_divideComplex, imaginary part

Percentage Accurate: 62.0% → 96.3%

Time: 20.0s

Alternatives: 16

Speedup: 0.7×

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: 62.0% 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: 96.3% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -4.2 \cdot 10^{+239}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(y.im \cdot \frac{x.re}{y.re} - x.im\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y.re}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{-x.re}{\mathsf{fma}\left(y.re, \frac{y.re}{y.im}, y.im\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.re -4.2e+239)
   (* (/ 1.0 (hypot y.re y.im)) (- (* y.im (/ x.re y.re)) x.im))
   (fma
    (/ y.re (hypot y.re y.im))
    (/ x.im (hypot y.re y.im))
    (/ (- x.re) (fma 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) {
	double tmp;
	if (y_46_re <= -4.2e+239) {
		tmp = (1.0 / hypot(y_46_re, y_46_im)) * ((y_46_im * (x_46_re / y_46_re)) - x_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)), (-x_46_re / fma(y_46_re, (y_46_re / y_46_im), y_46_im)));
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (y_46_re <= -4.2e+239)
		tmp = Float64(Float64(1.0 / hypot(y_46_re, y_46_im)) * Float64(Float64(y_46_im * Float64(x_46_re / y_46_re)) - x_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(-x_46_re) / fma(y_46_re, Float64(y_46_re / y_46_im), y_46_im)));
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[y$46$re, -4.2e+239], N[(N[(1.0 / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision] * N[(N[(y$46$im * N[(x$46$re / y$46$re), $MachinePrecision]), $MachinePrecision] - x$46$im), $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[((-x$46$re) / N[(y$46$re * N[(y$46$re / y$46$im), $MachinePrecision] + y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.re < -4.1999999999999998e239

   1. Initial program 32.7%

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

      1. *-un-lft-identity 32.7%

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

      2. add-sqr-sqrt 32.7%

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

      3. times-frac 32.7%

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

      4. hypot-def 32.7%

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

      5. hypot-def 53.4%

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

   4. Applied egg-rr 53.4%

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

   5. Taylor expanded in y.re around -inf 81.3%

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

      1. neg-mul-1 81.3%

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

      2. +-commutative 81.3%

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

      3. unsub-neg 81.3%

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

      4. *-lft-identity 81.3%

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

      5. times-frac 99.8%

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

      6. /-rgt-identity 99.8%

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

   7. Simplified 99.8%

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

   8. Taylor expanded in x.re around 0 81.3%

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

      1. associate-/l* 99.8%

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

      2. associate-/r/ 100.0%

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

   10. Simplified 100.0%

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

   if -4.1999999999999998e239 < y.re

   1. Initial program 57.7%

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

      1. div-sub 55.0%

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

      2. sub-neg 55.0%

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

      3. *-commutative 55.0%

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

      4. add-sqr-sqrt 55.0%

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

      5. times-frac 60.3%

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

      6. fma-def 60.3%

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

      7. hypot-def 60.3%

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

      8. hypot-def 75.4%

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

      9. associate-/l* 77.7%

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

      10. add-sqr-sqrt 77.7%

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

      11. pow2 77.7%

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

      12. hypot-def 77.7%

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

   4. Applied egg-rr 77.7%

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

   5. Taylor expanded in y.re around 0 93.2%

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

      1. +-commutative 93.2%

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

      2. unpow2 93.2%

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

      3. associate-*r/ 97.1%

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

      4. fma-def 97.1%

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

   7. Simplified 97.1%

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

4. Final simplification 97.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -4.2 \cdot 10^{+239}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(y.im \cdot \frac{x.re}{y.re} - x.im\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y.re}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{-x.re}{\mathsf{fma}\left(y.re, \frac{y.re}{y.im}, y.im\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 89.5% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y.re \cdot x.im - y.im \cdot x.re\\ t_1 := \frac{t_0}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{if}\;t_1 \leq -\infty \lor \neg \left(t_1 \leq 5 \cdot 10^{+237}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{y.re}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{-x.re}{y.im}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{t_0}{\mathsf{hypot}\left(y.re, y.im\right)}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (- (* y.re x.im) (* y.im x.re)))
        (t_1 (/ t_0 (+ (* y.re y.re) (* y.im y.im)))))
   (if (or (<= t_1 (- INFINITY)) (not (<= t_1 5e+237)))
     (fma
      (/ y.re (hypot y.re y.im))
      (/ x.im (hypot y.re y.im))
      (/ (- x.re) y.im))
     (/ (/ t_0 (hypot y.re y.im)) (hypot y.re y.im)))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = (y_46_re * x_46_im) - (y_46_im * x_46_re);
	double t_1 = t_0 / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	double tmp;
	if ((t_1 <= -((double) INFINITY)) || !(t_1 <= 5e+237)) {
		tmp = fma((y_46_re / hypot(y_46_re, y_46_im)), (x_46_im / hypot(y_46_re, y_46_im)), (-x_46_re / y_46_im));
	} else {
		tmp = (t_0 / hypot(y_46_re, y_46_im)) / hypot(y_46_re, y_46_im);
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(Float64(y_46_re * x_46_im) - Float64(y_46_im * x_46_re))
	t_1 = Float64(t_0 / Float64(Float64(y_46_re * y_46_re) + Float64(y_46_im * y_46_im)))
	tmp = 0.0
	if ((t_1 <= Float64(-Inf)) || !(t_1 <= 5e+237))
		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(-x_46_re) / y_46_im));
	else
		tmp = Float64(Float64(t_0 / hypot(y_46_re, y_46_im)) / hypot(y_46_re, y_46_im));
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[(y$46$re * x$46$im), $MachinePrecision] - N[(y$46$im * x$46$re), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 / N[(N[(y$46$re * y$46$re), $MachinePrecision] + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$1, (-Infinity)], N[Not[LessEqual[t$95$1, 5e+237]], $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[((-x$46$re) / y$46$im), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$0 / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision] / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 (*.f64 x.im y.re) (*.f64 x.re y.im)) (+.f64 (*.f64 y.re y.re) (*.f64 y.im y.im))) < -inf.0 or 5.0000000000000002e237 < (/.f64 (-.f64 (*.f64 x.im y.re) (*.f64 x.re y.im)) (+.f64 (*.f64 y.re y.re) (*.f64 y.im y.im)))

   1. Initial program 19.1%

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

      1. div-sub 12.1%

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

      2. sub-neg 12.1%

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

      3. *-commutative 12.1%

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

      4. add-sqr-sqrt 12.1%

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

      5. times-frac 27.1%

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

      6. fma-def 27.1%

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

      7. hypot-def 27.1%

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

      8. hypot-def 54.4%

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

      9. associate-/l* 61.4%

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

      10. add-sqr-sqrt 61.4%

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

      11. pow2 61.4%

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

      12. hypot-def 61.4%

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

   4. Applied egg-rr 61.4%

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

   5. Taylor expanded in y.re around 0 70.9%

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

   if -inf.0 < (/.f64 (-.f64 (*.f64 x.im y.re) (*.f64 x.re y.im)) (+.f64 (*.f64 y.re y.re) (*.f64 y.im y.im))) < 5.0000000000000002e237

   1. Initial program 75.1%

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

      1. *-un-lft-identity 75.1%

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

      2. add-sqr-sqrt 75.1%

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

      3. times-frac 75.2%

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

      4. hypot-def 75.2%

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

      5. hypot-def 99.1%

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

   4. Applied egg-rr 99.1%

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

      1. associate-*l/ 99.2%

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

      2. *-un-lft-identity 99.2%

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

      3. *-commutative 99.2%

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

      4. *-commutative 99.2%

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

   6. Applied egg-rr 99.2%

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

4. Final simplification 89.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im} \leq -\infty \lor \neg \left(\frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im} \leq 5 \cdot 10^{+237}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{y.re}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{-x.re}{y.im}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y.re \cdot x.im - y.im \cdot x.re}{\mathsf{hypot}\left(y.re, y.im\right)}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 84.5% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y.re \cdot x.im - y.im \cdot x.re\\ \mathbf{if}\;\frac{t_0}{y.re \cdot y.re + y.im \cdot y.im} \leq \infty:\\ \;\;\;\;\frac{\frac{t_0}{\mathsf{hypot}\left(y.re, y.im\right)}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-x.re}{\mathsf{fma}\left(y.re, \frac{y.re}{y.im}, y.im\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (- (* y.re x.im) (* y.im x.re))))
   (if (<= (/ t_0 (+ (* y.re y.re) (* y.im y.im))) INFINITY)
     (/ (/ t_0 (hypot y.re y.im)) (hypot y.re y.im))
     (/ (- x.re) (fma 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) {
	double t_0 = (y_46_re * x_46_im) - (y_46_im * x_46_re);
	double tmp;
	if ((t_0 / ((y_46_re * y_46_re) + (y_46_im * y_46_im))) <= ((double) INFINITY)) {
		tmp = (t_0 / hypot(y_46_re, y_46_im)) / hypot(y_46_re, y_46_im);
	} else {
		tmp = -x_46_re / fma(y_46_re, (y_46_re / y_46_im), y_46_im);
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(Float64(y_46_re * x_46_im) - Float64(y_46_im * x_46_re))
	tmp = 0.0
	if (Float64(t_0 / Float64(Float64(y_46_re * y_46_re) + Float64(y_46_im * y_46_im))) <= Inf)
		tmp = Float64(Float64(t_0 / hypot(y_46_re, y_46_im)) / hypot(y_46_re, y_46_im));
	else
		tmp = Float64(Float64(-x_46_re) / fma(y_46_re, Float64(y_46_re / y_46_im), y_46_im));
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[(y$46$re * x$46$im), $MachinePrecision] - N[(y$46$im * x$46$re), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t$95$0 / N[(N[(y$46$re * y$46$re), $MachinePrecision] + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], Infinity], N[(N[(t$95$0 / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision] / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision], N[((-x$46$re) / N[(y$46$re * N[(y$46$re / y$46$im), $MachinePrecision] + y$46$im), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 (*.f64 x.im y.re) (*.f64 x.re y.im)) (+.f64 (*.f64 y.re y.re) (*.f64 y.im y.im))) < +inf.0

   1. Initial program 70.2%

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

      1. *-un-lft-identity 70.2%

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

      2. add-sqr-sqrt 70.2%

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

      3. times-frac 70.2%

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

      4. hypot-def 70.2%

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

      5. hypot-def 92.6%

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

   4. Applied egg-rr 92.6%

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

      1. associate-*l/ 92.7%

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

      2. *-un-lft-identity 92.7%

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

      3. *-commutative 92.7%

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

      4. *-commutative 92.7%

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

   6. Applied egg-rr 92.7%

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

   if +inf.0 < (/.f64 (-.f64 (*.f64 x.im y.re) (*.f64 x.re y.im)) (+.f64 (*.f64 y.re y.re) (*.f64 y.im y.im)))

