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

Percentage Accurate: 88.6% → 98.2%

Time: 11.0s

Alternatives: 10

Speedup: 0.7×

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: 88.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{+108}:\\ \;\;\;\;\frac{\frac{1}{x\_m}}{y\_m \cdot \left(1 + z \cdot z\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{z} \cdot \frac{\frac{1}{x\_m}}{z}}{y\_m}\\ \end{array}\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (y_s x_s x_m y_m z)
 :precision binary64
 (*
  y_s
  (*
   x_s
   (if (<= (* z z) 2e+108)
     (/ (/ 1.0 x_m) (* y_m (+ 1.0 (* z z))))
     (/ (* (/ 1.0 z) (/ (/ 1.0 x_m) z)) y_m)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
assert(x_m < y_m && y_m < z);
double code(double y_s, double x_s, double x_m, double y_m, double z) {
	double tmp;
	if ((z * z) <= 2e+108) {
		tmp = (1.0 / x_m) / (y_m * (1.0 + (z * z)));
	} else {
		tmp = ((1.0 / z) * ((1.0 / x_m) / z)) / y_m;
	}
	return y_s * (x_s * tmp);
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
y\_m = abs(y)
y\_s = copysign(1.0d0, y)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
real(8) function code(y_s, x_s, x_m, y_m, z)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((z * z) <= 2d+108) then
        tmp = (1.0d0 / x_m) / (y_m * (1.0d0 + (z * z)))
    else
        tmp = ((1.0d0 / z) * ((1.0d0 / x_m) / z)) / y_m
    end if
    code = y_s * (x_s * tmp)
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
assert x_m < y_m && y_m < z;
public static double code(double y_s, double x_s, double x_m, double y_m, double z) {
	double tmp;
	if ((z * z) <= 2e+108) {
		tmp = (1.0 / x_m) / (y_m * (1.0 + (z * z)));
	} else {
		tmp = ((1.0 / z) * ((1.0 / x_m) / z)) / y_m;
	}
	return y_s * (x_s * tmp);
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
[x_m, y_m, z] = sort([x_m, y_m, z])
def code(y_s, x_s, x_m, y_m, z):
	tmp = 0
	if (z * z) <= 2e+108:
		tmp = (1.0 / x_m) / (y_m * (1.0 + (z * z)))
	else:
		tmp = ((1.0 / z) * ((1.0 / x_m) / z)) / y_m
	return y_s * (x_s * tmp)

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
y\_m = abs(y)
y\_s = copysign(1.0, y)
x_m, y_m, z = sort([x_m, y_m, z])
function code(y_s, x_s, x_m, y_m, z)
	tmp = 0.0
	if (Float64(z * z) <= 2e+108)
		tmp = Float64(Float64(1.0 / x_m) / Float64(y_m * Float64(1.0 + Float64(z * z))));
	else
		tmp = Float64(Float64(Float64(1.0 / z) * Float64(Float64(1.0 / x_m) / z)) / y_m);
	end
	return Float64(y_s * Float64(x_s * tmp))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
x_m, y_m, z = num2cell(sort([x_m, y_m, z])){:}
function tmp_2 = code(y_s, x_s, x_m, y_m, z)
	tmp = 0.0;
	if ((z * z) <= 2e+108)
		tmp = (1.0 / x_m) / (y_m * (1.0 + (z * z)));
	else
		tmp = ((1.0 / z) * ((1.0 / x_m) / z)) / y_m;
	end
	tmp_2 = y_s * (x_s * tmp);
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
code[y$95$s_, x$95$s_, x$95$m_, y$95$m_, z_] := N[(y$95$s * N[(x$95$s * If[LessEqual[N[(z * z), $MachinePrecision], 2e+108], N[(N[(1.0 / x$95$m), $MachinePrecision] / N[(y$95$m * N[(1.0 + N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 / z), $MachinePrecision] * N[(N[(1.0 / x$95$m), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision] / y$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 2.0000000000000001e108

   1. Initial program 99.8%

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

   if 2.0000000000000001e108 < (*.f64 z z)

   1. Initial program 78.4%

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

      1. associate-/l/ 78.4%

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

      2. associate-*l* 78.3%

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

      3. *-commutative 78.3%

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

      4. sqr-neg 78.3%

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

      5. +-commutative 78.3%

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

      6. sqr-neg 78.3%

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

      7. fma-define 78.3%

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

   3. Simplified 78.3%

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

      1. associate-*r* 79.8%

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

      2. *-commutative 79.8%

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

      3. *-commutative 79.8%

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

      4. add-sqr-sqrt 69.6%

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

      5. sqrt-div 40.8%

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

      6. metadata-eval 40.8%

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

      7. sqrt-prod 40.8%

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

      8. fma-undefine 40.8%

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

      9. +-commutative 40.8%

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

      10. hypot-1-def 40.8%

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

      11. sqrt-div 40.8%

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

      12. metadata-eval 40.8%

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

      13. sqrt-prod 40.8%

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

      14. fma-undefine 40.8%

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

      15. +-commutative 40.8%

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

      16. hypot-1-def 46.5%

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

   6. Applied egg-rr 46.5%

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

      1. unpow-1 46.5%

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

      2. unpow-1 46.5%

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

      3. pow-sqr 46.5%

         \[\leadsto \color{blue}{{\left(\mathsf{hypot}\left(1, z\right) \cdot \sqrt{x \cdot y}\right)}^{\left(2 \cdot -1\right)}} \]

      4. metadata-eval 46.5%

         \[\leadsto {\left(\mathsf{hypot}\left(1, z\right) \cdot \sqrt{x \cdot y}\right)}^{\color{blue}{-2}} \]

   8. Simplified 46.5%

      \[\leadsto \color{blue}{{\left(\mathsf{hypot}\left(1, z\right) \cdot \sqrt{x \cdot y}\right)}^{-2}} \]

