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

Percentage Accurate: 90.1% → 97.5%

Time: 8.6s

Alternatives: 9

Speedup: 0.8×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 90.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 97.5% accurate, 0.1× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 4 \cdot 10^{-151}:\\ \;\;\;\;\frac{\frac{1}{x}}{\mathsf{fma}\left(y \cdot z, z, y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(1, z\right)} \cdot \frac{\frac{\frac{1}{y}}{x}}{\mathsf{hypot}\left(1, z\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y 4e-151)
   (/ (/ 1.0 x) (fma (* y z) z y))
   (* (/ 1.0 (hypot 1.0 z)) (/ (/ (/ 1.0 y) x) (hypot 1.0 z)))))

C

assert(x < y);
double code(double x, double y, double z) {
	double tmp;
	if (y <= 4e-151) {
		tmp = (1.0 / x) / fma((y * z), z, y);
	} else {
		tmp = (1.0 / hypot(1.0, z)) * (((1.0 / y) / x) / hypot(1.0, z));
	}
	return tmp;
}

Julia

x, y = sort([x, y])
function code(x, y, z)
	tmp = 0.0
	if (y <= 4e-151)
		tmp = Float64(Float64(1.0 / x) / fma(Float64(y * z), z, y));
	else
		tmp = Float64(Float64(1.0 / hypot(1.0, z)) * Float64(Float64(Float64(1.0 / y) / x) / hypot(1.0, z)));
	end
	return tmp
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_, z_] := If[LessEqual[y, 4e-151], N[(N[(1.0 / x), $MachinePrecision] / N[(N[(y * z), $MachinePrecision] * z + y), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[Sqrt[1.0 ^ 2 + z ^ 2], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(1.0 / y), $MachinePrecision] / x), $MachinePrecision] / N[Sqrt[1.0 ^ 2 + z ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 3.9999999999999998e-151

   1. Initial program 92.0%

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

      1. distribute-lft-in 92.0%

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

      2. *-rgt-identity 92.0%

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

      3. +-commutative 92.0%

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

      4. associate-*r* 95.6%

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

      5. fma-def 95.6%

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

   3. Applied egg-rr 95.6%

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

   if 3.9999999999999998e-151 < y

   1. Initial program 89.6%

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

      1. associate-/r* 89.4%

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

      2. *-commutative 89.4%

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

      3. sqr-neg 89.4%

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

      4. +-commutative 89.4%

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

      5. distribute-lft1-in 89.4%

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

      6. *-commutative 89.4%

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

      7. fma-def 89.4%

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

      8. sqr-neg 89.4%

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

   3. Simplified 89.4%

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

      1. associate-/r* 89.7%

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

      2. fma-udef 89.6%

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

      3. *-rgt-identity 89.6%

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

      4. distribute-lft-in 89.6%

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

      5. +-commutative 89.6%

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

      6. associate-/r* 91.7%

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

      7. associate-/r* 91.6%

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

      8. *-un-lft-identity 91.6%

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

      9. +-commutative 91.6%

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

      10. fma-udef 91.6%

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

      11. add-sqr-sqrt 91.5%

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

      12. times-frac 91.5%

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

      13. fma-udef 91.5%

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

      14. +-commutative 91.5%

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

      15. hypot-1-def 91.5%

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

      16. *-commutative 91.5%

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

      17. associate-/r* 91.7%

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

      18. fma-udef 91.7%

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

      19. +-commutative 91.7%

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

      20. hypot-1-def 98.5%

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

   5. Applied egg-rr 98.5%

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

4. Final simplification 96.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 4 \cdot 10^{-151}:\\ \;\;\;\;\frac{\frac{1}{x}}{\mathsf{fma}\left(y \cdot z, z, y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(1, z\right)} \cdot \frac{\frac{\frac{1}{y}}{x}}{\mathsf{hypot}\left(1, z\right)}\\ \end{array} \]

Alternative 2: 97.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{+43}:\\ \;\;\;\;\frac{1}{x \cdot \left(y + y \cdot \left(z \cdot z\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{z}}{x \cdot \left(y \cdot z\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= (* z z) 2e+43)
   (/ 1.0 (* x (+ y (* y (* z z)))))
   (/ (/ 1.0 z) (* x (* y z)))))

C

assert(x < y);
double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 2e+43) {
		tmp = 1.0 / (x * (y + (y * (z * z))));
	} else {
		tmp = (1.0 / z) / (x * (y * z));
	}
	return tmp;
}

