_divideComplex, real part

Percentage Accurate: 62.2% → 88.1%

Time: 12.3s

Alternatives: 12

Speedup: 1.1×

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: 88.1% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -6.6 \cdot 10^{+108}:\\ \;\;\;\;\frac{x.re + x.im \cdot \left(\frac{1}{y.re} \cdot y.im\right)}{y.re}\\ \mathbf{elif}\;y.re \leq 1.9 \cdot 10^{+176}:\\ \;\;\;\;\frac{y.im \cdot \frac{\mathsf{fma}\left(x.re, \frac{y.re}{y.im}, x.im\right)}{\mathsf{hypot}\left(y.re, y.im\right)}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re + \left(x.im \cdot \frac{y.im}{y.re} - x.re \cdot {\left(\frac{y.im}{y.re}\right)}^{2}\right)}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.re -6.6e+108)
   (/ (+ x.re (* x.im (* (/ 1.0 y.re) y.im))) y.re)
   (if (<= y.re 1.9e+176)
     (/
      (* y.im (/ (fma x.re (/ y.re y.im) x.im) (hypot y.re y.im)))
      (hypot y.re y.im))
     (/
      (+ x.re (- (* x.im (/ y.im y.re)) (* x.re (pow (/ y.im y.re) 2.0))))
      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 <= -6.6e+108) {
		tmp = (x_46_re + (x_46_im * ((1.0 / y_46_re) * y_46_im))) / y_46_re;
	} else if (y_46_re <= 1.9e+176) {
		tmp = (y_46_im * (fma(x_46_re, (y_46_re / y_46_im), x_46_im) / hypot(y_46_re, y_46_im))) / hypot(y_46_re, y_46_im);
	} else {
		tmp = (x_46_re + ((x_46_im * (y_46_im / y_46_re)) - (x_46_re * pow((y_46_im / y_46_re), 2.0)))) / 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 <= -6.6e+108)
		tmp = Float64(Float64(x_46_re + Float64(x_46_im * Float64(Float64(1.0 / y_46_re) * y_46_im))) / y_46_re);
	elseif (y_46_re <= 1.9e+176)
		tmp = Float64(Float64(y_46_im * Float64(fma(x_46_re, Float64(y_46_re / y_46_im), x_46_im) / hypot(y_46_re, y_46_im))) / hypot(y_46_re, y_46_im));
	else
		tmp = Float64(Float64(x_46_re + Float64(Float64(x_46_im * Float64(y_46_im / y_46_re)) - Float64(x_46_re * (Float64(y_46_im / y_46_re) ^ 2.0)))) / y_46_re);
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[y$46$re, -6.6e+108], N[(N[(x$46$re + N[(x$46$im * N[(N[(1.0 / y$46$re), $MachinePrecision] * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision], If[LessEqual[y$46$re, 1.9e+176], N[(N[(y$46$im * N[(N[(x$46$re * N[(y$46$re / y$46$im), $MachinePrecision] + x$46$im), $MachinePrecision] / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision], N[(N[(x$46$re + N[(N[(x$46$im * N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision] - N[(x$46$re * N[Power[N[(y$46$im / y$46$re), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -6.60000000000000038e108

