

      7. *-commutative 89.8%

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

      8. div-sub 99.6%

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

   6. Simplified 99.6%

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

   if 2.0000000000000001e-13 < (*.f64 z z)

   1. Initial program 79.4%

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

      1. associate-/r* 79.4%

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

      2. *-commutative 79.4%

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

      3. sqr-neg 79.4%

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

      4. +-commutative 79.4%

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

      5. distribute-lft1-in 79.4%

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

      6. *-commutative 79.4%

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

      7. fma-def 79.4%

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

      8. sqr-neg 79.4%

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

   3. Simplified 79.4%

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

      1. fma-udef 79.4%

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

      2. +-commutative 79.4%

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

      3. *-commutative 79.4%

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

      4. distribute-rgt1-in 79.4%

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

      5. fma-udef 79.4%

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

      6. add-sqr-sqrt 79.4%

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

      7. associate-*l* 79.4%

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

      8. fma-udef 79.4%

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

      9. +-commutative 79.4%

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

      10. hypot-1-def 79.4%

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

      11. fma-udef 79.4%

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

      12. +-commutative 79.4%

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

      13. hypot-1-def 87.8%

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

   5. Applied egg-rr 87.8%

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

   6. Taylor expanded in z around inf 79.4%

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

      1. associate-/r* 79.4%

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

      2. unpow2 79.4%

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

      3. associate-*r* 87.8%

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

   8. Simplified 87.8%

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

      1. associate-/r* 94.5%

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

      2. div-inv 94.4%

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

   10. Applied egg-rr 94.4%

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

4. Final simplification 96.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{-13}:\\ \;\;\;\;\frac{1 - z \cdot z}{y \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x}}{y \cdot z} \cdot \frac{1}{z}\\ \end{array} \]

Alternative 7: 89.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} z = |z|\\ [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{-13}:\\ \;\;\;\;\frac{\frac{1}{x}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x \cdot \left(y \cdot \left(z \cdot z\right)\right)}\\ \end{array} \end{array} \]

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-13) (/ (/ 1.0 x) y) (/ 1.0 (* x (* y (* z z))))))

C

z = abs(z);
assert(x < y);
double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 2e-13) {
		tmp = (1.0 / x) / y;
	} else {
		tmp = 1.0 / (x * (y * (z * 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-13) then
        tmp = (1.0d0 / x) / y
    else
        tmp = 1.0d0 / (x * (y * (z * 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-13) {
		tmp = (1.0 / x) / y;
	} else {
		tmp = 1.0 / (x * (y * (z * z)));
	}
	return tmp;
}

Python

z = abs(z)
[x, y] = sort([x, y])
def code(x, y, z):
	tmp = 0
	if (z * z) <= 2e-13:
		tmp = (1.0 / x) / y
	else:
		tmp = 1.0 / (x * (y * (z * 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-13)
		tmp = Float64(Float64(1.0 / x) / y);
	else
		tmp = Float64(1.0 / Float64(x * Float64(y * Float64(z * 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-13)
		tmp = (1.0 / x) / y;
	else
		tmp = 1.0 / (x * (y * (z * 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-13], N[(N[(1.0 / x), $MachinePrecision] / y), $MachinePrecision], N[(1.0 / N[(x * N[(y * N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 2.0000000000000001e-13

   1. Initial program 99.6%

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

      1. associate-/r* 99.6%

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

      2. *-commutative 99.6%

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

      3. sqr-neg 99.6%

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

      4. +-commutative 99.6%

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

      5. distribute-lft1-in 99.6%

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

      6. *-commutative 99.6%

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

      7. fma-def 99.6%

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

      8. sqr-neg 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in z around 0 99.4%

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

      1. *-commutative 99.4%

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

   6. Simplified 99.4%

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

      1. associate-/r* 99.5%

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

      2. div-inv 99.3%

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

   8. Applied egg-rr 99.3%

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

      1. associate-*l/ 99.5%

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

      2. *-un-lft-identity 99.5%

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

   10. Applied egg-rr 99.5%

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

   if 2.0000000000000001e-13 < (*.f64 z z)

