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

Percentage Accurate: 90.7% → 98.9%

Time: 8.7s

Alternatives: 10

Speedup: 0.8×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 90.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.9% accurate, 0.1× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \cdot \left(1 + z \cdot z\right) \leq 5 \cdot 10^{+304}:\\ \;\;\;\;\frac{\frac{1}{\mathsf{fma}\left(z, y \cdot z, y\right)}}{x}\\ \mathbf{else}:\\ \;\;\;\;{z}^{-1} \cdot \frac{1}{y \cdot \left(z \cdot x\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= (* y (+ 1.0 (* z z))) 5e+304)
   (/ (/ 1.0 (fma z (* y z) y)) x)
   (* (pow z -1.0) (/ 1.0 (* y (* z x))))))

C

assert(x < y);
double code(double x, double y, double z) {
	double tmp;
	if ((y * (1.0 + (z * z))) <= 5e+304) {
		tmp = (1.0 / fma(z, (y * z), y)) / x;
	} else {
		tmp = pow(z, -1.0) * (1.0 / (y * (z * x)));
	}
	return tmp;
}

Julia

x, y = sort([x, y])
function code(x, y, z)
	tmp = 0.0
	if (Float64(y * Float64(1.0 + Float64(z * z))) <= 5e+304)
		tmp = Float64(Float64(1.0 / fma(z, Float64(y * z), y)) / x);
	else
		tmp = Float64((z ^ -1.0) * Float64(1.0 / Float64(y * Float64(z * x))));
	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[N[(y * N[(1.0 + N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 5e+304], N[(N[(1.0 / N[(z * N[(y * z), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], N[(N[Power[z, -1.0], $MachinePrecision] * N[(1.0 / N[(y * N[(z * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y (+.f64 1 (*.f64 z z))) < 4.9999999999999997e304

   1. Initial program 95.6%

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

      1. associate-/r* 95.4%

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

      2. +-commutative 95.4%

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

      3. fma-def 95.4%

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

   3. Simplified 95.4%

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

   4. Taylor expanded in x around 0 92.7%

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

      1. associate-*r* 95.4%

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

      2. *-commutative 95.4%

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

      3. associate-*r* 94.3%

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

      4. associate-/r* 94.5%

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

      5. unpow2 94.5%

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

      6. fma-udef 94.5%

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

      7. associate-/r* 95.6%

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

      8. associate-/l/ 95.6%

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

      9. fma-udef 95.6%

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

      10. distribute-lft-in 95.6%

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

      11. *-rgt-identity 95.6%

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

      12. fma-def 95.6%

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

      13. fma-def 95.6%

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

      14. *-commutative 95.6%

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

      15. associate-*r* 96.9%

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

      16. fma-udef 96.9%

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

   6. Simplified 96.9%

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

   if 4.9999999999999997e304 < (*.f64 y (+.f64 1 (*.f64 z z)))

   1. Initial program 64.0%

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

      1. associate-/r* 64.0%

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

      2. +-commutative 64.0%

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

      3. fma-def 64.0%

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

   3. Simplified 64.0%

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

   4. Taylor expanded in z around inf 68.0%

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

      1. *-commutative 68.0%

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

      2. unpow2 68.0%

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

      3. associate-*r* 67.7%

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

      4. associate-*l* 93.2%

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

      5. *-commutative 93.2%

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

      6. associate-*l* 95.8%

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

      7. *-commutative 95.8%

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

   6. Simplified 95.8%

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

      1. associate-/r* 95.8%

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

      2. div-inv 95.8%

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

      3. inv-pow 95.8%

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

      4. *-commutative 95.8%

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

      5. *-commutative 95.8%

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

      6. associate-*l* 99.7%

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

   8. Applied egg-rr 99.7%

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

4. Final simplification 97.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(1 + z \cdot z\right) \leq 5 \cdot 10^{+304}:\\ \;\;\;\;\frac{\frac{1}{\mathsf{fma}\left(z, y \cdot z, y\right)}}{x}\\ \mathbf{else}:\\ \;\;\;\;{z}^{-1} \cdot \frac{1}{y \cdot \left(z \cdot x\right)}\\ \end{array} \]