   1. Initial program 39.6%

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

   3. Taylor expanded in y.re around inf 79.9%

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

      1. associate-/l* 87.0%

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

   5. Simplified 87.0%

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

      1. clear-num 86.7%

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

      2. associate-/r/ 87.1%

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

   7. Applied egg-rr 87.1%

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

   if -6.60000000000000038e108 < y.re < 1.9000000000000001e176

   1. Initial program 69.9%

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

   3. Taylor expanded in y.im around inf 65.5%

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

      1. associate-/l* 64.0%

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

   5. Simplified 64.0%

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

      1. *-commutative 64.0%

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

      2. associate-*r/ 65.5%

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

      3. fma-define 65.5%

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

      4. add-sqr-sqrt 65.5%

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

      5. times-frac 68.4%

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

      6. associate-*r/ 68.6%

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

      7. +-commutative 68.6%

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

      8. fma-define 68.6%

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

      9. fma-define 68.5%

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

      10. hypot-define 68.6%

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

      11. fma-define 68.6%

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

      12. hypot-define 90.4%

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

   7. Applied egg-rr 90.4%

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

      1. *-commutative 90.4%

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

      2. associate-*l/ 90.5%

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

   9. Applied egg-rr 90.5%

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

   if 1.9000000000000001e176 < y.re

   1. Initial program 31.5%

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

   3. Taylor expanded in y.im around inf 21.6%

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

      1. associate-/l* 12.5%

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

   5. Simplified 12.5%

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

      1. *-commutative 12.5%

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

      2. associate-*r/ 21.6%

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

      3. fma-define 21.6%

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

      4. add-sqr-sqrt 21.6%

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

      5. times-frac 21.6%

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

      6. associate-*r/ 12.9%

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

      7. +-commutative 12.9%

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

      8. fma-define 12.9%

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

      9. fma-define 12.9%

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

      10. hypot-define 12.9%

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

      11. fma-define 12.9%

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

      12. hypot-define 32.4%

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

   7. Applied egg-rr 32.4%

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

      1. *-commutative 32.4%

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

      2. associate-*l/ 33.1%

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

   9. Applied egg-rr 33.1%

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

   10. Taylor expanded in y.re around inf 77.3%

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

       1. associate-*r/ 77.3%

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

       2. +-commutative 77.3%

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

       3. mul-1-neg 77.3%

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

       4. unsub-neg 77.3%

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

       5. associate-/l* 81.8%

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

       6. unpow2 81.8%

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

       7. unpow2 81.8%

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

       8. times-frac 99.9%

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

       9. unpow2 99.9%

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

   12. Simplified 99.9%

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

Alternative 2: 88.0% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -7.4 \cdot 10^{+108}:\\ \;\;\;\;\frac{x.re + x.im \cdot \left(\frac{1}{y.re} \cdot y.im\right)}{y.re}\\ \mathbf{elif}\;y.re \leq 1.95 \cdot 10^{+176}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x.re, \frac{y.re}{y.im}, x.im\right)}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{y.im}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re + \left(x.im \cdot \frac{y.im}{y.re} - x.re \cdot {\left(\frac{y.im}{y.re}\right)}^{2}\right)}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.re -7.4e+108)
   (/ (+ x.re (* x.im (* (/ 1.0 y.re) y.im))) y.re)
   (if (<= y.re 1.95e+176)
     (*
      (/ (fma x.re (/ y.re y.im) x.im) (hypot y.re y.im))
      (/ y.im (hypot y.re y.im)))
     (/
      (+ x.re (- (* x.im (/ y.im y.re)) (* x.re (pow (/ y.im y.re) 2.0))))
      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 <= -7.4e+108) {
		tmp = (x_46_re + (x_46_im * ((1.0 / y_46_re) * y_46_im))) / y_46_re;
	} else if (y_46_re <= 1.95e+176) {
		tmp = (fma(x_46_re, (y_46_re / y_46_im), x_46_im) / hypot(y_46_re, y_46_im)) * (y_46_im / hypot(y_46_re, y_46_im));
	} else {
		tmp = (x_46_re + ((x_46_im * (y_46_im / y_46_re)) - (x_46_re * pow((y_46_im / y_46_re), 2.0)))) / 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 <= -7.4e+108)
		tmp = Float64(Float64(x_46_re + Float64(x_46_im * Float64(Float64(1.0 / y_46_re) * y_46_im))) / y_46_re);
	elseif (y_46_re <= 1.95e+176)
		tmp = Float64(Float64(fma(x_46_re, Float64(y_46_re / y_46_im), x_46_im) / hypot(y_46_re, y_46_im)) * Float64(y_46_im / hypot(y_46_re, y_46_im)));
	else
		tmp = Float64(Float64(x_46_re + Float64(Float64(x_46_im * Float64(y_46_im / y_46_re)) - Float64(x_46_re * (Float64(y_46_im / y_46_re) ^ 2.0)))) / y_46_re);
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[y$46$re, -7.4e+108], N[(N[(x$46$re + N[(x$46$im * N[(N[(1.0 / y$46$re), $MachinePrecision] * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision], If[LessEqual[y$46$re, 1.95e+176], N[(N[(N[(x$46$re * N[(y$46$re / y$46$im), $MachinePrecision] + x$46$im), $MachinePrecision] / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision] * N[(y$46$im / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$re + N[(N[(x$46$im * N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision] - N[(x$46$re * N[Power[N[(y$46$im / y$46$re), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -7.3999999999999996e108

   1. Initial program 39.6%

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

   3. Taylor expanded in y.re around inf 79.9%

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

      1. associate-/l* 87.0%

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

   5. Simplified 87.0%

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

      1. clear-num 86.7%

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

      2. associate-/r/ 87.1%

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

   7. Applied egg-rr 87.1%

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

   if -7.3999999999999996e108 < y.re < 1.9500000000000001e176

   1. Initial program 69.9%

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

   3. Taylor expanded in y.im around inf 65.5%

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

      1. associate-/l* 64.0%

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

   5. Simplified 64.0%

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

      1. *-commutative 64.0%

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

      2. associate-*r/ 65.5%

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

      3. fma-define 65.5%

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

      4. add-sqr-sqrt 65.5%

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

      5. times-frac 68.4%

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

      6. associate-*r/ 68.6%

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

      7. +-commutative 68.6%

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

      8. fma-define 68.6%

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

      9. fma-define 68.5%

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

      10. hypot-define 68.6%

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

      11. fma-define 68.6%

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

      12. hypot-define 90.4%

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

   7. Applied egg-rr 90.4%

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

   if 1.9500000000000001e176 < y.re

   1. Initial program 31.5%

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

   3. Taylor expanded in y.im around inf 21.6%

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

      1. associate-/l* 12.5%

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

   5. Simplified 12.5%

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

      1. *-commutative 12.5%

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

      2. associate-*r/ 21.6%

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

      3. fma-define 21.6%

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

      4. add-sqr-sqrt 21.6%

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

      5. times-frac 21.6%

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

      6. associate-*r/ 12.9%

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

      7. +-commutative 12.9%

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

      8. fma-define 12.9%

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

      9. fma-define 12.9%

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

      10. hypot-define 12.9%

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

      11. fma-define 12.9%

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

      12. hypot-define 32.4%

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

   7. Applied egg-rr 32.4%

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

      1. *-commutative 32.4%

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

      2. associate-*l/ 33.1%

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

   9. Applied egg-rr 33.1%

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

   10. Taylor expanded in y.re around inf 77.3%

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

       1. associate-*r/ 77.3%

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

       2. +-commutative 77.3%

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

       3. mul-1-neg 77.3%

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

       4. unsub-neg 77.3%

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

       5. associate-/l* 81.8%

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

       6. unpow2 81.8%

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

       7. unpow2 81.8%

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

       8. times-frac 99.9%

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

       9. unpow2 99.9%

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

   12. Simplified 99.9%

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

Alternative 3: 82.4% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y.re \cdot x.re + x.im \cdot y.im\\ \mathbf{if}\;y.im \leq -4.1 \cdot 10^{+80}:\\ \;\;\;\;\frac{x.im + \left(\left(y.re \cdot \frac{x.re}{y.im} - x.re \cdot {\left(\frac{y.re}{y.im}\right)}^{3}\right) - x.im \cdot {\left(\frac{y.re}{y.im}\right)}^{2}\right)}{y.im}\\ \mathbf{elif}\;y.im \leq -1.82 \cdot 10^{-112}:\\ \;\;\;\;\frac{t\_0}{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)}\\ \mathbf{elif}\;y.im \leq 3.5 \cdot 10^{-164}:\\ \;\;\;\;\frac{x.re + \frac{x.im \cdot y.im}{y.re}}{y.re}\\ \mathbf{elif}\;y.im \leq 7000000000000:\\ \;\;\;\;\frac{t\_0}{y.im \cdot y.im + y.re \cdot y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x.re, \frac{y.re}{y.im}, x.im\right)}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (+ (* y.re x.re) (* x.im y.im))))
   (if (<= y.im -4.1e+80)
     (/
      (+
       x.im
       (-
        (- (* y.re (/ x.re y.im)) (* x.re (pow (/ y.re y.im) 3.0)))
        (* x.im (pow (/ y.re y.im) 2.0))))
      y.im)
     (if (<= y.im -1.82e-112)
       (/ t_0 (fma y.re y.re (* y.im y.im)))
       (if (<= y.im 3.5e-164)
         (/ (+ x.re (/ (* x.im y.im) y.re)) y.re)
         (if (<= y.im 7000000000000.0)
           (/ t_0 (+ (* y.im y.im) (* y.re y.re)))
           (/ (fma x.re (/ y.re y.im) x.im) (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 = (y_46_re * x_46_re) + (x_46_im * y_46_im);
	double tmp;
	if (y_46_im <= -4.1e+80) {
		tmp = (x_46_im + (((y_46_re * (x_46_re / y_46_im)) - (x_46_re * pow((y_46_re / y_46_im), 3.0))) - (x_46_im * pow((y_46_re / y_46_im), 2.0)))) / y_46_im;
	} else if (y_46_im <= -1.82e-112) {
		tmp = t_0 / fma(y_46_re, y_46_re, (y_46_im * y_46_im));
	} else if (y_46_im <= 3.5e-164) {
		tmp = (x_46_re + ((x_46_im * y_46_im) / y_46_re)) / y_46_re;
	} else if (y_46_im <= 7000000000000.0) {
		tmp = t_0 / ((y_46_im * y_46_im) + (y_46_re * y_46_re));
	} else {
		tmp = fma(x_46_re, (y_46_re / y_46_im), x_46_im) / 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(y_46_re * x_46_re) + Float64(x_46_im * y_46_im))
	tmp = 0.0
	if (y_46_im <= -4.1e+80)
		tmp = Float64(Float64(x_46_im + Float64(Float64(Float64(y_46_re * Float64(x_46_re / y_46_im)) - Float64(x_46_re * (Float64(y_46_re / y_46_im) ^ 3.0))) - Float64(x_46_im * (Float64(y_46_re / y_46_im) ^ 2.0)))) / y_46_im);
	elseif (y_46_im <= -1.82e-112)
		tmp = Float64(t_0 / fma(y_46_re, y_46_re, Float64(y_46_im * y_46_im)));
	elseif (y_46_im <= 3.5e-164)
		tmp = Float64(Float64(x_46_re + Float64(Float64(x_46_im * y_46_im) / y_46_re)) / y_46_re);
	elseif (y_46_im <= 7000000000000.0)
		tmp = Float64(t_0 / Float64(Float64(y_46_im * y_46_im) + Float64(y_46_re * y_46_re)));
	else
		tmp = Float64(fma(x_46_re, Float64(y_46_re / y_46_im), x_46_im) / hypot(y_46_re, y_46_im));
	end
	return 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 * x$46$re), $MachinePrecision] + N[(x$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$im, -4.1e+80], N[(N[(x$46$im + N[(N[(N[(y$46$re * N[(x$46$re / y$46$im), $MachinePrecision]), $MachinePrecision] - N[(x$46$re * N[Power[N[(y$46$re / y$46$im), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x$46$im * N[Power[N[(y$46$re / y$46$im), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$im), $MachinePrecision], If[LessEqual[y$46$im, -1.82e-112], N[(t$95$0 / N[(y$46$re * y$46$re + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, 3.5e-164], N[(N[(x$46$re + N[(N[(x$46$im * y$46$im), $MachinePrecision] / y$46$re), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision], If[LessEqual[y$46$im, 7000000000000.0], N[(t$95$0 / N[(N[(y$46$im * y$46$im), $MachinePrecision] + N[(y$46$re * y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$re * N[(y$46$re / y$46$im), $MachinePrecision] + x$46$im), $MachinePrecision] / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if y.im < -4.10000000000000001e80

