_divideComplex, imaginary part

Percentage Accurate: 61.5% → 82.2%

Time: 13.3s

Alternatives: 10

Speedup: 1.1×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 61.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 82.2% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y.re}{\frac{y.im}{x.im}}\\ \mathbf{if}\;y.im \leq -9.5 \cdot 10^{+31}:\\ \;\;\;\;\frac{t_0}{y.im} - \frac{x.re}{y.im}\\ \mathbf{elif}\;y.im \leq -2.1 \cdot 10^{-108}:\\ \;\;\;\;\frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.im \leq 1.3 \cdot 10^{-52}:\\ \;\;\;\;\frac{x.im}{y.re} - \frac{\frac{y.im \cdot x.re}{y.re}}{y.re}\\ \mathbf{elif}\;y.im \leq 2.2 \cdot 10^{+145}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x.im, y.re, y.im \cdot \left(-x.re\right)\right)}{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0 - x.re}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (/ y.re (/ y.im x.im))))
   (if (<= y.im -9.5e+31)
     (- (/ t_0 y.im) (/ x.re y.im))
     (if (<= y.im -2.1e-108)
       (/ (- (* y.re x.im) (* y.im x.re)) (+ (* y.re y.re) (* y.im y.im)))
       (if (<= y.im 1.3e-52)
         (- (/ x.im y.re) (/ (/ (* y.im x.re) y.re) y.re))
         (if (<= y.im 2.2e+145)
           (/ (fma x.im y.re (* y.im (- x.re))) (fma y.re y.re (* y.im y.im)))
           (/ (- t_0 x.re) y.im)))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = y_46_re / (y_46_im / x_46_im);
	double tmp;
	if (y_46_im <= -9.5e+31) {
		tmp = (t_0 / y_46_im) - (x_46_re / y_46_im);
	} else if (y_46_im <= -2.1e-108) {
		tmp = ((y_46_re * x_46_im) - (y_46_im * x_46_re)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	} else if (y_46_im <= 1.3e-52) {
		tmp = (x_46_im / y_46_re) - (((y_46_im * x_46_re) / y_46_re) / y_46_re);
	} else if (y_46_im <= 2.2e+145) {
		tmp = fma(x_46_im, y_46_re, (y_46_im * -x_46_re)) / fma(y_46_re, y_46_re, (y_46_im * y_46_im));
	} else {
		tmp = (t_0 - 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(y_46_re / Float64(y_46_im / x_46_im))
	tmp = 0.0
	if (y_46_im <= -9.5e+31)
		tmp = Float64(Float64(t_0 / y_46_im) - Float64(x_46_re / y_46_im));
	elseif (y_46_im <= -2.1e-108)
		tmp = Float64(Float64(Float64(y_46_re * x_46_im) - Float64(y_46_im * x_46_re)) / Float64(Float64(y_46_re * y_46_re) + Float64(y_46_im * y_46_im)));
	elseif (y_46_im <= 1.3e-52)
		tmp = Float64(Float64(x_46_im / y_46_re) - Float64(Float64(Float64(y_46_im * x_46_re) / y_46_re) / y_46_re));
	elseif (y_46_im <= 2.2e+145)
		tmp = Float64(fma(x_46_im, y_46_re, Float64(y_46_im * Float64(-x_46_re))) / fma(y_46_re, y_46_re, Float64(y_46_im * y_46_im)));
	else
		tmp = Float64(Float64(t_0 - 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[(y$46$re / N[(y$46$im / x$46$im), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$im, -9.5e+31], N[(N[(t$95$0 / y$46$im), $MachinePrecision] - N[(x$46$re / y$46$im), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, -2.1e-108], N[(N[(N[(y$46$re * x$46$im), $MachinePrecision] - N[(y$46$im * x$46$re), $MachinePrecision]), $MachinePrecision] / N[(N[(y$46$re * y$46$re), $MachinePrecision] + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, 1.3e-52], N[(N[(x$46$im / y$46$re), $MachinePrecision] - N[(N[(N[(y$46$im * x$46$re), $MachinePrecision] / y$46$re), $MachinePrecision] / y$46$re), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, 2.2e+145], N[(N[(x$46$im * y$46$re + N[(y$46$im * (-x$46$re)), $MachinePrecision]), $MachinePrecision] / N[(y$46$re * y$46$re + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$0 - x$46$re), $MachinePrecision] / y$46$im), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if y.im < -9.5000000000000008e31