Fortran

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+43) then
        tmp = 1.0d0 / (x * (y + (y * (z * z))))
    else
        tmp = (1.0d0 / z) / (x * (y * z))
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 2e+43) {
		tmp = 1.0 / (x * (y + (y * (z * z))));
	} else {
		tmp = (1.0 / z) / (x * (y * z));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z):
	tmp = 0
	if (z * z) <= 2e+43:
		tmp = 1.0 / (x * (y + (y * (z * z))))
	else:
		tmp = (1.0 / z) / (x * (y * z))
	return tmp

Julia

x, y = sort([x, y])
function code(x, y, z)
	tmp = 0.0
	if (Float64(z * z) <= 2e+43)
		tmp = Float64(1.0 / Float64(x * Float64(y + Float64(y * Float64(z * z)))));
	else
		tmp = Float64(Float64(1.0 / z) / Float64(x * Float64(y * z)));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z * z) <= 2e+43)
		tmp = 1.0 / (x * (y + (y * (z * z))));
	else
		tmp = (1.0 / z) / (x * (y * z));
	end
	tmp_2 = tmp;
end

Wolfram

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+43], N[(1.0 / N[(x * N[(y + N[(y * N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / z), $MachinePrecision] / N[(x * N[(y * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 2.00000000000000003e43

   1. Initial program 99.4%

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

      1. associate-/r* 99.4%

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

      2. *-commutative 99.4%

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

      3. sqr-neg 99.4%

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

      4. +-commutative 99.4%

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

      5. distribute-lft1-in 99.5%

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

      6. *-commutative 99.5%

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

      7. fma-def 99.5%

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

      8. sqr-neg 99.5%

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

   3. Simplified 99.5%

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

      1. fma-udef 99.5%

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

   5. Applied egg-rr 99.5%

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

   if 2.00000000000000003e43 < (*.f64 z z)

   1. Initial program 81.6%

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

      1. associate-/r* 81.7%

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

      2. *-commutative 81.7%

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

      3. sqr-neg 81.7%

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

      4. +-commutative 81.7%

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

      5. distribute-lft1-in 81.7%

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

      6. *-commutative 81.7%

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

      7. fma-def 81.7%

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

      8. sqr-neg 81.7%

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

   3. Simplified 81.7%

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

   4. Taylor expanded in z around inf 81.7%

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

      1. unpow2 81.7%

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

      2. associate-*r* 81.5%

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

      3. *-commutative 81.5%

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

   6. Simplified 81.5%

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

      1. frac-2neg 81.5%

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

      2. metadata-eval 81.5%

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

      3. div-inv 81.5%

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

      4. associate-*r* 89.5%

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

      5. distribute-rgt-neg-in 89.5%

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

      6. *-commutative 89.5%

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

      7. associate-*l* 95.7%

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

   8. Applied egg-rr 95.7%

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

      1. associate-*r/ 95.7%

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

      2. metadata-eval 95.7%

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

      3. *-commutative 95.7%

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

      4. associate-/r* 97.6%

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

      5. neg-mul-1 97.6%

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

      6. associate-/r* 97.6%

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

      7. metadata-eval 97.6%

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

   10. Simplified 97.6%

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

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{+43}:\\ \;\;\;\;\frac{1}{x \cdot \left(y + y \cdot \left(z \cdot z\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{z}}{x \cdot \left(y \cdot z\right)}\\ \end{array} \]

Alternative 3: 98.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{+43}:\\ \;\;\;\;\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{z}}{x \cdot \left(y \cdot z\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= (* z z) 2e+43)
   (/ (/ 1.0 x) (* y (+ 1.0 (* z z))))
   (/ (/ 1.0 z) (* x (* y z)))))

C

assert(x < y);
double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 2e+43) {
		tmp = (1.0 / x) / (y * (1.0 + (z * z)));
	} else {
		tmp = (1.0 / z) / (x * (y * z));
	}
	return tmp;
}

Fortran

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+43) then
        tmp = (1.0d0 / x) / (y * (1.0d0 + (z * z)))
    else
        tmp = (1.0d0 / z) / (x * (y * z))
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 2e+43) {
		tmp = (1.0 / x) / (y * (1.0 + (z * z)));
	} else {
		tmp = (1.0 / z) / (x * (y * z));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z):
	tmp = 0
	if (z * z) <= 2e+43:
		tmp = (1.0 / x) / (y * (1.0 + (z * z)))
	else:
		tmp = (1.0 / z) / (x * (y * z))
	return tmp

