_divideComplex, imaginary part

Percentage Accurate: 61.5% → 86.8%

Time: 10.1s

Alternatives: 11

Speedup: 1.1×

Specification

Math

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

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (/ (- (* x.im y.re) (* x.re 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_im * y_46_re) - (x_46_re * 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_46im * y_46re) - (x_46re * 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_im * y_46_re) - (x_46_re * 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_im * y_46_re) - (x_46_re * 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_im * y_46_re) - Float64(x_46_re * 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_im * y_46_re) - (x_46_re * 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$46$im * y$46$re), $MachinePrecision] - N[(x$46$re * 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: 61.5% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (/ (- (* x.im y.re) (* x.re 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_im * y_46_re) - (x_46_re * 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_46im * y_46re) - (x_46re * 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_im * y_46_re) - (x_46_re * 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_im * y_46_re) - (x_46_re * 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_im * y_46_re) - Float64(x_46_re * 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_im * y_46_re) - (x_46_re * 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$46$im * y$46$re), $MachinePrecision] - N[(x$46$re * 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: 86.8% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -1.32 \cdot 10^{+174}:\\ \;\;\;\;\frac{x.im - x.re \cdot \frac{y.im}{y.re}}{y.re}\\ \mathbf{elif}\;y.re \leq -3.1 \cdot 10^{-153} \lor \neg \left(y.re \leq 3.4 \cdot 10^{-136}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{y.re}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{y.im}{{\left(\mathsf{hypot}\left(y.re, y.im\right)\right)}^{2}} \cdot \left(-x.re\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im \cdot \frac{y.re}{y.im} - x.re}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.re -1.32e+174)
   (/ (- x.im (* x.re (/ y.im y.re))) y.re)
   (if (or (<= y.re -3.1e-153) (not (<= y.re 3.4e-136)))
     (fma
      (/ y.re (hypot y.re y.im))
      (/ x.im (hypot y.re y.im))
      (* (/ y.im (pow (hypot y.re y.im) 2.0)) (- x.re)))
     (/ (- (* x.im (/ y.re y.im)) x.re) 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 <= -1.32e+174) {
		tmp = (x_46_im - (x_46_re * (y_46_im / y_46_re))) / y_46_re;
	} else if ((y_46_re <= -3.1e-153) || !(y_46_re <= 3.4e-136)) {
		tmp = fma((y_46_re / hypot(y_46_re, y_46_im)), (x_46_im / hypot(y_46_re, y_46_im)), ((y_46_im / pow(hypot(y_46_re, y_46_im), 2.0)) * -x_46_re));
	} else {
		tmp = ((x_46_im * (y_46_re / y_46_im)) - x_46_re) / 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 <= -1.32e+174)
		tmp = Float64(Float64(x_46_im - Float64(x_46_re * Float64(y_46_im / y_46_re))) / y_46_re);
	elseif ((y_46_re <= -3.1e-153) || !(y_46_re <= 3.4e-136))
		tmp = fma(Float64(y_46_re / hypot(y_46_re, y_46_im)), Float64(x_46_im / hypot(y_46_re, y_46_im)), Float64(Float64(y_46_im / (hypot(y_46_re, y_46_im) ^ 2.0)) * Float64(-x_46_re)));
	else
		tmp = Float64(Float64(Float64(x_46_im * Float64(y_46_re / y_46_im)) - x_46_re) / y_46_im);
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[y$46$re, -1.32e+174], N[(N[(x$46$im - N[(x$46$re * N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision], If[Or[LessEqual[y$46$re, -3.1e-153], N[Not[LessEqual[y$46$re, 3.4e-136]], $MachinePrecision]], N[(N[(y$46$re / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision] * N[(x$46$im / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision] + N[(N[(y$46$im / N[Power[N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * (-x$46$re)), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x$46$im * N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision] - x$46$re), $MachinePrecision] / y$46$im), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -1.31999999999999999e174

   1. Initial program 32.2%

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

   3. Taylor expanded in y.re around inf 70.6%

      \[\leadsto \color{blue}{\frac{x.im + -1 \cdot \frac{x.re \cdot y.im}{y.re}}{y.re}} \]
   4. Step-by-step derivation

      1. remove-double-neg 70.6%

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

      2. mul-1-neg 70.6%

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

      3. neg-mul-1 70.6%

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

      4. distribute-lft-in 70.6%

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

      5. mul-1-neg 70.6%

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

      6. distribute-neg-in 70.6%

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

      7. mul-1-neg 70.6%

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

      8. remove-double-neg 70.6%

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

      9. unsub-neg 70.6%

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

      10. associate-/l* 96.3%

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

   5. Simplified 96.3%

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

   if -1.31999999999999999e174 < y.re < -3.09999999999999995e-153 or 3.4e-136 < y.re