   1. Initial program 0.0%

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

      1. div-sub 0.0%

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

      2. sub-neg 0.0%

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

      3. *-commutative 0.0%

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

      4. add-sqr-sqrt 0.0%

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

      5. times-frac 1.6%

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

      6. fma-def 1.6%

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

      7. hypot-def 1.6%

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

      8. hypot-def 41.5%

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

      9. associate-/l* 47.7%

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

      10. add-sqr-sqrt 47.7%

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

      11. pow2 47.7%

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

      12. hypot-def 47.7%

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

   4. Applied egg-rr 47.7%

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

   5. Taylor expanded in y.re around 0 80.3%

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

      1. +-commutative 80.3%

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

      2. unpow2 80.3%

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

      3. associate-*r/ 91.2%

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

      4. fma-def 91.2%

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

   7. Simplified 91.2%

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

   8. Taylor expanded in x.im around 0 36.8%

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

      1. associate-*r/ 36.8%

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

      2. neg-mul-1 36.8%

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

      3. unpow2 36.8%

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

      4. associate-*r/ 47.7%

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

      5. +-commutative 47.7%

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

      6. fma-def 47.7%

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

   10. Simplified 47.7%

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

4. Final simplification 83.4%

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

Alternative 4: 78.4% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{if}\;y.re \leq -1.8 \cdot 10^{+80}:\\ \;\;\;\;\frac{y.im \cdot \frac{x.re}{y.re} - x.im}{\mathsf{hypot}\left(y.im, y.re\right)}\\ \mathbf{elif}\;y.re \leq -1.15 \cdot 10^{-155}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq 6.4 \cdot 10^{-218}:\\ \;\;\;\;\frac{-x.re}{y.im}\\ \mathbf{elif}\;y.re \leq 4.1 \cdot 10^{+21}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(x.im - x.re \cdot \frac{y.im}{y.re}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0
         (/ (- (* y.re x.im) (* y.im x.re)) (+ (* y.re y.re) (* y.im y.im)))))
   (if (<= y.re -1.8e+80)
     (/ (- (* y.im (/ x.re y.re)) x.im) (hypot y.im y.re))
     (if (<= y.re -1.15e-155)
       t_0
       (if (<= y.re 6.4e-218)
         (/ (- x.re) y.im)
         (if (<= y.re 4.1e+21)
           t_0
           (* (/ 1.0 (hypot y.re y.im)) (- x.im (* x.re (/ y.im y.re))))))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = ((y_46_re * x_46_im) - (y_46_im * x_46_re)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	double tmp;
	if (y_46_re <= -1.8e+80) {
		tmp = ((y_46_im * (x_46_re / y_46_re)) - x_46_im) / hypot(y_46_im, y_46_re);
	} else if (y_46_re <= -1.15e-155) {
		tmp = t_0;
	} else if (y_46_re <= 6.4e-218) {
		tmp = -x_46_re / y_46_im;
	} else if (y_46_re <= 4.1e+21) {
		tmp = t_0;
	} else {
		tmp = (1.0 / hypot(y_46_re, y_46_im)) * (x_46_im - (x_46_re * (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 t_0 = ((y_46_re * x_46_im) - (y_46_im * x_46_re)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	double tmp;
	if (y_46_re <= -1.8e+80) {
		tmp = ((y_46_im * (x_46_re / y_46_re)) - x_46_im) / Math.hypot(y_46_im, y_46_re);
	} else if (y_46_re <= -1.15e-155) {
		tmp = t_0;
	} else if (y_46_re <= 6.4e-218) {
		tmp = -x_46_re / y_46_im;
	} else if (y_46_re <= 4.1e+21) {
		tmp = t_0;
	} else {
		tmp = (1.0 / Math.hypot(y_46_re, y_46_im)) * (x_46_im - (x_46_re * (y_46_im / y_46_re)));
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = ((y_46_re * x_46_im) - (y_46_im * x_46_re)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im))
	tmp = 0
	if y_46_re <= -1.8e+80:
		tmp = ((y_46_im * (x_46_re / y_46_re)) - x_46_im) / math.hypot(y_46_im, y_46_re)
	elif y_46_re <= -1.15e-155:
		tmp = t_0
	elif y_46_re <= 6.4e-218:
		tmp = -x_46_re / y_46_im
	elif y_46_re <= 4.1e+21:
		tmp = t_0
	else:
		tmp = (1.0 / math.hypot(y_46_re, y_46_im)) * (x_46_im - (x_46_re * (y_46_im / y_46_re)))
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(Float64(Float64(y_46_re * x_46_im) - Float64(y_46_im * x_46_re)) / Float64(Float64(y_46_re * y_46_re) + Float64(y_46_im * y_46_im)))
	tmp = 0.0
	if (y_46_re <= -1.8e+80)
		tmp = Float64(Float64(Float64(y_46_im * Float64(x_46_re / y_46_re)) - x_46_im) / hypot(y_46_im, y_46_re));
	elseif (y_46_re <= -1.15e-155)
		tmp = t_0;
	elseif (y_46_re <= 6.4e-218)
		tmp = Float64(Float64(-x_46_re) / y_46_im);
	elseif (y_46_re <= 4.1e+21)
		tmp = t_0;
	else
		tmp = Float64(Float64(1.0 / hypot(y_46_re, y_46_im)) * Float64(x_46_im - Float64(x_46_re * Float64(y_46_im / y_46_re))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = ((y_46_re * x_46_im) - (y_46_im * x_46_re)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	tmp = 0.0;
	if (y_46_re <= -1.8e+80)
		tmp = ((y_46_im * (x_46_re / y_46_re)) - x_46_im) / hypot(y_46_im, y_46_re);
	elseif (y_46_re <= -1.15e-155)
		tmp = t_0;
	elseif (y_46_re <= 6.4e-218)
		tmp = -x_46_re / y_46_im;
	elseif (y_46_re <= 4.1e+21)
		tmp = t_0;
	else
		tmp = (1.0 / hypot(y_46_re, y_46_im)) * (x_46_im - (x_46_re * (y_46_im / y_46_re)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[(N[(y$46$re * x$46$im), $MachinePrecision] - N[(y$46$im * x$46$re), $MachinePrecision]), $MachinePrecision] / N[(N[(y$46$re * y$46$re), $MachinePrecision] + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$re, -1.8e+80], N[(N[(N[(y$46$im * N[(x$46$re / y$46$re), $MachinePrecision]), $MachinePrecision] - x$46$im), $MachinePrecision] / N[Sqrt[y$46$im ^ 2 + y$46$re ^ 2], $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, -1.15e-155], t$95$0, If[LessEqual[y$46$re, 6.4e-218], N[((-x$46$re) / y$46$im), $MachinePrecision], If[LessEqual[y$46$re, 4.1e+21], t$95$0, N[(N[(1.0 / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision] * N[(x$46$im - N[(x$46$re * N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if y.re < -1.79999999999999997e80

   1. Initial program 32.6%

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

      1. *-un-lft-identity 32.6%

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

      2. add-sqr-sqrt 32.6%

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

      3. times-frac 32.7%

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

      4. hypot-def 32.7%

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

      5. hypot-def 58.9%

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

   4. Applied egg-rr 58.9%

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

   5. Taylor expanded in y.re around -inf 77.9%

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

      1. neg-mul-1 77.9%

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

      2. +-commutative 77.9%

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

      3. unsub-neg 77.9%

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

      4. *-lft-identity 77.9%

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

      5. times-frac 84.9%

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

      6. /-rgt-identity 84.9%

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

   7. Simplified 84.9%

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

      1. sub-neg 84.9%

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

      2. distribute-lft-in 84.9%

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

      3. associate-*r/ 77.9%

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

   9. Applied egg-rr 77.9%

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

       1. *-commutative 77.9%

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

       2. *-commutative 77.9%

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

       3. associate-*r/ 78.1%

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

       4. *-rgt-identity 78.1%

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

       5. distribute-frac-neg 78.1%

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

       6. associate-*r/ 78.1%

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

       7. *-rgt-identity 78.1%

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

       8. sub-neg 78.1%

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

       9. div-sub 78.1%

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

       10. associate-/l* 85.1%

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

       11. associate-/r/ 85.1%

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

       12. hypot-def 39.2%

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

       13. unpow2 39.2%

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

       14. unpow2 39.2%

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

       15. +-commutative 39.2%

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

       16. unpow2 39.2%

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

       17. unpow2 39.2%

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

       18. hypot-def 85.1%

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

   11. Simplified 85.1%

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

   if -1.79999999999999997e80 < y.re < -1.15000000000000003e-155 or 6.4000000000000002e-218 < y.re < 4.1e21