   9. Taylor expanded in z around inf 46.5%

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

       1. sqr-pow 46.5%

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

       2. pow2 46.5%

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

       3. metadata-eval 46.5%

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

       4. unpow-1 46.5%

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

       5. metadata-eval 46.5%

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

       6. add-sqr-sqrt 24.6%

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

       7. sqrt-prod 40.8%

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

       8. sqrt-prod 40.8%

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

       9. *-commutative 40.8%

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

       10. associate-*r* 38.2%

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

       11. sqrt-div 67.4%

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

       12. pow2 67.4%

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

       13. add-sqr-sqrt 78.3%

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

       14. *-commutative 78.3%

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

       15. associate-/r* 78.3%

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

       16. pow2 78.3%

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

   11. Applied egg-rr 78.3%

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

       1. associate-/r* 79.1%

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

       2. inv-pow 79.1%

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

       3. metadata-eval 79.1%

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

       4. sqrt-pow1 55.6%

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

       5. *-un-lft-identity 55.6%

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

       6. unpow2 55.6%

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

       7. times-frac 56.9%

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

       8. sqrt-pow1 92.3%

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

       9. metadata-eval 92.3%

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

       10. inv-pow 92.3%

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

   13. Applied egg-rr 92.3%

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

Alternative 2: 99.4% accurate, 0.0× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ \begin{array}{l} t_0 := \mathsf{hypot}\left(1, z\right) \cdot \sqrt{y\_m}\\ y\_s \cdot \left(x\_s \cdot \frac{\frac{1}{x\_m \cdot t\_0}}{t\_0}\right) \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (y_s x_s x_m y_m z)
 :precision binary64
 (let* ((t_0 (* (hypot 1.0 z) (sqrt y_m))))
   (* y_s (* x_s (/ (/ 1.0 (* x_m t_0)) t_0)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
assert(x_m < y_m && y_m < z);
double code(double y_s, double x_s, double x_m, double y_m, double z) {
	double t_0 = hypot(1.0, z) * sqrt(y_m);
	return y_s * (x_s * ((1.0 / (x_m * t_0)) / t_0));
}

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
assert x_m < y_m && y_m < z;
public static double code(double y_s, double x_s, double x_m, double y_m, double z) {
	double t_0 = Math.hypot(1.0, z) * Math.sqrt(y_m);
	return y_s * (x_s * ((1.0 / (x_m * t_0)) / t_0));
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
[x_m, y_m, z] = sort([x_m, y_m, z])
def code(y_s, x_s, x_m, y_m, z):
	t_0 = math.hypot(1.0, z) * math.sqrt(y_m)
	return y_s * (x_s * ((1.0 / (x_m * t_0)) / t_0))

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
y\_m = abs(y)
y\_s = copysign(1.0, y)
x_m, y_m, z = sort([x_m, y_m, z])
function code(y_s, x_s, x_m, y_m, z)
	t_0 = Float64(hypot(1.0, z) * sqrt(y_m))
	return Float64(y_s * Float64(x_s * Float64(Float64(1.0 / Float64(x_m * t_0)) / t_0)))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
x_m, y_m, z = num2cell(sort([x_m, y_m, z])){:}
function tmp = code(y_s, x_s, x_m, y_m, z)
	t_0 = hypot(1.0, z) * sqrt(y_m);
	tmp = y_s * (x_s * ((1.0 / (x_m * t_0)) / t_0));
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
code[y$95$s_, x$95$s_, x$95$m_, y$95$m_, z_] := Block[{t$95$0 = N[(N[Sqrt[1.0 ^ 2 + z ^ 2], $MachinePrecision] * N[Sqrt[y$95$m], $MachinePrecision]), $MachinePrecision]}, N[(y$95$s * N[(x$95$s * N[(N[(1.0 / N[(x$95$m * t$95$0), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 91.3%

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

   1. associate-/l/ 91.0%

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

   2. associate-*l* 91.0%

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

   3. *-commutative 91.0%

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

   4. sqr-neg 91.0%

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

   5. +-commutative 91.0%

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

   6. sqr-neg 91.0%

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

   7. fma-define 91.0%

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

3. Simplified 91.0%

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

   1. associate-*r* 91.2%

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

   2. *-commutative 91.2%

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

   3. associate-/r* 90.5%

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

   4. *-commutative 90.5%

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

   5. associate-/l/ 90.8%

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

   6. fma-undefine 90.8%

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

   7. +-commutative 90.8%

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

   8. associate-/r* 91.3%

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

   9. *-un-lft-identity 91.3%

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

   10. add-sqr-sqrt 44.1%

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

   11. times-frac 44.1%

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

   12. +-commutative 44.1%

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

   13. fma-undefine 44.1%

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

   14. *-commutative 44.1%

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

   15. sqrt-prod 44.1%

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

   16. fma-undefine 44.1%

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

   17. +-commutative 44.1%

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

   18. hypot-1-def 44.1%

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

   19. +-commutative 44.1%

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

6. Applied egg-rr 48.5%

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

   1. associate-*l/ 48.5%

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

   2. *-lft-identity 48.5%

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

   3. associate-/l/ 48.6%

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

   4. *-commutative 48.6%

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

8. Simplified 48.6%

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

Alternative 3: 97.9% accurate, 0.0× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ y\_s \cdot \left(x\_s \cdot \frac{1}{y\_m \cdot {\left(\mathsf{hypot}\left(1, z\right) \cdot \sqrt{x\_m}\right)}^{2}}\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (y_s x_s x_m y_m z)
 :precision binary64
 (* y_s (* x_s (/ 1.0 (* y_m (pow (* (hypot 1.0 z) (sqrt x_m)) 2.0))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
assert(x_m < y_m && y_m < z);
double code(double y_s, double x_s, double x_m, double y_m, double z) {
	return y_s * (x_s * (1.0 / (y_m * pow((hypot(1.0, z) * sqrt(x_m)), 2.0))));
}

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
assert x_m < y_m && y_m < z;
public static double code(double y_s, double x_s, double x_m, double y_m, double z) {
	return y_s * (x_s * (1.0 / (y_m * Math.pow((Math.hypot(1.0, z) * Math.sqrt(x_m)), 2.0))));
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
[x_m, y_m, z] = sort([x_m, y_m, z])
def code(y_s, x_s, x_m, y_m, z):
	return y_s * (x_s * (1.0 / (y_m * math.pow((math.hypot(1.0, z) * math.sqrt(x_m)), 2.0))))

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
y\_m = abs(y)
y\_s = copysign(1.0, y)
x_m, y_m, z = sort([x_m, y_m, z])
function code(y_s, x_s, x_m, y_m, z)
	return Float64(y_s * Float64(x_s * Float64(1.0 / Float64(y_m * (Float64(hypot(1.0, z) * sqrt(x_m)) ^ 2.0)))))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
x_m, y_m, z = num2cell(sort([x_m, y_m, z])){:}
function tmp = code(y_s, x_s, x_m, y_m, z)
	tmp = y_s * (x_s * (1.0 / (y_m * ((hypot(1.0, z) * sqrt(x_m)) ^ 2.0))));
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
code[y$95$s_, x$95$s_, x$95$m_, y$95$m_, z_] := N[(y$95$s * N[(x$95$s * N[(1.0 / N[(y$95$m * N[Power[N[(N[Sqrt[1.0 ^ 2 + z ^ 2], $MachinePrecision] * N[Sqrt[x$95$m], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 91.3%