   1. Initial program 67.9%

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

      1. div-sub 67.9%

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

      2. *-commutative 67.9%

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

      3. add-sqr-sqrt 67.9%

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

      4. times-frac 73.4%

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

      5. fma-neg 73.4%

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

      6. hypot-define 73.4%

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

      7. hypot-define 87.3%

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

      8. associate-/l* 87.5%

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

      9. add-sqr-sqrt 87.5%

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

      10. pow2 87.5%

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

      11. hypot-define 87.5%

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

   4. Applied egg-rr 87.5%

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

   if -3.09999999999999995e-153 < y.re < 3.4e-136

   1. Initial program 73.9%

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

      1. div-sub 68.1%

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

      2. *-commutative 68.1%

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

      3. add-sqr-sqrt 68.1%

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

      4. times-frac 66.4%

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

      5. fma-neg 66.4%

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

      6. hypot-define 66.4%

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

      7. hypot-define 67.8%

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

      8. associate-/l* 69.3%

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

      9. add-sqr-sqrt 69.3%

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

      10. pow2 69.3%

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

      11. hypot-define 69.3%

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

   4. Applied egg-rr 69.3%

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

   5. Taylor expanded in y.im around inf 93.7%

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

      1. associate-/l* 95.0%

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

   7. Simplified 95.0%

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

4. Final simplification 90.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -1.32 \cdot 10^{+174}:\\ \;\;\;\;\frac{x.im - x.re \cdot \frac{y.im}{y.re}}{y.re}\\ \mathbf{elif}\;y.re \leq -3.1 \cdot 10^{-153} \lor \neg \left(y.re \leq 3.4 \cdot 10^{-136}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{y.re}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{y.im}{{\left(\mathsf{hypot}\left(y.re, y.im\right)\right)}^{2}} \cdot \left(-x.re\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im \cdot \frac{y.re}{y.im} - x.re}{y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 80.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -6.2 \cdot 10^{-47}:\\ \;\;\;\;\frac{x.im - x.re \cdot \frac{y.im}{y.re}}{y.re}\\ \mathbf{elif}\;y.re \leq 1.7 \cdot 10^{-136}:\\ \;\;\;\;\frac{x.im \cdot \frac{y.re}{y.im} - x.re}{y.im}\\ \mathbf{elif}\;y.re \leq 1.7 \cdot 10^{+74}:\\ \;\;\;\;\frac{y.re \cdot x.im - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im - \frac{x.re}{\frac{y.re}{y.im}}}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.re -6.2e-47)
   (/ (- x.im (* x.re (/ y.im y.re))) y.re)
   (if (<= y.re 1.7e-136)
     (/ (- (* x.im (/ y.re y.im)) x.re) y.im)
     (if (<= y.re 1.7e+74)
       (/ (- (* y.re x.im) (* x.re y.im)) (+ (* y.re y.re) (* y.im y.im)))
       (/ (- x.im (/ x.re (/ 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 <= -6.2e-47) {
		tmp = (x_46_im - (x_46_re * (y_46_im / y_46_re))) / y_46_re;
	} else if (y_46_re <= 1.7e-136) {
		tmp = ((x_46_im * (y_46_re / y_46_im)) - x_46_re) / y_46_im;
	} else if (y_46_re <= 1.7e+74) {
		tmp = ((y_46_re * x_46_im) - (x_46_re * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	} else {
		tmp = (x_46_im - (x_46_re / (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 <= (-6.2d-47)) then
        tmp = (x_46im - (x_46re * (y_46im / y_46re))) / y_46re
    else if (y_46re <= 1.7d-136) then
        tmp = ((x_46im * (y_46re / y_46im)) - x_46re) / y_46im
    else if (y_46re <= 1.7d+74) then
        tmp = ((y_46re * x_46im) - (x_46re * y_46im)) / ((y_46re * y_46re) + (y_46im * y_46im))
    else
        tmp = (x_46im - (x_46re / (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 <= -6.2e-47) {
		tmp = (x_46_im - (x_46_re * (y_46_im / y_46_re))) / y_46_re;
	} else if (y_46_re <= 1.7e-136) {
		tmp = ((x_46_im * (y_46_re / y_46_im)) - x_46_re) / y_46_im;
	} else if (y_46_re <= 1.7e+74) {
		tmp = ((y_46_re * x_46_im) - (x_46_re * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	} else {
		tmp = (x_46_im - (x_46_re / (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 <= -6.2e-47:
		tmp = (x_46_im - (x_46_re * (y_46_im / y_46_re))) / y_46_re
	elif y_46_re <= 1.7e-136:
		tmp = ((x_46_im * (y_46_re / y_46_im)) - x_46_re) / y_46_im
	elif y_46_re <= 1.7e+74:
		tmp = ((y_46_re * x_46_im) - (x_46_re * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im))
	else:
		tmp = (x_46_im - (x_46_re / (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 <= -6.2e-47)
		tmp = Float64(Float64(x_46_im - Float64(x_46_re * Float64(y_46_im / y_46_re))) / y_46_re);
	elseif (y_46_re <= 1.7e-136)
		tmp = Float64(Float64(Float64(x_46_im * Float64(y_46_re / y_46_im)) - x_46_re) / y_46_im);
	elseif (y_46_re <= 1.7e+74)
		tmp = Float64(Float64(Float64(y_46_re * x_46_im) - Float64(x_46_re * y_46_im)) / Float64(Float64(y_46_re * y_46_re) + Float64(y_46_im * y_46_im)));
	else
		tmp = Float64(Float64(x_46_im - Float64(x_46_re / 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 <= -6.2e-47)
		tmp = (x_46_im - (x_46_re * (y_46_im / y_46_re))) / y_46_re;
	elseif (y_46_re <= 1.7e-136)
		tmp = ((x_46_im * (y_46_re / y_46_im)) - x_46_re) / y_46_im;
	elseif (y_46_re <= 1.7e+74)
		tmp = ((y_46_re * x_46_im) - (x_46_re * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	else
		tmp = (x_46_im - (x_46_re / (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, -6.2e-47], N[(N[(x$46$im - N[(x$46$re * N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision], If[LessEqual[y$46$re, 1.7e-136], N[(N[(N[(x$46$im * N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision] - x$46$re), $MachinePrecision] / y$46$im), $MachinePrecision], If[LessEqual[y$46$re, 1.7e+74], N[(N[(N[(y$46$re * x$46$im), $MachinePrecision] - N[(x$46$re * y$46$im), $MachinePrecision]), $MachinePrecision] / N[(N[(y$46$re * y$46$re), $MachinePrecision] + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$im - N[(x$46$re / N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if y.re < -6.1999999999999996e-47

   1. Initial program 53.1%

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

   3. Taylor expanded in y.re around inf 71.7%

      \[\leadsto \color{blue}{\frac{x.im + -1 \cdot \frac{x.re \cdot y.im}{y.re}}{y.re}} \]
   4. Step-by-step derivation

      1. remove-double-neg 71.7%

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

      2. mul-1-neg 71.7%

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

      3. neg-mul-1 71.7%

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

      4. distribute-lft-in 71.7%

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

      5. mul-1-neg 71.7%

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

      6. distribute-neg-in 71.7%

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

      7. mul-1-neg 71.7%

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

      8. remove-double-neg 71.7%

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

      9. unsub-neg 71.7%

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

      10. associate-/l* 83.5%

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

   5. Simplified 83.5%

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

   if -6.1999999999999996e-47 < y.re < 1.7e-136

   1. Initial program 71.9%

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

      1. div-sub 67.4%

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

      2. *-commutative 67.4%

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

      3. add-sqr-sqrt 67.4%

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

      4. times-frac 67.1%

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

      5. fma-neg 67.1%

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

      6. hypot-define 67.1%

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

      7. hypot-define 69.8%

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

      8. associate-/l* 71.9%

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

      9. add-sqr-sqrt 71.9%

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

      10. pow2 71.9%

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

      11. hypot-define 71.9%

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

   4. Applied egg-rr 71.9%

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

   5. Taylor expanded in y.im around inf 90.2%

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

      1. associate-/l* 90.8%

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

   7. Simplified 90.8%

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

   if 1.7e-136 < y.re < 1.7e74

   1. Initial program 85.7%

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

   if 1.7e74 < y.re

   1. Initial program 49.8%

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

   3. Taylor expanded in y.re around inf 83.9%

      \[\leadsto \color{blue}{\frac{x.im + -1 \cdot \frac{x.re \cdot y.im}{y.re}}{y.re}} \]
   4. Step-by-step derivation

