_divideComplex, imaginary part

Percentage Accurate: 61.8% → 83.6%

Time: 10.8s

Alternatives: 11

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 61.8% 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: 83.6% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y.im \cdot y.im + y.re \cdot y.re\\ \mathbf{if}\;y.im \leq -3.2 \cdot 10^{+151}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x.im}{y.im}, \frac{y.re}{y.im}, \frac{x.re}{0 - y.im}\right)\\ \mathbf{elif}\;y.im \leq -4.5 \cdot 10^{-73}:\\ \;\;\;\;x.re \cdot \left(\frac{x.im \cdot y.re}{x.re \cdot t\_0} - \frac{y.im}{t\_0}\right)\\ \mathbf{elif}\;y.im \leq 8.2 \cdot 10^{-102}:\\ \;\;\;\;\frac{x.im - \frac{x.re}{\frac{y.re}{y.im}}}{y.re}\\ \mathbf{elif}\;y.im \leq 1.95 \cdot 10^{+79}:\\ \;\;\;\;\frac{1}{t\_0} \cdot \left(x.im \cdot y.re - y.im \cdot x.re\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y.re}{\frac{y.im}{x.im}} - x.re}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (+ (* y.im y.im) (* y.re y.re))))
   (if (<= y.im -3.2e+151)
     (fma (/ x.im y.im) (/ y.re y.im) (/ x.re (- 0.0 y.im)))
     (if (<= y.im -4.5e-73)
       (* x.re (- (/ (* x.im y.re) (* x.re t_0)) (/ y.im t_0)))
       (if (<= y.im 8.2e-102)
         (/ (- x.im (/ x.re (/ y.re y.im))) y.re)
         (if (<= y.im 1.95e+79)
           (* (/ 1.0 t_0) (- (* x.im y.re) (* y.im x.re)))
           (/ (- (/ y.re (/ y.im x.im)) x.re) y.im)))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = (y_46_im * y_46_im) + (y_46_re * y_46_re);
	double tmp;
	if (y_46_im <= -3.2e+151) {
		tmp = fma((x_46_im / y_46_im), (y_46_re / y_46_im), (x_46_re / (0.0 - y_46_im)));
	} else if (y_46_im <= -4.5e-73) {
		tmp = x_46_re * (((x_46_im * y_46_re) / (x_46_re * t_0)) - (y_46_im / t_0));
	} else if (y_46_im <= 8.2e-102) {
		tmp = (x_46_im - (x_46_re / (y_46_re / y_46_im))) / y_46_re;
	} else if (y_46_im <= 1.95e+79) {
		tmp = (1.0 / t_0) * ((x_46_im * y_46_re) - (y_46_im * x_46_re));
	} else {
		tmp = ((y_46_re / (y_46_im / x_46_im)) - x_46_re) / y_46_im;
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(Float64(y_46_im * y_46_im) + Float64(y_46_re * y_46_re))
	tmp = 0.0
	if (y_46_im <= -3.2e+151)
		tmp = fma(Float64(x_46_im / y_46_im), Float64(y_46_re / y_46_im), Float64(x_46_re / Float64(0.0 - y_46_im)));
	elseif (y_46_im <= -4.5e-73)
		tmp = Float64(x_46_re * Float64(Float64(Float64(x_46_im * y_46_re) / Float64(x_46_re * t_0)) - Float64(y_46_im / t_0)));
	elseif (y_46_im <= 8.2e-102)
		tmp = Float64(Float64(x_46_im - Float64(x_46_re / Float64(y_46_re / y_46_im))) / y_46_re);
	elseif (y_46_im <= 1.95e+79)
		tmp = Float64(Float64(1.0 / t_0) * Float64(Float64(x_46_im * y_46_re) - Float64(y_46_im * x_46_re)));
	else
		tmp = Float64(Float64(Float64(y_46_re / Float64(y_46_im / x_46_im)) - x_46_re) / y_46_im);
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[(y$46$im * y$46$im), $MachinePrecision] + N[(y$46$re * y$46$re), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$im, -3.2e+151], N[(N[(x$46$im / y$46$im), $MachinePrecision] * N[(y$46$re / y$46$im), $MachinePrecision] + N[(x$46$re / N[(0.0 - y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, -4.5e-73], N[(x$46$re * N[(N[(N[(x$46$im * y$46$re), $MachinePrecision] / N[(x$46$re * t$95$0), $MachinePrecision]), $MachinePrecision] - N[(y$46$im / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, 8.2e-102], N[(N[(x$46$im - N[(x$46$re / N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision], If[LessEqual[y$46$im, 1.95e+79], N[(N[(1.0 / t$95$0), $MachinePrecision] * N[(N[(x$46$im * y$46$re), $MachinePrecision] - N[(y$46$im * x$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(y$46$re / N[(y$46$im / x$46$im), $MachinePrecision]), $MachinePrecision] - x$46$re), $MachinePrecision] / y$46$im), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if y.im < -3.19999999999999994e151

   1. Initial program 35.4%

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

   3. Taylor expanded in y.re around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. unpow2 N/A

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

      5. associate-/r* N/A

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

      6. div-sub N/A

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

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{x.im \cdot y.re}{y.im} - x.re\right), \color{blue}{y.im}\right) \]

      8. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(\frac{x.im \cdot y.re}{y.im}\right), x.re\right), y.im\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(\frac{y.re \cdot x.im}{y.im}\right), x.re\right), y.im\right) \]

      10. associate-/l* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(y.re \cdot \frac{x.im}{y.im}\right), x.re\right), y.im\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(y.re, \left(\frac{x.im}{y.im}\right)\right), x.re\right), y.im\right) \]

      12. /-lowering-/.f64 86.2%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{/.f64}\left(x.im, y.im\right)\right), x.re\right), y.im\right) \]

   5. Simplified 86.2%

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

      1. div-sub N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. fmm-def N/A

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

      5. fma-lowering-fma.f64 N/A

         \[\leadsto \mathsf{fma.f64}\left(\left(\frac{x.im}{y.im}\right), \color{blue}{\left(\frac{y.re}{y.im}\right)}, \left(\mathsf{neg}\left(\frac{x.re}{y.im}\right)\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{fma.f64}\left(\mathsf{/.f64}\left(x.im, y.im\right), \left(\frac{\color{blue}{y.re}}{y.im}\right), \left(\mathsf{neg}\left(\frac{x.re}{y.im}\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{fma.f64}\left(\mathsf{/.f64}\left(x.im, y.im\right), \mathsf{/.f64}\left(y.re, \color{blue}{y.im}\right), \left(\mathsf{neg}\left(\frac{x.re}{y.im}\right)\right)\right) \]

      8. neg-lowering-neg.f64 N/A

         \[\leadsto \mathsf{fma.f64}\left(\mathsf{/.f64}\left(x.im, y.im\right), \mathsf{/.f64}\left(y.re, y.im\right), \mathsf{neg.f64}\left(\left(\frac{x.re}{y.im}\right)\right)\right) \]

      9. /-lowering-/.f64 86.4%

         \[\leadsto \mathsf{fma.f64}\left(\mathsf{/.f64}\left(x.im, y.im\right), \mathsf{/.f64}\left(y.re, y.im\right), \mathsf{neg.f64}\left(\mathsf{/.f64}\left(x.re, y.im\right)\right)\right) \]

   7. Applied egg-rr 86.4%

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

   if -3.19999999999999994e151 < y.im < -4.5e-73

   1. Initial program 72.9%

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

   3. Taylor expanded in x.re around inf

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

         \[\leadsto x.re \cdot \left(\left(0 - \frac{y.im}{{y.im}^{2} + {y.re}^{2}}\right) + \frac{\color{blue}{x.im \cdot y.re}}{x.re \cdot \left({y.im}^{2} + {y.re}^{2}\right)}\right) \]

      3. associate-+l- N/A

         \[\leadsto x.re \cdot \left(0 - \color{blue}{\left(\frac{y.im}{{y.im}^{2} + {y.re}^{2}} - \frac{x.im \cdot y.re}{x.re \cdot \left({y.im}^{2} + {y.re}^{2}\right)}\right)}\right) \]

      4. unsub-neg N/A

         \[\leadsto x.re \cdot \left(0 - \left(\frac{y.im}{{y.im}^{2} + {y.re}^{2}} + \color{blue}{\left(\mathsf{neg}\left(\frac{x.im \cdot y.re}{x.re \cdot \left({y.im}^{2} + {y.re}^{2}\right)}\right)\right)}\right)\right) \]

      5. mul-1-neg N/A

         \[\leadsto x.re \cdot \left(0 - \left(\frac{y.im}{{y.im}^{2} + {y.re}^{2}} + -1 \cdot \color{blue}{\frac{x.im \cdot y.re}{x.re \cdot \left({y.im}^{2} + {y.re}^{2}\right)}}\right)\right) \]

      6. +-commutative N/A

         \[\leadsto x.re \cdot \left(0 - \left(-1 \cdot \frac{x.im \cdot y.re}{x.re \cdot \left({y.im}^{2} + {y.re}^{2}\right)} + \color{blue}{\frac{y.im}{{y.im}^{2} + {y.re}^{2}}}\right)\right) \]

      7. neg-sub0 N/A

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

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x.re, \color{blue}{\left(\mathsf{neg}\left(\left(-1 \cdot \frac{x.im \cdot y.re}{x.re \cdot \left({y.im}^{2} + {y.re}^{2}\right)} + \frac{y.im}{{y.im}^{2} + {y.re}^{2}}\right)\right)\right)}\right) \]

      9. neg-sub0 N/A

         \[\leadsto \mathsf{*.f64}\left(x.re, \left(0 - \color{blue}{\left(-1 \cdot \frac{x.im \cdot y.re}{x.re \cdot \left({y.im}^{2} + {y.re}^{2}\right)} + \frac{y.im}{{y.im}^{2} + {y.re}^{2}}\right)}\right)\right) \]

   5. Simplified 82.9%

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

   if -4.5e-73 < y.im < 8.2000000000000005e-102

   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
   3. Step-by-step derivation

      1. div-inv N/A

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

      2. *-commutative N/A

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

      3. flip3-+ N/A

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

      4. clear-num N/A

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

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\left(y.re \cdot y.re\right) \cdot \left(y.re \cdot y.re\right) + \left(\left(y.im \cdot y.im\right) \cdot \left(y.im \cdot y.im\right) - \left(y.re \cdot y.re\right) \cdot \left(y.im \cdot y.im\right)\right)}{{\left(y.re \cdot y.re\right)}^{3} + {\left(y.im \cdot y.im\right)}^{3}}\right), \color{blue}{\left(x.im \cdot y.re - x.re \cdot y.im\right)}\right) \]

   4. Applied egg-rr 72.8%

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

      1. associate-*l/ N/A

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

      2. clear-num N/A

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

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{y.re \cdot y.re + y.im \cdot y.im}{1 \cdot \left(x.im \cdot y.re - x.re \cdot y.im\right)}\right)}\right) \]

      4. *-lft-identity N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(\frac{y.re \cdot y.re + y.im \cdot y.im}{x.im \cdot y.re - \color{blue}{x.re \cdot y.im}}\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\left(y.re \cdot y.re + y.im \cdot y.im\right), \color{blue}{\left(x.im \cdot y.re - x.re \cdot y.im\right)}\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(y.re \cdot y.re\right), \left(y.im \cdot y.im\right)\right), \left(\color{blue}{x.im \cdot y.re} - x.re \cdot y.im\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(y.re, y.re\right), \left(y.im \cdot y.im\right)\right), \left(\color{blue}{x.im} \cdot y.re - x.re \cdot y.im\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(y.re, y.re\right), \mathsf{*.f64}\left(y.im, y.im\right)\right), \left(x.im \cdot \color{blue}{y.re} - x.re \cdot y.im\right)\right)\right) \]

      9. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(y.re, y.re\right), \mathsf{*.f64}\left(y.im, y.im\right)\right), \mathsf{\_.f64}\left(\left(x.im \cdot y.re\right), \color{blue}{\left(x.re \cdot y.im\right)}\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(y.re, y.re\right), \mathsf{*.f64}\left(y.im, y.im\right)\right), \mathsf{\_.f64}\left(\left(y.re \cdot x.im\right), \left(\color{blue}{x.re} \cdot y.im\right)\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(y.re, y.re\right), \mathsf{*.f64}\left(y.im, y.im\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(y.re, x.im\right), \left(\color{blue}{x.re} \cdot y.im\right)\right)\right)\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(y.re, y.re\right), \mathsf{*.f64}\left(y.im, y.im\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(y.re, x.im\right), \left(y.im \cdot \color{blue}{x.re}\right)\right)\right)\right) \]

      13. *-lowering-*.f64 72.9%

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(y.re, y.re\right), \mathsf{*.f64}\left(y.im, y.im\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(y.re, x.im\right), \mathsf{*.f64}\left(y.im, \color{blue}{x.re}\right)\right)\right)\right) \]

   6. Applied egg-rr 72.9%

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

   7. Taylor expanded in y.re around inf

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(x.im + -1 \cdot \frac{x.re \cdot y.im}{y.re}\right), \color{blue}{y.re}\right) \]

      2. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\left(x.im + \left(\mathsf{neg}\left(\frac{x.re \cdot y.im}{y.re}\right)\right)\right), y.re\right) \]

      3. unsub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\left(x.im - \frac{x.re \cdot y.im}{y.re}\right), y.re\right) \]

      4. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(x.im, \left(\frac{x.re \cdot y.im}{y.re}\right)\right), y.re\right) \]

      5. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(x.im, \left(x.re \cdot \frac{y.im}{y.re}\right)\right), y.re\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(x.im, \mathsf{*.f64}\left(x.re, \left(\frac{y.im}{y.re}\right)\right)\right), y.re\right) \]

      7. /-lowering-/.f64 95.1%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(x.im, \mathsf{*.f64}\left(x.re, \mathsf{/.f64}\left(y.im, y.re\right)\right)\right), y.re\right) \]

   9. Simplified 95.1%

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

       1. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(x.im - x.re \cdot \frac{y.im}{y.re}\right), \color{blue}{y.re}\right) \]

       2. --lowering--.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(x.im, \left(x.re \cdot \frac{y.im}{y.re}\right)\right), y.re\right) \]

       3. clear-num N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(x.im, \left(x.re \cdot \frac{1}{\frac{y.re}{y.im}}\right)\right), y.re\right) \]

       4. un-div-inv N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(x.im, \left(\frac{x.re}{\frac{y.re}{y.im}}\right)\right), y.re\right) \]

       5. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(x.im, \mathsf{/.f64}\left(x.re, \left(\frac{y.re}{y.im}\right)\right)\right), y.re\right) \]

       6. /-lowering-/.f64 95.1%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(x.im, \mathsf{/.f64}\left(x.re, \mathsf{/.f64}\left(y.re, y.im\right)\right)\right), y.re\right) \]

   11. Applied egg-rr 95.1%

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

   if 8.2000000000000005e-102 < y.im < 1.9499999999999999e79

   1. Initial program 78.9%

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

      1. div-inv N/A

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

      2. *-commutative N/A

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

      3. flip3-+ N/A

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

      4. clear-num N/A

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

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\left(y.re \cdot y.re\right) \cdot \left(y.re \cdot y.re\right) + \left(\left(y.im \cdot y.im\right) \cdot \left(y.im \cdot y.im\right) - \left(y.re \cdot y.re\right) \cdot \left(y.im \cdot y.im\right)\right)}{{\left(y.re \cdot y.re\right)}^{3} + {\left(y.im \cdot y.im\right)}^{3}}\right), \color{blue}{\left(x.im \cdot y.re - x.re \cdot y.im\right)}\right) \]

   4. Applied egg-rr 79.0%

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

   if 1.9499999999999999e79 < y.im

   1. Initial program 44.2%

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

   3. Taylor expanded in y.re around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. unpow2 N/A

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

      5. associate-/r* N/A

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

      6. div-sub N/A

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

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{x.im \cdot y.re}{y.im} - x.re\right), \color{blue}{y.im}\right) \]

      8. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(\frac{x.im \cdot y.re}{y.im}\right), x.re\right), y.im\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(\frac{y.re \cdot x.im}{y.im}\right), x.re\right), y.im\right) \]

      10. associate-/l* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(y.re \cdot \frac{x.im}{y.im}\right), x.re\right), y.im\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(y.re, \left(\frac{x.im}{y.im}\right)\right), x.re\right), y.im\right) \]