Alternative 2: 97.5% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 89.8%

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

   1. associate-/r* 89.6%

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

   2. +-commutative 89.6%

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

   3. fma-def 89.6%

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

3. Simplified 89.6%

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

   1. associate-/r* 89.8%

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

   2. *-un-lft-identity 89.8%

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

   3. fma-udef 89.8%

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

   4. +-commutative 89.8%

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

   5. add-sqr-sqrt 43.6%

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

   6. times-frac 43.5%

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

   7. *-commutative 43.5%

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

   8. sqrt-prod 43.5%

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

   9. hypot-1-def 43.5%

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

   10. *-commutative 43.5%

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

   11. sqrt-prod 44.3%

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

   12. hypot-1-def 50.1%

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

5. Applied egg-rr 50.1%

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

   1. associate-*l/ 50.1%

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

   2. *-lft-identity 50.1%

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

   3. *-commutative 50.1%

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

   4. associate-/r* 49.7%

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

   5. associate-/r* 49.4%

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

   6. associate-/l/ 49.4%

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

   7. rem-square-sqrt 98.5%

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

   8. associate-/r* 97.6%

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

   9. associate-/l/ 97.5%

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

   10. *-commutative 97.5%

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

7. Simplified 97.5%

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

8. Final simplification 97.5%

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

Alternative 3: 98.9% accurate, 0.1× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \cdot \left(1 + z \cdot z\right) \leq 5 \cdot 10^{+304}:\\ \;\;\;\;\frac{\frac{1}{\mathsf{fma}\left(z, y \cdot z, y\right)}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{y \cdot \left(z \cdot x\right)}}{z}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= (* y (+ 1.0 (* z z))) 5e+304)
   (/ (/ 1.0 (fma z (* y z) y)) x)
   (/ (/ 1.0 (* y (* z x))) z)))

C

assert(x < y);
double code(double x, double y, double z) {
	double tmp;
	if ((y * (1.0 + (z * z))) <= 5e+304) {
		tmp = (1.0 / fma(z, (y * z), y)) / x;
	} else {
		tmp = (1.0 / (y * (z * x))) / z;
	}
	return tmp;
}

Julia

x, y = sort([x, y])
function code(x, y, z)
	tmp = 0.0
	if (Float64(y * Float64(1.0 + Float64(z * z))) <= 5e+304)
		tmp = Float64(Float64(1.0 / fma(z, Float64(y * z), y)) / x);
	else
		tmp = Float64(Float64(1.0 / Float64(y * Float64(z * x))) / 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[N[(y * N[(1.0 + N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 5e+304], N[(N[(1.0 / N[(z * N[(y * z), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], N[(N[(1.0 / N[(y * N[(z * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y (+.f64 1 (*.f64 z z))) < 4.9999999999999997e304

   1. Initial program 95.6%

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

      1. associate-/r* 95.4%

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

      2. +-commutative 95.4%

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

      3. fma-def 95.4%

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

   3. Simplified 95.4%

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

   4. Taylor expanded in x around 0 92.7%

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

      1. associate-*r* 95.4%

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

      2. *-commutative 95.4%

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

      3. associate-*r* 94.3%

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

      4. associate-/r* 94.5%

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

      5. unpow2 94.5%

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

      6. fma-udef 94.5%

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

      7. associate-/r* 95.6%

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

      8. associate-/l/ 95.6%

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

      9. fma-udef 95.6%

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

      10. distribute-lft-in 95.6%

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

      11. *-rgt-identity 95.6%

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

      12. fma-def 95.6%

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

      13. fma-def 95.6%

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

      14. *-commutative 95.6%

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

      15. associate-*r* 96.9%

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

      16. fma-udef 96.9%

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

   6. Simplified 96.9%

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

   if 4.9999999999999997e304 < (*.f64 y (+.f64 1 (*.f64 z z)))