      1. remove-double-neg 83.9%

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

      2. mul-1-neg 83.9%

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

      3. neg-mul-1 83.9%

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

      4. distribute-lft-in 83.9%

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

      5. mul-1-neg 83.9%

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

      6. distribute-neg-in 83.9%

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

      7. mul-1-neg 83.9%

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

      8. remove-double-neg 83.9%

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

      9. unsub-neg 83.9%

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

      10. associate-/l* 88.5%

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

   5. Simplified 88.5%

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

      1. clear-num 88.5%

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

      2. un-div-inv 88.6%

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

   7. Applied egg-rr 88.6%

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

4. Final simplification 87.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -6.2 \cdot 10^{-47}:\\ \;\;\;\;\frac{x.im - x.re \cdot \frac{y.im}{y.re}}{y.re}\\ \mathbf{elif}\;y.re \leq 1.7 \cdot 10^{-136}:\\ \;\;\;\;\frac{x.im \cdot \frac{y.re}{y.im} - x.re}{y.im}\\ \mathbf{elif}\;y.re \leq 1.7 \cdot 10^{+74}:\\ \;\;\;\;\frac{y.re \cdot x.im - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im - \frac{x.re}{\frac{y.re}{y.im}}}{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 60.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -2.8 \cdot 10^{+215}:\\ \;\;\;\;\frac{y.im}{y.re} \cdot \frac{x.re}{-y.re}\\ \mathbf{elif}\;y.re \leq -1.4 \cdot 10^{-64} \lor \neg \left(y.re \leq 5.5 \cdot 10^{-57}\right):\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{-y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.re -2.8e+215)
   (* (/ y.im y.re) (/ x.re (- y.re)))
   (if (or (<= y.re -1.4e-64) (not (<= y.re 5.5e-57)))
     (/ x.im y.re)
     (/ x.re (- 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 <= -2.8e+215) {
		tmp = (y_46_im / y_46_re) * (x_46_re / -y_46_re);
	} else if ((y_46_re <= -1.4e-64) || !(y_46_re <= 5.5e-57)) {
		tmp = x_46_im / y_46_re;
	} else {
		tmp = x_46_re / -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 <= (-2.8d+215)) then
        tmp = (y_46im / y_46re) * (x_46re / -y_46re)
    else if ((y_46re <= (-1.4d-64)) .or. (.not. (y_46re <= 5.5d-57))) then
        tmp = x_46im / y_46re
    else
        tmp = x_46re / -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 <= -2.8e+215) {
		tmp = (y_46_im / y_46_re) * (x_46_re / -y_46_re);
	} else if ((y_46_re <= -1.4e-64) || !(y_46_re <= 5.5e-57)) {
		tmp = x_46_im / y_46_re;
	} else {
		tmp = x_46_re / -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 <= -2.8e+215:
		tmp = (y_46_im / y_46_re) * (x_46_re / -y_46_re)
	elif (y_46_re <= -1.4e-64) or not (y_46_re <= 5.5e-57):
		tmp = x_46_im / y_46_re
	else:
		tmp = x_46_re / -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 <= -2.8e+215)
		tmp = Float64(Float64(y_46_im / y_46_re) * Float64(x_46_re / Float64(-y_46_re)));
	elseif ((y_46_re <= -1.4e-64) || !(y_46_re <= 5.5e-57))
		tmp = Float64(x_46_im / y_46_re);
	else
		tmp = Float64(x_46_re / Float64(-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 <= -2.8e+215)
		tmp = (y_46_im / y_46_re) * (x_46_re / -y_46_re);
	elseif ((y_46_re <= -1.4e-64) || ~((y_46_re <= 5.5e-57)))
		tmp = x_46_im / y_46_re;
	else
		tmp = x_46_re / -y_46_im;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[y$46$re, -2.8e+215], N[(N[(y$46$im / y$46$re), $MachinePrecision] * N[(x$46$re / (-y$46$re)), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[y$46$re, -1.4e-64], N[Not[LessEqual[y$46$re, 5.5e-57]], $MachinePrecision]], N[(x$46$im / y$46$re), $MachinePrecision], N[(x$46$re / (-y$46$im)), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -2.8e215

   1. Initial program 45.7%

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

   3. Taylor expanded in y.re around inf 62.0%

      \[\leadsto \color{blue}{\frac{x.im + -1 \cdot \frac{x.re \cdot y.im}{y.re}}{y.re}} \]
   4. Step-by-step derivation