      12. /-lowering-/.f64 87.1%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{/.f64}\left(x.im, y.im\right)\right), x.re\right), y.im\right) \]

   5. Simplified 87.1%

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

      1. clear-num N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(y.re \cdot \frac{1}{\frac{y.im}{x.im}}\right), x.re\right), y.im\right) \]

      2. un-div-inv N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(\frac{y.re}{\frac{y.im}{x.im}}\right), x.re\right), y.im\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(y.re, \left(\frac{y.im}{x.im}\right)\right), x.re\right), y.im\right) \]

      4. /-lowering-/.f64 89.5%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(y.re, \mathsf{/.f64}\left(y.im, x.im\right)\right), x.re\right), y.im\right) \]

   7. Applied egg-rr 89.5%

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

4. Final simplification 88.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -3.2 \cdot 10^{+151}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x.im}{y.im}, \frac{y.re}{y.im}, \frac{x.re}{0 - y.im}\right)\\ \mathbf{elif}\;y.im \leq -4.5 \cdot 10^{-73}:\\ \;\;\;\;x.re \cdot \left(\frac{x.im \cdot y.re}{x.re \cdot \left(y.im \cdot y.im + y.re \cdot y.re\right)} - \frac{y.im}{y.im \cdot y.im + y.re \cdot y.re}\right)\\ \mathbf{elif}\;y.im \leq 8.2 \cdot 10^{-102}:\\ \;\;\;\;\frac{x.im - \frac{x.re}{\frac{y.re}{y.im}}}{y.re}\\ \mathbf{elif}\;y.im \leq 1.95 \cdot 10^{+79}:\\ \;\;\;\;\frac{1}{y.im \cdot y.im + y.re \cdot y.re} \cdot \left(x.im \cdot y.re - y.im \cdot x.re\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y.re}{\frac{y.im}{x.im}} - x.re}{y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 83.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y.im \cdot y.im + y.re \cdot y.re\\ \mathbf{if}\;y.im \leq -8.5 \cdot 10^{+150}:\\ \;\;\;\;\frac{x.im \cdot \frac{y.re}{y.im} - x.re}{y.im}\\ \mathbf{elif}\;y.im \leq -4.5 \cdot 10^{-73}:\\ \;\;\;\;x.re \cdot \left(\frac{x.im \cdot y.re}{x.re \cdot t\_0} - \frac{y.im}{t\_0}\right)\\ \mathbf{elif}\;y.im \leq 5.3 \cdot 10^{-102}:\\ \;\;\;\;\frac{x.im - \frac{x.re}{\frac{y.re}{y.im}}}{y.re}\\ \mathbf{elif}\;y.im \leq 8.8 \cdot 10^{+77}:\\ \;\;\;\;\frac{1}{t\_0} \cdot \left(x.im \cdot y.re - y.im \cdot x.re\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y.re}{\frac{y.im}{x.im}} - x.re}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (+ (* y.im y.im) (* y.re y.re))))
   (if (<= y.im -8.5e+150)
     (/ (- (* x.im (/ y.re y.im)) x.re) y.im)
     (if (<= y.im -4.5e-73)
       (* x.re (- (/ (* x.im y.re) (* x.re t_0)) (/ y.im t_0)))
       (if (<= y.im 5.3e-102)
         (/ (- x.im (/ x.re (/ y.re y.im))) y.re)
         (if (<= y.im 8.8e+77)
           (* (/ 1.0 t_0) (- (* x.im y.re) (* y.im x.re)))
           (/ (- (/ y.re (/ y.im x.im)) x.re) y.im)))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = (y_46_im * y_46_im) + (y_46_re * y_46_re);
	double tmp;
	if (y_46_im <= -8.5e+150) {
		tmp = ((x_46_im * (y_46_re / y_46_im)) - x_46_re) / y_46_im;
	} else if (y_46_im <= -4.5e-73) {
		tmp = x_46_re * (((x_46_im * y_46_re) / (x_46_re * t_0)) - (y_46_im / t_0));
	} else if (y_46_im <= 5.3e-102) {
		tmp = (x_46_im - (x_46_re / (y_46_re / y_46_im))) / y_46_re;
	} else if (y_46_im <= 8.8e+77) {
		tmp = (1.0 / t_0) * ((x_46_im * y_46_re) - (y_46_im * x_46_re));
	} else {
		tmp = ((y_46_re / (y_46_im / x_46_im)) - x_46_re) / y_46_im;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (y_46im * y_46im) + (y_46re * y_46re)
    if (y_46im <= (-8.5d+150)) then
        tmp = ((x_46im * (y_46re / y_46im)) - x_46re) / y_46im
    else if (y_46im <= (-4.5d-73)) then
        tmp = x_46re * (((x_46im * y_46re) / (x_46re * t_0)) - (y_46im / t_0))
    else if (y_46im <= 5.3d-102) then
        tmp = (x_46im - (x_46re / (y_46re / y_46im))) / y_46re
    else if (y_46im <= 8.8d+77) then
        tmp = (1.0d0 / t_0) * ((x_46im * y_46re) - (y_46im * x_46re))
    else
        tmp = ((y_46re / (y_46im / x_46im)) - x_46re) / y_46im
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = (y_46_im * y_46_im) + (y_46_re * y_46_re);
	double tmp;
	if (y_46_im <= -8.5e+150) {
		tmp = ((x_46_im * (y_46_re / y_46_im)) - x_46_re) / y_46_im;
	} else if (y_46_im <= -4.5e-73) {
		tmp = x_46_re * (((x_46_im * y_46_re) / (x_46_re * t_0)) - (y_46_im / t_0));
	} else if (y_46_im <= 5.3e-102) {
		tmp = (x_46_im - (x_46_re / (y_46_re / y_46_im))) / y_46_re;
	} else if (y_46_im <= 8.8e+77) {
		tmp = (1.0 / t_0) * ((x_46_im * y_46_re) - (y_46_im * x_46_re));
	} else {
		tmp = ((y_46_re / (y_46_im / x_46_im)) - x_46_re) / y_46_im;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = (y_46_im * y_46_im) + (y_46_re * y_46_re)
	tmp = 0
	if y_46_im <= -8.5e+150:
		tmp = ((x_46_im * (y_46_re / y_46_im)) - x_46_re) / y_46_im
	elif y_46_im <= -4.5e-73:
		tmp = x_46_re * (((x_46_im * y_46_re) / (x_46_re * t_0)) - (y_46_im / t_0))
	elif y_46_im <= 5.3e-102:
		tmp = (x_46_im - (x_46_re / (y_46_re / y_46_im))) / y_46_re
	elif y_46_im <= 8.8e+77:
		tmp = (1.0 / t_0) * ((x_46_im * y_46_re) - (y_46_im * x_46_re))
	else:
		tmp = ((y_46_re / (y_46_im / x_46_im)) - x_46_re) / y_46_im
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(Float64(y_46_im * y_46_im) + Float64(y_46_re * y_46_re))
	tmp = 0.0
	if (y_46_im <= -8.5e+150)
		tmp = Float64(Float64(Float64(x_46_im * Float64(y_46_re / y_46_im)) - x_46_re) / y_46_im);
	elseif (y_46_im <= -4.5e-73)
		tmp = Float64(x_46_re * Float64(Float64(Float64(x_46_im * y_46_re) / Float64(x_46_re * t_0)) - Float64(y_46_im / t_0)));
	elseif (y_46_im <= 5.3e-102)
		tmp = Float64(Float64(x_46_im - Float64(x_46_re / Float64(y_46_re / y_46_im))) / y_46_re);
	elseif (y_46_im <= 8.8e+77)
		tmp = Float64(Float64(1.0 / t_0) * Float64(Float64(x_46_im * y_46_re) - Float64(y_46_im * x_46_re)));
	else
		tmp = Float64(Float64(Float64(y_46_re / Float64(y_46_im / x_46_im)) - x_46_re) / y_46_im);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = (y_46_im * y_46_im) + (y_46_re * y_46_re);
	tmp = 0.0;
	if (y_46_im <= -8.5e+150)
		tmp = ((x_46_im * (y_46_re / y_46_im)) - x_46_re) / y_46_im;
	elseif (y_46_im <= -4.5e-73)
		tmp = x_46_re * (((x_46_im * y_46_re) / (x_46_re * t_0)) - (y_46_im / t_0));
	elseif (y_46_im <= 5.3e-102)
		tmp = (x_46_im - (x_46_re / (y_46_re / y_46_im))) / y_46_re;
	elseif (y_46_im <= 8.8e+77)
		tmp = (1.0 / t_0) * ((x_46_im * y_46_re) - (y_46_im * x_46_re));
	else
		tmp = ((y_46_re / (y_46_im / x_46_im)) - x_46_re) / y_46_im;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[(y$46$im * y$46$im), $MachinePrecision] + N[(y$46$re * y$46$re), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$im, -8.5e+150], N[(N[(N[(x$46$im * N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision] - x$46$re), $MachinePrecision] / y$46$im), $MachinePrecision], If[LessEqual[y$46$im, -4.5e-73], N[(x$46$re * N[(N[(N[(x$46$im * y$46$re), $MachinePrecision] / N[(x$46$re * t$95$0), $MachinePrecision]), $MachinePrecision] - N[(y$46$im / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, 5.3e-102], N[(N[(x$46$im - N[(x$46$re / N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision], If[LessEqual[y$46$im, 8.8e+77], N[(N[(1.0 / t$95$0), $MachinePrecision] * N[(N[(x$46$im * y$46$re), $MachinePrecision] - N[(y$46$im * x$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(y$46$re / N[(y$46$im / x$46$im), $MachinePrecision]), $MachinePrecision] - x$46$re), $MachinePrecision] / y$46$im), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if y.im < -8.4999999999999999e150

   1. Initial program 35.4%

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

   3. Taylor expanded in y.re around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. unpow2 N/A

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

      5. associate-/r* N/A

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

      6. div-sub N/A

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

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{x.im \cdot y.re}{y.im} - x.re\right), \color{blue}{y.im}\right) \]

      8. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(\frac{x.im \cdot y.re}{y.im}\right), x.re\right), y.im\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(\frac{y.re \cdot x.im}{y.im}\right), x.re\right), y.im\right) \]

      10. associate-/l* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(y.re \cdot \frac{x.im}{y.im}\right), x.re\right), y.im\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(y.re, \left(\frac{x.im}{y.im}\right)\right), x.re\right), y.im\right) \]

      12. /-lowering-/.f64 86.2%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{/.f64}\left(x.im, y.im\right)\right), x.re\right), y.im\right) \]

   5. Simplified 86.2%

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

      1. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(\frac{y.re \cdot x.im}{y.im}\right), x.re\right), y.im\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(\frac{x.im \cdot y.re}{y.im}\right), x.re\right), y.im\right) \]

      3. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(x.im \cdot \frac{y.re}{y.im}\right), x.re\right), y.im\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(x.im, \left(\frac{y.re}{y.im}\right)\right), x.re\right), y.im\right) \]

      5. /-lowering-/.f64 86.3%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(x.im, \mathsf{/.f64}\left(y.re, y.im\right)\right), x.re\right), y.im\right) \]

   7. Applied egg-rr 86.3%

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

   if -8.4999999999999999e150 < y.im < -4.5e-73

   1. Initial program 72.9%

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

   3. Taylor expanded in x.re around inf

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

         \[\leadsto x.re \cdot \left(\left(0 - \frac{y.im}{{y.im}^{2} + {y.re}^{2}}\right) + \frac{\color{blue}{x.im \cdot y.re}}{x.re \cdot \left({y.im}^{2} + {y.re}^{2}\right)}\right) \]

      3. associate-+l- N/A

         \[\leadsto x.re \cdot \left(0 - \color{blue}{\left(\frac{y.im}{{y.im}^{2} + {y.re}^{2}} - \frac{x.im \cdot y.re}{x.re \cdot \left({y.im}^{2} + {y.re}^{2}\right)}\right)}\right) \]

      4. unsub-neg N/A

         \[\leadsto x.re \cdot \left(0 - \left(\frac{y.im}{{y.im}^{2} + {y.re}^{2}} + \color{blue}{\left(\mathsf{neg}\left(\frac{x.im \cdot y.re}{x.re \cdot \left({y.im}^{2} + {y.re}^{2}\right)}\right)\right)}\right)\right) \]

      5. mul-1-neg N/A

         \[\leadsto x.re \cdot \left(0 - \left(\frac{y.im}{{y.im}^{2} + {y.re}^{2}} + -1 \cdot \color{blue}{\frac{x.im \cdot y.re}{x.re \cdot \left({y.im}^{2} + {y.re}^{2}\right)}}\right)\right) \]

      6. +-commutative N/A

         \[\leadsto x.re \cdot \left(0 - \left(-1 \cdot \frac{x.im \cdot y.re}{x.re \cdot \left({y.im}^{2} + {y.re}^{2}\right)} + \color{blue}{\frac{y.im}{{y.im}^{2} + {y.re}^{2}}}\right)\right) \]

      7. neg-sub0 N/A

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

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x.re, \color{blue}{\left(\mathsf{neg}\left(\left(-1 \cdot \frac{x.im \cdot y.re}{x.re \cdot \left({y.im}^{2} + {y.re}^{2}\right)} + \frac{y.im}{{y.im}^{2} + {y.re}^{2}}\right)\right)\right)}\right) \]

      9. neg-sub0 N/A

         \[\leadsto \mathsf{*.f64}\left(x.re, \left(0 - \color{blue}{\left(-1 \cdot \frac{x.im \cdot y.re}{x.re \cdot \left({y.im}^{2} + {y.re}^{2}\right)} + \frac{y.im}{{y.im}^{2} + {y.re}^{2}}\right)}\right)\right) \]

   5. Simplified 82.9%

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

   if -4.5e-73 < y.im < 5.3000000000000003e-102

   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
   3. Step-by-step derivation

      1. div-inv N/A

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

      2. *-commutative N/A

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

      3. flip3-+ N/A

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

      4. clear-num N/A

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

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\left(y.re \cdot y.re\right) \cdot \left(y.re \cdot y.re\right) + \left(\left(y.im \cdot y.im\right) \cdot \left(y.im \cdot y.im\right) - \left(y.re \cdot y.re\right) \cdot \left(y.im \cdot y.im\right)\right)}{{\left(y.re \cdot y.re\right)}^{3} + {\left(y.im \cdot y.im\right)}^{3}}\right), \color{blue}{\left(x.im \cdot y.re - x.re \cdot y.im\right)}\right) \]

   4. Applied egg-rr 72.8%

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

      1. associate-*l/ N/A

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

      2. clear-num N/A

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

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{y.re \cdot y.re + y.im \cdot y.im}{1 \cdot \left(x.im \cdot y.re - x.re \cdot y.im\right)}\right)}\right) \]

      4. *-lft-identity N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(\frac{y.re \cdot y.re + y.im \cdot y.im}{x.im \cdot y.re - \color{blue}{x.re \cdot y.im}}\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\left(y.re \cdot y.re + y.im \cdot y.im\right), \color{blue}{\left(x.im \cdot y.re - x.re \cdot y.im\right)}\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(y.re \cdot y.re\right), \left(y.im \cdot y.im\right)\right), \left(\color{blue}{x.im \cdot y.re} - x.re \cdot y.im\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(y.re, y.re\right), \left(y.im \cdot y.im\right)\right), \left(\color{blue}{x.im} \cdot y.re - x.re \cdot y.im\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(y.re, y.re\right), \mathsf{*.f64}\left(y.im, y.im\right)\right), \left(x.im \cdot \color{blue}{y.re} - x.re \cdot y.im\right)\right)\right) \]

      9. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(y.re, y.re\right), \mathsf{*.f64}\left(y.im, y.im\right)\right), \mathsf{\_.f64}\left(\left(x.im \cdot y.re\right), \color{blue}{\left(x.re \cdot y.im\right)}\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(y.re, y.re\right), \mathsf{*.f64}\left(y.im, y.im\right)\right), \mathsf{\_.f64}\left(\left(y.re \cdot x.im\right), \left(\color{blue}{x.re} \cdot y.im\right)\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(y.re, y.re\right), \mathsf{*.f64}\left(y.im, y.im\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(y.re, x.im\right), \left(\color{blue}{x.re} \cdot y.im\right)\right)\right)\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(y.re, y.re\right), \mathsf{*.f64}\left(y.im, y.im\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(y.re, x.im\right), \left(y.im \cdot \color{blue}{x.re}\right)\right)\right)\right) \]

