_divideComplex, imaginary part

Percentage Accurate: 61.4% → 90.3%

Time: 14.4s

Alternatives: 14

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: 61.4% 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: 90.3% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x.re \leq -7.4 \cdot 10^{+33} \lor \neg \left(x.re \leq 5 \cdot 10^{+46}\right):\\ \;\;\;\;\frac{x.im \cdot \frac{y.re}{x.re} - y.im}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{x.re}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y.re}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{y.im}{{\left(\mathsf{hypot}\left(y.re, y.im\right)\right)}^{2}} \cdot \left(-x.re\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= x.re -7.4e+33) (not (<= x.re 5e+46)))
   (*
    (/ (- (* x.im (/ y.re x.re)) y.im) (hypot y.re y.im))
    (/ x.re (hypot y.re y.im)))
   (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)))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((x_46_re <= -7.4e+33) || !(x_46_re <= 5e+46)) {
		tmp = (((x_46_im * (y_46_re / x_46_re)) - y_46_im) / hypot(y_46_re, y_46_im)) * (x_46_re / hypot(y_46_re, y_46_im));
	} else {
		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));
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if ((x_46_re <= -7.4e+33) || !(x_46_re <= 5e+46))
		tmp = Float64(Float64(Float64(Float64(x_46_im * Float64(y_46_re / x_46_re)) - y_46_im) / hypot(y_46_re, y_46_im)) * Float64(x_46_re / hypot(y_46_re, y_46_im)));
	else
		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)));
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[Or[LessEqual[x$46$re, -7.4e+33], N[Not[LessEqual[x$46$re, 5e+46]], $MachinePrecision]], N[(N[(N[(N[(x$46$im * N[(y$46$re / x$46$re), $MachinePrecision]), $MachinePrecision] - y$46$im), $MachinePrecision] / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision] * N[(x$46$re / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision]), $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]]

Derivation

1. Split input into 2 regimes

2. if x.re < -7.3999999999999997e33 or 5.0000000000000002e46 < x.re

   1. Initial program 57.4%

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

   3. Taylor expanded in x.re around inf 57.4%

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

      1. *-commutative 57.4%

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

      2. +-commutative 57.4%

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

      3. add-sqr-sqrt 57.4%

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

      4. hypot-undefine 57.4%

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

      5. hypot-undefine 57.4%

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

      6. times-frac 92.1%

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

      7. associate-/l* 96.9%

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

      8. hypot-undefine 60.1%

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

      9. +-commutative 60.1%

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

      10. hypot-define 96.9%

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

      11. hypot-undefine 60.1%

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

      12. +-commutative 60.1%

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

      13. hypot-define 96.9%

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

   5. Applied egg-rr 96.9%

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

   if -7.3999999999999997e33 < x.re < 5.0000000000000002e46

   1. Initial program 72.2%

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

      1. div-sub 71.3%

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

      2. *-commutative 71.3%

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

      3. fma-define 71.3%

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

      4. add-sqr-sqrt 71.3%

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

      5. times-frac 74.8%

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

      6. fma-neg 74.8%

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

      7. fma-define 74.8%

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

      8. hypot-define 74.9%

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

      9. fma-define 74.9%

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

      10. hypot-define 91.0%

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

      11. associate-/l* 91.9%

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

      12. fma-define 91.9%

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

      13. add-sqr-sqrt 91.9%

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

      14. pow2 91.9%

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

   4. Applied egg-rr 91.9%

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

4. Final simplification 93.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \leq -7.4 \cdot 10^{+33} \lor \neg \left(x.re \leq 5 \cdot 10^{+46}\right):\\ \;\;\;\;\frac{x.im \cdot \frac{y.re}{x.re} - y.im}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{x.re}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y.re}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{y.im}{{\left(\mathsf{hypot}\left(y.re, y.im\right)\right)}^{2}} \cdot \left(-x.re\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 89.9% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{fma}\left(y.re, \frac{x.im}{y.im}, -x.re\right)}{\mathsf{hypot}\left(y.im, y.re\right)}\\ \mathbf{if}\;y.im \leq -7.4 \cdot 10^{-165}:\\ \;\;\;\;t\_0 \cdot \frac{y.im}{\mathsf{hypot}\left(y.im, y.re\right)}\\ \mathbf{elif}\;y.im \leq 1.15 \cdot 10^{-106}:\\ \;\;\;\;\frac{x.im - x.re \cdot \frac{y.im}{y.re}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot {\left(\sqrt{\frac{y.im}{\mathsf{hypot}\left(y.re, y.im\right)}}\right)}^{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (/ (fma y.re (/ x.im y.im) (- x.re)) (hypot y.im y.re))))
   (if (<= y.im -7.4e-165)
     (* t_0 (/ y.im (hypot y.im y.re)))
     (if (<= y.im 1.15e-106)
       (/ (- x.im (* x.re (/ y.im y.re))) y.re)
       (* t_0 (pow (sqrt (/ y.im (hypot y.re y.im))) 2.0))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = fma(y_46_re, (x_46_im / y_46_im), -x_46_re) / hypot(y_46_im, y_46_re);
	double tmp;
	if (y_46_im <= -7.4e-165) {
		tmp = t_0 * (y_46_im / hypot(y_46_im, y_46_re));
	} else if (y_46_im <= 1.15e-106) {
		tmp = (x_46_im - (x_46_re * (y_46_im / y_46_re))) / y_46_re;
	} else {
		tmp = t_0 * pow(sqrt((y_46_im / hypot(y_46_re, y_46_im))), 2.0);
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(fma(y_46_re, Float64(x_46_im / y_46_im), Float64(-x_46_re)) / hypot(y_46_im, y_46_re))
	tmp = 0.0
	if (y_46_im <= -7.4e-165)
		tmp = Float64(t_0 * Float64(y_46_im / hypot(y_46_im, y_46_re)));
	elseif (y_46_im <= 1.15e-106)
		tmp = Float64(Float64(x_46_im - Float64(x_46_re * Float64(y_46_im / y_46_re))) / y_46_re);
	else
		tmp = Float64(t_0 * (sqrt(Float64(y_46_im / hypot(y_46_re, y_46_im))) ^ 2.0));
	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 * N[(x$46$im / y$46$im), $MachinePrecision] + (-x$46$re)), $MachinePrecision] / N[Sqrt[y$46$im ^ 2 + y$46$re ^ 2], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$im, -7.4e-165], N[(t$95$0 * N[(y$46$im / N[Sqrt[y$46$im ^ 2 + y$46$re ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, 1.15e-106], N[(N[(x$46$im - N[(x$46$re * N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision], N[(t$95$0 * N[Power[N[Sqrt[N[(y$46$im / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y.im < -7.40000000000000003e-165