   1. Initial program 80.6%

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

   if -1.15000000000000003e-155 < y.re < 6.4000000000000002e-218

   1. Initial program 64.2%

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

   3. Taylor expanded in y.re around 0 88.1%

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

      1. associate-*r/ 88.1%

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

      2. neg-mul-1 88.1%

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

   5. Simplified 88.1%

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

   if 4.1e21 < y.re

   1. Initial program 28.6%

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

      1. *-un-lft-identity 28.6%

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

      2. add-sqr-sqrt 28.6%

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

      3. times-frac 28.6%

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

      4. hypot-def 28.6%

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

      5. hypot-def 52.5%

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

   4. Applied egg-rr 52.5%

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

   5. Taylor expanded in y.re around inf 69.3%

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

      1. mul-1-neg 69.3%

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

      2. unsub-neg 69.3%

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

      3. *-lft-identity 69.3%

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

      4. times-frac 73.3%

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

      5. /-rgt-identity 73.3%

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

   7. Simplified 73.3%

      \[\leadsto \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \color{blue}{\left(x.im - x.re \cdot \frac{y.im}{y.re}\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 81.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -1.8 \cdot 10^{+80}:\\ \;\;\;\;\frac{y.im \cdot \frac{x.re}{y.re} - x.im}{\mathsf{hypot}\left(y.im, y.re\right)}\\ \mathbf{elif}\;y.re \leq -1.15 \cdot 10^{-155}:\\ \;\;\;\;\frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.re \leq 6.4 \cdot 10^{-218}:\\ \;\;\;\;\frac{-x.re}{y.im}\\ \mathbf{elif}\;y.re \leq 4.1 \cdot 10^{+21}:\\ \;\;\;\;\frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(x.im - x.re \cdot \frac{y.im}{y.re}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 82.1% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{if}\;y.im \leq -9.5 \cdot 10^{+63}:\\ \;\;\;\;\frac{x.re - y.re \cdot \frac{x.im}{y.im}}{\mathsf{hypot}\left(y.im, y.re\right)}\\ \mathbf{elif}\;y.im \leq -5.1 \cdot 10^{-126}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq 4 \cdot 10^{-104}:\\ \;\;\;\;\frac{x.im}{y.re} - \frac{x.re}{y.re \cdot \frac{y.re}{y.im}}\\ \mathbf{elif}\;y.im \leq 7.8 \cdot 10^{+139}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(\frac{x.im}{\frac{y.im}{y.re}} - x.re\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0
         (/ (- (* y.re x.im) (* y.im x.re)) (+ (* y.re y.re) (* y.im y.im)))))
   (if (<= y.im -9.5e+63)
     (/ (- x.re (* y.re (/ x.im y.im))) (hypot y.im y.re))
     (if (<= y.im -5.1e-126)
       t_0
       (if (<= y.im 4e-104)
         (- (/ x.im y.re) (/ x.re (* y.re (/ y.re y.im))))
         (if (<= y.im 7.8e+139)
           t_0
           (* (/ 1.0 (hypot y.re y.im)) (- (/ x.im (/ y.im y.re)) x.re))))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = ((y_46_re * x_46_im) - (y_46_im * x_46_re)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	double tmp;
	if (y_46_im <= -9.5e+63) {
		tmp = (x_46_re - (y_46_re * (x_46_im / y_46_im))) / hypot(y_46_im, y_46_re);
	} else if (y_46_im <= -5.1e-126) {
		tmp = t_0;
	} else if (y_46_im <= 4e-104) {
		tmp = (x_46_im / y_46_re) - (x_46_re / (y_46_re * (y_46_re / y_46_im)));
	} else if (y_46_im <= 7.8e+139) {
		tmp = t_0;
	} else {
		tmp = (1.0 / hypot(y_46_re, y_46_im)) * ((x_46_im / (y_46_im / y_46_re)) - x_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 t_0 = ((y_46_re * x_46_im) - (y_46_im * x_46_re)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	double tmp;
	if (y_46_im <= -9.5e+63) {
		tmp = (x_46_re - (y_46_re * (x_46_im / y_46_im))) / Math.hypot(y_46_im, y_46_re);
	} else if (y_46_im <= -5.1e-126) {
		tmp = t_0;
	} else if (y_46_im <= 4e-104) {
		tmp = (x_46_im / y_46_re) - (x_46_re / (y_46_re * (y_46_re / y_46_im)));
	} else if (y_46_im <= 7.8e+139) {
		tmp = t_0;
	} else {
		tmp = (1.0 / Math.hypot(y_46_re, y_46_im)) * ((x_46_im / (y_46_im / y_46_re)) - x_46_re);
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = ((y_46_re * x_46_im) - (y_46_im * x_46_re)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im))
	tmp = 0
	if y_46_im <= -9.5e+63:
		tmp = (x_46_re - (y_46_re * (x_46_im / y_46_im))) / math.hypot(y_46_im, y_46_re)
	elif y_46_im <= -5.1e-126:
		tmp = t_0
	elif y_46_im <= 4e-104:
		tmp = (x_46_im / y_46_re) - (x_46_re / (y_46_re * (y_46_re / y_46_im)))
	elif y_46_im <= 7.8e+139:
		tmp = t_0
	else:
		tmp = (1.0 / math.hypot(y_46_re, y_46_im)) * ((x_46_im / (y_46_im / y_46_re)) - x_46_re)
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(Float64(Float64(y_46_re * x_46_im) - Float64(y_46_im * x_46_re)) / Float64(Float64(y_46_re * y_46_re) + Float64(y_46_im * y_46_im)))
	tmp = 0.0
	if (y_46_im <= -9.5e+63)
		tmp = Float64(Float64(x_46_re - Float64(y_46_re * Float64(x_46_im / y_46_im))) / hypot(y_46_im, y_46_re));
	elseif (y_46_im <= -5.1e-126)
		tmp = t_0;
	elseif (y_46_im <= 4e-104)
		tmp = Float64(Float64(x_46_im / y_46_re) - Float64(x_46_re / Float64(y_46_re * Float64(y_46_re / y_46_im))));
	elseif (y_46_im <= 7.8e+139)
		tmp = t_0;
	else
		tmp = Float64(Float64(1.0 / hypot(y_46_re, y_46_im)) * Float64(Float64(x_46_im / Float64(y_46_im / y_46_re)) - x_46_re));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = ((y_46_re * x_46_im) - (y_46_im * x_46_re)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	tmp = 0.0;
	if (y_46_im <= -9.5e+63)
		tmp = (x_46_re - (y_46_re * (x_46_im / y_46_im))) / hypot(y_46_im, y_46_re);
	elseif (y_46_im <= -5.1e-126)
		tmp = t_0;
	elseif (y_46_im <= 4e-104)
		tmp = (x_46_im / y_46_re) - (x_46_re / (y_46_re * (y_46_re / y_46_im)));
	elseif (y_46_im <= 7.8e+139)
		tmp = t_0;
	else
		tmp = (1.0 / hypot(y_46_re, y_46_im)) * ((x_46_im / (y_46_im / y_46_re)) - x_46_re);
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[(N[(y$46$re * x$46$im), $MachinePrecision] - N[(y$46$im * x$46$re), $MachinePrecision]), $MachinePrecision] / N[(N[(y$46$re * y$46$re), $MachinePrecision] + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$im, -9.5e+63], N[(N[(x$46$re - N[(y$46$re * N[(x$46$im / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[y$46$im ^ 2 + y$46$re ^ 2], $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, -5.1e-126], t$95$0, If[LessEqual[y$46$im, 4e-104], N[(N[(x$46$im / y$46$re), $MachinePrecision] - N[(x$46$re / N[(y$46$re * N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, 7.8e+139], t$95$0, N[(N[(1.0 / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision] * N[(N[(x$46$im / N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision] - x$46$re), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if y.im < -9.5000000000000003e63

   1. Initial program 35.1%

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

      1. *-un-lft-identity 35.1%

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

      2. add-sqr-sqrt 35.1%

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

      3. times-frac 35.1%

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

      4. hypot-def 35.1%

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

      5. hypot-def 59.4%

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

   4. Applied egg-rr 59.4%

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

   5. Taylor expanded in y.im around -inf 74.1%

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

      1. mul-1-neg 74.1%

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

      2. unsub-neg 74.1%

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

      3. associate-/l* 80.3%

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

   7. Simplified 80.3%

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

      1. sub-neg 80.3%

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

      2. distribute-lft-in 80.3%

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

      3. associate-/r/ 80.3%

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

      4. distribute-rgt-neg-in 80.3%

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

   9. Applied egg-rr 80.3%

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

       1. associate-*l/ 80.4%

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

       2. *-lft-identity 80.4%

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

       3. associate-*l/ 80.4%

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

       4. *-lft-identity 80.4%

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

       5. distribute-rgt-neg-out 80.4%

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

       6. associate-*l/ 74.3%

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

       7. distribute-neg-frac 74.3%

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

       8. sub-neg 74.3%

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

       9. div-sub 74.3%

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

       10. associate-*l/ 80.4%

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

       11. *-commutative 80.4%

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

       12. hypot-def 40.4%

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

       13. unpow2 40.4%

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

       14. unpow2 40.4%

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

       15. +-commutative 40.4%

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

       16. unpow2 40.4%

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

       17. unpow2 40.4%

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

       18. hypot-def 80.4%

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

   11. Simplified 80.4%

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

   if -9.5000000000000003e63 < y.im < -5.10000000000000002e-126 or 3.99999999999999971e-104 < y.im < 7.80000000000000012e139

   1. Initial program 77.1%

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

   if -5.10000000000000002e-126 < y.im < 3.99999999999999971e-104

   1. Initial program 63.5%

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

   3. Taylor expanded in y.re around inf 84.5%

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

      1. +-commutative 84.5%

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

      2. mul-1-neg 84.5%

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

      3. unsub-neg 84.5%

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

      4. associate-/l* 85.7%

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

   5. Simplified 85.7%

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

      1. unpow2 85.7%

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

      2. *-un-lft-identity 85.7%

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

      3. times-frac 88.0%

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

   7. Applied egg-rr 88.0%

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

   if 7.80000000000000012e139 < y.im

   1. Initial program 18.8%

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

      1. *-un-lft-identity 18.8%

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

      2. add-sqr-sqrt 18.8%

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

      3. times-frac 18.8%

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

      4. hypot-def 18.8%

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

      5. hypot-def 50.0%

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

   4. Applied egg-rr 50.0%

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

   5. Taylor expanded in y.re around 0 68.0%

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

      1. neg-mul-1 68.0%

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

      2. +-commutative 68.0%

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

      3. unsub-neg 68.0%

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

      4. associate-/l* 83.0%

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

   7. Simplified 83.0%

      \[\leadsto \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \color{blue}{\left(\frac{x.im}{\frac{y.im}{y.re}} - x.re\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 82.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -9.5 \cdot 10^{+63}:\\ \;\;\;\;\frac{x.re - y.re \cdot \frac{x.im}{y.im}}{\mathsf{hypot}\left(y.im, y.re\right)}\\ \mathbf{elif}\;y.im \leq -5.1 \cdot 10^{-126}:\\ \;\;\;\;\frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.im \leq 4 \cdot 10^{-104}:\\ \;\;\;\;\frac{x.im}{y.re} - \frac{x.re}{y.re \cdot \frac{y.re}{y.im}}\\ \mathbf{elif}\;y.im \leq 7.8 \cdot 10^{+139}:\\ \;\;\;\;\frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(\frac{x.im}{\frac{y.im}{y.re}} - x.re\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 79.3% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ t_1 := \frac{-x.re}{\mathsf{fma}\left(y.re, \frac{y.re}{y.im}, y.im\right)}\\ \mathbf{if}\;y.im \leq -8.6 \cdot 10^{+152}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.im \leq -8 \cdot 10^{-125}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq 6.2 \cdot 10^{-103}:\\ \;\;\;\;\frac{x.im}{y.re} - \frac{x.re}{y.re \cdot \frac{y.re}{y.im}}\\ \mathbf{elif}\;y.im \leq 2.65 \cdot 10^{+59}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0
         (/ (- (* y.re x.im) (* y.im x.re)) (+ (* y.re y.re) (* y.im y.im))))
        (t_1 (/ (- x.re) (fma y.re (/ y.re y.im) y.im))))
   (if (<= y.im -8.6e+152)
     t_1
     (if (<= y.im -8e-125)
       t_0
       (if (<= y.im 6.2e-103)
         (- (/ x.im y.re) (/ x.re (* y.re (/ y.re y.im))))
         (if (<= y.im 2.65e+59) t_0 t_1))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = ((y_46_re * x_46_im) - (y_46_im * x_46_re)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	double t_1 = -x_46_re / fma(y_46_re, (y_46_re / y_46_im), y_46_im);
	double tmp;
	if (y_46_im <= -8.6e+152) {
		tmp = t_1;
	} else if (y_46_im <= -8e-125) {
		tmp = t_0;
	} else if (y_46_im <= 6.2e-103) {
		tmp = (x_46_im / y_46_re) - (x_46_re / (y_46_re * (y_46_re / y_46_im)));
	} else if (y_46_im <= 2.65e+59) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(Float64(Float64(y_46_re * x_46_im) - Float64(y_46_im * x_46_re)) / Float64(Float64(y_46_re * y_46_re) + Float64(y_46_im * y_46_im)))
	t_1 = Float64(Float64(-x_46_re) / fma(y_46_re, Float64(y_46_re / y_46_im), y_46_im))
	tmp = 0.0
	if (y_46_im <= -8.6e+152)
		tmp = t_1;
	elseif (y_46_im <= -8e-125)
		tmp = t_0;
	elseif (y_46_im <= 6.2e-103)
		tmp = Float64(Float64(x_46_im / y_46_re) - Float64(x_46_re / Float64(y_46_re * Float64(y_46_re / y_46_im))));
	elseif (y_46_im <= 2.65e+59)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[(N[(y$46$re * x$46$im), $MachinePrecision] - N[(y$46$im * x$46$re), $MachinePrecision]), $MachinePrecision] / N[(N[(y$46$re * y$46$re), $MachinePrecision] + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[((-x$46$re) / N[(y$46$re * N[(y$46$re / y$46$im), $MachinePrecision] + y$46$im), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$im, -8.6e+152], t$95$1, If[LessEqual[y$46$im, -8e-125], t$95$0, If[LessEqual[y$46$im, 6.2e-103], N[(N[(x$46$im / y$46$re), $MachinePrecision] - N[(x$46$re / N[(y$46$re * N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, 2.65e+59], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if y.im < -8.59999999999999989e152 or 2.6499999999999998e59 < y.im