      13. *-lowering-*.f64 72.9%

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(y.re, y.re\right), \mathsf{*.f64}\left(y.im, y.im\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(y.re, x.im\right), \mathsf{*.f64}\left(y.im, \color{blue}{x.re}\right)\right)\right)\right) \]

   6. Applied egg-rr 72.9%

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

   7. Taylor expanded in y.re around inf

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(x.im + -1 \cdot \frac{x.re \cdot y.im}{y.re}\right), \color{blue}{y.re}\right) \]

      2. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\left(x.im + \left(\mathsf{neg}\left(\frac{x.re \cdot y.im}{y.re}\right)\right)\right), y.re\right) \]

      3. unsub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\left(x.im - \frac{x.re \cdot y.im}{y.re}\right), y.re\right) \]

      4. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(x.im, \left(\frac{x.re \cdot y.im}{y.re}\right)\right), y.re\right) \]

      5. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(x.im, \left(x.re \cdot \frac{y.im}{y.re}\right)\right), y.re\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(x.im, \mathsf{*.f64}\left(x.re, \left(\frac{y.im}{y.re}\right)\right)\right), y.re\right) \]

      7. /-lowering-/.f64 95.1%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(x.im, \mathsf{*.f64}\left(x.re, \mathsf{/.f64}\left(y.im, y.re\right)\right)\right), y.re\right) \]

   9. Simplified 95.1%

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

       1. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(x.im - x.re \cdot \frac{y.im}{y.re}\right), \color{blue}{y.re}\right) \]

       2. --lowering--.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(x.im, \left(x.re \cdot \frac{y.im}{y.re}\right)\right), y.re\right) \]

       3. clear-num N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(x.im, \left(x.re \cdot \frac{1}{\frac{y.re}{y.im}}\right)\right), y.re\right) \]

       4. un-div-inv N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(x.im, \left(\frac{x.re}{\frac{y.re}{y.im}}\right)\right), y.re\right) \]

       5. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(x.im, \mathsf{/.f64}\left(x.re, \left(\frac{y.re}{y.im}\right)\right)\right), y.re\right) \]

       6. /-lowering-/.f64 95.1%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(x.im, \mathsf{/.f64}\left(x.re, \mathsf{/.f64}\left(y.re, y.im\right)\right)\right), y.re\right) \]

   11. Applied egg-rr 95.1%

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

   if 5.3000000000000003e-102 < y.im < 8.8000000000000002e77

   1. Initial program 78.9%

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

      1. div-inv N/A

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

      2. *-commutative N/A

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

      3. flip3-+ N/A

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

      4. clear-num N/A

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

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\left(y.re \cdot y.re\right) \cdot \left(y.re \cdot y.re\right) + \left(\left(y.im \cdot y.im\right) \cdot \left(y.im \cdot y.im\right) - \left(y.re \cdot y.re\right) \cdot \left(y.im \cdot y.im\right)\right)}{{\left(y.re \cdot y.re\right)}^{3} + {\left(y.im \cdot y.im\right)}^{3}}\right), \color{blue}{\left(x.im \cdot y.re - x.re \cdot y.im\right)}\right) \]

   4. Applied egg-rr 79.0%

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

   if 8.8000000000000002e77 < y.im

   1. Initial program 44.2%

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

   3. Taylor expanded in y.re around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. unpow2 N/A

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

      5. associate-/r* N/A

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

      6. div-sub N/A

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

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{x.im \cdot y.re}{y.im} - x.re\right), \color{blue}{y.im}\right) \]

      8. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(\frac{x.im \cdot y.re}{y.im}\right), x.re\right), y.im\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(\frac{y.re \cdot x.im}{y.im}\right), x.re\right), y.im\right) \]

      10. associate-/l* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(y.re \cdot \frac{x.im}{y.im}\right), x.re\right), y.im\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(y.re, \left(\frac{x.im}{y.im}\right)\right), x.re\right), y.im\right) \]

      12. /-lowering-/.f64 87.1%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{/.f64}\left(x.im, y.im\right)\right), x.re\right), y.im\right) \]

   5. Simplified 87.1%

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

      1. clear-num N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(y.re \cdot \frac{1}{\frac{y.im}{x.im}}\right), x.re\right), y.im\right) \]

      2. un-div-inv N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(\frac{y.re}{\frac{y.im}{x.im}}\right), x.re\right), y.im\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(y.re, \left(\frac{y.im}{x.im}\right)\right), x.re\right), y.im\right) \]

      4. /-lowering-/.f64 89.5%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(y.re, \mathsf{/.f64}\left(y.im, x.im\right)\right), x.re\right), y.im\right) \]

   7. Applied egg-rr 89.5%

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

4. Final simplification 88.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -8.5 \cdot 10^{+150}:\\ \;\;\;\;\frac{x.im \cdot \frac{y.re}{y.im} - x.re}{y.im}\\ \mathbf{elif}\;y.im \leq -4.5 \cdot 10^{-73}:\\ \;\;\;\;x.re \cdot \left(\frac{x.im \cdot y.re}{x.re \cdot \left(y.im \cdot y.im + y.re \cdot y.re\right)} - \frac{y.im}{y.im \cdot y.im + y.re \cdot y.re}\right)\\ \mathbf{elif}\;y.im \leq 5.3 \cdot 10^{-102}:\\ \;\;\;\;\frac{x.im - \frac{x.re}{\frac{y.re}{y.im}}}{y.re}\\ \mathbf{elif}\;y.im \leq 8.8 \cdot 10^{+77}:\\ \;\;\;\;\frac{1}{y.im \cdot y.im + y.re \cdot y.re} \cdot \left(x.im \cdot y.re - y.im \cdot x.re\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y.re}{\frac{y.im}{x.im}} - x.re}{y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 83.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{y.im \cdot y.im + y.re \cdot y.re} \cdot \left(x.im \cdot y.re - y.im \cdot x.re\right)\\ \mathbf{if}\;y.im \leq -1.9 \cdot 10^{+46}:\\ \;\;\;\;\frac{x.im \cdot \frac{y.re}{y.im} - x.re}{y.im}\\ \mathbf{elif}\;y.im \leq -3 \cdot 10^{-106}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.im \leq 5.4 \cdot 10^{-102}:\\ \;\;\;\;\frac{x.im - \frac{x.re}{\frac{y.re}{y.im}}}{y.re}\\ \mathbf{elif}\;y.im \leq 4.3 \cdot 10^{+77}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y.re}{\frac{y.im}{x.im}} - x.re}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0
         (*
          (/ 1.0 (+ (* y.im y.im) (* y.re y.re)))
          (- (* x.im y.re) (* y.im x.re)))))
   (if (<= y.im -1.9e+46)
     (/ (- (* x.im (/ y.re y.im)) x.re) y.im)
     (if (<= y.im -3e-106)
       t_0
       (if (<= y.im 5.4e-102)
         (/ (- x.im (/ x.re (/ y.re y.im))) y.re)
         (if (<= y.im 4.3e+77)
           t_0
           (/ (- (/ y.re (/ y.im x.im)) x.re) y.im)))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = (1.0 / ((y_46_im * y_46_im) + (y_46_re * y_46_re))) * ((x_46_im * y_46_re) - (y_46_im * x_46_re));
	double tmp;
	if (y_46_im <= -1.9e+46) {
		tmp = ((x_46_im * (y_46_re / y_46_im)) - x_46_re) / y_46_im;
	} else if (y_46_im <= -3e-106) {
		tmp = t_0;
	} else if (y_46_im <= 5.4e-102) {
		tmp = (x_46_im - (x_46_re / (y_46_re / y_46_im))) / y_46_re;
	} else if (y_46_im <= 4.3e+77) {
		tmp = t_0;
	} else {
		tmp = ((y_46_re / (y_46_im / x_46_im)) - x_46_re) / y_46_im;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (1.0d0 / ((y_46im * y_46im) + (y_46re * y_46re))) * ((x_46im * y_46re) - (y_46im * x_46re))
    if (y_46im <= (-1.9d+46)) then
        tmp = ((x_46im * (y_46re / y_46im)) - x_46re) / y_46im
    else if (y_46im <= (-3d-106)) then
        tmp = t_0
    else if (y_46im <= 5.4d-102) then
        tmp = (x_46im - (x_46re / (y_46re / y_46im))) / y_46re
    else if (y_46im <= 4.3d+77) then
        tmp = t_0
    else
        tmp = ((y_46re / (y_46im / x_46im)) - x_46re) / y_46im
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = (1.0 / ((y_46_im * y_46_im) + (y_46_re * y_46_re))) * ((x_46_im * y_46_re) - (y_46_im * x_46_re));
	double tmp;
	if (y_46_im <= -1.9e+46) {
		tmp = ((x_46_im * (y_46_re / y_46_im)) - x_46_re) / y_46_im;
	} else if (y_46_im <= -3e-106) {
		tmp = t_0;
	} else if (y_46_im <= 5.4e-102) {
		tmp = (x_46_im - (x_46_re / (y_46_re / y_46_im))) / y_46_re;
	} else if (y_46_im <= 4.3e+77) {
		tmp = t_0;
	} else {
		tmp = ((y_46_re / (y_46_im / x_46_im)) - x_46_re) / y_46_im;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = (1.0 / ((y_46_im * y_46_im) + (y_46_re * y_46_re))) * ((x_46_im * y_46_re) - (y_46_im * x_46_re))
	tmp = 0
	if y_46_im <= -1.9e+46:
		tmp = ((x_46_im * (y_46_re / y_46_im)) - x_46_re) / y_46_im
	elif y_46_im <= -3e-106:
		tmp = t_0
	elif y_46_im <= 5.4e-102:
		tmp = (x_46_im - (x_46_re / (y_46_re / y_46_im))) / y_46_re
	elif y_46_im <= 4.3e+77:
		tmp = t_0
	else:
		tmp = ((y_46_re / (y_46_im / x_46_im)) - x_46_re) / y_46_im
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(Float64(1.0 / Float64(Float64(y_46_im * y_46_im) + Float64(y_46_re * y_46_re))) * Float64(Float64(x_46_im * y_46_re) - Float64(y_46_im * x_46_re)))
	tmp = 0.0
	if (y_46_im <= -1.9e+46)
		tmp = Float64(Float64(Float64(x_46_im * Float64(y_46_re / y_46_im)) - x_46_re) / y_46_im);
	elseif (y_46_im <= -3e-106)
		tmp = t_0;
	elseif (y_46_im <= 5.4e-102)
		tmp = Float64(Float64(x_46_im - Float64(x_46_re / Float64(y_46_re / y_46_im))) / y_46_re);
	elseif (y_46_im <= 4.3e+77)
		tmp = t_0;
	else
		tmp = Float64(Float64(Float64(y_46_re / Float64(y_46_im / x_46_im)) - x_46_re) / y_46_im);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = (1.0 / ((y_46_im * y_46_im) + (y_46_re * y_46_re))) * ((x_46_im * y_46_re) - (y_46_im * x_46_re));
	tmp = 0.0;
	if (y_46_im <= -1.9e+46)
		tmp = ((x_46_im * (y_46_re / y_46_im)) - x_46_re) / y_46_im;
	elseif (y_46_im <= -3e-106)
		tmp = t_0;
	elseif (y_46_im <= 5.4e-102)
		tmp = (x_46_im - (x_46_re / (y_46_re / y_46_im))) / y_46_re;
	elseif (y_46_im <= 4.3e+77)
		tmp = t_0;
	else
		tmp = ((y_46_re / (y_46_im / x_46_im)) - x_46_re) / y_46_im;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[(1.0 / N[(N[(y$46$im * y$46$im), $MachinePrecision] + N[(y$46$re * y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(x$46$im * y$46$re), $MachinePrecision] - N[(y$46$im * x$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$im, -1.9e+46], N[(N[(N[(x$46$im * N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision] - x$46$re), $MachinePrecision] / y$46$im), $MachinePrecision], If[LessEqual[y$46$im, -3e-106], t$95$0, If[LessEqual[y$46$im, 5.4e-102], N[(N[(x$46$im - N[(x$46$re / N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision], If[LessEqual[y$46$im, 4.3e+77], t$95$0, N[(N[(N[(y$46$re / N[(y$46$im / x$46$im), $MachinePrecision]), $MachinePrecision] - x$46$re), $MachinePrecision] / y$46$im), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if y.im < -1.9e46

   1. Initial program 49.3%

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

   3. Taylor expanded in y.re around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. unpow2 N/A

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

      5. associate-/r* N/A

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

      6. div-sub N/A

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

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{x.im \cdot y.re}{y.im} - x.re\right), \color{blue}{y.im}\right) \]

      8. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(\frac{x.im \cdot y.re}{y.im}\right), x.re\right), y.im\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(\frac{y.re \cdot x.im}{y.im}\right), x.re\right), y.im\right) \]

      10. associate-/l* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(y.re \cdot \frac{x.im}{y.im}\right), x.re\right), y.im\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(y.re, \left(\frac{x.im}{y.im}\right)\right), x.re\right), y.im\right) \]

      12. /-lowering-/.f64 82.1%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{/.f64}\left(x.im, y.im\right)\right), x.re\right), y.im\right) \]

   5. Simplified 82.1%

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

      1. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(\frac{y.re \cdot x.im}{y.im}\right), x.re\right), y.im\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(\frac{x.im \cdot y.re}{y.im}\right), x.re\right), y.im\right) \]

      3. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(x.im \cdot \frac{y.re}{y.im}\right), x.re\right), y.im\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(x.im, \left(\frac{y.re}{y.im}\right)\right), x.re\right), y.im\right) \]

      5. /-lowering-/.f64 82.2%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(x.im, \mathsf{/.f64}\left(y.re, y.im\right)\right), x.re\right), y.im\right) \]

   7. Applied egg-rr 82.2%

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

   if -1.9e46 < y.im < -3.00000000000000019e-106 or 5.4e-102 < y.im < 4.29999999999999991e77

   1. Initial program 79.6%

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

      1. div-inv N/A

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

      2. *-commutative N/A

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

      3. flip3-+ N/A

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

      4. clear-num N/A

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

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\left(y.re \cdot y.re\right) \cdot \left(y.re \cdot y.re\right) + \left(\left(y.im \cdot y.im\right) \cdot \left(y.im \cdot y.im\right) - \left(y.re \cdot y.re\right) \cdot \left(y.im \cdot y.im\right)\right)}{{\left(y.re \cdot y.re\right)}^{3} + {\left(y.im \cdot y.im\right)}^{3}}\right), \color{blue}{\left(x.im \cdot y.re - x.re \cdot y.im\right)}\right) \]

   4. Applied egg-rr 79.7%

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

   if -3.00000000000000019e-106 < y.im < 5.4e-102

   1. Initial program 70.3%

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

      1. div-inv N/A

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

      2. *-commutative N/A

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

      3. flip3-+ N/A

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

      4. clear-num N/A

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

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\left(y.re \cdot y.re\right) \cdot \left(y.re \cdot y.re\right) + \left(\left(y.im \cdot y.im\right) \cdot \left(y.im \cdot y.im\right) - \left(y.re \cdot y.re\right) \cdot \left(y.im \cdot y.im\right)\right)}{{\left(y.re \cdot y.re\right)}^{3} + {\left(y.im \cdot y.im\right)}^{3}}\right), \color{blue}{\left(x.im \cdot y.re - x.re \cdot y.im\right)}\right) \]

   4. Applied egg-rr 70.3%

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

      1. associate-*l/ N/A

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

      2. clear-num N/A

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

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{y.re \cdot y.re + y.im \cdot y.im}{1 \cdot \left(x.im \cdot y.re - x.re \cdot y.im\right)}\right)}\right) \]

      4. *-lft-identity N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(\frac{y.re \cdot y.re + y.im \cdot y.im}{x.im \cdot y.re - \color{blue}{x.re \cdot y.im}}\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\left(y.re \cdot y.re + y.im \cdot y.im\right), \color{blue}{\left(x.im \cdot y.re - x.re \cdot y.im\right)}\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(y.re \cdot y.re\right), \left(y.im \cdot y.im\right)\right), \left(\color{blue}{x.im \cdot y.re} - x.re \cdot y.im\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(y.re, y.re\right), \left(y.im \cdot y.im\right)\right), \left(\color{blue}{x.im} \cdot y.re - x.re \cdot y.im\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(y.re, y.re\right), \mathsf{*.f64}\left(y.im, y.im\right)\right), \left(x.im \cdot \color{blue}{y.re} - x.re \cdot y.im\right)\right)\right) \]

