_divideComplex, real part

Percentage Accurate: 62.2% → 84.4%

Time: 11.2s

Alternatives: 10

Speedup: 2.1×

Specification

Math

\[\begin{array}{l} \\ \frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (/ (+ (* x.re y.re) (* x.im y.im)) (+ (* y.re y.re) (* y.im y.im))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	return ((x_46_re * y_46_re) + (x_46_im * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    code = ((x_46re * y_46re) + (x_46im * y_46im)) / ((y_46re * y_46re) + (y_46im * y_46im))
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	return ((x_46_re * y_46_re) + (x_46_im * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	return ((x_46_re * y_46_re) + (x_46_im * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im))

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	return Float64(Float64(Float64(x_46_re * y_46_re) + Float64(x_46_im * y_46_im)) / Float64(Float64(y_46_re * y_46_re) + Float64(y_46_im * y_46_im)))
end

MATLAB

function tmp = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = ((x_46_re * y_46_re) + (x_46_im * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := N[(N[(N[(x$46$re * y$46$re), $MachinePrecision] + N[(x$46$im * y$46$im), $MachinePrecision]), $MachinePrecision] / N[(N[(y$46$re * y$46$re), $MachinePrecision] + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 62.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (/ (+ (* x.re y.re) (* x.im y.im)) (+ (* y.re y.re) (* y.im y.im))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	return ((x_46_re * y_46_re) + (x_46_im * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    code = ((x_46re * y_46re) + (x_46im * y_46im)) / ((y_46re * y_46re) + (y_46im * y_46im))
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	return ((x_46_re * y_46_re) + (x_46_im * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	return ((x_46_re * y_46_re) + (x_46_im * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im))

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	return Float64(Float64(Float64(x_46_re * y_46_re) + Float64(x_46_im * y_46_im)) / Float64(Float64(y_46_re * y_46_re) + Float64(y_46_im * y_46_im)))
end

MATLAB

function tmp = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = ((x_46_re * y_46_re) + (x_46_im * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := N[(N[(N[(x$46$re * y$46$re), $MachinePrecision] + N[(x$46$im * y$46$im), $MachinePrecision]), $MachinePrecision] / N[(N[(y$46$re * y$46$re), $MachinePrecision] + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 84.4% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\frac{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}{\mathsf{hypot}\left(y.re, y.im\right)}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{if}\;y.re \leq -1.15 \cdot 10^{+110}:\\ \;\;\;\;\frac{\left(-x.re\right) - \frac{x.im}{\frac{y.re}{y.im}}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.re \leq -7.8 \cdot 10^{-98}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq 9.5 \cdot 10^{-89}:\\ \;\;\;\;\frac{1}{y.im} \cdot \left(x.im + \frac{x.re}{\frac{y.im}{y.re}}\right)\\ \mathbf{elif}\;y.re \leq 8.8 \cdot 10^{+26}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\frac{x.im}{y.re}}{y.re}, y.im, \frac{x.re}{y.re}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0
         (/
          (/ (fma x.re y.re (* x.im y.im)) (hypot y.re y.im))
          (hypot y.re y.im))))
   (if (<= y.re -1.15e+110)
     (/ (- (- x.re) (/ x.im (/ y.re y.im))) (hypot y.re y.im))
     (if (<= y.re -7.8e-98)
       t_0
       (if (<= y.re 9.5e-89)
         (* (/ 1.0 y.im) (+ x.im (/ x.re (/ y.im y.re))))
         (if (<= y.re 8.8e+26)
           t_0
           (fma (/ (/ x.im y.re) y.re) y.im (/ x.re y.re))))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = (fma(x_46_re, y_46_re, (x_46_im * y_46_im)) / hypot(y_46_re, y_46_im)) / hypot(y_46_re, y_46_im);
	double tmp;
	if (y_46_re <= -1.15e+110) {
		tmp = (-x_46_re - (x_46_im / (y_46_re / y_46_im))) / hypot(y_46_re, y_46_im);
	} else if (y_46_re <= -7.8e-98) {
		tmp = t_0;
	} else if (y_46_re <= 9.5e-89) {
		tmp = (1.0 / y_46_im) * (x_46_im + (x_46_re / (y_46_im / y_46_re)));
	} else if (y_46_re <= 8.8e+26) {
		tmp = t_0;
	} else {
		tmp = fma(((x_46_im / y_46_re) / y_46_re), y_46_im, (x_46_re / y_46_re));
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(Float64(fma(x_46_re, y_46_re, Float64(x_46_im * y_46_im)) / hypot(y_46_re, y_46_im)) / hypot(y_46_re, y_46_im))
	tmp = 0.0
	if (y_46_re <= -1.15e+110)
		tmp = Float64(Float64(Float64(-x_46_re) - Float64(x_46_im / Float64(y_46_re / y_46_im))) / hypot(y_46_re, y_46_im));
	elseif (y_46_re <= -7.8e-98)
		tmp = t_0;
	elseif (y_46_re <= 9.5e-89)
		tmp = Float64(Float64(1.0 / y_46_im) * Float64(x_46_im + Float64(x_46_re / Float64(y_46_im / y_46_re))));
	elseif (y_46_re <= 8.8e+26)
		tmp = t_0;
	else
		tmp = fma(Float64(Float64(x_46_im / y_46_re) / y_46_re), y_46_im, Float64(x_46_re / y_46_re));
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[(N[(x$46$re * y$46$re + N[(x$46$im * y$46$im), $MachinePrecision]), $MachinePrecision] / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision] / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$re, -1.15e+110], N[(N[((-x$46$re) - N[(x$46$im / N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, -7.8e-98], t$95$0, If[LessEqual[y$46$re, 9.5e-89], N[(N[(1.0 / y$46$im), $MachinePrecision] * N[(x$46$im + N[(x$46$re / N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 8.8e+26], t$95$0, N[(N[(N[(x$46$im / y$46$re), $MachinePrecision] / y$46$re), $MachinePrecision] * y$46$im + N[(x$46$re / y$46$re), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if y.re < -1.15e110

   1. Initial program 36.1%

      \[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   2. Step-by-step derivation

      1. add-sqr-sqrt 36.1%

         \[\leadsto \frac{x.re \cdot y.re + x.im \cdot y.im}{\color{blue}{\sqrt{y.re \cdot y.re + y.im \cdot y.im} \cdot \sqrt{y.re \cdot y.re + y.im \cdot y.im}}} \]

      2. *-un-lft-identity 36.1%

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

      3. times-frac 36.1%

         \[\leadsto \color{blue}{\frac{1}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}} \cdot \frac{x.re \cdot y.re + x.im \cdot y.im}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}} \]

      4. hypot-def 36.1%

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

      5. fma-def 36.1%

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

      6. hypot-def 52.2%

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

   3. Applied egg-rr 52.2%

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

      1. associate-*l/ 52.2%

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

      2. *-un-lft-identity 52.2%

         \[\leadsto \frac{\color{blue}{\frac{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}{\mathsf{hypot}\left(y.re, y.im\right)}}}{\mathsf{hypot}\left(y.re, y.im\right)} \]

   5. Applied egg-rr 52.2%

      \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}{\mathsf{hypot}\left(y.re, y.im\right)}}{\mathsf{hypot}\left(y.re, y.im\right)}} \]

   6. Taylor expanded in y.re around -inf 78.5%

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

      1. distribute-lft-out 78.5%

         \[\leadsto \frac{\color{blue}{-1 \cdot \left(x.re + \frac{x.im \cdot y.im}{y.re}\right)}}{\mathsf{hypot}\left(y.re, y.im\right)} \]

      2. associate-/l* 87.6%

         \[\leadsto \frac{-1 \cdot \left(x.re + \color{blue}{\frac{x.im}{\frac{y.re}{y.im}}}\right)}{\mathsf{hypot}\left(y.re, y.im\right)} \]

   8. Simplified 87.6%

      \[\leadsto \frac{\color{blue}{-1 \cdot \left(x.re + \frac{x.im}{\frac{y.re}{y.im}}\right)}}{\mathsf{hypot}\left(y.re, y.im\right)} \]

   if -1.15e110 < y.re < -7.79999999999999943e-98 or 9.50000000000000028e-89 < y.re < 8.80000000000000028e26

   1. Initial program 78.8%

      \[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   2. Step-by-step derivation

      1. add-sqr-sqrt 78.8%

         \[\leadsto \frac{x.re \cdot y.re + x.im \cdot y.im}{\color{blue}{\sqrt{y.re \cdot y.re + y.im \cdot y.im} \cdot \sqrt{y.re \cdot y.re + y.im \cdot y.im}}} \]

      2. *-un-lft-identity 78.8%

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

      3. times-frac 78.9%

         \[\leadsto \color{blue}{\frac{1}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}} \cdot \frac{x.re \cdot y.re + x.im \cdot y.im}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}} \]

      4. hypot-def 78.9%

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

      5. fma-def 78.9%

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

      6. hypot-def 89.9%

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

   3. Applied egg-rr 89.9%

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

      1. associate-*l/ 90.1%

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

      2. *-un-lft-identity 90.1%

         \[\leadsto \frac{\color{blue}{\frac{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}{\mathsf{hypot}\left(y.re, y.im\right)}}}{\mathsf{hypot}\left(y.re, y.im\right)} \]

   5. Applied egg-rr 90.1%

      \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}{\mathsf{hypot}\left(y.re, y.im\right)}}{\mathsf{hypot}\left(y.re, y.im\right)}} \]

   if -7.79999999999999943e-98 < y.re < 9.50000000000000028e-89

   1. Initial program 75.9%

      \[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]

   2. Taylor expanded in y.re around 0 87.7%

      \[\leadsto \color{blue}{\frac{x.im}{y.im} + \frac{x.re \cdot y.re}{{y.im}^{2}}} \]
   3. Step-by-step derivation

      1. associate-/l* 90.2%

         \[\leadsto \frac{x.im}{y.im} + \color{blue}{\frac{x.re}{\frac{{y.im}^{2}}{y.re}}} \]

      2. associate-/r/ 87.9%

         \[\leadsto \frac{x.im}{y.im} + \color{blue}{\frac{x.re}{{y.im}^{2}} \cdot y.re} \]