   1. Initial program 79.4%

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

      1. associate-/r* 79.4%

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

      2. *-commutative 79.4%

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

      3. sqr-neg 79.4%

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

      4. +-commutative 79.4%

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

      5. distribute-lft1-in 79.4%

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

      6. *-commutative 79.4%

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

      7. fma-def 79.4%

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

      8. sqr-neg 79.4%

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

   3. Simplified 79.4%

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

   4. Taylor expanded in z around inf 79.4%

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

      1. unpow2 79.4%

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

   6. Simplified 79.4%

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

4. Final simplification 88.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{-13}:\\ \;\;\;\;\frac{\frac{1}{x}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x \cdot \left(y \cdot \left(z \cdot z\right)\right)}\\ \end{array} \]

Alternative 8: 92.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} z = |z|\\ [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{-13}:\\ \;\;\;\;\frac{\frac{1}{x}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y \cdot \left(z \cdot \left(z \cdot x\right)\right)}\\ \end{array} \end{array} \]

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-13) (/ (/ 1.0 x) y) (/ 1.0 (* y (* z (* z x))))))

C

z = abs(z);
assert(x < y);
double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 2e-13) {
		tmp = (1.0 / x) / y;
	} else {
		tmp = 1.0 / (y * (z * (z * 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-13) then
        tmp = (1.0d0 / x) / y
    else
        tmp = 1.0d0 / (y * (z * (z * 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-13) {
		tmp = (1.0 / x) / y;
	} else {
		tmp = 1.0 / (y * (z * (z * x)));
	}
	return tmp;
}

Python

z = abs(z)
[x, y] = sort([x, y])
def code(x, y, z):
	tmp = 0
	if (z * z) <= 2e-13:
		tmp = (1.0 / x) / y
	else:
		tmp = 1.0 / (y * (z * (z * 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-13)
		tmp = Float64(Float64(1.0 / x) / y);
	else
		tmp = Float64(1.0 / Float64(y * Float64(z * Float64(z * 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-13)
		tmp = (1.0 / x) / y;
	else
		tmp = 1.0 / (y * (z * (z * 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-13], N[(N[(1.0 / x), $MachinePrecision] / y), $MachinePrecision], N[(1.0 / N[(y * N[(z * N[(z * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 2.0000000000000001e-13

   1. Initial program 99.6%

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

      1. associate-/r* 99.6%

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

      2. *-commutative 99.6%

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

      3. sqr-neg 99.6%

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

      4. +-commutative 99.6%

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

      5. distribute-lft1-in 99.6%

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

      6. *-commutative 99.6%

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

      7. fma-def 99.6%

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

      8. sqr-neg 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in z around 0 99.4%

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

      1. *-commutative 99.4%

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

   6. Simplified 99.4%

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

      1. associate-/r* 99.5%

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

      2. div-inv 99.3%

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

   8. Applied egg-rr 99.3%

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

      1. associate-*l/ 99.5%

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

      2. *-un-lft-identity 99.5%

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

   10. Applied egg-rr 99.5%

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

   if 2.0000000000000001e-13 < (*.f64 z z)

   1. Initial program 79.4%

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

      1. associate-/r* 79.4%

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

      2. *-commutative 79.4%

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

      3. sqr-neg 79.4%

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

      4. +-commutative 79.4%

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

      5. distribute-lft1-in 79.4%

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

      6. *-commutative 79.4%

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

      7. fma-def 79.4%

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

      8. sqr-neg 79.4%

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

   3. Simplified 79.4%

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

   4. Taylor expanded in z around inf 79.4%

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

      1. *-commutative 79.4%

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

      2. unpow2 79.4%

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

      3. associate-*l* 81.4%

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

      4. associate-*l* 91.9%

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

   6. Simplified 91.9%

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

4. Final simplification 95.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{-13}:\\ \;\;\;\;\frac{\frac{1}{x}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y \cdot \left(z \cdot \left(z \cdot x\right)\right)}\\ \end{array} \]

Alternative 9: 96.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} z = |z|\\ [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{-13}:\\ \;\;\;\;\frac{\frac{1}{x}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(y \cdot z\right) \cdot \left(z \cdot x\right)}\\ \end{array} \end{array} \]