      9. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(y.re, y.re\right), \mathsf{*.f64}\left(y.im, y.im\right)\right), \mathsf{\_.f64}\left(\left(x.im \cdot y.re\right), \color{blue}{\left(x.re \cdot y.im\right)}\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(y.re, y.re\right), \mathsf{*.f64}\left(y.im, y.im\right)\right), \mathsf{\_.f64}\left(\left(y.re \cdot x.im\right), \left(\color{blue}{x.re} \cdot y.im\right)\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(y.re, y.re\right), \mathsf{*.f64}\left(y.im, y.im\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(y.re, x.im\right), \left(\color{blue}{x.re} \cdot y.im\right)\right)\right)\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(y.re, y.re\right), \mathsf{*.f64}\left(y.im, y.im\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(y.re, x.im\right), \left(y.im \cdot \color{blue}{x.re}\right)\right)\right)\right) \]

      13. *-lowering-*.f64 70.4%

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(y.re, y.re\right), \mathsf{*.f64}\left(y.im, y.im\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(y.re, x.im\right), \mathsf{*.f64}\left(y.im, \color{blue}{x.re}\right)\right)\right)\right) \]

   6. Applied egg-rr 70.4%

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

   7. Taylor expanded in y.re around inf

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(x.im + -1 \cdot \frac{x.re \cdot y.im}{y.re}\right), \color{blue}{y.re}\right) \]

      2. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\left(x.im + \left(\mathsf{neg}\left(\frac{x.re \cdot y.im}{y.re}\right)\right)\right), y.re\right) \]

      3. unsub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\left(x.im - \frac{x.re \cdot y.im}{y.re}\right), y.re\right) \]

      4. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(x.im, \left(\frac{x.re \cdot y.im}{y.re}\right)\right), y.re\right) \]

      5. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(x.im, \left(x.re \cdot \frac{y.im}{y.re}\right)\right), y.re\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(x.im, \mathsf{*.f64}\left(x.re, \left(\frac{y.im}{y.re}\right)\right)\right), y.re\right) \]

      7. /-lowering-/.f64 96.6%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(x.im, \mathsf{*.f64}\left(x.re, \mathsf{/.f64}\left(y.im, y.re\right)\right)\right), y.re\right) \]

   9. Simplified 96.6%

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

       1. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(x.im - x.re \cdot \frac{y.im}{y.re}\right), \color{blue}{y.re}\right) \]

       2. --lowering--.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(x.im, \left(x.re \cdot \frac{y.im}{y.re}\right)\right), y.re\right) \]

       3. clear-num N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(x.im, \left(x.re \cdot \frac{1}{\frac{y.re}{y.im}}\right)\right), y.re\right) \]

       4. un-div-inv N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(x.im, \left(\frac{x.re}{\frac{y.re}{y.im}}\right)\right), y.re\right) \]

       5. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(x.im, \mathsf{/.f64}\left(x.re, \left(\frac{y.re}{y.im}\right)\right)\right), y.re\right) \]

       6. /-lowering-/.f64 96.7%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(x.im, \mathsf{/.f64}\left(x.re, \mathsf{/.f64}\left(y.re, y.im\right)\right)\right), y.re\right) \]

   11. Applied egg-rr 96.7%

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

   if 4.29999999999999991e77 < y.im

   1. Initial program 44.2%

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

   3. Taylor expanded in y.re around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. unpow2 N/A

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

      5. associate-/r* N/A

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

      6. div-sub N/A

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

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{x.im \cdot y.re}{y.im} - x.re\right), \color{blue}{y.im}\right) \]

      8. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(\frac{x.im \cdot y.re}{y.im}\right), x.re\right), y.im\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(\frac{y.re \cdot x.im}{y.im}\right), x.re\right), y.im\right) \]

      10. associate-/l* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(y.re \cdot \frac{x.im}{y.im}\right), x.re\right), y.im\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(y.re, \left(\frac{x.im}{y.im}\right)\right), x.re\right), y.im\right) \]

      12. /-lowering-/.f64 87.1%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{/.f64}\left(x.im, y.im\right)\right), x.re\right), y.im\right) \]

   5. Simplified 87.1%

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

      1. clear-num N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(y.re \cdot \frac{1}{\frac{y.im}{x.im}}\right), x.re\right), y.im\right) \]

      2. un-div-inv N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(\frac{y.re}{\frac{y.im}{x.im}}\right), x.re\right), y.im\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(y.re, \left(\frac{y.im}{x.im}\right)\right), x.re\right), y.im\right) \]

      4. /-lowering-/.f64 89.5%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(y.re, \mathsf{/.f64}\left(y.im, x.im\right)\right), x.re\right), y.im\right) \]

   7. Applied egg-rr 89.5%

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

4. Final simplification 87.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -1.9 \cdot 10^{+46}:\\ \;\;\;\;\frac{x.im \cdot \frac{y.re}{y.im} - x.re}{y.im}\\ \mathbf{elif}\;y.im \leq -3 \cdot 10^{-106}:\\ \;\;\;\;\frac{1}{y.im \cdot y.im + y.re \cdot y.re} \cdot \left(x.im \cdot y.re - y.im \cdot x.re\right)\\ \mathbf{elif}\;y.im \leq 5.4 \cdot 10^{-102}:\\ \;\;\;\;\frac{x.im - \frac{x.re}{\frac{y.re}{y.im}}}{y.re}\\ \mathbf{elif}\;y.im \leq 4.3 \cdot 10^{+77}:\\ \;\;\;\;\frac{1}{y.im \cdot y.im + y.re \cdot y.re} \cdot \left(x.im \cdot y.re - y.im \cdot x.re\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y.re}{\frac{y.im}{x.im}} - x.re}{y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 83.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x.im \cdot y.re - y.im \cdot x.re}{y.im \cdot y.im + y.re \cdot y.re}\\ \mathbf{if}\;y.im \leq -1.25 \cdot 10^{+46}:\\ \;\;\;\;\frac{x.im \cdot \frac{y.re}{y.im} - x.re}{y.im}\\ \mathbf{elif}\;y.im \leq -3.7 \cdot 10^{-106}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.im \leq 8.5 \cdot 10^{-102}:\\ \;\;\;\;\frac{x.im - \frac{x.re}{\frac{y.re}{y.im}}}{y.re}\\ \mathbf{elif}\;y.im \leq 2.45 \cdot 10^{+78}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y.re}{\frac{y.im}{x.im}} - x.re}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0
         (/ (- (* x.im y.re) (* y.im x.re)) (+ (* y.im y.im) (* y.re y.re)))))
   (if (<= y.im -1.25e+46)
     (/ (- (* x.im (/ y.re y.im)) x.re) y.im)
     (if (<= y.im -3.7e-106)
       t_0
       (if (<= y.im 8.5e-102)
         (/ (- x.im (/ x.re (/ y.re y.im))) y.re)
         (if (<= y.im 2.45e+78)
           t_0
           (/ (- (/ y.re (/ y.im x.im)) x.re) y.im)))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = ((x_46_im * y_46_re) - (y_46_im * x_46_re)) / ((y_46_im * y_46_im) + (y_46_re * y_46_re));
	double tmp;
	if (y_46_im <= -1.25e+46) {
		tmp = ((x_46_im * (y_46_re / y_46_im)) - x_46_re) / y_46_im;
	} else if (y_46_im <= -3.7e-106) {
		tmp = t_0;
	} else if (y_46_im <= 8.5e-102) {
		tmp = (x_46_im - (x_46_re / (y_46_re / y_46_im))) / y_46_re;
	} else if (y_46_im <= 2.45e+78) {
		tmp = t_0;
	} else {
		tmp = ((y_46_re / (y_46_im / x_46_im)) - x_46_re) / y_46_im;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ((x_46im * y_46re) - (y_46im * x_46re)) / ((y_46im * y_46im) + (y_46re * y_46re))
    if (y_46im <= (-1.25d+46)) then
        tmp = ((x_46im * (y_46re / y_46im)) - x_46re) / y_46im
    else if (y_46im <= (-3.7d-106)) then
        tmp = t_0
    else if (y_46im <= 8.5d-102) then
        tmp = (x_46im - (x_46re / (y_46re / y_46im))) / y_46re
    else if (y_46im <= 2.45d+78) then
        tmp = t_0
    else
        tmp = ((y_46re / (y_46im / x_46im)) - x_46re) / y_46im
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = ((x_46_im * y_46_re) - (y_46_im * x_46_re)) / ((y_46_im * y_46_im) + (y_46_re * y_46_re));
	double tmp;
	if (y_46_im <= -1.25e+46) {
		tmp = ((x_46_im * (y_46_re / y_46_im)) - x_46_re) / y_46_im;
	} else if (y_46_im <= -3.7e-106) {
		tmp = t_0;
	} else if (y_46_im <= 8.5e-102) {
		tmp = (x_46_im - (x_46_re / (y_46_re / y_46_im))) / y_46_re;
	} else if (y_46_im <= 2.45e+78) {
		tmp = t_0;
	} else {
		tmp = ((y_46_re / (y_46_im / x_46_im)) - x_46_re) / y_46_im;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = ((x_46_im * y_46_re) - (y_46_im * x_46_re)) / ((y_46_im * y_46_im) + (y_46_re * y_46_re))
	tmp = 0
	if y_46_im <= -1.25e+46:
		tmp = ((x_46_im * (y_46_re / y_46_im)) - x_46_re) / y_46_im
	elif y_46_im <= -3.7e-106:
		tmp = t_0
	elif y_46_im <= 8.5e-102:
		tmp = (x_46_im - (x_46_re / (y_46_re / y_46_im))) / y_46_re
	elif y_46_im <= 2.45e+78:
		tmp = t_0
	else:
		tmp = ((y_46_re / (y_46_im / x_46_im)) - x_46_re) / y_46_im
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(Float64(Float64(x_46_im * y_46_re) - Float64(y_46_im * x_46_re)) / Float64(Float64(y_46_im * y_46_im) + Float64(y_46_re * y_46_re)))
	tmp = 0.0
	if (y_46_im <= -1.25e+46)
		tmp = Float64(Float64(Float64(x_46_im * Float64(y_46_re / y_46_im)) - x_46_re) / y_46_im);
	elseif (y_46_im <= -3.7e-106)
		tmp = t_0;
	elseif (y_46_im <= 8.5e-102)
		tmp = Float64(Float64(x_46_im - Float64(x_46_re / Float64(y_46_re / y_46_im))) / y_46_re);
	elseif (y_46_im <= 2.45e+78)
		tmp = t_0;
	else
		tmp = Float64(Float64(Float64(y_46_re / Float64(y_46_im / x_46_im)) - x_46_re) / y_46_im);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = ((x_46_im * y_46_re) - (y_46_im * x_46_re)) / ((y_46_im * y_46_im) + (y_46_re * y_46_re));
	tmp = 0.0;
	if (y_46_im <= -1.25e+46)
		tmp = ((x_46_im * (y_46_re / y_46_im)) - x_46_re) / y_46_im;
	elseif (y_46_im <= -3.7e-106)
		tmp = t_0;
	elseif (y_46_im <= 8.5e-102)
		tmp = (x_46_im - (x_46_re / (y_46_re / y_46_im))) / y_46_re;
	elseif (y_46_im <= 2.45e+78)
		tmp = t_0;
	else
		tmp = ((y_46_re / (y_46_im / x_46_im)) - x_46_re) / y_46_im;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[(N[(x$46$im * y$46$re), $MachinePrecision] - N[(y$46$im * x$46$re), $MachinePrecision]), $MachinePrecision] / N[(N[(y$46$im * y$46$im), $MachinePrecision] + N[(y$46$re * y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$im, -1.25e+46], N[(N[(N[(x$46$im * N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision] - x$46$re), $MachinePrecision] / y$46$im), $MachinePrecision], If[LessEqual[y$46$im, -3.7e-106], t$95$0, If[LessEqual[y$46$im, 8.5e-102], N[(N[(x$46$im - N[(x$46$re / N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision], If[LessEqual[y$46$im, 2.45e+78], t$95$0, N[(N[(N[(y$46$re / N[(y$46$im / x$46$im), $MachinePrecision]), $MachinePrecision] - x$46$re), $MachinePrecision] / y$46$im), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if y.im < -1.2500000000000001e46

   1. Initial program 49.3%

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

   3. Taylor expanded in y.re around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. unpow2 N/A

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

      5. associate-/r* N/A

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

      6. div-sub N/A

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

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{x.im \cdot y.re}{y.im} - x.re\right), \color{blue}{y.im}\right) \]

      8. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(\frac{x.im \cdot y.re}{y.im}\right), x.re\right), y.im\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(\frac{y.re \cdot x.im}{y.im}\right), x.re\right), y.im\right) \]

      10. associate-/l* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(y.re \cdot \frac{x.im}{y.im}\right), x.re\right), y.im\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(y.re, \left(\frac{x.im}{y.im}\right)\right), x.re\right), y.im\right) \]

      12. /-lowering-/.f64 82.1%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{/.f64}\left(x.im, y.im\right)\right), x.re\right), y.im\right) \]

   5. Simplified 82.1%

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

      1. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(\frac{y.re \cdot x.im}{y.im}\right), x.re\right), y.im\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(\frac{x.im \cdot y.re}{y.im}\right), x.re\right), y.im\right) \]

      3. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(x.im \cdot \frac{y.re}{y.im}\right), x.re\right), y.im\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(x.im, \left(\frac{y.re}{y.im}\right)\right), x.re\right), y.im\right) \]

      5. /-lowering-/.f64 82.2%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(x.im, \mathsf{/.f64}\left(y.re, y.im\right)\right), x.re\right), y.im\right) \]

   7. Applied egg-rr 82.2%

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

   if -1.2500000000000001e46 < y.im < -3.69999999999999979e-106 or 8.49999999999999973e-102 < y.im < 2.4500000000000001e78

   1. Initial program 79.6%

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

   if -3.69999999999999979e-106 < y.im < 8.49999999999999973e-102

   1. Initial program 70.3%

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

      1. div-inv N/A

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

      2. *-commutative N/A

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

      3. flip3-+ N/A

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

      4. clear-num N/A

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

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\left(y.re \cdot y.re\right) \cdot \left(y.re \cdot y.re\right) + \left(\left(y.im \cdot y.im\right) \cdot \left(y.im \cdot y.im\right) - \left(y.re \cdot y.re\right) \cdot \left(y.im \cdot y.im\right)\right)}{{\left(y.re \cdot y.re\right)}^{3} + {\left(y.im \cdot y.im\right)}^{3}}\right), \color{blue}{\left(x.im \cdot y.re - x.re \cdot y.im\right)}\right) \]

   4. Applied egg-rr 70.3%

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

      1. associate-*l/ N/A

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

      2. clear-num N/A

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

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{y.re \cdot y.re + y.im \cdot y.im}{1 \cdot \left(x.im \cdot y.re - x.re \cdot y.im\right)}\right)}\right) \]

      4. *-lft-identity N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(\frac{y.re \cdot y.re + y.im \cdot y.im}{x.im \cdot y.re - \color{blue}{x.re \cdot y.im}}\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\left(y.re \cdot y.re + y.im \cdot y.im\right), \color{blue}{\left(x.im \cdot y.re - x.re \cdot y.im\right)}\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(y.re \cdot y.re\right), \left(y.im \cdot y.im\right)\right), \left(\color{blue}{x.im \cdot y.re} - x.re \cdot y.im\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(y.re, y.re\right), \left(y.im \cdot y.im\right)\right), \left(\color{blue}{x.im} \cdot y.re - x.re \cdot y.im\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(y.re, y.re\right), \mathsf{*.f64}\left(y.im, y.im\right)\right), \left(x.im \cdot \color{blue}{y.re} - x.re \cdot y.im\right)\right)\right) \]