   4. Simplified 87.9%

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

      1. pow2 87.9%

         \[\leadsto \frac{x.im}{y.im} + \frac{x.re}{\color{blue}{y.im \cdot y.im}} \cdot y.re \]

      2. associate-*l/ 87.7%

         \[\leadsto \frac{x.im}{y.im} + \color{blue}{\frac{x.re \cdot y.re}{y.im \cdot y.im}} \]

      3. associate-/r* 92.5%

         \[\leadsto \frac{x.im}{y.im} + \color{blue}{\frac{\frac{x.re \cdot y.re}{y.im}}{y.im}} \]

   6. Applied egg-rr 92.5%

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

      1. +-commutative 92.5%

         \[\leadsto \color{blue}{\frac{\frac{x.re \cdot y.re}{y.im}}{y.im} + \frac{x.im}{y.im}} \]

      2. div-inv 92.4%

         \[\leadsto \color{blue}{\frac{x.re \cdot y.re}{y.im} \cdot \frac{1}{y.im}} + \frac{x.im}{y.im} \]

      3. associate-/l* 92.4%

         \[\leadsto \color{blue}{\frac{x.re}{\frac{y.im}{y.re}}} \cdot \frac{1}{y.im} + \frac{x.im}{y.im} \]

      4. div-inv 92.3%

         \[\leadsto \frac{x.re}{\frac{y.im}{y.re}} \cdot \frac{1}{y.im} + \color{blue}{x.im \cdot \frac{1}{y.im}} \]

      5. distribute-rgt-out 93.4%

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

   8. Applied egg-rr 93.4%

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

   if 8.80000000000000028e26 < y.re

   1. Initial program 38.7%

      \[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]

   2. Taylor expanded in y.re around inf 77.2%

      \[\leadsto \color{blue}{\frac{x.re}{y.re} + \frac{x.im \cdot y.im}{{y.re}^{2}}} \]
   3. Step-by-step derivation

      1. +-commutative 77.2%

         \[\leadsto \color{blue}{\frac{x.im \cdot y.im}{{y.re}^{2}} + \frac{x.re}{y.re}} \]

      2. associate-/l* 77.5%

         \[\leadsto \color{blue}{\frac{x.im}{\frac{{y.re}^{2}}{y.im}}} + \frac{x.re}{y.re} \]

      3. associate-/r/ 79.3%

         \[\leadsto \color{blue}{\frac{x.im}{{y.re}^{2}} \cdot y.im} + \frac{x.re}{y.re} \]

      4. fma-def 79.3%

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{x.im}{{y.re}^{2}}, y.im, \frac{x.re}{y.re}\right)} \]

   4. Simplified 79.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{x.im}{{y.re}^{2}}, y.im, \frac{x.re}{y.re}\right)} \]
   5. Step-by-step derivation

      1. *-un-lft-identity 79.3%

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

      2. pow2 79.3%

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

      3. times-frac 82.8%

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

   6. Applied egg-rr 82.8%

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

      1. associate-*l/ 82.8%

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

      2. *-un-lft-identity 82.8%

         \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{\frac{x.im}{y.re}}}{y.re}, y.im, \frac{x.re}{y.re}\right) \]

   8. Applied egg-rr 82.8%

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{\frac{x.im}{y.re}}{y.re}}, y.im, \frac{x.re}{y.re}\right) \]
3. Recombined 4 regimes into one program.

4. Final simplification 89.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -1.15 \cdot 10^{+110}:\\ \;\;\;\;\frac{\left(-x.re\right) - \frac{x.im}{\frac{y.re}{y.im}}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.re \leq -7.8 \cdot 10^{-98}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}{\mathsf{hypot}\left(y.re, y.im\right)}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.re \leq 9.5 \cdot 10^{-89}:\\ \;\;\;\;\frac{1}{y.im} \cdot \left(x.im + \frac{x.re}{\frac{y.im}{y.re}}\right)\\ \mathbf{elif}\;y.re \leq 8.8 \cdot 10^{+26}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}{\mathsf{hypot}\left(y.re, y.im\right)}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\frac{x.im}{y.re}}{y.re}, y.im, \frac{x.re}{y.re}\right)\\ \end{array} \]

Alternative 2: 81.7% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x.im \cdot y.im + y.re \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ t_1 := \mathsf{fma}\left(\frac{\frac{x.im}{y.re}}{y.re}, y.im, \frac{x.re}{y.re}\right)\\ \mathbf{if}\;y.re \leq -3.2 \cdot 10^{+56}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.re \leq -1.6 \cdot 10^{-97}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq 7.5 \cdot 10^{-70}:\\ \;\;\;\;\frac{1}{y.im} \cdot \left(x.im + \frac{x.re}{\frac{y.im}{y.re}}\right)\\ \mathbf{elif}\;y.re \leq 1.4 \cdot 10^{+25}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0
         (/ (+ (* x.im y.im) (* y.re x.re)) (+ (* y.re y.re) (* y.im y.im))))
        (t_1 (fma (/ (/ x.im y.re) y.re) y.im (/ x.re y.re))))
   (if (<= y.re -3.2e+56)
     t_1
     (if (<= y.re -1.6e-97)
       t_0
       (if (<= y.re 7.5e-70)
         (* (/ 1.0 y.im) (+ x.im (/ x.re (/ y.im y.re))))
         (if (<= y.re 1.4e+25) t_0 t_1))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = ((x_46_im * y_46_im) + (y_46_re * x_46_re)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	double t_1 = fma(((x_46_im / y_46_re) / y_46_re), y_46_im, (x_46_re / y_46_re));
	double tmp;
	if (y_46_re <= -3.2e+56) {
		tmp = t_1;
	} else if (y_46_re <= -1.6e-97) {
		tmp = t_0;
	} else if (y_46_re <= 7.5e-70) {
		tmp = (1.0 / y_46_im) * (x_46_im + (x_46_re / (y_46_im / y_46_re)));
	} else if (y_46_re <= 1.4e+25) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(Float64(Float64(x_46_im * y_46_im) + Float64(y_46_re * x_46_re)) / Float64(Float64(y_46_re * y_46_re) + Float64(y_46_im * y_46_im)))
	t_1 = fma(Float64(Float64(x_46_im / y_46_re) / y_46_re), y_46_im, Float64(x_46_re / y_46_re))
	tmp = 0.0
	if (y_46_re <= -3.2e+56)
		tmp = t_1;
	elseif (y_46_re <= -1.6e-97)
		tmp = t_0;
	elseif (y_46_re <= 7.5e-70)
		tmp = Float64(Float64(1.0 / y_46_im) * Float64(x_46_im + Float64(x_46_re / Float64(y_46_im / y_46_re))));
	elseif (y_46_re <= 1.4e+25)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[(N[(x$46$im * y$46$im), $MachinePrecision] + N[(y$46$re * x$46$re), $MachinePrecision]), $MachinePrecision] / N[(N[(y$46$re * y$46$re), $MachinePrecision] + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(x$46$im / y$46$re), $MachinePrecision] / y$46$re), $MachinePrecision] * y$46$im + N[(x$46$re / y$46$re), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$re, -3.2e+56], t$95$1, If[LessEqual[y$46$re, -1.6e-97], t$95$0, If[LessEqual[y$46$re, 7.5e-70], N[(N[(1.0 / y$46$im), $MachinePrecision] * N[(x$46$im + N[(x$46$re / N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 1.4e+25], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -3.20000000000000003e56 or 1.4000000000000001e25 < y.re

   1. Initial program 39.1%

      \[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]

   2. Taylor expanded in y.re around inf 73.2%

      \[\leadsto \color{blue}{\frac{x.re}{y.re} + \frac{x.im \cdot y.im}{{y.re}^{2}}} \]
   3. Step-by-step derivation

      1. +-commutative 73.2%

         \[\leadsto \color{blue}{\frac{x.im \cdot y.im}{{y.re}^{2}} + \frac{x.re}{y.re}} \]

      2. associate-/l* 75.2%

         \[\leadsto \color{blue}{\frac{x.im}{\frac{{y.re}^{2}}{y.im}}} + \frac{x.re}{y.re} \]

      3. associate-/r/ 75.3%

         \[\leadsto \color{blue}{\frac{x.im}{{y.re}^{2}} \cdot y.im} + \frac{x.re}{y.re} \]

      4. fma-def 75.3%

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{x.im}{{y.re}^{2}}, y.im, \frac{x.re}{y.re}\right)} \]

   4. Simplified 75.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{x.im}{{y.re}^{2}}, y.im, \frac{x.re}{y.re}\right)} \]
   5. Step-by-step derivation

      1. *-un-lft-identity 75.3%

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

      2. pow2 75.3%

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

      3. times-frac 79.6%

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

   6. Applied egg-rr 79.6%

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

      1. associate-*l/ 79.6%

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

      2. *-un-lft-identity 79.6%

         \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{\frac{x.im}{y.re}}}{y.re}, y.im, \frac{x.re}{y.re}\right) \]

   8. Applied egg-rr 79.6%

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{\frac{x.im}{y.re}}{y.re}}, y.im, \frac{x.re}{y.re}\right) \]

   if -3.20000000000000003e56 < y.re < -1.5999999999999999e-97 or 7.49999999999999973e-70 < y.re < 1.4000000000000001e25

   1. Initial program 88.0%

      \[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]

   if -1.5999999999999999e-97 < y.re < 7.49999999999999973e-70

   1. Initial program 74.6%

      \[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]

   2. Taylor expanded in y.re around 0 85.6%

      \[\leadsto \color{blue}{\frac{x.im}{y.im} + \frac{x.re \cdot y.re}{{y.im}^{2}}} \]
   3. Step-by-step derivation

      1. associate-/l* 87.9%

         \[\leadsto \frac{x.im}{y.im} + \color{blue}{\frac{x.re}{\frac{{y.im}^{2}}{y.re}}} \]

      2. associate-/r/ 85.8%

         \[\leadsto \frac{x.im}{y.im} + \color{blue}{\frac{x.re}{{y.im}^{2}} \cdot y.re} \]

   4. Simplified 85.8%

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

      1. pow2 85.8%

         \[\leadsto \frac{x.im}{y.im} + \frac{x.re}{\color{blue}{y.im \cdot y.im}} \cdot y.re \]