   1. Initial program 42.3%

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

      1. fma-define 42.3%

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

      2. fma-define 42.3%

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

   3. Simplified 42.3%

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

      1. fma-define 42.3%

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

   6. Applied egg-rr 42.3%

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

   7. Taylor expanded in y.im around inf 75.6%

      \[\leadsto \color{blue}{\frac{\left(x.im + \left(-1 \cdot \frac{x.re \cdot {y.re}^{3}}{{y.im}^{3}} + \frac{x.re \cdot y.re}{y.im}\right)\right) - \frac{x.im \cdot {y.re}^{2}}{{y.im}^{2}}}{y.im}} \]
   8. Step-by-step derivation

   9. Simplified 87.9%

      \[\leadsto \color{blue}{\frac{x.im + \left(\left(y.re \cdot \frac{x.re}{y.im} - x.re \cdot {\left(\frac{y.re}{y.im}\right)}^{3}\right) - x.im \cdot {\left(\frac{y.re}{y.im}\right)}^{2}\right)}{y.im}} \]

   if -4.10000000000000001e80 < y.im < -1.81999999999999994e-112

   1. Initial program 94.3%

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

      1. fma-define 94.3%

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

      2. fma-define 94.3%

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

   3. Simplified 94.3%

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

      1. fma-define 94.3%

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

   6. Applied egg-rr 94.3%

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

   if -1.81999999999999994e-112 < y.im < 3.5e-164

   1. Initial program 59.4%

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

   3. Taylor expanded in y.re around inf 92.7%

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

   if 3.5e-164 < y.im < 7e12

   1. Initial program 79.1%

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

   if 7e12 < y.im

   1. Initial program 40.6%

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

   3. Taylor expanded in y.im around inf 40.6%

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

      1. associate-/l* 40.6%

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

   5. Simplified 40.6%

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

      1. *-commutative 40.6%

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

      2. associate-*r/ 40.6%

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

      3. fma-define 40.6%

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

      4. add-sqr-sqrt 40.6%

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

      5. times-frac 50.8%

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

      6. associate-*r/ 50.8%

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

      7. +-commutative 50.8%

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

      8. fma-define 50.8%

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

      9. fma-define 50.8%

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

      10. hypot-define 50.8%

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

      11. fma-define 50.8%

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

      12. hypot-define 90.6%

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

   7. Applied egg-rr 90.6%

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

      1. *-commutative 90.6%

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

      2. associate-*l/ 90.7%

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

   9. Applied egg-rr 90.7%

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

   10. Taylor expanded in y.im around inf 79.1%

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

       1. +-commutative 79.1%

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

       2. associate-/l* 79.1%

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

       3. fma-undefine 79.1%

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

   12. Simplified 79.1%

       \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x.re, \frac{y.re}{y.im}, x.im\right)}}{\mathsf{hypot}\left(y.re, y.im\right)} \]
3. Recombined 5 regimes into one program.