      9. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(y.re, y.re\right), \mathsf{*.f64}\left(y.im, y.im\right)\right), \mathsf{\_.f64}\left(\left(x.im \cdot y.re\right), \color{blue}{\left(x.re \cdot y.im\right)}\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(y.re, y.re\right), \mathsf{*.f64}\left(y.im, y.im\right)\right), \mathsf{\_.f64}\left(\left(y.re \cdot x.im\right), \left(\color{blue}{x.re} \cdot y.im\right)\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(y.re, y.re\right), \mathsf{*.f64}\left(y.im, y.im\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(y.re, x.im\right), \left(\color{blue}{x.re} \cdot y.im\right)\right)\right)\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(y.re, y.re\right), \mathsf{*.f64}\left(y.im, y.im\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(y.re, x.im\right), \left(y.im \cdot \color{blue}{x.re}\right)\right)\right)\right) \]

      13. *-lowering-*.f64 70.4%

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(y.re, y.re\right), \mathsf{*.f64}\left(y.im, y.im\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(y.re, x.im\right), \mathsf{*.f64}\left(y.im, \color{blue}{x.re}\right)\right)\right)\right) \]

   6. Applied egg-rr 70.4%

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

   7. Taylor expanded in y.re around inf

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(x.im + -1 \cdot \frac{x.re \cdot y.im}{y.re}\right), \color{blue}{y.re}\right) \]

      2. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\left(x.im + \left(\mathsf{neg}\left(\frac{x.re \cdot y.im}{y.re}\right)\right)\right), y.re\right) \]

      3. unsub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\left(x.im - \frac{x.re \cdot y.im}{y.re}\right), y.re\right) \]

      4. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(x.im, \left(\frac{x.re \cdot y.im}{y.re}\right)\right), y.re\right) \]

      5. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(x.im, \left(x.re \cdot \frac{y.im}{y.re}\right)\right), y.re\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(x.im, \mathsf{*.f64}\left(x.re, \left(\frac{y.im}{y.re}\right)\right)\right), y.re\right) \]

      7. /-lowering-/.f64 96.6%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(x.im, \mathsf{*.f64}\left(x.re, \mathsf{/.f64}\left(y.im, y.re\right)\right)\right), y.re\right) \]

   9. Simplified 96.6%

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

       1. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(x.im - x.re \cdot \frac{y.im}{y.re}\right), \color{blue}{y.re}\right) \]

       2. --lowering--.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(x.im, \left(x.re \cdot \frac{y.im}{y.re}\right)\right), y.re\right) \]

       3. clear-num N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(x.im, \left(x.re \cdot \frac{1}{\frac{y.re}{y.im}}\right)\right), y.re\right) \]

       4. un-div-inv N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(x.im, \left(\frac{x.re}{\frac{y.re}{y.im}}\right)\right), y.re\right) \]

       5. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(x.im, \mathsf{/.f64}\left(x.re, \left(\frac{y.re}{y.im}\right)\right)\right), y.re\right) \]

       6. /-lowering-/.f64 96.7%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(x.im, \mathsf{/.f64}\left(x.re, \mathsf{/.f64}\left(y.re, y.im\right)\right)\right), y.re\right) \]

   11. Applied egg-rr 96.7%

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

   if 2.4500000000000001e78 < y.im

   1. Initial program 44.2%

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

   3. Taylor expanded in y.re around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. unpow2 N/A

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

      5. associate-/r* N/A

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

      6. div-sub N/A

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

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{x.im \cdot y.re}{y.im} - x.re\right), \color{blue}{y.im}\right) \]

      8. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(\frac{x.im \cdot y.re}{y.im}\right), x.re\right), y.im\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(\frac{y.re \cdot x.im}{y.im}\right), x.re\right), y.im\right) \]

      10. associate-/l* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(y.re \cdot \frac{x.im}{y.im}\right), x.re\right), y.im\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(y.re, \left(\frac{x.im}{y.im}\right)\right), x.re\right), y.im\right) \]

      12. /-lowering-/.f64 87.1%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{/.f64}\left(x.im, y.im\right)\right), x.re\right), y.im\right) \]

   5. Simplified 87.1%

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

      1. clear-num N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(y.re \cdot \frac{1}{\frac{y.im}{x.im}}\right), x.re\right), y.im\right) \]

      2. un-div-inv N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(\frac{y.re}{\frac{y.im}{x.im}}\right), x.re\right), y.im\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(y.re, \left(\frac{y.im}{x.im}\right)\right), x.re\right), y.im\right) \]

      4. /-lowering-/.f64 89.5%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(y.re, \mathsf{/.f64}\left(y.im, x.im\right)\right), x.re\right), y.im\right) \]

   7. Applied egg-rr 89.5%

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

4. Final simplification 87.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -1.25 \cdot 10^{+46}:\\ \;\;\;\;\frac{x.im \cdot \frac{y.re}{y.im} - x.re}{y.im}\\ \mathbf{elif}\;y.im \leq -3.7 \cdot 10^{-106}:\\ \;\;\;\;\frac{x.im \cdot y.re - y.im \cdot x.re}{y.im \cdot y.im + y.re \cdot y.re}\\ \mathbf{elif}\;y.im \leq 8.5 \cdot 10^{-102}:\\ \;\;\;\;\frac{x.im - \frac{x.re}{\frac{y.re}{y.im}}}{y.re}\\ \mathbf{elif}\;y.im \leq 2.45 \cdot 10^{+78}:\\ \;\;\;\;\frac{x.im \cdot y.re - y.im \cdot x.re}{y.im \cdot y.im + y.re \cdot y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y.re}{\frac{y.im}{x.im}} - x.re}{y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 77.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -1.26 \cdot 10^{-70}:\\ \;\;\;\;\frac{x.im \cdot \frac{y.re}{y.im} - x.re}{y.im}\\ \mathbf{elif}\;y.im \leq 3.5 \cdot 10^{-47}:\\ \;\;\;\;\frac{x.im - \frac{x.re}{\frac{y.re}{y.im}}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y.re}{\frac{y.im}{x.im}} - x.re}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.im -1.26e-70)
   (/ (- (* x.im (/ y.re y.im)) x.re) y.im)
   (if (<= y.im 3.5e-47)
     (/ (- x.im (/ x.re (/ y.re y.im))) y.re)
     (/ (- (/ y.re (/ y.im x.im)) x.re) y.im))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_im <= -1.26e-70) {
		tmp = ((x_46_im * (y_46_re / y_46_im)) - x_46_re) / y_46_im;
	} else if (y_46_im <= 3.5e-47) {
		tmp = (x_46_im - (x_46_re / (y_46_re / y_46_im))) / y_46_re;
	} else {
		tmp = ((y_46_re / (y_46_im / x_46_im)) - x_46_re) / y_46_im;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if (y_46im <= (-1.26d-70)) then
        tmp = ((x_46im * (y_46re / y_46im)) - x_46re) / y_46im
    else if (y_46im <= 3.5d-47) then
        tmp = (x_46im - (x_46re / (y_46re / y_46im))) / y_46re
    else
        tmp = ((y_46re / (y_46im / x_46im)) - x_46re) / y_46im
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_im <= -1.26e-70) {
		tmp = ((x_46_im * (y_46_re / y_46_im)) - x_46_re) / y_46_im;
	} else if (y_46_im <= 3.5e-47) {
		tmp = (x_46_im - (x_46_re / (y_46_re / y_46_im))) / y_46_re;
	} else {
		tmp = ((y_46_re / (y_46_im / x_46_im)) - x_46_re) / y_46_im;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if y_46_im <= -1.26e-70:
		tmp = ((x_46_im * (y_46_re / y_46_im)) - x_46_re) / y_46_im
	elif y_46_im <= 3.5e-47:
		tmp = (x_46_im - (x_46_re / (y_46_re / y_46_im))) / y_46_re
	else:
		tmp = ((y_46_re / (y_46_im / x_46_im)) - x_46_re) / y_46_im
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (y_46_im <= -1.26e-70)
		tmp = Float64(Float64(Float64(x_46_im * Float64(y_46_re / y_46_im)) - x_46_re) / y_46_im);
	elseif (y_46_im <= 3.5e-47)
		tmp = Float64(Float64(x_46_im - Float64(x_46_re / Float64(y_46_re / y_46_im))) / y_46_re);
	else
		tmp = Float64(Float64(Float64(y_46_re / Float64(y_46_im / x_46_im)) - x_46_re) / y_46_im);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if (y_46_im <= -1.26e-70)
		tmp = ((x_46_im * (y_46_re / y_46_im)) - x_46_re) / y_46_im;
	elseif (y_46_im <= 3.5e-47)
		tmp = (x_46_im - (x_46_re / (y_46_re / y_46_im))) / y_46_re;
	else
		tmp = ((y_46_re / (y_46_im / x_46_im)) - x_46_re) / y_46_im;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[y$46$im, -1.26e-70], N[(N[(N[(x$46$im * N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision] - x$46$re), $MachinePrecision] / y$46$im), $MachinePrecision], If[LessEqual[y$46$im, 3.5e-47], N[(N[(x$46$im - N[(x$46$re / N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision], N[(N[(N[(y$46$re / N[(y$46$im / x$46$im), $MachinePrecision]), $MachinePrecision] - x$46$re), $MachinePrecision] / y$46$im), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y.im < -1.2600000000000001e-70

   1. Initial program 56.8%

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

   3. Taylor expanded in y.re around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. unpow2 N/A

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

      5. associate-/r* N/A

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

      6. div-sub N/A

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

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{x.im \cdot y.re}{y.im} - x.re\right), \color{blue}{y.im}\right) \]

      8. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(\frac{x.im \cdot y.re}{y.im}\right), x.re\right), y.im\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(\frac{y.re \cdot x.im}{y.im}\right), x.re\right), y.im\right) \]

      10. associate-/l* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(y.re \cdot \frac{x.im}{y.im}\right), x.re\right), y.im\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(y.re, \left(\frac{x.im}{y.im}\right)\right), x.re\right), y.im\right) \]

      12. /-lowering-/.f64 74.5%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{/.f64}\left(x.im, y.im\right)\right), x.re\right), y.im\right) \]

   5. Simplified 74.5%

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

      1. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(\frac{y.re \cdot x.im}{y.im}\right), x.re\right), y.im\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(\frac{x.im \cdot y.re}{y.im}\right), x.re\right), y.im\right) \]

      3. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(x.im \cdot \frac{y.re}{y.im}\right), x.re\right), y.im\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(x.im, \left(\frac{y.re}{y.im}\right)\right), x.re\right), y.im\right) \]

      5. /-lowering-/.f64 74.5%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(x.im, \mathsf{/.f64}\left(y.re, y.im\right)\right), x.re\right), y.im\right) \]

   7. Applied egg-rr 74.5%

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

   if -1.2600000000000001e-70 < y.im < 3.4999999999999998e-47

   1. Initial program 73.9%

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

      1. div-inv N/A

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

      2. *-commutative N/A

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

      3. flip3-+ N/A

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

      4. clear-num N/A

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

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\left(y.re \cdot y.re\right) \cdot \left(y.re \cdot y.re\right) + \left(\left(y.im \cdot y.im\right) \cdot \left(y.im \cdot y.im\right) - \left(y.re \cdot y.re\right) \cdot \left(y.im \cdot y.im\right)\right)}{{\left(y.re \cdot y.re\right)}^{3} + {\left(y.im \cdot y.im\right)}^{3}}\right), \color{blue}{\left(x.im \cdot y.re - x.re \cdot y.im\right)}\right) \]

   4. Applied egg-rr 73.9%

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

      1. associate-*l/ N/A

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

      2. clear-num N/A

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

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{y.re \cdot y.re + y.im \cdot y.im}{1 \cdot \left(x.im \cdot y.re - x.re \cdot y.im\right)}\right)}\right) \]

      4. *-lft-identity N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(\frac{y.re \cdot y.re + y.im \cdot y.im}{x.im \cdot y.re - \color{blue}{x.re \cdot y.im}}\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\left(y.re \cdot y.re + y.im \cdot y.im\right), \color{blue}{\left(x.im \cdot y.re - x.re \cdot y.im\right)}\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(y.re \cdot y.re\right), \left(y.im \cdot y.im\right)\right), \left(\color{blue}{x.im \cdot y.re} - x.re \cdot y.im\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(y.re, y.re\right), \left(y.im \cdot y.im\right)\right), \left(\color{blue}{x.im} \cdot y.re - x.re \cdot y.im\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(y.re, y.re\right), \mathsf{*.f64}\left(y.im, y.im\right)\right), \left(x.im \cdot \color{blue}{y.re} - x.re \cdot y.im\right)\right)\right) \]

      9. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(y.re, y.re\right), \mathsf{*.f64}\left(y.im, y.im\right)\right), \mathsf{\_.f64}\left(\left(x.im \cdot y.re\right), \color{blue}{\left(x.re \cdot y.im\right)}\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(y.re, y.re\right), \mathsf{*.f64}\left(y.im, y.im\right)\right), \mathsf{\_.f64}\left(\left(y.re \cdot x.im\right), \left(\color{blue}{x.re} \cdot y.im\right)\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(y.re, y.re\right), \mathsf{*.f64}\left(y.im, y.im\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(y.re, x.im\right), \left(\color{blue}{x.re} \cdot y.im\right)\right)\right)\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(y.re, y.re\right), \mathsf{*.f64}\left(y.im, y.im\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(y.re, x.im\right), \left(y.im \cdot \color{blue}{x.re}\right)\right)\right)\right) \]

      13. *-lowering-*.f64 74.0%

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(y.re, y.re\right), \mathsf{*.f64}\left(y.im, y.im\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(y.re, x.im\right), \mathsf{*.f64}\left(y.im, \color{blue}{x.re}\right)\right)\right)\right) \]

   6. Applied egg-rr 74.0%

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

   7. Taylor expanded in y.re around inf

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(x.im + -1 \cdot \frac{x.re \cdot y.im}{y.re}\right), \color{blue}{y.re}\right) \]

      2. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\left(x.im + \left(\mathsf{neg}\left(\frac{x.re \cdot y.im}{y.re}\right)\right)\right), y.re\right) \]

      3. unsub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\left(x.im - \frac{x.re \cdot y.im}{y.re}\right), y.re\right) \]

      4. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(x.im, \left(\frac{x.re \cdot y.im}{y.re}\right)\right), y.re\right) \]

      5. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(x.im, \left(x.re \cdot \frac{y.im}{y.re}\right)\right), y.re\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(x.im, \mathsf{*.f64}\left(x.re, \left(\frac{y.im}{y.re}\right)\right)\right), y.re\right) \]

      7. /-lowering-/.f64 92.0%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(x.im, \mathsf{*.f64}\left(x.re, \mathsf{/.f64}\left(y.im, y.re\right)\right)\right), y.re\right) \]

   9. Simplified 92.0%

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

       1. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(x.im - x.re \cdot \frac{y.im}{y.re}\right), \color{blue}{y.re}\right) \]

       2. --lowering--.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(x.im, \left(x.re \cdot \frac{y.im}{y.re}\right)\right), y.re\right) \]

       3. clear-num N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(x.im, \left(x.re \cdot \frac{1}{\frac{y.re}{y.im}}\right)\right), y.re\right) \]

       4. un-div-inv N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(x.im, \left(\frac{x.re}{\frac{y.re}{y.im}}\right)\right), y.re\right) \]

       5. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(x.im, \mathsf{/.f64}\left(x.re, \left(\frac{y.re}{y.im}\right)\right)\right), y.re\right) \]

       6. /-lowering-/.f64 92.1%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(x.im, \mathsf{/.f64}\left(x.re, \mathsf{/.f64}\left(y.re, y.im\right)\right)\right), y.re\right) \]

   11. Applied egg-rr 92.1%

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

   if 3.4999999999999998e-47 < y.im

   1. Initial program 58.2%

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

   3. Taylor expanded in y.re around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. unpow2 N/A

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

      5. associate-/r* N/A

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

      6. div-sub N/A

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

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{x.im \cdot y.re}{y.im} - x.re\right), \color{blue}{y.im}\right) \]

      8. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(\frac{x.im \cdot y.re}{y.im}\right), x.re\right), y.im\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(\frac{y.re \cdot x.im}{y.im}\right), x.re\right), y.im\right) \]

      10. associate-/l* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(y.re \cdot \frac{x.im}{y.im}\right), x.re\right), y.im\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(y.re, \left(\frac{x.im}{y.im}\right)\right), x.re\right), y.im\right) \]