      2. associate-*l/ 85.6%

         \[\leadsto \frac{x.im}{y.im} + \color{blue}{\frac{x.re \cdot y.re}{y.im \cdot y.im}} \]

      3. associate-/r* 91.0%

         \[\leadsto \frac{x.im}{y.im} + \color{blue}{\frac{\frac{x.re \cdot y.re}{y.im}}{y.im}} \]

   6. Applied egg-rr 91.0%

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

      1. +-commutative 91.0%

         \[\leadsto \color{blue}{\frac{\frac{x.re \cdot y.re}{y.im}}{y.im} + \frac{x.im}{y.im}} \]

      2. div-inv 91.0%

         \[\leadsto \color{blue}{\frac{x.re \cdot y.re}{y.im} \cdot \frac{1}{y.im}} + \frac{x.im}{y.im} \]

      3. associate-/l* 91.0%

         \[\leadsto \color{blue}{\frac{x.re}{\frac{y.im}{y.re}}} \cdot \frac{1}{y.im} + \frac{x.im}{y.im} \]

      4. div-inv 90.8%

         \[\leadsto \frac{x.re}{\frac{y.im}{y.re}} \cdot \frac{1}{y.im} + \color{blue}{x.im \cdot \frac{1}{y.im}} \]

      5. distribute-rgt-out 91.8%

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

   8. Applied egg-rr 91.8%

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

4. Final simplification 85.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -3.2 \cdot 10^{+56}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\frac{x.im}{y.re}}{y.re}, y.im, \frac{x.re}{y.re}\right)\\ \mathbf{elif}\;y.re \leq -1.6 \cdot 10^{-97}:\\ \;\;\;\;\frac{x.im \cdot y.im + y.re \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.re \leq 7.5 \cdot 10^{-70}:\\ \;\;\;\;\frac{1}{y.im} \cdot \left(x.im + \frac{x.re}{\frac{y.im}{y.re}}\right)\\ \mathbf{elif}\;y.re \leq 1.4 \cdot 10^{+25}:\\ \;\;\;\;\frac{x.im \cdot y.im + y.re \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\frac{x.im}{y.re}}{y.re}, y.im, \frac{x.re}{y.re}\right)\\ \end{array} \]

Alternative 3: 82.2% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x.im \cdot y.im + y.re \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{if}\;y.re \leq -2.6 \cdot 10^{+56}:\\ \;\;\;\;\frac{\left(-x.re\right) - \frac{x.im}{\frac{y.re}{y.im}}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.re \leq -3.55 \cdot 10^{-96}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq 1.05 \cdot 10^{-69}:\\ \;\;\;\;\frac{1}{y.im} \cdot \left(x.im + \frac{x.re}{\frac{y.im}{y.re}}\right)\\ \mathbf{elif}\;y.re \leq 8.8 \cdot 10^{+26}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\frac{x.im}{y.re}}{y.re}, y.im, \frac{x.re}{y.re}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0
         (/ (+ (* x.im y.im) (* y.re x.re)) (+ (* y.re y.re) (* y.im y.im)))))
   (if (<= y.re -2.6e+56)
     (/ (- (- x.re) (/ x.im (/ y.re y.im))) (hypot y.re y.im))
     (if (<= y.re -3.55e-96)
       t_0
       (if (<= y.re 1.05e-69)
         (* (/ 1.0 y.im) (+ x.im (/ x.re (/ y.im y.re))))
         (if (<= y.re 8.8e+26)
           t_0
           (fma (/ (/ x.im y.re) y.re) y.im (/ x.re y.re))))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = ((x_46_im * y_46_im) + (y_46_re * x_46_re)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	double tmp;
	if (y_46_re <= -2.6e+56) {
		tmp = (-x_46_re - (x_46_im / (y_46_re / y_46_im))) / hypot(y_46_re, y_46_im);
	} else if (y_46_re <= -3.55e-96) {
		tmp = t_0;
	} else if (y_46_re <= 1.05e-69) {
		tmp = (1.0 / y_46_im) * (x_46_im + (x_46_re / (y_46_im / y_46_re)));
	} else if (y_46_re <= 8.8e+26) {
		tmp = t_0;
	} else {
		tmp = fma(((x_46_im / y_46_re) / y_46_re), y_46_im, (x_46_re / y_46_re));
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(Float64(Float64(x_46_im * y_46_im) + Float64(y_46_re * x_46_re)) / Float64(Float64(y_46_re * y_46_re) + Float64(y_46_im * y_46_im)))
	tmp = 0.0
	if (y_46_re <= -2.6e+56)
		tmp = Float64(Float64(Float64(-x_46_re) - Float64(x_46_im / Float64(y_46_re / y_46_im))) / hypot(y_46_re, y_46_im));
	elseif (y_46_re <= -3.55e-96)
		tmp = t_0;
	elseif (y_46_re <= 1.05e-69)
		tmp = Float64(Float64(1.0 / y_46_im) * Float64(x_46_im + Float64(x_46_re / Float64(y_46_im / y_46_re))));
	elseif (y_46_re <= 8.8e+26)
		tmp = t_0;
	else
		tmp = fma(Float64(Float64(x_46_im / y_46_re) / y_46_re), y_46_im, Float64(x_46_re / y_46_re));
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[(N[(x$46$im * y$46$im), $MachinePrecision] + N[(y$46$re * x$46$re), $MachinePrecision]), $MachinePrecision] / N[(N[(y$46$re * y$46$re), $MachinePrecision] + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$re, -2.6e+56], N[(N[((-x$46$re) - N[(x$46$im / N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, -3.55e-96], t$95$0, If[LessEqual[y$46$re, 1.05e-69], N[(N[(1.0 / y$46$im), $MachinePrecision] * N[(x$46$im + N[(x$46$re / N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 8.8e+26], t$95$0, N[(N[(N[(x$46$im / y$46$re), $MachinePrecision] / y$46$re), $MachinePrecision] * y$46$im + N[(x$46$re / y$46$re), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if y.re < -2.60000000000000011e56

   1. Initial program 39.5%

      \[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   2. Step-by-step derivation

      1. add-sqr-sqrt 39.5%

         \[\leadsto \frac{x.re \cdot y.re + x.im \cdot y.im}{\color{blue}{\sqrt{y.re \cdot y.re + y.im \cdot y.im} \cdot \sqrt{y.re \cdot y.re + y.im \cdot y.im}}} \]

      2. *-un-lft-identity 39.5%

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

      3. times-frac 39.5%

         \[\leadsto \color{blue}{\frac{1}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}} \cdot \frac{x.re \cdot y.re + x.im \cdot y.im}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}} \]

      4. hypot-def 39.5%

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

      5. fma-def 39.5%

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

      6. hypot-def 57.2%

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

   3. Applied egg-rr 57.2%

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

      1. associate-*l/ 57.3%

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

      2. *-un-lft-identity 57.3%

         \[\leadsto \frac{\color{blue}{\frac{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}{\mathsf{hypot}\left(y.re, y.im\right)}}}{\mathsf{hypot}\left(y.re, y.im\right)} \]

   5. Applied egg-rr 57.3%

      \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}{\mathsf{hypot}\left(y.re, y.im\right)}}{\mathsf{hypot}\left(y.re, y.im\right)}} \]

   6. Taylor expanded in y.re around -inf 74.8%

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

      1. distribute-lft-out 74.8%

         \[\leadsto \frac{\color{blue}{-1 \cdot \left(x.re + \frac{x.im \cdot y.im}{y.re}\right)}}{\mathsf{hypot}\left(y.re, y.im\right)} \]

      2. associate-/l* 82.0%

         \[\leadsto \frac{-1 \cdot \left(x.re + \color{blue}{\frac{x.im}{\frac{y.re}{y.im}}}\right)}{\mathsf{hypot}\left(y.re, y.im\right)} \]

   8. Simplified 82.0%

      \[\leadsto \frac{\color{blue}{-1 \cdot \left(x.re + \frac{x.im}{\frac{y.re}{y.im}}\right)}}{\mathsf{hypot}\left(y.re, y.im\right)} \]

   if -2.60000000000000011e56 < y.re < -3.55000000000000019e-96 or 1.05e-69 < y.re < 8.80000000000000028e26

   1. Initial program 88.0%

      \[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]

   if -3.55000000000000019e-96 < y.re < 1.05e-69

   1. Initial program 74.6%

      \[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]

   2. Taylor expanded in y.re around 0 85.6%

      \[\leadsto \color{blue}{\frac{x.im}{y.im} + \frac{x.re \cdot y.re}{{y.im}^{2}}} \]
   3. Step-by-step derivation

      1. associate-/l* 87.9%

         \[\leadsto \frac{x.im}{y.im} + \color{blue}{\frac{x.re}{\frac{{y.im}^{2}}{y.re}}} \]

      2. associate-/r/ 85.8%

         \[\leadsto \frac{x.im}{y.im} + \color{blue}{\frac{x.re}{{y.im}^{2}} \cdot y.re} \]

   4. Simplified 85.8%

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

      1. pow2 85.8%

         \[\leadsto \frac{x.im}{y.im} + \frac{x.re}{\color{blue}{y.im \cdot y.im}} \cdot y.re \]

      2. associate-*l/ 85.6%

         \[\leadsto \frac{x.im}{y.im} + \color{blue}{\frac{x.re \cdot y.re}{y.im \cdot y.im}} \]

      3. associate-/r* 91.0%

         \[\leadsto \frac{x.im}{y.im} + \color{blue}{\frac{\frac{x.re \cdot y.re}{y.im}}{y.im}} \]

   6. Applied egg-rr 91.0%

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

      1. +-commutative 91.0%

         \[\leadsto \color{blue}{\frac{\frac{x.re \cdot y.re}{y.im}}{y.im} + \frac{x.im}{y.im}} \]

      2. div-inv 91.0%

         \[\leadsto \color{blue}{\frac{x.re \cdot y.re}{y.im} \cdot \frac{1}{y.im}} + \frac{x.im}{y.im} \]

      3. associate-/l* 91.0%

         \[\leadsto \color{blue}{\frac{x.re}{\frac{y.im}{y.re}}} \cdot \frac{1}{y.im} + \frac{x.im}{y.im} \]