   1. Initial program 43.1%

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

   3. Taylor expanded in y.re around 0 77.1%

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

      1. +-commutative 77.1%

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

      2. mul-1-neg 77.1%

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

      3. unsub-neg 77.1%

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

      4. associate-/l* 79.0%

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

      5. associate-/r/ 80.5%

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

   5. Simplified 80.5%

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

      1. *-un-lft-identity 80.5%

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

      2. pow2 80.5%

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

      3. times-frac 85.0%

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

   7. Applied egg-rr 85.0%

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

      1. associate-*l/ 85.1%

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

      2. *-un-lft-identity 85.1%

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

   9. Applied egg-rr 85.1%

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

       1. associate-*l/ 87.4%

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

       2. *-commutative 87.4%

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

       3. clear-num 87.4%

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

       4. un-div-inv 87.4%

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

   11. Applied egg-rr 87.4%

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

   if -9.5000000000000008e31 < y.im < -2.0999999999999999e-108

   1. Initial program 96.7%

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

   if -2.0999999999999999e-108 < y.im < 1.2999999999999999e-52

   1. Initial program 70.5%

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

   3. Taylor expanded in y.re around inf 80.1%

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

      1. +-commutative 80.1%

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

      2. mul-1-neg 80.1%

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

      3. unsub-neg 80.1%

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

      4. associate-/l* 79.0%

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

      5. associate-/r/ 80.2%

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

   5. Simplified 80.2%

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

      1. associate-*l/ 80.1%

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

      2. pow2 80.1%

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

      3. associate-/r* 89.5%

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

      4. *-commutative 89.5%

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

   7. Applied egg-rr 89.5%

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

   if 1.2999999999999999e-52 < y.im < 2.20000000000000009e145

   1. Initial program 77.5%

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

      1. fma-neg 77.5%

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

      2. distribute-lft-neg-out 77.5%

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

      3. *-commutative 77.5%

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

      4. fma-def 77.5%

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

   3. Simplified 77.5%

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

   if 2.20000000000000009e145 < y.im

   1. Initial program 24.5%

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

   3. Taylor expanded in y.re around 0 64.4%

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

      1. +-commutative 64.4%

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

      2. mul-1-neg 64.4%

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

      3. unsub-neg 64.4%

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

      4. associate-/l* 70.5%

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

      5. associate-/r/ 70.5%

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

   5. Simplified 70.5%

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

      1. *-un-lft-identity 70.5%

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

      2. pow2 70.5%

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

      3. times-frac 81.2%

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

   7. Applied egg-rr 81.2%

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

      1. associate-*l* 86.8%

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

      2. fma-neg 86.8%

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

   9. Applied egg-rr 86.8%

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

       1. fma-udef 86.8%

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

       2. unsub-neg 86.8%

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

       3. associate-*r* 81.2%

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

       4. times-frac 70.5%

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

       5. *-un-lft-identity 70.5%

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

       6. associate-/l/ 81.2%

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

       7. associate-*l/ 86.8%

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

       8. sub-div 86.8%

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

       9. *-commutative 86.8%

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

       10. clear-num 86.8%

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

       11. un-div-inv 86.8%

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

