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

Percentage Accurate: 90.8% → 97.5%

Time: 9.9s

Alternatives: 15

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 90.8% 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

\[\begin{array}{l} z = |z|\\ [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_0 := y \cdot \left(1 + z \cdot z\right)\\ \mathbf{if}\;t_0 \leq 10^{+305}:\\ \;\;\;\;\frac{\frac{1}{x}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(1, z\right)} \cdot \frac{1}{y \cdot \left(z \cdot x\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: z should be positive before calling this function
NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* y (+ 1.0 (* z z)))))
   (if (<= t_0 1e+305)
     (/ (/ 1.0 x) t_0)
     (* (/ 1.0 (hypot 1.0 z)) (/ 1.0 (* y (* z x)))))))

C

z = abs(z);
assert(x < y);
double code(double x, double y, double z) {
	double t_0 = y * (1.0 + (z * z));
	double tmp;
	if (t_0 <= 1e+305) {
		tmp = (1.0 / x) / t_0;
	} else {
		tmp = (1.0 / hypot(1.0, z)) * (1.0 / (y * (z * x)));
	}
	return tmp;
}

Java

z = Math.abs(z);
assert x < y;
public static double code(double x, double y, double z) {
	double t_0 = y * (1.0 + (z * z));
	double tmp;
	if (t_0 <= 1e+305) {
		tmp = (1.0 / x) / t_0;
	} else {
		tmp = (1.0 / Math.hypot(1.0, z)) * (1.0 / (y * (z * x)));
	}
	return tmp;
}

Python

z = abs(z)
[x, y] = sort([x, y])
def code(x, y, z):
	t_0 = y * (1.0 + (z * z))
	tmp = 0
	if t_0 <= 1e+305:
		tmp = (1.0 / x) / t_0
	else:
		tmp = (1.0 / math.hypot(1.0, z)) * (1.0 / (y * (z * x)))
	return tmp

Julia

z = abs(z)
x, y = sort([x, y])
function code(x, y, z)
	t_0 = Float64(y * Float64(1.0 + Float64(z * z)))
	tmp = 0.0
	if (t_0 <= 1e+305)
		tmp = Float64(Float64(1.0 / x) / t_0);
	else
		tmp = Float64(Float64(1.0 / hypot(1.0, z)) * Float64(1.0 / Float64(y * Float64(z * x))));
	end
	return tmp
end

MATLAB

z = abs(z)
x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y, z)
	t_0 = y * (1.0 + (z * z));
	tmp = 0.0;
	if (t_0 <= 1e+305)
		tmp = (1.0 / x) / t_0;
	else
		tmp = (1.0 / hypot(1.0, z)) * (1.0 / (y * (z * x)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: z should be positive before calling this function
NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_, z_] := Block[{t$95$0 = N[(y * N[(1.0 + N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 1e+305], N[(N[(1.0 / x), $MachinePrecision] / t$95$0), $MachinePrecision], N[(N[(1.0 / N[Sqrt[1.0 ^ 2 + z ^ 2], $MachinePrecision]), $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))) < 9.9999999999999994e304

   1. Initial program 94.1%

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

   if 9.9999999999999994e304 < (*.f64 y (+.f64 1 (*.f64 z z)))

   1. Initial program 76.8%

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

      1. associate-/r* 76.8%

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

      2. *-commutative 76.8%

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

      3. sqr-neg 76.8%

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

      4. +-commutative 76.8%

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

      5. distribute-lft1-in 76.8%

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

      6. *-commutative 76.8%

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

      7. fma-def 76.8%

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

      8. sqr-neg 76.8%

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

   3. Simplified 76.8%

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

      1. associate-/r* 76.8%

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

      2. fma-udef 76.8%

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

      3. *-rgt-identity 76.8%

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

      4. distribute-lft-in 76.8%

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

      5. +-commutative 76.8%

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

      6. associate-/r* 81.6%

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

      7. associate-/r* 81.6%

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

      8. *-un-lft-identity 81.6%

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

      9. +-commutative 81.6%

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

      10. fma-udef 81.6%

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

      11. add-sqr-sqrt 81.6%

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

      12. times-frac 81.6%

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

      13. fma-udef 81.6%

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

      14. +-commutative 81.6%

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

      15. hypot-1-def 81.6%

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

      16. fma-udef 81.6%

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

      17. +-commutative 81.6%

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

      18. hypot-1-def 92.1%

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

   5. Applied egg-rr 92.1%

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

   6. Taylor expanded in z around inf 92.1%

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

4. Final simplification 93.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(1 + z \cdot z\right) \leq 10^{+305}:\\ \;\;\;\;\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(1, z\right)} \cdot \frac{1}{y \cdot \left(z \cdot x\right)}\\ \end{array} \]

Alternative 2: 97.6% accurate, 0.1× speedup

Math

\[\begin{array}{l} z = |z|\\ [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 10^{+168}:\\ \;\;\;\;\frac{1}{y} \cdot \frac{\frac{1}{x}}{\mathsf{fma}\left(z, z, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\frac{1}{z}}{y}}{x}}{\mathsf{hypot}\left(1, z\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: z should be positive before calling this function
NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= (* z z) 1e+168)
   (* (/ 1.0 y) (/ (/ 1.0 x) (fma z z 1.0)))
   (/ (/ (/ (/ 1.0 z) y) x) (hypot 1.0 z))))

C

z = abs(z);
assert(x < y);
double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 1e+168) {
		tmp = (1.0 / y) * ((1.0 / x) / fma(z, z, 1.0));
	} else {
		tmp = (((1.0 / z) / y) / x) / hypot(1.0, z);
	}
	return tmp;
}

Julia

z = abs(z)
x, y = sort([x, y])
function code(x, y, z)
	tmp = 0.0
	if (Float64(z * z) <= 1e+168)
		tmp = Float64(Float64(1.0 / y) * Float64(Float64(1.0 / x) / fma(z, z, 1.0)));
	else
		tmp = Float64(Float64(Float64(Float64(1.0 / z) / y) / x) / hypot(1.0, z));
	end
	return tmp
end

Wolfram

NOTE: z should be positive before calling this function
NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_, z_] := If[LessEqual[N[(z * z), $MachinePrecision], 1e+168], N[(N[(1.0 / y), $MachinePrecision] * N[(N[(1.0 / x), $MachinePrecision] / N[(z * z + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(1.0 / z), $MachinePrecision] / y), $MachinePrecision] / x), $MachinePrecision] / N[Sqrt[1.0 ^ 2 + z ^ 2], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 9.9999999999999993e167

   1. Initial program 98.9%

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

      1. associate-/r* 98.6%

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

      2. *-commutative 98.6%

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

      3. sqr-neg 98.6%

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

      4. +-commutative 98.6%

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

      5. distribute-lft1-in 98.6%

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

      6. *-commutative 98.6%

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

      7. fma-def 98.6%

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

      8. sqr-neg 98.6%

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

   3. Simplified 98.6%

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

      1. associate-/r* 98.9%

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

      2. fma-udef 98.9%

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

      3. *-rgt-identity 98.9%

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

      4. distribute-lft-in 98.9%

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

      5. +-commutative 98.9%

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

      6. *-un-lft-identity 98.9%

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

      7. +-commutative 98.9%

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

      8. fma-udef 98.9%

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

      9. times-frac 98.3%

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

   5. Applied egg-rr 98.3%

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

   if 9.9999999999999993e167 < (*.f64 z z)

