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

Percentage Accurate: 91.1% → 97.9%

Time: 10.6s

Alternatives: 15

Speedup: 0.8×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 91.1% 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: 97.9% accurate, 0.1× speedup

Math

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

FPCore

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

C

z = abs(z);
assert(x < y);
double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 5e+94) {
		tmp = ((1.0 / y) / x) * (1.0 / fma(z, z, 1.0));
	} else {
		tmp = ((1.0 / y) / hypot(1.0, z)) * ((1.0 / 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+94)
		tmp = Float64(Float64(Float64(1.0 / y) / x) * Float64(1.0 / fma(z, z, 1.0)));
	else
		tmp = Float64(Float64(Float64(1.0 / y) / hypot(1.0, z)) * Float64(Float64(1.0 / 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[(z * z), $MachinePrecision], 5e+94], N[(N[(N[(1.0 / y), $MachinePrecision] / x), $MachinePrecision] * N[(1.0 / N[(z * z + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 / y), $MachinePrecision] / N[Sqrt[1.0 ^ 2 + z ^ 2], $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 / z), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 97.8%

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

      1. associate-/r* 97.3%

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

      2. *-commutative 97.3%

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

      3. sqr-neg 97.3%

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

      4. +-commutative 97.3%

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

      5. distribute-lft1-in 97.3%

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

      6. *-commutative 97.3%

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

      7. fma-def 97.3%

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

      8. sqr-neg 97.3%

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

   3. Simplified 97.3%

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

      1. associate-/r* 97.8%

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

      2. fma-udef 97.8%

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

      3. *-rgt-identity 97.8%

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

      4. distribute-lft-in 97.8%

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

      5. +-commutative 97.8%

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

      6. associate-/r* 99.7%

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

      7. associate-/r* 99.6%

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

      8. +-commutative 99.6%

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

      9. fma-udef 99.6%

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

      10. div-inv 99.6%

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

      11. *-commutative 99.6%

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

      12. associate-/r* 99.7%

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

   5. Applied egg-rr 99.7%

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

   if 5.0000000000000001e94 < (*.f64 z z)

   1. Initial program 77.7%

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

      1. associate-/r* 75.8%

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

      2. associate-/r* 75.9%

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

      3. sqr-neg 75.9%

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

      4. +-commutative 75.9%

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

      5. sqr-neg 75.9%

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

      6. fma-def 75.9%

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

   3. Simplified 75.9%

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

      1. associate-/r* 75.8%

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

      2. div-inv 75.8%

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

   5. Applied egg-rr 75.8%

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

      1. *-commutative 75.8%

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

      2. add-sqr-sqrt 75.8%

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

      3. times-frac 78.8%

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

      4. fma-udef 78.8%

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

      5. +-commutative 78.8%

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

      6. hypot-1-def 78.8%

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

      7. fma-udef 78.8%

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

      8. +-commutative 78.8%

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

      9. hypot-1-def 96.5%

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

   7. Applied egg-rr 96.5%

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

   8. Taylor expanded in z around inf 78.1%

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

      1. *-commutative 78.1%

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

      2. associate-/r* 78.1%

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

   10. Simplified 78.1%

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

4. Final simplification 90.6%

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

Alternative 2: 98.0% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 89.4%

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

   1. associate-/r* 89.7%

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

   2. associate-/r* 89.7%

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

   3. sqr-neg 89.7%

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

   4. +-commutative 89.7%

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

   5. sqr-neg 89.7%

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

   6. fma-def 89.7%

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

3. Simplified 89.7%

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

   1. associate-/r* 89.7%

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

   2. div-inv 89.7%

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

5. Applied egg-rr 89.7%

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

   1. *-commutative 89.7%

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

   2. add-sqr-sqrt 89.6%

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

   3. times-frac 90.6%

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

   4. fma-udef 90.6%

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

   5. +-commutative 90.6%

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

   6. hypot-1-def 90.6%

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

   7. fma-udef 90.6%

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

   8. +-commutative 90.6%

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

   9. hypot-1-def 98.1%

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

7. Applied egg-rr 98.1%

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

8. Final simplification 98.1%

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

Alternative 3: 98.8% accurate, 0.1× speedup

Math

\[\begin{array}{l} z = |z|\\ [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \cdot \left(1 + z \cdot z\right) \leq 4 \cdot 10^{+303}:\\ \;\;\;\;\frac{\frac{1}{x}}{\mathsf{fma}\left(y \cdot z, z, y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{z} \cdot \frac{\frac{1}{y}}{z \cdot x}\\ \end{array} \end{array} \]

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))) 4e+303)
   (/ (/ 1.0 x) (fma (* y z) z y))
   (* (/ 1.0 z) (/ (/ 1.0 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))) <= 4e+303) {
		tmp = (1.0 / x) / fma((y * z), z, y);
	} else {
		tmp = (1.0 / z) * ((1.0 / 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))) <= 4e+303)
		tmp = Float64(Float64(1.0 / x) / fma(Float64(y * z), z, y));
	else
		tmp = Float64(Float64(1.0 / z) * Float64(Float64(1.0 / 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], 4e+303], N[(N[(1.0 / x), $MachinePrecision] / N[(N[(y * z), $MachinePrecision] * z + y), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / z), $MachinePrecision] * N[(N[(1.0 / y), $MachinePrecision] / N[(z * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y (+.f64 1 (*.f64 z z))) < 4e303