      12. /-lowering-/.f64 72.7%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{/.f64}\left(x.im, y.im\right)\right), x.re\right), y.im\right) \]

   5. Simplified 72.7%

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

      1. clear-num N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(y.re \cdot \frac{1}{\frac{y.im}{x.im}}\right), x.re\right), y.im\right) \]

      2. un-div-inv N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(\frac{y.re}{\frac{y.im}{x.im}}\right), x.re\right), y.im\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(y.re, \left(\frac{y.im}{x.im}\right)\right), x.re\right), y.im\right) \]

      4. /-lowering-/.f64 74.1%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(y.re, \mathsf{/.f64}\left(y.im, x.im\right)\right), x.re\right), y.im\right) \]

   7. Applied egg-rr 74.1%

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

Alternative 6: 76.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x.im \cdot \frac{y.re}{y.im} - x.re}{y.im}\\ \mathbf{if}\;y.im \leq -1.2 \cdot 10^{-70}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.im \leq 2.95 \cdot 10^{-96}:\\ \;\;\;\;\frac{x.im - \frac{x.re}{\frac{y.re}{y.im}}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (/ (- (* x.im (/ y.re y.im)) x.re) y.im)))
   (if (<= y.im -1.2e-70)
     t_0
     (if (<= y.im 2.95e-96) (/ (- x.im (/ x.re (/ y.re y.im))) y.re) 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_im * (y_46_re / y_46_im)) - x_46_re) / y_46_im;
	double tmp;
	if (y_46_im <= -1.2e-70) {
		tmp = t_0;
	} else if (y_46_im <= 2.95e-96) {
		tmp = (x_46_im - (x_46_re / (y_46_re / y_46_im))) / y_46_re;
	} 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) :: tmp
    t_0 = ((x_46im * (y_46re / y_46im)) - x_46re) / y_46im
    if (y_46im <= (-1.2d-70)) then
        tmp = t_0
    else if (y_46im <= 2.95d-96) then
        tmp = (x_46im - (x_46re / (y_46re / y_46im))) / y_46re
    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_im * (y_46_re / y_46_im)) - x_46_re) / y_46_im;
	double tmp;
	if (y_46_im <= -1.2e-70) {
		tmp = t_0;
	} else if (y_46_im <= 2.95e-96) {
		tmp = (x_46_im - (x_46_re / (y_46_re / y_46_im))) / y_46_re;
	} 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_im * (y_46_re / y_46_im)) - x_46_re) / y_46_im
	tmp = 0
	if y_46_im <= -1.2e-70:
		tmp = t_0
	elif y_46_im <= 2.95e-96:
		tmp = (x_46_im - (x_46_re / (y_46_re / y_46_im))) / y_46_re
	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(Float64(x_46_im * Float64(y_46_re / y_46_im)) - x_46_re) / y_46_im)
	tmp = 0.0
	if (y_46_im <= -1.2e-70)
		tmp = t_0;
	elseif (y_46_im <= 2.95e-96)
		tmp = Float64(Float64(x_46_im - Float64(x_46_re / Float64(y_46_re / y_46_im))) / y_46_re);
	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_im * (y_46_re / y_46_im)) - x_46_re) / y_46_im;
	tmp = 0.0;
	if (y_46_im <= -1.2e-70)
		tmp = t_0;
	elseif (y_46_im <= 2.95e-96)
		tmp = (x_46_im - (x_46_re / (y_46_re / y_46_im))) / y_46_re;
	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[(N[(N[(x$46$im * N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision] - x$46$re), $MachinePrecision] / y$46$im), $MachinePrecision]}, If[LessEqual[y$46$im, -1.2e-70], t$95$0, If[LessEqual[y$46$im, 2.95e-96], N[(N[(x$46$im - N[(x$46$re / N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y.im < -1.2000000000000001e-70 or 2.94999999999999983e-96 < y.im

   1. Initial program 58.5%

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

   3. Taylor expanded in y.re around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. unpow2 N/A

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

      5. associate-/r* N/A

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

      6. div-sub N/A

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

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{x.im \cdot y.re}{y.im} - x.re\right), \color{blue}{y.im}\right) \]

      8. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(\frac{x.im \cdot y.re}{y.im}\right), x.re\right), y.im\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(\frac{y.re \cdot x.im}{y.im}\right), x.re\right), y.im\right) \]

      10. associate-/l* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(y.re \cdot \frac{x.im}{y.im}\right), x.re\right), y.im\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(y.re, \left(\frac{x.im}{y.im}\right)\right), x.re\right), y.im\right) \]

      12. /-lowering-/.f64 72.8%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{/.f64}\left(x.im, y.im\right)\right), x.re\right), y.im\right) \]

   5. Simplified 72.8%

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

      1. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(\frac{y.re \cdot x.im}{y.im}\right), x.re\right), y.im\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(\frac{x.im \cdot y.re}{y.im}\right), x.re\right), y.im\right) \]

      3. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(x.im \cdot \frac{y.re}{y.im}\right), x.re\right), y.im\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(x.im, \left(\frac{y.re}{y.im}\right)\right), x.re\right), y.im\right) \]

      5. /-lowering-/.f64 73.5%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(x.im, \mathsf{/.f64}\left(y.re, y.im\right)\right), x.re\right), y.im\right) \]

   7. Applied egg-rr 73.5%

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

   if -1.2000000000000001e-70 < y.im < 2.94999999999999983e-96

   1. Initial program 73.3%

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

      1. div-inv N/A

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

      2. *-commutative N/A

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

      3. flip3-+ N/A

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

      4. clear-num N/A

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

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\left(y.re \cdot y.re\right) \cdot \left(y.re \cdot y.re\right) + \left(\left(y.im \cdot y.im\right) \cdot \left(y.im \cdot y.im\right) - \left(y.re \cdot y.re\right) \cdot \left(y.im \cdot y.im\right)\right)}{{\left(y.re \cdot y.re\right)}^{3} + {\left(y.im \cdot y.im\right)}^{3}}\right), \color{blue}{\left(x.im \cdot y.re - x.re \cdot y.im\right)}\right) \]

   4. Applied egg-rr 73.3%

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

      1. associate-*l/ N/A

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

      2. clear-num N/A

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

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{y.re \cdot y.re + y.im \cdot y.im}{1 \cdot \left(x.im \cdot y.re - x.re \cdot y.im\right)}\right)}\right) \]

      4. *-lft-identity N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(\frac{y.re \cdot y.re + y.im \cdot y.im}{x.im \cdot y.re - \color{blue}{x.re \cdot y.im}}\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\left(y.re \cdot y.re + y.im \cdot y.im\right), \color{blue}{\left(x.im \cdot y.re - x.re \cdot y.im\right)}\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(y.re \cdot y.re\right), \left(y.im \cdot y.im\right)\right), \left(\color{blue}{x.im \cdot y.re} - x.re \cdot y.im\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(y.re, y.re\right), \left(y.im \cdot y.im\right)\right), \left(\color{blue}{x.im} \cdot y.re - x.re \cdot y.im\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(y.re, y.re\right), \mathsf{*.f64}\left(y.im, y.im\right)\right), \left(x.im \cdot \color{blue}{y.re} - x.re \cdot y.im\right)\right)\right) \]

      9. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(y.re, y.re\right), \mathsf{*.f64}\left(y.im, y.im\right)\right), \mathsf{\_.f64}\left(\left(x.im \cdot y.re\right), \color{blue}{\left(x.re \cdot y.im\right)}\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(y.re, y.re\right), \mathsf{*.f64}\left(y.im, y.im\right)\right), \mathsf{\_.f64}\left(\left(y.re \cdot x.im\right), \left(\color{blue}{x.re} \cdot y.im\right)\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(y.re, y.re\right), \mathsf{*.f64}\left(y.im, y.im\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(y.re, x.im\right), \left(\color{blue}{x.re} \cdot y.im\right)\right)\right)\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(y.re, y.re\right), \mathsf{*.f64}\left(y.im, y.im\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(y.re, x.im\right), \left(y.im \cdot \color{blue}{x.re}\right)\right)\right)\right) \]

      13. *-lowering-*.f64 73.4%

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(y.re, y.re\right), \mathsf{*.f64}\left(y.im, y.im\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(y.re, x.im\right), \mathsf{*.f64}\left(y.im, \color{blue}{x.re}\right)\right)\right)\right) \]

   6. Applied egg-rr 73.4%

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

   7. Taylor expanded in y.re around inf

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(x.im + -1 \cdot \frac{x.re \cdot y.im}{y.re}\right), \color{blue}{y.re}\right) \]

      2. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\left(x.im + \left(\mathsf{neg}\left(\frac{x.re \cdot y.im}{y.re}\right)\right)\right), y.re\right) \]

      3. unsub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\left(x.im - \frac{x.re \cdot y.im}{y.re}\right), y.re\right) \]

      4. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(x.im, \left(\frac{x.re \cdot y.im}{y.re}\right)\right), y.re\right) \]

      5. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(x.im, \left(x.re \cdot \frac{y.im}{y.re}\right)\right), y.re\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(x.im, \mathsf{*.f64}\left(x.re, \left(\frac{y.im}{y.re}\right)\right)\right), y.re\right) \]

      7. /-lowering-/.f64 94.2%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(x.im, \mathsf{*.f64}\left(x.re, \mathsf{/.f64}\left(y.im, y.re\right)\right)\right), y.re\right) \]

   9. Simplified 94.2%

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

       1. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(x.im - x.re \cdot \frac{y.im}{y.re}\right), \color{blue}{y.re}\right) \]

       2. --lowering--.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(x.im, \left(x.re \cdot \frac{y.im}{y.re}\right)\right), y.re\right) \]

       3. clear-num N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(x.im, \left(x.re \cdot \frac{1}{\frac{y.re}{y.im}}\right)\right), y.re\right) \]

       4. un-div-inv N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(x.im, \left(\frac{x.re}{\frac{y.re}{y.im}}\right)\right), y.re\right) \]

       5. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(x.im, \mathsf{/.f64}\left(x.re, \left(\frac{y.re}{y.im}\right)\right)\right), y.re\right) \]

       6. /-lowering-/.f64 94.3%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(x.im, \mathsf{/.f64}\left(x.re, \mathsf{/.f64}\left(y.re, y.im\right)\right)\right), y.re\right) \]

   11. Applied egg-rr 94.3%

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

Alternative 7: 71.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -1.35 \cdot 10^{-70}:\\ \;\;\;\;\frac{-1}{\frac{y.im}{x.re}}\\ \mathbf{elif}\;y.im \leq 1.1 \cdot 10^{+81}:\\ \;\;\;\;\frac{x.im - \frac{x.re}{\frac{y.re}{y.im}}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{0 - y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.im -1.35e-70)
   (/ -1.0 (/ y.im x.re))
   (if (<= y.im 1.1e+81)
     (/ (- x.im (/ x.re (/ y.re y.im))) y.re)
     (/ x.re (- 0.0 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 <= -1.35e-70) {
		tmp = -1.0 / (y_46_im / x_46_re);
	} else if (y_46_im <= 1.1e+81) {
		tmp = (x_46_im - (x_46_re / (y_46_re / y_46_im))) / y_46_re;
	} else {
		tmp = x_46_re / (0.0 - 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 <= (-1.35d-70)) then
        tmp = (-1.0d0) / (y_46im / x_46re)
    else if (y_46im <= 1.1d+81) then
        tmp = (x_46im - (x_46re / (y_46re / y_46im))) / y_46re
    else
        tmp = x_46re / (0.0d0 - 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 <= -1.35e-70) {
		tmp = -1.0 / (y_46_im / x_46_re);
	} else if (y_46_im <= 1.1e+81) {
		tmp = (x_46_im - (x_46_re / (y_46_re / y_46_im))) / y_46_re;
	} else {
		tmp = x_46_re / (0.0 - 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 <= -1.35e-70:
		tmp = -1.0 / (y_46_im / x_46_re)
	elif y_46_im <= 1.1e+81:
		tmp = (x_46_im - (x_46_re / (y_46_re / y_46_im))) / y_46_re
	else:
		tmp = x_46_re / (0.0 - 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 <= -1.35e-70)
		tmp = Float64(-1.0 / Float64(y_46_im / x_46_re));
	elseif (y_46_im <= 1.1e+81)
		tmp = Float64(Float64(x_46_im - Float64(x_46_re / Float64(y_46_re / y_46_im))) / y_46_re);
	else
		tmp = Float64(x_46_re / Float64(0.0 - 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 <= -1.35e-70)
		tmp = -1.0 / (y_46_im / x_46_re);
	elseif (y_46_im <= 1.1e+81)
		tmp = (x_46_im - (x_46_re / (y_46_re / y_46_im))) / y_46_re;
	else
		tmp = x_46_re / (0.0 - y_46_im);
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[y$46$im, -1.35e-70], N[(-1.0 / N[(y$46$im / x$46$re), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, 1.1e+81], N[(N[(x$46$im - N[(x$46$re / N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision], N[(x$46$re / N[(0.0 - y$46$im), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y.im < -1.3500000000000001e-70

   1. Initial program 56.8%

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

      1. div-inv N/A

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

      2. *-commutative N/A

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

      3. flip3-+ N/A

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

      4. clear-num N/A

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

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\left(y.re \cdot y.re\right) \cdot \left(y.re \cdot y.re\right) + \left(\left(y.im \cdot y.im\right) \cdot \left(y.im \cdot y.im\right) - \left(y.re \cdot y.re\right) \cdot \left(y.im \cdot y.im\right)\right)}{{\left(y.re \cdot y.re\right)}^{3} + {\left(y.im \cdot y.im\right)}^{3}}\right), \color{blue}{\left(x.im \cdot y.re - x.re \cdot y.im\right)}\right) \]

   4. Applied egg-rr 56.9%

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

      1. associate-*l/ N/A

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

      2. clear-num N/A

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

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{y.re \cdot y.re + y.im \cdot y.im}{1 \cdot \left(x.im \cdot y.re - x.re \cdot y.im\right)}\right)}\right) \]

      4. *-lft-identity N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(\frac{y.re \cdot y.re + y.im \cdot y.im}{x.im \cdot y.re - \color{blue}{x.re \cdot y.im}}\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\left(y.re \cdot y.re + y.im \cdot y.im\right), \color{blue}{\left(x.im \cdot y.re - x.re \cdot y.im\right)}\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(y.re \cdot y.re\right), \left(y.im \cdot y.im\right)\right), \left(\color{blue}{x.im \cdot y.re} - x.re \cdot y.im\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(y.re, y.re\right), \left(y.im \cdot y.im\right)\right), \left(\color{blue}{x.im} \cdot y.re - x.re \cdot y.im\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(y.re, y.re\right), \mathsf{*.f64}\left(y.im, y.im\right)\right), \left(x.im \cdot \color{blue}{y.re} - x.re \cdot y.im\right)\right)\right) \]

      9. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(y.re, y.re\right), \mathsf{*.f64}\left(y.im, y.im\right)\right), \mathsf{\_.f64}\left(\left(x.im \cdot y.re\right), \color{blue}{\left(x.re \cdot y.im\right)}\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(y.re, y.re\right), \mathsf{*.f64}\left(y.im, y.im\right)\right), \mathsf{\_.f64}\left(\left(y.re \cdot x.im\right), \left(\color{blue}{x.re} \cdot y.im\right)\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(y.re, y.re\right), \mathsf{*.f64}\left(y.im, y.im\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(y.re, x.im\right), \left(\color{blue}{x.re} \cdot y.im\right)\right)\right)\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(y.re, y.re\right), \mathsf{*.f64}\left(y.im, y.im\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(y.re, x.im\right), \left(y.im \cdot \color{blue}{x.re}\right)\right)\right)\right) \]

      13. *-lowering-*.f64 56.8%

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(y.re, y.re\right), \mathsf{*.f64}\left(y.im, y.im\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(y.re, x.im\right), \mathsf{*.f64}\left(y.im, \color{blue}{x.re}\right)\right)\right)\right) \]

   6. Applied egg-rr 56.8%

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

   7. Taylor expanded in y.re around 0

      \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(-1 \cdot \frac{y.im}{x.re}\right)}\right) \]
   8. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(\mathsf{neg}\left(\frac{y.im}{x.re}\right)\right)\right) \]

      2. neg-sub0 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(0 - \color{blue}{\frac{y.im}{x.re}}\right)\right) \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(0, \color{blue}{\left(\frac{y.im}{x.re}\right)}\right)\right) \]