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-13) (/ (/ 1.0 x) y) (/ 1.0 (* (* y z) (* z x)))))

C

z = abs(z);
assert(x < y);
double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 2e-13) {
		tmp = (1.0 / x) / y;
	} else {
		tmp = 1.0 / ((y * z) * (z * 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-13) then
        tmp = (1.0d0 / x) / y
    else
        tmp = 1.0d0 / ((y * z) * (z * 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-13) {
		tmp = (1.0 / x) / y;
	} else {
		tmp = 1.0 / ((y * z) * (z * x));
	}
	return tmp;
}

Python

z = abs(z)
[x, y] = sort([x, y])
def code(x, y, z):
	tmp = 0
	if (z * z) <= 2e-13:
		tmp = (1.0 / x) / y
	else:
		tmp = 1.0 / ((y * z) * (z * 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-13)
		tmp = Float64(Float64(1.0 / x) / y);
	else
		tmp = Float64(1.0 / Float64(Float64(y * z) * Float64(z * 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-13)
		tmp = (1.0 / x) / y;
	else
		tmp = 1.0 / ((y * z) * (z * 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-13], N[(N[(1.0 / x), $MachinePrecision] / y), $MachinePrecision], N[(1.0 / N[(N[(y * z), $MachinePrecision] * N[(z * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 2.0000000000000001e-13

   1. Initial program 99.6%

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

      1. associate-/r* 99.6%

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

      2. *-commutative 99.6%

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

      3. sqr-neg 99.6%

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

      4. +-commutative 99.6%

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

      5. distribute-lft1-in 99.6%

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

      6. *-commutative 99.6%

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

      7. fma-def 99.6%

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

      8. sqr-neg 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in z around 0 99.4%

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

      1. *-commutative 99.4%

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

   6. Simplified 99.4%

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

      1. associate-/r* 99.5%

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

      2. div-inv 99.3%

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

   8. Applied egg-rr 99.3%

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

      1. associate-*l/ 99.5%

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

      2. *-un-lft-identity 99.5%

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

   10. Applied egg-rr 99.5%

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

   if 2.0000000000000001e-13 < (*.f64 z z)

   1. Initial program 79.4%

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

      1. associate-/r* 79.4%

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

      2. *-commutative 79.4%

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

      3. sqr-neg 79.4%

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

      4. +-commutative 79.4%

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

      5. distribute-lft1-in 79.4%

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

      6. *-commutative 79.4%

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

      7. fma-def 79.4%

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

      8. sqr-neg 79.4%

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

   3. Simplified 79.4%

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

      1. fma-udef 79.4%

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

      2. +-commutative 79.4%

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

      3. *-commutative 79.4%

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

      4. distribute-rgt1-in 79.4%

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

      5. fma-udef 79.4%

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

      6. add-sqr-sqrt 79.4%

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

      7. associate-*l* 79.4%

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

      8. fma-udef 79.4%

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

      9. +-commutative 79.4%

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

      10. hypot-1-def 79.4%

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

      11. fma-udef 79.4%

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

      12. +-commutative 79.4%

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

      13. hypot-1-def 87.8%

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

   5. Applied egg-rr 87.8%

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

   6. Taylor expanded in z around inf 79.4%

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

      1. *-commutative 79.4%

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

      2. unpow2 79.4%

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

      3. associate-*r* 87.8%

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

      4. *-commutative 87.8%

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

      5. associate-*l* 93.1%

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

      6. *-commutative 93.1%

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

   8. Simplified 93.1%

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

4. Final simplification 95.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{-13}:\\ \;\;\;\;\frac{\frac{1}{x}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(y \cdot z\right) \cdot \left(z \cdot x\right)}\\ \end{array} \]

Alternative 10: 96.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} z = |z|\\ [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{-13}:\\ \;\;\;\;\frac{1 - z \cdot z}{y \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(y \cdot z\right) \cdot \left(z \cdot x\right)}\\ \end{array} \end{array} \]

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-13)
   (/ (- 1.0 (* z z)) (* y x))
   (/ 1.0 (* (* y z) (* z x)))))

