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

Percentage Accurate: 90.7% → 97.6%

Time: 9.3s

Alternatives: 15

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 90.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 97.6% accurate, 0.1× speedup

Math

\[\begin{array}{l} z = |z|\\ [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{+145}:\\ \;\;\;\;\frac{\frac{\frac{1}{\mathsf{fma}\left(z, z, 1\right)}}{x}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x}}{\mathsf{hypot}\left(1, z\right)} \cdot \frac{1}{z \cdot y}\\ \end{array} \end{array} \]

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+145)
   (/ (/ (/ 1.0 (fma z z 1.0)) x) y)
   (* (/ (/ 1.0 x) (hypot 1.0 z)) (/ 1.0 (* z y)))))

C

z = abs(z);
assert(x < y);
double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 2e+145) {
		tmp = ((1.0 / fma(z, z, 1.0)) / x) / y;
	} else {
		tmp = ((1.0 / x) / hypot(1.0, z)) * (1.0 / (z * y));
	}
	return tmp;
}

Julia

z = abs(z)
x, y = sort([x, y])
function code(x, y, z)
	tmp = 0.0
	if (Float64(z * z) <= 2e+145)
		tmp = Float64(Float64(Float64(1.0 / fma(z, z, 1.0)) / x) / y);
	else
		tmp = Float64(Float64(Float64(1.0 / x) / hypot(1.0, z)) * Float64(1.0 / Float64(z * y)));
	end
	return tmp
end

Wolfram

NOTE: z should be positive before calling this function
NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_, z_] := If[LessEqual[N[(z * z), $MachinePrecision], 2e+145], N[(N[(N[(1.0 / N[(z * z + 1.0), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] / y), $MachinePrecision], N[(N[(N[(1.0 / x), $MachinePrecision] / N[Sqrt[1.0 ^ 2 + z ^ 2], $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(z * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 2e145

   1. Initial program 98.5%

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

      1. associate-/r* 98.0%

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

      2. +-commutative 98.0%

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

      3. fma-def 98.0%

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

   3. Simplified 98.0%

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

      1. fma-udef 98.0%

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

      2. +-commutative 98.0%

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

      3. associate-/r* 98.5%

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

      4. associate-/r* 97.3%

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

      5. div-inv 97.2%

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

      6. add-sqr-sqrt 97.1%

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

      7. times-frac 98.9%

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

      8. hypot-1-def 98.9%

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

      9. hypot-1-def 98.9%

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

   5. Applied egg-rr 98.9%

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

   6. Taylor expanded in x around 0 99.2%

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

      1. associate-*r* 98.0%

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

      2. *-commutative 98.0%

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

      3. associate-*r* 97.4%

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

      4. associate-/r* 97.6%

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

      5. unpow2 97.6%

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

      6. fma-udef 97.6%

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

      7. *-commutative 97.6%

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

      8. associate-/r* 99.7%

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

   8. Simplified 99.7%

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

   if 2e145 < (*.f64 z z)

   1. Initial program 76.8%

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

      1. associate-/r* 76.7%

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

      2. +-commutative 76.7%

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

      3. fma-def 76.7%

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

   3. Simplified 76.7%

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

      1. fma-udef 76.7%

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

      2. +-commutative 76.7%

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

      3. associate-/r* 76.8%

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

      4. associate-/r* 78.5%

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

      5. div-inv 78.5%

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

      6. add-sqr-sqrt 78.5%

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

      7. times-frac 81.0%

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

      8. hypot-1-def 81.0%

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

      9. hypot-1-def 95.7%

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

   5. Applied egg-rr 95.7%

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

   6. Taylor expanded in z around inf 75.6%

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

4. Final simplification 90.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{+145}:\\ \;\;\;\;\frac{\frac{\frac{1}{\mathsf{fma}\left(z, z, 1\right)}}{x}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x}}{\mathsf{hypot}\left(1, z\right)} \cdot \frac{1}{z \cdot y}\\ \end{array} \]

Alternative 2: 97.7% accurate, 0.1× speedup

Math

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

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 x) (hypot 1.0 z)) (/ (/ 1.0 y) (hypot 1.0 z))))

C

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

Python

z = abs(z)
[x, y] = sort([x, y])
def code(x, y, z):
	return ((1.0 / x) / math.hypot(1.0, z)) * ((1.0 / y) / 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 / x) / hypot(1.0, z)) * Float64(Float64(1.0 / y) / hypot(1.0, z)))
end

MATLAB

z = abs(z)
x, y = num2cell(sort([x, y])){:}
function tmp = code(x, y, z)
	tmp = ((1.0 / x) / hypot(1.0, z)) * ((1.0 / y) / 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 / x), $MachinePrecision] / N[Sqrt[1.0 ^ 2 + z ^ 2], $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 / y), $MachinePrecision] / N[Sqrt[1.0 ^ 2 + z ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 90.2%

