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

Percentage Accurate: 90.7% → 97.7%

Time: 12.8s

Alternatives: 10

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: 90.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 97.7% accurate, 0.1× speedup

Math

\[\begin{array}{l} z = |z|\\ [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq 2 \cdot 10^{+91}:\\ \;\;\;\;\frac{\frac{--1}{x \cdot y}}{\mathsf{fma}\left(z, z, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{z}}{x \cdot \left(z \cdot y\right)}\\ \end{array} \end{array} \]

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+91)
   (/ (/ (- -1.0) (* x y)) (fma z z 1.0))
   (/ (/ 1.0 z) (* x (* z y)))))

C

z = abs(z);
assert(x < y);
double code(double x, double y, double z) {
	double tmp;
	if (z <= 2e+91) {
		tmp = (-(-1.0) / (x * y)) / fma(z, z, 1.0);
	} else {
		tmp = (1.0 / z) / (x * (z * y));
	}
	return tmp;
}

Julia

z = abs(z)
x, y = sort([x, y])
function code(x, y, z)
	tmp = 0.0
	if (z <= 2e+91)
		tmp = Float64(Float64(Float64(-(-1.0)) / Float64(x * y)) / fma(z, z, 1.0));
	else
		tmp = Float64(Float64(1.0 / z) / Float64(x * Float64(z * y)));
	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[z, 2e+91], N[(N[((--1.0) / N[(x * y), $MachinePrecision]), $MachinePrecision] / N[(z * z + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / z), $MachinePrecision] / N[(x * N[(z * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 2.00000000000000016e91

   1. Initial program 95.3%

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

      1. associate-/l/ 94.6%

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

      2. associate-*l* 92.9%

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

      3. *-commutative 92.9%

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

      4. sqr-neg 92.9%

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

      5. +-commutative 92.9%

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

      6. sqr-neg 92.9%

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

      7. fma-def 92.9%

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

   3. Simplified 92.9%

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

      1. frac-2neg 92.9%

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

      2. div-inv 92.9%

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

      3. metadata-eval 92.9%

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

      4. mul-1-neg 92.9%

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

      5. neg-sub0 92.9%

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

      6. *-commutative 92.9%

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

      7. distribute-rgt-neg-in 92.9%

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

      8. associate-/r* 92.9%

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

      9. associate-/r* 93.4%

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

      10. associate-/r* 95.3%

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

      11. div-inv 95.3%

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

      12. times-frac 95.3%

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

      13. remove-double-neg 95.3%

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

      14. distribute-neg-frac 95.3%

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

      15. frac-2neg 95.3%

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

      16. cancel-sign-sub-inv 95.3%

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

   5. Applied egg-rr 95.3%

      \[\leadsto \color{blue}{0 + \left(-\frac{1}{\mathsf{fma}\left(z, z, 1\right)}\right) \cdot \frac{\frac{-1}{x}}{y}} \]
   6. Step-by-step derivation