      1. remove-double-neg 62.0%

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

      2. mul-1-neg 62.0%

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

      3. neg-mul-1 62.0%

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

      4. distribute-lft-in 62.0%

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

      5. mul-1-neg 62.0%

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

      6. distribute-neg-in 62.0%

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

      7. mul-1-neg 62.0%

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

      8. remove-double-neg 62.0%

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

      9. unsub-neg 62.0%

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

      10. associate-/l* 95.8%

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

   5. Simplified 95.8%

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

   6. Taylor expanded in x.im around 0 53.8%

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

      1. mul-1-neg 53.8%

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

      2. distribute-frac-neg2 53.8%

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

      3. associate-/l* 83.5%

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

   8. Simplified 83.5%

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

      1. frac-2neg 83.5%

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

      2. distribute-frac-neg2 83.5%

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

      3. add-sqr-sqrt 0.0%

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

      4. sqrt-unprod 47.3%

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

      5. sqr-neg 47.3%

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

      6. sqrt-unprod 46.7%

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

      7. add-sqr-sqrt 46.7%

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

      8. frac-2neg 46.7%

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

      9. div-inv 46.7%

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

      10. *-commutative 46.7%

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

      11. associate-*l* 46.7%

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

      12. add-sqr-sqrt 46.7%

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

      13. sqrt-unprod 47.3%

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

      14. sqr-neg 47.3%

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

      15. sqrt-unprod 0.0%

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

      16. add-sqr-sqrt 83.5%

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

      17. div-inv 83.4%

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

   10. Applied egg-rr 83.4%

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

   if -2.8e215 < y.re < -1.40000000000000002e-64 or 5.50000000000000011e-57 < y.re

   1. Initial program 61.1%

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

   3. Taylor expanded in y.re around inf 62.7%

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

   if -1.40000000000000002e-64 < y.re < 5.50000000000000011e-57

   1. Initial program 73.2%

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

   3. Taylor expanded in y.re around 0 70.2%

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

      1. associate-*r/ 70.2%

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

      2. neg-mul-1 70.2%

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

   5. Simplified 70.2%

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

4. Final simplification 67.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -2.8 \cdot 10^{+215}:\\ \;\;\;\;\frac{y.im}{y.re} \cdot \frac{x.re}{-y.re}\\ \mathbf{elif}\;y.re \leq -1.4 \cdot 10^{-64} \lor \neg \left(y.re \leq 5.5 \cdot 10^{-57}\right):\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{-y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 60.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -2.75 \cdot 10^{+215}:\\ \;\;\;\;\frac{x.re \cdot \frac{-y.im}{y.re}}{y.re}\\ \mathbf{elif}\;y.re \leq -1.45 \cdot 10^{-64} \lor \neg \left(y.re \leq 8.4 \cdot 10^{-57}\right):\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{-y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.re -2.75e+215)
   (/ (* x.re (/ (- y.im) y.re)) y.re)
   (if (or (<= y.re -1.45e-64) (not (<= y.re 8.4e-57)))
     (/ x.im y.re)
     (/ x.re (- 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 <= -2.75e+215) {
		tmp = (x_46_re * (-y_46_im / y_46_re)) / y_46_re;
	} else if ((y_46_re <= -1.45e-64) || !(y_46_re <= 8.4e-57)) {
		tmp = x_46_im / y_46_re;
	} else {
		tmp = x_46_re / -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 <= (-2.75d+215)) then
        tmp = (x_46re * (-y_46im / y_46re)) / y_46re
    else if ((y_46re <= (-1.45d-64)) .or. (.not. (y_46re <= 8.4d-57))) then
        tmp = x_46im / y_46re
    else
        tmp = x_46re / -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 <= -2.75e+215) {
		tmp = (x_46_re * (-y_46_im / y_46_re)) / y_46_re;
	} else if ((y_46_re <= -1.45e-64) || !(y_46_re <= 8.4e-57)) {
		tmp = x_46_im / y_46_re;
	} else {
		tmp = x_46_re / -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 <= -2.75e+215:
		tmp = (x_46_re * (-y_46_im / y_46_re)) / y_46_re
	elif (y_46_re <= -1.45e-64) or not (y_46_re <= 8.4e-57):
		tmp = x_46_im / y_46_re
	else:
		tmp = x_46_re / -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 <= -2.75e+215)
		tmp = Float64(Float64(x_46_re * Float64(Float64(-y_46_im) / y_46_re)) / y_46_re);
	elseif ((y_46_re <= -1.45e-64) || !(y_46_re <= 8.4e-57))
		tmp = Float64(x_46_im / y_46_re);
	else
		tmp = Float64(x_46_re / Float64(-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 <= -2.75e+215)
		tmp = (x_46_re * (-y_46_im / y_46_re)) / y_46_re;
	elseif ((y_46_re <= -1.45e-64) || ~((y_46_re <= 8.4e-57)))
		tmp = x_46_im / y_46_re;
	else
		tmp = x_46_re / -y_46_im;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[y$46$re, -2.75e+215], N[(N[(x$46$re * N[((-y$46$im) / y$46$re), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision], If[Or[LessEqual[y$46$re, -1.45e-64], N[Not[LessEqual[y$46$re, 8.4e-57]], $MachinePrecision]], N[(x$46$im / y$46$re), $MachinePrecision], N[(x$46$re / (-y$46$im)), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -2.75e215

   1. Initial program 45.7%

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

   3. Taylor expanded in y.re around inf 62.0%

      \[\leadsto \color{blue}{\frac{x.im + -1 \cdot \frac{x.re \cdot y.im}{y.re}}{y.re}} \]
   4. Step-by-step derivation