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

   1. associate-/r* 89.8%

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

   2. +-commutative 89.8%

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

   3. fma-def 89.8%

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

3. Simplified 89.8%

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

   1. fma-udef 89.8%

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

   2. +-commutative 89.8%

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

   3. associate-/r* 90.2%

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

   4. associate-/r* 90.1%

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

   5. div-inv 90.0%

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

   6. add-sqr-sqrt 90.0%

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

   7. times-frac 92.0%

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

   8. hypot-1-def 92.0%

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

   9. hypot-1-def 97.7%

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

5. Applied egg-rr 97.7%

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

6. Final simplification 97.7%

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

Alternative 3: 97.5% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 90.2%

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

   1. associate-/r* 89.8%

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

   2. +-commutative 89.8%

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

   3. fma-def 89.8%

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

3. Simplified 89.8%

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

   1. fma-udef 89.8%

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

   2. +-commutative 89.8%

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

   3. associate-/r* 90.2%

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

   4. associate-/r* 90.1%

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

   5. div-inv 90.0%

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

   6. add-sqr-sqrt 90.0%

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

   7. times-frac 92.0%

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

   8. hypot-1-def 92.0%

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

   9. hypot-1-def 97.7%

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

5. Applied egg-rr 97.7%

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

   1. associate-/l/ 97.6%

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

   2. un-div-inv 97.7%

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

   3. associate-/l/ 97.5%

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

7. Applied egg-rr 97.5%

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

8. Final simplification 97.5%

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

Alternative 4: 97.6% accurate, 0.1× speedup

Math

\[\begin{array}{l} z = |z|\\ [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 4 \cdot 10^{+184}:\\ \;\;\;\;\frac{\frac{\frac{1}{\mathsf{fma}\left(z, z, 1\right)}}{x}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{z \cdot y}}{x \cdot z}\\ \end{array} \end{array} \]

FPCore

NOTE: z should be positive before calling this function
NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= (* z z) 4e+184)
   (/ (/ (/ 1.0 (fma z z 1.0)) x) y)
   (/ (/ 1.0 (* z y)) (* x z))))

C

z = abs(z);
assert(x < y);
double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 4e+184) {
		tmp = ((1.0 / fma(z, z, 1.0)) / x) / y;
	} else {
		tmp = (1.0 / (z * y)) / (x * z);
	}
	return tmp;
}

Julia

z = abs(z)
x, y = sort([x, y])
function code(x, y, z)
	tmp = 0.0
	if (Float64(z * z) <= 4e+184)
		tmp = Float64(Float64(Float64(1.0 / fma(z, z, 1.0)) / x) / y);
	else
		tmp = Float64(Float64(1.0 / Float64(z * y)) / Float64(x * z));
	end
	return tmp
end

Wolfram

NOTE: z should be positive before calling this function
NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_, z_] := If[LessEqual[N[(z * z), $MachinePrecision], 4e+184], N[(N[(N[(1.0 / N[(z * z + 1.0), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] / y), $MachinePrecision], N[(N[(1.0 / N[(z * y), $MachinePrecision]), $MachinePrecision] / N[(x * z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 4.00000000000000007e184

   1. Initial program 97.4%

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

      1. associate-/r* 97.0%

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

      2. +-commutative 97.0%

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

      3. fma-def 97.0%

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

   3. Simplified 97.0%

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

      1. fma-udef 97.0%

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

      2. +-commutative 97.0%

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

      3. associate-/r* 97.4%

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

      4. associate-/r* 95.8%

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

      5. div-inv 95.7%

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

      6. add-sqr-sqrt 95.6%

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

      7. times-frac 98.0%

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

      8. hypot-1-def 98.0%

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

      9. hypot-1-def 98.0%

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

   5. Applied egg-rr 98.0%

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

   6. Taylor expanded in x around 0 98.7%

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

      1. associate-*r* 97.0%

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

      2. *-commutative 97.0%

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

      3. associate-*r* 95.9%

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

      4. associate-/r* 96.1%

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

      5. unpow2 96.1%

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

      6. fma-udef 96.1%

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

      7. *-commutative 96.1%

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

      8. associate-/r* 99.1%

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

   8. Simplified 99.1%

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

   if 4.00000000000000007e184 < (*.f64 z z)

   1. Initial program 75.8%

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

      1. associate-/r* 75.6%

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

      2. +-commutative 75.6%

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

      3. fma-def 75.6%

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

   3. Simplified 75.6%

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

   4. Taylor expanded in z around inf 75.6%

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

      1. unpow2 75.6%

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

      2. associate-*r* 87.8%

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

      3. *-commutative 87.8%

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

   6. Simplified 87.8%

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

      1. inv-pow 87.8%

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

      2. associate-*r* 93.6%

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

      3. unpow-prod-down 93.4%

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

   8. Applied egg-rr 93.4%

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

      1. unpow-1 93.4%

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

      2. *-commutative 93.4%

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

      3. unpow-1 93.4%

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

   10. Simplified 93.4%

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

       1. un-div-inv 93.4%

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

       2. associate-/r* 93.4%

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

       3. associate-/l/ 88.0%

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

       4. associate-/r* 87.8%

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

       5. associate-*l* 96.4%

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

       6. *-commutative 96.4%

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

       7. associate-/r* 96.6%

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

   12. Applied egg-rr 96.6%

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

4. Final simplification 98.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 4 \cdot 10^{+184}:\\ \;\;\;\;\frac{\frac{\frac{1}{\mathsf{fma}\left(z, z, 1\right)}}{x}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{z \cdot y}}{x \cdot z}\\ \end{array} \]

Alternative 5: 97.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} z = |z|\\ [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{+44}:\\ \;\;\;\;\frac{1}{x \cdot \left(y + y \cdot \left(z \cdot z\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{z \cdot y}}{x \cdot z}\\ \end{array} \end{array} \]

FPCore

NOTE: z should be positive before calling this function
NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= (* z z) 2e+44)
   (/ 1.0 (* x (+ y (* y (* z z)))))
   (/ (/ 1.0 (* z y)) (* x z))))

C

z = abs(z);
assert(x < y);
double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 2e+44) {
		tmp = 1.0 / (x * (y + (y * (z * z))));
	} else {
		tmp = (1.0 / (z * y)) / (x * z);
	}
	return tmp;
}

Fortran

NOTE: z should be positive before calling this function
NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((z * z) <= 2d+44) then
        tmp = 1.0d0 / (x * (y + (y * (z * z))))
    else
        tmp = (1.0d0 / (z * y)) / (x * z)
    end if
    code = tmp
end function