   1. Initial program 91.2%

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

      1. distribute-lft-in 91.2%

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

      2. *-rgt-identity 91.2%

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

      3. +-commutative 91.2%

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

      4. associate-*r* 93.9%

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

      5. fma-def 93.9%

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

   3. Applied egg-rr 93.9%

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

   if 4e303 < (*.f64 y (+.f64 1 (*.f64 z z)))

   1. Initial program 80.8%

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

      1. associate-/r* 80.8%

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

      2. *-commutative 80.8%

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

      3. sqr-neg 80.8%

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

      4. +-commutative 80.8%

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

      5. distribute-lft1-in 80.8%

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

      6. *-commutative 80.8%

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

      7. fma-def 80.8%

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

      8. sqr-neg 80.8%

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

   3. Simplified 80.8%

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

   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. *-commutative 80.8%

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

      2. unpow2 80.8%

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

      3. associate-*l* 84.9%

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

      4. associate-*l* 93.3%

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

   6. Simplified 93.3%

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

      1. associate-/r* 93.4%

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

      2. *-un-lft-identity 93.4%

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

      3. times-frac 99.9%

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

   8. Applied egg-rr 99.9%

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

4. Final simplification 95.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(1 + z \cdot z\right) \leq 4 \cdot 10^{+303}:\\ \;\;\;\;\frac{\frac{1}{x}}{\mathsf{fma}\left(y \cdot z, z, y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{z} \cdot \frac{\frac{1}{y}}{z \cdot x}\\ \end{array} \]

Alternative 4: 97.8% accurate, 0.1× speedup

Math

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

FPCore

NOTE: z should be positive before calling this function
NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= (* z z) 5e+94)
   (* (/ (/ 1.0 y) x) (/ 1.0 (fma z z 1.0)))
   (/ (/ (/ 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+94) {
		tmp = ((1.0 / y) / x) * (1.0 / fma(z, z, 1.0));
	} 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+94)
		tmp = Float64(Float64(Float64(1.0 / y) / x) * Float64(1.0 / fma(z, z, 1.0)));
	else
		tmp = Float64(Float64(Float64(1.0 / z) / 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[(z * z), $MachinePrecision], 5e+94], N[(N[(N[(1.0 / y), $MachinePrecision] / x), $MachinePrecision] * N[(1.0 / N[(z * z + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 / z), $MachinePrecision] / y), $MachinePrecision] / N[(z * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 97.8%

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

      1. associate-/r* 97.3%

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

      2. *-commutative 97.3%

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

      3. sqr-neg 97.3%

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

      4. +-commutative 97.3%

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

      5. distribute-lft1-in 97.3%

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

      6. *-commutative 97.3%

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

      7. fma-def 97.3%

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

      8. sqr-neg 97.3%

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

   3. Simplified 97.3%

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

      1. associate-/r* 97.8%

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

      2. fma-udef 97.8%

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

      3. *-rgt-identity 97.8%

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

      4. distribute-lft-in 97.8%

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

      5. +-commutative 97.8%

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

      6. associate-/r* 99.7%

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

      7. associate-/r* 99.6%

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

      8. +-commutative 99.6%

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

      9. fma-udef 99.6%

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

      10. div-inv 99.6%

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

      11. *-commutative 99.6%

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

      12. associate-/r* 99.7%

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

   5. Applied egg-rr 99.7%

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

   if 5.0000000000000001e94 < (*.f64 z z)

   1. Initial program 77.7%

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

      1. associate-/r* 77.7%

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

      2. *-commutative 77.7%

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

      3. sqr-neg 77.7%

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

      4. +-commutative 77.7%

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

      5. distribute-lft1-in 77.7%

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

      6. *-commutative 77.7%

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

      7. fma-def 77.7%

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

      8. sqr-neg 77.7%

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

   3. Simplified 77.7%

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

   4. Taylor expanded in z around inf 77.7%

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

      1. *-commutative 77.7%

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

      2. unpow2 77.7%

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

      3. associate-*l* 76.8%

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

      4. associate-*l* 90.0%

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

   6. Simplified 90.0%

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

      1. associate-/r* 90.0%

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

      2. *-un-lft-identity 90.0%

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

      3. times-frac 94.6%

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

   8. Applied egg-rr 94.6%

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

      1. associate-*r/ 96.5%

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

      2. un-div-inv 96.5%

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

   10. Applied egg-rr 96.5%

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

4. Final simplification 98.3%

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

Alternative 5: 98.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} z = |z|\\ [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \cdot \left(1 + z \cdot z\right) \leq 4 \cdot 10^{+303}:\\ \;\;\;\;\frac{\frac{1}{x}}{y + z \cdot \left(y \cdot z\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{z} \cdot \frac{\frac{1}{y}}{z \cdot x}\\ \end{array} \end{array} \]

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))) 4e+303)
   (/ (/ 1.0 x) (+ y (* z (* y z))))
   (* (/ 1.0 z) (/ (/ 1.0 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))) <= 4e+303) {
		tmp = (1.0 / x) / (y + (z * (y * z)));
	} else {
		tmp = (1.0 / z) * ((1.0 / 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 ((y * (1.0d0 + (z * z))) <= 4d+303) then
        tmp = (1.0d0 / x) / (y + (z * (y * z)))
    else
        tmp = (1.0d0 / z) * ((1.0d0 / 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 ((y * (1.0 + (z * z))) <= 4e+303) {
		tmp = (1.0 / x) / (y + (z * (y * z)));
	} else {
		tmp = (1.0 / z) * ((1.0 / y) / (z * x));
	}
	return tmp;
}