C

z = abs(z);
assert(x < y);
double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 2e-13) {
		tmp = (1.0 - (z * z)) / (y * x);
	} else {
		tmp = 1.0 / ((y * z) * (z * 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-13) then
        tmp = (1.0d0 - (z * z)) / (y * x)
    else
        tmp = 1.0d0 / ((y * z) * (z * 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-13) {
		tmp = (1.0 - (z * z)) / (y * x);
	} else {
		tmp = 1.0 / ((y * z) * (z * x));
	}
	return tmp;
}

Python

z = abs(z)
[x, y] = sort([x, y])
def code(x, y, z):
	tmp = 0
	if (z * z) <= 2e-13:
		tmp = (1.0 - (z * z)) / (y * x)
	else:
		tmp = 1.0 / ((y * z) * (z * 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-13)
		tmp = Float64(Float64(1.0 - Float64(z * z)) / Float64(y * x));
	else
		tmp = Float64(1.0 / Float64(Float64(y * z) * Float64(z * 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-13)
		tmp = (1.0 - (z * z)) / (y * x);
	else
		tmp = 1.0 / ((y * z) * (z * 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-13], N[(N[(1.0 - N[(z * z), $MachinePrecision]), $MachinePrecision] / N[(y * x), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(y * z), $MachinePrecision] * N[(z * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 2.0000000000000001e-13

   1. Initial program 99.6%

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

      1. associate-/r* 99.6%

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

      2. *-commutative 99.6%

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

      3. sqr-neg 99.6%

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

      4. +-commutative 99.6%

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

      5. distribute-lft1-in 99.6%

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

      6. *-commutative 99.6%

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

      7. fma-def 99.6%

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

      8. sqr-neg 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in z around 0 89.8%

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

      1. associate-/l/ 89.8%

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

      2. +-commutative 89.8%

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

      3. mul-1-neg 89.8%

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

      4. unsub-neg 89.8%

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

      5. associate-/r* 89.8%

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

      6. unpow2 89.8%

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

      7. *-commutative 89.8%

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

      8. div-sub 99.6%

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

   6. Simplified 99.6%

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

   if 2.0000000000000001e-13 < (*.f64 z z)

   1. Initial program 79.4%

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

      1. associate-/r* 79.4%

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

      2. *-commutative 79.4%

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

      3. sqr-neg 79.4%

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

      4. +-commutative 79.4%

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

      5. distribute-lft1-in 79.4%

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

      6. *-commutative 79.4%

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

      7. fma-def 79.4%

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

      8. sqr-neg 79.4%

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

   3. Simplified 79.4%

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

      1. fma-udef 79.4%

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

      2. +-commutative 79.4%

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

      3. *-commutative 79.4%

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

      4. distribute-rgt1-in 79.4%

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

      5. fma-udef 79.4%

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

      6. add-sqr-sqrt 79.4%

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

      7. associate-*l* 79.4%

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

      8. fma-udef 79.4%

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

      9. +-commutative 79.4%

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

      10. hypot-1-def 79.4%

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

      11. fma-udef 79.4%

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

      12. +-commutative 79.4%

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

      13. hypot-1-def 87.8%

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

   5. Applied egg-rr 87.8%

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

   6. Taylor expanded in z around inf 79.4%

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

      1. *-commutative 79.4%

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

      2. unpow2 79.4%

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

      3. associate-*r* 87.8%

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

      4. *-commutative 87.8%

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

      5. associate-*l* 93.1%

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

      6. *-commutative 93.1%

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

   8. Simplified 93.1%

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

4. Final simplification 96.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{-13}:\\ \;\;\;\;\frac{1 - z \cdot z}{y \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(y \cdot z\right) \cdot \left(z \cdot x\right)}\\ \end{array} \]

Alternative 11: 69.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} z = |z|\\ [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq 2 \cdot 10^{+15}:\\ \;\;\;\;\frac{\frac{1}{x}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x \cdot \left(y \cdot z\right)}\\ \end{array} \end{array} \]

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 2e+15) (/ (/ 1.0 x) y) (/ 1.0 (* x (* y z)))))