Julia

x, y = sort([x, y])
function code(x, y, z)
	tmp = 0.0
	if (Float64(z * z) <= 2e+43)
		tmp = Float64(Float64(1.0 / x) / Float64(y * Float64(1.0 + Float64(z * z))));
	else
		tmp = Float64(Float64(1.0 / z) / Float64(x * Float64(y * z)));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z * z) <= 2e+43)
		tmp = (1.0 / x) / (y * (1.0 + (z * z)));
	else
		tmp = (1.0 / z) / (x * (y * z));
	end
	tmp_2 = tmp;
end

Wolfram

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+43], N[(N[(1.0 / x), $MachinePrecision] / N[(y * N[(1.0 + N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / z), $MachinePrecision] / N[(x * N[(y * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 2.00000000000000003e43

   1. Initial program 99.4%

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

   if 2.00000000000000003e43 < (*.f64 z z)

   1. Initial program 81.6%

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

      1. associate-/r* 81.7%

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

      2. *-commutative 81.7%

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

      3. sqr-neg 81.7%

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

      4. +-commutative 81.7%

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

      5. distribute-lft1-in 81.7%

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

      6. *-commutative 81.7%

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

      7. fma-def 81.7%

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

      8. sqr-neg 81.7%

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

   3. Simplified 81.7%

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

   4. Taylor expanded in z around inf 81.7%

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

      1. unpow2 81.7%

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

      2. associate-*r* 81.5%

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

      3. *-commutative 81.5%

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

   6. Simplified 81.5%

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

      1. frac-2neg 81.5%

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

      2. metadata-eval 81.5%

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

      3. div-inv 81.5%

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

      4. associate-*r* 89.5%

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

      5. distribute-rgt-neg-in 89.5%

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

      6. *-commutative 89.5%

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

      7. associate-*l* 95.7%

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

   8. Applied egg-rr 95.7%

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

      1. associate-*r/ 95.7%

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

      2. metadata-eval 95.7%

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

      3. *-commutative 95.7%

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

      4. associate-/r* 97.6%

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

      5. neg-mul-1 97.6%

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

      6. associate-/r* 97.6%

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

      7. metadata-eval 97.6%

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

   10. Simplified 97.6%

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

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{+43}:\\ \;\;\;\;\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{z}}{x \cdot \left(y \cdot z\right)}\\ \end{array} \]

Alternative 4: 98.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{+43}:\\ \;\;\;\;\frac{\frac{1}{x}}{y + y \cdot \left(z \cdot z\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{z}}{x \cdot \left(y \cdot z\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= (* z z) 2e+43)
   (/ (/ 1.0 x) (+ y (* y (* z z))))
   (/ (/ 1.0 z) (* x (* y z)))))

C

assert(x < y);
double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 2e+43) {
		tmp = (1.0 / x) / (y + (y * (z * z)));
	} else {
		tmp = (1.0 / z) / (x * (y * z));
	}
	return tmp;
}

Fortran

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+43) then
        tmp = (1.0d0 / x) / (y + (y * (z * z)))
    else
        tmp = (1.0d0 / z) / (x * (y * z))
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 2e+43) {
		tmp = (1.0 / x) / (y + (y * (z * z)));
	} else {
		tmp = (1.0 / z) / (x * (y * z));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z):
	tmp = 0
	if (z * z) <= 2e+43:
		tmp = (1.0 / x) / (y + (y * (z * z)))
	else:
		tmp = (1.0 / z) / (x * (y * z))
	return tmp

Julia

x, y = sort([x, y])
function code(x, y, z)
	tmp = 0.0
	if (Float64(z * z) <= 2e+43)
		tmp = Float64(Float64(1.0 / x) / Float64(y + Float64(y * Float64(z * z))));
	else
		tmp = Float64(Float64(1.0 / z) / Float64(x * Float64(y * z)));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z * z) <= 2e+43)
		tmp = (1.0 / x) / (y + (y * (z * z)));
	else
		tmp = (1.0 / z) / (x * (y * z));
	end
	tmp_2 = tmp;
end

Wolfram

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+43], N[(N[(1.0 / x), $MachinePrecision] / N[(y + N[(y * N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / z), $MachinePrecision] / N[(x * N[(y * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 2.00000000000000003e43