   1. Initial program 64.0%

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

      1. associate-/r* 64.0%

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

      2. +-commutative 64.0%

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

      3. fma-def 64.0%

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

   3. Simplified 64.0%

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

   4. Taylor expanded in z around inf 68.0%

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

      1. *-commutative 68.0%

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

      2. unpow2 68.0%

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

      3. associate-*r* 67.7%

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

      4. associate-*l* 93.2%

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

      5. *-commutative 93.2%

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

      6. associate-*l* 95.8%

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

      7. *-commutative 95.8%

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

   6. Simplified 95.8%

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

      1. expm1-log1p-u 95.8%

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

      2. expm1-udef 65.1%

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

      3. associate-*r* 65.1%

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

      4. *-commutative 65.1%

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

      5. associate-*r* 65.1%

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

   8. Applied egg-rr 65.1%

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

      1. expm1-def 99.7%

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

      2. expm1-log1p 99.7%

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

      3. associate-/r* 99.7%

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

      4. *-commutative 99.7%

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

      5. associate-/r* 99.7%

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

      6. unpow-1 99.7%

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

      7. *-commutative 99.7%

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

      8. associate-/r* 91.6%

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

      9. unpow-1 91.6%

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

      10. associate-/r* 91.5%

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

   10. Simplified 91.5%

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

   11. Taylor expanded in y around 0 99.7%

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

4. Final simplification 97.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(1 + z \cdot z\right) \leq 5 \cdot 10^{+304}:\\ \;\;\;\;\frac{\frac{1}{\mathsf{fma}\left(z, y \cdot z, y\right)}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{y \cdot \left(z \cdot x\right)}}{z}\\ \end{array} \]

Alternative 4: 97.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 4 \cdot 10^{+67}:\\ \;\;\;\;\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{y \cdot \left(z \cdot x\right)}}{z}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= (* z z) 4e+67)
   (/ (/ 1.0 x) (* y (+ 1.0 (* z z))))
   (/ (/ 1.0 (* y (* z x))) z)))

C

assert(x < y);
double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 4e+67) {
		tmp = (1.0 / x) / (y * (1.0 + (z * z)));
	} else {
		tmp = (1.0 / (y * (z * x))) / 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) <= 4d+67) then
        tmp = (1.0d0 / x) / (y * (1.0d0 + (z * z)))
    else
        tmp = (1.0d0 / (y * (z * x))) / 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) <= 4e+67) {
		tmp = (1.0 / x) / (y * (1.0 + (z * z)));
	} else {
		tmp = (1.0 / (y * (z * x))) / z;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z):
	tmp = 0
	if (z * z) <= 4e+67:
		tmp = (1.0 / x) / (y * (1.0 + (z * z)))
	else:
		tmp = (1.0 / (y * (z * x))) / z
	return tmp

Julia

x, y = sort([x, y])
function code(x, y, z)
	tmp = 0.0
	if (Float64(z * z) <= 4e+67)
		tmp = Float64(Float64(1.0 / x) / Float64(y * Float64(1.0 + Float64(z * z))));
	else
		tmp = Float64(Float64(1.0 / Float64(y * Float64(z * x))) / 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) <= 4e+67)
		tmp = (1.0 / x) / (y * (1.0 + (z * z)));
	else
		tmp = (1.0 / (y * (z * x))) / 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], 4e+67], N[(N[(1.0 / x), $MachinePrecision] / N[(y * N[(1.0 + N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[(y * N[(z * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 3.99999999999999993e67

   1. Initial program 99.7%

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

   if 3.99999999999999993e67 < (*.f64 z z)

   1. Initial program 79.1%

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

      1. associate-/r* 78.9%

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

      2. +-commutative 78.9%

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

      3. fma-def 78.9%

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

   3. Simplified 78.9%

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

   4. Taylor expanded in z around inf 75.9%

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

      1. *-commutative 75.9%

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

      2. unpow2 75.9%

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

      3. associate-*r* 78.3%

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

      4. associate-*l* 91.8%

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

      5. *-commutative 91.8%

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

      6. associate-*l* 94.9%

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

      7. *-commutative 94.9%

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

   6. Simplified 94.9%

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

      1. expm1-log1p-u 90.1%

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

      2. expm1-udef 58.9%

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

      3. associate-*r* 58.9%

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

      4. *-commutative 58.9%

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

      5. associate-*r* 58.9%

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

   8. Applied egg-rr 58.9%

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

      1. expm1-def 90.9%

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

      2. expm1-log1p 95.7%

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

      3. associate-/r* 96.7%

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

      4. *-commutative 96.7%

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

      5. associate-/r* 96.7%

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

      6. unpow-1 96.7%

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

      7. *-commutative 96.7%

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

      8. associate-/r* 91.2%

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

      9. unpow-1 91.2%

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

      10. associate-/r* 91.2%

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

   10. Simplified 91.2%

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

   11. Taylor expanded in y around 0 96.7%

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

4. Final simplification 98.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 4 \cdot 10^{+67}:\\ \;\;\;\;\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{y \cdot \left(z \cdot x\right)}}{z}\\ \end{array} \]

Alternative 5: 97.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 5 \cdot 10^{+251}:\\ \;\;\;\;\frac{\frac{\frac{1}{x}}{y}}{1 + z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{1}{x}}{y \cdot z}}{z}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= (* z z) 5e+251)
   (/ (/ (/ 1.0 x) y) (+ 1.0 (* z z)))
   (/ (/ (/ 1.0 x) (* y z)) z)))