   1. Initial program 78.6%

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

      1. associate-/r* 78.0%

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

      2. *-commutative 78.0%

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

      3. sqr-neg 78.0%

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

      4. +-commutative 78.0%

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

      5. distribute-lft1-in 78.0%

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

      6. *-commutative 78.0%

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

      7. fma-def 78.0%

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

      8. sqr-neg 78.0%

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

   3. Simplified 78.0%

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

      1. associate-/r* 78.6%

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

      2. fma-udef 78.6%

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

      3. *-rgt-identity 78.6%

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

      4. distribute-lft-in 78.6%

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

      5. +-commutative 78.6%

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

      6. associate-/r* 83.2%

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

      7. associate-/r* 83.2%

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

      8. *-un-lft-identity 83.2%

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

      9. +-commutative 83.2%

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

      10. fma-udef 83.2%

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

      11. add-sqr-sqrt 83.2%

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

      12. times-frac 83.1%

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

      13. fma-udef 83.1%

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

      14. +-commutative 83.1%

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

      15. hypot-1-def 83.1%

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

      16. fma-udef 83.1%

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

      17. +-commutative 83.1%

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

      18. hypot-1-def 89.4%

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

   5. Applied egg-rr 89.4%

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

   6. Taylor expanded in z around inf 85.7%

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

      1. associate-*r* 84.6%

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

      2. *-commutative 84.6%

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

   8. Simplified 84.6%

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

      1. associate-*l/ 84.6%

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

      2. *-un-lft-identity 84.6%

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

      3. associate-/r* 84.6%

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

      4. associate-/r* 84.6%

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

   10. Applied egg-rr 84.6%

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

4. Final simplification 93.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 10^{+168}:\\ \;\;\;\;\frac{1}{y} \cdot \frac{\frac{1}{x}}{\mathsf{fma}\left(z, z, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\frac{1}{z}}{y}}{x}}{\mathsf{hypot}\left(1, z\right)}\\ \end{array} \]

Alternative 3: 97.3% accurate, 0.1× speedup

Math

\[\begin{array}{l} z = |z|\\ [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 3 \cdot 10^{+289}:\\ \;\;\;\;\frac{\frac{1}{y \cdot x}}{1 + z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\frac{1}{z}}{y}}{x}}{\mathsf{hypot}\left(1, z\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: z should be positive before calling this function
NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= (* z z) 3e+289)
   (/ (/ 1.0 (* y x)) (+ 1.0 (* z z)))
   (/ (/ (/ (/ 1.0 z) y) x) (hypot 1.0 z))))

C

z = abs(z);
assert(x < y);
double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 3e+289) {
		tmp = (1.0 / (y * x)) / (1.0 + (z * z));
	} else {
		tmp = (((1.0 / z) / y) / x) / hypot(1.0, z);
	}
	return tmp;
}

Java

z = Math.abs(z);
assert x < y;
public static double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 3e+289) {
		tmp = (1.0 / (y * x)) / (1.0 + (z * z));
	} else {
		tmp = (((1.0 / z) / y) / x) / Math.hypot(1.0, z);
	}
	return tmp;
}

Python

z = abs(z)
[x, y] = sort([x, y])
def code(x, y, z):
	tmp = 0
	if (z * z) <= 3e+289:
		tmp = (1.0 / (y * x)) / (1.0 + (z * z))
	else:
		tmp = (((1.0 / z) / y) / x) / math.hypot(1.0, z)
	return tmp

Julia

z = abs(z)
x, y = sort([x, y])
function code(x, y, z)
	tmp = 0.0
	if (Float64(z * z) <= 3e+289)
		tmp = Float64(Float64(1.0 / Float64(y * x)) / Float64(1.0 + Float64(z * z)));
	else
		tmp = Float64(Float64(Float64(Float64(1.0 / z) / y) / x) / hypot(1.0, z));
	end
	return tmp
end

MATLAB

z = abs(z)
x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z * z) <= 3e+289)
		tmp = (1.0 / (y * x)) / (1.0 + (z * z));
	else
		tmp = (((1.0 / z) / y) / x) / hypot(1.0, z);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: z should be positive before calling this function
NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_, z_] := If[LessEqual[N[(z * z), $MachinePrecision], 3e+289], N[(N[(1.0 / N[(y * x), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(1.0 / z), $MachinePrecision] / y), $MachinePrecision] / x), $MachinePrecision] / N[Sqrt[1.0 ^ 2 + z ^ 2], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 3.0000000000000002e289

   1. Initial program 96.6%

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

      1. associate-/r* 96.1%

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

      2. *-commutative 96.1%

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

      3. sqr-neg 96.1%

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

      4. +-commutative 96.1%

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

      5. distribute-lft1-in 96.0%

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

      6. *-commutative 96.0%

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

      7. fma-def 96.0%

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

      8. sqr-neg 96.0%

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

   3. Simplified 96.0%

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

      1. associate-/r* 96.6%

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

      2. fma-udef 96.6%

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

      3. *-rgt-identity 96.6%

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

      4. distribute-lft-in 96.6%

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

      5. +-commutative 96.6%

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

      6. associate-/r* 98.6%

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

      7. associate-/r* 98.4%

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

      8. *-un-lft-identity 98.4%

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

      9. +-commutative 98.4%

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

      10. fma-udef 98.4%

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

      11. add-sqr-sqrt 98.3%

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

      12. times-frac 98.3%

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

      13. fma-udef 98.3%

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

      14. +-commutative 98.3%

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

      15. hypot-1-def 98.3%

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

      16. fma-udef 98.3%

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

      17. +-commutative 98.3%

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

      18. hypot-1-def 98.3%

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

   5. Applied egg-rr 98.3%

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

      1. frac-times 98.3%

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

      2. *-un-lft-identity 98.3%

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

      3. hypot-udef 98.4%

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

      4. hypot-udef 98.3%

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

      5. add-sqr-sqrt 98.4%

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

      6. metadata-eval 98.4%

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

   7. Applied egg-rr 98.4%

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

   if 3.0000000000000002e289 < (*.f64 z z)

   1. Initial program 75.7%

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

      1. associate-/r* 75.7%

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

      2. *-commutative 75.7%

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

      3. sqr-neg 75.7%

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

      4. +-commutative 75.7%

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

      5. distribute-lft1-in 75.7%

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

      6. *-commutative 75.7%

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

      7. fma-def 75.7%

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

      8. sqr-neg 75.7%

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

   3. Simplified 75.7%

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

      1. associate-/r* 75.7%

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

      2. fma-udef 75.7%

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

      3. *-rgt-identity 75.7%

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

      4. distribute-lft-in 75.7%

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

      5. +-commutative 75.7%

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

      6. associate-/r* 76.4%

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

      7. associate-/r* 76.4%

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

      8. *-un-lft-identity 76.4%

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

      9. +-commutative 76.4%

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

      10. fma-udef 76.4%

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

      11. add-sqr-sqrt 76.4%

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

      12. times-frac 76.4%

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

      13. fma-udef 76.4%

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

      14. +-commutative 76.4%

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

      15. hypot-1-def 76.4%

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

      16. fma-udef 76.4%

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

      17. +-commutative 76.4%

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

      18. hypot-1-def 85.9%

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

   5. Applied egg-rr 85.9%

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

   6. Taylor expanded in z around inf 90.5%

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

      1. associate-*r* 87.8%

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

      2. *-commutative 87.8%

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

   8. Simplified 87.8%

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

      1. associate-*l/ 87.8%

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

      2. *-un-lft-identity 87.8%

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

      3. associate-/r* 87.8%

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

      4. associate-/r* 87.7%

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

   10. Applied egg-rr 87.7%

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

4. Final simplification 95.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 3 \cdot 10^{+289}:\\ \;\;\;\;\frac{\frac{1}{y \cdot x}}{1 + z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\frac{1}{z}}{y}}{x}}{\mathsf{hypot}\left(1, z\right)}\\ \end{array} \]

Alternative 4: 96.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} z = |z|\\ [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_0 := y \cdot \left(1 + z \cdot z\right)\\ \mathbf{if}\;t_0 \leq 10^{+305}:\\ \;\;\;\;\frac{\frac{1}{x}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{z}}{x} \cdot \frac{\frac{1}{y}}{z}\\ \end{array} \end{array} \]

FPCore

NOTE: z should be positive before calling this function
NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* y (+ 1.0 (* z z)))))
   (if (<= t_0 1e+305) (/ (/ 1.0 x) t_0) (* (/ (/ 1.0 z) x) (/ (/ 1.0 y) z)))))

C

z = abs(z);
assert(x < y);
double code(double x, double y, double z) {
	double t_0 = y * (1.0 + (z * z));
	double tmp;
	if (t_0 <= 1e+305) {
		tmp = (1.0 / x) / t_0;
	} else {
		tmp = ((1.0 / z) / x) * ((1.0 / y) / z);
	}
	return tmp;
}