   1. Initial program 99.4%

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

      1. distribute-lft-in 99.5%

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

      2. *-rgt-identity 99.5%

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

      3. +-commutative 99.5%

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

      4. associate-*r* 99.5%

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

      5. fma-def 99.5%

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

   3. Applied egg-rr 99.5%

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

      1. fma-udef 99.5%

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

      2. associate-*l* 99.5%

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

   5. Applied egg-rr 99.5%

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

   if 2.00000000000000003e43 < (*.f64 z z)

   1. Initial program 81.6%

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

      1. associate-/r* 81.7%

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

      2. *-commutative 81.7%

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

      3. sqr-neg 81.7%

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

      4. +-commutative 81.7%

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

      5. distribute-lft1-in 81.7%

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

      6. *-commutative 81.7%

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

      7. fma-def 81.7%

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

      8. sqr-neg 81.7%

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

   3. Simplified 81.7%

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

   4. Taylor expanded in z around inf 81.7%

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

      1. unpow2 81.7%

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

      2. associate-*r* 81.5%

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

      3. *-commutative 81.5%

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

   6. Simplified 81.5%

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

      1. frac-2neg 81.5%

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

      2. metadata-eval 81.5%

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

      3. div-inv 81.5%

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

      4. associate-*r* 89.5%

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

      5. distribute-rgt-neg-in 89.5%

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

      6. *-commutative 89.5%

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

      7. associate-*l* 95.7%

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

   8. Applied egg-rr 95.7%

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

      1. associate-*r/ 95.7%

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

      2. metadata-eval 95.7%

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

      3. *-commutative 95.7%

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

      4. associate-/r* 97.6%

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

      5. neg-mul-1 97.6%

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

      6. associate-/r* 97.6%

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

      7. metadata-eval 97.6%

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

   10. Simplified 97.6%

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

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{+43}:\\ \;\;\;\;\frac{\frac{1}{x}}{y + y \cdot \left(z \cdot z\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{z}}{x \cdot \left(y \cdot z\right)}\\ \end{array} \]

Alternative 5: 89.2% accurate, 0.8× speedup

Math

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

FPCore

NOTE: 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

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: 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

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

[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

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

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: 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 99.5%

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

      1. distribute-lft-in 99.5%

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

      2. *-rgt-identity 99.5%

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

      3. +-commutative 99.5%

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

      4. associate-*r* 99.5%

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

      5. fma-def 99.5%

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

   3. Applied egg-rr 99.5%

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

   4. Taylor expanded in z around 0 97.6%

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

   if 1 < (*.f64 z z)

   1. Initial program 83.3%

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

      1. associate-/r* 83.4%

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

      2. *-commutative 83.4%

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

      3. sqr-neg 83.4%

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

      4. +-commutative 83.4%

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

      5. distribute-lft1-in 83.4%

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

      6. *-commutative 83.4%

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

      7. fma-def 83.4%

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

      8. sqr-neg 83.4%

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

   3. Simplified 83.4%

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

   4. Taylor expanded in z around inf 81.3%

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

      1. unpow2 81.3%

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

   6. Simplified 81.3%

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

4. Final simplification 89.1%

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

Alternative 6: 89.6% accurate, 0.8× speedup

Math

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

FPCore

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 (* y (* x (* z z))))))

C

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 / (y * (x * (z * z)));
	}
	return tmp;
}

Fortran

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 / (y * (x * (z * z)))
    end if
    code = tmp
end function

Java

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 / (y * (x * (z * z)));
	}
	return tmp;
}

Python

[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 / (y * (x * (z * z)))
	return tmp

Julia

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(y * Float64(x * Float64(z * z))));
	end
	return tmp
end

MATLAB

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 / (y * (x * (z * z)));
	end
	tmp_2 = tmp;
end

Wolfram

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[(y * N[(x * N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 99.5%

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

      1. distribute-lft-in 99.5%

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

      2. *-rgt-identity 99.5%

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

      3. +-commutative 99.5%

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

      4. associate-*r* 99.5%

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

      5. fma-def 99.5%

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

   3. Applied egg-rr 99.5%

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

   4. Taylor expanded in z around 0 97.6%

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

   if 1 < (*.f64 z z)