      4. /-lowering-/.f64 65.5%

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(y.im, \color{blue}{x.re}\right)\right)\right) \]

   9. Simplified 65.5%

      \[\leadsto \frac{1}{\color{blue}{0 - \frac{y.im}{x.re}}} \]

   if -1.3500000000000001e-70 < y.im < 1.09999999999999993e81

   1. Initial program 74.1%

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

      1. div-inv N/A

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

      2. *-commutative N/A

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

      3. flip3-+ N/A

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

      4. clear-num N/A

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

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\left(y.re \cdot y.re\right) \cdot \left(y.re \cdot y.re\right) + \left(\left(y.im \cdot y.im\right) \cdot \left(y.im \cdot y.im\right) - \left(y.re \cdot y.re\right) \cdot \left(y.im \cdot y.im\right)\right)}{{\left(y.re \cdot y.re\right)}^{3} + {\left(y.im \cdot y.im\right)}^{3}}\right), \color{blue}{\left(x.im \cdot y.re - x.re \cdot y.im\right)}\right) \]

   4. Applied egg-rr 74.1%

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

      1. associate-*l/ N/A

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

      2. clear-num N/A

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

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{y.re \cdot y.re + y.im \cdot y.im}{1 \cdot \left(x.im \cdot y.re - x.re \cdot y.im\right)}\right)}\right) \]

      4. *-lft-identity N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(\frac{y.re \cdot y.re + y.im \cdot y.im}{x.im \cdot y.re - \color{blue}{x.re \cdot y.im}}\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\left(y.re \cdot y.re + y.im \cdot y.im\right), \color{blue}{\left(x.im \cdot y.re - x.re \cdot y.im\right)}\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(y.re \cdot y.re\right), \left(y.im \cdot y.im\right)\right), \left(\color{blue}{x.im \cdot y.re} - x.re \cdot y.im\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(y.re, y.re\right), \left(y.im \cdot y.im\right)\right), \left(\color{blue}{x.im} \cdot y.re - x.re \cdot y.im\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(y.re, y.re\right), \mathsf{*.f64}\left(y.im, y.im\right)\right), \left(x.im \cdot \color{blue}{y.re} - x.re \cdot y.im\right)\right)\right) \]

      9. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(y.re, y.re\right), \mathsf{*.f64}\left(y.im, y.im\right)\right), \mathsf{\_.f64}\left(\left(x.im \cdot y.re\right), \color{blue}{\left(x.re \cdot y.im\right)}\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(y.re, y.re\right), \mathsf{*.f64}\left(y.im, y.im\right)\right), \mathsf{\_.f64}\left(\left(y.re \cdot x.im\right), \left(\color{blue}{x.re} \cdot y.im\right)\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(y.re, y.re\right), \mathsf{*.f64}\left(y.im, y.im\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(y.re, x.im\right), \left(\color{blue}{x.re} \cdot y.im\right)\right)\right)\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(y.re, y.re\right), \mathsf{*.f64}\left(y.im, y.im\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(y.re, x.im\right), \left(y.im \cdot \color{blue}{x.re}\right)\right)\right)\right) \]

      13. *-lowering-*.f64 74.1%

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(y.re, y.re\right), \mathsf{*.f64}\left(y.im, y.im\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(y.re, x.im\right), \mathsf{*.f64}\left(y.im, \color{blue}{x.re}\right)\right)\right)\right) \]

   6. Applied egg-rr 74.1%

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

   7. Taylor expanded in y.re around inf

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(x.im + -1 \cdot \frac{x.re \cdot y.im}{y.re}\right), \color{blue}{y.re}\right) \]

      2. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\left(x.im + \left(\mathsf{neg}\left(\frac{x.re \cdot y.im}{y.re}\right)\right)\right), y.re\right) \]

      3. unsub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\left(x.im - \frac{x.re \cdot y.im}{y.re}\right), y.re\right) \]

      4. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(x.im, \left(\frac{x.re \cdot y.im}{y.re}\right)\right), y.re\right) \]

      5. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(x.im, \left(x.re \cdot \frac{y.im}{y.re}\right)\right), y.re\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(x.im, \mathsf{*.f64}\left(x.re, \left(\frac{y.im}{y.re}\right)\right)\right), y.re\right) \]

      7. /-lowering-/.f64 82.2%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(x.im, \mathsf{*.f64}\left(x.re, \mathsf{/.f64}\left(y.im, y.re\right)\right)\right), y.re\right) \]

   9. Simplified 82.2%

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

       1. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(x.im - x.re \cdot \frac{y.im}{y.re}\right), \color{blue}{y.re}\right) \]

       2. --lowering--.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(x.im, \left(x.re \cdot \frac{y.im}{y.re}\right)\right), y.re\right) \]

       3. clear-num N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(x.im, \left(x.re \cdot \frac{1}{\frac{y.re}{y.im}}\right)\right), y.re\right) \]

       4. un-div-inv N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(x.im, \left(\frac{x.re}{\frac{y.re}{y.im}}\right)\right), y.re\right) \]

       5. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(x.im, \mathsf{/.f64}\left(x.re, \left(\frac{y.re}{y.im}\right)\right)\right), y.re\right) \]

       6. /-lowering-/.f64 82.2%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(x.im, \mathsf{/.f64}\left(x.re, \mathsf{/.f64}\left(y.re, y.im\right)\right)\right), y.re\right) \]

   11. Applied egg-rr 82.2%

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

   if 1.09999999999999993e81 < y.im

   1. Initial program 43.8%

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

   3. Taylor expanded in y.re around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. unpow2 N/A

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

      5. associate-/r* N/A

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

      6. div-sub N/A

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

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{x.im \cdot y.re}{y.im} - x.re\right), \color{blue}{y.im}\right) \]

      8. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(\frac{x.im \cdot y.re}{y.im}\right), x.re\right), y.im\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(\frac{y.re \cdot x.im}{y.im}\right), x.re\right), y.im\right) \]

      10. associate-/l* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(y.re \cdot \frac{x.im}{y.im}\right), x.re\right), y.im\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(y.re, \left(\frac{x.im}{y.im}\right)\right), x.re\right), y.im\right) \]

      12. /-lowering-/.f64 89.0%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{/.f64}\left(x.im, y.im\right)\right), x.re\right), y.im\right) \]

   5. Simplified 89.0%

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

   6. Taylor expanded in y.re around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(-1 \cdot x.re\right)}, y.im\right) \]
   7. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\mathsf{neg}\left(x.re\right)\right), y.im\right) \]

      2. neg-sub0 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(0 - x.re\right), y.im\right) \]

      3. --lowering--.f64 76.4%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, x.re\right), y.im\right) \]

   8. Simplified 76.4%

      \[\leadsto \frac{\color{blue}{0 - x.re}}{y.im} \]
   9. Step-by-step derivation

      1. sub0-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\mathsf{neg}\left(x.re\right)\right), y.im\right) \]

      2. neg-lowering-neg.f64 76.4%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{neg.f64}\left(x.re\right), y.im\right) \]

   10. Applied egg-rr 76.4%

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

4. Final simplification 76.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -1.35 \cdot 10^{-70}:\\ \;\;\;\;\frac{-1}{\frac{y.im}{x.re}}\\ \mathbf{elif}\;y.im \leq 1.1 \cdot 10^{+81}:\\ \;\;\;\;\frac{x.im - \frac{x.re}{\frac{y.re}{y.im}}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{0 - y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 71.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -1.26 \cdot 10^{-70}:\\ \;\;\;\;\frac{-1}{\frac{y.im}{x.re}}\\ \mathbf{elif}\;y.im \leq 3.3 \cdot 10^{+79}:\\ \;\;\;\;\frac{x.im - x.re \cdot \frac{y.im}{y.re}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{0 - y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.im -1.26e-70)
   (/ -1.0 (/ y.im x.re))
   (if (<= y.im 3.3e+79)
     (/ (- x.im (* x.re (/ y.im y.re))) y.re)
     (/ x.re (- 0.0 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 <= -1.26e-70) {
		tmp = -1.0 / (y_46_im / x_46_re);
	} else if (y_46_im <= 3.3e+79) {
		tmp = (x_46_im - (x_46_re * (y_46_im / y_46_re))) / y_46_re;
	} else {
		tmp = x_46_re / (0.0 - 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 <= (-1.26d-70)) then
        tmp = (-1.0d0) / (y_46im / x_46re)
    else if (y_46im <= 3.3d+79) then
        tmp = (x_46im - (x_46re * (y_46im / y_46re))) / y_46re
    else
        tmp = x_46re / (0.0d0 - 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 <= -1.26e-70) {
		tmp = -1.0 / (y_46_im / x_46_re);
	} else if (y_46_im <= 3.3e+79) {
		tmp = (x_46_im - (x_46_re * (y_46_im / y_46_re))) / y_46_re;
	} else {
		tmp = x_46_re / (0.0 - 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 <= -1.26e-70:
		tmp = -1.0 / (y_46_im / x_46_re)
	elif y_46_im <= 3.3e+79:
		tmp = (x_46_im - (x_46_re * (y_46_im / y_46_re))) / y_46_re
	else:
		tmp = x_46_re / (0.0 - 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 <= -1.26e-70)
		tmp = Float64(-1.0 / Float64(y_46_im / x_46_re));
	elseif (y_46_im <= 3.3e+79)
		tmp = Float64(Float64(x_46_im - Float64(x_46_re * Float64(y_46_im / y_46_re))) / y_46_re);
	else
		tmp = Float64(x_46_re / Float64(0.0 - 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 <= -1.26e-70)
		tmp = -1.0 / (y_46_im / x_46_re);
	elseif (y_46_im <= 3.3e+79)
		tmp = (x_46_im - (x_46_re * (y_46_im / y_46_re))) / y_46_re;
	else
		tmp = x_46_re / (0.0 - y_46_im);
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[y$46$im, -1.26e-70], N[(-1.0 / N[(y$46$im / x$46$re), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, 3.3e+79], N[(N[(x$46$im - N[(x$46$re * N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision], N[(x$46$re / N[(0.0 - y$46$im), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y.im < -1.2600000000000001e-70

   1. Initial program 56.8%

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

      1. div-inv N/A

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

      2. *-commutative N/A

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

      3. flip3-+ N/A

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

      4. clear-num N/A

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

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\left(y.re \cdot y.re\right) \cdot \left(y.re \cdot y.re\right) + \left(\left(y.im \cdot y.im\right) \cdot \left(y.im \cdot y.im\right) - \left(y.re \cdot y.re\right) \cdot \left(y.im \cdot y.im\right)\right)}{{\left(y.re \cdot y.re\right)}^{3} + {\left(y.im \cdot y.im\right)}^{3}}\right), \color{blue}{\left(x.im \cdot y.re - x.re \cdot y.im\right)}\right) \]

   4. Applied egg-rr 56.9%

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

      1. associate-*l/ N/A

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

      2. clear-num N/A

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

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{y.re \cdot y.re + y.im \cdot y.im}{1 \cdot \left(x.im \cdot y.re - x.re \cdot y.im\right)}\right)}\right) \]

      4. *-lft-identity N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(\frac{y.re \cdot y.re + y.im \cdot y.im}{x.im \cdot y.re - \color{blue}{x.re \cdot y.im}}\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\left(y.re \cdot y.re + y.im \cdot y.im\right), \color{blue}{\left(x.im \cdot y.re - x.re \cdot y.im\right)}\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(y.re \cdot y.re\right), \left(y.im \cdot y.im\right)\right), \left(\color{blue}{x.im \cdot y.re} - x.re \cdot y.im\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(y.re, y.re\right), \left(y.im \cdot y.im\right)\right), \left(\color{blue}{x.im} \cdot y.re - x.re \cdot y.im\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(y.re, y.re\right), \mathsf{*.f64}\left(y.im, y.im\right)\right), \left(x.im \cdot \color{blue}{y.re} - x.re \cdot y.im\right)\right)\right) \]

      9. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(y.re, y.re\right), \mathsf{*.f64}\left(y.im, y.im\right)\right), \mathsf{\_.f64}\left(\left(x.im \cdot y.re\right), \color{blue}{\left(x.re \cdot y.im\right)}\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(y.re, y.re\right), \mathsf{*.f64}\left(y.im, y.im\right)\right), \mathsf{\_.f64}\left(\left(y.re \cdot x.im\right), \left(\color{blue}{x.re} \cdot y.im\right)\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(y.re, y.re\right), \mathsf{*.f64}\left(y.im, y.im\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(y.re, x.im\right), \left(\color{blue}{x.re} \cdot y.im\right)\right)\right)\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(y.re, y.re\right), \mathsf{*.f64}\left(y.im, y.im\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(y.re, x.im\right), \left(y.im \cdot \color{blue}{x.re}\right)\right)\right)\right) \]

      13. *-lowering-*.f64 56.8%

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(y.re, y.re\right), \mathsf{*.f64}\left(y.im, y.im\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(y.re, x.im\right), \mathsf{*.f64}\left(y.im, \color{blue}{x.re}\right)\right)\right)\right) \]

   6. Applied egg-rr 56.8%

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

   7. Taylor expanded in y.re around 0

      \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(-1 \cdot \frac{y.im}{x.re}\right)}\right) \]
   8. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(\mathsf{neg}\left(\frac{y.im}{x.re}\right)\right)\right) \]

      2. neg-sub0 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(0 - \color{blue}{\frac{y.im}{x.re}}\right)\right) \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(0, \color{blue}{\left(\frac{y.im}{x.re}\right)}\right)\right) \]

      4. /-lowering-/.f64 65.5%

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(y.im, \color{blue}{x.re}\right)\right)\right) \]

   9. Simplified 65.5%

      \[\leadsto \frac{1}{\color{blue}{0 - \frac{y.im}{x.re}}} \]

   if -1.2600000000000001e-70 < y.im < 3.3000000000000002e79

   1. Initial program 74.1%

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

      1. div-inv N/A

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

      2. *-commutative N/A

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

      3. flip3-+ N/A

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

      4. clear-num N/A

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

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\left(y.re \cdot y.re\right) \cdot \left(y.re \cdot y.re\right) + \left(\left(y.im \cdot y.im\right) \cdot \left(y.im \cdot y.im\right) - \left(y.re \cdot y.re\right) \cdot \left(y.im \cdot y.im\right)\right)}{{\left(y.re \cdot y.re\right)}^{3} + {\left(y.im \cdot y.im\right)}^{3}}\right), \color{blue}{\left(x.im \cdot y.re - x.re \cdot y.im\right)}\right) \]

   4. Applied egg-rr 74.1%

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

      1. associate-*l/ N/A

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

      2. clear-num N/A

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

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{y.re \cdot y.re + y.im \cdot y.im}{1 \cdot \left(x.im \cdot y.re - x.re \cdot y.im\right)}\right)}\right) \]

      4. *-lft-identity N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(\frac{y.re \cdot y.re + y.im \cdot y.im}{x.im \cdot y.re - \color{blue}{x.re \cdot y.im}}\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\left(y.re \cdot y.re + y.im \cdot y.im\right), \color{blue}{\left(x.im \cdot y.re - x.re \cdot y.im\right)}\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(y.re \cdot y.re\right), \left(y.im \cdot y.im\right)\right), \left(\color{blue}{x.im \cdot y.re} - x.re \cdot y.im\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(y.re, y.re\right), \left(y.im \cdot y.im\right)\right), \left(\color{blue}{x.im} \cdot y.re - x.re \cdot y.im\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(y.re, y.re\right), \mathsf{*.f64}\left(y.im, y.im\right)\right), \left(x.im \cdot \color{blue}{y.re} - x.re \cdot y.im\right)\right)\right) \]

      9. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(y.re, y.re\right), \mathsf{*.f64}\left(y.im, y.im\right)\right), \mathsf{\_.f64}\left(\left(x.im \cdot y.re\right), \color{blue}{\left(x.re \cdot y.im\right)}\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(y.re, y.re\right), \mathsf{*.f64}\left(y.im, y.im\right)\right), \mathsf{\_.f64}\left(\left(y.re \cdot x.im\right), \left(\color{blue}{x.re} \cdot y.im\right)\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(y.re, y.re\right), \mathsf{*.f64}\left(y.im, y.im\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(y.re, x.im\right), \left(\color{blue}{x.re} \cdot y.im\right)\right)\right)\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(y.re, y.re\right), \mathsf{*.f64}\left(y.im, y.im\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(y.re, x.im\right), \left(y.im \cdot \color{blue}{x.re}\right)\right)\right)\right) \]