      4. div-inv 90.8%

         \[\leadsto \frac{x.re}{\frac{y.im}{y.re}} \cdot \frac{1}{y.im} + \color{blue}{x.im \cdot \frac{1}{y.im}} \]

      5. distribute-rgt-out 91.8%

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

   8. Applied egg-rr 91.8%

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

   if 8.80000000000000028e26 < y.re

   1. Initial program 38.7%

      \[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]

   2. Taylor expanded in y.re around inf 77.2%

      \[\leadsto \color{blue}{\frac{x.re}{y.re} + \frac{x.im \cdot y.im}{{y.re}^{2}}} \]
   3. Step-by-step derivation

      1. +-commutative 77.2%

         \[\leadsto \color{blue}{\frac{x.im \cdot y.im}{{y.re}^{2}} + \frac{x.re}{y.re}} \]

      2. associate-/l* 77.5%

         \[\leadsto \color{blue}{\frac{x.im}{\frac{{y.re}^{2}}{y.im}}} + \frac{x.re}{y.re} \]

      3. associate-/r/ 79.3%

         \[\leadsto \color{blue}{\frac{x.im}{{y.re}^{2}} \cdot y.im} + \frac{x.re}{y.re} \]

      4. fma-def 79.3%

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{x.im}{{y.re}^{2}}, y.im, \frac{x.re}{y.re}\right)} \]

   4. Simplified 79.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{x.im}{{y.re}^{2}}, y.im, \frac{x.re}{y.re}\right)} \]
   5. Step-by-step derivation

      1. *-un-lft-identity 79.3%

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

      2. pow2 79.3%

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

      3. times-frac 82.8%

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

   6. Applied egg-rr 82.8%

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

      1. associate-*l/ 82.8%

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

      2. *-un-lft-identity 82.8%

         \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{\frac{x.im}{y.re}}}{y.re}, y.im, \frac{x.re}{y.re}\right) \]

   8. Applied egg-rr 82.8%

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{\frac{x.im}{y.re}}{y.re}}, y.im, \frac{x.re}{y.re}\right) \]
3. Recombined 4 regimes into one program.

4. Final simplification 87.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -2.6 \cdot 10^{+56}:\\ \;\;\;\;\frac{\left(-x.re\right) - \frac{x.im}{\frac{y.re}{y.im}}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.re \leq -3.55 \cdot 10^{-96}:\\ \;\;\;\;\frac{x.im \cdot y.im + y.re \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.re \leq 1.05 \cdot 10^{-69}:\\ \;\;\;\;\frac{1}{y.im} \cdot \left(x.im + \frac{x.re}{\frac{y.im}{y.re}}\right)\\ \mathbf{elif}\;y.re \leq 8.8 \cdot 10^{+26}:\\ \;\;\;\;\frac{x.im \cdot y.im + y.re \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\frac{x.im}{y.re}}{y.re}, y.im, \frac{x.re}{y.re}\right)\\ \end{array} \]

Alternative 4: 79.1% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x.im \cdot y.im + y.re \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{if}\;y.re \leq -1 \cdot 10^{+158}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{elif}\;y.re \leq -5.25 \cdot 10^{-98}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq 1.06 \cdot 10^{-69}:\\ \;\;\;\;\frac{1}{y.im} \cdot \left(x.im + \frac{x.re}{\frac{y.im}{y.re}}\right)\\ \mathbf{elif}\;y.re \leq 7.8 \cdot 10^{+76}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0
         (/ (+ (* x.im y.im) (* y.re x.re)) (+ (* y.re y.re) (* y.im y.im)))))
   (if (<= y.re -1e+158)
     (/ x.re y.re)
     (if (<= y.re -5.25e-98)
       t_0
       (if (<= y.re 1.06e-69)
         (* (/ 1.0 y.im) (+ x.im (/ x.re (/ y.im y.re))))
         (if (<= y.re 7.8e+76) t_0 (/ x.re (hypot y.re y.im))))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = ((x_46_im * y_46_im) + (y_46_re * x_46_re)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	double tmp;
	if (y_46_re <= -1e+158) {
		tmp = x_46_re / y_46_re;
	} else if (y_46_re <= -5.25e-98) {
		tmp = t_0;
	} else if (y_46_re <= 1.06e-69) {
		tmp = (1.0 / y_46_im) * (x_46_im + (x_46_re / (y_46_im / y_46_re)));
	} else if (y_46_re <= 7.8e+76) {
		tmp = t_0;
	} else {
		tmp = x_46_re / hypot(y_46_re, y_46_im);
	}
	return tmp;
}

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = ((x_46_im * y_46_im) + (y_46_re * x_46_re)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	double tmp;
	if (y_46_re <= -1e+158) {
		tmp = x_46_re / y_46_re;
	} else if (y_46_re <= -5.25e-98) {
		tmp = t_0;
	} else if (y_46_re <= 1.06e-69) {
		tmp = (1.0 / y_46_im) * (x_46_im + (x_46_re / (y_46_im / y_46_re)));
	} else if (y_46_re <= 7.8e+76) {
		tmp = t_0;
	} else {
		tmp = x_46_re / Math.hypot(y_46_re, y_46_im);
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = ((x_46_im * y_46_im) + (y_46_re * x_46_re)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im))
	tmp = 0
	if y_46_re <= -1e+158:
		tmp = x_46_re / y_46_re
	elif y_46_re <= -5.25e-98:
		tmp = t_0
	elif y_46_re <= 1.06e-69:
		tmp = (1.0 / y_46_im) * (x_46_im + (x_46_re / (y_46_im / y_46_re)))
	elif y_46_re <= 7.8e+76:
		tmp = t_0
	else:
		tmp = x_46_re / math.hypot(y_46_re, y_46_im)
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(Float64(Float64(x_46_im * y_46_im) + Float64(y_46_re * x_46_re)) / Float64(Float64(y_46_re * y_46_re) + Float64(y_46_im * y_46_im)))
	tmp = 0.0
	if (y_46_re <= -1e+158)
		tmp = Float64(x_46_re / y_46_re);
	elseif (y_46_re <= -5.25e-98)
		tmp = t_0;
	elseif (y_46_re <= 1.06e-69)
		tmp = Float64(Float64(1.0 / y_46_im) * Float64(x_46_im + Float64(x_46_re / Float64(y_46_im / y_46_re))));
	elseif (y_46_re <= 7.8e+76)
		tmp = t_0;
	else
		tmp = Float64(x_46_re / hypot(y_46_re, y_46_im));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = ((x_46_im * y_46_im) + (y_46_re * x_46_re)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	tmp = 0.0;
	if (y_46_re <= -1e+158)
		tmp = x_46_re / y_46_re;
	elseif (y_46_re <= -5.25e-98)
		tmp = t_0;
	elseif (y_46_re <= 1.06e-69)
		tmp = (1.0 / y_46_im) * (x_46_im + (x_46_re / (y_46_im / y_46_re)));
	elseif (y_46_re <= 7.8e+76)
		tmp = t_0;
	else
		tmp = x_46_re / hypot(y_46_re, y_46_im);
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[(N[(x$46$im * y$46$im), $MachinePrecision] + N[(y$46$re * x$46$re), $MachinePrecision]), $MachinePrecision] / N[(N[(y$46$re * y$46$re), $MachinePrecision] + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$re, -1e+158], N[(x$46$re / y$46$re), $MachinePrecision], If[LessEqual[y$46$re, -5.25e-98], t$95$0, If[LessEqual[y$46$re, 1.06e-69], N[(N[(1.0 / y$46$im), $MachinePrecision] * N[(x$46$im + N[(x$46$re / N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 7.8e+76], t$95$0, N[(x$46$re / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if y.re < -9.99999999999999953e157

   1. Initial program 22.9%

      \[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]

   2. Taylor expanded in y.re around inf 81.1%

      \[\leadsto \color{blue}{\frac{x.re}{y.re}} \]

   if -9.99999999999999953e157 < y.re < -5.2500000000000003e-98 or 1.05999999999999997e-69 < y.re < 7.79999999999999979e76

   1. Initial program 78.8%

      \[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]

   if -5.2500000000000003e-98 < y.re < 1.05999999999999997e-69

   1. Initial program 74.6%

      \[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]

   2. Taylor expanded in y.re around 0 85.6%

      \[\leadsto \color{blue}{\frac{x.im}{y.im} + \frac{x.re \cdot y.re}{{y.im}^{2}}} \]
   3. Step-by-step derivation

      1. associate-/l* 87.9%

         \[\leadsto \frac{x.im}{y.im} + \color{blue}{\frac{x.re}{\frac{{y.im}^{2}}{y.re}}} \]

      2. associate-/r/ 85.8%

         \[\leadsto \frac{x.im}{y.im} + \color{blue}{\frac{x.re}{{y.im}^{2}} \cdot y.re} \]

   4. Simplified 85.8%

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

      1. pow2 85.8%

         \[\leadsto \frac{x.im}{y.im} + \frac{x.re}{\color{blue}{y.im \cdot y.im}} \cdot y.re \]

      2. associate-*l/ 85.6%

         \[\leadsto \frac{x.im}{y.im} + \color{blue}{\frac{x.re \cdot y.re}{y.im \cdot y.im}} \]

      3. associate-/r* 91.0%

         \[\leadsto \frac{x.im}{y.im} + \color{blue}{\frac{\frac{x.re \cdot y.re}{y.im}}{y.im}} \]

   6. Applied egg-rr 91.0%

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

      1. +-commutative 91.0%

         \[\leadsto \color{blue}{\frac{\frac{x.re \cdot y.re}{y.im}}{y.im} + \frac{x.im}{y.im}} \]

      2. div-inv 91.0%

         \[\leadsto \color{blue}{\frac{x.re \cdot y.re}{y.im} \cdot \frac{1}{y.im}} + \frac{x.im}{y.im} \]

      3. associate-/l* 91.0%

         \[\leadsto \color{blue}{\frac{x.re}{\frac{y.im}{y.re}}} \cdot \frac{1}{y.im} + \frac{x.im}{y.im} \]