      1. +-lft-identity 95.3%

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

      2. distribute-neg-frac 95.3%

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

      3. metadata-eval 95.3%

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

      4. associate-*l/ 95.3%

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

      5. mul-1-neg 95.3%

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

      6. associate-/l/ 95.3%

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

      7. *-commutative 95.3%

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

   7. Simplified 95.3%

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

   if 2.00000000000000016e91 < z

   1. Initial program 74.9%

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

      1. associate-/l/ 75.0%

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

      2. associate-*l* 77.3%

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

      3. *-commutative 77.3%

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

      4. sqr-neg 77.3%

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

      5. +-commutative 77.3%

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

      6. sqr-neg 77.3%

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

      7. fma-def 77.3%

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

   3. Simplified 77.3%

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

   4. Taylor expanded in z around inf 77.3%

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

      1. unpow2 77.3%

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

   6. Simplified 77.3%

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

      1. associate-/r* 77.1%

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

      2. associate-*r* 84.1%

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

      3. associate-/r* 93.3%

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

      4. div-inv 93.2%

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

      5. associate-/l/ 93.2%

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

      6. *-commutative 93.2%

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

      7. associate-*r* 80.0%

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

      8. *-commutative 80.0%

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

      9. associate-*l* 93.2%

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

   8. Applied egg-rr 93.2%

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

      1. unpow-1 93.2%

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

      2. associate-*l/ 93.2%

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

      3. *-lft-identity 93.2%

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

      4. unpow-1 93.2%

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

   10. Simplified 93.2%

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

4. Final simplification 95.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 2 \cdot 10^{+91}:\\ \;\;\;\;\frac{\frac{--1}{x \cdot y}}{\mathsf{fma}\left(z, z, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{z}}{x \cdot \left(z \cdot y\right)}\\ \end{array} \]

Alternative 2: 97.3% accurate, 0.1× speedup

Math

\[\begin{array}{l} z = |z|\\ [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq 6.8 \cdot 10^{+140}:\\ \;\;\;\;\frac{\frac{1}{x}}{\mathsf{fma}\left(z \cdot y, z, y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{z \cdot x}{\frac{\frac{1}{y}}{z}}}\\ \end{array} \end{array} \]

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 6.8e+140)
   (/ (/ 1.0 x) (fma (* z y) z y))
   (/ 1.0 (/ (* z x) (/ (/ 1.0 y) z)))))

C

z = abs(z);
assert(x < y);
double code(double x, double y, double z) {
	double tmp;
	if (z <= 6.8e+140) {
		tmp = (1.0 / x) / fma((z * y), z, y);
	} else {
		tmp = 1.0 / ((z * x) / ((1.0 / y) / z));
	}
	return tmp;
}

Julia

z = abs(z)
x, y = sort([x, y])
function code(x, y, z)
	tmp = 0.0
	if (z <= 6.8e+140)
		tmp = Float64(Float64(1.0 / x) / fma(Float64(z * y), z, y));
	else
		tmp = Float64(1.0 / Float64(Float64(z * x) / Float64(Float64(1.0 / y) / 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[z, 6.8e+140], N[(N[(1.0 / x), $MachinePrecision] / N[(N[(z * y), $MachinePrecision] * z + y), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(z * x), $MachinePrecision] / N[(N[(1.0 / y), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 6.8e140

   1. Initial program 94.7%

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

   2. Taylor expanded in y around 0 94.7%

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

      1. unpow2 94.7%

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

      2. distribute-lft-in 94.7%

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

      3. +-commutative 94.7%

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

      4. *-rgt-identity 94.7%

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

      5. associate-*r* 95.6%

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

      6. fma-udef 95.6%

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

   4. Simplified 95.6%

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

   if 6.8e140 < z

   1. Initial program 71.4%

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

      1. associate-/l/ 71.3%

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

      2. associate-*l* 71.3%

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

      3. *-commutative 71.3%

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

      4. sqr-neg 71.3%

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

      5. +-commutative 71.3%

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

      6. sqr-neg 71.3%

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

      7. fma-def 71.3%

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

   3. Simplified 71.3%

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

   4. Taylor expanded in z around inf 71.3%

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

      1. unpow2 71.3%

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

   6. Simplified 71.3%

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

      1. /-rgt-identity 71.3%

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

      2. *-commutative 71.3%

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

      3. associate-/l* 71.3%

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

      4. associate-*r* 80.8%

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

      5. associate-/l* 96.6%

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

   8. Applied egg-rr 96.6%

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

4. Final simplification 95.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 6.8 \cdot 10^{+140}:\\ \;\;\;\;\frac{\frac{1}{x}}{\mathsf{fma}\left(z \cdot y, z, y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{z \cdot x}{\frac{\frac{1}{y}}{z}}}\\ \end{array} \]

Alternative 3: 97.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} z = |z|\\ [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 10^{+40}:\\ \;\;\;\;\frac{1}{y \cdot \left(x + x \cdot \left(z \cdot z\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{z}}{x \cdot \left(z \cdot y\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: z should be positive before calling this function
NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= (* z z) 1e+40)
   (/ 1.0 (* y (+ x (* x (* z z)))))
   (/ (/ 1.0 z) (* x (* z y)))))