Python

z = abs(z)
[x, y] = sort([x, y])
def code(x, y, z):
	tmp = 0
	if (y * (1.0 + (z * z))) <= 4e+303:
		tmp = (1.0 / x) / (y + (z * (y * z)))
	else:
		tmp = (1.0 / z) * ((1.0 / 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))) <= 4e+303)
		tmp = Float64(Float64(1.0 / x) / Float64(y + Float64(z * Float64(y * z))));
	else
		tmp = Float64(Float64(1.0 / z) * Float64(Float64(1.0 / 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 ((y * (1.0 + (z * z))) <= 4e+303)
		tmp = (1.0 / x) / (y + (z * (y * z)));
	else
		tmp = (1.0 / z) * ((1.0 / 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[(y * N[(1.0 + N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 4e+303], N[(N[(1.0 / x), $MachinePrecision] / N[(y + N[(z * N[(y * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / z), $MachinePrecision] * N[(N[(1.0 / y), $MachinePrecision] / N[(z * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y (+.f64 1 (*.f64 z z))) < 4e303

   1. Initial program 91.2%

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

      1. distribute-lft-in 91.2%

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

      2. *-rgt-identity 91.2%

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

      3. +-commutative 91.2%

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

      4. associate-*r* 93.9%

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

      5. fma-def 93.9%

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

   3. Applied egg-rr 93.9%

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

      1. fma-udef 93.9%

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

      2. *-commutative 93.9%

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

   5. Applied egg-rr 93.9%

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

   if 4e303 < (*.f64 y (+.f64 1 (*.f64 z z)))

   1. Initial program 80.8%

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

      1. associate-/r* 80.8%

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

      2. *-commutative 80.8%

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

      3. sqr-neg 80.8%

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

      4. +-commutative 80.8%

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

      5. distribute-lft1-in 80.8%

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

      6. *-commutative 80.8%

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

      7. fma-def 80.8%

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

      8. sqr-neg 80.8%

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

   3. Simplified 80.8%

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

   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. *-commutative 80.8%

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

      2. unpow2 80.8%

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

      3. associate-*l* 84.9%

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

      4. associate-*l* 93.3%

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

   6. Simplified 93.3%

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

      1. associate-/r* 93.4%

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

      2. *-un-lft-identity 93.4%

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

      3. times-frac 99.9%

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

   8. Applied egg-rr 99.9%

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

4. Final simplification 95.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(1 + z \cdot z\right) \leq 4 \cdot 10^{+303}:\\ \;\;\;\;\frac{\frac{1}{x}}{y + z \cdot \left(y \cdot z\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{z} \cdot \frac{\frac{1}{y}}{z \cdot x}\\ \end{array} \]

Alternative 6: 97.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} z = |z|\\ [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{+47}:\\ \;\;\;\;\frac{1}{x \cdot \left(y + y \cdot \left(z \cdot z\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{1}{z}}{y}}{z \cdot x}\\ \end{array} \end{array} \]

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+47)
   (/ 1.0 (* x (+ y (* y (* z z)))))
   (/ (/ (/ 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) <= 2e+47) {
		tmp = 1.0 / (x * (y + (y * (z * z))));
	} 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) <= 2d+47) then
        tmp = 1.0d0 / (x * (y + (y * (z * z))))
    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) <= 2e+47) {
		tmp = 1.0 / (x * (y + (y * (z * z))));
	} 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) <= 2e+47:
		tmp = 1.0 / (x * (y + (y * (z * z))))
	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) <= 2e+47)
		tmp = Float64(1.0 / Float64(x * Float64(y + Float64(y * Float64(z * z)))));
	else
		tmp = Float64(Float64(Float64(1.0 / 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) <= 2e+47)
		tmp = 1.0 / (x * (y + (y * (z * z))));
	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], 2e+47], N[(1.0 / N[(x * N[(y + N[(y * N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 / z), $MachinePrecision] / y), $MachinePrecision] / N[(z * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 2.0000000000000001e47

   1. Initial program 99.7%

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

      1. associate-/r* 99.6%

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

      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. Step-by-step derivation

      1. fma-udef 99.6%

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

   5. Applied egg-rr 99.6%

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

   if 2.0000000000000001e47 < (*.f64 z z)

   1. Initial program 77.5%

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

      1. associate-/r* 77.0%

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

      2. *-commutative 77.0%

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

      3. sqr-neg 77.0%

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

      4. +-commutative 77.0%

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

      5. distribute-lft1-in 77.0%

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

      6. *-commutative 77.0%

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

      7. fma-def 77.0%

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

      8. sqr-neg 77.0%

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

   3. Simplified 77.0%

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

   4. Taylor expanded in z around inf 77.0%

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

      1. *-commutative 77.0%

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

      2. unpow2 77.0%

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

      3. associate-*l* 78.6%

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

      4. associate-*l* 90.4%

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

   6. Simplified 90.4%

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

      1. associate-/r* 91.0%

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

      2. *-un-lft-identity 91.0%

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

      3. times-frac 95.1%

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

   8. Applied egg-rr 95.1%

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

      1. associate-*r/ 96.5%

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

      2. un-div-inv 96.5%

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

   10. Applied egg-rr 96.5%

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

4. Final simplification 98.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{+47}:\\ \;\;\;\;\frac{1}{x \cdot \left(y + y \cdot \left(z \cdot z\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{1}{z}}{y}}{z \cdot x}\\ \end{array} \]