   1. Initial program 32.7%

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

      1. div-sub 32.7%

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

      2. sub-neg 32.7%

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

      3. *-commutative 32.7%

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

      4. add-sqr-sqrt 32.7%

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

      5. times-frac 34.0%

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

      6. fma-def 34.0%

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

      7. hypot-def 34.0%

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

      8. hypot-def 48.6%

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

      9. associate-/l* 53.3%

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

      10. add-sqr-sqrt 53.3%

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

      11. pow2 53.3%

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

      12. hypot-def 53.3%

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

   4. Applied egg-rr 53.3%

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

   5. Taylor expanded in y.re around 0 85.8%

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

      1. +-commutative 85.8%

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

      2. unpow2 85.8%

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

      3. associate-*r/ 95.4%

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

      4. fma-def 95.4%

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

   7. Simplified 95.4%

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

   8. Taylor expanded in x.im around 0 65.3%

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

      1. associate-*r/ 65.3%

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

      2. neg-mul-1 65.3%

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

      3. unpow2 65.3%

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

      4. associate-*r/ 75.1%

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

      5. +-commutative 75.1%

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

      6. fma-def 75.1%

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

   10. Simplified 75.1%

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

   if -8.59999999999999989e152 < y.im < -8.0000000000000001e-125 or 6.2000000000000003e-103 < y.im < 2.6499999999999998e59

   1. Initial program 72.8%

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

   if -8.0000000000000001e-125 < y.im < 6.2000000000000003e-103

   1. Initial program 63.5%

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

   3. Taylor expanded in y.re around inf 84.5%

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

      1. +-commutative 84.5%

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

      2. mul-1-neg 84.5%

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

      3. unsub-neg 84.5%

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

      4. associate-/l* 85.7%

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

   5. Simplified 85.7%

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

      1. unpow2 85.7%

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

      2. *-un-lft-identity 85.7%

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

      3. times-frac 88.0%

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

   7. Applied egg-rr 88.0%

      \[\leadsto \frac{x.im}{y.re} - \frac{x.re}{\color{blue}{\frac{y.re}{1} \cdot \frac{y.re}{y.im}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 78.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -8.6 \cdot 10^{+152}:\\ \;\;\;\;\frac{-x.re}{\mathsf{fma}\left(y.re, \frac{y.re}{y.im}, y.im\right)}\\ \mathbf{elif}\;y.im \leq -8 \cdot 10^{-125}:\\ \;\;\;\;\frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.im \leq 6.2 \cdot 10^{-103}:\\ \;\;\;\;\frac{x.im}{y.re} - \frac{x.re}{y.re \cdot \frac{y.re}{y.im}}\\ \mathbf{elif}\;y.im \leq 2.65 \cdot 10^{+59}:\\ \;\;\;\;\frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{-x.re}{\mathsf{fma}\left(y.re, \frac{y.re}{y.im}, y.im\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 81.0% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{if}\;y.im \leq -7.4 \cdot 10^{+61}:\\ \;\;\;\;\frac{x.re - y.re \cdot \frac{x.im}{y.im}}{\mathsf{hypot}\left(y.im, y.re\right)}\\ \mathbf{elif}\;y.im \leq -1.32 \cdot 10^{-126}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq 3.8 \cdot 10^{-105}:\\ \;\;\;\;\frac{x.im}{y.re} - \frac{x.re}{y.re \cdot \frac{y.re}{y.im}}\\ \mathbf{elif}\;y.im \leq 2.15 \cdot 10^{+59}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{-x.re}{\mathsf{fma}\left(y.re, \frac{y.re}{y.im}, y.im\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0
         (/ (- (* y.re x.im) (* y.im x.re)) (+ (* y.re y.re) (* y.im y.im)))))
   (if (<= y.im -7.4e+61)
     (/ (- x.re (* y.re (/ x.im y.im))) (hypot y.im y.re))
     (if (<= y.im -1.32e-126)
       t_0
       (if (<= y.im 3.8e-105)
         (- (/ x.im y.re) (/ x.re (* y.re (/ y.re y.im))))
         (if (<= y.im 2.15e+59)
           t_0
           (/ (- x.re) (fma 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) {
	double t_0 = ((y_46_re * x_46_im) - (y_46_im * x_46_re)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	double tmp;
	if (y_46_im <= -7.4e+61) {
		tmp = (x_46_re - (y_46_re * (x_46_im / y_46_im))) / hypot(y_46_im, y_46_re);
	} else if (y_46_im <= -1.32e-126) {
		tmp = t_0;
	} else if (y_46_im <= 3.8e-105) {
		tmp = (x_46_im / y_46_re) - (x_46_re / (y_46_re * (y_46_re / y_46_im)));
	} else if (y_46_im <= 2.15e+59) {
		tmp = t_0;
	} else {
		tmp = -x_46_re / fma(y_46_re, (y_46_re / y_46_im), y_46_im);
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(Float64(Float64(y_46_re * x_46_im) - Float64(y_46_im * x_46_re)) / Float64(Float64(y_46_re * y_46_re) + Float64(y_46_im * y_46_im)))
	tmp = 0.0
	if (y_46_im <= -7.4e+61)
		tmp = Float64(Float64(x_46_re - Float64(y_46_re * Float64(x_46_im / y_46_im))) / hypot(y_46_im, y_46_re));
	elseif (y_46_im <= -1.32e-126)
		tmp = t_0;
	elseif (y_46_im <= 3.8e-105)
		tmp = Float64(Float64(x_46_im / y_46_re) - Float64(x_46_re / Float64(y_46_re * Float64(y_46_re / y_46_im))));
	elseif (y_46_im <= 2.15e+59)
		tmp = t_0;
	else
		tmp = Float64(Float64(-x_46_re) / fma(y_46_re, Float64(y_46_re / y_46_im), 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[(N[(y$46$re * x$46$im), $MachinePrecision] - N[(y$46$im * x$46$re), $MachinePrecision]), $MachinePrecision] / N[(N[(y$46$re * y$46$re), $MachinePrecision] + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$im, -7.4e+61], N[(N[(x$46$re - N[(y$46$re * N[(x$46$im / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[y$46$im ^ 2 + y$46$re ^ 2], $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, -1.32e-126], t$95$0, If[LessEqual[y$46$im, 3.8e-105], N[(N[(x$46$im / y$46$re), $MachinePrecision] - N[(x$46$re / N[(y$46$re * N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, 2.15e+59], t$95$0, N[((-x$46$re) / N[(y$46$re * N[(y$46$re / y$46$im), $MachinePrecision] + y$46$im), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if y.im < -7.40000000000000005e61

   1. Initial program 35.1%

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

      1. *-un-lft-identity 35.1%

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

      2. add-sqr-sqrt 35.1%

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

      3. times-frac 35.1%

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

      4. hypot-def 35.1%

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

      5. hypot-def 59.4%

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

   4. Applied egg-rr 59.4%

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

   5. Taylor expanded in y.im around -inf 74.1%

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

      1. mul-1-neg 74.1%

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

      2. unsub-neg 74.1%

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

      3. associate-/l* 80.3%

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

   7. Simplified 80.3%

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

      1. sub-neg 80.3%

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

      2. distribute-lft-in 80.3%

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

      3. associate-/r/ 80.3%

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

      4. distribute-rgt-neg-in 80.3%

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

   9. Applied egg-rr 80.3%

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

       1. associate-*l/ 80.4%

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

       2. *-lft-identity 80.4%

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

       3. associate-*l/ 80.4%

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

       4. *-lft-identity 80.4%

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

       5. distribute-rgt-neg-out 80.4%

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

       6. associate-*l/ 74.3%

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

       7. distribute-neg-frac 74.3%

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

       8. sub-neg 74.3%

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

       9. div-sub 74.3%

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

       10. associate-*l/ 80.4%

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

       11. *-commutative 80.4%

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

       12. hypot-def 40.4%

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

       13. unpow2 40.4%

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

       14. unpow2 40.4%

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

       15. +-commutative 40.4%

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

       16. unpow2 40.4%

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

       17. unpow2 40.4%

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

       18. hypot-def 80.4%

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

   11. Simplified 80.4%

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

   if -7.40000000000000005e61 < y.im < -1.31999999999999992e-126 or 3.7999999999999998e-105 < y.im < 2.15000000000000012e59