Java

z = Math.abs(z);
assert x < y;
public static double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 2e+44) {
		tmp = 1.0 / (x * (y + (y * (z * z))));
	} else {
		tmp = (1.0 / (z * y)) / (x * z);
	}
	return tmp;
}

Python

z = abs(z)
[x, y] = sort([x, y])
def code(x, y, z):
	tmp = 0
	if (z * z) <= 2e+44:
		tmp = 1.0 / (x * (y + (y * (z * z))))
	else:
		tmp = (1.0 / (z * y)) / (x * z)
	return tmp

Julia

z = abs(z)
x, y = sort([x, y])
function code(x, y, z)
	tmp = 0.0
	if (Float64(z * z) <= 2e+44)
		tmp = Float64(1.0 / Float64(x * Float64(y + Float64(y * Float64(z * z)))));
	else
		tmp = Float64(Float64(1.0 / Float64(z * y)) / Float64(x * z));
	end
	return tmp
end

MATLAB

z = abs(z)
x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z * z) <= 2e+44)
		tmp = 1.0 / (x * (y + (y * (z * z))));
	else
		tmp = (1.0 / (z * y)) / (x * z);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: z should be positive before calling this function
NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_, z_] := If[LessEqual[N[(z * z), $MachinePrecision], 2e+44], N[(1.0 / N[(x * N[(y + N[(y * N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[(z * y), $MachinePrecision]), $MachinePrecision] / N[(x * z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 2.0000000000000002e44

   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* 99.3%

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

      2. +-commutative 99.3%

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

      3. fma-def 99.3%

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

   3. Simplified 99.3%

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

      1. fma-udef 99.3%

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

      2. distribute-lft-in 99.3%

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

      3. *-rgt-identity 99.3%

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

   5. Applied egg-rr 99.3%

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

   if 2.0000000000000002e44 < (*.f64 z z)

   1. Initial program 80.2%

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

      1. associate-/r* 79.9%

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

      2. +-commutative 79.9%

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

      3. fma-def 79.9%

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

   3. Simplified 79.9%

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

   4. Taylor expanded in z around inf 79.9%

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

      1. unpow2 79.9%

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

      2. associate-*r* 88.3%

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

      3. *-commutative 88.3%

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

   6. Simplified 88.3%

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

      1. inv-pow 88.3%

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

      2. associate-*r* 93.1%

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

      3. unpow-prod-down 93.0%

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

   8. Applied egg-rr 93.0%

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

      1. unpow-1 93.0%

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

      2. *-commutative 93.0%

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

      3. unpow-1 93.0%

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

   10. Simplified 93.0%

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

       1. un-div-inv 93.0%

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

       2. associate-/r* 93.0%

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

       3. associate-/l/ 88.6%

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

       4. associate-/r* 88.3%

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

       5. associate-*l* 95.0%

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

       6. *-commutative 95.0%

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

       7. associate-/r* 95.3%

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

   12. Applied egg-rr 95.3%

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

4. Final simplification 97.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{+44}:\\ \;\;\;\;\frac{1}{x \cdot \left(y + y \cdot \left(z \cdot z\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{z \cdot y}}{x \cdot z}\\ \end{array} \]

Alternative 6: 97.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} z = |z|\\ [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{+44}:\\ \;\;\;\;\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{z \cdot y}}{x \cdot z}\\ \end{array} \end{array} \]

FPCore

NOTE: z should be positive before calling this function
NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= (* z z) 2e+44)
   (/ (/ 1.0 x) (* y (+ 1.0 (* z z))))
   (/ (/ 1.0 (* z y)) (* x z))))

C

z = abs(z);
assert(x < y);
double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 2e+44) {
		tmp = (1.0 / x) / (y * (1.0 + (z * z)));
	} else {
		tmp = (1.0 / (z * y)) / (x * z);
	}
	return tmp;
}

Fortran

NOTE: z should be positive before calling this function
NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((z * z) <= 2d+44) then
        tmp = (1.0d0 / x) / (y * (1.0d0 + (z * z)))
    else
        tmp = (1.0d0 / (z * y)) / (x * z)
    end if
    code = tmp
end function

Java

z = Math.abs(z);
assert x < y;
public static double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 2e+44) {
		tmp = (1.0 / x) / (y * (1.0 + (z * z)));
	} else {
		tmp = (1.0 / (z * y)) / (x * z);
	}
	return tmp;
}

Python

z = abs(z)
[x, y] = sort([x, y])
def code(x, y, z):
	tmp = 0
	if (z * z) <= 2e+44:
		tmp = (1.0 / x) / (y * (1.0 + (z * z)))
	else:
		tmp = (1.0 / (z * y)) / (x * z)
	return tmp

Julia

z = abs(z)
x, y = sort([x, y])
function code(x, y, z)
	tmp = 0.0
	if (Float64(z * z) <= 2e+44)
		tmp = Float64(Float64(1.0 / x) / Float64(y * Float64(1.0 + Float64(z * z))));
	else
		tmp = Float64(Float64(1.0 / Float64(z * y)) / Float64(x * z));
	end
	return tmp
end

MATLAB

z = abs(z)
x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z * z) <= 2e+44)
		tmp = (1.0 / x) / (y * (1.0 + (z * z)));
	else
		tmp = (1.0 / (z * y)) / (x * z);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: z should be positive before calling this function
NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_, z_] := If[LessEqual[N[(z * z), $MachinePrecision], 2e+44], N[(N[(1.0 / x), $MachinePrecision] / N[(y * N[(1.0 + N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[(z * y), $MachinePrecision]), $MachinePrecision] / N[(x * z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 2.0000000000000002e44

   1. Initial program 99.7%

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

   if 2.0000000000000002e44 < (*.f64 z z)