C

z = abs(z);
assert(x < y);
double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 1e+40) {
		tmp = 1.0 / (y * (x + (x * (z * z))));
	} else {
		tmp = (1.0 / z) / (x * (z * y));
	}
	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+40) then
        tmp = 1.0d0 / (y * (x + (x * (z * z))))
    else
        tmp = (1.0d0 / z) / (x * (z * y))
    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+40) {
		tmp = 1.0 / (y * (x + (x * (z * z))));
	} else {
		tmp = (1.0 / z) / (x * (z * y));
	}
	return tmp;
}

Python

z = abs(z)
[x, y] = sort([x, y])
def code(x, y, z):
	tmp = 0
	if (z * z) <= 1e+40:
		tmp = 1.0 / (y * (x + (x * (z * z))))
	else:
		tmp = (1.0 / z) / (x * (z * y))
	return tmp

Julia

z = abs(z)
x, y = sort([x, y])
function code(x, y, z)
	tmp = 0.0
	if (Float64(z * z) <= 1e+40)
		tmp = Float64(1.0 / Float64(y * Float64(x + Float64(x * Float64(z * z)))));
	else
		tmp = Float64(Float64(1.0 / z) / Float64(x * Float64(z * y)));
	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+40)
		tmp = 1.0 / (y * (x + (x * (z * z))));
	else
		tmp = (1.0 / z) / (x * (z * y));
	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+40], N[(1.0 / N[(y * N[(x + N[(x * N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / z), $MachinePrecision] / N[(x * N[(z * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 1.00000000000000003e40

   1. Initial program 99.7%

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

      1. associate-/l/ 99.7%

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

      2. associate-*l* 99.7%

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

      3. *-commutative 99.7%

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

      4. sqr-neg 99.7%

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

      5. +-commutative 99.7%

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

      6. sqr-neg 99.7%

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

      7. fma-def 99.7%

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

   3. Simplified 99.7%

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

      1. fma-udef 99.7%

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

      2. distribute-lft-in 99.7%

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

      3. *-rgt-identity 99.7%

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

   5. Applied egg-rr 99.7%

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

   if 1.00000000000000003e40 < (*.f64 z z)

   1. Initial program 82.1%

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

      1. associate-/l/ 80.7%

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

      2. associate-*l* 78.3%

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

      3. *-commutative 78.3%

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

      4. sqr-neg 78.3%

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

      5. +-commutative 78.3%

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

      6. sqr-neg 78.3%

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

      7. fma-def 78.3%

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

   3. Simplified 78.3%

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

   4. Taylor expanded in z around inf 78.3%

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

      1. unpow2 78.3%

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

   6. Simplified 78.3%

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

      1. associate-/r* 78.4%

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

      2. associate-*r* 83.5%

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

      3. associate-/r* 92.5%

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

      4. div-inv 92.5%

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

      5. associate-/l/ 92.5%

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

      6. *-commutative 92.5%

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

      7. associate-*r* 88.8%

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

      8. *-commutative 88.8%

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

      9. associate-*l* 96.4%

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

   8. Applied egg-rr 96.4%

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

      1. unpow-1 96.4%

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

      2. associate-*l/ 96.4%

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

      3. *-lft-identity 96.4%

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

      4. unpow-1 96.4%

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

   10. Simplified 96.4%

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

4. Final simplification 98.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 10^{+40}:\\ \;\;\;\;\frac{1}{y \cdot \left(x + x \cdot \left(z \cdot z\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{z}}{x \cdot \left(z \cdot y\right)}\\ \end{array} \]

Alternative 4: 97.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} z = |z|\\ [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{+281}:\\ \;\;\;\;\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{z \cdot x}{\frac{\frac{1}{y}}{z}}}\\ \end{array} \end{array} \]