Fortran

NOTE: z should be positive before calling this function
NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = y * (1.0d0 + (z * z))
    if (t_0 <= 1d+305) then
        tmp = (1.0d0 / x) / t_0
    else
        tmp = ((1.0d0 / z) / x) * ((1.0d0 / y) / z)
    end if
    code = tmp
end function

Java

z = Math.abs(z);
assert x < y;
public static double code(double x, double y, double z) {
	double t_0 = y * (1.0 + (z * z));
	double tmp;
	if (t_0 <= 1e+305) {
		tmp = (1.0 / x) / t_0;
	} else {
		tmp = ((1.0 / z) / x) * ((1.0 / y) / z);
	}
	return tmp;
}

Python

z = abs(z)
[x, y] = sort([x, y])
def code(x, y, z):
	t_0 = y * (1.0 + (z * z))
	tmp = 0
	if t_0 <= 1e+305:
		tmp = (1.0 / x) / t_0
	else:
		tmp = ((1.0 / z) / x) * ((1.0 / y) / z)
	return tmp

Julia

z = abs(z)
x, y = sort([x, y])
function code(x, y, z)
	t_0 = Float64(y * Float64(1.0 + Float64(z * z)))
	tmp = 0.0
	if (t_0 <= 1e+305)
		tmp = Float64(Float64(1.0 / x) / t_0);
	else
		tmp = Float64(Float64(Float64(1.0 / z) / x) * Float64(Float64(1.0 / y) / z));
	end
	return tmp
end

MATLAB

z = abs(z)
x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y, z)
	t_0 = y * (1.0 + (z * z));
	tmp = 0.0;
	if (t_0 <= 1e+305)
		tmp = (1.0 / x) / t_0;
	else
		tmp = ((1.0 / z) / x) * ((1.0 / y) / z);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: z should be positive before calling this function
NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_, z_] := Block[{t$95$0 = N[(y * N[(1.0 + N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 1e+305], N[(N[(1.0 / x), $MachinePrecision] / t$95$0), $MachinePrecision], N[(N[(N[(1.0 / z), $MachinePrecision] / x), $MachinePrecision] * N[(N[(1.0 / y), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y (+.f64 1 (*.f64 z z))) < 9.9999999999999994e304

   1. Initial program 94.1%

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

   if 9.9999999999999994e304 < (*.f64 y (+.f64 1 (*.f64 z z)))

   1. Initial program 76.8%

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

      1. associate-/r* 76.8%

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

      2. *-commutative 76.8%

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

      3. sqr-neg 76.8%

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

      4. +-commutative 76.8%

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

      5. distribute-lft1-in 76.8%

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

      6. *-commutative 76.8%

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

      7. fma-def 76.8%

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

      8. sqr-neg 76.8%

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

   3. Simplified 76.8%

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

   4. Taylor expanded in z around inf 81.9%

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

      1. unpow2 81.9%

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

      2. associate-*l* 89.8%

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

   6. Simplified 89.8%

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

      1. frac-2neg 89.8%

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

      2. metadata-eval 89.8%

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

      3. associate-*r* 81.9%

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

      4. associate-*r* 76.8%

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

      5. remove-double-neg 76.8%

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

      6. distribute-rgt-neg-out 76.8%

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

      7. distribute-lft-neg-out 76.8%

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

      8. distribute-rgt-neg-out 76.8%

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

      9. remove-double-neg 76.8%

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

      10. distribute-rgt-neg-out 76.8%

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

      11. associate-/l/ 76.8%

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

      12. frac-2neg 76.8%

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

      13. div-inv 76.8%

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

      14. distribute-neg-frac 76.8%

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

      15. metadata-eval 76.8%

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

      16. metadata-eval 76.8%

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

      17. distribute-rgt-neg-out 76.8%

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

      18. remove-double-neg 76.8%

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

      19. associate-*r* 87.1%

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

   8. Applied egg-rr 97.2%

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

4. Final simplification 94.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(1 + z \cdot z\right) \leq 10^{+305}:\\ \;\;\;\;\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{z}}{x} \cdot \frac{\frac{1}{y}}{z}\\ \end{array} \]

Alternative 5: 94.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} z = |z|\\ [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{-6}:\\ \;\;\;\;\frac{\frac{1}{y}}{x}\\ \mathbf{elif}\;z \cdot z \leq 5 \cdot 10^{+289}:\\ \;\;\;\;\frac{1}{\left(z \cdot z\right) \cdot \left(y \cdot x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y \cdot \left(z \cdot \left(z \cdot x\right)\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: z should be positive before calling this function
NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= (* z z) 2e-6)
   (/ (/ 1.0 y) x)
   (if (<= (* z z) 5e+289)
     (/ 1.0 (* (* z z) (* y x)))
     (/ 1.0 (* y (* z (* z x)))))))

C

z = abs(z);
assert(x < y);
double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 2e-6) {
		tmp = (1.0 / y) / x;
	} else if ((z * z) <= 5e+289) {
		tmp = 1.0 / ((z * z) * (y * x));
	} else {
		tmp = 1.0 / (y * (z * (z * x)));
	}
	return tmp;
}

Fortran

NOTE: z should be positive before calling this function
NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((z * z) <= 2d-6) then
        tmp = (1.0d0 / y) / x
    else if ((z * z) <= 5d+289) then
        tmp = 1.0d0 / ((z * z) * (y * x))
    else
        tmp = 1.0d0 / (y * (z * (z * x)))
    end if
    code = tmp
end function

Java

z = Math.abs(z);
assert x < y;
public static double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 2e-6) {
		tmp = (1.0 / y) / x;
	} else if ((z * z) <= 5e+289) {
		tmp = 1.0 / ((z * z) * (y * x));
	} else {
		tmp = 1.0 / (y * (z * (z * x)));
	}
	return tmp;
}

Python

z = abs(z)
[x, y] = sort([x, y])
def code(x, y, z):
	tmp = 0
	if (z * z) <= 2e-6:
		tmp = (1.0 / y) / x
	elif (z * z) <= 5e+289:
		tmp = 1.0 / ((z * z) * (y * x))
	else:
		tmp = 1.0 / (y * (z * (z * x)))
	return tmp

Julia

z = abs(z)
x, y = sort([x, y])
function code(x, y, z)
	tmp = 0.0
	if (Float64(z * z) <= 2e-6)
		tmp = Float64(Float64(1.0 / y) / x);
	elseif (Float64(z * z) <= 5e+289)
		tmp = Float64(1.0 / Float64(Float64(z * z) * Float64(y * x)));
	else
		tmp = Float64(1.0 / Float64(y * Float64(z * Float64(z * x))));
	end
	return tmp
end

MATLAB

z = abs(z)
x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z * z) <= 2e-6)
		tmp = (1.0 / y) / x;
	elseif ((z * z) <= 5e+289)
		tmp = 1.0 / ((z * z) * (y * x));
	else
		tmp = 1.0 / (y * (z * (z * x)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: z should be positive before calling this function
NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_, z_] := If[LessEqual[N[(z * z), $MachinePrecision], 2e-6], N[(N[(1.0 / y), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[N[(z * z), $MachinePrecision], 5e+289], N[(1.0 / N[(N[(z * z), $MachinePrecision] * N[(y * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(y * N[(z * N[(z * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 z z) < 1.99999999999999991e-6