Alternative 7: 97.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} z = |z|\\ [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 10^{+206}:\\ \;\;\;\;\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{z \cdot \left(x \cdot \left(y \cdot z\right)\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: z should be positive before calling this function
NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= (* z z) 1e+206)
   (/ (/ 1.0 x) (* y (+ 1.0 (* z z))))
   (/ 1.0 (* z (* x (* y z))))))

C

z = abs(z);
assert(x < y);
double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 1e+206) {
		tmp = (1.0 / x) / (y * (1.0 + (z * z)));
	} else {
		tmp = 1.0 / (z * (x * (y * z)));
	}
	return tmp;
}

Fortran

NOTE: z should be positive before calling this function
NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((z * z) <= 1d+206) then
        tmp = (1.0d0 / x) / (y * (1.0d0 + (z * z)))
    else
        tmp = 1.0d0 / (z * (x * (y * z)))
    end if
    code = tmp
end function

Java

z = Math.abs(z);
assert x < y;
public static double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 1e+206) {
		tmp = (1.0 / x) / (y * (1.0 + (z * z)));
	} else {
		tmp = 1.0 / (z * (x * (y * z)));
	}
	return tmp;
}

Python

z = abs(z)
[x, y] = sort([x, y])
def code(x, y, z):
	tmp = 0
	if (z * z) <= 1e+206:
		tmp = (1.0 / x) / (y * (1.0 + (z * z)))
	else:
		tmp = 1.0 / (z * (x * (y * z)))
	return tmp

Julia

z = abs(z)
x, y = sort([x, y])
function code(x, y, z)
	tmp = 0.0
	if (Float64(z * z) <= 1e+206)
		tmp = Float64(Float64(1.0 / x) / Float64(y * Float64(1.0 + Float64(z * z))));
	else
		tmp = Float64(1.0 / Float64(z * 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 * z) <= 1e+206)
		tmp = (1.0 / x) / (y * (1.0 + (z * z)));
	else
		tmp = 1.0 / (z * (x * (y * z)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: z should be positive before calling this function
NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_, z_] := If[LessEqual[N[(z * z), $MachinePrecision], 1e+206], N[(N[(1.0 / x), $MachinePrecision] / N[(y * N[(1.0 + N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(z * N[(x * N[(y * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 1e206

   1. Initial program 96.8%

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

   if 1e206 < (*.f64 z z)

   1. Initial program 75.4%

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

      1. associate-/r* 75.4%

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

      2. *-commutative 75.4%

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

      3. sqr-neg 75.4%

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

      4. +-commutative 75.4%

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

      5. distribute-lft1-in 75.4%

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

      6. *-commutative 75.4%

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

      7. fma-def 75.4%

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

      8. sqr-neg 75.4%

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

   3. Simplified 75.4%

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

   4. Taylor expanded in z around inf 75.4%

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

      1. *-commutative 75.4%

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

      2. unpow2 75.4%

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

      3. associate-*l* 73.3%

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

      4. associate-*l* 89.1%

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

   6. Simplified 89.1%

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

   7. Taylor expanded in y around 0 75.4%

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

      1. *-commutative 75.4%

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

      2. unpow2 75.4%

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

      3. associate-*r* 85.0%

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

      4. *-commutative 85.0%

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

   9. Simplified 85.0%

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

   10. Taylor expanded in x around 0 75.4%

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

       1. *-commutative 75.4%

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

       2. unpow2 75.4%

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

       3. associate-*r* 85.0%

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

       4. *-commutative 85.0%

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

       5. associate-*r* 85.0%

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

       6. associate-*r* 97.5%

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

       7. *-commutative 97.5%

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

   12. Simplified 97.5%

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

4. Final simplification 97.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 10^{+206}:\\ \;\;\;\;\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{z \cdot \left(x \cdot \left(y \cdot z\right)\right)}\\ \end{array} \]

Alternative 8: 90.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} z = |z|\\ [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 0.0005:\\ \;\;\;\;\frac{\frac{1}{x}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x \cdot \left(y \cdot \left(z \cdot z\right)\right)}\\ \end{array} \end{array} \]

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) 0.0005) (/ (/ 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) <= 0.0005) {
		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) <= 0.0005d0) 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) <= 0.0005) {
		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) <= 0.0005:
		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) <= 0.0005)
		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) <= 0.0005)
		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], 0.0005], 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) < 5.0000000000000001e-4

   1. Initial program 99.7%

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

      1. distribute-lft-in 99.7%

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

      2. *-rgt-identity 99.7%

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

      3. +-commutative 99.7%

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

      4. associate-*r* 99.7%

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

      5. fma-def 99.7%

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

   3. Applied egg-rr 99.7%

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

   4. Taylor expanded in z around 0 99.0%

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

   if 5.0000000000000001e-4 < (*.f64 z z)