4. Final simplification 86.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -4.1 \cdot 10^{+80}:\\ \;\;\;\;\frac{x.im + \left(\left(y.re \cdot \frac{x.re}{y.im} - x.re \cdot {\left(\frac{y.re}{y.im}\right)}^{3}\right) - x.im \cdot {\left(\frac{y.re}{y.im}\right)}^{2}\right)}{y.im}\\ \mathbf{elif}\;y.im \leq -1.82 \cdot 10^{-112}:\\ \;\;\;\;\frac{y.re \cdot x.re + x.im \cdot y.im}{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)}\\ \mathbf{elif}\;y.im \leq 3.5 \cdot 10^{-164}:\\ \;\;\;\;\frac{x.re + \frac{x.im \cdot y.im}{y.re}}{y.re}\\ \mathbf{elif}\;y.im \leq 7000000000000:\\ \;\;\;\;\frac{y.re \cdot x.re + x.im \cdot y.im}{y.im \cdot y.im + y.re \cdot y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x.re, \frac{y.re}{y.im}, x.im\right)}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 82.5% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y.re \cdot x.re + x.im \cdot y.im\\ \mathbf{if}\;y.im \leq -1.45 \cdot 10^{+80}:\\ \;\;\;\;\frac{x.im + x.re \cdot \frac{y.re}{y.im}}{y.im}\\ \mathbf{elif}\;y.im \leq -1.62 \cdot 10^{-111}:\\ \;\;\;\;\frac{t\_0}{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)}\\ \mathbf{elif}\;y.im \leq 3.5 \cdot 10^{-164}:\\ \;\;\;\;\frac{x.re + \frac{x.im \cdot y.im}{y.re}}{y.re}\\ \mathbf{elif}\;y.im \leq 7000000000000:\\ \;\;\;\;\frac{t\_0}{y.im \cdot y.im + y.re \cdot y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x.re, \frac{y.re}{y.im}, x.im\right)}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (+ (* y.re x.re) (* x.im y.im))))
   (if (<= y.im -1.45e+80)
     (/ (+ x.im (* x.re (/ y.re y.im))) y.im)
     (if (<= y.im -1.62e-111)
       (/ t_0 (fma y.re y.re (* y.im y.im)))
       (if (<= y.im 3.5e-164)
         (/ (+ x.re (/ (* x.im y.im) y.re)) y.re)
         (if (<= y.im 7000000000000.0)
           (/ t_0 (+ (* y.im y.im) (* y.re y.re)))
           (/ (fma x.re (/ y.re y.im) x.im) (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 = (y_46_re * x_46_re) + (x_46_im * y_46_im);
	double tmp;
	if (y_46_im <= -1.45e+80) {
		tmp = (x_46_im + (x_46_re * (y_46_re / y_46_im))) / y_46_im;
	} else if (y_46_im <= -1.62e-111) {
		tmp = t_0 / fma(y_46_re, y_46_re, (y_46_im * y_46_im));
	} else if (y_46_im <= 3.5e-164) {
		tmp = (x_46_re + ((x_46_im * y_46_im) / y_46_re)) / y_46_re;
	} else if (y_46_im <= 7000000000000.0) {
		tmp = t_0 / ((y_46_im * y_46_im) + (y_46_re * y_46_re));
	} else {
		tmp = fma(x_46_re, (y_46_re / y_46_im), x_46_im) / 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(y_46_re * x_46_re) + Float64(x_46_im * y_46_im))
	tmp = 0.0
	if (y_46_im <= -1.45e+80)
		tmp = Float64(Float64(x_46_im + Float64(x_46_re * Float64(y_46_re / y_46_im))) / y_46_im);
	elseif (y_46_im <= -1.62e-111)
		tmp = Float64(t_0 / fma(y_46_re, y_46_re, Float64(y_46_im * y_46_im)));
	elseif (y_46_im <= 3.5e-164)
		tmp = Float64(Float64(x_46_re + Float64(Float64(x_46_im * y_46_im) / y_46_re)) / y_46_re);
	elseif (y_46_im <= 7000000000000.0)
		tmp = Float64(t_0 / Float64(Float64(y_46_im * y_46_im) + Float64(y_46_re * y_46_re)));
	else
		tmp = Float64(fma(x_46_re, Float64(y_46_re / y_46_im), x_46_im) / hypot(y_46_re, y_46_im));
	end
	return 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 * x$46$re), $MachinePrecision] + N[(x$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$im, -1.45e+80], N[(N[(x$46$im + N[(x$46$re * N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$im), $MachinePrecision], If[LessEqual[y$46$im, -1.62e-111], N[(t$95$0 / N[(y$46$re * y$46$re + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, 3.5e-164], N[(N[(x$46$re + N[(N[(x$46$im * y$46$im), $MachinePrecision] / y$46$re), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision], If[LessEqual[y$46$im, 7000000000000.0], N[(t$95$0 / N[(N[(y$46$im * y$46$im), $MachinePrecision] + N[(y$46$re * y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$re * N[(y$46$re / y$46$im), $MachinePrecision] + x$46$im), $MachinePrecision] / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if y.im < -1.44999999999999993e80

   1. Initial program 42.3%

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

   3. Taylor expanded in y.im around inf 83.4%

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

      1. associate-/l* 86.4%

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

   5. Simplified 86.4%

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

   if -1.44999999999999993e80 < y.im < -1.62000000000000004e-111

   1. Initial program 94.3%

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

      1. fma-define 94.3%

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

      2. fma-define 94.3%

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

   3. Simplified 94.3%

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

      1. fma-define 94.3%

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

   6. Applied egg-rr 94.3%

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

   if -1.62000000000000004e-111 < y.im < 3.5e-164

   1. Initial program 59.4%

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

   3. Taylor expanded in y.re around inf 92.7%

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

   if 3.5e-164 < y.im < 7e12

   1. Initial program 79.1%

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

   if 7e12 < y.im

   1. Initial program 40.6%

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

   3. Taylor expanded in y.im around inf 40.6%

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

      1. associate-/l* 40.6%

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

   5. Simplified 40.6%

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

      1. *-commutative 40.6%

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

      2. associate-*r/ 40.6%

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

      3. fma-define 40.6%

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

      4. add-sqr-sqrt 40.6%

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

      5. times-frac 50.8%

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

      6. associate-*r/ 50.8%

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

      7. +-commutative 50.8%

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

      8. fma-define 50.8%

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

      9. fma-define 50.8%

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

      10. hypot-define 50.8%

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

      11. fma-define 50.8%

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

      12. hypot-define 90.6%

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

   7. Applied egg-rr 90.6%

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

      1. *-commutative 90.6%

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

      2. associate-*l/ 90.7%

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

   9. Applied egg-rr 90.7%

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

   10. Taylor expanded in y.im around inf 79.1%

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

       1. +-commutative 79.1%

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

       2. associate-/l* 79.1%

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

       3. fma-undefine 79.1%

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

   12. Simplified 79.1%

       \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x.re, \frac{y.re}{y.im}, x.im\right)}}{\mathsf{hypot}\left(y.re, y.im\right)} \]
3. Recombined 5 regimes into one program.

4. Final simplification 86.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -1.45 \cdot 10^{+80}:\\ \;\;\;\;\frac{x.im + x.re \cdot \frac{y.re}{y.im}}{y.im}\\ \mathbf{elif}\;y.im \leq -1.62 \cdot 10^{-111}:\\ \;\;\;\;\frac{y.re \cdot x.re + x.im \cdot y.im}{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)}\\ \mathbf{elif}\;y.im \leq 3.5 \cdot 10^{-164}:\\ \;\;\;\;\frac{x.re + \frac{x.im \cdot y.im}{y.re}}{y.re}\\ \mathbf{elif}\;y.im \leq 7000000000000:\\ \;\;\;\;\frac{y.re \cdot x.re + x.im \cdot y.im}{y.im \cdot y.im + y.re \cdot y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x.re, \frac{y.re}{y.im}, x.im\right)}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 82.5% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y.re \cdot x.re + x.im \cdot y.im\\ t_1 := \frac{x.im + x.re \cdot \frac{y.re}{y.im}}{y.im}\\ \mathbf{if}\;y.im \leq -4.1 \cdot 10^{+79}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y.im \leq -3.45 \cdot 10^{-112}:\\ \;\;\;\;\frac{t\_0}{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)}\\ \mathbf{elif}\;y.im \leq 3.5 \cdot 10^{-164}:\\ \;\;\;\;\frac{x.re + \frac{x.im \cdot y.im}{y.re}}{y.re}\\ \mathbf{elif}\;y.im \leq 2.8 \cdot 10^{+25}:\\ \;\;\;\;\frac{t\_0}{y.im \cdot y.im + y.re \cdot y.re}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (+ (* y.re x.re) (* x.im y.im)))
        (t_1 (/ (+ x.im (* x.re (/ y.re y.im))) y.im)))
   (if (<= y.im -4.1e+79)
     t_1
     (if (<= y.im -3.45e-112)
       (/ t_0 (fma y.re y.re (* y.im y.im)))
       (if (<= y.im 3.5e-164)
         (/ (+ x.re (/ (* x.im y.im) y.re)) y.re)
         (if (<= y.im 2.8e+25)
           (/ t_0 (+ (* y.im y.im) (* y.re y.re)))
           t_1))))))