      4. div-inv 90.8%

         \[\leadsto \frac{x.re}{\frac{y.im}{y.re}} \cdot \frac{1}{y.im} + \color{blue}{x.im \cdot \frac{1}{y.im}} \]

      5. distribute-rgt-out 91.8%

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

   8. Applied egg-rr 91.8%

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

   if 7.79999999999999979e76 < y.re

   1. Initial program 31.7%

      \[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   2. Step-by-step derivation

      1. add-sqr-sqrt 31.7%

         \[\leadsto \frac{x.re \cdot y.re + x.im \cdot y.im}{\color{blue}{\sqrt{y.re \cdot y.re + y.im \cdot y.im} \cdot \sqrt{y.re \cdot y.re + y.im \cdot y.im}}} \]

      2. *-un-lft-identity 31.7%

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

      3. times-frac 31.7%

         \[\leadsto \color{blue}{\frac{1}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}} \cdot \frac{x.re \cdot y.re + x.im \cdot y.im}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}} \]

      4. hypot-def 31.7%

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

      5. fma-def 31.7%

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

      6. hypot-def 52.4%

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

   3. Applied egg-rr 52.4%

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

   4. Taylor expanded in y.re around inf 76.3%

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

      1. expm1-log1p-u 62.8%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot x.re\right)\right)} \]

      2. expm1-udef 27.6%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot x.re\right)} - 1} \]

      3. associate-*l/ 27.6%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{1 \cdot x.re}{\mathsf{hypot}\left(y.re, y.im\right)}}\right)} - 1 \]

      4. *-un-lft-identity 27.6%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{x.re}}{\mathsf{hypot}\left(y.re, y.im\right)}\right)} - 1 \]

   6. Applied egg-rr 27.6%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{x.re}{\mathsf{hypot}\left(y.re, y.im\right)}\right)} - 1} \]
   7. Step-by-step derivation

      1. expm1-def 63.1%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{x.re}{\mathsf{hypot}\left(y.re, y.im\right)}\right)\right)} \]

      2. expm1-log1p 76.6%

         \[\leadsto \color{blue}{\frac{x.re}{\mathsf{hypot}\left(y.re, y.im\right)}} \]

   8. Simplified 76.6%

      \[\leadsto \color{blue}{\frac{x.re}{\mathsf{hypot}\left(y.re, y.im\right)}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 83.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -1 \cdot 10^{+158}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{elif}\;y.re \leq -5.25 \cdot 10^{-98}:\\ \;\;\;\;\frac{x.im \cdot y.im + y.re \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.re \leq 1.06 \cdot 10^{-69}:\\ \;\;\;\;\frac{1}{y.im} \cdot \left(x.im + \frac{x.re}{\frac{y.im}{y.re}}\right)\\ \mathbf{elif}\;y.re \leq 7.8 \cdot 10^{+76}:\\ \;\;\;\;\frac{x.im \cdot y.im + y.re \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \end{array} \]

Alternative 5: 79.2% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x.im \cdot y.im + y.re \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{if}\;y.re \leq -5.8 \cdot 10^{+86}:\\ \;\;\;\;x.re \cdot \frac{-1}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.re \leq -1.22 \cdot 10^{-97}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq 2.1 \cdot 10^{-69}:\\ \;\;\;\;\frac{1}{y.im} \cdot \left(x.im + \frac{x.re}{\frac{y.im}{y.re}}\right)\\ \mathbf{elif}\;y.re \leq 6.2 \cdot 10^{+76}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0
         (/ (+ (* x.im y.im) (* y.re x.re)) (+ (* y.re y.re) (* y.im y.im)))))
   (if (<= y.re -5.8e+86)
     (* x.re (/ -1.0 (hypot y.re y.im)))
     (if (<= y.re -1.22e-97)
       t_0
       (if (<= y.re 2.1e-69)
         (* (/ 1.0 y.im) (+ x.im (/ x.re (/ y.im y.re))))
         (if (<= y.re 6.2e+76) t_0 (/ x.re (hypot y.re y.im))))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = ((x_46_im * y_46_im) + (y_46_re * x_46_re)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	double tmp;
	if (y_46_re <= -5.8e+86) {
		tmp = x_46_re * (-1.0 / hypot(y_46_re, y_46_im));
	} else if (y_46_re <= -1.22e-97) {
		tmp = t_0;
	} else if (y_46_re <= 2.1e-69) {
		tmp = (1.0 / y_46_im) * (x_46_im + (x_46_re / (y_46_im / y_46_re)));
	} else if (y_46_re <= 6.2e+76) {
		tmp = t_0;
	} else {
		tmp = x_46_re / hypot(y_46_re, y_46_im);
	}
	return tmp;
}

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = ((x_46_im * y_46_im) + (y_46_re * x_46_re)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	double tmp;
	if (y_46_re <= -5.8e+86) {
		tmp = x_46_re * (-1.0 / Math.hypot(y_46_re, y_46_im));
	} else if (y_46_re <= -1.22e-97) {
		tmp = t_0;
	} else if (y_46_re <= 2.1e-69) {
		tmp = (1.0 / y_46_im) * (x_46_im + (x_46_re / (y_46_im / y_46_re)));
	} else if (y_46_re <= 6.2e+76) {
		tmp = t_0;
	} else {
		tmp = x_46_re / Math.hypot(y_46_re, y_46_im);
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = ((x_46_im * y_46_im) + (y_46_re * x_46_re)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im))
	tmp = 0
	if y_46_re <= -5.8e+86:
		tmp = x_46_re * (-1.0 / math.hypot(y_46_re, y_46_im))
	elif y_46_re <= -1.22e-97:
		tmp = t_0
	elif y_46_re <= 2.1e-69:
		tmp = (1.0 / y_46_im) * (x_46_im + (x_46_re / (y_46_im / y_46_re)))
	elif y_46_re <= 6.2e+76:
		tmp = t_0
	else:
		tmp = x_46_re / math.hypot(y_46_re, y_46_im)
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(Float64(Float64(x_46_im * y_46_im) + Float64(y_46_re * x_46_re)) / Float64(Float64(y_46_re * y_46_re) + Float64(y_46_im * y_46_im)))
	tmp = 0.0
	if (y_46_re <= -5.8e+86)
		tmp = Float64(x_46_re * Float64(-1.0 / hypot(y_46_re, y_46_im)));
	elseif (y_46_re <= -1.22e-97)
		tmp = t_0;
	elseif (y_46_re <= 2.1e-69)
		tmp = Float64(Float64(1.0 / y_46_im) * Float64(x_46_im + Float64(x_46_re / Float64(y_46_im / y_46_re))));
	elseif (y_46_re <= 6.2e+76)
		tmp = t_0;
	else
		tmp = Float64(x_46_re / hypot(y_46_re, y_46_im));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = ((x_46_im * y_46_im) + (y_46_re * x_46_re)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	tmp = 0.0;
	if (y_46_re <= -5.8e+86)
		tmp = x_46_re * (-1.0 / hypot(y_46_re, y_46_im));
	elseif (y_46_re <= -1.22e-97)
		tmp = t_0;
	elseif (y_46_re <= 2.1e-69)
		tmp = (1.0 / y_46_im) * (x_46_im + (x_46_re / (y_46_im / y_46_re)));
	elseif (y_46_re <= 6.2e+76)
		tmp = t_0;
	else
		tmp = x_46_re / hypot(y_46_re, y_46_im);
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[(N[(x$46$im * y$46$im), $MachinePrecision] + N[(y$46$re * x$46$re), $MachinePrecision]), $MachinePrecision] / N[(N[(y$46$re * y$46$re), $MachinePrecision] + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$re, -5.8e+86], N[(x$46$re * N[(-1.0 / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, -1.22e-97], t$95$0, If[LessEqual[y$46$re, 2.1e-69], N[(N[(1.0 / y$46$im), $MachinePrecision] * N[(x$46$im + N[(x$46$re / N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 6.2e+76], t$95$0, N[(x$46$re / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if y.re < -5.79999999999999981e86

   1. Initial program 34.1%

      \[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   2. Step-by-step derivation

      1. add-sqr-sqrt 34.1%

         \[\leadsto \frac{x.re \cdot y.re + x.im \cdot y.im}{\color{blue}{\sqrt{y.re \cdot y.re + y.im \cdot y.im} \cdot \sqrt{y.re \cdot y.re + y.im \cdot y.im}}} \]

      2. *-un-lft-identity 34.1%

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

      3. times-frac 34.1%

         \[\leadsto \color{blue}{\frac{1}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}} \cdot \frac{x.re \cdot y.re + x.im \cdot y.im}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}} \]

      4. hypot-def 34.1%

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

      5. fma-def 34.1%

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

      6. hypot-def 53.1%

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

   3. Applied egg-rr 53.1%

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

   4. Taylor expanded in y.re around -inf 74.0%

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

      1. neg-mul-1 74.0%

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

   6. Simplified 74.0%

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

   if -5.79999999999999981e86 < y.re < -1.22e-97 or 2.1e-69 < y.re < 6.20000000000000023e76

   1. Initial program 85.0%

      \[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]

   if -1.22e-97 < y.re < 2.1e-69

   1. Initial program 74.6%

      \[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]

   2. Taylor expanded in y.re around 0 85.6%

      \[\leadsto \color{blue}{\frac{x.im}{y.im} + \frac{x.re \cdot y.re}{{y.im}^{2}}} \]
   3. Step-by-step derivation

      1. associate-/l* 87.9%

         \[\leadsto \frac{x.im}{y.im} + \color{blue}{\frac{x.re}{\frac{{y.im}^{2}}{y.re}}} \]

      2. associate-/r/ 85.8%

         \[\leadsto \frac{x.im}{y.im} + \color{blue}{\frac{x.re}{{y.im}^{2}} \cdot y.re} \]

   4. Simplified 85.8%

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

      1. pow2 85.8%

         \[\leadsto \frac{x.im}{y.im} + \frac{x.re}{\color{blue}{y.im \cdot y.im}} \cdot y.re \]

      2. associate-*l/ 85.6%

         \[\leadsto \frac{x.im}{y.im} + \color{blue}{\frac{x.re \cdot y.re}{y.im \cdot y.im}} \]