   1. Initial program 80.3%

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

      1. associate-/r* 79.8%

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

      2. *-commutative 79.8%

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

      3. sqr-neg 79.8%

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

      4. +-commutative 79.8%

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

      5. distribute-lft1-in 79.8%

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

      6. *-commutative 79.8%

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

      7. fma-def 79.8%

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

      8. sqr-neg 79.8%

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

   3. Simplified 79.8%

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

   4. Taylor expanded in z around inf 78.1%

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

      1. unpow2 78.1%

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

   6. Simplified 78.1%

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

4. Final simplification 87.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 0.0005:\\ \;\;\;\;\frac{\frac{1}{x}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x \cdot \left(y \cdot \left(z \cdot z\right)\right)}\\ \end{array} \]

Alternative 9: 93.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} z = |z|\\ [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 0.0005:\\ \;\;\;\;\frac{\frac{1}{x}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x \cdot \left(z \cdot \left(y \cdot z\right)\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: z should be positive before calling this function
NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= (* z z) 0.0005) (/ (/ 1.0 x) y) (/ 1.0 (* x (* z (* y z))))))

C

z = abs(z);
assert(x < y);
double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 0.0005) {
		tmp = (1.0 / x) / y;
	} else {
		tmp = 1.0 / (x * (z * (y * z)));
	}
	return tmp;
}

Fortran

NOTE: z should be positive before calling this function
NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((z * z) <= 0.0005d0) then
        tmp = (1.0d0 / x) / y
    else
        tmp = 1.0d0 / (x * (z * (y * z)))
    end if
    code = tmp
end function

Java

z = Math.abs(z);
assert x < y;
public static double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 0.0005) {
		tmp = (1.0 / x) / y;
	} else {
		tmp = 1.0 / (x * (z * (y * z)));
	}
	return tmp;
}

Python

z = abs(z)
[x, y] = sort([x, y])
def code(x, y, z):
	tmp = 0
	if (z * z) <= 0.0005:
		tmp = (1.0 / x) / y
	else:
		tmp = 1.0 / (x * (z * (y * z)))
	return tmp

Julia

z = abs(z)
x, y = sort([x, y])
function code(x, y, z)
	tmp = 0.0
	if (Float64(z * z) <= 0.0005)
		tmp = Float64(Float64(1.0 / x) / y);
	else
		tmp = Float64(1.0 / Float64(x * Float64(z * Float64(y * z))));
	end
	return tmp
end

MATLAB

z = abs(z)
x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z * z) <= 0.0005)
		tmp = (1.0 / x) / y;
	else
		tmp = 1.0 / (x * (z * (y * z)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: z should be positive before calling this function
NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_, z_] := If[LessEqual[N[(z * z), $MachinePrecision], 0.0005], N[(N[(1.0 / x), $MachinePrecision] / y), $MachinePrecision], N[(1.0 / N[(x * N[(z * N[(y * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 5.0000000000000001e-4

   1. Initial program 99.7%

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

      1. distribute-lft-in 99.7%

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

      2. *-rgt-identity 99.7%

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

      3. +-commutative 99.7%

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

      4. associate-*r* 99.7%

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

      5. fma-def 99.7%

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

   3. Applied egg-rr 99.7%

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

   4. Taylor expanded in z around 0 99.0%

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

   if 5.0000000000000001e-4 < (*.f64 z z)

   1. Initial program 80.3%

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

      1. associate-/r* 79.8%

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

      2. *-commutative 79.8%

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

      3. sqr-neg 79.8%

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

      4. +-commutative 79.8%

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

      5. distribute-lft1-in 79.8%

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

      6. *-commutative 79.8%

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

      7. fma-def 79.8%

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

      8. sqr-neg 79.8%

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

   3. Simplified 79.8%

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

   4. Taylor expanded in z around inf 78.1%

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

      1. *-commutative 78.1%

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

      2. unpow2 78.1%

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

      3. associate-*l* 78.4%

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

      4. associate-*l* 88.8%

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

   6. Simplified 88.8%

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

   7. Taylor expanded in y around 0 78.1%

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

      1. *-commutative 78.1%

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

      2. unpow2 78.1%

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

      3. associate-*r* 84.3%

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

      4. *-commutative 84.3%

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

   9. Simplified 84.3%

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

4. Final simplification 91.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 0.0005:\\ \;\;\;\;\frac{\frac{1}{x}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x \cdot \left(z \cdot \left(y \cdot z\right)\right)}\\ \end{array} \]

Alternative 10: 97.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} z = |z|\\ [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 0.0005:\\ \;\;\;\;\frac{\frac{1}{x}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{z \cdot \left(x \cdot \left(y \cdot z\right)\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: z should be positive before calling this function
NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= (* z z) 0.0005) (/ (/ 1.0 x) y) (/ 1.0 (* z (* x (* y z))))))

C

z = abs(z);
assert(x < y);
double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 0.0005) {
		tmp = (1.0 / x) / y;
	} else {
		tmp = 1.0 / (z * (x * (y * z)));
	}
	return tmp;
}

Fortran

NOTE: z should be positive before calling this function
NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((z * z) <= 0.0005d0) then
        tmp = (1.0d0 / x) / y
    else
        tmp = 1.0d0 / (z * (x * (y * z)))
    end if
    code = tmp
end function