   11. Applied egg-rr 86.8%

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

4. Final simplification 87.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -9.5 \cdot 10^{+31}:\\ \;\;\;\;\frac{\frac{y.re}{\frac{y.im}{x.im}}}{y.im} - \frac{x.re}{y.im}\\ \mathbf{elif}\;y.im \leq -2.1 \cdot 10^{-108}:\\ \;\;\;\;\frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.im \leq 1.3 \cdot 10^{-52}:\\ \;\;\;\;\frac{x.im}{y.re} - \frac{\frac{y.im \cdot x.re}{y.re}}{y.re}\\ \mathbf{elif}\;y.im \leq 2.2 \cdot 10^{+145}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x.im, y.re, y.im \cdot \left(-x.re\right)\right)}{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y.re}{\frac{y.im}{x.im}} - x.re}{y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 82.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ t_1 := \frac{y.re}{\frac{y.im}{x.im}}\\ \mathbf{if}\;y.im \leq -9.5 \cdot 10^{+31}:\\ \;\;\;\;\frac{t_1}{y.im} - \frac{x.re}{y.im}\\ \mathbf{elif}\;y.im \leq -1.7 \cdot 10^{-112}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq 1.8 \cdot 10^{-46}:\\ \;\;\;\;\frac{x.im}{y.re} - \frac{\frac{y.im \cdot x.re}{y.re}}{y.re}\\ \mathbf{elif}\;y.im \leq 1.65 \cdot 10^{+145}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{t_1 - x.re}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0
         (/ (- (* y.re x.im) (* y.im x.re)) (+ (* y.re y.re) (* y.im y.im))))
        (t_1 (/ y.re (/ y.im x.im))))
   (if (<= y.im -9.5e+31)
     (- (/ t_1 y.im) (/ x.re y.im))
     (if (<= y.im -1.7e-112)
       t_0
       (if (<= y.im 1.8e-46)
         (- (/ x.im y.re) (/ (/ (* y.im x.re) y.re) y.re))
         (if (<= y.im 1.65e+145) t_0 (/ (- t_1 x.re) y.im)))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = ((y_46_re * x_46_im) - (y_46_im * x_46_re)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	double t_1 = y_46_re / (y_46_im / x_46_im);
	double tmp;
	if (y_46_im <= -9.5e+31) {
		tmp = (t_1 / y_46_im) - (x_46_re / y_46_im);
	} else if (y_46_im <= -1.7e-112) {
		tmp = t_0;
	} else if (y_46_im <= 1.8e-46) {
		tmp = (x_46_im / y_46_re) - (((y_46_im * x_46_re) / y_46_re) / y_46_re);
	} else if (y_46_im <= 1.65e+145) {
		tmp = t_0;
	} else {
		tmp = (t_1 - 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) :: t_1
    real(8) :: tmp
    t_0 = ((y_46re * x_46im) - (y_46im * x_46re)) / ((y_46re * y_46re) + (y_46im * y_46im))
    t_1 = y_46re / (y_46im / x_46im)
    if (y_46im <= (-9.5d+31)) then
        tmp = (t_1 / y_46im) - (x_46re / y_46im)
    else if (y_46im <= (-1.7d-112)) then
        tmp = t_0
    else if (y_46im <= 1.8d-46) then
        tmp = (x_46im / y_46re) - (((y_46im * x_46re) / y_46re) / y_46re)
    else if (y_46im <= 1.65d+145) then
        tmp = t_0
    else
        tmp = (t_1 - 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_re * x_46_im) - (y_46_im * x_46_re)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	double t_1 = y_46_re / (y_46_im / x_46_im);
	double tmp;
	if (y_46_im <= -9.5e+31) {
		tmp = (t_1 / y_46_im) - (x_46_re / y_46_im);
	} else if (y_46_im <= -1.7e-112) {
		tmp = t_0;
	} else if (y_46_im <= 1.8e-46) {
		tmp = (x_46_im / y_46_re) - (((y_46_im * x_46_re) / y_46_re) / y_46_re);
	} else if (y_46_im <= 1.65e+145) {
		tmp = t_0;
	} else {
		tmp = (t_1 - x_46_re) / y_46_im;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = ((y_46_re * x_46_im) - (y_46_im * x_46_re)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im))
	t_1 = y_46_re / (y_46_im / x_46_im)
	tmp = 0
	if y_46_im <= -9.5e+31:
		tmp = (t_1 / y_46_im) - (x_46_re / y_46_im)
	elif y_46_im <= -1.7e-112:
		tmp = t_0
	elif y_46_im <= 1.8e-46:
		tmp = (x_46_im / y_46_re) - (((y_46_im * x_46_re) / y_46_re) / y_46_re)
	elif y_46_im <= 1.65e+145:
		tmp = t_0
	else:
		tmp = (t_1 - x_46_re) / y_46_im
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(Float64(Float64(y_46_re * x_46_im) - Float64(y_46_im * x_46_re)) / Float64(Float64(y_46_re * y_46_re) + Float64(y_46_im * y_46_im)))
	t_1 = Float64(y_46_re / Float64(y_46_im / x_46_im))
	tmp = 0.0
	if (y_46_im <= -9.5e+31)
		tmp = Float64(Float64(t_1 / y_46_im) - Float64(x_46_re / y_46_im));
	elseif (y_46_im <= -1.7e-112)
		tmp = t_0;
	elseif (y_46_im <= 1.8e-46)
		tmp = Float64(Float64(x_46_im / y_46_re) - Float64(Float64(Float64(y_46_im * x_46_re) / y_46_re) / y_46_re));
	elseif (y_46_im <= 1.65e+145)
		tmp = t_0;
	else
		tmp = Float64(Float64(t_1 - x_46_re) / y_46_im);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = ((y_46_re * x_46_im) - (y_46_im * x_46_re)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	t_1 = y_46_re / (y_46_im / x_46_im);
	tmp = 0.0;
	if (y_46_im <= -9.5e+31)
		tmp = (t_1 / y_46_im) - (x_46_re / y_46_im);
	elseif (y_46_im <= -1.7e-112)
		tmp = t_0;
	elseif (y_46_im <= 1.8e-46)
		tmp = (x_46_im / y_46_re) - (((y_46_im * x_46_re) / y_46_re) / y_46_re);
	elseif (y_46_im <= 1.65e+145)
		tmp = t_0;
	else
		tmp = (t_1 - x_46_re) / y_46_im;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[(N[(y$46$re * x$46$im), $MachinePrecision] - N[(y$46$im * x$46$re), $MachinePrecision]), $MachinePrecision] / N[(N[(y$46$re * y$46$re), $MachinePrecision] + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(y$46$re / N[(y$46$im / x$46$im), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$im, -9.5e+31], N[(N[(t$95$1 / y$46$im), $MachinePrecision] - N[(x$46$re / y$46$im), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, -1.7e-112], t$95$0, If[LessEqual[y$46$im, 1.8e-46], N[(N[(x$46$im / y$46$re), $MachinePrecision] - N[(N[(N[(y$46$im * x$46$re), $MachinePrecision] / y$46$re), $MachinePrecision] / y$46$re), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, 1.65e+145], t$95$0, N[(N[(t$95$1 - x$46$re), $MachinePrecision] / y$46$im), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if y.im < -9.5000000000000008e31