   1. Initial program 80.2%

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

      1. associate-/r* 79.9%

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

      2. +-commutative 79.9%

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

      3. fma-def 79.9%

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

   3. Simplified 79.9%

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

   4. Taylor expanded in z around inf 79.9%

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

      1. unpow2 79.9%

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

      2. associate-*r* 88.3%

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

      3. *-commutative 88.3%

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

   6. Simplified 88.3%

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

      1. inv-pow 88.3%

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

      2. associate-*r* 93.1%

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

      3. unpow-prod-down 93.0%

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

   8. Applied egg-rr 93.0%

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

      1. unpow-1 93.0%

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

      2. *-commutative 93.0%

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

      3. unpow-1 93.0%

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

   10. Simplified 93.0%

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

       1. un-div-inv 93.0%

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

       2. associate-/r* 93.0%

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

       3. associate-/l/ 88.6%

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

       4. associate-/r* 88.3%

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

       5. associate-*l* 95.0%

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

       6. *-commutative 95.0%

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

       7. associate-/r* 95.3%

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

   12. Applied egg-rr 95.3%

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

4. Final simplification 97.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{+44}:\\ \;\;\;\;\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{z \cdot y}}{x \cdot z}\\ \end{array} \]

Alternative 7: 97.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} z = |z|\\ [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{+44}:\\ \;\;\;\;\frac{\frac{\frac{1}{x}}{y}}{1 + z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{z \cdot y}}{x \cdot z}\\ \end{array} \end{array} \]

FPCore

NOTE: z should be positive before calling this function
NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= (* z z) 2e+44)
   (/ (/ (/ 1.0 x) y) (+ 1.0 (* z z)))
   (/ (/ 1.0 (* z y)) (* x z))))

C

z = abs(z);
assert(x < y);
double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 2e+44) {
		tmp = ((1.0 / x) / y) / (1.0 + (z * z));
	} else {
		tmp = (1.0 / (z * y)) / (x * z);
	}
	return tmp;
}

Fortran

NOTE: z should be positive before calling this function
NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((z * z) <= 2d+44) then
        tmp = ((1.0d0 / x) / y) / (1.0d0 + (z * z))
    else
        tmp = (1.0d0 / (z * y)) / (x * z)
    end if
    code = tmp
end function

Java

z = Math.abs(z);
assert x < y;
public static double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 2e+44) {
		tmp = ((1.0 / x) / y) / (1.0 + (z * z));
	} else {
		tmp = (1.0 / (z * y)) / (x * z);
	}
	return tmp;
}

Python

z = abs(z)
[x, y] = sort([x, y])
def code(x, y, z):
	tmp = 0
	if (z * z) <= 2e+44:
		tmp = ((1.0 / x) / y) / (1.0 + (z * z))
	else:
		tmp = (1.0 / (z * y)) / (x * z)
	return tmp

Julia

z = abs(z)
x, y = sort([x, y])
function code(x, y, z)
	tmp = 0.0
	if (Float64(z * z) <= 2e+44)
		tmp = Float64(Float64(Float64(1.0 / x) / y) / Float64(1.0 + Float64(z * z)));
	else
		tmp = Float64(Float64(1.0 / Float64(z * y)) / Float64(x * z));
	end
	return tmp
end

MATLAB

z = abs(z)
x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z * z) <= 2e+44)
		tmp = ((1.0 / x) / y) / (1.0 + (z * z));
	else
		tmp = (1.0 / (z * y)) / (x * z);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: z should be positive before calling this function
NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_, z_] := If[LessEqual[N[(z * z), $MachinePrecision], 2e+44], N[(N[(N[(1.0 / x), $MachinePrecision] / y), $MachinePrecision] / N[(1.0 + N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[(z * y), $MachinePrecision]), $MachinePrecision] / N[(x * z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 2.0000000000000002e44

   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* 99.7%

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

   3. Simplified 99.7%

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

   if 2.0000000000000002e44 < (*.f64 z z)

   1. Initial program 80.2%

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

      1. associate-/r* 79.9%

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

      2. +-commutative 79.9%

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

      3. fma-def 79.9%

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

   3. Simplified 79.9%

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

   4. Taylor expanded in z around inf 79.9%

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

      1. unpow2 79.9%

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

      2. associate-*r* 88.3%

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

      3. *-commutative 88.3%

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

   6. Simplified 88.3%

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

      1. inv-pow 88.3%

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

      2. associate-*r* 93.1%

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

      3. unpow-prod-down 93.0%

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

   8. Applied egg-rr 93.0%

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

      1. unpow-1 93.0%

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

      2. *-commutative 93.0%

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

      3. unpow-1 93.0%

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

   10. Simplified 93.0%

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

       1. un-div-inv 93.0%

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

       2. associate-/r* 93.0%

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

       3. associate-/l/ 88.6%

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

       4. associate-/r* 88.3%

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

       5. associate-*l* 95.0%

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

       6. *-commutative 95.0%

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

       7. associate-/r* 95.3%

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

   12. Applied egg-rr 95.3%

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

4. Final simplification 97.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{+44}:\\ \;\;\;\;\frac{\frac{\frac{1}{x}}{y}}{1 + z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{z \cdot y}}{x \cdot z}\\ \end{array} \]

Alternative 8: 93.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} z = |z|\\ [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq 1:\\ \;\;\;\;\frac{\frac{1}{x}}{y}\\ \mathbf{elif}\;z \leq 4.6 \cdot 10^{+181}:\\ \;\;\;\;\frac{1}{x \cdot \left(z \cdot \left(z \cdot y\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{z \cdot \left(z \cdot \left(x \cdot y\right)\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: z should be positive before calling this function
NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= z 1.0)
   (/ (/ 1.0 x) y)
   (if (<= z 4.6e+181)
     (/ 1.0 (* x (* z (* z y))))
     (/ 1.0 (* z (* z (* x y)))))))