Java

z = Math.abs(z);
assert x < y;
public static double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 0.0005) {
		tmp = (1.0 / x) / y;
	} else {
		tmp = 1.0 / (z * (x * (y * z)));
	}
	return tmp;
}

Python

z = abs(z)
[x, y] = sort([x, y])
def code(x, y, z):
	tmp = 0
	if (z * z) <= 0.0005:
		tmp = (1.0 / x) / y
	else:
		tmp = 1.0 / (z * (x * (y * z)))
	return tmp

Julia

z = abs(z)
x, y = sort([x, y])
function code(x, y, z)
	tmp = 0.0
	if (Float64(z * z) <= 0.0005)
		tmp = Float64(Float64(1.0 / x) / y);
	else
		tmp = Float64(1.0 / Float64(z * 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 * z) <= 0.0005)
		tmp = (1.0 / x) / y;
	else
		tmp = 1.0 / (z * (x * (y * z)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: z should be positive before calling this function
NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_, z_] := If[LessEqual[N[(z * z), $MachinePrecision], 0.0005], N[(N[(1.0 / x), $MachinePrecision] / y), $MachinePrecision], N[(1.0 / N[(z * N[(x * N[(y * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 5.0000000000000001e-4

   1. Initial program 99.7%

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

      1. distribute-lft-in 99.7%

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

      2. *-rgt-identity 99.7%

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

      3. +-commutative 99.7%

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

      4. associate-*r* 99.7%

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

      5. fma-def 99.7%

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

   3. Applied egg-rr 99.7%

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

   4. Taylor expanded in z around 0 99.0%

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

   if 5.0000000000000001e-4 < (*.f64 z z)

   1. Initial program 80.3%

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

      1. associate-/r* 79.8%

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

      2. *-commutative 79.8%

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

      3. sqr-neg 79.8%

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

      4. +-commutative 79.8%

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

      5. distribute-lft1-in 79.8%

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

      6. *-commutative 79.8%

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

      7. fma-def 79.8%

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

      8. sqr-neg 79.8%

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

   3. Simplified 79.8%

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

   4. Taylor expanded in z around inf 78.1%

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

      1. *-commutative 78.1%

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

      2. unpow2 78.1%

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

      3. associate-*l* 78.4%

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

      4. associate-*l* 88.8%

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

   6. Simplified 88.8%

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

   7. Taylor expanded in y around 0 78.1%

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

      1. *-commutative 78.1%

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

      2. unpow2 78.1%

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

      3. associate-*r* 84.3%

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

      4. *-commutative 84.3%

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

   9. Simplified 84.3%

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

   10. Taylor expanded in x around 0 78.1%

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

       1. *-commutative 78.1%

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

       2. unpow2 78.1%

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

       3. associate-*r* 84.3%

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

       4. *-commutative 84.3%

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

       5. associate-*r* 84.3%

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

       6. associate-*r* 92.7%

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

       7. *-commutative 92.7%

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

   12. Simplified 92.7%

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

4. Final simplification 95.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 0.0005:\\ \;\;\;\;\frac{\frac{1}{x}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{z \cdot \left(x \cdot \left(y \cdot z\right)\right)}\\ \end{array} \]

Alternative 11: 97.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} z = |z|\\ [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 0.0005:\\ \;\;\;\;\frac{1 - z \cdot z}{y \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{z \cdot \left(x \cdot \left(y \cdot z\right)\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: z should be positive before calling this function
NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= (* z z) 0.0005)
   (/ (- 1.0 (* z z)) (* y x))
   (/ 1.0 (* z (* x (* y z))))))

C

z = abs(z);
assert(x < y);
double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 0.0005) {
		tmp = (1.0 - (z * z)) / (y * x);
	} else {
		tmp = 1.0 / (z * (x * (y * z)));
	}
	return tmp;
}

Fortran

NOTE: z should be positive before calling this function
NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((z * z) <= 0.0005d0) then
        tmp = (1.0d0 - (z * z)) / (y * x)
    else
        tmp = 1.0d0 / (z * (x * (y * z)))
    end if
    code = tmp
end function

Java

z = Math.abs(z);
assert x < y;
public static double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 0.0005) {
		tmp = (1.0 - (z * z)) / (y * x);
	} else {
		tmp = 1.0 / (z * (x * (y * z)));
	}
	return tmp;
}

Python

z = abs(z)
[x, y] = sort([x, y])
def code(x, y, z):
	tmp = 0
	if (z * z) <= 0.0005:
		tmp = (1.0 - (z * z)) / (y * x)
	else:
		tmp = 1.0 / (z * (x * (y * z)))
	return tmp

Julia

z = abs(z)
x, y = sort([x, y])
function code(x, y, z)
	tmp = 0.0
	if (Float64(z * z) <= 0.0005)
		tmp = Float64(Float64(1.0 - Float64(z * z)) / Float64(y * x));
	else
		tmp = Float64(1.0 / Float64(z * 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 * z) <= 0.0005)
		tmp = (1.0 - (z * z)) / (y * x);
	else
		tmp = 1.0 / (z * (x * (y * z)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: z should be positive before calling this function
NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_, z_] := If[LessEqual[N[(z * z), $MachinePrecision], 0.0005], N[(N[(1.0 - N[(z * z), $MachinePrecision]), $MachinePrecision] / N[(y * x), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(z * N[(x * N[(y * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 5.0000000000000001e-4