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+281)
   (/ (/ 1.0 x) (* y (+ 1.0 (* z z))))
   (/ 1.0 (/ (* z x) (/ (/ 1.0 y) z)))))

C

z = abs(z);
assert(x < y);
double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 2e+281) {
		tmp = (1.0 / x) / (y * (1.0 + (z * z)));
	} else {
		tmp = 1.0 / ((z * x) / ((1.0 / y) / z));
	}
	return tmp;
}

Fortran

NOTE: z should be positive before calling this function
NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((z * z) <= 2d+281) then
        tmp = (1.0d0 / x) / (y * (1.0d0 + (z * z)))
    else
        tmp = 1.0d0 / ((z * x) / ((1.0d0 / y) / z))
    end if
    code = tmp
end function

Java

z = Math.abs(z);
assert x < y;
public static double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 2e+281) {
		tmp = (1.0 / x) / (y * (1.0 + (z * z)));
	} else {
		tmp = 1.0 / ((z * x) / ((1.0 / y) / z));
	}
	return tmp;
}

Python

z = abs(z)
[x, y] = sort([x, y])
def code(x, y, z):
	tmp = 0
	if (z * z) <= 2e+281:
		tmp = (1.0 / x) / (y * (1.0 + (z * z)))
	else:
		tmp = 1.0 / ((z * x) / ((1.0 / y) / z))
	return tmp

Julia

z = abs(z)
x, y = sort([x, y])
function code(x, y, z)
	tmp = 0.0
	if (Float64(z * z) <= 2e+281)
		tmp = Float64(Float64(1.0 / x) / Float64(y * Float64(1.0 + Float64(z * z))));
	else
		tmp = Float64(1.0 / Float64(Float64(z * x) / Float64(Float64(1.0 / y) / z)));
	end
	return tmp
end

MATLAB

z = abs(z)
x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z * z) <= 2e+281)
		tmp = (1.0 / x) / (y * (1.0 + (z * z)));
	else
		tmp = 1.0 / ((z * x) / ((1.0 / y) / z));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: z should be positive before calling this function
NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_, z_] := If[LessEqual[N[(z * z), $MachinePrecision], 2e+281], N[(N[(1.0 / x), $MachinePrecision] / N[(y * N[(1.0 + N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(z * x), $MachinePrecision] / N[(N[(1.0 / y), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 2.0000000000000001e281

   1. Initial program 98.2%

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

   if 2.0000000000000001e281 < (*.f64 z z)

   1. Initial program 72.5%

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

      1. associate-/l/ 72.5%

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

      2. associate-*l* 72.5%

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

      3. *-commutative 72.5%

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

      4. sqr-neg 72.5%

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

      5. +-commutative 72.5%

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

      6. sqr-neg 72.5%

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

      7. fma-def 72.5%

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

   3. Simplified 72.5%

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

   4. Taylor expanded in z around inf 72.5%

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

      1. unpow2 72.5%

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

   6. Simplified 72.5%

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

      1. /-rgt-identity 72.5%

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

      2. *-commutative 72.5%

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

      3. associate-/l* 72.5%

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

      4. associate-*r* 81.7%

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

      5. associate-/l* 96.8%

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

   8. Applied egg-rr 96.8%

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

4. Final simplification 97.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{+281}:\\ \;\;\;\;\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{z \cdot x}{\frac{\frac{1}{y}}{z}}}\\ \end{array} \]

Alternative 5: 93.3% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

NOTE: z should be positive before calling this function
NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_, z_] := If[LessEqual[N[(z * z), $MachinePrecision], 5e-5], N[(1.0 / N[(x * y), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(x * N[(z * N[(z * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 5.00000000000000024e-5

   1. Initial program 99.7%

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

      1. associate-/l/ 99.8%

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

      2. associate-*l* 99.8%

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

      3. *-commutative 99.8%

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

      4. sqr-neg 99.8%

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

      5. +-commutative 99.8%

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

      6. sqr-neg 99.8%

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

      7. fma-def 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in z around 0 99.3%

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

   if 5.00000000000000024e-5 < (*.f64 z z)