C

z = abs(z);
assert(x < y);
double code(double x, double y, double z) {
	double tmp;
	if (z <= 1.0) {
		tmp = (1.0 / x) / y;
	} else if (z <= 4.6e+181) {
		tmp = 1.0 / (x * (z * (z * y)));
	} else {
		tmp = 1.0 / (z * (z * (x * y)));
	}
	return tmp;
}

Fortran

NOTE: z should be positive before calling this function
NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (z <= 1.0d0) then
        tmp = (1.0d0 / x) / y
    else if (z <= 4.6d+181) then
        tmp = 1.0d0 / (x * (z * (z * y)))
    else
        tmp = 1.0d0 / (z * (z * (x * y)))
    end if
    code = tmp
end function

Java

z = Math.abs(z);
assert x < y;
public static double code(double x, double y, double z) {
	double tmp;
	if (z <= 1.0) {
		tmp = (1.0 / x) / y;
	} else if (z <= 4.6e+181) {
		tmp = 1.0 / (x * (z * (z * y)));
	} else {
		tmp = 1.0 / (z * (z * (x * y)));
	}
	return tmp;
}

Python

z = abs(z)
[x, y] = sort([x, y])
def code(x, y, z):
	tmp = 0
	if z <= 1.0:
		tmp = (1.0 / x) / y
	elif z <= 4.6e+181:
		tmp = 1.0 / (x * (z * (z * y)))
	else:
		tmp = 1.0 / (z * (z * (x * y)))
	return tmp

Julia

z = abs(z)
x, y = sort([x, y])
function code(x, y, z)
	tmp = 0.0
	if (z <= 1.0)
		tmp = Float64(Float64(1.0 / x) / y);
	elseif (z <= 4.6e+181)
		tmp = Float64(1.0 / Float64(x * Float64(z * Float64(z * y))));
	else
		tmp = Float64(1.0 / Float64(z * Float64(z * Float64(x * y))));
	end
	return tmp
end

MATLAB

z = abs(z)
x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= 1.0)
		tmp = (1.0 / x) / y;
	elseif (z <= 4.6e+181)
		tmp = 1.0 / (x * (z * (z * y)));
	else
		tmp = 1.0 / (z * (z * (x * y)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: z should be positive before calling this function
NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_, z_] := If[LessEqual[z, 1.0], N[(N[(1.0 / x), $MachinePrecision] / y), $MachinePrecision], If[LessEqual[z, 4.6e+181], N[(1.0 / N[(x * N[(z * N[(z * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(z * N[(z * N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < 1

   1. Initial program 91.2%

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

      1. associate-/r* 91.0%

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

      2. +-commutative 91.0%

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

      3. fma-def 91.0%

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

   3. Simplified 91.0%

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

      1. fma-udef 91.0%

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

      2. +-commutative 91.0%

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

      3. associate-/r* 91.2%

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

      4. associate-/r* 91.6%

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

      5. div-inv 91.5%

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

      6. add-sqr-sqrt 91.5%

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

      7. times-frac 93.4%

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

      8. hypot-1-def 93.4%

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

      9. hypot-1-def 98.2%

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

   5. Applied egg-rr 98.2%

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

      1. associate-/l/ 98.2%

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

      2. un-div-inv 98.4%

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

      3. associate-/l/ 98.3%

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

   7. Applied egg-rr 98.3%

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

   8. Taylor expanded in z around 0 67.7%

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

      1. *-commutative 67.7%

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

      2. associate-/r* 67.9%

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

   10. Simplified 67.9%

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

   if 1 < z < 4.5999999999999998e181

   1. Initial program 92.0%

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

      1. associate-/r* 91.0%

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

      2. +-commutative 91.0%

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

      3. fma-def 91.0%

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

   3. Simplified 91.0%

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

   4. Taylor expanded in z around inf 90.6%

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

      1. unpow2 90.6%

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

      2. associate-*r* 93.1%

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

      3. *-commutative 93.1%

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

   6. Simplified 93.1%

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

   if 4.5999999999999998e181 < z

   1. Initial program 75.6%

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

      1. associate-/r* 75.6%

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

      2. +-commutative 75.6%

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

      3. fma-def 75.6%

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

   3. Simplified 75.6%

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

   4. Taylor expanded in z around inf 75.6%

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

      1. unpow2 75.6%

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

      2. *-commutative 75.6%

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

      3. associate-*l* 75.4%

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

      4. *-commutative 75.4%

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

      5. associate-*l* 94.8%

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

   6. Simplified 94.8%

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

4. Final simplification 73.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 1:\\ \;\;\;\;\frac{\frac{1}{x}}{y}\\ \mathbf{elif}\;z \leq 4.6 \cdot 10^{+181}:\\ \;\;\;\;\frac{1}{x \cdot \left(z \cdot \left(z \cdot y\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{z \cdot \left(z \cdot \left(x \cdot y\right)\right)}\\ \end{array} \]

Alternative 9: 89.7% accurate, 0.8× speedup

Math

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

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

   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* 99.3%

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

      2. +-commutative 99.3%

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

      3. fma-def 99.3%

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

   3. Simplified 99.3%

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

      1. fma-udef 99.3%

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

      2. +-commutative 99.3%

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

      3. associate-/r* 99.7%

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

      4. associate-/r* 99.7%

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

      5. div-inv 99.5%

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

      6. add-sqr-sqrt 99.5%

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

      7. times-frac 99.5%

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

      8. hypot-1-def 99.6%

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

      9. hypot-1-def 99.5%

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

   5. Applied egg-rr 99.5%

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

      1. associate-/l/ 99.6%

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

      2. un-div-inv 99.7%

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

      3. associate-/l/ 99.7%

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

   7. Applied egg-rr 99.7%

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

   8. Taylor expanded in z around 0 97.7%

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

      1. *-commutative 97.7%

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

      2. associate-/r* 98.1%

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

   10. Simplified 98.1%

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

   if 1 < (*.f64 z z)