   1. Initial program 99.6%

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

      1. associate-/r* 99.2%

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

      2. *-commutative 99.2%

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

      3. sqr-neg 99.2%

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

      4. +-commutative 99.2%

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

      5. distribute-lft1-in 99.2%

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

      6. *-commutative 99.2%

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

      7. fma-def 99.2%

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

      8. sqr-neg 99.2%

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

   3. Simplified 99.2%

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

      1. associate-/r* 99.6%

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

      2. fma-udef 99.6%

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

      3. *-rgt-identity 99.6%

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

      4. distribute-lft-in 99.6%

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

      5. +-commutative 99.6%

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

      6. associate-/r* 99.6%

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

      7. associate-/r* 99.2%

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

      8. *-un-lft-identity 99.2%

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

      9. +-commutative 99.2%

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

      10. fma-udef 99.2%

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

      11. add-sqr-sqrt 99.1%

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

      12. times-frac 99.2%

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

      13. fma-udef 99.2%

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

      14. +-commutative 99.2%

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

      15. hypot-1-def 99.2%

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

      16. fma-udef 99.2%

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

      17. +-commutative 99.2%

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

      18. hypot-1-def 99.1%

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

   5. Applied egg-rr 99.1%

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

      1. frac-times 99.1%

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

      2. *-un-lft-identity 99.1%

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

      3. hypot-udef 99.2%

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

      4. hypot-udef 99.1%

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

      5. add-sqr-sqrt 99.2%

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

      6. metadata-eval 99.2%

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

   7. Applied egg-rr 99.2%

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

   8. Taylor expanded in z around 0 97.8%

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

      1. associate-/r* 98.3%

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

   10. Simplified 98.3%

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

   if 1.99999999999999991e-6 < (*.f64 z z) < 5.00000000000000031e289

   1. Initial program 90.3%

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

      1. associate-/r* 89.5%

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

      2. *-commutative 89.5%

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

      3. sqr-neg 89.5%

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

      4. +-commutative 89.5%

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

      5. distribute-lft1-in 89.5%

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

      6. *-commutative 89.5%

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

      7. fma-def 89.5%

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

      8. sqr-neg 89.5%

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

   3. Simplified 89.5%

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

      1. associate-/r* 90.3%

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

      2. fma-udef 90.3%

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

      3. *-rgt-identity 90.3%

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

      4. distribute-lft-in 90.3%

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

      5. +-commutative 90.3%

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

      6. associate-/r* 97.0%

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

      7. associate-/r* 97.1%

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

      8. *-un-lft-identity 97.1%

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

      9. +-commutative 97.1%

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

      10. fma-udef 97.1%

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

      11. add-sqr-sqrt 97.0%

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

      12. times-frac 96.8%

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

      13. fma-udef 96.8%

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

      14. +-commutative 96.8%

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

      15. hypot-1-def 96.8%

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

      16. fma-udef 96.8%

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

      17. +-commutative 96.8%

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

      18. hypot-1-def 96.8%

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

   5. Applied egg-rr 96.8%

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

   6. Taylor expanded in z around inf 86.1%

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

      1. associate-*r* 87.3%

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

      2. unpow2 87.3%

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

      3. *-commutative 87.3%

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

      4. unpow2 87.3%

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

      5. associate-*r* 93.8%

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

      6. *-commutative 93.8%

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

      7. unpow2 93.8%

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

   8. Simplified 93.8%

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

   if 5.00000000000000031e289 < (*.f64 z z)

   1. Initial program 76.8%

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

      1. associate-/r* 76.8%

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

      2. *-commutative 76.8%

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

      3. sqr-neg 76.8%

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

      4. +-commutative 76.8%

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

      5. distribute-lft1-in 76.8%

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

      6. *-commutative 76.8%

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

      7. fma-def 76.8%

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

      8. sqr-neg 76.8%

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

   3. Simplified 76.8%

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

   4. Taylor expanded in z around inf 76.8%

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

      1. unpow2 76.8%

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

      2. associate-*l* 87.8%

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

   6. Simplified 87.8%

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

4. Final simplification 94.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{-6}:\\ \;\;\;\;\frac{\frac{1}{y}}{x}\\ \mathbf{elif}\;z \cdot z \leq 5 \cdot 10^{+289}:\\ \;\;\;\;\frac{1}{\left(z \cdot z\right) \cdot \left(y \cdot x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y \cdot \left(z \cdot \left(z \cdot x\right)\right)}\\ \end{array} \]

Alternative 6: 96.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} z = |z|\\ [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{-6}:\\ \;\;\;\;\frac{1 - z \cdot z}{y \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{z} \cdot \frac{\frac{\frac{1}{z}}{y}}{x}\\ \end{array} \end{array} \]

FPCore

NOTE: z should be positive before calling this function
NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= (* z z) 2e-6)
   (/ (- 1.0 (* z z)) (* y x))
   (* (/ 1.0 z) (/ (/ (/ 1.0 z) y) x))))

C

z = abs(z);
assert(x < y);
double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 2e-6) {
		tmp = (1.0 - (z * z)) / (y * x);
	} else {
		tmp = (1.0 / z) * (((1.0 / z) / y) / x);
	}
	return tmp;
}

Fortran

NOTE: z should be positive before calling this function
NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((z * z) <= 2d-6) then
        tmp = (1.0d0 - (z * z)) / (y * x)
    else
        tmp = (1.0d0 / z) * (((1.0d0 / z) / y) / x)
    end if
    code = tmp
end function

Java

z = Math.abs(z);
assert x < y;
public static double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 2e-6) {
		tmp = (1.0 - (z * z)) / (y * x);
	} else {
		tmp = (1.0 / z) * (((1.0 / z) / y) / x);
	}
	return tmp;
}

Python

z = abs(z)
[x, y] = sort([x, y])
def code(x, y, z):
	tmp = 0
	if (z * z) <= 2e-6:
		tmp = (1.0 - (z * z)) / (y * x)
	else:
		tmp = (1.0 / z) * (((1.0 / z) / y) / x)
	return tmp

Julia

z = abs(z)
x, y = sort([x, y])
function code(x, y, z)
	tmp = 0.0
	if (Float64(z * z) <= 2e-6)
		tmp = Float64(Float64(1.0 - Float64(z * z)) / Float64(y * x));
	else
		tmp = Float64(Float64(1.0 / z) * Float64(Float64(Float64(1.0 / z) / y) / x));
	end
	return tmp
end

MATLAB

z = abs(z)
x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z * z) <= 2e-6)
		tmp = (1.0 - (z * z)) / (y * x);
	else
		tmp = (1.0 / z) * (((1.0 / z) / y) / x);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: z should be positive before calling this function
NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_, z_] := If[LessEqual[N[(z * z), $MachinePrecision], 2e-6], N[(N[(1.0 - N[(z * z), $MachinePrecision]), $MachinePrecision] / N[(y * x), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / z), $MachinePrecision] * N[(N[(N[(1.0 / z), $MachinePrecision] / y), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 1.99999999999999991e-6

   1. Initial program 99.6%

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

      1. associate-/r* 99.2%

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

      2. *-commutative 99.2%

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

      3. sqr-neg 99.2%

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

      4. +-commutative 99.2%

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

      5. distribute-lft1-in 99.2%

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

      6. *-commutative 99.2%

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

      7. fma-def 99.2%

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

      8. sqr-neg 99.2%

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

   3. Simplified 99.2%

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

   4. Taylor expanded in z around 0 86.7%

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

      1. *-commutative 86.7%

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

      2. mul-1-neg 86.7%

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

      3. unsub-neg 86.7%

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

      4. unpow2 86.7%

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

      5. *-commutative 86.7%

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

      6. div-sub 98.9%

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

      7. *-commutative 98.9%

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

   6. Simplified 98.9%

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

   if 1.99999999999999991e-6 < (*.f64 z z)

   1. Initial program 84.1%

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

      1. associate-/r* 83.7%

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

      2. *-commutative 83.7%

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

      3. sqr-neg 83.7%

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

      4. +-commutative 83.7%

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

      5. distribute-lft1-in 83.7%

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

      6. *-commutative 83.7%

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

      7. fma-def 83.7%

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

      8. sqr-neg 83.7%

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

   3. Simplified 83.7%

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

   4. Taylor expanded in z around inf 81.9%

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

      1. unpow2 81.9%

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

      2. associate-*r* 82.5%

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

      3. *-commutative 82.5%

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

      4. associate-/r* 82.9%

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

      5. associate-*r* 88.7%

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

      6. *-commutative 88.7%

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

      7. associate-*r* 88.7%

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

   6. Simplified 88.7%

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

      1. *-un-lft-identity 88.7%

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

      2. *-commutative 88.7%

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

      3. times-frac 93.7%

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

      4. associate-/r* 93.7%

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

      5. associate-/l/ 93.6%

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

      6. associate-/r* 93.7%

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

   8. Applied egg-rr 93.7%

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

4. Final simplification 96.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{-6}:\\ \;\;\;\;\frac{1 - z \cdot z}{y \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{z} \cdot \frac{\frac{\frac{1}{z}}{y}}{x}\\ \end{array} \]