   1. Initial program 78.7%

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

   if -1.31999999999999992e-126 < y.im < 3.7999999999999998e-105

   1. Initial program 63.5%

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

   3. Taylor expanded in y.re around inf 84.5%

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

      1. +-commutative 84.5%

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

      2. mul-1-neg 84.5%

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

      3. unsub-neg 84.5%

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

      4. associate-/l* 85.7%

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

   5. Simplified 85.7%

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

      1. unpow2 85.7%

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

      2. *-un-lft-identity 85.7%

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

      3. times-frac 88.0%

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

   7. Applied egg-rr 88.0%

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

   if 2.15000000000000012e59 < y.im

   1. Initial program 34.8%

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

      1. div-sub 34.8%

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

      2. sub-neg 34.8%

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

      3. *-commutative 34.8%

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

      4. add-sqr-sqrt 34.8%

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

      5. times-frac 36.8%

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

      6. fma-def 36.8%

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

      7. hypot-def 36.8%

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

      8. hypot-def 53.2%

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

      9. associate-/l* 59.8%

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

      10. add-sqr-sqrt 59.8%

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

      11. pow2 59.8%

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

      12. hypot-def 59.8%

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

   4. Applied egg-rr 59.8%

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

   5. Taylor expanded in y.re around 0 84.2%

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

      1. +-commutative 84.2%

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

      2. unpow2 84.2%

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

      3. associate-*r/ 94.3%

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

      4. fma-def 94.3%

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

   7. Simplified 94.3%

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

   8. Taylor expanded in x.im around 0 58.4%

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

      1. associate-*r/ 58.4%

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

      2. neg-mul-1 58.4%

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

      3. unpow2 58.4%

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

      4. associate-*r/ 68.8%

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

      5. +-commutative 68.8%

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

      6. fma-def 68.8%

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

   10. Simplified 68.8%

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

4. Final simplification 80.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -7.4 \cdot 10^{+61}:\\ \;\;\;\;\frac{x.re - y.re \cdot \frac{x.im}{y.im}}{\mathsf{hypot}\left(y.im, y.re\right)}\\ \mathbf{elif}\;y.im \leq -1.32 \cdot 10^{-126}:\\ \;\;\;\;\frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.im \leq 3.8 \cdot 10^{-105}:\\ \;\;\;\;\frac{x.im}{y.re} - \frac{x.re}{y.re \cdot \frac{y.re}{y.im}}\\ \mathbf{elif}\;y.im \leq 2.15 \cdot 10^{+59}:\\ \;\;\;\;\frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{-x.re}{\mathsf{fma}\left(y.re, \frac{y.re}{y.im}, y.im\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 77.6% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{if}\;y.im \leq -9 \cdot 10^{+153}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot x.re\\ \mathbf{elif}\;y.im \leq -8.2 \cdot 10^{-126}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq 1.85 \cdot 10^{-104}:\\ \;\;\;\;\frac{x.im}{y.re} - \frac{x.re}{y.re \cdot \frac{y.re}{y.im}}\\ \mathbf{elif}\;y.im \leq 2.4 \cdot 10^{+189}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot \frac{-1}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0
         (/ (- (* y.re x.im) (* y.im x.re)) (+ (* y.re y.re) (* y.im y.im)))))
   (if (<= y.im -9e+153)
     (* (/ 1.0 (hypot y.re y.im)) x.re)
     (if (<= y.im -8.2e-126)
       t_0
       (if (<= y.im 1.85e-104)
         (- (/ x.im y.re) (/ x.re (* y.re (/ y.re y.im))))
         (if (<= y.im 2.4e+189) t_0 (* x.re (/ -1.0 (hypot y.re y.im)))))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = ((y_46_re * x_46_im) - (y_46_im * x_46_re)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	double tmp;
	if (y_46_im <= -9e+153) {
		tmp = (1.0 / hypot(y_46_re, y_46_im)) * x_46_re;
	} else if (y_46_im <= -8.2e-126) {
		tmp = t_0;
	} else if (y_46_im <= 1.85e-104) {
		tmp = (x_46_im / y_46_re) - (x_46_re / (y_46_re * (y_46_re / y_46_im)));
	} else if (y_46_im <= 2.4e+189) {
		tmp = t_0;
	} else {
		tmp = x_46_re * (-1.0 / hypot(y_46_re, y_46_im));
	}
	return tmp;
}

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = ((y_46_re * x_46_im) - (y_46_im * x_46_re)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	double tmp;
	if (y_46_im <= -9e+153) {
		tmp = (1.0 / Math.hypot(y_46_re, y_46_im)) * x_46_re;
	} else if (y_46_im <= -8.2e-126) {
		tmp = t_0;
	} else if (y_46_im <= 1.85e-104) {
		tmp = (x_46_im / y_46_re) - (x_46_re / (y_46_re * (y_46_re / y_46_im)));
	} else if (y_46_im <= 2.4e+189) {
		tmp = t_0;
	} else {
		tmp = x_46_re * (-1.0 / Math.hypot(y_46_re, y_46_im));
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = ((y_46_re * x_46_im) - (y_46_im * x_46_re)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im))
	tmp = 0
	if y_46_im <= -9e+153:
		tmp = (1.0 / math.hypot(y_46_re, y_46_im)) * x_46_re
	elif y_46_im <= -8.2e-126:
		tmp = t_0
	elif y_46_im <= 1.85e-104:
		tmp = (x_46_im / y_46_re) - (x_46_re / (y_46_re * (y_46_re / y_46_im)))
	elif y_46_im <= 2.4e+189:
		tmp = t_0
	else:
		tmp = x_46_re * (-1.0 / math.hypot(y_46_re, y_46_im))
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(Float64(Float64(y_46_re * x_46_im) - Float64(y_46_im * x_46_re)) / Float64(Float64(y_46_re * y_46_re) + Float64(y_46_im * y_46_im)))
	tmp = 0.0
	if (y_46_im <= -9e+153)
		tmp = Float64(Float64(1.0 / hypot(y_46_re, y_46_im)) * x_46_re);
	elseif (y_46_im <= -8.2e-126)
		tmp = t_0;
	elseif (y_46_im <= 1.85e-104)
		tmp = Float64(Float64(x_46_im / y_46_re) - Float64(x_46_re / Float64(y_46_re * Float64(y_46_re / y_46_im))));
	elseif (y_46_im <= 2.4e+189)
		tmp = t_0;
	else
		tmp = Float64(x_46_re * Float64(-1.0 / hypot(y_46_re, y_46_im)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = ((y_46_re * x_46_im) - (y_46_im * x_46_re)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	tmp = 0.0;
	if (y_46_im <= -9e+153)
		tmp = (1.0 / hypot(y_46_re, y_46_im)) * x_46_re;
	elseif (y_46_im <= -8.2e-126)
		tmp = t_0;
	elseif (y_46_im <= 1.85e-104)
		tmp = (x_46_im / y_46_re) - (x_46_re / (y_46_re * (y_46_re / y_46_im)));
	elseif (y_46_im <= 2.4e+189)
		tmp = t_0;
	else
		tmp = x_46_re * (-1.0 / hypot(y_46_re, y_46_im));
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[(N[(y$46$re * x$46$im), $MachinePrecision] - N[(y$46$im * x$46$re), $MachinePrecision]), $MachinePrecision] / N[(N[(y$46$re * y$46$re), $MachinePrecision] + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$im, -9e+153], N[(N[(1.0 / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision] * x$46$re), $MachinePrecision], If[LessEqual[y$46$im, -8.2e-126], t$95$0, If[LessEqual[y$46$im, 1.85e-104], N[(N[(x$46$im / y$46$re), $MachinePrecision] - N[(x$46$re / N[(y$46$re * N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, 2.4e+189], t$95$0, N[(x$46$re * N[(-1.0 / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if y.im < -9.0000000000000002e153

   1. Initial program 29.3%

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

      1. *-un-lft-identity 29.3%

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

      2. add-sqr-sqrt 29.3%

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

      3. times-frac 29.3%

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

      4. hypot-def 29.3%

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

      5. hypot-def 61.5%

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

   4. Applied egg-rr 61.5%

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

   5. Taylor expanded in y.im around -inf 79.8%

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

   if -9.0000000000000002e153 < y.im < -8.1999999999999995e-126 or 1.85e-104 < y.im < 2.4000000000000001e189

   1. Initial program 69.0%

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

   if -8.1999999999999995e-126 < y.im < 1.85e-104

   1. Initial program 63.5%

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

   3. Taylor expanded in y.re around inf 84.5%

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

      1. +-commutative 84.5%

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

      2. mul-1-neg 84.5%

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

      3. unsub-neg 84.5%

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

      4. associate-/l* 85.7%

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

   5. Simplified 85.7%

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

      1. unpow2 85.7%

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

      2. *-un-lft-identity 85.7%

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

      3. times-frac 88.0%

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

   7. Applied egg-rr 88.0%

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

   if 2.4000000000000001e189 < y.im

   1. Initial program 15.3%

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

      1. *-un-lft-identity 15.3%

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

      2. add-sqr-sqrt 15.3%

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

      3. times-frac 15.3%

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

      4. hypot-def 15.3%

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

      5. hypot-def 44.8%

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

   4. Applied egg-rr 44.8%

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

   5. Taylor expanded in y.re around 0 74.9%

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

      1. neg-mul-1 74.9%

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

   7. Simplified 74.9%

      \[\leadsto \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \color{blue}{\left(-x.re\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 77.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -9 \cdot 10^{+153}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot x.re\\ \mathbf{elif}\;y.im \leq -8.2 \cdot 10^{-126}:\\ \;\;\;\;\frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.im \leq 1.85 \cdot 10^{-104}:\\ \;\;\;\;\frac{x.im}{y.re} - \frac{x.re}{y.re \cdot \frac{y.re}{y.im}}\\ \mathbf{elif}\;y.im \leq 2.4 \cdot 10^{+189}:\\ \;\;\;\;\frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot \frac{-1}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 77.2% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{if}\;y.re \leq -3 \cdot 10^{+82}:\\ \;\;\;\;\frac{y.im \cdot \frac{x.re}{y.re} - x.im}{\mathsf{hypot}\left(y.im, y.re\right)}\\ \mathbf{elif}\;y.re \leq -1.75 \cdot 10^{-157}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq 6.5 \cdot 10^{-218}:\\ \;\;\;\;\frac{-x.re}{y.im}\\ \mathbf{elif}\;y.re \leq 4.3 \cdot 10^{+21}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re} - \frac{x.re}{y.re \cdot \frac{y.re}{y.im}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0
         (/ (- (* y.re x.im) (* y.im x.re)) (+ (* y.re y.re) (* y.im y.im)))))
   (if (<= y.re -3e+82)
     (/ (- (* y.im (/ x.re y.re)) x.im) (hypot y.im y.re))
     (if (<= y.re -1.75e-157)
       t_0
       (if (<= y.re 6.5e-218)
         (/ (- x.re) y.im)
         (if (<= y.re 4.3e+21)
           t_0
           (- (/ x.im y.re) (/ x.re (* y.re (/ y.re y.im))))))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = ((y_46_re * x_46_im) - (y_46_im * x_46_re)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	double tmp;
	if (y_46_re <= -3e+82) {
		tmp = ((y_46_im * (x_46_re / y_46_re)) - x_46_im) / hypot(y_46_im, y_46_re);
	} else if (y_46_re <= -1.75e-157) {
		tmp = t_0;
	} else if (y_46_re <= 6.5e-218) {
		tmp = -x_46_re / y_46_im;
	} else if (y_46_re <= 4.3e+21) {
		tmp = t_0;
	} else {
		tmp = (x_46_im / y_46_re) - (x_46_re / (y_46_re * (y_46_re / y_46_im)));
	}
	return tmp;
}