   1. Initial program 66.7%

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

   3. Taylor expanded in y.im around inf 64.8%

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

      1. *-commutative 64.8%

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

      2. associate-/l* 63.9%

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

   5. Simplified 63.9%

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

      1. *-commutative 63.9%

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

      2. add-sqr-sqrt 63.9%

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

      3. hypot-undefine 63.9%

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

      4. hypot-undefine 63.9%

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

      5. times-frac 92.0%

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

      6. fma-neg 92.0%

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

      7. hypot-undefine 67.4%

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

      8. +-commutative 67.4%

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

      9. hypot-undefine 92.0%

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

      10. hypot-undefine 67.4%

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

      11. +-commutative 67.4%

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

      12. hypot-undefine 92.0%

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

   7. Applied egg-rr 92.0%

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

   if -7.40000000000000003e-165 < y.im < 1.15e-106

   1. Initial program 76.3%

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

   3. Taylor expanded in y.re around inf 93.6%

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

      1. mul-1-neg 93.6%

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

      2. unsub-neg 93.6%

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

      3. unsub-neg 93.6%

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

      4. remove-double-neg 93.6%

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

      5. mul-1-neg 93.6%

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

      6. neg-mul-1 93.6%

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

      7. mul-1-neg 93.6%

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

      8. distribute-lft-in 93.6%

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

      9. distribute-lft-in 93.6%

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

      10. mul-1-neg 93.6%

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

      11. unsub-neg 93.6%

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

      12. neg-mul-1 93.6%

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

      13. mul-1-neg 93.6%

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

      14. remove-double-neg 93.6%

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

      15. associate-/l* 93.7%

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

   5. Simplified 93.7%

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

   if 1.15e-106 < y.im

   1. Initial program 56.2%

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

   3. Taylor expanded in y.im around inf 56.2%

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

      1. *-commutative 56.2%

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

      2. associate-/l* 56.2%

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

   5. Simplified 56.2%

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

      1. *-commutative 56.2%

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

      2. add-sqr-sqrt 56.2%

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

      3. hypot-undefine 56.2%

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

      4. hypot-undefine 56.2%

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

      5. times-frac 91.1%

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

      6. fma-neg 91.1%

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

      7. hypot-undefine 58.0%

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

      8. +-commutative 58.0%

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

      9. hypot-undefine 91.1%

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

      10. hypot-undefine 58.0%

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

      11. +-commutative 58.0%

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

      12. hypot-undefine 91.1%

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

   7. Applied egg-rr 91.1%

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

      1. add-sqr-sqrt 91.1%

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

      2. pow2 91.1%

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

      3. hypot-undefine 58.0%

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

      4. +-commutative 58.0%

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

      5. hypot-define 91.1%

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

   9. Applied egg-rr 91.1%

      \[\leadsto \frac{\mathsf{fma}\left(y.re, \frac{x.im}{y.im}, -x.re\right)}{\mathsf{hypot}\left(y.im, y.re\right)} \cdot \color{blue}{{\left(\sqrt{\frac{y.im}{\mathsf{hypot}\left(y.re, y.im\right)}}\right)}^{2}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 92.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -7.4 \cdot 10^{-165}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y.re, \frac{x.im}{y.im}, -x.re\right)}{\mathsf{hypot}\left(y.im, y.re\right)} \cdot \frac{y.im}{\mathsf{hypot}\left(y.im, y.re\right)}\\ \mathbf{elif}\;y.im \leq 1.15 \cdot 10^{-106}:\\ \;\;\;\;\frac{x.im - x.re \cdot \frac{y.im}{y.re}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y.re, \frac{x.im}{y.im}, -x.re\right)}{\mathsf{hypot}\left(y.im, y.re\right)} \cdot {\left(\sqrt{\frac{y.im}{\mathsf{hypot}\left(y.re, y.im\right)}}\right)}^{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 90.0% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -5.5 \cdot 10^{-165} \lor \neg \left(y.im \leq 9.6 \cdot 10^{-109}\right):\\ \;\;\;\;\frac{\mathsf{fma}\left(y.re, \frac{x.im}{y.im}, -x.re\right)}{\mathsf{hypot}\left(y.im, y.re\right)} \cdot \frac{y.im}{\mathsf{hypot}\left(y.im, y.re\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im - x.re \cdot \frac{y.im}{y.re}}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.im -5.5e-165) (not (<= y.im 9.6e-109)))
   (*
    (/ (fma y.re (/ x.im y.im) (- x.re)) (hypot y.im y.re))
    (/ y.im (hypot y.im y.re)))
   (/ (- x.im (* x.re (/ y.im y.re))) 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 <= -5.5e-165) || !(y_46_im <= 9.6e-109)) {
		tmp = (fma(y_46_re, (x_46_im / y_46_im), -x_46_re) / hypot(y_46_im, y_46_re)) * (y_46_im / hypot(y_46_im, y_46_re));
	} else {
		tmp = (x_46_im - (x_46_re * (y_46_im / y_46_re))) / 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 <= -5.5e-165) || !(y_46_im <= 9.6e-109))
		tmp = Float64(Float64(fma(y_46_re, Float64(x_46_im / y_46_im), Float64(-x_46_re)) / hypot(y_46_im, y_46_re)) * Float64(y_46_im / hypot(y_46_im, y_46_re)));
	else
		tmp = Float64(Float64(x_46_im - Float64(x_46_re * Float64(y_46_im / y_46_re))) / y_46_re);
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[Or[LessEqual[y$46$im, -5.5e-165], N[Not[LessEqual[y$46$im, 9.6e-109]], $MachinePrecision]], N[(N[(N[(y$46$re * N[(x$46$im / y$46$im), $MachinePrecision] + (-x$46$re)), $MachinePrecision] / N[Sqrt[y$46$im ^ 2 + y$46$re ^ 2], $MachinePrecision]), $MachinePrecision] * N[(y$46$im / N[Sqrt[y$46$im ^ 2 + y$46$re ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$im - N[(x$46$re * N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.im < -5.49999999999999969e-165 or 9.59999999999999955e-109 < y.im