   1. Initial program 99.7%

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

      1. associate-/r* 99.6%

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

      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 90.1%

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

      1. associate-/l/ 90.1%

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

      2. +-commutative 90.1%

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

      3. mul-1-neg 90.1%

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

      4. unsub-neg 90.1%

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

      5. associate-/r* 90.1%

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

      6. unpow2 90.1%

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

      7. *-commutative 90.1%

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

      8. div-sub 99.3%

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

   6. Simplified 99.3%

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

   if 5.0000000000000001e-4 < (*.f64 z z)

   1. Initial program 80.3%

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

      1. associate-/r* 79.8%

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

      2. *-commutative 79.8%

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

      3. sqr-neg 79.8%

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

      4. +-commutative 79.8%

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

      5. distribute-lft1-in 79.8%

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

      6. *-commutative 79.8%

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

      7. fma-def 79.8%

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

      8. sqr-neg 79.8%

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

   3. Simplified 79.8%

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

   4. Taylor expanded in z around inf 78.1%

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

      1. *-commutative 78.1%

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

      2. unpow2 78.1%

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

      3. associate-*l* 78.4%

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

      4. associate-*l* 88.8%

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

   6. Simplified 88.8%

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

   7. Taylor expanded in y around 0 78.1%

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

      1. *-commutative 78.1%

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

      2. unpow2 78.1%

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

      3. associate-*r* 84.3%

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

      4. *-commutative 84.3%

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

   9. Simplified 84.3%

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

   10. Taylor expanded in x around 0 78.1%

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

       1. *-commutative 78.1%

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

       2. unpow2 78.1%

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

       3. associate-*r* 84.3%

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

       4. *-commutative 84.3%

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

       5. associate-*r* 84.3%

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

       6. associate-*r* 92.7%

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

       7. *-commutative 92.7%

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

   12. Simplified 92.7%

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

4. Final simplification 95.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 0.0005:\\ \;\;\;\;\frac{1 - z \cdot z}{y \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{z \cdot \left(x \cdot \left(y \cdot z\right)\right)}\\ \end{array} \]

Alternative 12: 97.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} z = |z|\\ [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 0.0005:\\ \;\;\;\;\frac{1 - z \cdot z}{y \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{1}{y}}{z}}{z \cdot x}\\ \end{array} \end{array} \]

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) 0.0005)
   (/ (- 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) <= 0.0005) {
		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) <= 0.0005d0) 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) <= 0.0005) {
		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) <= 0.0005:
		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) <= 0.0005)
		tmp = Float64(Float64(1.0 - Float64(z * z)) / Float64(y * x));
	else
		tmp = Float64(Float64(Float64(1.0 / 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) <= 0.0005)
		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], 0.0005], N[(N[(1.0 - N[(z * z), $MachinePrecision]), $MachinePrecision] / N[(y * x), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 / y), $MachinePrecision] / z), $MachinePrecision] / N[(z * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 5.0000000000000001e-4

   1. Initial program 99.7%

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

      1. associate-/r* 99.6%

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

      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 90.1%

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

      1. associate-/l/ 90.1%

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

      2. +-commutative 90.1%

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

      3. mul-1-neg 90.1%

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

      4. unsub-neg 90.1%

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

      5. associate-/r* 90.1%

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

      6. unpow2 90.1%

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

      7. *-commutative 90.1%

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

      8. div-sub 99.3%

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

   6. Simplified 99.3%

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

   if 5.0000000000000001e-4 < (*.f64 z z)

   1. Initial program 80.3%

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

      1. associate-/r* 80.9%

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

      2. associate-/r* 80.9%

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

      3. sqr-neg 80.9%

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

      4. +-commutative 80.9%

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

      5. sqr-neg 80.9%

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

      6. fma-def 80.9%

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

   3. Simplified 80.9%

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

      1. associate-/r* 80.9%

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

      2. div-inv 80.9%

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

   5. Applied egg-rr 80.9%

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

      1. *-commutative 80.9%

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

      2. add-sqr-sqrt 80.9%

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

      3. times-frac 82.8%

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

      4. fma-udef 82.8%

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

      5. +-commutative 82.8%

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

      6. hypot-1-def 82.8%

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

      7. fma-udef 82.8%

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

      8. +-commutative 82.8%

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

      9. hypot-1-def 96.7%

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

   7. Applied egg-rr 96.7%

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

   8. Taylor expanded in z around inf 78.1%

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

      1. *-commutative 78.1%

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

      2. associate-*r* 78.4%

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

      3. unpow2 78.4%

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

      4. associate-/r* 78.8%

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

      5. *-rgt-identity 78.8%

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

      6. associate-*l* 89.2%

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

      7. *-commutative 89.2%

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

      8. times-frac 93.2%

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

      9. associate-/r* 93.8%

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

      10. associate-/r* 93.1%

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

      11. times-frac 93.9%

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

      12. *-lft-identity 93.9%

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

      13. associate-/r* 94.5%

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

      14. *-commutative 94.5%

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

   10. Simplified 94.5%

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

4. Final simplification 96.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 0.0005:\\ \;\;\;\;\frac{1 - z \cdot z}{y \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{1}{y}}{z}}{z \cdot x}\\ \end{array} \]