      3. associate-/r* 91.0%

         \[\leadsto \frac{x.im}{y.im} + \color{blue}{\frac{\frac{x.re \cdot y.re}{y.im}}{y.im}} \]

   6. Applied egg-rr 91.0%

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

      1. +-commutative 91.0%

         \[\leadsto \color{blue}{\frac{\frac{x.re \cdot y.re}{y.im}}{y.im} + \frac{x.im}{y.im}} \]

      2. div-inv 91.0%

         \[\leadsto \color{blue}{\frac{x.re \cdot y.re}{y.im} \cdot \frac{1}{y.im}} + \frac{x.im}{y.im} \]

      3. associate-/l* 91.0%

         \[\leadsto \color{blue}{\frac{x.re}{\frac{y.im}{y.re}}} \cdot \frac{1}{y.im} + \frac{x.im}{y.im} \]

      4. div-inv 90.8%

         \[\leadsto \frac{x.re}{\frac{y.im}{y.re}} \cdot \frac{1}{y.im} + \color{blue}{x.im \cdot \frac{1}{y.im}} \]

      5. distribute-rgt-out 91.8%

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

   8. Applied egg-rr 91.8%

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

   if 6.20000000000000023e76 < y.re

   1. Initial program 31.7%

      \[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   2. Step-by-step derivation

      1. add-sqr-sqrt 31.7%

         \[\leadsto \frac{x.re \cdot y.re + x.im \cdot y.im}{\color{blue}{\sqrt{y.re \cdot y.re + y.im \cdot y.im} \cdot \sqrt{y.re \cdot y.re + y.im \cdot y.im}}} \]

      2. *-un-lft-identity 31.7%

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

      3. times-frac 31.7%

         \[\leadsto \color{blue}{\frac{1}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}} \cdot \frac{x.re \cdot y.re + x.im \cdot y.im}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}} \]

      4. hypot-def 31.7%

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

      5. fma-def 31.7%

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

      6. hypot-def 52.4%

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

   3. Applied egg-rr 52.4%

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

   4. Taylor expanded in y.re around inf 76.3%

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

      1. expm1-log1p-u 62.8%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot x.re\right)\right)} \]

      2. expm1-udef 27.6%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot x.re\right)} - 1} \]

      3. associate-*l/ 27.6%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{1 \cdot x.re}{\mathsf{hypot}\left(y.re, y.im\right)}}\right)} - 1 \]

      4. *-un-lft-identity 27.6%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{x.re}}{\mathsf{hypot}\left(y.re, y.im\right)}\right)} - 1 \]

   6. Applied egg-rr 27.6%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{x.re}{\mathsf{hypot}\left(y.re, y.im\right)}\right)} - 1} \]
   7. Step-by-step derivation

      1. expm1-def 63.1%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{x.re}{\mathsf{hypot}\left(y.re, y.im\right)}\right)\right)} \]

      2. expm1-log1p 76.6%

         \[\leadsto \color{blue}{\frac{x.re}{\mathsf{hypot}\left(y.re, y.im\right)}} \]

   8. Simplified 76.6%

      \[\leadsto \color{blue}{\frac{x.re}{\mathsf{hypot}\left(y.re, y.im\right)}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 84.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -5.8 \cdot 10^{+86}:\\ \;\;\;\;x.re \cdot \frac{-1}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.re \leq -1.22 \cdot 10^{-97}:\\ \;\;\;\;\frac{x.im \cdot y.im + y.re \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.re \leq 2.1 \cdot 10^{-69}:\\ \;\;\;\;\frac{1}{y.im} \cdot \left(x.im + \frac{x.re}{\frac{y.im}{y.re}}\right)\\ \mathbf{elif}\;y.re \leq 6.2 \cdot 10^{+76}:\\ \;\;\;\;\frac{x.im \cdot y.im + y.re \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \end{array} \]

Alternative 6: 78.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x.im \cdot y.im + y.re \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{if}\;y.re \leq -1 \cdot 10^{+158}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{elif}\;y.re \leq -7.8 \cdot 10^{-98}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq 8.2 \cdot 10^{-70}:\\ \;\;\;\;\frac{1}{y.im} \cdot \left(x.im + \frac{x.re}{\frac{y.im}{y.re}}\right)\\ \mathbf{elif}\;y.re \leq 7.8 \cdot 10^{+76}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0
         (/ (+ (* x.im y.im) (* y.re x.re)) (+ (* y.re y.re) (* y.im y.im)))))
   (if (<= y.re -1e+158)
     (/ x.re y.re)
     (if (<= y.re -7.8e-98)
       t_0
       (if (<= y.re 8.2e-70)
         (* (/ 1.0 y.im) (+ x.im (/ x.re (/ y.im y.re))))
         (if (<= y.re 7.8e+76) t_0 (/ x.re y.re)))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = ((x_46_im * y_46_im) + (y_46_re * x_46_re)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	double tmp;
	if (y_46_re <= -1e+158) {
		tmp = x_46_re / y_46_re;
	} else if (y_46_re <= -7.8e-98) {
		tmp = t_0;
	} else if (y_46_re <= 8.2e-70) {
		tmp = (1.0 / y_46_im) * (x_46_im + (x_46_re / (y_46_im / y_46_re)));
	} else if (y_46_re <= 7.8e+76) {
		tmp = t_0;
	} else {
		tmp = x_46_re / y_46_re;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ((x_46im * y_46im) + (y_46re * x_46re)) / ((y_46re * y_46re) + (y_46im * y_46im))
    if (y_46re <= (-1d+158)) then
        tmp = x_46re / y_46re
    else if (y_46re <= (-7.8d-98)) then
        tmp = t_0
    else if (y_46re <= 8.2d-70) then
        tmp = (1.0d0 / y_46im) * (x_46im + (x_46re / (y_46im / y_46re)))
    else if (y_46re <= 7.8d+76) then
        tmp = t_0
    else
        tmp = x_46re / y_46re
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = ((x_46_im * y_46_im) + (y_46_re * x_46_re)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	double tmp;
	if (y_46_re <= -1e+158) {
		tmp = x_46_re / y_46_re;
	} else if (y_46_re <= -7.8e-98) {
		tmp = t_0;
	} else if (y_46_re <= 8.2e-70) {
		tmp = (1.0 / y_46_im) * (x_46_im + (x_46_re / (y_46_im / y_46_re)));
	} else if (y_46_re <= 7.8e+76) {
		tmp = t_0;
	} else {
		tmp = x_46_re / y_46_re;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = ((x_46_im * y_46_im) + (y_46_re * x_46_re)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im))
	tmp = 0
	if y_46_re <= -1e+158:
		tmp = x_46_re / y_46_re
	elif y_46_re <= -7.8e-98:
		tmp = t_0
	elif y_46_re <= 8.2e-70:
		tmp = (1.0 / y_46_im) * (x_46_im + (x_46_re / (y_46_im / y_46_re)))
	elif y_46_re <= 7.8e+76:
		tmp = t_0
	else:
		tmp = x_46_re / y_46_re
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(Float64(Float64(x_46_im * y_46_im) + Float64(y_46_re * x_46_re)) / Float64(Float64(y_46_re * y_46_re) + Float64(y_46_im * y_46_im)))
	tmp = 0.0
	if (y_46_re <= -1e+158)
		tmp = Float64(x_46_re / y_46_re);
	elseif (y_46_re <= -7.8e-98)
		tmp = t_0;
	elseif (y_46_re <= 8.2e-70)
		tmp = Float64(Float64(1.0 / y_46_im) * Float64(x_46_im + Float64(x_46_re / Float64(y_46_im / y_46_re))));
	elseif (y_46_re <= 7.8e+76)
		tmp = t_0;
	else
		tmp = Float64(x_46_re / y_46_re);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = ((x_46_im * y_46_im) + (y_46_re * x_46_re)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	tmp = 0.0;
	if (y_46_re <= -1e+158)
		tmp = x_46_re / y_46_re;
	elseif (y_46_re <= -7.8e-98)
		tmp = t_0;
	elseif (y_46_re <= 8.2e-70)
		tmp = (1.0 / y_46_im) * (x_46_im + (x_46_re / (y_46_im / y_46_re)));
	elseif (y_46_re <= 7.8e+76)
		tmp = t_0;
	else
		tmp = x_46_re / y_46_re;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[(N[(x$46$im * y$46$im), $MachinePrecision] + N[(y$46$re * x$46$re), $MachinePrecision]), $MachinePrecision] / N[(N[(y$46$re * y$46$re), $MachinePrecision] + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$re, -1e+158], N[(x$46$re / y$46$re), $MachinePrecision], If[LessEqual[y$46$re, -7.8e-98], t$95$0, If[LessEqual[y$46$re, 8.2e-70], N[(N[(1.0 / y$46$im), $MachinePrecision] * N[(x$46$im + N[(x$46$re / N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 7.8e+76], t$95$0, N[(x$46$re / y$46$re), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -9.99999999999999953e157 or 7.79999999999999979e76 < y.re

   1. Initial program 28.2%

      \[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]

   2. Taylor expanded in y.re around inf 78.2%

      \[\leadsto \color{blue}{\frac{x.re}{y.re}} \]

   if -9.99999999999999953e157 < y.re < -7.79999999999999943e-98 or 8.19999999999999955e-70 < y.re < 7.79999999999999979e76

   1. Initial program 78.8%

      \[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]

   if -7.79999999999999943e-98 < y.re < 8.19999999999999955e-70

   1. Initial program 74.6%

      \[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]

   2. Taylor expanded in y.re around 0 85.6%

      \[\leadsto \color{blue}{\frac{x.im}{y.im} + \frac{x.re \cdot y.re}{{y.im}^{2}}} \]
   3. Step-by-step derivation

      1. associate-/l* 87.9%

         \[\leadsto \frac{x.im}{y.im} + \color{blue}{\frac{x.re}{\frac{{y.im}^{2}}{y.re}}} \]

      2. associate-/r/ 85.8%

         \[\leadsto \frac{x.im}{y.im} + \color{blue}{\frac{x.re}{{y.im}^{2}} \cdot y.re} \]

   4. Simplified 85.8%

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

      1. pow2 85.8%

         \[\leadsto \frac{x.im}{y.im} + \frac{x.re}{\color{blue}{y.im \cdot y.im}} \cdot y.re \]