C

assert(x < y);
double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 5e+251) {
		tmp = ((1.0 / x) / y) / (1.0 + (z * z));
	} 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) <= 5d+251) then
        tmp = ((1.0d0 / x) / y) / (1.0d0 + (z * z))
    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) <= 5e+251) {
		tmp = ((1.0 / x) / y) / (1.0 + (z * z));
	} 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) <= 5e+251:
		tmp = ((1.0 / x) / y) / (1.0 + (z * z))
	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) <= 5e+251)
		tmp = Float64(Float64(Float64(1.0 / x) / y) / Float64(1.0 + Float64(z * z)));
	else
		tmp = Float64(Float64(Float64(1.0 / x) / Float64(y * 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) <= 5e+251)
		tmp = ((1.0 / x) / y) / (1.0 + (z * z));
	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], 5e+251], N[(N[(N[(1.0 / x), $MachinePrecision] / y), $MachinePrecision] / N[(1.0 + N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 / x), $MachinePrecision] / N[(y * z), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 5.0000000000000005e251

   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* 98.6%

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

   3. Simplified 98.6%

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

   if 5.0000000000000005e251 < (*.f64 z z)

   1. Initial program 71.5%

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

      1. associate-/r* 71.6%

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

      2. +-commutative 71.6%

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

      3. fma-def 71.6%

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

   3. Simplified 71.6%

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

   4. Taylor expanded in z around inf 70.5%

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

      1. *-commutative 70.5%

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

      2. unpow2 70.5%

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

      3. associate-*r* 70.5%

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

      4. associate-*l* 90.6%

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

      5. *-commutative 90.6%

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

      6. associate-*l* 95.2%

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

      7. *-commutative 95.2%

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

   6. Simplified 95.2%

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

      1. expm1-log1p-u 92.5%

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

      2. expm1-udef 67.3%

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

      3. associate-*r* 67.3%

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

      4. *-commutative 67.3%

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

      5. associate-*r* 67.3%

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

   8. Applied egg-rr 67.3%

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

      1. expm1-def 95.9%

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

      2. expm1-log1p 98.5%

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

      3. associate-/r* 99.7%

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

      4. *-commutative 99.7%

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

      5. associate-/r* 99.7%

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

      6. unpow-1 99.7%

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

      7. *-commutative 99.7%

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

      8. associate-/r* 89.3%

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

      9. unpow-1 89.3%

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

      10. associate-/r* 89.3%

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

   10. Simplified 89.3%

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

   11. Taylor expanded in y around 0 99.7%

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

       1. associate-*r* 95.6%

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

       2. *-commutative 95.6%

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

       3. associate-/l/ 95.6%

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

       4. *-commutative 95.6%

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

   13. Simplified 95.6%

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

4. Final simplification 97.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 5 \cdot 10^{+251}:\\ \;\;\;\;\frac{\frac{\frac{1}{x}}{y}}{1 + z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{1}{x}}{y \cdot z}}{z}\\ \end{array} \]

Alternative 6: 89.8% 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 99.7%

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

      1. associate-/r* 99.6%

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

      2. +-commutative 99.6%

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

      3. fma-def 99.6%

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

   3. Simplified 99.6%

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

      1. fma-udef 99.6%

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

      2. +-commutative 99.6%

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

      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. add-sqr-sqrt 99.6%

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

      6. div-inv 99.5%

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

      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 99.5%

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

      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. Taylor expanded in z around 0 98.8%

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

      1. *-commutative 98.8%

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

      2. associate-/r* 98.9%

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

   8. Simplified 98.9%

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

   if 1 < (*.f64 z z)

   1. Initial program 81.4%

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

      1. associate-/r* 81.3%

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

      2. +-commutative 81.3%

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

      3. fma-def 81.3%

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

   3. Simplified 81.3%

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

   4. Taylor expanded in z around inf 79.8%

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

      1. unpow2 79.8%

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

   6. Simplified 79.8%

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

4. Final simplification 88.5%

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

Alternative 7: 93.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 0.001:\\ \;\;\;\;\frac{\frac{1}{x}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x \cdot \left(z \cdot \left(y \cdot z\right)\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= (* z z) 0.001) (/ (/ 1.0 x) y) (/ 1.0 (* x (* z (* y z))))))