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = ((y_46_re * x_46_im) - (y_46_im * x_46_re)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	double tmp;
	if (y_46_re <= -3e+82) {
		tmp = ((y_46_im * (x_46_re / y_46_re)) - x_46_im) / Math.hypot(y_46_im, y_46_re);
	} else if (y_46_re <= -1.75e-157) {
		tmp = t_0;
	} else if (y_46_re <= 6.5e-218) {
		tmp = -x_46_re / y_46_im;
	} else if (y_46_re <= 4.3e+21) {
		tmp = t_0;
	} else {
		tmp = (x_46_im / y_46_re) - (x_46_re / (y_46_re * (y_46_re / y_46_im)));
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = ((y_46_re * x_46_im) - (y_46_im * x_46_re)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im))
	tmp = 0
	if y_46_re <= -3e+82:
		tmp = ((y_46_im * (x_46_re / y_46_re)) - x_46_im) / math.hypot(y_46_im, y_46_re)
	elif y_46_re <= -1.75e-157:
		tmp = t_0
	elif y_46_re <= 6.5e-218:
		tmp = -x_46_re / y_46_im
	elif y_46_re <= 4.3e+21:
		tmp = t_0
	else:
		tmp = (x_46_im / y_46_re) - (x_46_re / (y_46_re * (y_46_re / y_46_im)))
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(Float64(Float64(y_46_re * x_46_im) - Float64(y_46_im * x_46_re)) / Float64(Float64(y_46_re * y_46_re) + Float64(y_46_im * y_46_im)))
	tmp = 0.0
	if (y_46_re <= -3e+82)
		tmp = Float64(Float64(Float64(y_46_im * Float64(x_46_re / y_46_re)) - x_46_im) / hypot(y_46_im, y_46_re));
	elseif (y_46_re <= -1.75e-157)
		tmp = t_0;
	elseif (y_46_re <= 6.5e-218)
		tmp = Float64(Float64(-x_46_re) / y_46_im);
	elseif (y_46_re <= 4.3e+21)
		tmp = t_0;
	else
		tmp = Float64(Float64(x_46_im / y_46_re) - Float64(x_46_re / Float64(y_46_re * Float64(y_46_re / y_46_im))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = ((y_46_re * x_46_im) - (y_46_im * x_46_re)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	tmp = 0.0;
	if (y_46_re <= -3e+82)
		tmp = ((y_46_im * (x_46_re / y_46_re)) - x_46_im) / hypot(y_46_im, y_46_re);
	elseif (y_46_re <= -1.75e-157)
		tmp = t_0;
	elseif (y_46_re <= 6.5e-218)
		tmp = -x_46_re / y_46_im;
	elseif (y_46_re <= 4.3e+21)
		tmp = t_0;
	else
		tmp = (x_46_im / y_46_re) - (x_46_re / (y_46_re * (y_46_re / y_46_im)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[(N[(y$46$re * x$46$im), $MachinePrecision] - N[(y$46$im * x$46$re), $MachinePrecision]), $MachinePrecision] / N[(N[(y$46$re * y$46$re), $MachinePrecision] + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$re, -3e+82], N[(N[(N[(y$46$im * N[(x$46$re / y$46$re), $MachinePrecision]), $MachinePrecision] - x$46$im), $MachinePrecision] / N[Sqrt[y$46$im ^ 2 + y$46$re ^ 2], $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, -1.75e-157], t$95$0, If[LessEqual[y$46$re, 6.5e-218], N[((-x$46$re) / y$46$im), $MachinePrecision], If[LessEqual[y$46$re, 4.3e+21], t$95$0, N[(N[(x$46$im / y$46$re), $MachinePrecision] - N[(x$46$re / N[(y$46$re * N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if y.re < -2.99999999999999989e82

   1. Initial program 32.6%

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

      1. *-un-lft-identity 32.6%

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

      2. add-sqr-sqrt 32.6%

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

      3. times-frac 32.7%

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

      4. hypot-def 32.7%

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

      5. hypot-def 58.9%

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

   4. Applied egg-rr 58.9%

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

   5. Taylor expanded in y.re around -inf 77.9%

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

      1. neg-mul-1 77.9%

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

      2. +-commutative 77.9%

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

      3. unsub-neg 77.9%

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

      4. *-lft-identity 77.9%

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

      5. times-frac 84.9%

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

      6. /-rgt-identity 84.9%

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

   7. Simplified 84.9%

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

      1. sub-neg 84.9%

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

      2. distribute-lft-in 84.9%

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

      3. associate-*r/ 77.9%

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

   9. Applied egg-rr 77.9%

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

       1. *-commutative 77.9%

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

       2. *-commutative 77.9%

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

       3. associate-*r/ 78.1%

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

       4. *-rgt-identity 78.1%

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

       5. distribute-frac-neg 78.1%

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

       6. associate-*r/ 78.1%

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

       7. *-rgt-identity 78.1%

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

       8. sub-neg 78.1%

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

       9. div-sub 78.1%

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

       10. associate-/l* 85.1%

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

       11. associate-/r/ 85.1%

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

       12. hypot-def 39.2%

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

       13. unpow2 39.2%

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

       14. unpow2 39.2%

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

       15. +-commutative 39.2%

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

       16. unpow2 39.2%

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

       17. unpow2 39.2%

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

       18. hypot-def 85.1%

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

   11. Simplified 85.1%

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

   if -2.99999999999999989e82 < y.re < -1.7500000000000001e-157 or 6.49999999999999983e-218 < y.re < 4.3e21

   1. Initial program 80.6%

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

   if -1.7500000000000001e-157 < y.re < 6.49999999999999983e-218

   1. Initial program 64.2%

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

   3. Taylor expanded in y.re around 0 88.1%

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

      1. associate-*r/ 88.1%

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

      2. neg-mul-1 88.1%

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

   5. Simplified 88.1%

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

   if 4.3e21 < y.re

   1. Initial program 28.6%

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

   3. Taylor expanded in y.re around inf 62.1%

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

      1. +-commutative 62.1%

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

      2. mul-1-neg 62.1%

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

      3. unsub-neg 62.1%

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

      4. associate-/l* 62.5%

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

   5. Simplified 62.5%

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

      1. unpow2 62.5%

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

      2. *-un-lft-identity 62.5%

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

      3. times-frac 67.9%

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

   7. Applied egg-rr 67.9%

      \[\leadsto \frac{x.im}{y.re} - \frac{x.re}{\color{blue}{\frac{y.re}{1} \cdot \frac{y.re}{y.im}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 80.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -3 \cdot 10^{+82}:\\ \;\;\;\;\frac{y.im \cdot \frac{x.re}{y.re} - x.im}{\mathsf{hypot}\left(y.im, y.re\right)}\\ \mathbf{elif}\;y.re \leq -1.75 \cdot 10^{-157}:\\ \;\;\;\;\frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.re \leq 6.5 \cdot 10^{-218}:\\ \;\;\;\;\frac{-x.re}{y.im}\\ \mathbf{elif}\;y.re \leq 4.3 \cdot 10^{+21}:\\ \;\;\;\;\frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re} - \frac{x.re}{y.re \cdot \frac{y.re}{y.im}}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 77.6% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{if}\;y.im \leq -1.05 \cdot 10^{+153}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot x.re\\ \mathbf{elif}\;y.im \leq -4.1 \cdot 10^{-126}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq 5.8 \cdot 10^{-109}:\\ \;\;\;\;\frac{x.im}{y.re} - \frac{x.re}{y.re \cdot \frac{y.re}{y.im}}\\ \mathbf{elif}\;y.im \leq 2.4 \cdot 10^{+189}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{-x.re}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0
         (/ (- (* y.re x.im) (* y.im x.re)) (+ (* y.re y.re) (* y.im y.im)))))
   (if (<= y.im -1.05e+153)
     (* (/ 1.0 (hypot y.re y.im)) x.re)
     (if (<= y.im -4.1e-126)
       t_0
       (if (<= y.im 5.8e-109)
         (- (/ x.im y.re) (/ x.re (* y.re (/ y.re y.im))))
         (if (<= y.im 2.4e+189) t_0 (/ (- 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 = ((y_46_re * x_46_im) - (y_46_im * x_46_re)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	double tmp;
	if (y_46_im <= -1.05e+153) {
		tmp = (1.0 / hypot(y_46_re, y_46_im)) * x_46_re;
	} else if (y_46_im <= -4.1e-126) {
		tmp = t_0;
	} else if (y_46_im <= 5.8e-109) {
		tmp = (x_46_im / y_46_re) - (x_46_re / (y_46_re * (y_46_re / y_46_im)));
	} else if (y_46_im <= 2.4e+189) {
		tmp = t_0;
	} else {
		tmp = -x_46_re / y_46_im;
	}
	return tmp;
}

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = ((y_46_re * x_46_im) - (y_46_im * x_46_re)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	double tmp;
	if (y_46_im <= -1.05e+153) {
		tmp = (1.0 / Math.hypot(y_46_re, y_46_im)) * x_46_re;
	} else if (y_46_im <= -4.1e-126) {
		tmp = t_0;
	} else if (y_46_im <= 5.8e-109) {
		tmp = (x_46_im / y_46_re) - (x_46_re / (y_46_re * (y_46_re / y_46_im)));
	} else if (y_46_im <= 2.4e+189) {
		tmp = t_0;
	} else {
		tmp = -x_46_re / y_46_im;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = ((y_46_re * x_46_im) - (y_46_im * x_46_re)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im))
	tmp = 0
	if y_46_im <= -1.05e+153:
		tmp = (1.0 / math.hypot(y_46_re, y_46_im)) * x_46_re
	elif y_46_im <= -4.1e-126:
		tmp = t_0
	elif y_46_im <= 5.8e-109:
		tmp = (x_46_im / y_46_re) - (x_46_re / (y_46_re * (y_46_re / y_46_im)))
	elif y_46_im <= 2.4e+189:
		tmp = t_0
	else:
		tmp = -x_46_re / y_46_im
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(Float64(Float64(y_46_re * x_46_im) - Float64(y_46_im * x_46_re)) / Float64(Float64(y_46_re * y_46_re) + Float64(y_46_im * y_46_im)))
	tmp = 0.0
	if (y_46_im <= -1.05e+153)
		tmp = Float64(Float64(1.0 / hypot(y_46_re, y_46_im)) * x_46_re);
	elseif (y_46_im <= -4.1e-126)
		tmp = t_0;
	elseif (y_46_im <= 5.8e-109)
		tmp = Float64(Float64(x_46_im / y_46_re) - Float64(x_46_re / Float64(y_46_re * Float64(y_46_re / y_46_im))));
	elseif (y_46_im <= 2.4e+189)
		tmp = t_0;
	else
		tmp = Float64(Float64(-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 = ((y_46_re * x_46_im) - (y_46_im * x_46_re)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	tmp = 0.0;
	if (y_46_im <= -1.05e+153)
		tmp = (1.0 / hypot(y_46_re, y_46_im)) * x_46_re;
	elseif (y_46_im <= -4.1e-126)
		tmp = t_0;
	elseif (y_46_im <= 5.8e-109)
		tmp = (x_46_im / y_46_re) - (x_46_re / (y_46_re * (y_46_re / y_46_im)));
	elseif (y_46_im <= 2.4e+189)
		tmp = t_0;
	else
		tmp = -x_46_re / y_46_im;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[(N[(y$46$re * x$46$im), $MachinePrecision] - N[(y$46$im * x$46$re), $MachinePrecision]), $MachinePrecision] / N[(N[(y$46$re * y$46$re), $MachinePrecision] + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$im, -1.05e+153], N[(N[(1.0 / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision] * x$46$re), $MachinePrecision], If[LessEqual[y$46$im, -4.1e-126], t$95$0, If[LessEqual[y$46$im, 5.8e-109], N[(N[(x$46$im / y$46$re), $MachinePrecision] - N[(x$46$re / N[(y$46$re * N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, 2.4e+189], t$95$0, N[((-x$46$re) / y$46$im), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if y.im < -1.05000000000000008e153

   1. Initial program 29.3%

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

      1. *-un-lft-identity 29.3%

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

      2. add-sqr-sqrt 29.3%

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

      3. times-frac 29.3%

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

      4. hypot-def 29.3%

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

      5. hypot-def 61.5%

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

   4. Applied egg-rr 61.5%

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

   5. Taylor expanded in y.im around -inf 79.8%

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

   if -1.05000000000000008e153 < y.im < -4.0999999999999997e-126 or 5.8e-109 < y.im < 2.4000000000000001e189