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 * x_46_re) + (x_46_im * y_46_im);
	double t_1 = (x_46_im + (x_46_re * (y_46_re / y_46_im))) / y_46_im;
	double tmp;
	if (y_46_im <= -4.1e+79) {
		tmp = t_1;
	} else if (y_46_im <= -3.45e-112) {
		tmp = t_0 / fma(y_46_re, y_46_re, (y_46_im * y_46_im));
	} else if (y_46_im <= 3.5e-164) {
		tmp = (x_46_re + ((x_46_im * y_46_im) / y_46_re)) / y_46_re;
	} else if (y_46_im <= 2.8e+25) {
		tmp = t_0 / ((y_46_im * y_46_im) + (y_46_re * y_46_re));
	} 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(y_46_re * x_46_re) + Float64(x_46_im * y_46_im))
	t_1 = Float64(Float64(x_46_im + Float64(x_46_re * Float64(y_46_re / y_46_im))) / y_46_im)
	tmp = 0.0
	if (y_46_im <= -4.1e+79)
		tmp = t_1;
	elseif (y_46_im <= -3.45e-112)
		tmp = Float64(t_0 / fma(y_46_re, y_46_re, Float64(y_46_im * y_46_im)));
	elseif (y_46_im <= 3.5e-164)
		tmp = Float64(Float64(x_46_re + Float64(Float64(x_46_im * y_46_im) / y_46_re)) / y_46_re);
	elseif (y_46_im <= 2.8e+25)
		tmp = Float64(t_0 / Float64(Float64(y_46_im * y_46_im) + Float64(y_46_re * y_46_re)));
	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[(y$46$re * x$46$re), $MachinePrecision] + N[(x$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x$46$im + N[(x$46$re * N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$im), $MachinePrecision]}, If[LessEqual[y$46$im, -4.1e+79], t$95$1, If[LessEqual[y$46$im, -3.45e-112], N[(t$95$0 / N[(y$46$re * y$46$re + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, 3.5e-164], N[(N[(x$46$re + N[(N[(x$46$im * y$46$im), $MachinePrecision] / y$46$re), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision], If[LessEqual[y$46$im, 2.8e+25], N[(t$95$0 / N[(N[(y$46$im * y$46$im), $MachinePrecision] + N[(y$46$re * y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]]

Derivation

1. Split input into 4 regimes

2. if y.im < -4.1e79 or 2.8000000000000002e25 < y.im

   1. Initial program 40.9%

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

   3. Taylor expanded in y.im around inf 80.7%

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

      1. associate-/l* 82.1%

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

   5. Simplified 82.1%

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

   if -4.1e79 < y.im < -3.45000000000000009e-112

   1. Initial program 94.3%

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

      1. fma-define 94.3%

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

      2. fma-define 94.3%

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

   3. Simplified 94.3%

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

      1. fma-define 94.3%

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

   6. Applied egg-rr 94.3%

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

   if -3.45000000000000009e-112 < y.im < 3.5e-164

   1. Initial program 59.4%

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

   3. Taylor expanded in y.re around inf 92.7%

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

   if 3.5e-164 < y.im < 2.8000000000000002e25

   1. Initial program 76.9%

      \[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   2. Add Preprocessing
3. Recombined 4 regimes into one program.