Alternative 7: 97.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} z = |z|\\ [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 5 \cdot 10^{+73}:\\ \;\;\;\;\frac{1}{x \cdot \left(y + y \cdot \left(z \cdot z\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{z} \cdot \frac{\frac{\frac{1}{z}}{y}}{x}\\ \end{array} \end{array} \]

FPCore

NOTE: z should be positive before calling this function
NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= (* z z) 5e+73)
   (/ 1.0 (* x (+ y (* y (* z z)))))
   (* (/ 1.0 z) (/ (/ (/ 1.0 z) y) x))))

C

z = abs(z);
assert(x < y);
double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 5e+73) {
		tmp = 1.0 / (x * (y + (y * (z * z))));
	} else {
		tmp = (1.0 / z) * (((1.0 / z) / y) / x);
	}
	return tmp;
}

Fortran

NOTE: z should be positive before calling this function
NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((z * z) <= 5d+73) then
        tmp = 1.0d0 / (x * (y + (y * (z * z))))
    else
        tmp = (1.0d0 / z) * (((1.0d0 / z) / y) / x)
    end if
    code = tmp
end function

Java

z = Math.abs(z);
assert x < y;
public static double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 5e+73) {
		tmp = 1.0 / (x * (y + (y * (z * z))));
	} else {
		tmp = (1.0 / z) * (((1.0 / z) / y) / x);
	}
	return tmp;
}

Python

z = abs(z)
[x, y] = sort([x, y])
def code(x, y, z):
	tmp = 0
	if (z * z) <= 5e+73:
		tmp = 1.0 / (x * (y + (y * (z * z))))
	else:
		tmp = (1.0 / z) * (((1.0 / z) / y) / x)
	return tmp

Julia

z = abs(z)
x, y = sort([x, y])
function code(x, y, z)
	tmp = 0.0
	if (Float64(z * z) <= 5e+73)
		tmp = Float64(1.0 / Float64(x * Float64(y + Float64(y * Float64(z * z)))));
	else
		tmp = Float64(Float64(1.0 / z) * Float64(Float64(Float64(1.0 / z) / y) / x));
	end
	return tmp
end

MATLAB

z = abs(z)
x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z * z) <= 5e+73)
		tmp = 1.0 / (x * (y + (y * (z * z))));
	else
		tmp = (1.0 / z) * (((1.0 / z) / y) / x);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: z should be positive before calling this function
NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_, z_] := If[LessEqual[N[(z * z), $MachinePrecision], 5e+73], N[(1.0 / N[(x * N[(y + N[(y * N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / z), $MachinePrecision] * N[(N[(N[(1.0 / z), $MachinePrecision] / y), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 4.99999999999999976e73

   1. Initial program 99.6%

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

      1. associate-/r* 99.2%

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

      2. *-commutative 99.2%

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

      3. sqr-neg 99.2%

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

      4. +-commutative 99.2%

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

      5. distribute-lft1-in 99.2%

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

      6. *-commutative 99.2%

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

      7. fma-def 99.2%

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

      8. sqr-neg 99.2%

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

   3. Simplified 99.2%

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

      1. fma-udef 99.2%

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

   5. Applied egg-rr 99.2%

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

   if 4.99999999999999976e73 < (*.f64 z z)

   1. Initial program 82.0%

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

      1. associate-/r* 81.5%

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

      2. *-commutative 81.5%

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

      3. sqr-neg 81.5%

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

      4. +-commutative 81.5%

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

      5. distribute-lft1-in 81.5%

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

      6. *-commutative 81.5%

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

      7. fma-def 81.5%

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

      8. sqr-neg 81.5%

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

   3. Simplified 81.5%

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

   4. Taylor expanded in z around inf 80.8%

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

      1. unpow2 80.8%

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

      2. associate-*r* 81.5%

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

      3. *-commutative 81.5%

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

      4. associate-/r* 82.0%

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

      5. associate-*r* 88.6%

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

      6. *-commutative 88.6%

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

      7. associate-*r* 88.6%

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

   6. Simplified 88.6%

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

      1. *-un-lft-identity 88.6%

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

      2. *-commutative 88.6%

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

      3. times-frac 95.1%

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

      4. associate-/r* 95.0%

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

      5. associate-/l/ 95.0%

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

      6. associate-/r* 95.1%

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

   8. Applied egg-rr 95.1%

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

4. Final simplification 97.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 5 \cdot 10^{+73}:\\ \;\;\;\;\frac{1}{x \cdot \left(y + y \cdot \left(z \cdot z\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{z} \cdot \frac{\frac{\frac{1}{z}}{y}}{x}\\ \end{array} \]

Alternative 8: 97.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} z = |z|\\ [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{+307}:\\ \;\;\;\;\frac{\frac{1}{y \cdot x}}{1 + z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{z} \cdot \frac{\frac{\frac{1}{z}}{y}}{x}\\ \end{array} \end{array} \]

FPCore

NOTE: z should be positive before calling this function
NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= (* z z) 2e+307)
   (/ (/ 1.0 (* y x)) (+ 1.0 (* z z)))
   (* (/ 1.0 z) (/ (/ (/ 1.0 z) y) x))))

C

z = abs(z);
assert(x < y);
double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 2e+307) {
		tmp = (1.0 / (y * x)) / (1.0 + (z * z));
	} else {
		tmp = (1.0 / z) * (((1.0 / z) / y) / x);
	}
	return tmp;
}

Fortran

NOTE: z should be positive before calling this function
NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((z * z) <= 2d+307) then
        tmp = (1.0d0 / (y * x)) / (1.0d0 + (z * z))
    else
        tmp = (1.0d0 / z) * (((1.0d0 / z) / y) / x)
    end if
    code = tmp
end function

Java

z = Math.abs(z);
assert x < y;
public static double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 2e+307) {
		tmp = (1.0 / (y * x)) / (1.0 + (z * z));
	} else {
		tmp = (1.0 / z) * (((1.0 / z) / y) / x);
	}
	return tmp;
}

Python

z = abs(z)
[x, y] = sort([x, y])
def code(x, y, z):
	tmp = 0
	if (z * z) <= 2e+307:
		tmp = (1.0 / (y * x)) / (1.0 + (z * z))
	else:
		tmp = (1.0 / z) * (((1.0 / z) / y) / x)
	return tmp

Julia

z = abs(z)
x, y = sort([x, y])
function code(x, y, z)
	tmp = 0.0
	if (Float64(z * z) <= 2e+307)
		tmp = Float64(Float64(1.0 / Float64(y * x)) / Float64(1.0 + Float64(z * z)));
	else
		tmp = Float64(Float64(1.0 / z) * Float64(Float64(Float64(1.0 / z) / y) / x));
	end
	return tmp
end

MATLAB

z = abs(z)
x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z * z) <= 2e+307)
		tmp = (1.0 / (y * x)) / (1.0 + (z * z));
	else
		tmp = (1.0 / z) * (((1.0 / z) / y) / x);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: z should be positive before calling this function
NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_, z_] := If[LessEqual[N[(z * z), $MachinePrecision], 2e+307], N[(N[(1.0 / N[(y * x), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / z), $MachinePrecision] * N[(N[(N[(1.0 / z), $MachinePrecision] / y), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 1.99999999999999997e307