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

   1. associate-/l/ 91.0%

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

   2. associate-*l* 91.0%

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

   3. *-commutative 91.0%

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

   4. sqr-neg 91.0%

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

   5. +-commutative 91.0%

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

   6. sqr-neg 91.0%

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

   7. fma-define 91.0%

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

3. Simplified 91.0%

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

   1. add-sqr-sqrt 44.0%

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

   2. pow2 44.0%

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

   3. *-commutative 44.0%

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

   4. sqrt-prod 44.0%

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

   5. fma-undefine 44.0%

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

   6. +-commutative 44.0%

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

   7. hypot-1-def 46.2%

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

6. Applied egg-rr 46.2%

   \[\leadsto \frac{1}{y \cdot \color{blue}{{\left(\mathsf{hypot}\left(1, z\right) \cdot \sqrt{x}\right)}^{2}}} \]
7. Add Preprocessing

Alternative 4: 98.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;z \cdot z \leq 5 \cdot 10^{-5}:\\ \;\;\;\;\frac{\frac{1}{x\_m}}{y\_m}\\ \mathbf{elif}\;z \cdot z \leq 5 \cdot 10^{+301}:\\ \;\;\;\;\frac{\frac{1}{x\_m \cdot \left(z \cdot z\right)}}{y\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x\_m \cdot \left(z \cdot y\_m\right)}}{z}\\ \end{array}\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (y_s x_s x_m y_m z)
 :precision binary64
 (*
  y_s
  (*
   x_s
   (if (<= (* z z) 5e-5)
     (/ (/ 1.0 x_m) y_m)
     (if (<= (* z z) 5e+301)
       (/ (/ 1.0 (* x_m (* z z))) y_m)
       (/ (/ 1.0 (* x_m (* z y_m))) z))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
assert(x_m < y_m && y_m < z);
double code(double y_s, double x_s, double x_m, double y_m, double z) {
	double tmp;
	if ((z * z) <= 5e-5) {
		tmp = (1.0 / x_m) / y_m;
	} else if ((z * z) <= 5e+301) {
		tmp = (1.0 / (x_m * (z * z))) / y_m;
	} else {
		tmp = (1.0 / (x_m * (z * y_m))) / z;
	}
	return y_s * (x_s * tmp);
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
y\_m = abs(y)
y\_s = copysign(1.0d0, y)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
real(8) function code(y_s, x_s, x_m, y_m, z)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((z * z) <= 5d-5) then
        tmp = (1.0d0 / x_m) / y_m
    else if ((z * z) <= 5d+301) then
        tmp = (1.0d0 / (x_m * (z * z))) / y_m
    else
        tmp = (1.0d0 / (x_m * (z * y_m))) / z
    end if
    code = y_s * (x_s * tmp)
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
assert x_m < y_m && y_m < z;
public static double code(double y_s, double x_s, double x_m, double y_m, double z) {
	double tmp;
	if ((z * z) <= 5e-5) {
		tmp = (1.0 / x_m) / y_m;
	} else if ((z * z) <= 5e+301) {
		tmp = (1.0 / (x_m * (z * z))) / y_m;
	} else {
		tmp = (1.0 / (x_m * (z * y_m))) / z;
	}
	return y_s * (x_s * tmp);
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
[x_m, y_m, z] = sort([x_m, y_m, z])
def code(y_s, x_s, x_m, y_m, z):
	tmp = 0
	if (z * z) <= 5e-5:
		tmp = (1.0 / x_m) / y_m
	elif (z * z) <= 5e+301:
		tmp = (1.0 / (x_m * (z * z))) / y_m
	else:
		tmp = (1.0 / (x_m * (z * y_m))) / z
	return y_s * (x_s * tmp)

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
y\_m = abs(y)
y\_s = copysign(1.0, y)
x_m, y_m, z = sort([x_m, y_m, z])
function code(y_s, x_s, x_m, y_m, z)
	tmp = 0.0
	if (Float64(z * z) <= 5e-5)
		tmp = Float64(Float64(1.0 / x_m) / y_m);
	elseif (Float64(z * z) <= 5e+301)
		tmp = Float64(Float64(1.0 / Float64(x_m * Float64(z * z))) / y_m);
	else
		tmp = Float64(Float64(1.0 / Float64(x_m * Float64(z * y_m))) / z);
	end
	return Float64(y_s * Float64(x_s * tmp))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
x_m, y_m, z = num2cell(sort([x_m, y_m, z])){:}
function tmp_2 = code(y_s, x_s, x_m, y_m, z)
	tmp = 0.0;
	if ((z * z) <= 5e-5)
		tmp = (1.0 / x_m) / y_m;
	elseif ((z * z) <= 5e+301)
		tmp = (1.0 / (x_m * (z * z))) / y_m;
	else
		tmp = (1.0 / (x_m * (z * y_m))) / z;
	end
	tmp_2 = y_s * (x_s * tmp);
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
code[y$95$s_, x$95$s_, x$95$m_, y$95$m_, z_] := N[(y$95$s * N[(x$95$s * If[LessEqual[N[(z * z), $MachinePrecision], 5e-5], N[(N[(1.0 / x$95$m), $MachinePrecision] / y$95$m), $MachinePrecision], If[LessEqual[N[(z * z), $MachinePrecision], 5e+301], N[(N[(1.0 / N[(x$95$m * N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$95$m), $MachinePrecision], N[(N[(1.0 / N[(x$95$m * N[(z * y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if (*.f64 z z) < 5.00000000000000024e-5

   1. Initial program 99.8%

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

   3. Taylor expanded in z around 0 98.7%

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

   if 5.00000000000000024e-5 < (*.f64 z z) < 5.0000000000000004e301

   1. Initial program 91.2%

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

      1. associate-/l/ 91.1%

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

      2. associate-*l* 89.5%

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

      3. *-commutative 89.5%

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

      4. sqr-neg 89.5%

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

      5. +-commutative 89.5%

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

      6. sqr-neg 89.5%

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

      7. fma-define 89.5%

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

   3. Simplified 89.5%

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

      1. associate-*r* 91.0%

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

      2. *-commutative 91.0%

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

      3. *-commutative 91.0%

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

      4. add-sqr-sqrt 62.6%

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

      5. sqrt-div 55.1%

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

      6. metadata-eval 55.1%

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

      7. sqrt-prod 55.1%

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

      8. fma-undefine 55.1%

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

      9. +-commutative 55.1%

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

      10. hypot-1-def 55.1%

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

      11. sqrt-div 55.0%

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

      12. metadata-eval 55.0%

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

      13. sqrt-prod 55.0%

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

      14. fma-undefine 55.0%

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

      15. +-commutative 55.0%

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

      16. hypot-1-def 55.0%

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

   6. Applied egg-rr 55.0%

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

      1. unpow-1 55.0%

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

      2. unpow-1 55.0%

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

      3. pow-sqr 54.9%

         \[\leadsto \color{blue}{{\left(\mathsf{hypot}\left(1, z\right) \cdot \sqrt{x \cdot y}\right)}^{\left(2 \cdot -1\right)}} \]