   1. Initial program 84.0%

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

      1. associate-/l/ 82.7%

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

      2. associate-*l* 80.6%

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

      3. *-commutative 80.6%

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

      4. sqr-neg 80.6%

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

      5. +-commutative 80.6%

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

      6. sqr-neg 80.6%

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

      7. fma-def 80.6%

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

   3. Simplified 80.6%

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

      1. fma-udef 80.6%

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

      2. distribute-lft-in 80.6%

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

      3. *-rgt-identity 80.6%

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

   5. Applied egg-rr 80.6%

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

   6. Taylor expanded in z around inf 81.7%

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

      1. unpow2 81.7%

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

      2. associate-*r* 86.9%

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

   8. Simplified 86.9%

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

4. Final simplification 93.2%

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

Alternative 6: 94.1% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

NOTE: z should be positive before calling this function
NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_, z_] := If[LessEqual[N[(z * z), $MachinePrecision], 5e-5], N[(1.0 / N[(x * y), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(z * N[(z * N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 5.00000000000000024e-5

   1. Initial program 99.7%

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

      1. associate-/l/ 99.8%

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

      2. associate-*l* 99.8%

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

      3. *-commutative 99.8%

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

      4. sqr-neg 99.8%

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

      5. +-commutative 99.8%

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

      6. sqr-neg 99.8%

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

      7. fma-def 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in z around 0 99.3%

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

   if 5.00000000000000024e-5 < (*.f64 z z)

   1. Initial program 84.0%

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

      1. associate-/l/ 82.7%

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

      2. associate-*l* 80.6%

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

      3. *-commutative 80.6%

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

      4. sqr-neg 80.6%

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

      5. +-commutative 80.6%

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

      6. sqr-neg 80.6%

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

      7. fma-def 80.6%

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

   3. Simplified 80.6%

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

      1. fma-udef 80.6%

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

      2. distribute-lft-in 80.6%

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

      3. *-rgt-identity 80.6%

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

   5. Applied egg-rr 80.6%

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

   6. Taylor expanded in z around inf 81.7%

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

      1. unpow2 81.7%

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

      2. associate-*r* 86.9%

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

   8. Simplified 86.9%

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

      1. /-rgt-identity 86.9%

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

      2. frac-2neg 86.9%

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

      3. neg-sub0 86.1%

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

      4. metadata-eval 86.1%

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

      5. div-sub 86.9%

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

      6. metadata-eval 86.1%

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

      7. remove-double-div 85.3%

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

      8. associate-/r* 85.4%

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

      9. clear-num 86.1%

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

      10. associate-/l* 92.0%

          \[\leadsto \frac{1}{0 - \frac{\color{blue}{\frac{y \cdot z}{\frac{\frac{1}{x}}{z}}}}{-1}} \]

      11. associate-/r* 90.5%

          \[\leadsto \frac{1}{0 - \frac{\frac{y \cdot z}{\color{blue}{\frac{1}{x \cdot z}}}}{-1}} \]

      12. metadata-eval 90.5%

          \[\leadsto \frac{1}{0 - \frac{\frac{y \cdot z}{\frac{\color{blue}{--1}}{x \cdot z}}}{-1}} \]

      13. distribute-neg-frac 90.5%

          \[\leadsto \frac{1}{0 - \frac{\frac{y \cdot z}{\color{blue}{-\frac{-1}{x \cdot z}}}}{-1}} \]

      14. associate-/r* 90.5%

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

      15. *-commutative 90.5%

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

      16. mul-1-neg 90.5%

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

      17. remove-double-neg 90.5%

          \[\leadsto \frac{1}{0 - \frac{y \cdot z}{\color{blue}{\frac{-1}{x \cdot z}}}} \]

      18. frac-2neg 90.5%

          \[\leadsto \frac{1}{0 - \frac{y \cdot z}{\color{blue}{\frac{--1}{-x \cdot z}}}} \]

      19. metadata-eval 90.5%

          \[\leadsto \frac{1}{0 - \frac{y \cdot z}{\frac{\color{blue}{1}}{-x \cdot z}}} \]