   1. Initial program 43.1%

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

   3. Taylor expanded in y.re around 0 77.1%

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

      1. +-commutative 77.1%

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

      2. mul-1-neg 77.1%

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

      3. unsub-neg 77.1%

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

      4. associate-/l* 79.0%

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

      5. associate-/r/ 80.5%

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

   5. Simplified 80.5%

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

      1. *-un-lft-identity 80.5%

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

      2. pow2 80.5%

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

      3. times-frac 85.0%

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

   7. Applied egg-rr 85.0%

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

      1. associate-*l/ 85.1%

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

      2. *-un-lft-identity 85.1%

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

   9. Applied egg-rr 85.1%

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

       1. associate-*l/ 87.4%

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

       2. *-commutative 87.4%

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

       3. clear-num 87.4%

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

       4. un-div-inv 87.4%

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

   11. Applied egg-rr 87.4%

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

   if -9.5000000000000008e31 < y.im < -1.6999999999999999e-112 or 1.8e-46 < y.im < 1.65000000000000013e145

   1. Initial program 86.2%

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

   if -1.6999999999999999e-112 < y.im < 1.8e-46

   1. Initial program 70.5%

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

   3. Taylor expanded in y.re around inf 80.1%

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

      1. +-commutative 80.1%

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

      2. mul-1-neg 80.1%

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

      3. unsub-neg 80.1%

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

      4. associate-/l* 79.0%

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

      5. associate-/r/ 80.2%

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

   5. Simplified 80.2%

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

      1. associate-*l/ 80.1%

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

      2. pow2 80.1%

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

      3. associate-/r* 89.5%

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

      4. *-commutative 89.5%

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

   7. Applied egg-rr 89.5%

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

   if 1.65000000000000013e145 < y.im

   1. Initial program 24.5%

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

   3. Taylor expanded in y.re around 0 64.4%

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

      1. +-commutative 64.4%

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

      2. mul-1-neg 64.4%

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

      3. unsub-neg 64.4%

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

      4. associate-/l* 70.5%

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

      5. associate-/r/ 70.5%

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

   5. Simplified 70.5%

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

      1. *-un-lft-identity 70.5%

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

      2. pow2 70.5%

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

      3. times-frac 81.2%

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

   7. Applied egg-rr 81.2%

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

      1. associate-*l* 86.8%

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

      2. fma-neg 86.8%

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

   9. Applied egg-rr 86.8%

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

       1. fma-udef 86.8%

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

       2. unsub-neg 86.8%

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

       3. associate-*r* 81.2%

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

       4. times-frac 70.5%

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

       5. *-un-lft-identity 70.5%

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

       6. associate-/l/ 81.2%

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

       7. associate-*l/ 86.8%

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

       8. sub-div 86.8%

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

       9. *-commutative 86.8%

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

       10. clear-num 86.8%

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