   1. Initial program 62.2%

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

   3. Taylor expanded in y.im around inf 61.1%

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

      1. *-commutative 61.1%

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

      2. associate-/l* 60.5%

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

   5. Simplified 60.5%

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

      1. *-commutative 60.5%

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

      2. add-sqr-sqrt 60.5%

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

      3. hypot-undefine 60.5%

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

      4. hypot-undefine 60.5%

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

      5. times-frac 91.6%

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

      6. fma-neg 91.6%

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

      7. hypot-undefine 63.3%

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

      8. +-commutative 63.3%

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

      9. hypot-undefine 91.6%

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

      10. hypot-undefine 63.3%

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

      11. +-commutative 63.3%

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

      12. hypot-undefine 91.6%

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

   7. Applied egg-rr 91.6%

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

   if -5.49999999999999969e-165 < y.im < 9.59999999999999955e-109

   1. Initial program 76.3%

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

   3. Taylor expanded in y.re around inf 93.6%

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

      1. mul-1-neg 93.6%

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

      2. unsub-neg 93.6%

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

      3. unsub-neg 93.6%

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

      4. remove-double-neg 93.6%

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

      5. mul-1-neg 93.6%

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

      6. neg-mul-1 93.6%

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

      7. mul-1-neg 93.6%

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

      8. distribute-lft-in 93.6%

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

      9. distribute-lft-in 93.6%

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

      10. mul-1-neg 93.6%

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

      11. unsub-neg 93.6%

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

      12. neg-mul-1 93.6%

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

      13. mul-1-neg 93.6%

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

      14. remove-double-neg 93.6%

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

      15. associate-/l* 93.7%

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

   5. Simplified 93.7%

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

4. Final simplification 92.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -5.5 \cdot 10^{-165} \lor \neg \left(y.im \leq 9.6 \cdot 10^{-109}\right):\\ \;\;\;\;\frac{\mathsf{fma}\left(y.re, \frac{x.im}{y.im}, -x.re\right)}{\mathsf{hypot}\left(y.im, y.re\right)} \cdot \frac{y.im}{\mathsf{hypot}\left(y.im, y.re\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im - x.re \cdot \frac{y.im}{y.re}}{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 83.0% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x.im \cdot y.re - x.re \cdot y.im\\ \mathbf{if}\;y.im \leq -5.8 \cdot 10^{+85}:\\ \;\;\;\;\frac{y.re \cdot \frac{x.im}{y.im} - x.re}{y.im}\\ \mathbf{elif}\;y.im \leq -1.3 \cdot 10^{-159}:\\ \;\;\;\;\frac{t\_0}{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)}\\ \mathbf{elif}\;y.im \leq 2.05 \cdot 10^{-46}:\\ \;\;\;\;\frac{x.im - x.re \cdot \frac{y.im}{y.re}}{y.re}\\ \mathbf{elif}\;y.im \leq 1.65 \cdot 10^{+47}:\\ \;\;\;\;\frac{t\_0}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y.re, \frac{x.im}{y.im}, -x.re\right)}{\mathsf{hypot}\left(y.im, y.re\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (- (* x.im y.re) (* x.re y.im))))
   (if (<= y.im -5.8e+85)
     (/ (- (* y.re (/ x.im y.im)) x.re) y.im)
     (if (<= y.im -1.3e-159)
       (/ t_0 (fma y.im y.im (* y.re y.re)))
       (if (<= y.im 2.05e-46)
         (/ (- x.im (* x.re (/ y.im y.re))) y.re)
         (if (<= y.im 1.65e+47)
           (/ t_0 (+ (* y.re y.re) (* y.im y.im)))
           (/ (fma y.re (/ x.im y.im) (- x.re)) (hypot y.im y.re))))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = (x_46_im * y_46_re) - (x_46_re * y_46_im);
	double tmp;
	if (y_46_im <= -5.8e+85) {
		tmp = ((y_46_re * (x_46_im / y_46_im)) - x_46_re) / y_46_im;
	} else if (y_46_im <= -1.3e-159) {
		tmp = t_0 / fma(y_46_im, y_46_im, (y_46_re * y_46_re));
	} else if (y_46_im <= 2.05e-46) {
		tmp = (x_46_im - (x_46_re * (y_46_im / y_46_re))) / y_46_re;
	} else if (y_46_im <= 1.65e+47) {
		tmp = t_0 / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	} else {
		tmp = fma(y_46_re, (x_46_im / y_46_im), -x_46_re) / hypot(y_46_im, y_46_re);
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(Float64(x_46_im * y_46_re) - Float64(x_46_re * y_46_im))
	tmp = 0.0
	if (y_46_im <= -5.8e+85)
		tmp = Float64(Float64(Float64(y_46_re * Float64(x_46_im / y_46_im)) - x_46_re) / y_46_im);
	elseif (y_46_im <= -1.3e-159)
		tmp = Float64(t_0 / fma(y_46_im, y_46_im, Float64(y_46_re * y_46_re)));
	elseif (y_46_im <= 2.05e-46)
		tmp = Float64(Float64(x_46_im - Float64(x_46_re * Float64(y_46_im / y_46_re))) / y_46_re);
	elseif (y_46_im <= 1.65e+47)
		tmp = Float64(t_0 / Float64(Float64(y_46_re * y_46_re) + Float64(y_46_im * y_46_im)));
	else
		tmp = Float64(fma(y_46_re, Float64(x_46_im / y_46_im), Float64(-x_46_re)) / hypot(y_46_im, y_46_re));
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[(x$46$im * y$46$re), $MachinePrecision] - N[(x$46$re * y$46$im), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$im, -5.8e+85], N[(N[(N[(y$46$re * N[(x$46$im / y$46$im), $MachinePrecision]), $MachinePrecision] - x$46$re), $MachinePrecision] / y$46$im), $MachinePrecision], If[LessEqual[y$46$im, -1.3e-159], N[(t$95$0 / N[(y$46$im * y$46$im + N[(y$46$re * y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, 2.05e-46], 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$46$im, 1.65e+47], N[(t$95$0 / N[(N[(y$46$re * y$46$re), $MachinePrecision] + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y$46$re * N[(x$46$im / y$46$im), $MachinePrecision] + (-x$46$re)), $MachinePrecision] / N[Sqrt[y$46$im ^ 2 + y$46$re ^ 2], $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if y.im < -5.79999999999999995e85

   1. Initial program 43.7%

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

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

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

      1. +-commutative 72.4%

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

      2. mul-1-neg 72.4%

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

      3. unsub-neg 72.4%

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

      4. unpow2 72.4%

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

      5. associate-/r* 78.3%

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

      6. div-sub 78.3%

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

      7. *-commutative 78.3%

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

      8. associate-/l* 88.3%

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

   5. Simplified 88.3%

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

   if -5.79999999999999995e85 < y.im < -1.2999999999999999e-159

   1. Initial program 90.2%

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

      1. fma-neg 90.2%

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

      2. distribute-rgt-neg-out 90.2%

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

      3. +-commutative 90.2%

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

      4. fma-define 90.2%

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

   3. Simplified 90.2%

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

      1. distribute-rgt-neg-out 90.2%

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

      2. fma-neg 90.2%

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

   6. Applied egg-rr 90.2%

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

   if -1.2999999999999999e-159 < y.im < 2.05e-46

   1. Initial program 75.0%

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

   3. Taylor expanded in y.re around inf 90.2%

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

      1. mul-1-neg 90.2%

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

      2. unsub-neg 90.2%

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

      3. unsub-neg 90.2%

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

      4. remove-double-neg 90.2%

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

      5. mul-1-neg 90.2%

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

      6. neg-mul-1 90.2%

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

      7. mul-1-neg 90.2%

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

      8. distribute-lft-in 90.2%

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

      9. distribute-lft-in 90.2%

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

      10. mul-1-neg 90.2%

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

      11. unsub-neg 90.2%

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

      12. neg-mul-1 90.2%

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

      13. mul-1-neg 90.2%

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

      14. remove-double-neg 90.2%

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

      15. associate-/l* 90.2%

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

   5. Simplified 90.2%

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

   if 2.05e-46 < y.im < 1.65e47

   1. Initial program 82.6%

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

   if 1.65e47 < y.im

   1. Initial program 38.3%

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

   3. Taylor expanded in y.im around inf 38.3%

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

      1. *-commutative 38.3%

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

      2. associate-/l* 38.3%

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

   5. Simplified 38.3%

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

      1. *-commutative 38.3%

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

      2. add-sqr-sqrt 38.3%

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

      3. hypot-undefine 38.3%

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

      4. hypot-undefine 38.3%

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

      5. times-frac 95.5%

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

      6. fma-neg 95.5%

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

      7. hypot-undefine 41.4%

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

      8. +-commutative 41.4%

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

      9. hypot-undefine 95.5%

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

      10. hypot-undefine 41.4%

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

      11. +-commutative 41.4%

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

      12. hypot-undefine 95.5%

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

   7. Applied egg-rr 95.5%

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

   8. Taylor expanded in y.im around inf 80.2%

      \[\leadsto \frac{\mathsf{fma}\left(y.re, \frac{x.im}{y.im}, -x.re\right)}{\mathsf{hypot}\left(y.im, y.re\right)} \cdot \color{blue}{1} \]
3. Recombined 5 regimes into one program.

4. Final simplification 87.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -5.8 \cdot 10^{+85}:\\ \;\;\;\;\frac{y.re \cdot \frac{x.im}{y.im} - x.re}{y.im}\\ \mathbf{elif}\;y.im \leq -1.3 \cdot 10^{-159}:\\ \;\;\;\;\frac{x.im \cdot y.re - x.re \cdot y.im}{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)}\\ \mathbf{elif}\;y.im \leq 2.05 \cdot 10^{-46}:\\ \;\;\;\;\frac{x.im - x.re \cdot \frac{y.im}{y.re}}{y.re}\\ \mathbf{elif}\;y.im \leq 1.65 \cdot 10^{+47}:\\ \;\;\;\;\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y.re, \frac{x.im}{y.im}, -x.re\right)}{\mathsf{hypot}\left(y.im, y.re\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 88.6% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x.re \leq -3 \cdot 10^{-64} \lor \neg \left(x.re \leq 7.2 \cdot 10^{-35}\right):\\ \;\;\;\;\frac{x.im \cdot \frac{y.re}{x.re} - y.im}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{x.re}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{y.re}{\mathsf{hypot}\left(y.im, y.re\right)} \cdot \frac{x.im}{\mathsf{hypot}\left(y.im, y.re\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= x.re -3e-64) (not (<= x.re 7.2e-35)))
   (*
    (/ (- (* x.im (/ y.re x.re)) y.im) (hypot y.re y.im))
    (/ x.re (hypot y.re y.im)))
   (* (/ y.re (hypot y.im y.re)) (/ x.im (hypot 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 ((x_46_re <= -3e-64) || !(x_46_re <= 7.2e-35)) {
		tmp = (((x_46_im * (y_46_re / x_46_re)) - y_46_im) / hypot(y_46_re, y_46_im)) * (x_46_re / hypot(y_46_re, y_46_im));
	} else {
		tmp = (y_46_re / hypot(y_46_im, y_46_re)) * (x_46_im / hypot(y_46_im, y_46_re));
	}
	return tmp;
}