      1. remove-double-neg 62.0%

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

      2. mul-1-neg 62.0%

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

      3. neg-mul-1 62.0%

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

      4. distribute-lft-in 62.0%

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

      5. mul-1-neg 62.0%

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

      6. distribute-neg-in 62.0%

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

      7. mul-1-neg 62.0%

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

      8. remove-double-neg 62.0%

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

      9. unsub-neg 62.0%

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

      10. associate-/l* 95.8%

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

   5. Simplified 95.8%

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

   6. Taylor expanded in x.im around 0 53.8%

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

      1. mul-1-neg 53.8%

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

      2. distribute-frac-neg2 53.8%

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

      3. associate-/l* 83.5%

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

   8. Simplified 83.5%

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

   if -2.75e215 < y.re < -1.4499999999999999e-64 or 8.3999999999999998e-57 < y.re

   1. Initial program 61.1%

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

   3. Taylor expanded in y.re around inf 62.7%

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

   if -1.4499999999999999e-64 < y.re < 8.3999999999999998e-57

   1. Initial program 73.2%

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

   3. Taylor expanded in y.re around 0 70.2%

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

      1. associate-*r/ 70.2%

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

      2. neg-mul-1 70.2%

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

   5. Simplified 70.2%

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

4. Final simplification 67.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -2.75 \cdot 10^{+215}:\\ \;\;\;\;\frac{x.re \cdot \frac{-y.im}{y.re}}{y.re}\\ \mathbf{elif}\;y.re \leq -1.45 \cdot 10^{-64} \lor \neg \left(y.re \leq 8.4 \cdot 10^{-57}\right):\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{-y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 60.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -2.8 \cdot 10^{+215}:\\ \;\;\;\;\frac{\frac{x.re}{\frac{y.re}{-y.im}}}{y.re}\\ \mathbf{elif}\;y.re \leq -6 \cdot 10^{-65} \lor \neg \left(y.re \leq 1.4 \cdot 10^{-56}\right):\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{-y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.re -2.8e+215)
   (/ (/ x.re (/ y.re (- y.im))) y.re)
   (if (or (<= y.re -6e-65) (not (<= y.re 1.4e-56)))
     (/ x.im y.re)
     (/ x.re (- 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 <= -2.8e+215) {
		tmp = (x_46_re / (y_46_re / -y_46_im)) / y_46_re;
	} else if ((y_46_re <= -6e-65) || !(y_46_re <= 1.4e-56)) {
		tmp = x_46_im / y_46_re;
	} else {
		tmp = x_46_re / -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 <= (-2.8d+215)) then
        tmp = (x_46re / (y_46re / -y_46im)) / y_46re
    else if ((y_46re <= (-6d-65)) .or. (.not. (y_46re <= 1.4d-56))) then
        tmp = x_46im / y_46re
    else
        tmp = x_46re / -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 <= -2.8e+215) {
		tmp = (x_46_re / (y_46_re / -y_46_im)) / y_46_re;
	} else if ((y_46_re <= -6e-65) || !(y_46_re <= 1.4e-56)) {
		tmp = x_46_im / y_46_re;
	} else {
		tmp = x_46_re / -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 <= -2.8e+215:
		tmp = (x_46_re / (y_46_re / -y_46_im)) / y_46_re
	elif (y_46_re <= -6e-65) or not (y_46_re <= 1.4e-56):
		tmp = x_46_im / y_46_re
	else:
		tmp = x_46_re / -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 <= -2.8e+215)
		tmp = Float64(Float64(x_46_re / Float64(y_46_re / Float64(-y_46_im))) / y_46_re);
	elseif ((y_46_re <= -6e-65) || !(y_46_re <= 1.4e-56))
		tmp = Float64(x_46_im / y_46_re);
	else
		tmp = Float64(x_46_re / Float64(-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 <= -2.8e+215)
		tmp = (x_46_re / (y_46_re / -y_46_im)) / y_46_re;
	elseif ((y_46_re <= -6e-65) || ~((y_46_re <= 1.4e-56)))
		tmp = x_46_im / y_46_re;
	else
		tmp = x_46_re / -y_46_im;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[y$46$re, -2.8e+215], N[(N[(x$46$re / N[(y$46$re / (-y$46$im)), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision], If[Or[LessEqual[y$46$re, -6e-65], N[Not[LessEqual[y$46$re, 1.4e-56]], $MachinePrecision]], N[(x$46$im / y$46$re), $MachinePrecision], N[(x$46$re / (-y$46$im)), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -2.8e215

   1. Initial program 45.7%

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

   3. Taylor expanded in y.re around inf 62.0%

      \[\leadsto \color{blue}{\frac{x.im + -1 \cdot \frac{x.re \cdot y.im}{y.re}}{y.re}} \]
   4. Step-by-step derivation

      1. remove-double-neg 62.0%

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

      2. mul-1-neg 62.0%

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

      3. neg-mul-1 62.0%

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

      4. distribute-lft-in 62.0%

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

      5. mul-1-neg 62.0%

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

      6. distribute-neg-in 62.0%

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

      7. mul-1-neg 62.0%

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

      8. remove-double-neg 62.0%

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

      9. unsub-neg 62.0%

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

      10. associate-/l* 95.8%

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

   5. Simplified 95.8%

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

   6. Taylor expanded in x.im around 0 53.8%

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

      1. mul-1-neg 53.8%

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

      2. distribute-frac-neg2 53.8%

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

      3. associate-/l* 83.5%

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

   8. Simplified 83.5%

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

      1. distribute-frac-neg2 83.5%

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

      2. distribute-rgt-neg-in 83.5%

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

      3. clear-num 83.5%

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

      4. div-inv 83.5%

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

      5. distribute-neg-frac 83.5%

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

   10. Applied egg-rr 83.5%

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

   if -2.8e215 < y.re < -5.99999999999999996e-65 or 1.39999999999999997e-56 < y.re

   1. Initial program 61.1%

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

   3. Taylor expanded in y.re around inf 62.7%

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

   if -5.99999999999999996e-65 < y.re < 1.39999999999999997e-56

   1. Initial program 73.2%

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

   3. Taylor expanded in y.re around 0 70.2%

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

      1. associate-*r/ 70.2%

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

      2. neg-mul-1 70.2%

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

   5. Simplified 70.2%

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

4. Final simplification 67.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -2.8 \cdot 10^{+215}:\\ \;\;\;\;\frac{\frac{x.re}{\frac{y.re}{-y.im}}}{y.re}\\ \mathbf{elif}\;y.re \leq -6 \cdot 10^{-65} \lor \neg \left(y.re \leq 1.4 \cdot 10^{-56}\right):\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{-y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 69.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -4.5 \cdot 10^{-96} \lor \neg \left(y.re \leq 2.3 \cdot 10^{-63}\right):\\ \;\;\;\;\frac{x.im - x.re \cdot \frac{y.im}{y.re}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{-y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.re -4.5e-96) (not (<= y.re 2.3e-63)))
   (/ (- x.im (* x.re (/ y.im y.re))) y.re)
   (/ x.re (- 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 <= -4.5e-96) || !(y_46_re <= 2.3e-63)) {
		tmp = (x_46_im - (x_46_re * (y_46_im / y_46_re))) / y_46_re;
	} else {
		tmp = x_46_re / -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 <= (-4.5d-96)) .or. (.not. (y_46re <= 2.3d-63))) then
        tmp = (x_46im - (x_46re * (y_46im / y_46re))) / y_46re
    else
        tmp = x_46re / -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 <= -4.5e-96) || !(y_46_re <= 2.3e-63)) {
		tmp = (x_46_im - (x_46_re * (y_46_im / y_46_re))) / y_46_re;
	} else {
		tmp = x_46_re / -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 <= -4.5e-96) or not (y_46_re <= 2.3e-63):
		tmp = (x_46_im - (x_46_re * (y_46_im / y_46_re))) / y_46_re
	else:
		tmp = x_46_re / -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 <= -4.5e-96) || !(y_46_re <= 2.3e-63))
		tmp = Float64(Float64(x_46_im - Float64(x_46_re * Float64(y_46_im / y_46_re))) / y_46_re);
	else
		tmp = Float64(x_46_re / Float64(-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 <= -4.5e-96) || ~((y_46_re <= 2.3e-63)))
		tmp = (x_46_im - (x_46_re * (y_46_im / y_46_re))) / y_46_re;
	else
		tmp = x_46_re / -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, -4.5e-96], N[Not[LessEqual[y$46$re, 2.3e-63]], $MachinePrecision]], N[(N[(x$46$im - N[(x$46$re * N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision], N[(x$46$re / (-y$46$im)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.re < -4.5e-96 or 2.3e-63 < y.re