C

assert(x < y);
double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 0.001) {
		tmp = (1.0 / x) / y;
	} else {
		tmp = 1.0 / (x * (z * (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) <= 0.001d0) then
        tmp = (1.0d0 / x) / y
    else
        tmp = 1.0d0 / (x * (z * (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) <= 0.001) {
		tmp = (1.0 / x) / y;
	} else {
		tmp = 1.0 / (x * (z * (y * z)));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z):
	tmp = 0
	if (z * z) <= 0.001:
		tmp = (1.0 / x) / y
	else:
		tmp = 1.0 / (x * (z * (y * z)))
	return tmp

Julia

x, y = sort([x, y])
function code(x, y, z)
	tmp = 0.0
	if (Float64(z * z) <= 0.001)
		tmp = Float64(Float64(1.0 / x) / y);
	else
		tmp = Float64(1.0 / Float64(x * Float64(z * 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) <= 0.001)
		tmp = (1.0 / x) / y;
	else
		tmp = 1.0 / (x * (z * (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], 0.001], N[(N[(1.0 / x), $MachinePrecision] / y), $MachinePrecision], N[(1.0 / N[(x * N[(z * N[(y * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 1e-3

   1. Initial program 99.7%

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

      1. associate-/r* 99.6%

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

      2. +-commutative 99.6%

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

      3. fma-def 99.6%

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

   3. Simplified 99.6%

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

      1. fma-udef 99.6%

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

      2. +-commutative 99.6%

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

      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. add-sqr-sqrt 99.6%

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

      6. div-inv 99.5%

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

      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 99.5%

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

      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. Taylor expanded in z around 0 98.8%

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

      1. *-commutative 98.8%

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

      2. associate-/r* 98.9%

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

   8. Simplified 98.9%

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

   if 1e-3 < (*.f64 z z)

   1. Initial program 81.4%

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

      1. associate-/r* 81.3%

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

      2. +-commutative 81.3%

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

      3. fma-def 81.3%

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

   3. Simplified 81.3%

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

   4. Taylor expanded in z around inf 79.8%

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

      1. unpow2 79.8%

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

      2. *-commutative 79.8%

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

      3. associate-*r* 86.5%

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

   6. Simplified 86.5%

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

4. Final simplification 92.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 0.001:\\ \;\;\;\;\frac{\frac{1}{x}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x \cdot \left(z \cdot \left(y \cdot z\right)\right)}\\ \end{array} \]

Alternative 8: 96.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 0.001:\\ \;\;\;\;\frac{\frac{1}{x}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{z \cdot \left(x \cdot \left(y \cdot z\right)\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= (* z z) 0.001) (/ (/ 1.0 x) y) (/ 1.0 (* z (* x (* y z))))))

C

assert(x < y);
double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 0.001) {
		tmp = (1.0 / x) / y;
	} 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) <= 0.001d0) then
        tmp = (1.0d0 / x) / y
    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) <= 0.001) {
		tmp = (1.0 / x) / y;
	} 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) <= 0.001:
		tmp = (1.0 / x) / y
	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) <= 0.001)
		tmp = Float64(Float64(1.0 / x) / y);
	else
		tmp = Float64(1.0 / Float64(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) <= 0.001)
		tmp = (1.0 / x) / y;
	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], 0.001], N[(N[(1.0 / x), $MachinePrecision] / y), $MachinePrecision], N[(1.0 / N[(z * N[(x * N[(y * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 1e-3

   1. Initial program 99.7%

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

      1. associate-/r* 99.6%

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

      2. +-commutative 99.6%

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

      3. fma-def 99.6%

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

   3. Simplified 99.6%

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

      1. fma-udef 99.6%

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

      2. +-commutative 99.6%

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

      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. add-sqr-sqrt 99.6%

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

      6. div-inv 99.5%

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

      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 99.5%

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

      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. Taylor expanded in z around 0 98.8%

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

      1. *-commutative 98.8%

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

      2. associate-/r* 98.9%

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

   8. Simplified 98.9%

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

   if 1e-3 < (*.f64 z z)

   1. Initial program 81.4%

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

      1. associate-/r* 81.3%

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

      2. +-commutative 81.3%

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

      3. fma-def 81.3%

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

   3. Simplified 81.3%

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

   4. Taylor expanded in z around inf 77.1%

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

      1. *-commutative 77.1%

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

      2. unpow2 77.1%

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

      3. associate-*r* 79.3%

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

      4. associate-*l* 91.2%

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

      5. *-commutative 91.2%

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

      6. associate-*l* 93.9%

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

      7. *-commutative 93.9%

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

   6. Simplified 93.9%

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

4. Final simplification 96.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 0.001:\\ \;\;\;\;\frac{\frac{1}{x}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{z \cdot \left(x \cdot \left(y \cdot z\right)\right)}\\ \end{array} \]