   1. Initial program 80.9%

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

      1. associate-/r* 80.6%

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

      2. +-commutative 80.6%

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

      3. fma-def 80.6%

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

   3. Simplified 80.6%

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

      1. fma-udef 80.6%

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

      2. +-commutative 80.6%

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

      3. associate-/r* 80.9%

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

      4. associate-/r* 80.8%

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

      5. div-inv 80.8%

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

      6. add-sqr-sqrt 80.8%

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

      7. times-frac 84.7%

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

      8. hypot-1-def 84.7%

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

      9. hypot-1-def 95.9%

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

   5. Applied egg-rr 95.9%

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

   6. Taylor expanded in z around inf 85.8%

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

      1. unpow2 85.8%

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

      2. associate-*r* 80.1%

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

      3. associate-*l* 88.1%

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

      4. *-commutative 88.1%

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

      5. associate-*l* 80.1%

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

   8. Simplified 80.1%

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

4. Final simplification 89.0%

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

Alternative 10: 93.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} z = |z|\\ [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 0.0004:\\ \;\;\;\;\frac{\frac{1}{x}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x \cdot \left(z \cdot \left(z \cdot y\right)\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: z should be positive before calling this function
NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= (* z z) 0.0004) (/ (/ 1.0 x) y) (/ 1.0 (* x (* z (* z y))))))

C

z = abs(z);
assert(x < y);
double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 0.0004) {
		tmp = (1.0 / x) / y;
	} else {
		tmp = 1.0 / (x * (z * (z * y)));
	}
	return tmp;
}

Fortran

NOTE: z should be positive before calling this function
NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((z * z) <= 0.0004d0) then
        tmp = (1.0d0 / x) / y
    else
        tmp = 1.0d0 / (x * (z * (z * y)))
    end if
    code = tmp
end function

Java

z = Math.abs(z);
assert x < y;
public static double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 0.0004) {
		tmp = (1.0 / x) / y;
	} else {
		tmp = 1.0 / (x * (z * (z * y)));
	}
	return tmp;
}

Python

z = abs(z)
[x, y] = sort([x, y])
def code(x, y, z):
	tmp = 0
	if (z * z) <= 0.0004:
		tmp = (1.0 / x) / y
	else:
		tmp = 1.0 / (x * (z * (z * y)))
	return tmp

Julia

z = abs(z)
x, y = sort([x, y])
function code(x, y, z)
	tmp = 0.0
	if (Float64(z * z) <= 0.0004)
		tmp = Float64(Float64(1.0 / x) / y);
	else
		tmp = Float64(1.0 / Float64(x * Float64(z * Float64(z * y))));
	end
	return tmp
end

MATLAB

z = abs(z)
x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z * z) <= 0.0004)
		tmp = (1.0 / x) / y;
	else
		tmp = 1.0 / (x * (z * (z * y)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: z should be positive before calling this function
NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_, z_] := If[LessEqual[N[(z * z), $MachinePrecision], 0.0004], N[(N[(1.0 / x), $MachinePrecision] / y), $MachinePrecision], N[(1.0 / N[(x * N[(z * N[(z * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 4.00000000000000019e-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* 99.3%

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

      2. +-commutative 99.3%

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

      3. fma-def 99.3%

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

   3. Simplified 99.3%

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

      1. fma-udef 99.3%

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

      2. +-commutative 99.3%

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

      3. associate-/r* 99.7%

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

      4. associate-/r* 99.7%

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

      5. div-inv 99.5%

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

      6. add-sqr-sqrt 99.5%

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

      7. times-frac 99.5%

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

      8. hypot-1-def 99.6%

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

      9. hypot-1-def 99.5%

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

   5. Applied egg-rr 99.5%

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

      1. associate-/l/ 99.6%

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

      2. un-div-inv 99.7%

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

      3. associate-/l/ 99.7%

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

   7. Applied egg-rr 99.7%

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

   8. Taylor expanded in z around 0 97.7%

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

      1. *-commutative 97.7%

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

      2. associate-/r* 98.1%

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

   10. Simplified 98.1%

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

   if 4.00000000000000019e-4 < (*.f64 z z)

   1. Initial program 80.9%

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

      1. associate-/r* 80.6%

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

      2. +-commutative 80.6%

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

      3. fma-def 80.6%

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

   3. Simplified 80.6%

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

   4. Taylor expanded in z around inf 80.1%

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

      1. unpow2 80.1%

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

      2. associate-*r* 88.1%

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

      3. *-commutative 88.1%

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

   6. Simplified 88.1%

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

4. Final simplification 93.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 0.0004:\\ \;\;\;\;\frac{\frac{1}{x}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x \cdot \left(z \cdot \left(z \cdot y\right)\right)}\\ \end{array} \]

Alternative 11: 96.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} z = |z|\\ [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 0.0004:\\ \;\;\;\;\frac{\frac{1}{x}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(z \cdot y\right) \cdot \left(x \cdot z\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: z should be positive before calling this function
NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= (* z z) 0.0004) (/ (/ 1.0 x) y) (/ 1.0 (* (* z y) (* x z)))))

C

z = abs(z);
assert(x < y);
double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 0.0004) {
		tmp = (1.0 / x) / y;
	} else {
		tmp = 1.0 / ((z * y) * (x * z));
	}
	return tmp;
}

Fortran

NOTE: z should be positive before calling this function
NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((z * z) <= 0.0004d0) then
        tmp = (1.0d0 / x) / y
    else
        tmp = 1.0d0 / ((z * y) * (x * z))
    end if
    code = tmp
end function