   1. Initial program 95.9%

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

      1. associate-/r* 95.3%

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

      2. *-commutative 95.3%

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

      3. sqr-neg 95.3%

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

      4. +-commutative 95.3%

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

      5. distribute-lft1-in 95.3%

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

      6. *-commutative 95.3%

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

      7. fma-def 95.3%

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

      8. sqr-neg 95.3%

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

   3. Simplified 95.3%

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

      1. associate-/r* 95.9%

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

      2. fma-udef 95.9%

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

      3. *-rgt-identity 95.9%

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

      4. distribute-lft-in 95.9%

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

      5. +-commutative 95.9%

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

      6. associate-/r* 98.2%

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

      7. associate-/r* 98.0%

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

      8. *-un-lft-identity 98.0%

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

      9. +-commutative 98.0%

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

      10. fma-udef 98.0%

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

      11. add-sqr-sqrt 97.9%

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

      12. times-frac 97.9%

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

      13. fma-udef 97.9%

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

      14. +-commutative 97.9%

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

      15. hypot-1-def 97.9%

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

      16. fma-udef 97.9%

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

      17. +-commutative 97.9%

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

      18. hypot-1-def 97.8%

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

   5. Applied egg-rr 97.8%

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

      1. frac-times 97.9%

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

      2. *-un-lft-identity 97.9%

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

      3. hypot-udef 97.9%

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

      4. hypot-udef 97.9%

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

      5. add-sqr-sqrt 98.0%

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

      6. metadata-eval 98.0%

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

   7. Applied egg-rr 98.0%

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

   if 1.99999999999999997e307 < (*.f64 z z)

   1. Initial program 76.1%

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

      1. associate-/r* 76.1%

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

      2. *-commutative 76.1%

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

      3. sqr-neg 76.1%

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

      4. +-commutative 76.1%

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

      5. distribute-lft1-in 76.1%

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

      6. *-commutative 76.1%

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

      7. fma-def 76.1%

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

      8. sqr-neg 76.1%

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

   3. Simplified 76.1%

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

   4. Taylor expanded in z around inf 76.1%

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

      1. unpow2 76.1%

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

      2. associate-*r* 76.1%

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

      3. *-commutative 76.1%

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

      4. associate-/r* 76.1%

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

      5. associate-*r* 89.7%

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

      6. *-commutative 89.7%

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

      7. associate-*r* 89.7%

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

   6. Simplified 89.7%

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

      1. *-un-lft-identity 89.7%

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

      2. *-commutative 89.7%

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

      3. times-frac 98.2%

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

      4. associate-/r* 98.1%

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

      5. associate-/l/ 98.1%

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

      6. associate-/r* 98.5%

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

   8. Applied egg-rr 98.5%

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

4. Final simplification 98.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{+307}:\\ \;\;\;\;\frac{\frac{1}{y \cdot x}}{1 + z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{z} \cdot \frac{\frac{\frac{1}{z}}{y}}{x}\\ \end{array} \]

Alternative 9: 94.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} z = |z|\\ [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq 0.86:\\ \;\;\;\;\frac{1 - z \cdot z}{y \cdot x}\\ \mathbf{elif}\;z \leq 2.5 \cdot 10^{+145}:\\ \;\;\;\;\frac{1}{\left(z \cdot z\right) \cdot \left(y \cdot x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y \cdot \left(z \cdot \left(z \cdot x\right)\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: z should be positive before calling this function
NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= z 0.86)
   (/ (- 1.0 (* z z)) (* y x))
   (if (<= z 2.5e+145)
     (/ 1.0 (* (* z z) (* y x)))
     (/ 1.0 (* y (* z (* z x)))))))

C

z = abs(z);
assert(x < y);
double code(double x, double y, double z) {
	double tmp;
	if (z <= 0.86) {
		tmp = (1.0 - (z * z)) / (y * x);
	} else if (z <= 2.5e+145) {
		tmp = 1.0 / ((z * z) * (y * x));
	} else {
		tmp = 1.0 / (y * (z * (z * x)));
	}
	return tmp;
}

Fortran

NOTE: z should be positive before calling this function
NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (z <= 0.86d0) then
        tmp = (1.0d0 - (z * z)) / (y * x)
    else if (z <= 2.5d+145) then
        tmp = 1.0d0 / ((z * z) * (y * x))
    else
        tmp = 1.0d0 / (y * (z * (z * x)))
    end if
    code = tmp
end function

Java

z = Math.abs(z);
assert x < y;
public static double code(double x, double y, double z) {
	double tmp;
	if (z <= 0.86) {
		tmp = (1.0 - (z * z)) / (y * x);
	} else if (z <= 2.5e+145) {
		tmp = 1.0 / ((z * z) * (y * x));
	} else {
		tmp = 1.0 / (y * (z * (z * x)));
	}
	return tmp;
}

Python

z = abs(z)
[x, y] = sort([x, y])
def code(x, y, z):
	tmp = 0
	if z <= 0.86:
		tmp = (1.0 - (z * z)) / (y * x)
	elif z <= 2.5e+145:
		tmp = 1.0 / ((z * z) * (y * x))
	else:
		tmp = 1.0 / (y * (z * (z * x)))
	return tmp

Julia

z = abs(z)
x, y = sort([x, y])
function code(x, y, z)
	tmp = 0.0
	if (z <= 0.86)
		tmp = Float64(Float64(1.0 - Float64(z * z)) / Float64(y * x));
	elseif (z <= 2.5e+145)
		tmp = Float64(1.0 / Float64(Float64(z * z) * Float64(y * x)));
	else
		tmp = Float64(1.0 / Float64(y * Float64(z * Float64(z * x))));
	end
	return tmp
end

MATLAB

z = abs(z)
x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= 0.86)
		tmp = (1.0 - (z * z)) / (y * x);
	elseif (z <= 2.5e+145)
		tmp = 1.0 / ((z * z) * (y * x));
	else
		tmp = 1.0 / (y * (z * (z * x)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: z should be positive before calling this function
NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_, z_] := If[LessEqual[z, 0.86], N[(N[(1.0 - N[(z * z), $MachinePrecision]), $MachinePrecision] / N[(y * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 2.5e+145], N[(1.0 / N[(N[(z * z), $MachinePrecision] * N[(y * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(y * N[(z * N[(z * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < 0.859999999999999987

   1. Initial program 96.5%

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

      1. associate-/r* 96.2%

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

      2. *-commutative 96.2%

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

      3. sqr-neg 96.2%

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

      4. +-commutative 96.2%

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

      5. distribute-lft1-in 96.2%

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

      6. *-commutative 96.2%

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

      7. fma-def 96.2%

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

      8. sqr-neg 96.2%

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

   3. Simplified 96.2%

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

   4. Taylor expanded in z around 0 61.0%

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

      1. *-commutative 61.0%

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

      2. mul-1-neg 61.0%

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

      3. unsub-neg 61.0%

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

      4. unpow2 61.0%

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

      5. *-commutative 61.0%

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

      6. div-sub 69.3%

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

      7. *-commutative 69.3%

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

   6. Simplified 69.3%

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

   if 0.859999999999999987 < z < 2.49999999999999983e145

   1. Initial program 87.1%

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

      1. associate-/r* 85.8%

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

      2. *-commutative 85.8%

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

      3. sqr-neg 85.8%

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

      4. +-commutative 85.8%

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

      5. distribute-lft1-in 85.8%

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

      6. *-commutative 85.8%

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

      7. fma-def 85.8%

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

      8. sqr-neg 85.8%

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

   3. Simplified 85.8%

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

      1. associate-/r* 87.1%

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

      2. fma-udef 87.1%

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

      3. *-rgt-identity 87.1%

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

      4. distribute-lft-in 87.1%

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

      5. +-commutative 87.1%

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

      6. associate-/r* 95.6%

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

      7. associate-/r* 95.7%

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

      8. *-un-lft-identity 95.7%

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

      9. +-commutative 95.7%

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

      10. fma-udef 95.7%

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

      11. add-sqr-sqrt 95.6%

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

      12. times-frac 95.4%

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

      13. fma-udef 95.4%

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

      14. +-commutative 95.4%

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

      15. hypot-1-def 95.4%

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

      16. fma-udef 95.4%

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

      17. +-commutative 95.4%

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

      18. hypot-1-def 95.4%

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

   5. Applied egg-rr 95.4%

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

   6. Taylor expanded in z around inf 86.4%

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

      1. associate-*r* 82.3%

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

      2. unpow2 82.3%

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

      3. *-commutative 82.3%

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

      4. unpow2 82.3%

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

      5. associate-*r* 90.6%

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

      6. *-commutative 90.6%

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

      7. unpow2 90.6%

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

   8. Simplified 90.6%

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

   if 2.49999999999999983e145 < z

   1. Initial program 68.7%

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

      1. associate-/r* 68.7%

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

      2. *-commutative 68.7%

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

      3. sqr-neg 68.7%

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

      4. +-commutative 68.7%

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

      5. distribute-lft1-in 68.7%

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

      6. *-commutative 68.7%

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

      7. fma-def 68.7%

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

      8. sqr-neg 68.7%

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

   3. Simplified 68.7%

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

   4. Taylor expanded in z around inf 68.7%

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

      1. unpow2 68.7%

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

      2. associate-*l* 81.5%

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

   6. Simplified 81.5%

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

4. Final simplification 74.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 0.86:\\ \;\;\;\;\frac{1 - z \cdot z}{y \cdot x}\\ \mathbf{elif}\;z \leq 2.5 \cdot 10^{+145}:\\ \;\;\;\;\frac{1}{\left(z \cdot z\right) \cdot \left(y \cdot x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y \cdot \left(z \cdot \left(z \cdot x\right)\right)}\\ \end{array} \]