      4. metadata-eval 54.9%

         \[\leadsto {\left(\mathsf{hypot}\left(1, z\right) \cdot \sqrt{x \cdot y}\right)}^{\color{blue}{-2}} \]

   8. Simplified 54.9%

      \[\leadsto \color{blue}{{\left(\mathsf{hypot}\left(1, z\right) \cdot \sqrt{x \cdot y}\right)}^{-2}} \]

   9. Taylor expanded in z around inf 53.9%

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

       1. sqr-pow 54.0%

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

       2. pow2 54.0%

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

       3. metadata-eval 54.0%

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

       4. unpow-1 54.0%

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

       5. metadata-eval 54.0%

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

       6. add-sqr-sqrt 33.4%

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

       7. sqrt-prod 54.0%

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

       8. sqrt-prod 53.9%

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

       9. *-commutative 53.9%

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

       10. associate-*r* 49.6%

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

       11. sqrt-div 57.7%

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

       12. pow2 57.7%

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

       13. add-sqr-sqrt 88.4%

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

       14. *-commutative 88.4%

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

       15. associate-/r* 88.4%

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

       16. pow2 88.4%

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

   11. Applied egg-rr 88.4%

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

       1. unpow2 88.4%

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

   13. Applied egg-rr 88.4%

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

   if 5.0000000000000004e301 < (*.f64 z z)

   1. Initial program 73.9%

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

      1. associate-/l/ 73.9%

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

      2. associate-*l* 75.5%

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

      3. *-commutative 75.5%

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

      4. sqr-neg 75.5%

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

      5. +-commutative 75.5%

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

      6. sqr-neg 75.5%

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

      7. fma-define 75.5%

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

   3. Simplified 75.5%

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

      1. associate-*r* 74.9%

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

      2. *-commutative 74.9%

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

      3. *-commutative 74.9%

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

      4. add-sqr-sqrt 74.9%

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

      5. sqrt-div 31.2%

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

      6. metadata-eval 31.2%

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

      7. sqrt-prod 31.2%

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

      8. fma-undefine 31.2%

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

      9. +-commutative 31.2%

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

      10. hypot-1-def 31.2%

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

      11. sqrt-div 31.2%

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

      12. metadata-eval 31.2%

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

      13. sqrt-prod 31.2%

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

      14. fma-undefine 31.2%

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

      15. +-commutative 31.2%

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

      16. hypot-1-def 40.4%

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

   6. Applied egg-rr 40.4%

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

      1. unpow-1 40.4%

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

      2. unpow-1 40.4%

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

      3. pow-sqr 40.5%

         \[\leadsto \color{blue}{{\left(\mathsf{hypot}\left(1, z\right) \cdot \sqrt{x \cdot y}\right)}^{\left(2 \cdot -1\right)}} \]

      4. metadata-eval 40.5%

         \[\leadsto {\left(\mathsf{hypot}\left(1, z\right) \cdot \sqrt{x \cdot y}\right)}^{\color{blue}{-2}} \]

   8. Simplified 40.5%

      \[\leadsto \color{blue}{{\left(\mathsf{hypot}\left(1, z\right) \cdot \sqrt{x \cdot y}\right)}^{-2}} \]

   9. Taylor expanded in z around inf 40.5%

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

       1. sqr-pow 40.4%

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

       2. pow2 40.4%

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

       3. metadata-eval 40.4%

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

       4. unpow-1 40.4%

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

       5. metadata-eval 40.4%

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

       6. add-sqr-sqrt 20.9%

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

       7. sqrt-prod 31.2%

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

       8. sqrt-prod 31.2%

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

       9. *-commutative 31.2%

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

       10. associate-*r* 31.5%

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

       11. sqrt-div 75.5%

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

       12. pow2 75.5%

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

       13. add-sqr-sqrt 75.5%

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

       14. associate-/r* 75.5%

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

       15. associate-*r* 96.8%

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

       16. associate-/r* 99.8%

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

   11. Applied egg-rr 99.8%

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

   12. Taylor expanded in y around 0 99.8%

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

4. Final simplification 96.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 5 \cdot 10^{-5}:\\ \;\;\;\;\frac{\frac{1}{x}}{y}\\ \mathbf{elif}\;z \cdot z \leq 5 \cdot 10^{+301}:\\ \;\;\;\;\frac{\frac{1}{x \cdot \left(z \cdot z\right)}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x \cdot \left(z \cdot y\right)}}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 98.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;z \cdot z \leq 5 \cdot 10^{-5}:\\ \;\;\;\;\frac{\frac{1}{x\_m}}{y\_m}\\ \mathbf{elif}\;z \cdot z \leq 5 \cdot 10^{+301}:\\ \;\;\;\;\frac{1}{y\_m \cdot \left(x\_m \cdot \left(z \cdot z\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x\_m \cdot \left(z \cdot y\_m\right)}}{z}\\ \end{array}\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (y_s x_s x_m y_m z)
 :precision binary64
 (*
  y_s
  (*
   x_s
   (if (<= (* z z) 5e-5)
     (/ (/ 1.0 x_m) y_m)
     (if (<= (* z z) 5e+301)
       (/ 1.0 (* y_m (* x_m (* z z))))
       (/ (/ 1.0 (* x_m (* z y_m))) z))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
assert(x_m < y_m && y_m < z);
double code(double y_s, double x_s, double x_m, double y_m, double z) {
	double tmp;
	if ((z * z) <= 5e-5) {
		tmp = (1.0 / x_m) / y_m;
	} else if ((z * z) <= 5e+301) {
		tmp = 1.0 / (y_m * (x_m * (z * z)));
	} else {
		tmp = (1.0 / (x_m * (z * y_m))) / z;
	}
	return y_s * (x_s * tmp);
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
y\_m = abs(y)
y\_s = copysign(1.0d0, y)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
real(8) function code(y_s, x_s, x_m, y_m, z)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((z * z) <= 5d-5) then
        tmp = (1.0d0 / x_m) / y_m
    else if ((z * z) <= 5d+301) then
        tmp = 1.0d0 / (y_m * (x_m * (z * z)))
    else
        tmp = (1.0d0 / (x_m * (z * y_m))) / z
    end if
    code = y_s * (x_s * tmp)
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
assert x_m < y_m && y_m < z;
public static double code(double y_s, double x_s, double x_m, double y_m, double z) {
	double tmp;
	if ((z * z) <= 5e-5) {
		tmp = (1.0 / x_m) / y_m;
	} else if ((z * z) <= 5e+301) {
		tmp = 1.0 / (y_m * (x_m * (z * z)));
	} else {
		tmp = (1.0 / (x_m * (z * y_m))) / z;
	}
	return y_s * (x_s * tmp);
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
[x_m, y_m, z] = sort([x_m, y_m, z])
def code(y_s, x_s, x_m, y_m, z):
	tmp = 0
	if (z * z) <= 5e-5:
		tmp = (1.0 / x_m) / y_m
	elif (z * z) <= 5e+301:
		tmp = 1.0 / (y_m * (x_m * (z * z)))
	else:
		tmp = (1.0 / (x_m * (z * y_m))) / z
	return y_s * (x_s * tmp)