       11. un-div-inv 86.8%

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

   11. Applied egg-rr 86.8%

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

4. Final simplification 87.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -9.5 \cdot 10^{+31}:\\ \;\;\;\;\frac{\frac{y.re}{\frac{y.im}{x.im}}}{y.im} - \frac{x.re}{y.im}\\ \mathbf{elif}\;y.im \leq -1.7 \cdot 10^{-112}:\\ \;\;\;\;\frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.im \leq 1.8 \cdot 10^{-46}:\\ \;\;\;\;\frac{x.im}{y.re} - \frac{\frac{y.im \cdot x.re}{y.re}}{y.re}\\ \mathbf{elif}\;y.im \leq 1.65 \cdot 10^{+145}:\\ \;\;\;\;\frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y.re}{\frac{y.im}{x.im}} - x.re}{y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 77.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x.im \cdot \frac{y.re}{y.im} - x.re}{y.im}\\ t_1 := \frac{x.im}{y.re} + y.im \cdot \left(\frac{x.re}{y.re} \cdot \frac{-1}{y.re}\right)\\ \mathbf{if}\;y.re \leq -2.2 \cdot 10^{-20}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.re \leq 4.8 \cdot 10^{-35}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq 2.7 \cdot 10^{+19}:\\ \;\;\;\;\frac{x.im}{y.re} - \frac{\frac{y.im \cdot x.re}{y.re}}{y.re}\\ \mathbf{elif}\;y.re \leq 5.9 \cdot 10^{+75}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (/ (- (* x.im (/ y.re y.im)) x.re) y.im))
        (t_1 (+ (/ x.im y.re) (* y.im (* (/ x.re y.re) (/ -1.0 y.re))))))
   (if (<= y.re -2.2e-20)
     t_1
     (if (<= y.re 4.8e-35)
       t_0
       (if (<= y.re 2.7e+19)
         (- (/ x.im y.re) (/ (/ (* y.im x.re) y.re) y.re))
         (if (<= y.re 5.9e+75) t_0 t_1))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = ((x_46_im * (y_46_re / y_46_im)) - x_46_re) / y_46_im;
	double t_1 = (x_46_im / y_46_re) + (y_46_im * ((x_46_re / y_46_re) * (-1.0 / y_46_re)));
	double tmp;
	if (y_46_re <= -2.2e-20) {
		tmp = t_1;
	} else if (y_46_re <= 4.8e-35) {
		tmp = t_0;
	} else if (y_46_re <= 2.7e+19) {
		tmp = (x_46_im / y_46_re) - (((y_46_im * x_46_re) / y_46_re) / y_46_re);
	} else if (y_46_re <= 5.9e+75) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = ((x_46im * (y_46re / y_46im)) - x_46re) / y_46im
    t_1 = (x_46im / y_46re) + (y_46im * ((x_46re / y_46re) * ((-1.0d0) / y_46re)))
    if (y_46re <= (-2.2d-20)) then
        tmp = t_1
    else if (y_46re <= 4.8d-35) then
        tmp = t_0
    else if (y_46re <= 2.7d+19) then
        tmp = (x_46im / y_46re) - (((y_46im * x_46re) / y_46re) / y_46re)
    else if (y_46re <= 5.9d+75) then
        tmp = t_0
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = ((x_46_im * (y_46_re / y_46_im)) - x_46_re) / y_46_im;
	double t_1 = (x_46_im / y_46_re) + (y_46_im * ((x_46_re / y_46_re) * (-1.0 / y_46_re)));
	double tmp;
	if (y_46_re <= -2.2e-20) {
		tmp = t_1;
	} else if (y_46_re <= 4.8e-35) {
		tmp = t_0;
	} else if (y_46_re <= 2.7e+19) {
		tmp = (x_46_im / y_46_re) - (((y_46_im * x_46_re) / y_46_re) / y_46_re);
	} else if (y_46_re <= 5.9e+75) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = ((x_46_im * (y_46_re / y_46_im)) - x_46_re) / y_46_im
	t_1 = (x_46_im / y_46_re) + (y_46_im * ((x_46_re / y_46_re) * (-1.0 / y_46_re)))
	tmp = 0
	if y_46_re <= -2.2e-20:
		tmp = t_1
	elif y_46_re <= 4.8e-35:
		tmp = t_0
	elif y_46_re <= 2.7e+19:
		tmp = (x_46_im / y_46_re) - (((y_46_im * x_46_re) / y_46_re) / y_46_re)
	elif y_46_re <= 5.9e+75:
		tmp = t_0
	else:
		tmp = t_1
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(Float64(Float64(x_46_im * Float64(y_46_re / y_46_im)) - x_46_re) / y_46_im)
	t_1 = Float64(Float64(x_46_im / y_46_re) + Float64(y_46_im * Float64(Float64(x_46_re / y_46_re) * Float64(-1.0 / y_46_re))))
	tmp = 0.0
	if (y_46_re <= -2.2e-20)
		tmp = t_1;
	elseif (y_46_re <= 4.8e-35)
		tmp = t_0;
	elseif (y_46_re <= 2.7e+19)
		tmp = Float64(Float64(x_46_im / y_46_re) - Float64(Float64(Float64(y_46_im * x_46_re) / y_46_re) / y_46_re));
	elseif (y_46_re <= 5.9e+75)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = ((x_46_im * (y_46_re / y_46_im)) - x_46_re) / y_46_im;
	t_1 = (x_46_im / y_46_re) + (y_46_im * ((x_46_re / y_46_re) * (-1.0 / y_46_re)));
	tmp = 0.0;
	if (y_46_re <= -2.2e-20)
		tmp = t_1;
	elseif (y_46_re <= 4.8e-35)
		tmp = t_0;
	elseif (y_46_re <= 2.7e+19)
		tmp = (x_46_im / y_46_re) - (((y_46_im * x_46_re) / y_46_re) / y_46_re);
	elseif (y_46_re <= 5.9e+75)
		tmp = t_0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[(N[(x$46$im * N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision] - x$46$re), $MachinePrecision] / y$46$im), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x$46$im / y$46$re), $MachinePrecision] + N[(y$46$im * N[(N[(x$46$re / y$46$re), $MachinePrecision] * N[(-1.0 / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$re, -2.2e-20], t$95$1, If[LessEqual[y$46$re, 4.8e-35], t$95$0, If[LessEqual[y$46$re, 2.7e+19], N[(N[(x$46$im / y$46$re), $MachinePrecision] - N[(N[(N[(y$46$im * x$46$re), $MachinePrecision] / y$46$re), $MachinePrecision] / y$46$re), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 5.9e+75], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -2.19999999999999991e-20 or 5.89999999999999983e75 < y.re