Alternative 10: 94.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} z = |z|\\ [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq 0.86:\\ \;\;\;\;\frac{1 - z \cdot z}{y \cdot x}\\ \mathbf{elif}\;z \leq 6.6 \cdot 10^{+145}:\\ \;\;\;\;\frac{\frac{\frac{1}{y}}{x}}{z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y \cdot \left(z \cdot \left(z \cdot x\right)\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: z should be positive before calling this function
NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= z 0.86)
   (/ (- 1.0 (* z z)) (* y x))
   (if (<= z 6.6e+145)
     (/ (/ (/ 1.0 y) x) (* z z))
     (/ 1.0 (* y (* z (* z x)))))))

C

z = abs(z);
assert(x < y);
double code(double x, double y, double z) {
	double tmp;
	if (z <= 0.86) {
		tmp = (1.0 - (z * z)) / (y * x);
	} else if (z <= 6.6e+145) {
		tmp = ((1.0 / y) / x) / (z * z);
	} else {
		tmp = 1.0 / (y * (z * (z * x)));
	}
	return tmp;
}

Fortran

NOTE: z should be positive before calling this function
NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (z <= 0.86d0) then
        tmp = (1.0d0 - (z * z)) / (y * x)
    else if (z <= 6.6d+145) then
        tmp = ((1.0d0 / y) / x) / (z * z)
    else
        tmp = 1.0d0 / (y * (z * (z * x)))
    end if
    code = tmp
end function

Java

z = Math.abs(z);
assert x < y;
public static double code(double x, double y, double z) {
	double tmp;
	if (z <= 0.86) {
		tmp = (1.0 - (z * z)) / (y * x);
	} else if (z <= 6.6e+145) {
		tmp = ((1.0 / y) / x) / (z * z);
	} else {
		tmp = 1.0 / (y * (z * (z * x)));
	}
	return tmp;
}

Python

z = abs(z)
[x, y] = sort([x, y])
def code(x, y, z):
	tmp = 0
	if z <= 0.86:
		tmp = (1.0 - (z * z)) / (y * x)
	elif z <= 6.6e+145:
		tmp = ((1.0 / y) / x) / (z * z)
	else:
		tmp = 1.0 / (y * (z * (z * x)))
	return tmp

Julia

z = abs(z)
x, y = sort([x, y])
function code(x, y, z)
	tmp = 0.0
	if (z <= 0.86)
		tmp = Float64(Float64(1.0 - Float64(z * z)) / Float64(y * x));
	elseif (z <= 6.6e+145)
		tmp = Float64(Float64(Float64(1.0 / y) / x) / Float64(z * z));
	else
		tmp = Float64(1.0 / Float64(y * Float64(z * Float64(z * x))));
	end
	return tmp
end

MATLAB

z = abs(z)
x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= 0.86)
		tmp = (1.0 - (z * z)) / (y * x);
	elseif (z <= 6.6e+145)
		tmp = ((1.0 / y) / x) / (z * z);
	else
		tmp = 1.0 / (y * (z * (z * x)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: z should be positive before calling this function
NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_, z_] := If[LessEqual[z, 0.86], N[(N[(1.0 - N[(z * z), $MachinePrecision]), $MachinePrecision] / N[(y * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 6.6e+145], N[(N[(N[(1.0 / y), $MachinePrecision] / x), $MachinePrecision] / N[(z * z), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(y * N[(z * N[(z * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < 0.859999999999999987

   1. Initial program 96.5%

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

      1. associate-/r* 96.2%

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

      2. *-commutative 96.2%

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

      3. sqr-neg 96.2%

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

      4. +-commutative 96.2%

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

      5. distribute-lft1-in 96.2%

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

      6. *-commutative 96.2%

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

      7. fma-def 96.2%

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

      8. sqr-neg 96.2%

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

   3. Simplified 96.2%

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

   4. Taylor expanded in z around 0 61.0%

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

      1. *-commutative 61.0%

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

      2. mul-1-neg 61.0%

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

      3. unsub-neg 61.0%

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

      4. unpow2 61.0%

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

      5. *-commutative 61.0%

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

      6. div-sub 69.3%

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

      7. *-commutative 69.3%

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

   6. Simplified 69.3%

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

   if 0.859999999999999987 < z < 6.60000000000000054e145

   1. Initial program 87.1%

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

      1. associate-/r* 85.8%

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

      2. *-commutative 85.8%

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

      3. sqr-neg 85.8%

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

      4. +-commutative 85.8%

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

      5. distribute-lft1-in 85.8%

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

      6. *-commutative 85.8%

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

      7. fma-def 85.8%

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

      8. sqr-neg 85.8%

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

   3. Simplified 85.8%

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

      1. associate-/r* 87.1%

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

      2. fma-udef 87.1%

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

      3. *-rgt-identity 87.1%

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

      4. distribute-lft-in 87.1%

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

      5. +-commutative 87.1%

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

      6. associate-/r* 95.6%

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

      7. associate-/r* 95.7%

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

      8. *-un-lft-identity 95.7%

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

      9. +-commutative 95.7%

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

      10. fma-udef 95.7%

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

      11. add-sqr-sqrt 95.6%

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

      12. times-frac 95.4%

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

      13. fma-udef 95.4%

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

      14. +-commutative 95.4%

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

      15. hypot-1-def 95.4%

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

      16. fma-udef 95.4%

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

      17. +-commutative 95.4%

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

      18. hypot-1-def 95.4%

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

   5. Applied egg-rr 95.4%

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

      1. frac-times 95.6%

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

      2. *-un-lft-identity 95.6%

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

      3. hypot-udef 95.6%

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

      4. hypot-udef 95.6%

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

      5. add-sqr-sqrt 95.7%

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

      6. metadata-eval 95.7%

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

   7. Applied egg-rr 95.7%

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

   8. Taylor expanded in z around inf 86.4%

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

      1. associate-/r* 86.3%

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

      2. unpow2 86.3%

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

      3. *-commutative 86.3%

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

      4. associate-/r* 92.1%

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

   10. Simplified 92.1%

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

   if 6.60000000000000054e145 < z

   1. Initial program 68.7%

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

      1. associate-/r* 68.7%

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

      2. *-commutative 68.7%

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

      3. sqr-neg 68.7%

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

      4. +-commutative 68.7%

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

      5. distribute-lft1-in 68.7%

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

      6. *-commutative 68.7%

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

      7. fma-def 68.7%

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

      8. sqr-neg 68.7%

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

   3. Simplified 68.7%

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

   4. Taylor expanded in z around inf 68.7%

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

      1. unpow2 68.7%

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

      2. associate-*l* 81.5%

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

   6. Simplified 81.5%

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

4. Final simplification 74.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 0.86:\\ \;\;\;\;\frac{1 - z \cdot z}{y \cdot x}\\ \mathbf{elif}\;z \leq 6.6 \cdot 10^{+145}:\\ \;\;\;\;\frac{\frac{\frac{1}{y}}{x}}{z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y \cdot \left(z \cdot \left(z \cdot x\right)\right)}\\ \end{array} \]