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((x_46_re <= -3e-64) || !(x_46_re <= 7.2e-35)) {
		tmp = (((x_46_im * (y_46_re / x_46_re)) - y_46_im) / Math.hypot(y_46_re, y_46_im)) * (x_46_re / Math.hypot(y_46_re, y_46_im));
	} else {
		tmp = (y_46_re / Math.hypot(y_46_im, y_46_re)) * (x_46_im / Math.hypot(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 (x_46_re <= -3e-64) or not (x_46_re <= 7.2e-35):
		tmp = (((x_46_im * (y_46_re / x_46_re)) - y_46_im) / math.hypot(y_46_re, y_46_im)) * (x_46_re / math.hypot(y_46_re, y_46_im))
	else:
		tmp = (y_46_re / math.hypot(y_46_im, y_46_re)) * (x_46_im / math.hypot(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 ((x_46_re <= -3e-64) || !(x_46_re <= 7.2e-35))
		tmp = Float64(Float64(Float64(Float64(x_46_im * Float64(y_46_re / x_46_re)) - y_46_im) / hypot(y_46_re, y_46_im)) * Float64(x_46_re / hypot(y_46_re, y_46_im)));
	else
		tmp = Float64(Float64(y_46_re / hypot(y_46_im, y_46_re)) * Float64(x_46_im / hypot(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 ((x_46_re <= -3e-64) || ~((x_46_re <= 7.2e-35)))
		tmp = (((x_46_im * (y_46_re / x_46_re)) - y_46_im) / hypot(y_46_re, y_46_im)) * (x_46_re / hypot(y_46_re, y_46_im));
	else
		tmp = (y_46_re / hypot(y_46_im, y_46_re)) * (x_46_im / hypot(y_46_im, y_46_re));
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[Or[LessEqual[x$46$re, -3e-64], N[Not[LessEqual[x$46$re, 7.2e-35]], $MachinePrecision]], N[(N[(N[(N[(x$46$im * N[(y$46$re / x$46$re), $MachinePrecision]), $MachinePrecision] - y$46$im), $MachinePrecision] / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision] * N[(x$46$re / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y$46$re / N[Sqrt[y$46$im ^ 2 + y$46$re ^ 2], $MachinePrecision]), $MachinePrecision] * N[(x$46$im / N[Sqrt[y$46$im ^ 2 + y$46$re ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x.re < -3.0000000000000001e-64 or 7.20000000000000038e-35 < x.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 x.re around inf 60.1%

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

      1. *-commutative 60.1%

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

      2. +-commutative 60.1%

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

      3. add-sqr-sqrt 60.1%

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

      4. hypot-undefine 60.1%

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

      5. hypot-undefine 60.1%

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

      6. times-frac 89.8%

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

      7. associate-/l* 93.4%

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

      8. hypot-undefine 62.1%

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

      9. +-commutative 62.1%

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

      10. hypot-define 93.4%

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

      11. hypot-undefine 62.1%

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

      12. +-commutative 62.1%

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

      13. hypot-define 93.4%

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

   5. Applied egg-rr 93.4%

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

   if -3.0000000000000001e-64 < x.re < 7.20000000000000038e-35

   1. Initial program 73.6%

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

   3. Taylor expanded in x.im around inf 64.9%

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

      1. *-commutative 64.9%

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

   5. Simplified 64.9%

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

      1. add-sqr-sqrt 64.9%

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

      2. hypot-undefine 64.9%

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

      3. hypot-undefine 64.9%

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

      4. frac-times 87.9%

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

      5. hypot-undefine 67.9%

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

      6. +-commutative 67.9%

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

      7. hypot-undefine 87.9%

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

      8. hypot-undefine 67.9%

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

      9. +-commutative 67.9%

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

      10. hypot-undefine 87.9%

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

   7. Applied egg-rr 87.9%

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

4. Final simplification 90.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \leq -3 \cdot 10^{-64} \lor \neg \left(x.re \leq 7.2 \cdot 10^{-35}\right):\\ \;\;\;\;\frac{x.im \cdot \frac{y.re}{x.re} - y.im}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{x.re}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{y.re}{\mathsf{hypot}\left(y.im, y.re\right)} \cdot \frac{x.im}{\mathsf{hypot}\left(y.im, y.re\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 82.8% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x.im \cdot y.re - x.re \cdot y.im\\ \mathbf{if}\;y.im \leq -8 \cdot 10^{+85}:\\ \;\;\;\;\frac{y.re \cdot \frac{x.im}{y.im} - x.re}{y.im}\\ \mathbf{elif}\;y.im \leq -9 \cdot 10^{-160}:\\ \;\;\;\;\frac{t\_0}{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)}\\ \mathbf{elif}\;y.im \leq 4.1 \cdot 10^{-47}:\\ \;\;\;\;\frac{x.im - x.re \cdot \frac{y.im}{y.re}}{y.re}\\ \mathbf{elif}\;y.im \leq 4.1 \cdot 10^{+148}:\\ \;\;\;\;\frac{t\_0}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y.re}{\frac{y.im}{x.im}} - x.re}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (- (* x.im y.re) (* x.re y.im))))
   (if (<= y.im -8e+85)
     (/ (- (* y.re (/ x.im y.im)) x.re) y.im)
     (if (<= y.im -9e-160)
       (/ t_0 (fma y.im y.im (* y.re y.re)))
       (if (<= y.im 4.1e-47)
         (/ (- x.im (* x.re (/ y.im y.re))) y.re)
         (if (<= y.im 4.1e+148)
           (/ t_0 (+ (* y.re y.re) (* y.im y.im)))
           (/ (- (/ y.re (/ y.im x.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 t_0 = (x_46_im * y_46_re) - (x_46_re * y_46_im);
	double tmp;
	if (y_46_im <= -8e+85) {
		tmp = ((y_46_re * (x_46_im / y_46_im)) - x_46_re) / y_46_im;
	} else if (y_46_im <= -9e-160) {
		tmp = t_0 / fma(y_46_im, y_46_im, (y_46_re * y_46_re));
	} else if (y_46_im <= 4.1e-47) {
		tmp = (x_46_im - (x_46_re * (y_46_im / y_46_re))) / y_46_re;
	} else if (y_46_im <= 4.1e+148) {
		tmp = t_0 / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	} else {
		tmp = ((y_46_re / (y_46_im / x_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)
	t_0 = Float64(Float64(x_46_im * y_46_re) - Float64(x_46_re * y_46_im))
	tmp = 0.0
	if (y_46_im <= -8e+85)
		tmp = Float64(Float64(Float64(y_46_re * Float64(x_46_im / y_46_im)) - x_46_re) / y_46_im);
	elseif (y_46_im <= -9e-160)
		tmp = Float64(t_0 / fma(y_46_im, y_46_im, Float64(y_46_re * y_46_re)));
	elseif (y_46_im <= 4.1e-47)
		tmp = Float64(Float64(x_46_im - Float64(x_46_re * Float64(y_46_im / y_46_re))) / y_46_re);
	elseif (y_46_im <= 4.1e+148)
		tmp = Float64(t_0 / Float64(Float64(y_46_re * y_46_re) + Float64(y_46_im * y_46_im)));
	else
		tmp = Float64(Float64(Float64(y_46_re / Float64(y_46_im / x_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_] := Block[{t$95$0 = N[(N[(x$46$im * y$46$re), $MachinePrecision] - N[(x$46$re * y$46$im), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$im, -8e+85], N[(N[(N[(y$46$re * N[(x$46$im / y$46$im), $MachinePrecision]), $MachinePrecision] - x$46$re), $MachinePrecision] / y$46$im), $MachinePrecision], If[LessEqual[y$46$im, -9e-160], N[(t$95$0 / N[(y$46$im * y$46$im + N[(y$46$re * y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, 4.1e-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$46$im, 4.1e+148], N[(t$95$0 / N[(N[(y$46$re * y$46$re), $MachinePrecision] + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(y$46$re / N[(y$46$im / x$46$im), $MachinePrecision]), $MachinePrecision] - x$46$re), $MachinePrecision] / y$46$im), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if y.im < -8.0000000000000001e85