Java

z = Math.abs(z);
assert x < y;
public static double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 0.0004) {
		tmp = (1.0 / x) / y;
	} else {
		tmp = 1.0 / ((z * y) * (x * z));
	}
	return tmp;
}

Python

z = abs(z)
[x, y] = sort([x, y])
def code(x, y, z):
	tmp = 0
	if (z * z) <= 0.0004:
		tmp = (1.0 / x) / y
	else:
		tmp = 1.0 / ((z * y) * (x * z))
	return tmp

Julia

z = abs(z)
x, y = sort([x, y])
function code(x, y, z)
	tmp = 0.0
	if (Float64(z * z) <= 0.0004)
		tmp = Float64(Float64(1.0 / x) / y);
	else
		tmp = Float64(1.0 / Float64(Float64(z * y) * Float64(x * z)));
	end
	return tmp
end

MATLAB

z = abs(z)
x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z * z) <= 0.0004)
		tmp = (1.0 / x) / y;
	else
		tmp = 1.0 / ((z * y) * (x * z));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: z should be positive before calling this function
NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_, z_] := If[LessEqual[N[(z * z), $MachinePrecision], 0.0004], N[(N[(1.0 / x), $MachinePrecision] / y), $MachinePrecision], N[(1.0 / N[(N[(z * y), $MachinePrecision] * N[(x * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 4.00000000000000019e-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* 99.3%

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

      2. +-commutative 99.3%

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

      3. fma-def 99.3%

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

   3. Simplified 99.3%

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

      1. fma-udef 99.3%

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

      2. +-commutative 99.3%

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

      3. associate-/r* 99.7%

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

      4. associate-/r* 99.7%

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

      5. div-inv 99.5%

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

      6. add-sqr-sqrt 99.5%

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

      7. times-frac 99.5%

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

      8. hypot-1-def 99.6%

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

      9. hypot-1-def 99.5%

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

   5. Applied egg-rr 99.5%

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

      1. associate-/l/ 99.6%

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

      2. un-div-inv 99.7%

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

      3. associate-/l/ 99.7%

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

   7. Applied egg-rr 99.7%

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

   8. Taylor expanded in z around 0 97.7%

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

      1. *-commutative 97.7%

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

      2. associate-/r* 98.1%

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

   10. Simplified 98.1%

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

   if 4.00000000000000019e-4 < (*.f64 z z)

   1. Initial program 80.9%

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

      1. associate-/r* 80.6%

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

      2. +-commutative 80.6%

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

      3. fma-def 80.6%

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

   3. Simplified 80.6%

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

      1. fma-udef 80.6%

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

      2. +-commutative 80.6%

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

      3. associate-/r* 80.9%

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

      4. associate-/r* 80.8%

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

      5. div-inv 80.8%

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

      6. add-sqr-sqrt 80.8%

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

      7. times-frac 84.7%

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

      8. hypot-1-def 84.7%

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

      9. hypot-1-def 95.9%

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

   5. Applied egg-rr 95.9%

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

      1. associate-/l/ 95.7%

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

      2. un-div-inv 95.8%

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

      3. associate-/l/ 95.5%

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

   7. Applied egg-rr 95.5%

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

   8. Taylor expanded in z around inf 85.8%

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

      1. *-commutative 85.8%

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

      2. unpow2 85.8%

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

      3. *-commutative 85.8%

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

      4. associate-*r* 80.1%

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

      5. associate-*r* 88.1%

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

      6. associate-*r* 94.6%

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

      7. *-commutative 94.6%

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

   10. Simplified 94.6%

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

4. Final simplification 96.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 0.0004:\\ \;\;\;\;\frac{\frac{1}{x}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(z \cdot y\right) \cdot \left(x \cdot z\right)}\\ \end{array} \]

Alternative 12: 96.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} z = |z|\\ [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 0.0004:\\ \;\;\;\;\frac{\frac{1}{x}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{z \cdot y}}{x \cdot z}\\ \end{array} \end{array} \]

FPCore

NOTE: z should be positive before calling this function
NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= (* z z) 0.0004) (/ (/ 1.0 x) y) (/ (/ 1.0 (* z y)) (* x z))))

C

z = abs(z);
assert(x < y);
double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 0.0004) {
		tmp = (1.0 / x) / y;
	} else {
		tmp = (1.0 / (z * y)) / (x * z);
	}
	return tmp;
}

Fortran

NOTE: z should be positive before calling this function
NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((z * z) <= 0.0004d0) then
        tmp = (1.0d0 / x) / y
    else
        tmp = (1.0d0 / (z * y)) / (x * z)
    end if
    code = tmp
end function

Java

z = Math.abs(z);
assert x < y;
public static double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 0.0004) {
		tmp = (1.0 / x) / y;
	} else {
		tmp = (1.0 / (z * y)) / (x * z);
	}
	return tmp;
}

Python

z = abs(z)
[x, y] = sort([x, y])
def code(x, y, z):
	tmp = 0
	if (z * z) <= 0.0004:
		tmp = (1.0 / x) / y
	else:
		tmp = (1.0 / (z * y)) / (x * z)
	return tmp

Julia

z = abs(z)
x, y = sort([x, y])
function code(x, y, z)
	tmp = 0.0
	if (Float64(z * z) <= 0.0004)
		tmp = Float64(Float64(1.0 / x) / y);
	else
		tmp = Float64(Float64(1.0 / Float64(z * y)) / Float64(x * z));
	end
	return tmp
end

MATLAB

z = abs(z)
x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z * z) <= 0.0004)
		tmp = (1.0 / x) / y;
	else
		tmp = (1.0 / (z * y)) / (x * z);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: z should be positive before calling this function
NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_, z_] := If[LessEqual[N[(z * z), $MachinePrecision], 0.0004], N[(N[(1.0 / x), $MachinePrecision] / y), $MachinePrecision], N[(N[(1.0 / N[(z * y), $MachinePrecision]), $MachinePrecision] / N[(x * z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 4.00000000000000019e-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* 99.3%