Alternative 11: 93.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if z < 1

   1. Initial program 96.5%

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

      1. associate-/r* 96.2%

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

      2. *-commutative 96.2%

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

      3. sqr-neg 96.2%

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

      4. +-commutative 96.2%

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

      5. distribute-lft1-in 96.2%

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

      6. *-commutative 96.2%

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

      7. fma-def 96.2%

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

      8. sqr-neg 96.2%

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

   3. Simplified 96.2%

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

      1. associate-/r* 96.5%

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

      2. fma-udef 96.5%

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

      3. *-rgt-identity 96.5%

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

      4. distribute-lft-in 96.5%

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

      5. +-commutative 96.5%

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

      6. associate-/r* 96.5%

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

      7. associate-/r* 96.2%

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

      8. *-un-lft-identity 96.2%

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

      9. +-commutative 96.2%

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

      10. fma-udef 96.2%

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

      11. add-sqr-sqrt 96.2%

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

      12. times-frac 96.1%

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

      13. fma-udef 96.1%

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

      14. +-commutative 96.1%

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

      15. hypot-1-def 96.1%

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

      16. fma-udef 96.1%

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

      17. +-commutative 96.1%

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

      18. hypot-1-def 98.2%

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

   5. Applied egg-rr 98.2%

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

      1. frac-times 96.2%

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

      2. *-un-lft-identity 96.2%

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

      3. hypot-udef 96.2%

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

      4. hypot-udef 96.2%

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

      5. add-sqr-sqrt 96.2%

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

      6. metadata-eval 96.2%

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

   7. Applied egg-rr 96.2%

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

   8. Taylor expanded in z around 0 72.2%

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

      1. associate-/r* 72.5%

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

   10. Simplified 72.5%

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

   if 1 < z

   1. Initial program 79.7%

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

      1. associate-/r* 79.0%

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

      2. *-commutative 79.0%

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

      3. sqr-neg 79.0%

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

      4. +-commutative 79.0%

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

      5. distribute-lft1-in 79.0%

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

      6. *-commutative 79.0%

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

      7. fma-def 79.0%

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

      8. sqr-neg 79.0%

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

   3. Simplified 79.0%

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

   4. Taylor expanded in z around inf 79.3%

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

      1. unpow2 79.3%

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

      2. associate-*l* 84.4%

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

   6. Simplified 84.4%

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

4. Final simplification 76.0%

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

Alternative 12: 70.1% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if z < 1

   1. Initial program 96.5%

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

      1. associate-/r* 96.2%

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

      2. *-commutative 96.2%

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

      3. sqr-neg 96.2%

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

      4. +-commutative 96.2%

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

      5. distribute-lft1-in 96.2%

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

      6. *-commutative 96.2%

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

      7. fma-def 96.2%

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

      8. sqr-neg 96.2%

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

   3. Simplified 96.2%

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

      1. associate-/r* 96.5%

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

      2. fma-udef 96.5%

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

      3. *-rgt-identity 96.5%

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

      4. distribute-lft-in 96.5%

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

      5. +-commutative 96.5%

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

      6. associate-/r* 96.5%

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

      7. associate-/r* 96.2%

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

      8. *-un-lft-identity 96.2%

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

      9. +-commutative 96.2%

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

      10. fma-udef 96.2%

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

      11. add-sqr-sqrt 96.2%

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

      12. times-frac 96.1%

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

      13. fma-udef 96.1%

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

      14. +-commutative 96.1%

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

      15. hypot-1-def 96.1%

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

      16. fma-udef 96.1%

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

      17. +-commutative 96.1%

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

      18. hypot-1-def 98.2%

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

   5. Applied egg-rr 98.2%

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

      1. frac-times 96.2%

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

      2. *-un-lft-identity 96.2%

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

      3. hypot-udef 96.2%

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

      4. hypot-udef 96.2%

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

      5. add-sqr-sqrt 96.2%

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

      6. metadata-eval 96.2%

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

   7. Applied egg-rr 96.2%

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

   8. Taylor expanded in z around 0 72.2%

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

      1. associate-/r* 72.5%

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

   10. Simplified 72.5%

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

   if 1 < z

   1. Initial program 79.7%

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

      1. associate-/r* 79.0%

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

      2. *-commutative 79.0%

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

      3. sqr-neg 79.0%

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

      4. +-commutative 79.0%

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

      5. distribute-lft1-in 79.0%

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

      6. *-commutative 79.0%

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

      7. fma-def 79.0%

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

      8. sqr-neg 79.0%

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

   3. Simplified 79.0%

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

      1. associate-/r* 79.7%

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

      2. fma-udef 79.7%

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

      3. *-rgt-identity 79.7%

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

      4. distribute-lft-in 79.7%

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

      5. +-commutative 79.7%

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

      6. associate-/r* 85.5%

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

      7. associate-/r* 85.6%

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

      8. *-un-lft-identity 85.6%

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

      9. +-commutative 85.6%

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

      10. fma-udef 85.6%

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

      11. add-sqr-sqrt 85.5%

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

      12. times-frac 85.4%

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

      13. fma-udef 85.4%

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

      14. +-commutative 85.4%

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

      15. hypot-1-def 85.4%

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

      16. fma-udef 85.4%

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

      17. +-commutative 85.4%

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

      18. hypot-1-def 88.1%

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

   5. Applied egg-rr 88.1%

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

   6. Taylor expanded in z around inf 91.8%

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

      1. associate-*r* 91.6%

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

      2. *-commutative 91.6%

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

   8. Simplified 91.6%

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

   9. Taylor expanded in z around 0 34.4%

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

4. Final simplification 61.3%

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

Alternative 13: 59.1% accurate, 2.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 91.6%

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

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

5. Final simplification 56.2%

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

Alternative 14: 59.1% accurate, 2.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 91.6%

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

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

   1. associate-/r* 56.3%

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

   2. div-inv 56.3%

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

6. Applied egg-rr 56.3%

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

   1. associate-*l/ 56.3%

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

   2. *-un-lft-identity 56.3%

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

8. Applied egg-rr 56.3%

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

9. Final simplification 56.3%

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

Alternative 15: 59.1% accurate, 2.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 91.6%

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

   1. associate-/r* 91.6%

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

   2. fma-udef 91.6%

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

   3. *-rgt-identity 91.6%

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

   4. distribute-lft-in 91.6%

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

   5. +-commutative 91.6%

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

   6. associate-/r* 93.3%

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

   7. associate-/r* 93.1%

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

   8. *-un-lft-identity 93.1%

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

   9. +-commutative 93.1%

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

   10. fma-udef 93.1%

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

   11. add-sqr-sqrt 93.0%

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

   12. times-frac 93.0%

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

   13. fma-udef 93.0%

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

   14. +-commutative 93.0%

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

   15. hypot-1-def 93.0%

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

   16. fma-udef 93.0%

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

   17. +-commutative 93.0%

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

   18. hypot-1-def 95.3%

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

5. Applied egg-rr 95.3%

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

   1. frac-times 93.0%

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

   2. *-un-lft-identity 93.0%

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

   3. hypot-udef 93.1%

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

   4. hypot-udef 93.0%

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

   5. add-sqr-sqrt 93.1%

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

   6. metadata-eval 93.1%

      \[\leadsto \frac{\frac{1}{x \cdot y}}{\color{blue}{1} + z \cdot z} \]

7. Applied egg-rr 93.1%

   \[\leadsto \color{blue}{\frac{\frac{1}{x \cdot y}}{1 + z \cdot z}} \]

8. Taylor expanded in z around 0 56.2%

   \[\leadsto \color{blue}{\frac{1}{y \cdot x}} \]
9. Step-by-step derivation

   1. associate-/r* 56.3%

      \[\leadsto \color{blue}{\frac{\frac{1}{y}}{x}} \]

10. Simplified 56.3%

    \[\leadsto \color{blue}{\frac{\frac{1}{y}}{x}} \]

11. Final simplification 56.3%

    \[\leadsto \frac{\frac{1}{y}}{x} \]

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 2023309 
(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)))))