   1. Initial program 43.7%

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

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

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

      1. +-commutative 72.4%

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

      2. mul-1-neg 72.4%

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

      3. unsub-neg 72.4%

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

      4. unpow2 72.4%

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

      5. associate-/r* 78.3%

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

      6. div-sub 78.3%

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

      7. *-commutative 78.3%

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

      8. associate-/l* 88.3%

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

   5. Simplified 88.3%

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

   if -8.0000000000000001e85 < y.im < -9.00000000000000053e-160

   1. Initial program 90.2%

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

      1. fma-neg 90.2%

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

      2. distribute-rgt-neg-out 90.2%

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

      3. +-commutative 90.2%

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

      4. fma-define 90.2%

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

   3. Simplified 90.2%

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

      1. distribute-rgt-neg-out 90.2%

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

      2. fma-neg 90.2%

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

   6. Applied egg-rr 90.2%

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

   if -9.00000000000000053e-160 < y.im < 4.10000000000000002e-47

   1. Initial program 75.0%

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

   3. Taylor expanded in y.re around inf 90.2%

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

      1. mul-1-neg 90.2%

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

      2. unsub-neg 90.2%

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

      3. unsub-neg 90.2%

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

      4. remove-double-neg 90.2%

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

      5. mul-1-neg 90.2%

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

      6. neg-mul-1 90.2%

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

      7. mul-1-neg 90.2%

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

      8. distribute-lft-in 90.2%

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

      9. distribute-lft-in 90.2%

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

      10. mul-1-neg 90.2%

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

      11. unsub-neg 90.2%

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

      12. neg-mul-1 90.2%

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

      13. mul-1-neg 90.2%

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

      14. remove-double-neg 90.2%

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

      15. associate-/l* 90.2%

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

   5. Simplified 90.2%

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

   if 4.10000000000000002e-47 < y.im < 4.0999999999999998e148

   1. Initial program 75.4%

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

   if 4.0999999999999998e148 < y.im

   1. Initial program 28.1%

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

   3. Taylor expanded in y.re around 0 66.8%

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

      1. +-commutative 66.8%

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

      2. mul-1-neg 66.8%

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

      3. unsub-neg 66.8%

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

      4. unpow2 66.8%

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

      5. associate-/r* 79.1%

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

      6. div-sub 79.1%

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

      7. *-commutative 79.1%

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

      8. associate-/l* 85.8%

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

   5. Simplified 85.8%

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

      1. clear-num 85.8%

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

      2. un-div-inv 85.8%

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

   7. Applied egg-rr 85.8%

      \[\leadsto \frac{\color{blue}{\frac{y.re}{\frac{y.im}{x.im}}} - x.re}{y.im} \]
3. Recombined 5 regimes into one program.

4. Final simplification 87.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -8 \cdot 10^{+85}:\\ \;\;\;\;\frac{y.re \cdot \frac{x.im}{y.im} - x.re}{y.im}\\ \mathbf{elif}\;y.im \leq -9 \cdot 10^{-160}:\\ \;\;\;\;\frac{x.im \cdot y.re - x.re \cdot y.im}{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)}\\ \mathbf{elif}\;y.im \leq 4.1 \cdot 10^{-47}:\\ \;\;\;\;\frac{x.im - x.re \cdot \frac{y.im}{y.re}}{y.re}\\ \mathbf{elif}\;y.im \leq 4.1 \cdot 10^{+148}:\\ \;\;\;\;\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y.re}{\frac{y.im}{x.im}} - x.re}{y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 82.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{if}\;y.im \leq -8.8 \cdot 10^{+85}:\\ \;\;\;\;\frac{y.re \cdot \frac{x.im}{y.im} - x.re}{y.im}\\ \mathbf{elif}\;y.im \leq -3.8 \cdot 10^{-160}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.im \leq 3.75 \cdot 10^{-47}:\\ \;\;\;\;\frac{x.im - x.re \cdot \frac{y.im}{y.re}}{y.re}\\ \mathbf{elif}\;y.im \leq 9.5 \cdot 10^{+145}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y.re}{\frac{y.im}{x.im}} - x.re}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0
         (/ (- (* x.im y.re) (* x.re y.im)) (+ (* y.re y.re) (* y.im y.im)))))
   (if (<= y.im -8.8e+85)
     (/ (- (* y.re (/ x.im y.im)) x.re) y.im)
     (if (<= y.im -3.8e-160)
       t_0
       (if (<= y.im 3.75e-47)
         (/ (- x.im (* x.re (/ y.im y.re))) y.re)
         (if (<= y.im 9.5e+145)
           t_0
           (/ (- (/ y.re (/ y.im x.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 t_0 = ((x_46_im * y_46_re) - (x_46_re * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	double tmp;
	if (y_46_im <= -8.8e+85) {
		tmp = ((y_46_re * (x_46_im / y_46_im)) - x_46_re) / y_46_im;
	} else if (y_46_im <= -3.8e-160) {
		tmp = t_0;
	} else if (y_46_im <= 3.75e-47) {
		tmp = (x_46_im - (x_46_re * (y_46_im / y_46_re))) / y_46_re;
	} else if (y_46_im <= 9.5e+145) {
		tmp = t_0;
	} else {
		tmp = ((y_46_re / (y_46_im / x_46_im)) - 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) :: t_0
    real(8) :: tmp
    t_0 = ((x_46im * y_46re) - (x_46re * y_46im)) / ((y_46re * y_46re) + (y_46im * y_46im))
    if (y_46im <= (-8.8d+85)) then
        tmp = ((y_46re * (x_46im / y_46im)) - x_46re) / y_46im
    else if (y_46im <= (-3.8d-160)) then
        tmp = t_0
    else if (y_46im <= 3.75d-47) then
        tmp = (x_46im - (x_46re * (y_46im / y_46re))) / y_46re
    else if (y_46im <= 9.5d+145) then
        tmp = t_0
    else
        tmp = ((y_46re / (y_46im / x_46im)) - 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 t_0 = ((x_46_im * y_46_re) - (x_46_re * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	double tmp;
	if (y_46_im <= -8.8e+85) {
		tmp = ((y_46_re * (x_46_im / y_46_im)) - x_46_re) / y_46_im;
	} else if (y_46_im <= -3.8e-160) {
		tmp = t_0;
	} else if (y_46_im <= 3.75e-47) {
		tmp = (x_46_im - (x_46_re * (y_46_im / y_46_re))) / y_46_re;
	} else if (y_46_im <= 9.5e+145) {
		tmp = t_0;
	} else {
		tmp = ((y_46_re / (y_46_im / x_46_im)) - x_46_re) / y_46_im;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = ((x_46_im * y_46_re) - (x_46_re * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im))
	tmp = 0
	if y_46_im <= -8.8e+85:
		tmp = ((y_46_re * (x_46_im / y_46_im)) - x_46_re) / y_46_im
	elif y_46_im <= -3.8e-160:
		tmp = t_0
	elif y_46_im <= 3.75e-47:
		tmp = (x_46_im - (x_46_re * (y_46_im / y_46_re))) / y_46_re
	elif y_46_im <= 9.5e+145:
		tmp = t_0
	else:
		tmp = ((y_46_re / (y_46_im / x_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)
	t_0 = 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)))
	tmp = 0.0
	if (y_46_im <= -8.8e+85)
		tmp = Float64(Float64(Float64(y_46_re * Float64(x_46_im / y_46_im)) - x_46_re) / y_46_im);
	elseif (y_46_im <= -3.8e-160)
		tmp = t_0;
	elseif (y_46_im <= 3.75e-47)
		tmp = Float64(Float64(x_46_im - Float64(x_46_re * Float64(y_46_im / y_46_re))) / y_46_re);
	elseif (y_46_im <= 9.5e+145)
		tmp = t_0;
	else
		tmp = Float64(Float64(Float64(y_46_re / Float64(y_46_im / x_46_im)) - x_46_re) / y_46_im);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = ((x_46_im * y_46_re) - (x_46_re * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	tmp = 0.0;
	if (y_46_im <= -8.8e+85)
		tmp = ((y_46_re * (x_46_im / y_46_im)) - x_46_re) / y_46_im;
	elseif (y_46_im <= -3.8e-160)
		tmp = t_0;
	elseif (y_46_im <= 3.75e-47)
		tmp = (x_46_im - (x_46_re * (y_46_im / y_46_re))) / y_46_re;
	elseif (y_46_im <= 9.5e+145)
		tmp = t_0;
	else
		tmp = ((y_46_re / (y_46_im / x_46_im)) - 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_] := Block[{t$95$0 = 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]}, If[LessEqual[y$46$im, -8.8e+85], N[(N[(N[(y$46$re * N[(x$46$im / y$46$im), $MachinePrecision]), $MachinePrecision] - x$46$re), $MachinePrecision] / y$46$im), $MachinePrecision], If[LessEqual[y$46$im, -3.8e-160], t$95$0, If[LessEqual[y$46$im, 3.75e-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$46$im, 9.5e+145], t$95$0, N[(N[(N[(y$46$re / N[(y$46$im / x$46$im), $MachinePrecision]), $MachinePrecision] - x$46$re), $MachinePrecision] / y$46$im), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if y.im < -8.8000000000000007e85