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
y\_m = abs(y)
y\_s = copysign(1.0, y)
x_m, y_m, z = sort([x_m, y_m, z])
function code(y_s, x_s, x_m, y_m, z)
	tmp = 0.0
	if (Float64(z * z) <= 5e-5)
		tmp = Float64(Float64(1.0 / x_m) / y_m);
	elseif (Float64(z * z) <= 5e+301)
		tmp = Float64(1.0 / Float64(y_m * Float64(x_m * Float64(z * z))));
	else
		tmp = Float64(Float64(1.0 / Float64(x_m * Float64(z * y_m))) / z);
	end
	return Float64(y_s * Float64(x_s * tmp))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
x_m, y_m, z = num2cell(sort([x_m, y_m, z])){:}
function tmp_2 = code(y_s, x_s, x_m, y_m, z)
	tmp = 0.0;
	if ((z * z) <= 5e-5)
		tmp = (1.0 / x_m) / y_m;
	elseif ((z * z) <= 5e+301)
		tmp = 1.0 / (y_m * (x_m * (z * z)));
	else
		tmp = (1.0 / (x_m * (z * y_m))) / z;
	end
	tmp_2 = y_s * (x_s * tmp);
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
code[y$95$s_, x$95$s_, x$95$m_, y$95$m_, z_] := N[(y$95$s * N[(x$95$s * If[LessEqual[N[(z * z), $MachinePrecision], 5e-5], N[(N[(1.0 / x$95$m), $MachinePrecision] / y$95$m), $MachinePrecision], If[LessEqual[N[(z * z), $MachinePrecision], 5e+301], N[(1.0 / N[(y$95$m * N[(x$95$m * N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[(x$95$m * N[(z * y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if (*.f64 z z) < 5.00000000000000024e-5

   1. Initial program 99.8%

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

   3. Taylor expanded in z around 0 98.7%

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

   if 5.00000000000000024e-5 < (*.f64 z z) < 5.0000000000000004e301

   1. Initial program 91.2%

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

      1. associate-/l/ 91.1%

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

      2. associate-*l* 89.5%

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

      3. *-commutative 89.5%

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

      4. sqr-neg 89.5%

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

      5. +-commutative 89.5%

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

      6. sqr-neg 89.5%

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

      7. fma-define 89.5%

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

   3. Simplified 89.5%

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

   5. Taylor expanded in z around inf 88.4%

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

      1. unpow2 88.4%

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

   7. Applied egg-rr 88.4%

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

   if 5.0000000000000004e301 < (*.f64 z z)

   1. Initial program 73.9%

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

      1. associate-/l/ 73.9%

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

      2. associate-*l* 75.5%

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

      3. *-commutative 75.5%

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

      4. sqr-neg 75.5%

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

      5. +-commutative 75.5%

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

      6. sqr-neg 75.5%

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

      7. fma-define 75.5%

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

   3. Simplified 75.5%

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

      1. associate-*r* 74.9%

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

      2. *-commutative 74.9%

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

      3. *-commutative 74.9%

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

      4. add-sqr-sqrt 74.9%

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

      5. sqrt-div 31.2%

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

      6. metadata-eval 31.2%

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

      7. sqrt-prod 31.2%

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

      8. fma-undefine 31.2%

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

      9. +-commutative 31.2%

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

      10. hypot-1-def 31.2%

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

      11. sqrt-div 31.2%

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

      12. metadata-eval 31.2%

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

      13. sqrt-prod 31.2%

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

      14. fma-undefine 31.2%

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

      15. +-commutative 31.2%

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

      16. hypot-1-def 40.4%

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

   6. Applied egg-rr 40.4%

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

      1. unpow-1 40.4%

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

      2. unpow-1 40.4%

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

      3. pow-sqr 40.5%

         \[\leadsto \color{blue}{{\left(\mathsf{hypot}\left(1, z\right) \cdot \sqrt{x \cdot y}\right)}^{\left(2 \cdot -1\right)}} \]

      4. metadata-eval 40.5%

         \[\leadsto {\left(\mathsf{hypot}\left(1, z\right) \cdot \sqrt{x \cdot y}\right)}^{\color{blue}{-2}} \]

   8. Simplified 40.5%

      \[\leadsto \color{blue}{{\left(\mathsf{hypot}\left(1, z\right) \cdot \sqrt{x \cdot y}\right)}^{-2}} \]

   9. Taylor expanded in z around inf 40.5%

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

       1. sqr-pow 40.4%

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

       2. pow2 40.4%

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

       3. metadata-eval 40.4%

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

       4. unpow-1 40.4%

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

       5. metadata-eval 40.4%

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

       6. add-sqr-sqrt 20.9%

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

       7. sqrt-prod 31.2%

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

       8. sqrt-prod 31.2%

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

       9. *-commutative 31.2%

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

       10. associate-*r* 31.5%

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

       11. sqrt-div 75.5%

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

       12. pow2 75.5%

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

       13. add-sqr-sqrt 75.5%

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

       14. associate-/r* 75.5%

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

       15. associate-*r* 96.8%

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

       16. associate-/r* 99.8%

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

   11. Applied egg-rr 99.8%

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

   12. Taylor expanded in y around 0 99.8%

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

4. Final simplification 96.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 5 \cdot 10^{-5}:\\ \;\;\;\;\frac{\frac{1}{x}}{y}\\ \mathbf{elif}\;z \cdot z \leq 5 \cdot 10^{+301}:\\ \;\;\;\;\frac{1}{y \cdot \left(x \cdot \left(z \cdot z\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x \cdot \left(z \cdot y\right)}}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 98.1% accurate, 0.6× speedup