      2. associate-*l/ 85.6%

         \[\leadsto \frac{x.im}{y.im} + \color{blue}{\frac{x.re \cdot y.re}{y.im \cdot y.im}} \]

      3. associate-/r* 91.0%

         \[\leadsto \frac{x.im}{y.im} + \color{blue}{\frac{\frac{x.re \cdot y.re}{y.im}}{y.im}} \]

   6. Applied egg-rr 91.0%

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

      1. +-commutative 91.0%

         \[\leadsto \color{blue}{\frac{\frac{x.re \cdot y.re}{y.im}}{y.im} + \frac{x.im}{y.im}} \]

      2. div-inv 91.0%

         \[\leadsto \color{blue}{\frac{x.re \cdot y.re}{y.im} \cdot \frac{1}{y.im}} + \frac{x.im}{y.im} \]

      3. associate-/l* 91.0%

         \[\leadsto \color{blue}{\frac{x.re}{\frac{y.im}{y.re}}} \cdot \frac{1}{y.im} + \frac{x.im}{y.im} \]

      4. div-inv 90.8%

         \[\leadsto \frac{x.re}{\frac{y.im}{y.re}} \cdot \frac{1}{y.im} + \color{blue}{x.im \cdot \frac{1}{y.im}} \]

      5. distribute-rgt-out 91.8%

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

   8. Applied egg-rr 91.8%

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

4. Final simplification 83.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -1 \cdot 10^{+158}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{elif}\;y.re \leq -7.8 \cdot 10^{-98}:\\ \;\;\;\;\frac{x.im \cdot y.im + y.re \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.re \leq 8.2 \cdot 10^{-70}:\\ \;\;\;\;\frac{1}{y.im} \cdot \left(x.im + \frac{x.re}{\frac{y.im}{y.re}}\right)\\ \mathbf{elif}\;y.re \leq 7.8 \cdot 10^{+76}:\\ \;\;\;\;\frac{x.im \cdot y.im + y.re \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \end{array} \]

Alternative 7: 71.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y.re \cdot y.re + y.im \cdot y.im\\ \mathbf{if}\;y.re \leq -1.6 \cdot 10^{+61}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{elif}\;y.re \leq 3.6 \cdot 10^{-68}:\\ \;\;\;\;\frac{1}{y.im} \cdot \left(x.im + \frac{x.re}{\frac{y.im}{y.re}}\right)\\ \mathbf{elif}\;y.re \leq 1.02 \cdot 10^{-13}:\\ \;\;\;\;\frac{y.re \cdot x.re}{t_0}\\ \mathbf{elif}\;y.re \leq 4 \cdot 10^{+76}:\\ \;\;\;\;\frac{x.im \cdot y.im}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (+ (* y.re y.re) (* y.im y.im))))
   (if (<= y.re -1.6e+61)
     (/ x.re y.re)
     (if (<= y.re 3.6e-68)
       (* (/ 1.0 y.im) (+ x.im (/ x.re (/ y.im y.re))))
       (if (<= y.re 1.02e-13)
         (/ (* y.re x.re) t_0)
         (if (<= y.re 4e+76) (/ (* x.im y.im) t_0) (/ x.re y.re)))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = (y_46_re * y_46_re) + (y_46_im * y_46_im);
	double tmp;
	if (y_46_re <= -1.6e+61) {
		tmp = x_46_re / y_46_re;
	} else if (y_46_re <= 3.6e-68) {
		tmp = (1.0 / y_46_im) * (x_46_im + (x_46_re / (y_46_im / y_46_re)));
	} else if (y_46_re <= 1.02e-13) {
		tmp = (y_46_re * x_46_re) / t_0;
	} else if (y_46_re <= 4e+76) {
		tmp = (x_46_im * y_46_im) / t_0;
	} else {
		tmp = x_46_re / y_46_re;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (y_46re * y_46re) + (y_46im * y_46im)
    if (y_46re <= (-1.6d+61)) then
        tmp = x_46re / y_46re
    else if (y_46re <= 3.6d-68) then
        tmp = (1.0d0 / y_46im) * (x_46im + (x_46re / (y_46im / y_46re)))
    else if (y_46re <= 1.02d-13) then
        tmp = (y_46re * x_46re) / t_0
    else if (y_46re <= 4d+76) then
        tmp = (x_46im * y_46im) / t_0
    else
        tmp = x_46re / y_46re
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = (y_46_re * y_46_re) + (y_46_im * y_46_im);
	double tmp;
	if (y_46_re <= -1.6e+61) {
		tmp = x_46_re / y_46_re;
	} else if (y_46_re <= 3.6e-68) {
		tmp = (1.0 / y_46_im) * (x_46_im + (x_46_re / (y_46_im / y_46_re)));
	} else if (y_46_re <= 1.02e-13) {
		tmp = (y_46_re * x_46_re) / t_0;
	} else if (y_46_re <= 4e+76) {
		tmp = (x_46_im * y_46_im) / t_0;
	} else {
		tmp = x_46_re / y_46_re;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = (y_46_re * y_46_re) + (y_46_im * y_46_im)
	tmp = 0
	if y_46_re <= -1.6e+61:
		tmp = x_46_re / y_46_re
	elif y_46_re <= 3.6e-68:
		tmp = (1.0 / y_46_im) * (x_46_im + (x_46_re / (y_46_im / y_46_re)))
	elif y_46_re <= 1.02e-13:
		tmp = (y_46_re * x_46_re) / t_0
	elif y_46_re <= 4e+76:
		tmp = (x_46_im * y_46_im) / t_0
	else:
		tmp = x_46_re / y_46_re
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(Float64(y_46_re * y_46_re) + Float64(y_46_im * y_46_im))
	tmp = 0.0
	if (y_46_re <= -1.6e+61)
		tmp = Float64(x_46_re / y_46_re);
	elseif (y_46_re <= 3.6e-68)
		tmp = Float64(Float64(1.0 / y_46_im) * Float64(x_46_im + Float64(x_46_re / Float64(y_46_im / y_46_re))));
	elseif (y_46_re <= 1.02e-13)
		tmp = Float64(Float64(y_46_re * x_46_re) / t_0);
	elseif (y_46_re <= 4e+76)
		tmp = Float64(Float64(x_46_im * y_46_im) / t_0);
	else
		tmp = Float64(x_46_re / y_46_re);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = (y_46_re * y_46_re) + (y_46_im * y_46_im);
	tmp = 0.0;
	if (y_46_re <= -1.6e+61)
		tmp = x_46_re / y_46_re;
	elseif (y_46_re <= 3.6e-68)
		tmp = (1.0 / y_46_im) * (x_46_im + (x_46_re / (y_46_im / y_46_re)));
	elseif (y_46_re <= 1.02e-13)
		tmp = (y_46_re * x_46_re) / t_0;
	elseif (y_46_re <= 4e+76)
		tmp = (x_46_im * y_46_im) / t_0;
	else
		tmp = x_46_re / y_46_re;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[(y$46$re * y$46$re), $MachinePrecision] + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$re, -1.6e+61], N[(x$46$re / y$46$re), $MachinePrecision], If[LessEqual[y$46$re, 3.6e-68], N[(N[(1.0 / y$46$im), $MachinePrecision] * N[(x$46$im + N[(x$46$re / N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 1.02e-13], N[(N[(y$46$re * x$46$re), $MachinePrecision] / t$95$0), $MachinePrecision], If[LessEqual[y$46$re, 4e+76], N[(N[(x$46$im * y$46$im), $MachinePrecision] / t$95$0), $MachinePrecision], N[(x$46$re / y$46$re), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if y.re < -1.5999999999999999e61 or 4.0000000000000002e76 < y.re

   1. Initial program 36.3%

      \[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]

   2. Taylor expanded in y.re around inf 72.6%

      \[\leadsto \color{blue}{\frac{x.re}{y.re}} \]

   if -1.5999999999999999e61 < y.re < 3.60000000000000007e-68

   1. Initial program 77.4%

      \[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]

   2. Taylor expanded in y.re around 0 75.8%

      \[\leadsto \color{blue}{\frac{x.im}{y.im} + \frac{x.re \cdot y.re}{{y.im}^{2}}} \]
   3. Step-by-step derivation

      1. associate-/l* 78.3%

         \[\leadsto \frac{x.im}{y.im} + \color{blue}{\frac{x.re}{\frac{{y.im}^{2}}{y.re}}} \]

      2. associate-/r/ 76.0%

         \[\leadsto \frac{x.im}{y.im} + \color{blue}{\frac{x.re}{{y.im}^{2}} \cdot y.re} \]

   4. Simplified 76.0%

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

      1. pow2 76.0%

         \[\leadsto \frac{x.im}{y.im} + \frac{x.re}{\color{blue}{y.im \cdot y.im}} \cdot y.re \]

      2. associate-*l/ 75.8%

         \[\leadsto \frac{x.im}{y.im} + \color{blue}{\frac{x.re \cdot y.re}{y.im \cdot y.im}} \]

      3. associate-/r* 80.7%

         \[\leadsto \frac{x.im}{y.im} + \color{blue}{\frac{\frac{x.re \cdot y.re}{y.im}}{y.im}} \]

   6. Applied egg-rr 80.7%

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

      1. +-commutative 80.7%

         \[\leadsto \color{blue}{\frac{\frac{x.re \cdot y.re}{y.im}}{y.im} + \frac{x.im}{y.im}} \]

      2. div-inv 80.7%

         \[\leadsto \color{blue}{\frac{x.re \cdot y.re}{y.im} \cdot \frac{1}{y.im}} + \frac{x.im}{y.im} \]

      3. associate-/l* 81.4%

         \[\leadsto \color{blue}{\frac{x.re}{\frac{y.im}{y.re}}} \cdot \frac{1}{y.im} + \frac{x.im}{y.im} \]

      4. div-inv 81.3%

         \[\leadsto \frac{x.re}{\frac{y.im}{y.re}} \cdot \frac{1}{y.im} + \color{blue}{x.im \cdot \frac{1}{y.im}} \]

      5. distribute-rgt-out 82.2%

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

   8. Applied egg-rr 82.2%

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

   if 3.60000000000000007e-68 < y.re < 1.0199999999999999e-13

   1. Initial program 86.9%

      \[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]

   2. Taylor expanded in x.re around inf 80.5%

      \[\leadsto \frac{\color{blue}{x.re \cdot y.re}}{y.re \cdot y.re + y.im \cdot y.im} \]
   3. Step-by-step derivation

      1. *-commutative 80.5%

         \[\leadsto \frac{\color{blue}{y.re \cdot x.re}}{y.re \cdot y.re + y.im \cdot y.im} \]

   4. Simplified 80.5%

      \[\leadsto \frac{\color{blue}{y.re \cdot x.re}}{y.re \cdot y.re + y.im \cdot y.im} \]

   if 1.0199999999999999e-13 < y.re < 4.0000000000000002e76

   1. Initial program 85.4%

      \[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]

   2. Taylor expanded in x.re around 0 63.3%

      \[\leadsto \frac{\color{blue}{x.im \cdot y.im}}{y.re \cdot y.re + y.im \cdot y.im} \]
3. Recombined 4 regimes into one program.