      20. associate-/r/ 90.6%

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

      21. /-rgt-identity 90.6%

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

      22. neg-mul-1 90.6%

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

      23. associate-*r* 90.6%

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

   10. Applied egg-rr 85.9%

       \[\leadsto \frac{1}{\color{blue}{0 - z \cdot \left(z \cdot \left(-y \cdot x\right)\right)}} \]
   11. Step-by-step derivation

       1. sub0-neg 87.6%

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

       2. distribute-rgt-neg-in 87.6%

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

       3. distribute-rgt-neg-out 87.6%

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

       4. remove-double-neg 87.6%

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

       5. *-commutative 87.6%

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

   12. Simplified 87.6%

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

4. Final simplification 93.6%

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

Alternative 7: 96.8% accurate, 0.8× speedup

Math

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

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) 5e-5) (/ 1.0 (* x y)) (/ 1.0 (* (* z y) (* z x)))))

C

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

Python

z = abs(z)
[x, y] = sort([x, y])
def code(x, y, z):
	tmp = 0
	if (z * z) <= 5e-5:
		tmp = 1.0 / (x * y)
	else:
		tmp = 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(z * z) <= 5e-5)
		tmp = Float64(1.0 / Float64(x * y));
	else
		tmp = Float64(1.0 / Float64(Float64(z * y) * 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) <= 5e-5)
		tmp = 1.0 / (x * y);
	else
		tmp = 1.0 / ((z * y) * (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], 5e-5], N[(1.0 / N[(x * y), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(z * y), $MachinePrecision] * N[(z * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 5.00000000000000024e-5

   1. Initial program 99.7%

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

      1. associate-/l/ 99.8%

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

      2. associate-*l* 99.8%

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

      3. *-commutative 99.8%

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

      4. sqr-neg 99.8%

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

      5. +-commutative 99.8%

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

      6. sqr-neg 99.8%

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

      7. fma-def 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in z around 0 99.3%

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

   if 5.00000000000000024e-5 < (*.f64 z z)

   1. Initial program 84.0%

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

      1. associate-/l/ 82.7%

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

      2. associate-*l* 80.6%

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

      3. *-commutative 80.6%

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

      4. sqr-neg 80.6%

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

      5. +-commutative 80.6%

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

      6. sqr-neg 80.6%

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

      7. fma-def 80.6%

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

   3. Simplified 80.6%

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

      1. fma-udef 80.6%

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

      2. distribute-lft-in 80.6%

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

      3. *-rgt-identity 80.6%

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

   5. Applied egg-rr 80.6%

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

      1. +-commutative 80.6%

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

      2. distribute-lft-in 80.6%

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

      3. /-rgt-identity 80.6%

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

      4. associate-/r/ 80.6%

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

      5. frac-2neg 80.6%

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

      6. /-rgt-identity 80.6%

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

      7. associate-/r/ 80.6%

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

      8. associate-/r* 81.3%

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

      9. associate-/r* 85.9%

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

      10. associate-/l* 93.9%

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

      11. frac-add 76.6%

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

      12. associate-/r* 76.6%

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

      13. distribute-neg-frac 76.6%

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

      14. metadata-eval 76.6%

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

      15. distribute-neg-frac 76.6%

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

      16. metadata-eval 76.6%

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

      17. associate-/r* 76.5%

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

   7. Applied egg-rr 76.5%

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

      1. associate-*r/ 76.5%

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

      2. *-rgt-identity 76.5%

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

      3. associate-/r/ 81.3%

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

   9. Simplified 81.3%

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

   10. Taylor expanded in z around inf 91.4%

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

4. Final simplification 95.4%

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

Alternative 8: 97.1% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

NOTE: z should be positive before calling this function
NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_, z_] := If[LessEqual[N[(z * z), $MachinePrecision], 5e-5], N[(1.0 / N[(x * y), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / z), $MachinePrecision] / N[(x * N[(z * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 5.00000000000000024e-5