Math

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

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (y_s x_s x_m y_m z)
 :precision binary64
 (*
  y_s
  (*
   x_s
   (if (<= (* z z) 5e-5)
     (/ (/ 1.0 x_m) y_m)
     (/ (* (/ 1.0 z) (/ (/ 1.0 x_m) z)) y_m)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
assert(x_m < y_m && y_m < z);
double code(double y_s, double x_s, double x_m, double y_m, double z) {
	double tmp;
	if ((z * z) <= 5e-5) {
		tmp = (1.0 / x_m) / y_m;
	} else {
		tmp = ((1.0 / z) * ((1.0 / x_m) / z)) / y_m;
	}
	return y_s * (x_s * tmp);
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
y\_m = abs(y)
y\_s = copysign(1.0d0, y)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
real(8) function code(y_s, x_s, x_m, y_m, z)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((z * z) <= 5d-5) then
        tmp = (1.0d0 / x_m) / y_m
    else
        tmp = ((1.0d0 / z) * ((1.0d0 / x_m) / z)) / y_m
    end if
    code = y_s * (x_s * tmp)
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
assert x_m < y_m && y_m < z;
public static double code(double y_s, double x_s, double x_m, double y_m, double z) {
	double tmp;
	if ((z * z) <= 5e-5) {
		tmp = (1.0 / x_m) / y_m;
	} else {
		tmp = ((1.0 / z) * ((1.0 / x_m) / z)) / y_m;
	}
	return y_s * (x_s * tmp);
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
[x_m, y_m, z] = sort([x_m, y_m, z])
def code(y_s, x_s, x_m, y_m, z):
	tmp = 0
	if (z * z) <= 5e-5:
		tmp = (1.0 / x_m) / y_m
	else:
		tmp = ((1.0 / z) * ((1.0 / x_m) / z)) / y_m
	return y_s * (x_s * tmp)

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
y\_m = abs(y)
y\_s = copysign(1.0, y)
x_m, y_m, z = sort([x_m, y_m, z])
function code(y_s, x_s, x_m, y_m, z)
	tmp = 0.0
	if (Float64(z * z) <= 5e-5)
		tmp = Float64(Float64(1.0 / x_m) / y_m);
	else
		tmp = Float64(Float64(Float64(1.0 / z) * Float64(Float64(1.0 / x_m) / z)) / y_m);
	end
	return Float64(y_s * Float64(x_s * tmp))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
x_m, y_m, z = num2cell(sort([x_m, y_m, z])){:}
function tmp_2 = code(y_s, x_s, x_m, y_m, z)
	tmp = 0.0;
	if ((z * z) <= 5e-5)
		tmp = (1.0 / x_m) / y_m;
	else
		tmp = ((1.0 / z) * ((1.0 / x_m) / z)) / y_m;
	end
	tmp_2 = y_s * (x_s * tmp);
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
code[y$95$s_, x$95$s_, x$95$m_, y$95$m_, z_] := N[(y$95$s * N[(x$95$s * If[LessEqual[N[(z * z), $MachinePrecision], 5e-5], N[(N[(1.0 / x$95$m), $MachinePrecision] / y$95$m), $MachinePrecision], N[(N[(N[(1.0 / z), $MachinePrecision] * N[(N[(1.0 / x$95$m), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision] / y$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 5.00000000000000024e-5

   1. Initial program 99.8%

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

   3. Taylor expanded in z around 0 98.7%

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

   if 5.00000000000000024e-5 < (*.f64 z z)

   1. Initial program 82.7%

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

      1. associate-/l/ 82.7%

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

      2. associate-*l* 82.6%

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

      3. *-commutative 82.6%

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

      4. sqr-neg 82.6%

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

      5. +-commutative 82.6%

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

      6. sqr-neg 82.6%

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

      7. fma-define 82.6%

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

   3. Simplified 82.6%

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

      1. associate-*r* 83.1%

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

      2. *-commutative 83.1%

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

      3. *-commutative 83.1%

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

      4. add-sqr-sqrt 68.6%

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

      5. sqrt-div 43.4%

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

      6. metadata-eval 43.4%

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

      7. sqrt-prod 43.4%

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

      8. fma-undefine 43.4%

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

      9. +-commutative 43.4%

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

      10. hypot-1-def 43.4%

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

      11. sqrt-div 43.4%

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

      12. metadata-eval 43.4%

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

      13. sqrt-prod 43.4%

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

      14. fma-undefine 43.4%

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

      15. +-commutative 43.4%

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

      16. hypot-1-def 47.9%

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

   6. Applied egg-rr 47.9%

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

      1. unpow-1 47.9%

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

      2. unpow-1 47.9%

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

      3. pow-sqr 47.9%

         \[\leadsto \color{blue}{{\left(\mathsf{hypot}\left(1, z\right) \cdot \sqrt{x \cdot y}\right)}^{\left(2 \cdot -1\right)}} \]

      4. metadata-eval 47.9%

         \[\leadsto {\left(\mathsf{hypot}\left(1, z\right) \cdot \sqrt{x \cdot y}\right)}^{\color{blue}{-2}} \]

   8. Simplified 47.9%

      \[\leadsto \color{blue}{{\left(\mathsf{hypot}\left(1, z\right) \cdot \sqrt{x \cdot y}\right)}^{-2}} \]

   9. Taylor expanded in z around inf 47.3%

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

       1. sqr-pow 47.4%

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

       2. pow2 47.4%

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

       3. metadata-eval 47.4%

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

       4. unpow-1 47.4%

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

       5. metadata-eval 47.4%

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

       6. add-sqr-sqrt 27.3%

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

       7. sqrt-prod 42.8%

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

       8. sqrt-prod 42.8%

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

       9. *-commutative 42.8%

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

       10. associate-*r* 40.7%

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

       11. sqrt-div 66.4%

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

       12. pow2 66.4%

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

       13. add-sqr-sqrt 82.1%

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

       14. *-commutative 82.1%

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

       15. associate-/r* 82.1%

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

       16. pow2 82.1%

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

   11. Applied egg-rr 82.1%

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

       1. associate-/r* 82.7%

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

       2. inv-pow 82.7%

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

       3. metadata-eval 82.7%

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

       4. sqrt-pow1 53.4%

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

       5. *-un-lft-identity 53.4%

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

       6. unpow2 53.4%

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

       7. times-frac 54.4%

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

       8. sqrt-pow1 93.3%

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

       9. metadata-eval 93.3%

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

       10. inv-pow 93.3%

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

   13. Applied egg-rr 93.3%

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

Alternative 7: 98.0% accurate, 0.7× speedup

Math

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

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (y_s x_s x_m y_m z)
 :precision binary64
 (*
  y_s
  (*
   x_s
   (if (<= (* z z) 5e-5)
     (/ (/ 1.0 x_m) y_m)
     (/ (/ (/ 1.0 y_m) (* x_m z)) z)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
assert(x_m < y_m && y_m < z);
double code(double y_s, double x_s, double x_m, double y_m, double z) {
	double tmp;
	if ((z * z) <= 5e-5) {
		tmp = (1.0 / x_m) / y_m;
	} else {
		tmp = ((1.0 / y_m) / (x_m * z)) / z;
	}
	return y_s * (x_s * tmp);
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
y\_m = abs(y)
y\_s = copysign(1.0d0, y)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
real(8) function code(y_s, x_s, x_m, y_m, z)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((z * z) <= 5d-5) then
        tmp = (1.0d0 / x_m) / y_m
    else
        tmp = ((1.0d0 / y_m) / (x_m * z)) / z
    end if
    code = y_s * (x_s * tmp)
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
assert x_m < y_m && y_m < z;
public static double code(double y_s, double x_s, double x_m, double y_m, double z) {
	double tmp;
	if ((z * z) <= 5e-5) {
		tmp = (1.0 / x_m) / y_m;
	} else {
		tmp = ((1.0 / y_m) / (x_m * z)) / z;
	}
	return y_s * (x_s * tmp);
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
[x_m, y_m, z] = sort([x_m, y_m, z])
def code(y_s, x_s, x_m, y_m, z):
	tmp = 0
	if (z * z) <= 5e-5:
		tmp = (1.0 / x_m) / y_m
	else:
		tmp = ((1.0 / y_m) / (x_m * z)) / z
	return y_s * (x_s * tmp)