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

      2. +-commutative 99.3%

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

      3. fma-def 99.3%

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

   3. Simplified 99.3%

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

      1. fma-udef 99.3%

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

      2. +-commutative 99.3%

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

      3. associate-/r* 99.7%

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

      4. associate-/r* 99.7%

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

      5. div-inv 99.5%

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

      6. add-sqr-sqrt 99.5%

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

      7. times-frac 99.5%

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

      8. hypot-1-def 99.6%

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

      9. hypot-1-def 99.5%

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

   5. Applied egg-rr 99.5%

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

      1. associate-/l/ 99.6%

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

      2. un-div-inv 99.7%

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

      3. associate-/l/ 99.7%

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

   7. Applied egg-rr 99.7%

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

   8. Taylor expanded in z around 0 97.7%

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

      1. *-commutative 97.7%

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

      2. associate-/r* 98.1%

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

   10. Simplified 98.1%

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

   if 4.00000000000000019e-4 < (*.f64 z z)

   1. Initial program 80.9%

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

      1. associate-/r* 80.6%

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

      2. +-commutative 80.6%

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

      3. fma-def 80.6%

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

   3. Simplified 80.6%

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

   4. Taylor expanded in z around inf 80.1%

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

      1. unpow2 80.1%

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

      2. associate-*r* 88.1%

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

      3. *-commutative 88.1%

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

   6. Simplified 88.1%

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

      1. inv-pow 88.1%

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

      2. associate-*r* 92.7%

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

      3. unpow-prod-down 92.6%

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

   8. Applied egg-rr 92.6%

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

      1. unpow-1 92.6%

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

      2. *-commutative 92.6%

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

      3. unpow-1 92.6%

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

   10. Simplified 92.6%

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

       1. un-div-inv 92.7%

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

       2. associate-/r* 92.6%

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

       3. associate-/l/ 88.4%

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

       4. associate-/r* 88.1%

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

       5. associate-*l* 94.6%

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

       6. *-commutative 94.6%

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

       7. associate-/r* 94.8%

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

   12. Applied egg-rr 94.8%

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

4. Final simplification 96.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 0.0004:\\ \;\;\;\;\frac{\frac{1}{x}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{z \cdot y}}{x \cdot z}\\ \end{array} \]

Alternative 13: 93.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if z < 1

   1. Initial program 91.2%

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

      1. associate-/r* 91.0%

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

      2. +-commutative 91.0%

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

      3. fma-def 91.0%

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

   3. Simplified 91.0%

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

      1. fma-udef 91.0%

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

      2. +-commutative 91.0%

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

      3. associate-/r* 91.2%

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

      4. associate-/r* 91.6%

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

      5. div-inv 91.5%

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

      6. add-sqr-sqrt 91.5%

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

      7. times-frac 93.4%

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

      8. hypot-1-def 93.4%

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

      9. hypot-1-def 98.2%

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

   5. Applied egg-rr 98.2%

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

      1. associate-/l/ 98.2%

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

      2. un-div-inv 98.4%

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

      3. associate-/l/ 98.3%

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

   7. Applied egg-rr 98.3%

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

   8. Taylor expanded in z around 0 67.7%

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

      1. *-commutative 67.7%

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

      2. associate-/r* 67.9%

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

   10. Simplified 67.9%

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

   if 1 < z

   1. Initial program 86.4%

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

      1. associate-/r* 85.8%

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

      2. +-commutative 85.8%

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

      3. fma-def 85.8%

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

   3. Simplified 85.8%

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

      1. fma-udef 85.8%

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

      2. +-commutative 85.8%

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

      3. associate-/r* 86.4%

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

      4. associate-/r* 84.7%

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

      5. div-inv 84.7%

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

      6. add-sqr-sqrt 84.7%

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

      7. times-frac 87.0%

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

      8. hypot-1-def 87.0%

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

      9. hypot-1-def 95.7%

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

   5. Applied egg-rr 95.7%

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

   6. Taylor expanded in z around inf 87.1%

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

      1. unpow2 87.1%

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

      2. *-commutative 87.1%

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

      3. associate-*r* 87.2%

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

      4. *-commutative 87.2%

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

   8. Simplified 87.2%

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

4. Final simplification 72.1%

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

Alternative 14: 59.0% accurate, 2.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 90.2%

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

   1. associate-/r* 89.8%

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

   2. +-commutative 89.8%

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

   3. fma-def 89.8%

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

3. Simplified 89.8%

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

4. Taylor expanded in z around 0 55.4%

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

5. Final simplification 55.4%

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

Alternative 15: 59.0% accurate, 2.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 90.2%

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

   1. associate-/r* 89.8%

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

   2. +-commutative 89.8%

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

   3. fma-def 89.8%

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

3. Simplified 89.8%

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

   1. fma-udef 89.8%

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

   2. +-commutative 89.8%

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

   3. associate-/r* 90.2%

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

   4. associate-/r* 90.1%

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

   5. div-inv 90.0%

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

   6. add-sqr-sqrt 90.0%

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

   7. times-frac 92.0%

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

   8. hypot-1-def 92.0%

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

   9. hypot-1-def 97.7%

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

5. Applied egg-rr 97.7%

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

   1. associate-/l/ 97.6%

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

   2. un-div-inv 97.7%

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

   3. associate-/l/ 97.5%

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

7. Applied egg-rr 97.5%

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

8. Taylor expanded in z around 0 55.4%

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

   1. *-commutative 55.4%

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

   2. associate-/r* 55.6%

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

10. Simplified 55.6%

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

11. Final simplification 55.6%

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

Developer target: 92.5% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2023229 
(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)))))