   1. Initial program 83.3%

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

      1. associate-/r* 83.4%

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

      2. *-commutative 83.4%

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

      3. sqr-neg 83.4%

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

      4. +-commutative 83.4%

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

      5. distribute-lft1-in 83.4%

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

      6. *-commutative 83.4%

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

      7. fma-def 83.4%

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

      8. sqr-neg 83.4%

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

   3. Simplified 83.4%

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

   4. Taylor expanded in z around inf 81.3%

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

      1. unpow2 81.3%

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

      2. associate-*r* 80.5%

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

      3. *-commutative 80.5%

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

      4. associate-*l* 79.1%

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

   6. Simplified 79.1%

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

4. Final simplification 88.0%

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

Alternative 7: 89.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 0.75:\\ \;\;\;\;\frac{1 - z \cdot z}{y \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y \cdot \left(x \cdot \left(z \cdot z\right)\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= (* z z) 0.75)
   (/ (- 1.0 (* z z)) (* y x))
   (/ 1.0 (* y (* x (* z z))))))

C

assert(x < y);
double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 0.75) {
		tmp = (1.0 - (z * z)) / (y * x);
	} else {
		tmp = 1.0 / (y * (x * (z * z)));
	}
	return tmp;
}

Fortran

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) <= 0.75d0) then
        tmp = (1.0d0 - (z * z)) / (y * x)
    else
        tmp = 1.0d0 / (y * (x * (z * z)))
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 0.75) {
		tmp = (1.0 - (z * z)) / (y * x);
	} else {
		tmp = 1.0 / (y * (x * (z * z)));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z):
	tmp = 0
	if (z * z) <= 0.75:
		tmp = (1.0 - (z * z)) / (y * x)
	else:
		tmp = 1.0 / (y * (x * (z * z)))
	return tmp

Julia

x, y = sort([x, y])
function code(x, y, z)
	tmp = 0.0
	if (Float64(z * z) <= 0.75)
		tmp = Float64(Float64(1.0 - Float64(z * z)) / Float64(y * x));
	else
		tmp = Float64(1.0 / Float64(y * Float64(x * Float64(z * z))));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z * z) <= 0.75)
		tmp = (1.0 - (z * z)) / (y * x);
	else
		tmp = 1.0 / (y * (x * (z * z)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_, z_] := If[LessEqual[N[(z * z), $MachinePrecision], 0.75], N[(N[(1.0 - N[(z * z), $MachinePrecision]), $MachinePrecision] / N[(y * x), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(y * N[(x * N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 0.75

   1. Initial program 99.5%

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

      1. associate-/r* 99.5%

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

      2. *-commutative 99.5%

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

      3. sqr-neg 99.5%

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

      4. +-commutative 99.5%

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

      5. distribute-lft1-in 99.5%

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

      6. *-commutative 99.5%

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

      7. fma-def 99.5%

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

      8. sqr-neg 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in z around 0 83.6%

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

      1. associate-/l/ 83.8%

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

      2. +-commutative 83.8%

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

      3. mul-1-neg 83.8%

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

      4. unsub-neg 83.8%

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

      5. associate-/r* 83.6%

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

      6. unpow2 83.6%

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

      7. *-commutative 83.6%

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

      8. div-sub 98.3%

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

   6. Simplified 98.3%

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

   if 0.75 < (*.f64 z z)

   1. Initial program 83.3%

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

      1. associate-/r* 83.4%

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

      2. *-commutative 83.4%

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

      3. sqr-neg 83.4%

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

      4. +-commutative 83.4%

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

      5. distribute-lft1-in 83.4%

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

      6. *-commutative 83.4%

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

      7. fma-def 83.4%

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

      8. sqr-neg 83.4%

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

   3. Simplified 83.4%

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

   4. Taylor expanded in z around inf 81.3%

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

      1. unpow2 81.3%

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

      2. associate-*r* 80.5%

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

      3. *-commutative 80.5%

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

      4. associate-*l* 79.1%

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

   6. Simplified 79.1%

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

4. Final simplification 88.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 0.75:\\ \;\;\;\;\frac{1 - z \cdot z}{y \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y \cdot \left(x \cdot \left(z \cdot z\right)\right)}\\ \end{array} \]

Alternative 8: 97.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 0.2:\\ \;\;\;\;\frac{1 - z \cdot z}{y \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{z}}{x \cdot \left(y \cdot z\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= (* z z) 0.2) (/ (- 1.0 (* z z)) (* y x)) (/ (/ 1.0 z) (* x (* y z)))))

C

assert(x < y);
double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 0.2) {
		tmp = (1.0 - (z * z)) / (y * x);
	} else {
		tmp = (1.0 / z) / (x * (y * z));
	}
	return tmp;
}