4. Final simplification 77.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -1.6 \cdot 10^{+61}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{elif}\;y.re \leq 3.6 \cdot 10^{-68}:\\ \;\;\;\;\frac{1}{y.im} \cdot \left(x.im + \frac{x.re}{\frac{y.im}{y.re}}\right)\\ \mathbf{elif}\;y.re \leq 1.02 \cdot 10^{-13}:\\ \;\;\;\;\frac{y.re \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.re \leq 4 \cdot 10^{+76}:\\ \;\;\;\;\frac{x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \end{array} \]

Alternative 8: 72.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -3.5 \cdot 10^{+56} \lor \neg \left(y.re \leq 9 \cdot 10^{-21}\right):\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y.im} \cdot \left(x.im + \frac{x.re}{\frac{y.im}{y.re}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.re -3.5e+56) (not (<= y.re 9e-21)))
   (/ x.re y.re)
   (* (/ 1.0 y.im) (+ x.im (/ x.re (/ y.im y.re))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_re <= -3.5e+56) || !(y_46_re <= 9e-21)) {
		tmp = x_46_re / y_46_re;
	} else {
		tmp = (1.0 / y_46_im) * (x_46_im + (x_46_re / (y_46_im / y_46_re)));
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if ((y_46re <= (-3.5d+56)) .or. (.not. (y_46re <= 9d-21))) then
        tmp = x_46re / y_46re
    else
        tmp = (1.0d0 / y_46im) * (x_46im + (x_46re / (y_46im / y_46re)))
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_re <= -3.5e+56) || !(y_46_re <= 9e-21)) {
		tmp = x_46_re / y_46_re;
	} else {
		tmp = (1.0 / y_46_im) * (x_46_im + (x_46_re / (y_46_im / y_46_re)));
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if (y_46_re <= -3.5e+56) or not (y_46_re <= 9e-21):
		tmp = x_46_re / y_46_re
	else:
		tmp = (1.0 / y_46_im) * (x_46_im + (x_46_re / (y_46_im / y_46_re)))
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if ((y_46_re <= -3.5e+56) || !(y_46_re <= 9e-21))
		tmp = Float64(x_46_re / y_46_re);
	else
		tmp = Float64(Float64(1.0 / y_46_im) * Float64(x_46_im + Float64(x_46_re / Float64(y_46_im / y_46_re))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if ((y_46_re <= -3.5e+56) || ~((y_46_re <= 9e-21)))
		tmp = x_46_re / y_46_re;
	else
		tmp = (1.0 / y_46_im) * (x_46_im + (x_46_re / (y_46_im / y_46_re)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[Or[LessEqual[y$46$re, -3.5e+56], N[Not[LessEqual[y$46$re, 9e-21]], $MachinePrecision]], N[(x$46$re / y$46$re), $MachinePrecision], N[(N[(1.0 / y$46$im), $MachinePrecision] * N[(x$46$im + N[(x$46$re / N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.re < -3.49999999999999999e56 or 8.99999999999999936e-21 < y.re

   1. Initial program 43.6%

      \[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]

   2. Taylor expanded in y.re around inf 67.8%

      \[\leadsto \color{blue}{\frac{x.re}{y.re}} \]

   if -3.49999999999999999e56 < y.re < 8.99999999999999936e-21

   1. Initial program 77.9%

      \[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]

   2. Taylor expanded in y.re around 0 74.5%

      \[\leadsto \color{blue}{\frac{x.im}{y.im} + \frac{x.re \cdot y.re}{{y.im}^{2}}} \]
   3. Step-by-step derivation

      1. associate-/l* 76.8%

         \[\leadsto \frac{x.im}{y.im} + \color{blue}{\frac{x.re}{\frac{{y.im}^{2}}{y.re}}} \]

      2. associate-/r/ 74.7%

         \[\leadsto \frac{x.im}{y.im} + \color{blue}{\frac{x.re}{{y.im}^{2}} \cdot y.re} \]

   4. Simplified 74.7%

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

      1. pow2 74.7%

         \[\leadsto \frac{x.im}{y.im} + \frac{x.re}{\color{blue}{y.im \cdot y.im}} \cdot y.re \]

      2. associate-*l/ 74.5%

         \[\leadsto \frac{x.im}{y.im} + \color{blue}{\frac{x.re \cdot y.re}{y.im \cdot y.im}} \]

      3. associate-/r* 79.0%

         \[\leadsto \frac{x.im}{y.im} + \color{blue}{\frac{\frac{x.re \cdot y.re}{y.im}}{y.im}} \]

   6. Applied egg-rr 79.0%

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

      1. +-commutative 79.0%

         \[\leadsto \color{blue}{\frac{\frac{x.re \cdot y.re}{y.im}}{y.im} + \frac{x.im}{y.im}} \]

      2. div-inv 78.9%

         \[\leadsto \color{blue}{\frac{x.re \cdot y.re}{y.im} \cdot \frac{1}{y.im}} + \frac{x.im}{y.im} \]

      3. associate-/l* 79.7%

         \[\leadsto \color{blue}{\frac{x.re}{\frac{y.im}{y.re}}} \cdot \frac{1}{y.im} + \frac{x.im}{y.im} \]

      4. div-inv 79.5%

         \[\leadsto \frac{x.re}{\frac{y.im}{y.re}} \cdot \frac{1}{y.im} + \color{blue}{x.im \cdot \frac{1}{y.im}} \]

      5. distribute-rgt-out 80.3%

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

   8. Applied egg-rr 80.3%

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

4. Final simplification 74.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -3.5 \cdot 10^{+56} \lor \neg \left(y.re \leq 9 \cdot 10^{-21}\right):\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y.im} \cdot \left(x.im + \frac{x.re}{\frac{y.im}{y.re}}\right)\\ \end{array} \]

Alternative 9: 62.7% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -3.5 \cdot 10^{-80} \lor \neg \left(y.re \leq 5.6 \cdot 10^{-21}\right):\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.re -3.5e-80) (not (<= y.re 5.6e-21)))
   (/ x.re y.re)
   (/ x.im y.im)))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_re <= -3.5e-80) || !(y_46_re <= 5.6e-21)) {
		tmp = x_46_re / y_46_re;
	} else {
		tmp = x_46_im / y_46_im;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if ((y_46re <= (-3.5d-80)) .or. (.not. (y_46re <= 5.6d-21))) then
        tmp = x_46re / y_46re
    else
        tmp = x_46im / y_46im
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_re <= -3.5e-80) || !(y_46_re <= 5.6e-21)) {
		tmp = x_46_re / y_46_re;
	} else {
		tmp = x_46_im / y_46_im;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if (y_46_re <= -3.5e-80) or not (y_46_re <= 5.6e-21):
		tmp = x_46_re / y_46_re
	else:
		tmp = x_46_im / y_46_im
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if ((y_46_re <= -3.5e-80) || !(y_46_re <= 5.6e-21))
		tmp = Float64(x_46_re / y_46_re);
	else
		tmp = Float64(x_46_im / y_46_im);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if ((y_46_re <= -3.5e-80) || ~((y_46_re <= 5.6e-21)))
		tmp = x_46_re / y_46_re;
	else
		tmp = x_46_im / y_46_im;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[Or[LessEqual[y$46$re, -3.5e-80], N[Not[LessEqual[y$46$re, 5.6e-21]], $MachinePrecision]], N[(x$46$re / y$46$re), $MachinePrecision], N[(x$46$im / y$46$im), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.re < -3.50000000000000015e-80 or 5.60000000000000008e-21 < y.re

   1. Initial program 50.3%

      \[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]

   2. Taylor expanded in y.re around inf 64.5%

      \[\leadsto \color{blue}{\frac{x.re}{y.re}} \]

   if -3.50000000000000015e-80 < y.re < 5.60000000000000008e-21

   1. Initial program 76.7%

      \[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]

   2. Taylor expanded in y.re around 0 67.4%

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

4. Final simplification 65.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -3.5 \cdot 10^{-80} \lor \neg \left(y.re \leq 5.6 \cdot 10^{-21}\right):\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \end{array} \]

Alternative 10: 42.6% accurate, 5.0× speedup

Math

\[\begin{array}{l} \\ \frac{x.im}{y.im} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im) :precision binary64 (/ x.im y.im))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	return x_46_im / y_46_im;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    code = x_46im / y_46im
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	return x_46_im / y_46_im;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	return x_46_im / y_46_im

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	return Float64(x_46_im / y_46_im)
end

MATLAB

function tmp = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = x_46_im / y_46_im;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := N[(x$46$im / y$46$im), $MachinePrecision]

Derivation

1. Initial program 61.8%

   \[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]

2. Taylor expanded in y.re around 0 38.7%

   \[\leadsto \color{blue}{\frac{x.im}{y.im}} \]

3. Final simplification 38.7%

   \[\leadsto \frac{x.im}{y.im} \]

Reproduce

herbie shell --seed 2023322 
(FPCore (x.re x.im y.re y.im)
  :name "_divideComplex, real part"
  :precision binary64
  (/ (+ (* x.re y.re) (* x.im y.im)) (+ (* y.re y.re) (* y.im y.im))))