   1. Initial program 60.1%

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

   3. Taylor expanded in y.re around inf 74.4%

      \[\leadsto \color{blue}{\frac{x.im + -1 \cdot \frac{x.re \cdot y.im}{y.re}}{y.re}} \]
   4. Step-by-step derivation

      1. remove-double-neg 74.4%

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

      2. mul-1-neg 74.4%

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

      3. neg-mul-1 74.4%

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

      4. distribute-lft-in 74.4%

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

      5. mul-1-neg 74.4%

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

      6. distribute-neg-in 74.4%

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

      7. mul-1-neg 74.4%

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

      8. remove-double-neg 74.4%

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

      9. unsub-neg 74.4%

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

      10. associate-/l* 81.6%

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

   5. Simplified 81.6%

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

   if -4.5e-96 < y.re < 2.3e-63

   1. Initial program 71.9%

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

   3. Taylor expanded in y.re around 0 72.4%

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

      1. associate-*r/ 72.4%

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

      2. neg-mul-1 72.4%

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

   5. Simplified 72.4%

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

4. Final simplification 77.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -4.5 \cdot 10^{-96} \lor \neg \left(y.re \leq 2.3 \cdot 10^{-63}\right):\\ \;\;\;\;\frac{x.im - x.re \cdot \frac{y.im}{y.re}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{-y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 69.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -5.8 \cdot 10^{-96}:\\ \;\;\;\;\frac{x.im - x.re \cdot \frac{y.im}{y.re}}{y.re}\\ \mathbf{elif}\;y.re \leq 4.9 \cdot 10^{-59}:\\ \;\;\;\;\frac{x.re}{-y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im - \frac{x.re}{\frac{y.re}{y.im}}}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.re -5.8e-96)
   (/ (- x.im (* x.re (/ y.im y.re))) y.re)
   (if (<= y.re 4.9e-59)
     (/ x.re (- y.im))
     (/ (- x.im (/ x.re (/ 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 <= -5.8e-96) {
		tmp = (x_46_im - (x_46_re * (y_46_im / y_46_re))) / y_46_re;
	} else if (y_46_re <= 4.9e-59) {
		tmp = x_46_re / -y_46_im;
	} else {
		tmp = (x_46_im - (x_46_re / (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 <= (-5.8d-96)) then
        tmp = (x_46im - (x_46re * (y_46im / y_46re))) / y_46re
    else if (y_46re <= 4.9d-59) then
        tmp = x_46re / -y_46im
    else
        tmp = (x_46im - (x_46re / (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 <= -5.8e-96) {
		tmp = (x_46_im - (x_46_re * (y_46_im / y_46_re))) / y_46_re;
	} else if (y_46_re <= 4.9e-59) {
		tmp = x_46_re / -y_46_im;
	} else {
		tmp = (x_46_im - (x_46_re / (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 <= -5.8e-96:
		tmp = (x_46_im - (x_46_re * (y_46_im / y_46_re))) / y_46_re
	elif y_46_re <= 4.9e-59:
		tmp = x_46_re / -y_46_im
	else:
		tmp = (x_46_im - (x_46_re / (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 <= -5.8e-96)
		tmp = Float64(Float64(x_46_im - Float64(x_46_re * Float64(y_46_im / y_46_re))) / y_46_re);
	elseif (y_46_re <= 4.9e-59)
		tmp = Float64(x_46_re / Float64(-y_46_im));
	else
		tmp = Float64(Float64(x_46_im - Float64(x_46_re / 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 <= -5.8e-96)
		tmp = (x_46_im - (x_46_re * (y_46_im / y_46_re))) / y_46_re;
	elseif (y_46_re <= 4.9e-59)
		tmp = x_46_re / -y_46_im;
	else
		tmp = (x_46_im - (x_46_re / (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, -5.8e-96], N[(N[(x$46$im - N[(x$46$re * N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision], If[LessEqual[y$46$re, 4.9e-59], N[(x$46$re / (-y$46$im)), $MachinePrecision], N[(N[(x$46$im - N[(x$46$re / 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 < -5.79999999999999987e-96

   1. Initial program 56.3%

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

   3. Taylor expanded in y.re around inf 69.8%

      \[\leadsto \color{blue}{\frac{x.im + -1 \cdot \frac{x.re \cdot y.im}{y.re}}{y.re}} \]
   4. Step-by-step derivation

      1. remove-double-neg 69.8%

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

      2. mul-1-neg 69.8%

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

      3. neg-mul-1 69.8%

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

      4. distribute-lft-in 69.8%

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

      5. mul-1-neg 69.8%

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

      6. distribute-neg-in 69.8%

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

      7. mul-1-neg 69.8%

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

      8. remove-double-neg 69.8%

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

      9. unsub-neg 69.8%

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

      10. associate-/l* 80.5%

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

   5. Simplified 80.5%

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

   if -5.79999999999999987e-96 < y.re < 4.89999999999999977e-59

   1. Initial program 71.9%

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

   3. Taylor expanded in y.re around 0 72.4%

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

      1. associate-*r/ 72.4%

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

      2. neg-mul-1 72.4%

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

   5. Simplified 72.4%

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

   if 4.89999999999999977e-59 < y.re

   1. Initial program 64.7%

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

   3. Taylor expanded in y.re around inf 80.0%

      \[\leadsto \color{blue}{\frac{x.im + -1 \cdot \frac{x.re \cdot y.im}{y.re}}{y.re}} \]
   4. Step-by-step derivation