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
y\_m = abs(y)
y\_s = copysign(1.0, y)
x_m, y_m, z = sort([x_m, y_m, z])
function code(y_s, x_s, x_m, y_m, z)
	tmp = 0.0
	if (Float64(z * z) <= 5e-5)
		tmp = Float64(Float64(1.0 / x_m) / y_m);
	else
		tmp = Float64(Float64(Float64(1.0 / y_m) / Float64(x_m * z)) / z);
	end
	return Float64(y_s * Float64(x_s * tmp))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
x_m, y_m, z = num2cell(sort([x_m, y_m, z])){:}
function tmp_2 = code(y_s, x_s, x_m, y_m, z)
	tmp = 0.0;
	if ((z * z) <= 5e-5)
		tmp = (1.0 / x_m) / y_m;
	else
		tmp = ((1.0 / y_m) / (x_m * z)) / z;
	end
	tmp_2 = y_s * (x_s * tmp);
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
code[y$95$s_, x$95$s_, x$95$m_, y$95$m_, z_] := N[(y$95$s * N[(x$95$s * If[LessEqual[N[(z * z), $MachinePrecision], 5e-5], N[(N[(1.0 / x$95$m), $MachinePrecision] / y$95$m), $MachinePrecision], N[(N[(N[(1.0 / y$95$m), $MachinePrecision] / N[(x$95$m * z), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 5.00000000000000024e-5

   1. Initial program 99.8%

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

   3. Taylor expanded in z around 0 98.7%

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

   if 5.00000000000000024e-5 < (*.f64 z z)

   1. Initial program 82.7%

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

      1. associate-/l/ 82.7%

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

      2. associate-*l* 82.6%

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

      3. *-commutative 82.6%

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

      4. sqr-neg 82.6%

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

      5. +-commutative 82.6%

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

      6. sqr-neg 82.6%

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

      7. fma-define 82.6%

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

   3. Simplified 82.6%

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

      1. associate-*r* 83.1%

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

      2. *-commutative 83.1%

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

      3. *-commutative 83.1%

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

      4. add-sqr-sqrt 68.6%

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

      5. sqrt-div 43.4%

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

      6. metadata-eval 43.4%

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

      7. sqrt-prod 43.4%

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

      8. fma-undefine 43.4%

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

      9. +-commutative 43.4%

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

      10. hypot-1-def 43.4%

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

      11. sqrt-div 43.4%

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

      12. metadata-eval 43.4%

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

      13. sqrt-prod 43.4%

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

      14. fma-undefine 43.4%

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

      15. +-commutative 43.4%

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

      16. hypot-1-def 47.9%

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

   6. Applied egg-rr 47.9%

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

      1. unpow-1 47.9%

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

      2. unpow-1 47.9%

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

      3. pow-sqr 47.9%

         \[\leadsto \color{blue}{{\left(\mathsf{hypot}\left(1, z\right) \cdot \sqrt{x \cdot y}\right)}^{\left(2 \cdot -1\right)}} \]

      4. metadata-eval 47.9%

         \[\leadsto {\left(\mathsf{hypot}\left(1, z\right) \cdot \sqrt{x \cdot y}\right)}^{\color{blue}{-2}} \]

   8. Simplified 47.9%

      \[\leadsto \color{blue}{{\left(\mathsf{hypot}\left(1, z\right) \cdot \sqrt{x \cdot y}\right)}^{-2}} \]

   9. Taylor expanded in z around inf 47.3%

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

       1. sqr-pow 47.4%

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

       2. pow2 47.4%

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

       3. metadata-eval 47.4%

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

       4. unpow-1 47.4%

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

       5. metadata-eval 47.4%

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

       6. add-sqr-sqrt 27.3%

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

       7. sqrt-prod 42.8%

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

       8. sqrt-prod 42.8%

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

       9. *-commutative 42.8%

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

       10. associate-*r* 40.7%

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

       11. sqrt-div 66.4%

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

       12. pow2 66.4%

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

       13. add-sqr-sqrt 82.1%

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

       14. associate-/r* 82.1%

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

       15. associate-*r* 92.5%

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

       16. associate-/r* 95.4%

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

   11. Applied egg-rr 95.4%

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

Alternative 8: 91.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;z \cdot z \leq 0.031:\\ \;\;\;\;\frac{\frac{1}{x\_m}}{y\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y\_m \cdot \left(x\_m \cdot \left(z \cdot z\right)\right)}\\ \end{array}\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (y_s x_s x_m y_m z)
 :precision binary64
 (*
  y_s
  (*
   x_s
   (if (<= (* z z) 0.031)
     (/ (/ 1.0 x_m) y_m)
     (/ 1.0 (* y_m (* x_m (* z z))))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
assert(x_m < y_m && y_m < z);
double code(double y_s, double x_s, double x_m, double y_m, double z) {
	double tmp;
	if ((z * z) <= 0.031) {
		tmp = (1.0 / x_m) / y_m;
	} else {
		tmp = 1.0 / (y_m * (x_m * (z * z)));
	}
	return y_s * (x_s * tmp);
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
y\_m = abs(y)
y\_s = copysign(1.0d0, y)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
real(8) function code(y_s, x_s, x_m, y_m, z)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((z * z) <= 0.031d0) then
        tmp = (1.0d0 / x_m) / y_m
    else
        tmp = 1.0d0 / (y_m * (x_m * (z * z)))
    end if
    code = y_s * (x_s * tmp)
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
assert x_m < y_m && y_m < z;
public static double code(double y_s, double x_s, double x_m, double y_m, double z) {
	double tmp;
	if ((z * z) <= 0.031) {
		tmp = (1.0 / x_m) / y_m;
	} else {
		tmp = 1.0 / (y_m * (x_m * (z * z)));
	}
	return y_s * (x_s * tmp);
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
[x_m, y_m, z] = sort([x_m, y_m, z])
def code(y_s, x_s, x_m, y_m, z):
	tmp = 0
	if (z * z) <= 0.031:
		tmp = (1.0 / x_m) / y_m
	else:
		tmp = 1.0 / (y_m * (x_m * (z * z)))
	return y_s * (x_s * tmp)