   1. Initial program 69.0%

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

   if -4.0999999999999997e-126 < y.im < 5.8e-109

   1. Initial program 63.5%

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

   3. Taylor expanded in y.re around inf 84.5%

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

      1. +-commutative 84.5%

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

      2. mul-1-neg 84.5%

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

      3. unsub-neg 84.5%

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

      4. associate-/l* 85.7%

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

   5. Simplified 85.7%

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

      1. unpow2 85.7%

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

      2. *-un-lft-identity 85.7%

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

      3. times-frac 88.0%

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

   7. Applied egg-rr 88.0%

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

   if 2.4000000000000001e189 < y.im

   1. Initial program 15.3%

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

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

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

      1. associate-*r/ 74.5%

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

      2. neg-mul-1 74.5%

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

   5. Simplified 74.5%

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

4. Final simplification 77.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -1.05 \cdot 10^{+153}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot x.re\\ \mathbf{elif}\;y.im \leq -4.1 \cdot 10^{-126}:\\ \;\;\;\;\frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.im \leq 5.8 \cdot 10^{-109}:\\ \;\;\;\;\frac{x.im}{y.re} - \frac{x.re}{y.re \cdot \frac{y.re}{y.im}}\\ \mathbf{elif}\;y.im \leq 2.4 \cdot 10^{+189}:\\ \;\;\;\;\frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{-x.re}{y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 77.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-x.re}{y.im}\\ t_1 := \frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{if}\;y.im \leq -1.4 \cdot 10^{+153}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq -4.1 \cdot 10^{-126}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.im \leq 2.2 \cdot 10^{-107}:\\ \;\;\;\;\frac{x.im}{y.re} - \frac{x.re}{y.re \cdot \frac{y.re}{y.im}}\\ \mathbf{elif}\;y.im \leq 2.4 \cdot 10^{+189}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (/ (- x.re) y.im))
        (t_1
         (/ (- (* y.re x.im) (* y.im x.re)) (+ (* y.re y.re) (* y.im y.im)))))
   (if (<= y.im -1.4e+153)
     t_0
     (if (<= y.im -4.1e-126)
       t_1
       (if (<= y.im 2.2e-107)
         (- (/ x.im y.re) (/ x.re (* y.re (/ y.re y.im))))
         (if (<= y.im 2.4e+189) t_1 t_0))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = -x_46_re / y_46_im;
	double t_1 = ((y_46_re * x_46_im) - (y_46_im * x_46_re)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	double tmp;
	if (y_46_im <= -1.4e+153) {
		tmp = t_0;
	} else if (y_46_im <= -4.1e-126) {
		tmp = t_1;
	} else if (y_46_im <= 2.2e-107) {
		tmp = (x_46_im / y_46_re) - (x_46_re / (y_46_re * (y_46_re / y_46_im)));
	} else if (y_46_im <= 2.4e+189) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = -x_46re / y_46im
    t_1 = ((y_46re * x_46im) - (y_46im * x_46re)) / ((y_46re * y_46re) + (y_46im * y_46im))
    if (y_46im <= (-1.4d+153)) then
        tmp = t_0
    else if (y_46im <= (-4.1d-126)) then
        tmp = t_1
    else if (y_46im <= 2.2d-107) then
        tmp = (x_46im / y_46re) - (x_46re / (y_46re * (y_46re / y_46im)))
    else if (y_46im <= 2.4d+189) then
        tmp = t_1
    else
        tmp = t_0
    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_re / y_46_im;
	double t_1 = ((y_46_re * x_46_im) - (y_46_im * x_46_re)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	double tmp;
	if (y_46_im <= -1.4e+153) {
		tmp = t_0;
	} else if (y_46_im <= -4.1e-126) {
		tmp = t_1;
	} else if (y_46_im <= 2.2e-107) {
		tmp = (x_46_im / y_46_re) - (x_46_re / (y_46_re * (y_46_re / y_46_im)));
	} else if (y_46_im <= 2.4e+189) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = -x_46_re / y_46_im
	t_1 = ((y_46_re * x_46_im) - (y_46_im * x_46_re)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im))
	tmp = 0
	if y_46_im <= -1.4e+153:
		tmp = t_0
	elif y_46_im <= -4.1e-126:
		tmp = t_1
	elif y_46_im <= 2.2e-107:
		tmp = (x_46_im / y_46_re) - (x_46_re / (y_46_re * (y_46_re / y_46_im)))
	elif y_46_im <= 2.4e+189:
		tmp = t_1
	else:
		tmp = t_0
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(Float64(-x_46_re) / y_46_im)
	t_1 = Float64(Float64(Float64(y_46_re * x_46_im) - Float64(y_46_im * x_46_re)) / Float64(Float64(y_46_re * y_46_re) + Float64(y_46_im * y_46_im)))
	tmp = 0.0
	if (y_46_im <= -1.4e+153)
		tmp = t_0;
	elseif (y_46_im <= -4.1e-126)
		tmp = t_1;
	elseif (y_46_im <= 2.2e-107)
		tmp = Float64(Float64(x_46_im / y_46_re) - Float64(x_46_re / Float64(y_46_re * Float64(y_46_re / y_46_im))));
	elseif (y_46_im <= 2.4e+189)
		tmp = t_1;
	else
		tmp = t_0;
	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_re / y_46_im;
	t_1 = ((y_46_re * x_46_im) - (y_46_im * x_46_re)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	tmp = 0.0;
	if (y_46_im <= -1.4e+153)
		tmp = t_0;
	elseif (y_46_im <= -4.1e-126)
		tmp = t_1;
	elseif (y_46_im <= 2.2e-107)
		tmp = (x_46_im / y_46_re) - (x_46_re / (y_46_re * (y_46_re / y_46_im)));
	elseif (y_46_im <= 2.4e+189)
		tmp = t_1;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[((-x$46$re) / y$46$im), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(y$46$re * x$46$im), $MachinePrecision] - N[(y$46$im * x$46$re), $MachinePrecision]), $MachinePrecision] / N[(N[(y$46$re * y$46$re), $MachinePrecision] + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$im, -1.4e+153], t$95$0, If[LessEqual[y$46$im, -4.1e-126], t$95$1, If[LessEqual[y$46$im, 2.2e-107], N[(N[(x$46$im / y$46$re), $MachinePrecision] - N[(x$46$re / N[(y$46$re * N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, 2.4e+189], t$95$1, t$95$0]]]]]]

Derivation

1. Split input into 3 regimes

2. if y.im < -1.39999999999999993e153 or 2.4000000000000001e189 < y.im

   1. Initial program 22.6%

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

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

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

      1. associate-*r/ 75.8%

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

      2. neg-mul-1 75.8%

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

   5. Simplified 75.8%

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

   if -1.39999999999999993e153 < y.im < -4.0999999999999997e-126 or 2.20000000000000012e-107 < y.im < 2.4000000000000001e189

   1. Initial program 69.0%

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

   if -4.0999999999999997e-126 < y.im < 2.20000000000000012e-107

   1. Initial program 63.5%

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

   3. Taylor expanded in y.re around inf 84.5%

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

      1. +-commutative 84.5%

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

      2. mul-1-neg 84.5%

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

      3. unsub-neg 84.5%

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

      4. associate-/l* 85.7%

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

   5. Simplified 85.7%

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

      1. unpow2 85.7%

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

      2. *-un-lft-identity 85.7%

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

      3. times-frac 88.0%

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

   7. Applied egg-rr 88.0%

      \[\leadsto \frac{x.im}{y.re} - \frac{x.re}{\color{blue}{\frac{y.re}{1} \cdot \frac{y.re}{y.im}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 77.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -1.4 \cdot 10^{+153}:\\ \;\;\;\;\frac{-x.re}{y.im}\\ \mathbf{elif}\;y.im \leq -4.1 \cdot 10^{-126}:\\ \;\;\;\;\frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.im \leq 2.2 \cdot 10^{-107}:\\ \;\;\;\;\frac{x.im}{y.re} - \frac{x.re}{y.re \cdot \frac{y.re}{y.im}}\\ \mathbf{elif}\;y.im \leq 2.4 \cdot 10^{+189}:\\ \;\;\;\;\frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{-x.re}{y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 64.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-x.re}{y.im}\\ t_1 := \frac{y.re \cdot x.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{if}\;y.re \leq -1.05 \cdot 10^{+109}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{elif}\;y.re \leq -5.8 \cdot 10^{-39}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.re \leq 2.05 \cdot 10^{-116}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq 7 \cdot 10^{-43}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.re \leq 5.5 \cdot 10^{+23}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (/ (- x.re) y.im))
        (t_1 (/ (* y.re x.im) (+ (* y.re y.re) (* y.im y.im)))))
   (if (<= y.re -1.05e+109)
     (/ x.im y.re)
     (if (<= y.re -5.8e-39)
       t_1
       (if (<= y.re 2.05e-116)
         t_0
         (if (<= y.re 7e-43) t_1 (if (<= y.re 5.5e+23) t_0 (/ x.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_re / y_46_im;
	double t_1 = (y_46_re * x_46_im) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	double tmp;
	if (y_46_re <= -1.05e+109) {
		tmp = x_46_im / y_46_re;
	} else if (y_46_re <= -5.8e-39) {
		tmp = t_1;
	} else if (y_46_re <= 2.05e-116) {
		tmp = t_0;
	} else if (y_46_re <= 7e-43) {
		tmp = t_1;
	} else if (y_46_re <= 5.5e+23) {
		tmp = t_0;
	} 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) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = -x_46re / y_46im
    t_1 = (y_46re * x_46im) / ((y_46re * y_46re) + (y_46im * y_46im))
    if (y_46re <= (-1.05d+109)) then
        tmp = x_46im / y_46re
    else if (y_46re <= (-5.8d-39)) then
        tmp = t_1
    else if (y_46re <= 2.05d-116) then
        tmp = t_0
    else if (y_46re <= 7d-43) then
        tmp = t_1
    else if (y_46re <= 5.5d+23) then
        tmp = t_0
    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 t_0 = -x_46_re / y_46_im;
	double t_1 = (y_46_re * x_46_im) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	double tmp;
	if (y_46_re <= -1.05e+109) {
		tmp = x_46_im / y_46_re;
	} else if (y_46_re <= -5.8e-39) {
		tmp = t_1;
	} else if (y_46_re <= 2.05e-116) {
		tmp = t_0;
	} else if (y_46_re <= 7e-43) {
		tmp = t_1;
	} else if (y_46_re <= 5.5e+23) {
		tmp = t_0;
	} 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):
	t_0 = -x_46_re / y_46_im
	t_1 = (y_46_re * x_46_im) / ((y_46_re * y_46_re) + (y_46_im * y_46_im))
	tmp = 0
	if y_46_re <= -1.05e+109:
		tmp = x_46_im / y_46_re
	elif y_46_re <= -5.8e-39:
		tmp = t_1
	elif y_46_re <= 2.05e-116:
		tmp = t_0
	elif y_46_re <= 7e-43:
		tmp = t_1
	elif y_46_re <= 5.5e+23:
		tmp = t_0
	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)
	t_0 = Float64(Float64(-x_46_re) / y_46_im)
	t_1 = Float64(Float64(y_46_re * x_46_im) / Float64(Float64(y_46_re * y_46_re) + Float64(y_46_im * y_46_im)))
	tmp = 0.0
	if (y_46_re <= -1.05e+109)
		tmp = Float64(x_46_im / y_46_re);
	elseif (y_46_re <= -5.8e-39)
		tmp = t_1;
	elseif (y_46_re <= 2.05e-116)
		tmp = t_0;
	elseif (y_46_re <= 7e-43)
		tmp = t_1;
	elseif (y_46_re <= 5.5e+23)
		tmp = t_0;
	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)
	t_0 = -x_46_re / y_46_im;
	t_1 = (y_46_re * x_46_im) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	tmp = 0.0;
	if (y_46_re <= -1.05e+109)
		tmp = x_46_im / y_46_re;
	elseif (y_46_re <= -5.8e-39)
		tmp = t_1;
	elseif (y_46_re <= 2.05e-116)
		tmp = t_0;
	elseif (y_46_re <= 7e-43)
		tmp = t_1;
	elseif (y_46_re <= 5.5e+23)
		tmp = t_0;
	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_] := Block[{t$95$0 = N[((-x$46$re) / y$46$im), $MachinePrecision]}, Block[{t$95$1 = N[(N[(y$46$re * x$46$im), $MachinePrecision] / N[(N[(y$46$re * y$46$re), $MachinePrecision] + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$re, -1.05e+109], N[(x$46$im / y$46$re), $MachinePrecision], If[LessEqual[y$46$re, -5.8e-39], t$95$1, If[LessEqual[y$46$re, 2.05e-116], t$95$0, If[LessEqual[y$46$re, 7e-43], t$95$1, If[LessEqual[y$46$re, 5.5e+23], t$95$0, N[(x$46$im / y$46$re), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -1.0500000000000001e109 or 5.50000000000000004e23 < y.re