   1. Initial program 43.7%

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

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

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

      1. +-commutative 72.4%

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

      2. mul-1-neg 72.4%

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

      3. unsub-neg 72.4%

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

      4. unpow2 72.4%

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

      5. associate-/r* 78.3%

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

      6. div-sub 78.3%

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

      7. *-commutative 78.3%

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

      8. associate-/l* 88.3%

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

   5. Simplified 88.3%

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

   if -8.8000000000000007e85 < y.im < -3.7999999999999998e-160 or 3.74999999999999984e-47 < y.im < 9.49999999999999948e145

   1. Initial program 84.1%

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

   if -3.7999999999999998e-160 < y.im < 3.74999999999999984e-47

   1. Initial program 75.0%

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

   3. Taylor expanded in y.re around inf 90.2%

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

      1. mul-1-neg 90.2%

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

      2. unsub-neg 90.2%

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

      3. unsub-neg 90.2%

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

      4. remove-double-neg 90.2%

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

      5. mul-1-neg 90.2%

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

      6. neg-mul-1 90.2%

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

      7. mul-1-neg 90.2%

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

      8. distribute-lft-in 90.2%

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

      9. distribute-lft-in 90.2%

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

      10. mul-1-neg 90.2%

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

      11. unsub-neg 90.2%

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

      12. neg-mul-1 90.2%

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

      13. mul-1-neg 90.2%

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

      14. remove-double-neg 90.2%

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

      15. associate-/l* 90.2%

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

   5. Simplified 90.2%

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

   if 9.49999999999999948e145 < y.im

   1. Initial program 28.1%

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

   3. Taylor expanded in y.re around 0 66.8%

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

      1. +-commutative 66.8%

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

      2. mul-1-neg 66.8%

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

      3. unsub-neg 66.8%

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

      4. unpow2 66.8%

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

      5. associate-/r* 79.1%

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

      6. div-sub 79.1%

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

      7. *-commutative 79.1%

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

      8. associate-/l* 85.8%

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

   5. Simplified 85.8%

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

      1. clear-num 85.8%

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

      2. un-div-inv 85.8%

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

   7. Applied egg-rr 85.8%

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

4. Final simplification 87.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -8.8 \cdot 10^{+85}:\\ \;\;\;\;\frac{y.re \cdot \frac{x.im}{y.im} - x.re}{y.im}\\ \mathbf{elif}\;y.im \leq -3.8 \cdot 10^{-160}:\\ \;\;\;\;\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.im \leq 3.75 \cdot 10^{-47}:\\ \;\;\;\;\frac{x.im - x.re \cdot \frac{y.im}{y.re}}{y.re}\\ \mathbf{elif}\;y.im \leq 9.5 \cdot 10^{+145}:\\ \;\;\;\;\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y.re}{\frac{y.im}{x.im}} - x.re}{y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 72.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -4.2 \cdot 10^{-20} \lor \neg \left(y.im \leq 3.2 \cdot 10^{+14}\right):\\ \;\;\;\;\frac{x.re}{-y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im - x.re \cdot \frac{y.im}{y.re}}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.im -4.2e-20) (not (<= y.im 3.2e+14)))
   (/ x.re (- y.im))
   (/ (- x.im (* x.re (/ y.im y.re))) 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 <= -4.2e-20) || !(y_46_im <= 3.2e+14)) {
		tmp = x_46_re / -y_46_im;
	} else {
		tmp = (x_46_im - (x_46_re * (y_46_im / y_46_re))) / y_46_re;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if ((y_46im <= (-4.2d-20)) .or. (.not. (y_46im <= 3.2d+14))) then
        tmp = x_46re / -y_46im
    else
        tmp = (x_46im - (x_46re * (y_46im / y_46re))) / y_46re
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_im <= -4.2e-20) || !(y_46_im <= 3.2e+14)) {
		tmp = x_46_re / -y_46_im;
	} else {
		tmp = (x_46_im - (x_46_re * (y_46_im / y_46_re))) / 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 <= -4.2e-20) or not (y_46_im <= 3.2e+14):
		tmp = x_46_re / -y_46_im
	else:
		tmp = (x_46_im - (x_46_re * (y_46_im / y_46_re))) / 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 <= -4.2e-20) || !(y_46_im <= 3.2e+14))
		tmp = Float64(x_46_re / Float64(-y_46_im));
	else
		tmp = Float64(Float64(x_46_im - Float64(x_46_re * Float64(y_46_im / y_46_re))) / y_46_re);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if ((y_46_im <= -4.2e-20) || ~((y_46_im <= 3.2e+14)))
		tmp = x_46_re / -y_46_im;
	else
		tmp = (x_46_im - (x_46_re * (y_46_im / y_46_re))) / 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, -4.2e-20], N[Not[LessEqual[y$46$im, 3.2e+14]], $MachinePrecision]], N[(x$46$re / (-y$46$im)), $MachinePrecision], N[(N[(x$46$im - N[(x$46$re * N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.im < -4.1999999999999998e-20 or 3.2e14 < y.im