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
y\_m = abs(y)
y\_s = copysign(1.0, y)
x_m, y_m, z = sort([x_m, y_m, z])
function code(y_s, x_s, x_m, y_m, z)
	tmp = 0.0
	if (Float64(z * z) <= 0.031)
		tmp = Float64(Float64(1.0 / x_m) / y_m);
	else
		tmp = Float64(1.0 / Float64(y_m * Float64(x_m * Float64(z * z))));
	end
	return Float64(y_s * Float64(x_s * tmp))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
x_m, y_m, z = num2cell(sort([x_m, y_m, z])){:}
function tmp_2 = code(y_s, x_s, x_m, y_m, z)
	tmp = 0.0;
	if ((z * z) <= 0.031)
		tmp = (1.0 / x_m) / y_m;
	else
		tmp = 1.0 / (y_m * (x_m * (z * z)));
	end
	tmp_2 = y_s * (x_s * tmp);
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
code[y$95$s_, x$95$s_, x$95$m_, y$95$m_, z_] := N[(y$95$s * N[(x$95$s * If[LessEqual[N[(z * z), $MachinePrecision], 0.031], N[(N[(1.0 / x$95$m), $MachinePrecision] / y$95$m), $MachinePrecision], N[(1.0 / N[(y$95$m * N[(x$95$m * N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 0.031

   1. Initial program 99.8%

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

   3. Taylor expanded in z around 0 98.7%

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

   if 0.031 < (*.f64 z z)

   1. Initial program 82.7%

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

      1. associate-/l/ 82.7%

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

      2. associate-*l* 82.6%

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

      3. *-commutative 82.6%

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

      4. sqr-neg 82.6%

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

      5. +-commutative 82.6%

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

      6. sqr-neg 82.6%

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

      7. fma-define 82.6%

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

   3. Simplified 82.6%

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

   5. Taylor expanded in z around inf 82.1%

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

      1. unpow2 82.1%

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

   7. Applied egg-rr 82.1%

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

Alternative 9: 58.9% accurate, 2.2× speedup

Math

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

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (y_s x_s x_m y_m z)
 :precision binary64
 (* y_s (* x_s (/ (/ 1.0 x_m) y_m))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
assert(x_m < y_m && y_m < z);
double code(double y_s, double x_s, double x_m, double y_m, double z) {
	return y_s * (x_s * ((1.0 / x_m) / y_m));
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
y\_m = abs(y)
y\_s = copysign(1.0d0, y)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
real(8) function code(y_s, x_s, x_m, y_m, z)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    code = y_s * (x_s * ((1.0d0 / x_m) / y_m))
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
assert x_m < y_m && y_m < z;
public static double code(double y_s, double x_s, double x_m, double y_m, double z) {
	return y_s * (x_s * ((1.0 / x_m) / y_m));
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
[x_m, y_m, z] = sort([x_m, y_m, z])
def code(y_s, x_s, x_m, y_m, z):
	return y_s * (x_s * ((1.0 / x_m) / y_m))

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
y\_m = abs(y)
y\_s = copysign(1.0, y)
x_m, y_m, z = sort([x_m, y_m, z])
function code(y_s, x_s, x_m, y_m, z)
	return Float64(y_s * Float64(x_s * Float64(Float64(1.0 / x_m) / y_m)))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
x_m, y_m, z = num2cell(sort([x_m, y_m, z])){:}
function tmp = code(y_s, x_s, x_m, y_m, z)
	tmp = y_s * (x_s * ((1.0 / x_m) / y_m));
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
code[y$95$s_, x$95$s_, x$95$m_, y$95$m_, z_] := N[(y$95$s * N[(x$95$s * N[(N[(1.0 / x$95$m), $MachinePrecision] / y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 91.3%

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

3. Taylor expanded in z around 0 59.1%

   \[\leadsto \frac{\frac{1}{x}}{\color{blue}{y}} \]
4. Add Preprocessing

Alternative 10: 58.9% accurate, 2.2× speedup

Math

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

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (y_s x_s x_m y_m z)
 :precision binary64
 (* y_s (* x_s (/ 1.0 (* x_m y_m)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
assert(x_m < y_m && y_m < z);
double code(double y_s, double x_s, double x_m, double y_m, double z) {
	return y_s * (x_s * (1.0 / (x_m * y_m)));
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
y\_m = abs(y)
y\_s = copysign(1.0d0, y)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
real(8) function code(y_s, x_s, x_m, y_m, z)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    code = y_s * (x_s * (1.0d0 / (x_m * y_m)))
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
assert x_m < y_m && y_m < z;
public static double code(double y_s, double x_s, double x_m, double y_m, double z) {
	return y_s * (x_s * (1.0 / (x_m * y_m)));
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
[x_m, y_m, z] = sort([x_m, y_m, z])
def code(y_s, x_s, x_m, y_m, z):
	return y_s * (x_s * (1.0 / (x_m * y_m)))

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
y\_m = abs(y)
y\_s = copysign(1.0, y)
x_m, y_m, z = sort([x_m, y_m, z])
function code(y_s, x_s, x_m, y_m, z)
	return Float64(y_s * Float64(x_s * Float64(1.0 / Float64(x_m * y_m))))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
x_m, y_m, z = num2cell(sort([x_m, y_m, z])){:}
function tmp = code(y_s, x_s, x_m, y_m, z)
	tmp = y_s * (x_s * (1.0 / (x_m * y_m)));
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
code[y$95$s_, x$95$s_, x$95$m_, y$95$m_, z_] := N[(y$95$s * N[(x$95$s * N[(1.0 / N[(x$95$m * y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 91.3%

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

   1. associate-/l/ 91.0%

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

   2. associate-*l* 91.0%

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

   3. *-commutative 91.0%

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

   4. sqr-neg 91.0%

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

   5. +-commutative 91.0%

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

   6. sqr-neg 91.0%

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

   7. fma-define 91.0%

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

3. Simplified 91.0%

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

5. Taylor expanded in z around 0 58.9%

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

6. Final simplification 58.9%

   \[\leadsto \frac{1}{x \cdot y} \]
7. Add Preprocessing

Developer Target 1: 92.6% accurate, 0.3× 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 2024121 
(FPCore (x y z)
  :name "Statistics.Distribution.CauchyLorentz:$cdensity from math-functions-0.1.5.2"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< (* y (+ 1 (* z z))) -inf.0) (/ (/ 1 y) (* (+ 1 (* z z)) x)) (if (< (* y (+ 1 (* z z))) 868074325056725200000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000) (/ (/ 1 x) (* (+ 1 (* z z)) y)) (/ (/ 1 y) (* (+ 1 (* z z)) x)))))

  (/ (/ 1.0 x) (* y (+ 1.0 (* z z)))))