C

z = abs(z);
assert(x < y);
double code(double x, double y, double z) {
	double tmp;
	if (z <= 2e+15) {
		tmp = (1.0 / x) / y;
	} else {
		tmp = 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 <= 2d+15) then
        tmp = (1.0d0 / x) / y
    else
        tmp = 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 <= 2e+15) {
		tmp = (1.0 / x) / y;
	} else {
		tmp = 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 <= 2e+15:
		tmp = (1.0 / x) / y
	else:
		tmp = 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 (z <= 2e+15)
		tmp = Float64(Float64(1.0 / x) / y);
	else
		tmp = Float64(1.0 / Float64(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 <= 2e+15)
		tmp = (1.0 / x) / y;
	else
		tmp = 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[z, 2e+15], N[(N[(1.0 / x), $MachinePrecision] / y), $MachinePrecision], N[(1.0 / N[(x * N[(y * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 2e15

   1. Initial program 90.6%

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

      1. associate-/r* 90.6%

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

      2. *-commutative 90.6%

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

      3. sqr-neg 90.6%

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

      4. +-commutative 90.6%

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

      5. distribute-lft1-in 90.6%

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

      6. *-commutative 90.6%

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

      7. fma-def 90.6%

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

      8. sqr-neg 90.6%

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

   3. Simplified 90.6%

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

   4. Taylor expanded in z around 0 68.3%

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

      1. *-commutative 68.3%

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

   6. Simplified 68.3%

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

      1. associate-/r* 68.0%

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

      2. div-inv 67.9%

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

   8. Applied egg-rr 67.9%

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

      1. associate-*l/ 68.0%

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

      2. *-un-lft-identity 68.0%

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

   10. Applied egg-rr 68.0%

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

   if 2e15 < z

   1. Initial program 81.9%

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

      1. associate-/r* 81.9%

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

      2. *-commutative 81.9%

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

      3. sqr-neg 81.9%

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

      4. +-commutative 81.9%

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

      5. distribute-lft1-in 81.9%

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

      6. *-commutative 81.9%

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

      7. fma-def 81.9%

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

      8. sqr-neg 81.9%

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

   3. Simplified 81.9%

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

      1. *-commutative 81.9%

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

      2. fma-udef 81.9%

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

      3. *-rgt-identity 81.9%

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

      4. distribute-lft-in 81.9%

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

      5. +-commutative 81.9%

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

      6. /-rgt-identity 81.9%

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

      7. clear-num 81.9%

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

      8. div-inv 81.9%

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

      9. +-commutative 81.9%

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

      10. fma-udef 81.9%

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

      11. *-commutative 81.9%

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

      12. associate-/l* 81.7%

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

      13. add-sqr-sqrt 81.7%

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

      14. associate-/r* 81.8%

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

      15. associate-/l* 81.8%

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

      16. fma-udef 81.8%

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

      17. +-commutative 81.8%

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

      18. hypot-1-def 81.8%

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

      19. *-commutative 81.8%

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

      20. associate-/r* 81.8%

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

      21. fma-udef 81.8%

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

      22. +-commutative 81.8%

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

      23. hypot-1-def 86.0%

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

   5. Applied egg-rr 86.0%

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

      1. associate-/r/ 86.1%

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

      2. associate-/r* 86.1%

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

   7. Simplified 86.1%

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

   8. Taylor expanded in z around inf 94.6%

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

   9. Taylor expanded in z around 0 41.2%

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

4. Final simplification 60.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 2 \cdot 10^{+15}:\\ \;\;\;\;\frac{\frac{1}{x}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x \cdot \left(y \cdot z\right)}\\ \end{array} \]

Alternative 12: 57.8% 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 88.2%

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

   1. associate-/r* 88.2%

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

   2. *-commutative 88.2%

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

   3. sqr-neg 88.2%

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

   4. +-commutative 88.2%

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

   5. distribute-lft1-in 88.2%

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

   6. *-commutative 88.2%

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

   7. fma-def 88.2%

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

   8. sqr-neg 88.2%

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

3. Simplified 88.2%

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

4. Taylor expanded in z around 0 56.1%

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

   1. *-commutative 56.1%

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

6. Simplified 56.1%

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

7. Final simplification 56.1%

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

Developer target: 92.2% 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 2023279 
(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)))))