Alternative 13: 97.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} z = |z|\\ [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 0.0005:\\ \;\;\;\;\frac{1 - z \cdot z}{y \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{1}{z}}{y}}{z \cdot x}\\ \end{array} \end{array} \]

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) 0.0005)
   (/ (- 1.0 (* z z)) (* y x))
   (/ (/ (/ 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) <= 0.0005) {
		tmp = (1.0 - (z * z)) / (y * x);
	} 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) <= 0.0005d0) then
        tmp = (1.0d0 - (z * z)) / (y * x)
    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) <= 0.0005) {
		tmp = (1.0 - (z * z)) / (y * x);
	} 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) <= 0.0005:
		tmp = (1.0 - (z * z)) / (y * x)
	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) <= 0.0005)
		tmp = Float64(Float64(1.0 - Float64(z * z)) / Float64(y * x));
	else
		tmp = Float64(Float64(Float64(1.0 / 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) <= 0.0005)
		tmp = (1.0 - (z * z)) / (y * x);
	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], 0.0005], N[(N[(1.0 - N[(z * z), $MachinePrecision]), $MachinePrecision] / N[(y * x), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 / z), $MachinePrecision] / y), $MachinePrecision] / N[(z * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 5.0000000000000001e-4

   1. Initial program 99.7%

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

      1. associate-/r* 99.6%

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

      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 90.1%

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

      1. associate-/l/ 90.1%

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

      2. +-commutative 90.1%

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

      3. mul-1-neg 90.1%

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

      4. unsub-neg 90.1%

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

      5. associate-/r* 90.1%

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

      6. unpow2 90.1%

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

      7. *-commutative 90.1%

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

      8. div-sub 99.3%

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

   6. Simplified 99.3%

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

   if 5.0000000000000001e-4 < (*.f64 z z)

   1. Initial program 80.3%

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

      1. associate-/r* 79.8%

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

      2. *-commutative 79.8%

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

      3. sqr-neg 79.8%

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

      4. +-commutative 79.8%

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

      5. distribute-lft1-in 79.8%

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

      6. *-commutative 79.8%

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

      7. fma-def 79.8%

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

      8. sqr-neg 79.8%

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

   3. Simplified 79.8%

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

   4. Taylor expanded in z around inf 78.1%

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

      1. *-commutative 78.1%

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

      2. unpow2 78.1%

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

      3. associate-*l* 78.4%

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

      4. associate-*l* 88.8%

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

   6. Simplified 88.8%

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

      1. associate-/r* 89.2%

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

      2. *-un-lft-identity 89.2%

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

      3. times-frac 93.2%

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

   8. Applied egg-rr 93.2%

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

      1. associate-*r/ 94.4%

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

      2. un-div-inv 94.4%

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

   10. Applied egg-rr 94.4%

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

4. Final simplification 96.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 0.0005:\\ \;\;\;\;\frac{1 - z \cdot z}{y \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{1}{z}}{y}}{z \cdot x}\\ \end{array} \]

Alternative 14: 70.5% accurate, 1.2× speedup

Math

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

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 1.0) (/ (/ 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 <= 1.0) {
		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 <= 1.0d0) 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 <= 1.0) {
		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 <= 1.0:
		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 <= 1.0)
		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 <= 1.0)
		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, 1.0], 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 < 1

   1. Initial program 94.1%

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

      1. distribute-lft-in 94.1%

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

      2. *-rgt-identity 94.1%

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

      3. +-commutative 94.1%

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

      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%

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

   4. Taylor expanded in z around 0 71.6%

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

   if 1 < z

   1. Initial program 77.4%

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

      1. associate-/r* 77.2%

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

      2. associate-/r* 77.3%

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

      3. sqr-neg 77.3%

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

      4. +-commutative 77.3%

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

      5. sqr-neg 77.3%

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

      6. fma-def 77.3%

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

   3. Simplified 77.3%

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

      1. associate-/r* 77.2%

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

      2. div-inv 77.3%

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

   5. Applied egg-rr 77.3%

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

      1. *-commutative 77.3%

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

      2. add-sqr-sqrt 77.3%

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

      3. times-frac 80.4%

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

      4. fma-udef 80.4%

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

      5. +-commutative 80.4%

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

      6. hypot-1-def 80.4%

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

      7. fma-udef 80.4%

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

      8. +-commutative 80.4%

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

      9. hypot-1-def 96.3%

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

   7. Applied egg-rr 96.3%

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

   8. Taylor expanded in z around inf 94.7%

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

      1. *-commutative 94.7%

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

      2. associate-/r* 95.5%

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

   10. Simplified 95.5%

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

   11. Taylor expanded in z around 0 42.8%

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

4. Final simplification 63.5%

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

Alternative 15: 58.7% 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 89.4%

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

   1. associate-/r* 89.1%

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

   2. *-commutative 89.1%

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

   3. sqr-neg 89.1%

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

   4. +-commutative 89.1%

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

   5. distribute-lft1-in 89.1%

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

   6. *-commutative 89.1%

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

   7. fma-def 89.1%

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

   8. sqr-neg 89.1%

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

3. Simplified 89.1%

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

4. Taylor expanded in z around 0 57.9%

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

   1. *-commutative 57.9%

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

6. Simplified 57.9%

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

7. Final simplification 57.9%

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

Developer target: 92.9% 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 2023272 
(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)))))