      13. *-lowering-*.f64 74.1%

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(y.re, y.re\right), \mathsf{*.f64}\left(y.im, y.im\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(y.re, x.im\right), \mathsf{*.f64}\left(y.im, \color{blue}{x.re}\right)\right)\right)\right) \]

   6. Applied egg-rr 74.1%

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

   7. Taylor expanded in y.re around inf

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(x.im + -1 \cdot \frac{x.re \cdot y.im}{y.re}\right), \color{blue}{y.re}\right) \]

      2. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\left(x.im + \left(\mathsf{neg}\left(\frac{x.re \cdot y.im}{y.re}\right)\right)\right), y.re\right) \]

      3. unsub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\left(x.im - \frac{x.re \cdot y.im}{y.re}\right), y.re\right) \]

      4. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(x.im, \left(\frac{x.re \cdot y.im}{y.re}\right)\right), y.re\right) \]

      5. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(x.im, \left(x.re \cdot \frac{y.im}{y.re}\right)\right), y.re\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(x.im, \mathsf{*.f64}\left(x.re, \left(\frac{y.im}{y.re}\right)\right)\right), y.re\right) \]

      7. /-lowering-/.f64 82.2%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(x.im, \mathsf{*.f64}\left(x.re, \mathsf{/.f64}\left(y.im, y.re\right)\right)\right), y.re\right) \]

   9. Simplified 82.2%

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

   if 3.3000000000000002e79 < y.im

   1. Initial program 43.8%

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

   3. Taylor expanded in y.re around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. unpow2 N/A

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

      5. associate-/r* N/A

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

      6. div-sub N/A

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

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{x.im \cdot y.re}{y.im} - x.re\right), \color{blue}{y.im}\right) \]

      8. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(\frac{x.im \cdot y.re}{y.im}\right), x.re\right), y.im\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(\frac{y.re \cdot x.im}{y.im}\right), x.re\right), y.im\right) \]

      10. associate-/l* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(y.re \cdot \frac{x.im}{y.im}\right), x.re\right), y.im\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(y.re, \left(\frac{x.im}{y.im}\right)\right), x.re\right), y.im\right) \]

      12. /-lowering-/.f64 89.0%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{/.f64}\left(x.im, y.im\right)\right), x.re\right), y.im\right) \]

   5. Simplified 89.0%

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

   6. Taylor expanded in y.re around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(-1 \cdot x.re\right)}, y.im\right) \]
   7. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\mathsf{neg}\left(x.re\right)\right), y.im\right) \]

      2. neg-sub0 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(0 - x.re\right), y.im\right) \]

      3. --lowering--.f64 76.4%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, x.re\right), y.im\right) \]

   8. Simplified 76.4%

      \[\leadsto \frac{\color{blue}{0 - x.re}}{y.im} \]
   9. Step-by-step derivation

      1. sub0-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\mathsf{neg}\left(x.re\right)\right), y.im\right) \]

      2. neg-lowering-neg.f64 76.4%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{neg.f64}\left(x.re\right), y.im\right) \]

   10. Applied egg-rr 76.4%

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

4. Final simplification 76.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -1.26 \cdot 10^{-70}:\\ \;\;\;\;\frac{-1}{\frac{y.im}{x.re}}\\ \mathbf{elif}\;y.im \leq 3.3 \cdot 10^{+79}:\\ \;\;\;\;\frac{x.im - x.re \cdot \frac{y.im}{y.re}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{0 - y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 62.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -1.35 \cdot 10^{-70}:\\ \;\;\;\;\frac{-1}{\frac{y.im}{x.re}}\\ \mathbf{elif}\;y.im \leq 4.5 \cdot 10^{+79}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{0 - y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.im -1.35e-70)
   (/ -1.0 (/ y.im x.re))
   (if (<= y.im 4.5e+79) (/ x.im y.re) (/ x.re (- 0.0 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 <= -1.35e-70) {
		tmp = -1.0 / (y_46_im / x_46_re);
	} else if (y_46_im <= 4.5e+79) {
		tmp = x_46_im / y_46_re;
	} else {
		tmp = x_46_re / (0.0 - 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 <= (-1.35d-70)) then
        tmp = (-1.0d0) / (y_46im / x_46re)
    else if (y_46im <= 4.5d+79) then
        tmp = x_46im / y_46re
    else
        tmp = x_46re / (0.0d0 - 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 <= -1.35e-70) {
		tmp = -1.0 / (y_46_im / x_46_re);
	} else if (y_46_im <= 4.5e+79) {
		tmp = x_46_im / y_46_re;
	} else {
		tmp = x_46_re / (0.0 - 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 <= -1.35e-70:
		tmp = -1.0 / (y_46_im / x_46_re)
	elif y_46_im <= 4.5e+79:
		tmp = x_46_im / y_46_re
	else:
		tmp = x_46_re / (0.0 - 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 <= -1.35e-70)
		tmp = Float64(-1.0 / Float64(y_46_im / x_46_re));
	elseif (y_46_im <= 4.5e+79)
		tmp = Float64(x_46_im / y_46_re);
	else
		tmp = Float64(x_46_re / Float64(0.0 - 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 <= -1.35e-70)
		tmp = -1.0 / (y_46_im / x_46_re);
	elseif (y_46_im <= 4.5e+79)
		tmp = x_46_im / y_46_re;
	else
		tmp = x_46_re / (0.0 - y_46_im);
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[y$46$im, -1.35e-70], N[(-1.0 / N[(y$46$im / x$46$re), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, 4.5e+79], N[(x$46$im / y$46$re), $MachinePrecision], N[(x$46$re / N[(0.0 - y$46$im), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y.im < -1.3500000000000001e-70

   1. Initial program 56.8%

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

      1. div-inv N/A

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

      2. *-commutative N/A

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

      3. flip3-+ N/A

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

      4. clear-num N/A

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

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\left(y.re \cdot y.re\right) \cdot \left(y.re \cdot y.re\right) + \left(\left(y.im \cdot y.im\right) \cdot \left(y.im \cdot y.im\right) - \left(y.re \cdot y.re\right) \cdot \left(y.im \cdot y.im\right)\right)}{{\left(y.re \cdot y.re\right)}^{3} + {\left(y.im \cdot y.im\right)}^{3}}\right), \color{blue}{\left(x.im \cdot y.re - x.re \cdot y.im\right)}\right) \]

   4. Applied egg-rr 56.9%

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

      1. associate-*l/ N/A

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

      2. clear-num N/A

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

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{y.re \cdot y.re + y.im \cdot y.im}{1 \cdot \left(x.im \cdot y.re - x.re \cdot y.im\right)}\right)}\right) \]

      4. *-lft-identity N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(\frac{y.re \cdot y.re + y.im \cdot y.im}{x.im \cdot y.re - \color{blue}{x.re \cdot y.im}}\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\left(y.re \cdot y.re + y.im \cdot y.im\right), \color{blue}{\left(x.im \cdot y.re - x.re \cdot y.im\right)}\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(y.re \cdot y.re\right), \left(y.im \cdot y.im\right)\right), \left(\color{blue}{x.im \cdot y.re} - x.re \cdot y.im\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(y.re, y.re\right), \left(y.im \cdot y.im\right)\right), \left(\color{blue}{x.im} \cdot y.re - x.re \cdot y.im\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(y.re, y.re\right), \mathsf{*.f64}\left(y.im, y.im\right)\right), \left(x.im \cdot \color{blue}{y.re} - x.re \cdot y.im\right)\right)\right) \]

      9. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(y.re, y.re\right), \mathsf{*.f64}\left(y.im, y.im\right)\right), \mathsf{\_.f64}\left(\left(x.im \cdot y.re\right), \color{blue}{\left(x.re \cdot y.im\right)}\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(y.re, y.re\right), \mathsf{*.f64}\left(y.im, y.im\right)\right), \mathsf{\_.f64}\left(\left(y.re \cdot x.im\right), \left(\color{blue}{x.re} \cdot y.im\right)\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(y.re, y.re\right), \mathsf{*.f64}\left(y.im, y.im\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(y.re, x.im\right), \left(\color{blue}{x.re} \cdot y.im\right)\right)\right)\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(y.re, y.re\right), \mathsf{*.f64}\left(y.im, y.im\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(y.re, x.im\right), \left(y.im \cdot \color{blue}{x.re}\right)\right)\right)\right) \]

      13. *-lowering-*.f64 56.8%

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(y.re, y.re\right), \mathsf{*.f64}\left(y.im, y.im\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(y.re, x.im\right), \mathsf{*.f64}\left(y.im, \color{blue}{x.re}\right)\right)\right)\right) \]

   6. Applied egg-rr 56.8%

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

   7. Taylor expanded in y.re around 0

      \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(-1 \cdot \frac{y.im}{x.re}\right)}\right) \]
   8. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(\mathsf{neg}\left(\frac{y.im}{x.re}\right)\right)\right) \]

      2. neg-sub0 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(0 - \color{blue}{\frac{y.im}{x.re}}\right)\right) \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(0, \color{blue}{\left(\frac{y.im}{x.re}\right)}\right)\right) \]

      4. /-lowering-/.f64 65.5%

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(y.im, \color{blue}{x.re}\right)\right)\right) \]

   9. Simplified 65.5%

      \[\leadsto \frac{1}{\color{blue}{0 - \frac{y.im}{x.re}}} \]

   if -1.3500000000000001e-70 < y.im < 4.49999999999999994e79

   1. Initial program 74.1%

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

   3. Taylor expanded in y.re around inf

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

      1. /-lowering-/.f64 65.8%

         \[\leadsto \mathsf{/.f64}\left(x.im, \color{blue}{y.re}\right) \]

   5. Simplified 65.8%

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

   if 4.49999999999999994e79 < y.im

   1. Initial program 43.8%

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

   3. Taylor expanded in y.re around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. unpow2 N/A

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

      5. associate-/r* N/A

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

      6. div-sub N/A

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

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{x.im \cdot y.re}{y.im} - x.re\right), \color{blue}{y.im}\right) \]

      8. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(\frac{x.im \cdot y.re}{y.im}\right), x.re\right), y.im\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(\frac{y.re \cdot x.im}{y.im}\right), x.re\right), y.im\right) \]

      10. associate-/l* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(y.re \cdot \frac{x.im}{y.im}\right), x.re\right), y.im\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(y.re, \left(\frac{x.im}{y.im}\right)\right), x.re\right), y.im\right) \]

      12. /-lowering-/.f64 89.0%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{/.f64}\left(x.im, y.im\right)\right), x.re\right), y.im\right) \]

   5. Simplified 89.0%

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

   6. Taylor expanded in y.re around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(-1 \cdot x.re\right)}, y.im\right) \]
   7. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\mathsf{neg}\left(x.re\right)\right), y.im\right) \]

      2. neg-sub0 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(0 - x.re\right), y.im\right) \]

      3. --lowering--.f64 76.4%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, x.re\right), y.im\right) \]

   8. Simplified 76.4%

      \[\leadsto \frac{\color{blue}{0 - x.re}}{y.im} \]
   9. Step-by-step derivation

      1. sub0-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\mathsf{neg}\left(x.re\right)\right), y.im\right) \]

      2. neg-lowering-neg.f64 76.4%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{neg.f64}\left(x.re\right), y.im\right) \]

   10. Applied egg-rr 76.4%

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

4. Final simplification 67.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -1.35 \cdot 10^{-70}:\\ \;\;\;\;\frac{-1}{\frac{y.im}{x.re}}\\ \mathbf{elif}\;y.im \leq 4.5 \cdot 10^{+79}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{0 - y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 62.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x.re}{0 - y.im}\\ \mathbf{if}\;y.im \leq -1.9 \cdot 10^{-72}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.im \leq 1.4 \cdot 10^{+81}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (/ x.re (- 0.0 y.im))))
   (if (<= y.im -1.9e-72) t_0 (if (<= y.im 1.4e+81) (/ x.im y.re) 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 / (0.0 - y_46_im);
	double tmp;
	if (y_46_im <= -1.9e-72) {
		tmp = t_0;
	} else if (y_46_im <= 1.4e+81) {
		tmp = x_46_im / y_46_re;
	} 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) :: tmp
    t_0 = x_46re / (0.0d0 - y_46im)
    if (y_46im <= (-1.9d-72)) then
        tmp = t_0
    else if (y_46im <= 1.4d+81) then
        tmp = x_46im / y_46re
    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 / (0.0 - y_46_im);
	double tmp;
	if (y_46_im <= -1.9e-72) {
		tmp = t_0;
	} else if (y_46_im <= 1.4e+81) {
		tmp = x_46_im / y_46_re;
	} 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 / (0.0 - y_46_im)
	tmp = 0
	if y_46_im <= -1.9e-72:
		tmp = t_0
	elif y_46_im <= 1.4e+81:
		tmp = x_46_im / y_46_re
	else:
		tmp = t_0
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(x_46_re / Float64(0.0 - y_46_im))
	tmp = 0.0
	if (y_46_im <= -1.9e-72)
		tmp = t_0;
	elseif (y_46_im <= 1.4e+81)
		tmp = Float64(x_46_im / y_46_re);
	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 / (0.0 - y_46_im);
	tmp = 0.0;
	if (y_46_im <= -1.9e-72)
		tmp = t_0;
	elseif (y_46_im <= 1.4e+81)
		tmp = x_46_im / y_46_re;
	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 / N[(0.0 - y$46$im), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$im, -1.9e-72], t$95$0, If[LessEqual[y$46$im, 1.4e+81], N[(x$46$im / y$46$re), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y.im < -1.90000000000000001e-72 or 1.39999999999999997e81 < y.im

   1. Initial program 52.9%

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

   3. Taylor expanded in y.re around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. unpow2 N/A

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

      5. associate-/r* N/A

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

      6. div-sub N/A

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

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{x.im \cdot y.re}{y.im} - x.re\right), \color{blue}{y.im}\right) \]

      8. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(\frac{x.im \cdot y.re}{y.im}\right), x.re\right), y.im\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(\frac{y.re \cdot x.im}{y.im}\right), x.re\right), y.im\right) \]

      10. associate-/l* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(y.re \cdot \frac{x.im}{y.im}\right), x.re\right), y.im\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(y.re, \left(\frac{x.im}{y.im}\right)\right), x.re\right), y.im\right) \]

      12. /-lowering-/.f64 78.9%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{/.f64}\left(x.im, y.im\right)\right), x.re\right), y.im\right) \]

   5. Simplified 78.9%

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

   6. Taylor expanded in y.re around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(-1 \cdot x.re\right)}, y.im\right) \]
   7. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\mathsf{neg}\left(x.re\right)\right), y.im\right) \]

      2. neg-sub0 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(0 - x.re\right), y.im\right) \]

      3. --lowering--.f64 68.3%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, x.re\right), y.im\right) \]

   8. Simplified 68.3%

      \[\leadsto \frac{\color{blue}{0 - x.re}}{y.im} \]
   9. Step-by-step derivation

      1. sub0-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\mathsf{neg}\left(x.re\right)\right), y.im\right) \]

      2. neg-lowering-neg.f64 68.3%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{neg.f64}\left(x.re\right), y.im\right) \]

   10. Applied egg-rr 68.3%

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

   if -1.90000000000000001e-72 < y.im < 1.39999999999999997e81

   1. Initial program 74.1%

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

   3. Taylor expanded in y.re around inf

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

      1. /-lowering-/.f64 65.8%

         \[\leadsto \mathsf{/.f64}\left(x.im, \color{blue}{y.re}\right) \]

   5. Simplified 65.8%

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

4. Final simplification 67.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -1.9 \cdot 10^{-72}:\\ \;\;\;\;\frac{x.re}{0 - y.im}\\ \mathbf{elif}\;y.im \leq 1.4 \cdot 10^{+81}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{0 - y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 42.3% accurate, 5.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 64.3%

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

3. Taylor expanded in y.re around inf

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

   1. /-lowering-/.f64 46.9%

      \[\leadsto \mathsf{/.f64}\left(x.im, \color{blue}{y.re}\right) \]

5. Simplified 46.9%

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

Reproduce

herbie shell --seed 2024138 
(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))))