4. Final simplification 86.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -4.1 \cdot 10^{+79}:\\ \;\;\;\;\frac{x.im + x.re \cdot \frac{y.re}{y.im}}{y.im}\\ \mathbf{elif}\;y.im \leq -3.45 \cdot 10^{-112}:\\ \;\;\;\;\frac{y.re \cdot x.re + x.im \cdot y.im}{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)}\\ \mathbf{elif}\;y.im \leq 3.5 \cdot 10^{-164}:\\ \;\;\;\;\frac{x.re + \frac{x.im \cdot y.im}{y.re}}{y.re}\\ \mathbf{elif}\;y.im \leq 2.8 \cdot 10^{+25}:\\ \;\;\;\;\frac{y.re \cdot x.re + x.im \cdot y.im}{y.im \cdot y.im + y.re \cdot y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im + x.re \cdot \frac{y.re}{y.im}}{y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 82.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y.re \cdot x.re + x.im \cdot y.im}{y.im \cdot y.im + y.re \cdot y.re}\\ t_1 := \frac{x.im + x.re \cdot \frac{y.re}{y.im}}{y.im}\\ \mathbf{if}\;y.im \leq -3.2 \cdot 10^{+79}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y.im \leq -1.75 \cdot 10^{-112}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.im \leq 3.5 \cdot 10^{-164}:\\ \;\;\;\;\frac{x.re + \frac{x.im \cdot y.im}{y.re}}{y.re}\\ \mathbf{elif}\;y.im \leq 2.8 \cdot 10^{+25}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0
         (/ (+ (* y.re x.re) (* x.im y.im)) (+ (* y.im y.im) (* y.re y.re))))
        (t_1 (/ (+ x.im (* x.re (/ y.re y.im))) y.im)))
   (if (<= y.im -3.2e+79)
     t_1
     (if (<= y.im -1.75e-112)
       t_0
       (if (<= y.im 3.5e-164)
         (/ (+ x.re (/ (* x.im y.im) y.re)) y.re)
         (if (<= y.im 2.8e+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 = ((y_46_re * x_46_re) + (x_46_im * y_46_im)) / ((y_46_im * y_46_im) + (y_46_re * y_46_re));
	double t_1 = (x_46_im + (x_46_re * (y_46_re / y_46_im))) / y_46_im;
	double tmp;
	if (y_46_im <= -3.2e+79) {
		tmp = t_1;
	} else if (y_46_im <= -1.75e-112) {
		tmp = t_0;
	} else if (y_46_im <= 3.5e-164) {
		tmp = (x_46_re + ((x_46_im * y_46_im) / y_46_re)) / y_46_re;
	} else if (y_46_im <= 2.8e+25) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	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) :: t_1
    real(8) :: tmp
    t_0 = ((y_46re * x_46re) + (x_46im * y_46im)) / ((y_46im * y_46im) + (y_46re * y_46re))
    t_1 = (x_46im + (x_46re * (y_46re / y_46im))) / y_46im
    if (y_46im <= (-3.2d+79)) then
        tmp = t_1
    else if (y_46im <= (-1.75d-112)) then
        tmp = t_0
    else if (y_46im <= 3.5d-164) then
        tmp = (x_46re + ((x_46im * y_46im) / y_46re)) / y_46re
    else if (y_46im <= 2.8d+25) then
        tmp = t_0
    else
        tmp = t_1
    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 * x_46_re) + (x_46_im * y_46_im)) / ((y_46_im * y_46_im) + (y_46_re * y_46_re));
	double t_1 = (x_46_im + (x_46_re * (y_46_re / y_46_im))) / y_46_im;
	double tmp;
	if (y_46_im <= -3.2e+79) {
		tmp = t_1;
	} else if (y_46_im <= -1.75e-112) {
		tmp = t_0;
	} else if (y_46_im <= 3.5e-164) {
		tmp = (x_46_re + ((x_46_im * y_46_im) / y_46_re)) / y_46_re;
	} else if (y_46_im <= 2.8e+25) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = ((y_46_re * x_46_re) + (x_46_im * y_46_im)) / ((y_46_im * y_46_im) + (y_46_re * y_46_re))
	t_1 = (x_46_im + (x_46_re * (y_46_re / y_46_im))) / y_46_im
	tmp = 0
	if y_46_im <= -3.2e+79:
		tmp = t_1
	elif y_46_im <= -1.75e-112:
		tmp = t_0
	elif y_46_im <= 3.5e-164:
		tmp = (x_46_re + ((x_46_im * y_46_im) / y_46_re)) / y_46_re
	elif y_46_im <= 2.8e+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(y_46_re * x_46_re) + Float64(x_46_im * y_46_im)) / Float64(Float64(y_46_im * y_46_im) + Float64(y_46_re * y_46_re)))
	t_1 = Float64(Float64(x_46_im + Float64(x_46_re * Float64(y_46_re / y_46_im))) / y_46_im)
	tmp = 0.0
	if (y_46_im <= -3.2e+79)
		tmp = t_1;
	elseif (y_46_im <= -1.75e-112)
		tmp = t_0;
	elseif (y_46_im <= 3.5e-164)
		tmp = Float64(Float64(x_46_re + Float64(Float64(x_46_im * y_46_im) / y_46_re)) / y_46_re);
	elseif (y_46_im <= 2.8e+25)
		tmp = t_0;
	else
		tmp = t_1;
	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 * x_46_re) + (x_46_im * y_46_im)) / ((y_46_im * y_46_im) + (y_46_re * y_46_re));
	t_1 = (x_46_im + (x_46_re * (y_46_re / y_46_im))) / y_46_im;
	tmp = 0.0;
	if (y_46_im <= -3.2e+79)
		tmp = t_1;
	elseif (y_46_im <= -1.75e-112)
		tmp = t_0;
	elseif (y_46_im <= 3.5e-164)
		tmp = (x_46_re + ((x_46_im * y_46_im) / y_46_re)) / y_46_re;
	elseif (y_46_im <= 2.8e+25)
		tmp = t_0;
	else
		tmp = t_1;
	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[(y$46$re * x$46$re), $MachinePrecision] + N[(x$46$im * y$46$im), $MachinePrecision]), $MachinePrecision] / N[(N[(y$46$im * y$46$im), $MachinePrecision] + N[(y$46$re * y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x$46$im + N[(x$46$re * N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$im), $MachinePrecision]}, If[LessEqual[y$46$im, -3.2e+79], t$95$1, If[LessEqual[y$46$im, -1.75e-112], t$95$0, If[LessEqual[y$46$im, 3.5e-164], N[(N[(x$46$re + N[(N[(x$46$im * y$46$im), $MachinePrecision] / y$46$re), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision], If[LessEqual[y$46$im, 2.8e+25], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if y.im < -3.20000000000000003e79 or 2.8000000000000002e25 < y.im

   1. Initial program 40.9%

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

   3. Taylor expanded in y.im around inf 80.7%

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

      1. associate-/l* 82.1%

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

   5. Simplified 82.1%

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

   if -3.20000000000000003e79 < y.im < -1.74999999999999997e-112 or 3.5e-164 < y.im < 2.8000000000000002e25

   1. Initial program 84.1%

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

   if -1.74999999999999997e-112 < y.im < 3.5e-164

   1. Initial program 59.4%

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

   3. Taylor expanded in y.re around inf 92.7%

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

4. Final simplification 86.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -3.2 \cdot 10^{+79}:\\ \;\;\;\;\frac{x.im + x.re \cdot \frac{y.re}{y.im}}{y.im}\\ \mathbf{elif}\;y.im \leq -1.75 \cdot 10^{-112}:\\ \;\;\;\;\frac{y.re \cdot x.re + x.im \cdot y.im}{y.im \cdot y.im + y.re \cdot y.re}\\ \mathbf{elif}\;y.im \leq 3.5 \cdot 10^{-164}:\\ \;\;\;\;\frac{x.re + \frac{x.im \cdot y.im}{y.re}}{y.re}\\ \mathbf{elif}\;y.im \leq 2.8 \cdot 10^{+25}:\\ \;\;\;\;\frac{y.re \cdot x.re + x.im \cdot y.im}{y.im \cdot y.im + y.re \cdot y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im + x.re \cdot \frac{y.re}{y.im}}{y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 77.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -1.4 \cdot 10^{+86} \lor \neg \left(y.re \leq 1.55 \cdot 10^{+34}\right):\\ \;\;\;\;\frac{x.re + x.im \cdot \frac{y.im}{y.re}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im + x.re \cdot \frac{y.re}{y.im}}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.re -1.4e+86) (not (<= y.re 1.55e+34)))
   (/ (+ x.re (* x.im (/ y.im y.re))) y.re)
   (/ (+ x.im (* x.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) {
	double tmp;
	if ((y_46_re <= -1.4e+86) || !(y_46_re <= 1.55e+34)) {
		tmp = (x_46_re + (x_46_im * (y_46_im / y_46_re))) / y_46_re;
	} else {
		tmp = (x_46_im + (x_46_re * (y_46_re / y_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 <= (-1.4d+86)) .or. (.not. (y_46re <= 1.55d+34))) then
        tmp = (x_46re + (x_46im * (y_46im / y_46re))) / y_46re
    else
        tmp = (x_46im + (x_46re * (y_46re / y_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 <= -1.4e+86) || !(y_46_re <= 1.55e+34)) {
		tmp = (x_46_re + (x_46_im * (y_46_im / y_46_re))) / y_46_re;
	} else {
		tmp = (x_46_im + (x_46_re * (y_46_re / y_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 <= -1.4e+86) or not (y_46_re <= 1.55e+34):
		tmp = (x_46_re + (x_46_im * (y_46_im / y_46_re))) / y_46_re
	else:
		tmp = (x_46_im + (x_46_re * (y_46_re / y_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 <= -1.4e+86) || !(y_46_re <= 1.55e+34))
		tmp = Float64(Float64(x_46_re + Float64(x_46_im * Float64(y_46_im / y_46_re))) / y_46_re);
	else
		tmp = Float64(Float64(x_46_im + Float64(x_46_re * Float64(y_46_re / y_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 <= -1.4e+86) || ~((y_46_re <= 1.55e+34)))
		tmp = (x_46_re + (x_46_im * (y_46_im / y_46_re))) / y_46_re;
	else
		tmp = (x_46_im + (x_46_re * (y_46_re / y_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, -1.4e+86], N[Not[LessEqual[y$46$re, 1.55e+34]], $MachinePrecision]], N[(N[(x$46$re + N[(x$46$im * N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision], N[(N[(x$46$im + N[(x$46$re * N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$im), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.re < -1.40000000000000002e86 or 1.54999999999999989e34 < y.re