   1. Initial program 29.5%

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

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

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

   if -1.0500000000000001e109 < y.re < -5.79999999999999975e-39 or 2.0499999999999999e-116 < y.re < 6.99999999999999994e-43

   1. Initial program 80.9%

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

   3. Taylor expanded in x.im around inf 58.4%

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

      1. *-commutative 58.4%

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

   5. Simplified 58.4%

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

   if -5.79999999999999975e-39 < y.re < 2.0499999999999999e-116 or 6.99999999999999994e-43 < y.re < 5.50000000000000004e23

   1. Initial program 70.4%

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

   3. Taylor expanded in y.re around 0 74.9%

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

      1. associate-*r/ 74.9%

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

      2. neg-mul-1 74.9%

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

   5. Simplified 74.9%

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

4. Final simplification 69.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -1.05 \cdot 10^{+109}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{elif}\;y.re \leq -5.8 \cdot 10^{-39}:\\ \;\;\;\;\frac{y.re \cdot x.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.re \leq 2.05 \cdot 10^{-116}:\\ \;\;\;\;\frac{-x.re}{y.im}\\ \mathbf{elif}\;y.re \leq 7 \cdot 10^{-43}:\\ \;\;\;\;\frac{y.re \cdot x.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.re \leq 5.5 \cdot 10^{+23}:\\ \;\;\;\;\frac{-x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 62.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -4.2 \cdot 10^{-34} \lor \neg \left(y.re \leq 5.4 \cdot 10^{-103} \lor \neg \left(y.re \leq 2.5 \cdot 10^{-44}\right) \land y.re \leq 4.8 \cdot 10^{+21}\right):\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{-x.re}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.re -4.2e-34)
         (not
          (or (<= y.re 5.4e-103)
              (and (not (<= y.re 2.5e-44)) (<= y.re 4.8e+21)))))
   (/ x.im y.re)
   (/ (- x.re) y.im)))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_re <= -4.2e-34) || !((y_46_re <= 5.4e-103) || (!(y_46_re <= 2.5e-44) && (y_46_re <= 4.8e+21)))) {
		tmp = x_46_im / y_46_re;
	} else {
		tmp = -x_46_re / y_46_im;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if ((y_46re <= (-4.2d-34)) .or. (.not. (y_46re <= 5.4d-103) .or. (.not. (y_46re <= 2.5d-44)) .and. (y_46re <= 4.8d+21))) then
        tmp = x_46im / y_46re
    else
        tmp = -x_46re / y_46im
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_re <= -4.2e-34) || !((y_46_re <= 5.4e-103) || (!(y_46_re <= 2.5e-44) && (y_46_re <= 4.8e+21)))) {
		tmp = x_46_im / y_46_re;
	} else {
		tmp = -x_46_re / y_46_im;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if (y_46_re <= -4.2e-34) or not ((y_46_re <= 5.4e-103) or (not (y_46_re <= 2.5e-44) and (y_46_re <= 4.8e+21))):
		tmp = x_46_im / y_46_re
	else:
		tmp = -x_46_re / y_46_im
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if ((y_46_re <= -4.2e-34) || !((y_46_re <= 5.4e-103) || (!(y_46_re <= 2.5e-44) && (y_46_re <= 4.8e+21))))
		tmp = Float64(x_46_im / y_46_re);
	else
		tmp = Float64(Float64(-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_re <= -4.2e-34) || ~(((y_46_re <= 5.4e-103) || (~((y_46_re <= 2.5e-44)) && (y_46_re <= 4.8e+21)))))
		tmp = x_46_im / y_46_re;
	else
		tmp = -x_46_re / y_46_im;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[Or[LessEqual[y$46$re, -4.2e-34], N[Not[Or[LessEqual[y$46$re, 5.4e-103], And[N[Not[LessEqual[y$46$re, 2.5e-44]], $MachinePrecision], LessEqual[y$46$re, 4.8e+21]]]], $MachinePrecision]], N[(x$46$im / y$46$re), $MachinePrecision], N[((-x$46$re) / y$46$im), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.re < -4.2000000000000002e-34 or 5.40000000000000019e-103 < y.re < 2.50000000000000019e-44 or 4.8e21 < y.re

   1. Initial program 44.5%

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

   3. Taylor expanded in y.re around inf 62.4%

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

   if -4.2000000000000002e-34 < y.re < 5.40000000000000019e-103 or 2.50000000000000019e-44 < y.re < 4.8e21

   1. Initial program 70.9%

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

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

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

      1. associate-*r/ 74.5%

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

      2. neg-mul-1 74.5%

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

   5. Simplified 74.5%

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

4. Final simplification 67.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -4.2 \cdot 10^{-34} \lor \neg \left(y.re \leq 5.4 \cdot 10^{-103} \lor \neg \left(y.re \leq 2.5 \cdot 10^{-44}\right) \land y.re \leq 4.8 \cdot 10^{+21}\right):\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{-x.re}{y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 70.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -9.5 \cdot 10^{-19} \lor \neg \left(y.im \leq 4.3 \cdot 10^{-10}\right):\\ \;\;\;\;\frac{-x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re} - \frac{x.re}{y.re \cdot \frac{y.re}{y.im}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.im -9.5e-19) (not (<= y.im 4.3e-10)))
   (/ (- x.re) y.im)
   (- (/ x.im y.re) (/ x.re (* y.re (/ y.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 <= -9.5e-19) || !(y_46_im <= 4.3e-10)) {
		tmp = -x_46_re / y_46_im;
	} else {
		tmp = (x_46_im / y_46_re) - (x_46_re / (y_46_re * (y_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 <= (-9.5d-19)) .or. (.not. (y_46im <= 4.3d-10))) then
        tmp = -x_46re / y_46im
    else
        tmp = (x_46im / y_46re) - (x_46re / (y_46re * (y_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 <= -9.5e-19) || !(y_46_im <= 4.3e-10)) {
		tmp = -x_46_re / y_46_im;
	} else {
		tmp = (x_46_im / y_46_re) - (x_46_re / (y_46_re * (y_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 <= -9.5e-19) or not (y_46_im <= 4.3e-10):
		tmp = -x_46_re / y_46_im
	else:
		tmp = (x_46_im / y_46_re) - (x_46_re / (y_46_re * (y_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 <= -9.5e-19) || !(y_46_im <= 4.3e-10))
		tmp = Float64(Float64(-x_46_re) / y_46_im);
	else
		tmp = Float64(Float64(x_46_im / y_46_re) - Float64(x_46_re / Float64(y_46_re * Float64(y_46_re / y_46_im))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if ((y_46_im <= -9.5e-19) || ~((y_46_im <= 4.3e-10)))
		tmp = -x_46_re / y_46_im;
	else
		tmp = (x_46_im / y_46_re) - (x_46_re / (y_46_re * (y_46_re / y_46_im)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[Or[LessEqual[y$46$im, -9.5e-19], N[Not[LessEqual[y$46$im, 4.3e-10]], $MachinePrecision]], N[((-x$46$re) / y$46$im), $MachinePrecision], N[(N[(x$46$im / y$46$re), $MachinePrecision] - N[(x$46$re / N[(y$46$re * N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.im < -9.4999999999999995e-19 or 4.30000000000000014e-10 < y.im

   1. Initial program 43.0%

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

   3. Taylor expanded in y.re around 0 62.0%

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

      1. associate-*r/ 62.0%

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

      2. neg-mul-1 62.0%

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

   5. Simplified 62.0%

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

   if -9.4999999999999995e-19 < y.im < 4.30000000000000014e-10

   1. Initial program 68.6%

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

   3. Taylor expanded in y.re around inf 76.4%

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

      1. +-commutative 76.4%

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

      2. mul-1-neg 76.4%

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

      3. unsub-neg 76.4%

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

      4. associate-/l* 76.5%

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

   5. Simplified 76.5%

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

      1. unpow2 76.5%

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

      2. *-un-lft-identity 76.5%

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

      3. times-frac 78.1%

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

   7. Applied egg-rr 78.1%

      \[\leadsto \frac{x.im}{y.re} - \frac{x.re}{\color{blue}{\frac{y.re}{1} \cdot \frac{y.re}{y.im}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 70.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -9.5 \cdot 10^{-19} \lor \neg \left(y.im \leq 4.3 \cdot 10^{-10}\right):\\ \;\;\;\;\frac{-x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re} - \frac{x.re}{y.re \cdot \frac{y.re}{y.im}}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 9.0% 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 55.7%

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

   1. *-un-lft-identity 55.7%

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

   2. add-sqr-sqrt 55.7%

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

   3. times-frac 55.7%

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

   4. hypot-def 55.7%

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

   5. hypot-def 74.0%

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

4. Applied egg-rr 74.0%

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

   1. clear-num 73.9%

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

   2. frac-times 73.7%

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

   3. metadata-eval 73.7%

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

   4. *-commutative 73.7%

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

   5. *-commutative 73.7%

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

6. Applied egg-rr 73.7%

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

7. Taylor expanded in y.re around -inf 30.5%

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

8. Taylor expanded in y.im around -inf 6.6%

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

9. Final simplification 6.6%

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

Alternative 16: 43.2% accurate, 5.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 55.7%

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

3. Taylor expanded in y.re around inf 42.6%

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

4. Final simplification 42.6%

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

Reproduce

herbie shell --seed 2024020 
(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))))