Fortran

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) <= 0.2d0) then
        tmp = (1.0d0 - (z * z)) / (y * x)
    else
        tmp = (1.0d0 / z) / (x * (y * z))
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 0.2) {
		tmp = (1.0 - (z * z)) / (y * x);
	} else {
		tmp = (1.0 / z) / (x * (y * z));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z):
	tmp = 0
	if (z * z) <= 0.2:
		tmp = (1.0 - (z * z)) / (y * x)
	else:
		tmp = (1.0 / z) / (x * (y * z))
	return tmp

Julia

x, y = sort([x, y])
function code(x, y, z)
	tmp = 0.0
	if (Float64(z * z) <= 0.2)
		tmp = Float64(Float64(1.0 - Float64(z * z)) / Float64(y * x));
	else
		tmp = Float64(Float64(1.0 / z) / Float64(x * Float64(y * z)));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z * z) <= 0.2)
		tmp = (1.0 - (z * z)) / (y * x);
	else
		tmp = (1.0 / z) / (x * (y * z));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_, z_] := If[LessEqual[N[(z * z), $MachinePrecision], 0.2], N[(N[(1.0 - N[(z * z), $MachinePrecision]), $MachinePrecision] / N[(y * x), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / z), $MachinePrecision] / N[(x * N[(y * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 0.20000000000000001

   1. Initial program 99.5%

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

      1. associate-/r* 99.5%

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

      2. *-commutative 99.5%

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

      3. sqr-neg 99.5%

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

      4. +-commutative 99.5%

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

      5. distribute-lft1-in 99.5%

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

      6. *-commutative 99.5%

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

      7. fma-def 99.5%

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

      8. sqr-neg 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in z around 0 83.6%

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

      1. associate-/l/ 83.8%

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

      2. +-commutative 83.8%

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

      3. mul-1-neg 83.8%

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

      4. unsub-neg 83.8%

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

      5. associate-/r* 83.6%

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

      6. unpow2 83.6%

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

      7. *-commutative 83.6%

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

      8. div-sub 98.3%

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

   6. Simplified 98.3%

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

   if 0.20000000000000001 < (*.f64 z z)

   1. Initial program 83.3%

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

      1. associate-/r* 83.4%

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

      2. *-commutative 83.4%

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

      3. sqr-neg 83.4%

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

      4. +-commutative 83.4%

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

      5. distribute-lft1-in 83.4%

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

      6. *-commutative 83.4%

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

      7. fma-def 83.4%

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

      8. sqr-neg 83.4%

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

   3. Simplified 83.4%

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

   4. Taylor expanded in z around inf 81.3%

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

      1. unpow2 81.3%

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

      2. associate-*r* 80.5%

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

      3. *-commutative 80.5%

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

   6. Simplified 80.5%

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

      1. frac-2neg 80.5%

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

      2. metadata-eval 80.5%

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

      3. div-inv 80.5%

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

      4. associate-*r* 87.7%

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

      5. distribute-rgt-neg-in 87.7%

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

      6. *-commutative 87.7%

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

      7. associate-*l* 93.9%

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

   8. Applied egg-rr 93.9%

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

      1. associate-*r/ 93.9%

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

      2. metadata-eval 93.9%

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

      3. *-commutative 93.9%

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

      4. associate-/r* 95.6%

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

      5. neg-mul-1 95.6%

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

      6. associate-/r* 95.6%

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

      7. metadata-eval 95.6%

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

   10. Simplified 95.6%

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

4. Final simplification 96.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 0.2:\\ \;\;\;\;\frac{1 - z \cdot z}{y \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{z}}{x \cdot \left(y \cdot z\right)}\\ \end{array} \]

Alternative 9: 58.6% accurate, 2.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 91.1%

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

   1. associate-/r* 91.1%

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

   2. *-commutative 91.1%

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

   3. sqr-neg 91.1%

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

   4. +-commutative 91.1%

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

   5. distribute-lft1-in 91.1%

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

   6. *-commutative 91.1%

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

   7. fma-def 91.1%

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

   8. sqr-neg 91.1%

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

3. Simplified 91.1%

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

4. Taylor expanded in z around 0 56.2%

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

   1. *-commutative 56.2%

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

6. Simplified 56.2%

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

7. Final simplification 56.2%

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

Developer target: 92.2% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2023271 
(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)))))