   1. Initial program 48.9%

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

   3. Taylor expanded in y.re around inf 78.4%

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

      1. associate-/l* 83.4%

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

   5. Simplified 83.4%

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

   if -1.40000000000000002e86 < y.re < 1.54999999999999989e34

   1. Initial program 69.5%

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

   3. Taylor expanded in y.im around inf 78.2%

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

      1. associate-/l* 78.4%

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

   5. Simplified 78.4%

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

4. Final simplification 80.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -1.4 \cdot 10^{+86} \lor \neg \left(y.re \leq 1.55 \cdot 10^{+34}\right):\\ \;\;\;\;\frac{x.re + x.im \cdot \frac{y.im}{y.re}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im + x.re \cdot \frac{y.re}{y.im}}{y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 72.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -1.22 \cdot 10^{+107} \lor \neg \left(y.re \leq 4.8 \cdot 10^{+46}\right):\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im + x.re \cdot \frac{y.re}{y.im}}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.re -1.22e+107) (not (<= y.re 4.8e+46)))
   (/ x.re y.re)
   (/ (+ x.im (* x.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) {
	double tmp;
	if ((y_46_re <= -1.22e+107) || !(y_46_re <= 4.8e+46)) {
		tmp = x_46_re / y_46_re;
	} else {
		tmp = (x_46_im + (x_46_re * (y_46_re / y_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 <= (-1.22d+107)) .or. (.not. (y_46re <= 4.8d+46))) then
        tmp = x_46re / y_46re
    else
        tmp = (x_46im + (x_46re * (y_46re / y_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 <= -1.22e+107) || !(y_46_re <= 4.8e+46)) {
		tmp = x_46_re / y_46_re;
	} else {
		tmp = (x_46_im + (x_46_re * (y_46_re / y_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 <= -1.22e+107) or not (y_46_re <= 4.8e+46):
		tmp = x_46_re / y_46_re
	else:
		tmp = (x_46_im + (x_46_re * (y_46_re / y_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 <= -1.22e+107) || !(y_46_re <= 4.8e+46))
		tmp = Float64(x_46_re / y_46_re);
	else
		tmp = Float64(Float64(x_46_im + Float64(x_46_re * Float64(y_46_re / y_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 <= -1.22e+107) || ~((y_46_re <= 4.8e+46)))
		tmp = x_46_re / y_46_re;
	else
		tmp = (x_46_im + (x_46_re * (y_46_re / y_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, -1.22e+107], N[Not[LessEqual[y$46$re, 4.8e+46]], $MachinePrecision]], N[(x$46$re / y$46$re), $MachinePrecision], N[(N[(x$46$im + N[(x$46$re * N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$im), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.re < -1.22e107 or 4.80000000000000017e46 < y.re

   1. Initial program 47.0%

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

   3. Taylor expanded in y.re around inf 72.9%

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

   if -1.22e107 < y.re < 4.80000000000000017e46

   1. Initial program 69.4%

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

   3. Taylor expanded in y.im around inf 75.9%

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

      1. associate-/l* 76.7%

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

   5. Simplified 76.7%

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

4. Final simplification 75.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -1.22 \cdot 10^{+107} \lor \neg \left(y.re \leq 4.8 \cdot 10^{+46}\right):\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im + x.re \cdot \frac{y.re}{y.im}}{y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 77.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -1 \cdot 10^{+86}:\\ \;\;\;\;\frac{x.re + x.im \cdot \left(\frac{1}{y.re} \cdot y.im\right)}{y.re}\\ \mathbf{elif}\;y.re \leq 4.6 \cdot 10^{+32}:\\ \;\;\;\;\frac{x.im + x.re \cdot \frac{y.re}{y.im}}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re + \frac{x.im}{\frac{y.re}{y.im}}}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.re -1e+86)
   (/ (+ x.re (* x.im (* (/ 1.0 y.re) y.im))) y.re)
   (if (<= y.re 4.6e+32)
     (/ (+ x.im (* x.re (/ y.re y.im))) y.im)
     (/ (+ x.re (/ x.im (/ y.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 <= -1e+86) {
		tmp = (x_46_re + (x_46_im * ((1.0 / y_46_re) * y_46_im))) / y_46_re;
	} else if (y_46_re <= 4.6e+32) {
		tmp = (x_46_im + (x_46_re * (y_46_re / y_46_im))) / y_46_im;
	} else {
		tmp = (x_46_re + (x_46_im / (y_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 <= (-1d+86)) then
        tmp = (x_46re + (x_46im * ((1.0d0 / y_46re) * y_46im))) / y_46re
    else if (y_46re <= 4.6d+32) then
        tmp = (x_46im + (x_46re * (y_46re / y_46im))) / y_46im
    else
        tmp = (x_46re + (x_46im / (y_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 <= -1e+86) {
		tmp = (x_46_re + (x_46_im * ((1.0 / y_46_re) * y_46_im))) / y_46_re;
	} else if (y_46_re <= 4.6e+32) {
		tmp = (x_46_im + (x_46_re * (y_46_re / y_46_im))) / y_46_im;
	} else {
		tmp = (x_46_re + (x_46_im / (y_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 <= -1e+86:
		tmp = (x_46_re + (x_46_im * ((1.0 / y_46_re) * y_46_im))) / y_46_re
	elif y_46_re <= 4.6e+32:
		tmp = (x_46_im + (x_46_re * (y_46_re / y_46_im))) / y_46_im
	else:
		tmp = (x_46_re + (x_46_im / (y_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 <= -1e+86)
		tmp = Float64(Float64(x_46_re + Float64(x_46_im * Float64(Float64(1.0 / y_46_re) * y_46_im))) / y_46_re);
	elseif (y_46_re <= 4.6e+32)
		tmp = Float64(Float64(x_46_im + Float64(x_46_re * Float64(y_46_re / y_46_im))) / y_46_im);
	else
		tmp = Float64(Float64(x_46_re + Float64(x_46_im / Float64(y_46_re / 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 <= -1e+86)
		tmp = (x_46_re + (x_46_im * ((1.0 / y_46_re) * y_46_im))) / y_46_re;
	elseif (y_46_re <= 4.6e+32)
		tmp = (x_46_im + (x_46_re * (y_46_re / y_46_im))) / y_46_im;
	else
		tmp = (x_46_re + (x_46_im / (y_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[LessEqual[y$46$re, -1e+86], N[(N[(x$46$re + N[(x$46$im * N[(N[(1.0 / y$46$re), $MachinePrecision] * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision], If[LessEqual[y$46$re, 4.6e+32], N[(N[(x$46$im + N[(x$46$re * N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$im), $MachinePrecision], N[(N[(x$46$re + N[(x$46$im / N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -1e86