   1. Initial program 53.6%

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

   3. Taylor expanded in y.re around 0 65.8%

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

      1. associate-*r/ 65.8%

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

      2. neg-mul-1 65.8%

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

   5. Simplified 65.8%

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

   if -4.1999999999999998e-20 < y.im < 3.2e14

   1. Initial program 79.5%

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

   3. Taylor expanded in y.re around inf 83.1%

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

      1. mul-1-neg 83.1%

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

      2. unsub-neg 83.1%

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

      3. unsub-neg 83.1%

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

      4. remove-double-neg 83.1%

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

      5. mul-1-neg 83.1%

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

      6. neg-mul-1 83.1%

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

      7. mul-1-neg 83.1%

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

      8. distribute-lft-in 83.1%

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

      9. distribute-lft-in 83.1%

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

      10. mul-1-neg 83.1%

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

      11. unsub-neg 83.1%

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

      12. neg-mul-1 83.1%

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

      13. mul-1-neg 83.1%

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

      14. remove-double-neg 83.1%

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

      15. associate-/l* 83.1%

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

   5. Simplified 83.1%

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

4. Final simplification 74.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -4.2 \cdot 10^{-20} \lor \neg \left(y.im \leq 3.2 \cdot 10^{+14}\right):\\ \;\;\;\;\frac{x.re}{-y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im - x.re \cdot \frac{y.im}{y.re}}{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 78.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -9.2 \cdot 10^{-24} \lor \neg \left(y.im \leq 1.25 \cdot 10^{-30}\right):\\ \;\;\;\;\frac{y.re \cdot \frac{x.im}{y.im} - x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im - x.re \cdot \frac{y.im}{y.re}}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.im -9.2e-24) (not (<= y.im 1.25e-30)))
   (/ (- (* y.re (/ x.im y.im)) x.re) y.im)
   (/ (- x.im (* x.re (/ y.im y.re))) 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 <= -9.2e-24) || !(y_46_im <= 1.25e-30)) {
		tmp = ((y_46_re * (x_46_im / y_46_im)) - x_46_re) / y_46_im;
	} else {
		tmp = (x_46_im - (x_46_re * (y_46_im / y_46_re))) / y_46_re;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if ((y_46im <= (-9.2d-24)) .or. (.not. (y_46im <= 1.25d-30))) then
        tmp = ((y_46re * (x_46im / y_46im)) - x_46re) / y_46im
    else
        tmp = (x_46im - (x_46re * (y_46im / y_46re))) / y_46re
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_im <= -9.2e-24) || !(y_46_im <= 1.25e-30)) {
		tmp = ((y_46_re * (x_46_im / y_46_im)) - x_46_re) / y_46_im;
	} else {
		tmp = (x_46_im - (x_46_re * (y_46_im / y_46_re))) / 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 <= -9.2e-24) or not (y_46_im <= 1.25e-30):
		tmp = ((y_46_re * (x_46_im / y_46_im)) - x_46_re) / y_46_im
	else:
		tmp = (x_46_im - (x_46_re * (y_46_im / y_46_re))) / 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 <= -9.2e-24) || !(y_46_im <= 1.25e-30))
		tmp = Float64(Float64(Float64(y_46_re * Float64(x_46_im / y_46_im)) - x_46_re) / y_46_im);
	else
		tmp = Float64(Float64(x_46_im - Float64(x_46_re * Float64(y_46_im / y_46_re))) / y_46_re);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if ((y_46_im <= -9.2e-24) || ~((y_46_im <= 1.25e-30)))
		tmp = ((y_46_re * (x_46_im / y_46_im)) - x_46_re) / y_46_im;
	else
		tmp = (x_46_im - (x_46_re * (y_46_im / y_46_re))) / 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, -9.2e-24], N[Not[LessEqual[y$46$im, 1.25e-30]], $MachinePrecision]], N[(N[(N[(y$46$re * N[(x$46$im / y$46$im), $MachinePrecision]), $MachinePrecision] - x$46$re), $MachinePrecision] / y$46$im), $MachinePrecision], N[(N[(x$46$im - N[(x$46$re * N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.im < -9.2000000000000004e-24 or 1.24999999999999993e-30 < y.im

   1. Initial program 56.5%

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

   3. Taylor expanded in y.re around 0 65.4%

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

      1. +-commutative 65.4%

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

      2. mul-1-neg 65.4%

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

      3. unsub-neg 65.4%

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

      4. unpow2 65.4%

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

      5. associate-/r* 70.1%

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

      6. div-sub 70.1%

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

      7. *-commutative 70.1%

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

      8. associate-/l* 75.0%

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

   5. Simplified 75.0%

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

   if -9.2000000000000004e-24 < y.im < 1.24999999999999993e-30

   1. Initial program 78.8%

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

   3. Taylor expanded in y.re around inf 87.0%

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

      1. mul-1-neg 87.0%

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

      2. unsub-neg 87.0%

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

      3. unsub-neg 87.0%

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

      4. remove-double-neg 87.0%

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

      5. mul-1-neg 87.0%

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

      6. neg-mul-1 87.0%

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

      7. mul-1-neg 87.0%

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

      8. distribute-lft-in 87.0%

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

      9. distribute-lft-in 87.0%

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

      10. mul-1-neg 87.0%

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

      11. unsub-neg 87.0%

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

      12. neg-mul-1 87.0%

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

      13. mul-1-neg 87.0%

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

      14. remove-double-neg 87.0%

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

      15. associate-/l* 87.0%

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

   5. Simplified 87.0%

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

4. Final simplification 80.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -9.2 \cdot 10^{-24} \lor \neg \left(y.im \leq 1.25 \cdot 10^{-30}\right):\\ \;\;\;\;\frac{y.re \cdot \frac{x.im}{y.im} - x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im - x.re \cdot \frac{y.im}{y.re}}{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 78.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -5.4 \cdot 10^{-25}:\\ \;\;\;\;\frac{y.re \cdot \frac{x.im}{y.im} - x.re}{y.im}\\ \mathbf{elif}\;y.im \leq 7.8 \cdot 10^{-31}:\\ \;\;\;\;\frac{x.im - x.re \cdot \frac{y.im}{y.re}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y.re}{\frac{y.im}{x.im}} - x.re}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.im -5.4e-25)
   (/ (- (* y.re (/ x.im y.im)) x.re) y.im)
   (if (<= y.im 7.8e-31)
     (/ (- x.im (* x.re (/ y.im y.re))) y.re)
     (/ (- (/ y.re (/ y.im x.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_im <= -5.4e-25) {
		tmp = ((y_46_re * (x_46_im / y_46_im)) - x_46_re) / y_46_im;
	} else if (y_46_im <= 7.8e-31) {
		tmp = (x_46_im - (x_46_re * (y_46_im / y_46_re))) / y_46_re;
	} else {
		tmp = ((y_46_re / (y_46_im / x_46_im)) - 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_46im <= (-5.4d-25)) then
        tmp = ((y_46re * (x_46im / y_46im)) - x_46re) / y_46im
    else if (y_46im <= 7.8d-31) then
        tmp = (x_46im - (x_46re * (y_46im / y_46re))) / y_46re
    else
        tmp = ((y_46re / (y_46im / x_46im)) - 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_im <= -5.4e-25) {
		tmp = ((y_46_re * (x_46_im / y_46_im)) - x_46_re) / y_46_im;
	} else if (y_46_im <= 7.8e-31) {
		tmp = (x_46_im - (x_46_re * (y_46_im / y_46_re))) / y_46_re;
	} else {
		tmp = ((y_46_re / (y_46_im / x_46_im)) - 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_im <= -5.4e-25:
		tmp = ((y_46_re * (x_46_im / y_46_im)) - x_46_re) / y_46_im
	elif y_46_im <= 7.8e-31:
		tmp = (x_46_im - (x_46_re * (y_46_im / y_46_re))) / y_46_re
	else:
		tmp = ((y_46_re / (y_46_im / x_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_im <= -5.4e-25)
		tmp = Float64(Float64(Float64(y_46_re * Float64(x_46_im / y_46_im)) - x_46_re) / y_46_im);
	elseif (y_46_im <= 7.8e-31)
		tmp = Float64(Float64(x_46_im - Float64(x_46_re * Float64(y_46_im / y_46_re))) / y_46_re);
	else
		tmp = Float64(Float64(Float64(y_46_re / Float64(y_46_im / x_46_im)) - x_46_re) / y_46_im);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if (y_46_im <= -5.4e-25)
		tmp = ((y_46_re * (x_46_im / y_46_im)) - x_46_re) / y_46_im;
	elseif (y_46_im <= 7.8e-31)
		tmp = (x_46_im - (x_46_re * (y_46_im / y_46_re))) / y_46_re;
	else
		tmp = ((y_46_re / (y_46_im / x_46_im)) - 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$46$im, -5.4e-25], N[(N[(N[(y$46$re * N[(x$46$im / y$46$im), $MachinePrecision]), $MachinePrecision] - x$46$re), $MachinePrecision] / y$46$im), $MachinePrecision], If[LessEqual[y$46$im, 7.8e-31], N[(N[(x$46$im - N[(x$46$re * N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision], N[(N[(N[(y$46$re / N[(y$46$im / x$46$im), $MachinePrecision]), $MachinePrecision] - x$46$re), $MachinePrecision] / y$46$im), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y.im < -5.40000000000000032e-25