Alternative 9: 96.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 0.001:\\ \;\;\;\;\frac{\frac{1}{x}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{y \cdot \left(z \cdot x\right)}}{z}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= (* z z) 0.001) (/ (/ 1.0 x) y) (/ (/ 1.0 (* y (* z x))) z)))

C

assert(x < y);
double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 0.001) {
		tmp = (1.0 / x) / y;
	} else {
		tmp = (1.0 / (y * (z * x))) / 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) <= 0.001d0) then
        tmp = (1.0d0 / x) / y
    else
        tmp = (1.0d0 / (y * (z * x))) / 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) <= 0.001) {
		tmp = (1.0 / x) / y;
	} else {
		tmp = (1.0 / (y * (z * x))) / z;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z):
	tmp = 0
	if (z * z) <= 0.001:
		tmp = (1.0 / x) / y
	else:
		tmp = (1.0 / (y * (z * x))) / z
	return tmp

Julia

x, y = sort([x, y])
function code(x, y, z)
	tmp = 0.0
	if (Float64(z * z) <= 0.001)
		tmp = Float64(Float64(1.0 / x) / y);
	else
		tmp = Float64(Float64(1.0 / Float64(y * Float64(z * x))) / 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) <= 0.001)
		tmp = (1.0 / x) / y;
	else
		tmp = (1.0 / (y * (z * x))) / 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], 0.001], N[(N[(1.0 / x), $MachinePrecision] / y), $MachinePrecision], N[(N[(1.0 / N[(y * N[(z * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 1e-3

   1. Initial program 99.7%

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

      1. associate-/r* 99.6%

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

      2. +-commutative 99.6%

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

      3. fma-def 99.6%

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

   3. Simplified 99.6%

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

      1. fma-udef 99.6%

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

      2. +-commutative 99.6%

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

      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. add-sqr-sqrt 99.6%

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

      6. div-inv 99.5%

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

      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 99.5%

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

      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. Taylor expanded in z around 0 98.8%

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

      1. *-commutative 98.8%

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

      2. associate-/r* 98.9%

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

   8. Simplified 98.9%

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

   if 1e-3 < (*.f64 z z)

   1. Initial program 81.4%

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

      1. associate-/r* 81.3%

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

      2. +-commutative 81.3%

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

      3. fma-def 81.3%

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

   3. Simplified 81.3%

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

   4. Taylor expanded in z around inf 77.1%

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

      1. *-commutative 77.1%

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

      2. unpow2 77.1%

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

      3. associate-*r* 79.3%

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

      4. associate-*l* 91.2%

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

      5. *-commutative 91.2%

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

      6. associate-*l* 93.9%

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

      7. *-commutative 93.9%

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

   6. Simplified 93.9%

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

      1. expm1-log1p-u 87.3%

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

      2. expm1-udef 55.5%

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

      3. associate-*r* 55.5%

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

      4. *-commutative 55.5%

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

      5. associate-*r* 55.5%

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

   8. Applied egg-rr 55.5%

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

      1. expm1-def 88.0%

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

      2. expm1-log1p 94.6%

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

      3. associate-/r* 95.5%

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

      4. *-commutative 95.5%

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

      5. associate-/r* 95.5%

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

      6. unpow-1 95.5%

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

      7. *-commutative 95.5%

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

      8. associate-/r* 90.7%

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

      9. unpow-1 90.7%

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

      10. associate-/r* 90.6%

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

   10. Simplified 90.6%

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

   11. Taylor expanded in y around 0 95.5%

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

4. Final simplification 97.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 0.001:\\ \;\;\;\;\frac{\frac{1}{x}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{y \cdot \left(z \cdot x\right)}}{z}\\ \end{array} \]

Alternative 10: 58.5% 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 89.8%

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

   1. associate-/r* 89.6%

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

   2. +-commutative 89.6%

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

   3. fma-def 89.6%

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

3. Simplified 89.6%

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

4. Taylor expanded in z around 0 52.6%

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

5. Final simplification 52.6%

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

Developer target: 92.8% 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 2023192 
(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)))))