   1. Initial program 41.8%

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

   3. Taylor expanded in y.re around inf 78.0%

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

      1. associate-/l* 84.3%

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

   5. Simplified 84.3%

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

      1. clear-num 84.1%

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

      2. associate-/r/ 84.4%

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

   7. Applied egg-rr 84.4%

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

   if -1e86 < y.re < 4.5999999999999999e32

   1. Initial program 69.5%

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

   3. Taylor expanded in y.im around inf 78.2%

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

      1. associate-/l* 78.4%

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

   5. Simplified 78.4%

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

   if 4.5999999999999999e32 < y.re

   1. Initial program 55.5%

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

   3. Taylor expanded in y.re around inf 78.7%

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

      1. associate-/l* 82.6%

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

   5. Simplified 82.6%

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

      1. clear-num 82.6%

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

      2. un-div-inv 82.7%

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

   7. Applied egg-rr 82.7%

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

Alternative 10: 77.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -1 \cdot 10^{+86}:\\ \;\;\;\;\frac{x.re + x.im \cdot \frac{y.im}{y.re}}{y.re}\\ \mathbf{elif}\;y.re \leq 1.3 \cdot 10^{+31}:\\ \;\;\;\;\frac{x.im + x.re \cdot \frac{y.re}{y.im}}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re + \frac{x.im}{\frac{y.re}{y.im}}}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.re -1e+86)
   (/ (+ x.re (* x.im (/ y.im y.re))) y.re)
   (if (<= y.re 1.3e+31)
     (/ (+ x.im (* x.re (/ y.re y.im))) y.im)
     (/ (+ x.re (/ x.im (/ y.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 <= -1e+86) {
		tmp = (x_46_re + (x_46_im * (y_46_im / y_46_re))) / y_46_re;
	} else if (y_46_re <= 1.3e+31) {
		tmp = (x_46_im + (x_46_re * (y_46_re / y_46_im))) / y_46_im;
	} else {
		tmp = (x_46_re + (x_46_im / (y_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 <= (-1d+86)) then
        tmp = (x_46re + (x_46im * (y_46im / y_46re))) / y_46re
    else if (y_46re <= 1.3d+31) then
        tmp = (x_46im + (x_46re * (y_46re / y_46im))) / y_46im
    else
        tmp = (x_46re + (x_46im / (y_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 <= -1e+86) {
		tmp = (x_46_re + (x_46_im * (y_46_im / y_46_re))) / y_46_re;
	} else if (y_46_re <= 1.3e+31) {
		tmp = (x_46_im + (x_46_re * (y_46_re / y_46_im))) / y_46_im;
	} else {
		tmp = (x_46_re + (x_46_im / (y_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 <= -1e+86:
		tmp = (x_46_re + (x_46_im * (y_46_im / y_46_re))) / y_46_re
	elif y_46_re <= 1.3e+31:
		tmp = (x_46_im + (x_46_re * (y_46_re / y_46_im))) / y_46_im
	else:
		tmp = (x_46_re + (x_46_im / (y_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 <= -1e+86)
		tmp = Float64(Float64(x_46_re + Float64(x_46_im * Float64(y_46_im / y_46_re))) / y_46_re);
	elseif (y_46_re <= 1.3e+31)
		tmp = Float64(Float64(x_46_im + Float64(x_46_re * Float64(y_46_re / y_46_im))) / y_46_im);
	else
		tmp = Float64(Float64(x_46_re + Float64(x_46_im / Float64(y_46_re / 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 <= -1e+86)
		tmp = (x_46_re + (x_46_im * (y_46_im / y_46_re))) / y_46_re;
	elseif (y_46_re <= 1.3e+31)
		tmp = (x_46_im + (x_46_re * (y_46_re / y_46_im))) / y_46_im;
	else
		tmp = (x_46_re + (x_46_im / (y_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[LessEqual[y$46$re, -1e+86], N[(N[(x$46$re + N[(x$46$im * N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision], If[LessEqual[y$46$re, 1.3e+31], N[(N[(x$46$im + N[(x$46$re * N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$im), $MachinePrecision], N[(N[(x$46$re + N[(x$46$im / N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -1e86

   1. Initial program 41.8%

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

   3. Taylor expanded in y.re around inf 78.0%

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

      1. associate-/l* 84.3%

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

   5. Simplified 84.3%

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

   if -1e86 < y.re < 1.3e31

   1. Initial program 69.5%

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

   3. Taylor expanded in y.im around inf 78.2%

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

      1. associate-/l* 78.4%

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

   5. Simplified 78.4%

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

   if 1.3e31 < y.re

   1. Initial program 55.5%

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

   3. Taylor expanded in y.re around inf 78.7%

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

      1. associate-/l* 82.6%

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

   5. Simplified 82.6%

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

      1. clear-num 82.6%

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

      2. un-div-inv 82.7%

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

   7. Applied egg-rr 82.7%

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

Alternative 11: 63.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -4.2 \cdot 10^{+72} \lor \neg \left(y.re \leq 1.4 \cdot 10^{+31}\right):\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.re -4.2e+72) (not (<= y.re 1.4e+31)))
   (/ 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 <= -4.2e+72) || !(y_46_re <= 1.4e+31)) {
		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 <= (-4.2d+72)) .or. (.not. (y_46re <= 1.4d+31))) 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 <= -4.2e+72) || !(y_46_re <= 1.4e+31)) {
		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 <= -4.2e+72) or not (y_46_re <= 1.4e+31):
		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 <= -4.2e+72) || !(y_46_re <= 1.4e+31))
		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 <= -4.2e+72) || ~((y_46_re <= 1.4e+31)))
		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, -4.2e+72], N[Not[LessEqual[y$46$re, 1.4e+31]], $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 < -4.2000000000000003e72 or 1.40000000000000008e31 < y.re

   1. Initial program 49.0%

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

   3. Taylor expanded in y.re around inf 68.0%

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

   if -4.2000000000000003e72 < y.re < 1.40000000000000008e31

   1. Initial program 69.9%

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

   3. Taylor expanded in y.re around 0 65.2%

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

4. Final simplification 66.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -4.2 \cdot 10^{+72} \lor \neg \left(y.re \leq 1.4 \cdot 10^{+31}\right):\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 42.2% 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 61.5%

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

3. Taylor expanded in y.re around 0 45.1%

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

Reproduce

herbie shell --seed 2024139 
(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))))