   1. Initial program 53.4%

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

   3. Taylor expanded in y.re around inf 74.7%

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

      1. +-commutative 74.7%

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

      2. mul-1-neg 74.7%

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

      3. unsub-neg 74.7%

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

      4. associate-/l* 76.0%

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

      5. associate-/r/ 77.6%

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

   5. Simplified 77.6%

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

      1. *-un-lft-identity 77.6%

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

      2. pow2 77.6%

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

      3. times-frac 81.4%

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

   7. Applied egg-rr 81.4%

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

   if -2.19999999999999991e-20 < y.re < 4.8000000000000003e-35 or 2.7e19 < y.re < 5.89999999999999983e75

   1. Initial program 67.2%

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

   3. Taylor expanded in y.re around 0 80.2%

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

      1. +-commutative 80.2%

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

      2. mul-1-neg 80.2%

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

      3. unsub-neg 80.2%

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

      4. associate-/l* 81.2%

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

      5. associate-/r/ 80.3%

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

   5. Simplified 80.3%

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

      1. *-un-lft-identity 80.3%

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

      2. pow2 80.3%

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

      3. times-frac 83.4%

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

   7. Applied egg-rr 83.4%

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

      1. associate-*l* 87.8%

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

      2. fma-neg 87.8%

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

   9. Applied egg-rr 87.8%

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

       1. expm1-log1p-u 69.2%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{fma}\left(\frac{1}{y.im}, \frac{x.im}{y.im} \cdot y.re, -\frac{x.re}{y.im}\right)\right)\right)} \]

       2. expm1-udef 34.4%

          \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\mathsf{fma}\left(\frac{1}{y.im}, \frac{x.im}{y.im} \cdot y.re, -\frac{x.re}{y.im}\right)\right)} - 1} \]