      1. remove-double-neg 80.0%

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

      2. mul-1-neg 80.0%

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

      3. neg-mul-1 80.0%

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

      4. distribute-lft-in 80.0%

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

      5. mul-1-neg 80.0%

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

      6. distribute-neg-in 80.0%

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

      7. mul-1-neg 80.0%

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

      8. remove-double-neg 80.0%

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

      9. unsub-neg 80.0%

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

      10. associate-/l* 82.9%

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

   5. Simplified 82.9%

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

      1. clear-num 82.8%

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

      2. un-div-inv 82.9%

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

   7. Applied egg-rr 82.9%

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

4. Final simplification 77.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -5.8 \cdot 10^{-96}:\\ \;\;\;\;\frac{x.im - x.re \cdot \frac{y.im}{y.re}}{y.re}\\ \mathbf{elif}\;y.re \leq 4.9 \cdot 10^{-59}:\\ \;\;\;\;\frac{x.re}{-y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im - \frac{x.re}{\frac{y.re}{y.im}}}{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 76.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -6.2 \cdot 10^{-47}:\\ \;\;\;\;\frac{x.im - x.re \cdot \frac{y.im}{y.re}}{y.re}\\ \mathbf{elif}\;y.re \leq 1.6 \cdot 10^{-58}:\\ \;\;\;\;\frac{x.im \cdot \frac{y.re}{y.im} - x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im - \frac{x.re}{\frac{y.re}{y.im}}}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.re -6.2e-47)
   (/ (- x.im (* x.re (/ y.im y.re))) y.re)
   (if (<= y.re 1.6e-58)
     (/ (- (* x.im (/ y.re y.im)) x.re) y.im)
     (/ (- x.im (/ x.re (/ 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 <= -6.2e-47) {
		tmp = (x_46_im - (x_46_re * (y_46_im / y_46_re))) / y_46_re;
	} else if (y_46_re <= 1.6e-58) {
		tmp = ((x_46_im * (y_46_re / y_46_im)) - x_46_re) / y_46_im;
	} else {
		tmp = (x_46_im - (x_46_re / (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 <= (-6.2d-47)) then
        tmp = (x_46im - (x_46re * (y_46im / y_46re))) / y_46re
    else if (y_46re <= 1.6d-58) then
        tmp = ((x_46im * (y_46re / y_46im)) - x_46re) / y_46im
    else
        tmp = (x_46im - (x_46re / (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 <= -6.2e-47) {
		tmp = (x_46_im - (x_46_re * (y_46_im / y_46_re))) / y_46_re;
	} else if (y_46_re <= 1.6e-58) {
		tmp = ((x_46_im * (y_46_re / y_46_im)) - x_46_re) / y_46_im;
	} else {
		tmp = (x_46_im - (x_46_re / (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 <= -6.2e-47:
		tmp = (x_46_im - (x_46_re * (y_46_im / y_46_re))) / y_46_re
	elif y_46_re <= 1.6e-58:
		tmp = ((x_46_im * (y_46_re / y_46_im)) - x_46_re) / y_46_im
	else:
		tmp = (x_46_im - (x_46_re / (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 <= -6.2e-47)
		tmp = Float64(Float64(x_46_im - Float64(x_46_re * Float64(y_46_im / y_46_re))) / y_46_re);
	elseif (y_46_re <= 1.6e-58)
		tmp = Float64(Float64(Float64(x_46_im * Float64(y_46_re / y_46_im)) - x_46_re) / y_46_im);
	else
		tmp = Float64(Float64(x_46_im - Float64(x_46_re / 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 <= -6.2e-47)
		tmp = (x_46_im - (x_46_re * (y_46_im / y_46_re))) / y_46_re;
	elseif (y_46_re <= 1.6e-58)
		tmp = ((x_46_im * (y_46_re / y_46_im)) - x_46_re) / y_46_im;
	else
		tmp = (x_46_im - (x_46_re / (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, -6.2e-47], N[(N[(x$46$im - N[(x$46$re * N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision], If[LessEqual[y$46$re, 1.6e-58], N[(N[(N[(x$46$im * N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision] - x$46$re), $MachinePrecision] / y$46$im), $MachinePrecision], N[(N[(x$46$im - N[(x$46$re / 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 < -6.1999999999999996e-47

   1. Initial program 53.1%

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

   3. Taylor expanded in y.re around inf 71.7%

      \[\leadsto \color{blue}{\frac{x.im + -1 \cdot \frac{x.re \cdot y.im}{y.re}}{y.re}} \]
   4. Step-by-step derivation

      1. remove-double-neg 71.7%

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

      2. mul-1-neg 71.7%

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

      3. neg-mul-1 71.7%

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

      4. distribute-lft-in 71.7%

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

      5. mul-1-neg 71.7%

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

      6. distribute-neg-in 71.7%

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

      7. mul-1-neg 71.7%

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

      8. remove-double-neg 71.7%

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

      9. unsub-neg 71.7%

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

      10. associate-/l* 83.5%

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

   5. Simplified 83.5%

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

   if -6.1999999999999996e-47 < y.re < 1.6e-58

   1. Initial program 73.0%

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

      1. div-sub 69.1%

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

      2. *-commutative 69.1%

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

      3. add-sqr-sqrt 69.1%

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

      4. times-frac 69.7%

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

      5. fma-neg 69.7%

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

      6. hypot-define 69.7%

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

      7. hypot-define 72.0%

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

      8. associate-/l* 73.9%

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

      9. add-sqr-sqrt 73.9%

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

      10. pow2 73.9%

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

      11. hypot-define 73.9%

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

   4. Applied egg-rr 73.9%

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

   5. Taylor expanded in y.im around inf 88.2%

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

      1. associate-/l* 88.7%

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

   7. Simplified 88.7%

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

   if 1.6e-58 < y.re

   1. Initial program 64.7%

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

   3. Taylor expanded in y.re around inf 80.0%

      \[\leadsto \color{blue}{\frac{x.im + -1 \cdot \frac{x.re \cdot y.im}{y.re}}{y.re}} \]
   4. Step-by-step derivation