   1. Initial program 60.7%

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

   3. Taylor expanded in y.re around 0 68.6%

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

      1. +-commutative 68.6%

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

      2. mul-1-neg 68.6%

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

      3. unsub-neg 68.6%

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

      4. unpow2 68.6%

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

      5. associate-/r* 72.3%

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

      6. div-sub 72.3%

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

      7. *-commutative 72.3%

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

      8. associate-/l* 78.6%

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

   5. Simplified 78.6%

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

   if -5.40000000000000032e-25 < y.im < 7.8000000000000003e-31

   1. Initial program 78.8%

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

   3. Taylor expanded in y.re around inf 87.0%

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

      1. mul-1-neg 87.0%

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

      2. unsub-neg 87.0%

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

      3. unsub-neg 87.0%

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

      4. remove-double-neg 87.0%

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

      5. mul-1-neg 87.0%

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

      6. neg-mul-1 87.0%

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

      7. mul-1-neg 87.0%

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

      8. distribute-lft-in 87.0%

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

      9. distribute-lft-in 87.0%

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

      10. mul-1-neg 87.0%

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

      11. unsub-neg 87.0%

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

      12. neg-mul-1 87.0%

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

      13. mul-1-neg 87.0%

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

      14. remove-double-neg 87.0%

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

      15. associate-/l* 87.0%

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

   5. Simplified 87.0%

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

   if 7.8000000000000003e-31 < y.im

   1. Initial program 51.3%

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

   3. Taylor expanded in y.re around 0 61.5%

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

      1. +-commutative 61.5%

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

      2. mul-1-neg 61.5%

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

      3. unsub-neg 61.5%

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

      4. unpow2 61.5%

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

      5. associate-/r* 67.5%

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

      6. div-sub 67.5%

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

      7. *-commutative 67.5%

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

      8. associate-/l* 70.7%

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

   5. Simplified 70.7%

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

      1. clear-num 70.7%

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

      2. un-div-inv 70.8%

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

   7. Applied egg-rr 70.8%

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

4. Final simplification 80.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -5.4 \cdot 10^{-25}:\\ \;\;\;\;\frac{y.re \cdot \frac{x.im}{y.im} - x.re}{y.im}\\ \mathbf{elif}\;y.im \leq 7.8 \cdot 10^{-31}:\\ \;\;\;\;\frac{x.im - x.re \cdot \frac{y.im}{y.re}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y.re}{\frac{y.im}{x.im}} - x.re}{y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 64.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -2.8 \cdot 10^{-32} \lor \neg \left(y.im \leq 3.8 \cdot 10^{-30}\right):\\ \;\;\;\;\frac{x.re}{-y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.im -2.8e-32) (not (<= y.im 3.8e-30)))
   (/ 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 <= -2.8e-32) || !(y_46_im <= 3.8e-30)) {
		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 <= (-2.8d-32)) .or. (.not. (y_46im <= 3.8d-30))) 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 <= -2.8e-32) || !(y_46_im <= 3.8e-30)) {
		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 <= -2.8e-32) or not (y_46_im <= 3.8e-30):
		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 <= -2.8e-32) || !(y_46_im <= 3.8e-30))
		tmp = Float64(x_46_re / Float64(-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 <= -2.8e-32) || ~((y_46_im <= 3.8e-30)))
		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, -2.8e-32], N[Not[LessEqual[y$46$im, 3.8e-30]], $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 < -2.7999999999999999e-32 or 3.8000000000000003e-30 < y.im

   1. Initial program 57.2%

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

   3. Taylor expanded in y.re around 0 62.9%

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

      1. associate-*r/ 62.9%

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

      2. neg-mul-1 62.9%

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

   5. Simplified 62.9%

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

   if -2.7999999999999999e-32 < y.im < 3.8000000000000003e-30

   1. Initial program 78.5%

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

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

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

4. Final simplification 66.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -2.8 \cdot 10^{-32} \lor \neg \left(y.im \leq 3.8 \cdot 10^{-30}\right):\\ \;\;\;\;\frac{x.re}{-y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 44.7% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -4 \cdot 10^{+192}:\\ \;\;\;\;\frac{x.im}{-y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.im -4e+192) (/ x.im (- 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 <= -4e+192) {
		tmp = x_46_im / -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 <= (-4d+192)) then
        tmp = x_46im / -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 <= -4e+192) {
		tmp = x_46_im / -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 <= -4e+192:
		tmp = x_46_im / -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 <= -4e+192)
		tmp = Float64(x_46_im / Float64(-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 <= -4e+192)
		tmp = x_46_im / -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[LessEqual[y$46$im, -4e+192], N[(x$46$im / (-y$46$im)), $MachinePrecision], N[(x$46$im / y$46$re), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.im < -4.00000000000000016e192

   1. Initial program 48.6%

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

   3. Taylor expanded in y.im around inf 48.6%

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

      1. *-commutative 48.6%

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

      2. associate-/l* 48.6%

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

   5. Simplified 48.6%

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

      1. *-commutative 48.6%

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

      2. add-sqr-sqrt 48.6%

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

      3. hypot-undefine 48.6%

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

      4. hypot-undefine 48.6%

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

      5. times-frac 99.9%

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

      6. fma-neg 99.9%

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

      7. hypot-undefine 49.9%

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

      8. +-commutative 49.9%

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

      9. hypot-undefine 99.9%

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

      10. hypot-undefine 49.9%

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

      11. +-commutative 49.9%

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

      12. hypot-undefine 99.9%

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

   7. Applied egg-rr 99.9%

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

   8. Taylor expanded in y.re around inf 29.7%

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

   9. Taylor expanded in y.im around -inf 29.7%

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

       1. neg-mul-1 29.7%

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

       2. distribute-neg-frac2 29.7%

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

   11. Simplified 29.7%

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

   if -4.00000000000000016e192 < y.im

   1. Initial program 68.5%

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

   3. Taylor expanded in y.re around inf 48.4%

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

4. Final simplification 46.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -4 \cdot 10^{+192}:\\ \;\;\;\;\frac{x.im}{-y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 44.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -1.6 \cdot 10^{+188}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.im -1.6e+188) (/ x.im 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 <= -1.6e+188) {
		tmp = x_46_im / 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 <= (-1.6d+188)) then
        tmp = x_46im / 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 <= -1.6e+188) {
		tmp = x_46_im / 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 <= -1.6e+188:
		tmp = x_46_im / 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 <= -1.6e+188)
		tmp = Float64(x_46_im / 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 <= -1.6e+188)
		tmp = x_46_im / 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[LessEqual[y$46$im, -1.6e+188], N[(x$46$im / y$46$im), $MachinePrecision], N[(x$46$im / y$46$re), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.im < -1.59999999999999985e188

   1. Initial program 48.6%

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

   3. Taylor expanded in y.im around inf 48.6%

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

      1. *-commutative 48.6%

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

      2. associate-/l* 48.6%

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

   5. Simplified 48.6%

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

      1. *-commutative 48.6%

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

      2. add-sqr-sqrt 48.6%

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

      3. hypot-undefine 48.6%

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

      4. hypot-undefine 48.6%

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

      5. times-frac 99.9%

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

      6. fma-neg 99.9%

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

      7. hypot-undefine 49.9%

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

      8. +-commutative 49.9%

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

      9. hypot-undefine 99.9%

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

      10. hypot-undefine 49.9%

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

      11. +-commutative 49.9%

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

      12. hypot-undefine 99.9%

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

   7. Applied egg-rr 99.9%

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

   8. Taylor expanded in y.re around inf 29.7%

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

   9. Taylor expanded in y.im around inf 29.1%

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

   if -1.59999999999999985e188 < y.im

   1. Initial program 68.5%

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

   3. Taylor expanded in y.re around inf 48.4%

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

4. Final simplification 46.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -1.6 \cdot 10^{+188}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 9.7% 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 66.4%

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

3. Taylor expanded in y.im around inf 60.6%

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

   1. *-commutative 60.6%

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

   2. associate-/l* 59.1%

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

5. Simplified 59.1%

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

   1. *-commutative 59.1%

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

   2. add-sqr-sqrt 59.1%

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

   3. hypot-undefine 59.1%

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

   4. hypot-undefine 59.1%

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

   5. times-frac 82.4%

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

   6. fma-neg 82.4%

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

   7. hypot-undefine 60.1%

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

   8. +-commutative 60.1%

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

   9. hypot-undefine 82.4%

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

   10. hypot-undefine 60.1%

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

   11. +-commutative 60.1%

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

   12. hypot-undefine 82.4%

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

7. Applied egg-rr 82.4%

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

8. Taylor expanded in y.re around inf 27.5%

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

9. Taylor expanded in y.im around inf 10.4%

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

10. Final simplification 10.4%

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

Reproduce

herbie shell --seed 2024055 
(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))))