   1. Initial program 99.7%

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

      1. associate-/l/ 99.8%

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

      2. associate-*l* 99.8%

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

      3. *-commutative 99.8%

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

      4. sqr-neg 99.8%

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

      5. +-commutative 99.8%

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

      6. sqr-neg 99.8%

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

      7. fma-def 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in z around 0 99.3%

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

   if 5.00000000000000024e-5 < (*.f64 z z)

   1. Initial program 84.0%

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

      1. associate-/l/ 82.7%

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

      2. associate-*l* 80.6%

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

      3. *-commutative 80.6%

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

      4. sqr-neg 80.6%

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

      5. +-commutative 80.6%

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

      6. sqr-neg 80.6%

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

      7. fma-def 80.6%

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

   3. Simplified 80.6%

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

   4. Taylor expanded in z around inf 79.5%

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

      1. unpow2 79.5%

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

   6. Simplified 79.5%

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

      1. associate-/r* 79.6%

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

      2. associate-*r* 84.1%

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

      3. associate-/r* 92.2%

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

      4. div-inv 92.2%

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

      5. associate-/l/ 92.1%

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

      6. *-commutative 92.1%

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

      7. associate-*r* 88.8%

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

      8. *-commutative 88.8%

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

      9. associate-*l* 95.6%

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

   8. Applied egg-rr 95.6%

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

      1. unpow-1 95.6%

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

      2. associate-*l/ 95.6%

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

      3. *-lft-identity 95.6%

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

      4. unpow-1 95.6%

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

   10. Simplified 95.6%

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

4. Final simplification 97.5%

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

Alternative 9: 89.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} z = |z|\\ [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq 0.3:\\ \;\;\;\;\frac{1}{x \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x \cdot \left(y \cdot \left(z \cdot z\right)\right)}\\ \end{array} \end{array} \]

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 0.3) (/ 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 <= 0.3) {
		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 <= 0.3d0) 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 <= 0.3) {
		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 <= 0.3:
		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 (z <= 0.3)
		tmp = Float64(1.0 / Float64(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 <= 0.3)
		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[z, 0.3], N[(1.0 / N[(x * y), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(x * N[(y * N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 0.299999999999999989

   1. Initial program 94.8%

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

      1. associate-/l/ 94.4%

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

      2. associate-*l* 92.5%

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

      3. *-commutative 92.5%

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

      4. sqr-neg 92.5%

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

      5. +-commutative 92.5%

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

      6. sqr-neg 92.5%

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

      7. fma-def 92.5%

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

   3. Simplified 92.5%

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

   4. Taylor expanded in z around 0 79.4%

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

   if 0.299999999999999989 < z

   1. Initial program 84.1%

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

      1. associate-/l/ 82.8%

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

      2. associate-*l* 84.3%

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

      3. *-commutative 84.3%

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

      4. sqr-neg 84.3%

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

      5. +-commutative 84.3%

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

      6. sqr-neg 84.3%

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

      7. fma-def 84.3%

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

   3. Simplified 84.3%

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

   4. Taylor expanded in z around inf 80.8%

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

      1. associate-*r* 80.5%

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

      2. unpow2 80.5%

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

      3. associate-*r* 80.8%

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

   6. Simplified 80.8%

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

4. Final simplification 79.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 0.3:\\ \;\;\;\;\frac{1}{x \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x \cdot \left(y \cdot \left(z \cdot z\right)\right)}\\ \end{array} \]

Alternative 10: 59.7% accurate, 2.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 92.0%

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

   1. associate-/l/ 91.4%

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

   2. associate-*l* 90.3%

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

   3. *-commutative 90.3%

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

   4. sqr-neg 90.3%

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

   5. +-commutative 90.3%

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

   6. sqr-neg 90.3%

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

   7. fma-def 90.3%

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

3. Simplified 90.3%

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

4. Taylor expanded in z around 0 65.2%

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

5. Final simplification 65.2%

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

Developer target: 92.6% 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 2023297 
(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)))))