      1. remove-double-neg 80.0%

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

      2. mul-1-neg 80.0%

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

      3. neg-mul-1 80.0%

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

      4. distribute-lft-in 80.0%

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

      5. mul-1-neg 80.0%

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

      6. distribute-neg-in 80.0%

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

      7. mul-1-neg 80.0%

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

      8. remove-double-neg 80.0%

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

      9. unsub-neg 80.0%

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

      10. associate-/l* 82.9%

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

   5. Simplified 82.9%

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

      1. clear-num 82.8%

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

      2. un-div-inv 82.9%

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

   7. Applied egg-rr 82.9%

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

4. Final simplification 85.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -6.2 \cdot 10^{-47}:\\ \;\;\;\;\frac{x.im - x.re \cdot \frac{y.im}{y.re}}{y.re}\\ \mathbf{elif}\;y.re \leq 1.6 \cdot 10^{-58}:\\ \;\;\;\;\frac{x.im \cdot \frac{y.re}{y.im} - x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im - \frac{x.re}{\frac{y.re}{y.im}}}{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 63.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -1.5 \cdot 10^{-65} \lor \neg \left(y.re \leq 1.05 \cdot 10^{-56}\right):\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{-y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.re -1.5e-65) (not (<= y.re 1.05e-56)))
   (/ x.im y.re)
   (/ x.re (- 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 <= -1.5e-65) || !(y_46_re <= 1.05e-56)) {
		tmp = x_46_im / y_46_re;
	} else {
		tmp = x_46_re / -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 <= (-1.5d-65)) .or. (.not. (y_46re <= 1.05d-56))) then
        tmp = x_46im / y_46re
    else
        tmp = x_46re / -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 <= -1.5e-65) || !(y_46_re <= 1.05e-56)) {
		tmp = x_46_im / y_46_re;
	} else {
		tmp = x_46_re / -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 <= -1.5e-65) or not (y_46_re <= 1.05e-56):
		tmp = x_46_im / y_46_re
	else:
		tmp = x_46_re / -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 <= -1.5e-65) || !(y_46_re <= 1.05e-56))
		tmp = Float64(x_46_im / y_46_re);
	else
		tmp = Float64(x_46_re / Float64(-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 <= -1.5e-65) || ~((y_46_re <= 1.05e-56)))
		tmp = x_46_im / y_46_re;
	else
		tmp = x_46_re / -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, -1.5e-65], N[Not[LessEqual[y$46$re, 1.05e-56]], $MachinePrecision]], N[(x$46$im / y$46$re), $MachinePrecision], N[(x$46$re / (-y$46$im)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.re < -1.49999999999999999e-65 or 1.05000000000000003e-56 < y.re

   1. Initial program 58.7%

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

   3. Taylor expanded in y.re around inf 62.2%

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

   if -1.49999999999999999e-65 < y.re < 1.05000000000000003e-56

   1. Initial program 73.2%

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

   3. Taylor expanded in y.re around 0 70.2%

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

      1. associate-*r/ 70.2%

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

      2. neg-mul-1 70.2%

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

   5. Simplified 70.2%

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

4. Final simplification 65.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -1.5 \cdot 10^{-65} \lor \neg \left(y.re \leq 1.05 \cdot 10^{-56}\right):\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{-y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 46.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -3.6 \cdot 10^{+116} \lor \neg \left(y.im \leq 7.2 \cdot 10^{+95}\right):\\ \;\;\;\;\frac{x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.im -3.6e+116) (not (<= y.im 7.2e+95)))
   (/ x.re y.im)
   (/ x.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_im <= -3.6e+116) || !(y_46_im <= 7.2e+95)) {
		tmp = x_46_re / y_46_im;
	} else {
		tmp = x_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_46im <= (-3.6d+116)) .or. (.not. (y_46im <= 7.2d+95))) then
        tmp = x_46re / y_46im
    else
        tmp = x_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_im <= -3.6e+116) || !(y_46_im <= 7.2e+95)) {
		tmp = x_46_re / y_46_im;
	} else {
		tmp = x_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_im <= -3.6e+116) or not (y_46_im <= 7.2e+95):
		tmp = x_46_re / y_46_im
	else:
		tmp = x_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_im <= -3.6e+116) || !(y_46_im <= 7.2e+95))
		tmp = Float64(x_46_re / y_46_im);
	else
		tmp = Float64(x_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_im <= -3.6e+116) || ~((y_46_im <= 7.2e+95)))
		tmp = x_46_re / y_46_im;
	else
		tmp = x_46_im / y_46_re;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[Or[LessEqual[y$46$im, -3.6e+116], N[Not[LessEqual[y$46$im, 7.2e+95]], $MachinePrecision]], N[(x$46$re / y$46$im), $MachinePrecision], N[(x$46$im / y$46$re), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.im < -3.59999999999999971e116 or 7.19999999999999955e95 < y.im

   1. Initial program 42.7%

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

      1. div-sub 42.7%

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

      2. *-commutative 42.7%

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

      3. add-sqr-sqrt 42.7%

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

      4. times-frac 42.8%

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

      5. fma-neg 42.8%

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

      6. hypot-define 42.8%

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

      7. hypot-define 55.6%

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

      8. associate-/l* 56.8%

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

      9. add-sqr-sqrt 56.8%

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

      10. pow2 56.8%

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

      11. hypot-define 56.8%

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

   4. Applied egg-rr 56.8%

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

   5. Taylor expanded in y.im around inf 82.7%

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

      1. div-inv 82.7%

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

      2. associate-/l* 84.7%

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

      3. fma-neg 84.7%

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

      4. add-sqr-sqrt 39.6%

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

      5. sqrt-unprod 55.8%

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

      6. sqr-neg 55.8%

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

      7. sqrt-unprod 20.1%

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

      8. add-sqr-sqrt 42.2%

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

   7. Applied egg-rr 42.2%

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

   8. Taylor expanded in x.im around 0 30.8%

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

   if -3.59999999999999971e116 < y.im < 7.19999999999999955e95

   1. Initial program 74.6%

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

   3. Taylor expanded in y.re around inf 53.4%

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

4. Final simplification 46.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -3.6 \cdot 10^{+116} \lor \neg \left(y.im \leq 7.2 \cdot 10^{+95}\right):\\ \;\;\;\;\frac{x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 43.5% accurate, 5.0× speedup

Math

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

FPCore

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

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_re;
}

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_46re
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_re;
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 64.9%

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

3. Taylor expanded in y.re around inf 40.8%

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

4. Final simplification 40.8%

   \[\leadsto \frac{x.im}{y.re} \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024059 
(FPCore (x.re x.im y.re y.im)
  :name "_divideComplex, imaginary part"
  :precision binary64
  (/ (- (* x.im y.re) (* x.re y.im)) (+ (* y.re y.re) (* y.im y.im))))