   11. Applied egg-rr 34.4%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\frac{y.re}{\frac{y.im}{x.im}} - x.re}{y.im}\right)} - 1} \]
   12. Step-by-step derivation

       1. expm1-def 69.2%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{y.re}{\frac{y.im}{x.im}} - x.re}{y.im}\right)\right)} \]

       2. expm1-log1p 88.7%

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

       3. associate-/r/ 89.3%

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

   13. Simplified 89.3%

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

   if 4.8000000000000003e-35 < y.re < 2.7e19

   1. Initial program 76.8%

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

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

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

      1. +-commutative 71.7%

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

      2. mul-1-neg 71.7%

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

      3. unsub-neg 71.7%

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

      4. associate-/l* 71.6%

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

      5. associate-/r/ 66.3%

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

   5. Simplified 66.3%

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

      1. associate-*l/ 71.7%

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

      2. pow2 71.7%

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

      3. associate-/r* 71.4%

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

      4. *-commutative 71.4%

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

   7. Applied egg-rr 71.4%

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

4. Final simplification 84.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -2.2 \cdot 10^{-20}:\\ \;\;\;\;\frac{x.im}{y.re} + y.im \cdot \left(\frac{x.re}{y.re} \cdot \frac{-1}{y.re}\right)\\ \mathbf{elif}\;y.re \leq 4.8 \cdot 10^{-35}:\\ \;\;\;\;\frac{x.im \cdot \frac{y.re}{y.im} - x.re}{y.im}\\ \mathbf{elif}\;y.re \leq 2.7 \cdot 10^{+19}:\\ \;\;\;\;\frac{x.im}{y.re} - \frac{\frac{y.im \cdot x.re}{y.re}}{y.re}\\ \mathbf{elif}\;y.re \leq 5.9 \cdot 10^{+75}:\\ \;\;\;\;\frac{x.im \cdot \frac{y.re}{y.im} - x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re} + y.im \cdot \left(\frac{x.re}{y.re} \cdot \frac{-1}{y.re}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 77.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x.im \cdot \frac{y.re}{y.im} - x.re}{y.im}\\ \mathbf{if}\;y.re \leq -8.2 \cdot 10^{-21}:\\ \;\;\;\;\frac{x.im}{y.re} + y.im \cdot \left(\frac{x.re}{y.re} \cdot \frac{-1}{y.re}\right)\\ \mathbf{elif}\;y.re \leq 6.6 \cdot 10^{-35}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq 4.8 \cdot 10^{+18}:\\ \;\;\;\;\frac{x.im}{y.re} - \frac{\frac{y.im \cdot x.re}{y.re}}{y.re}\\ \mathbf{elif}\;y.re \leq 9.2 \cdot 10^{+76}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re} - \frac{y.im}{y.re \cdot \frac{y.re}{x.re}}\\ \end{array} \end{array} \]

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.re -8.2e-21)
     (+ (/ x.im y.re) (* y.im (* (/ x.re y.re) (/ -1.0 y.re))))
     (if (<= y.re 6.6e-35)
       t_0
       (if (<= y.re 4.8e+18)
         (- (/ x.im y.re) (/ (/ (* y.im x.re) y.re) y.re))
         (if (<= y.re 9.2e+76)
           t_0
           (- (/ x.im y.re) (/ y.im (* y.re (/ y.re x.re))))))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = ((x_46_im * (y_46_re / y_46_im)) - x_46_re) / y_46_im;
	double tmp;
	if (y_46_re <= -8.2e-21) {
		tmp = (x_46_im / y_46_re) + (y_46_im * ((x_46_re / y_46_re) * (-1.0 / y_46_re)));
	} else if (y_46_re <= 6.6e-35) {
		tmp = t_0;
	} else if (y_46_re <= 4.8e+18) {
		tmp = (x_46_im / y_46_re) - (((y_46_im * x_46_re) / y_46_re) / y_46_re);
	} else if (y_46_re <= 9.2e+76) {
		tmp = t_0;
	} else {
		tmp = (x_46_im / y_46_re) - (y_46_im / (y_46_re * (y_46_re / x_46_re)));
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ((x_46im * (y_46re / y_46im)) - x_46re) / y_46im
    if (y_46re <= (-8.2d-21)) then
        tmp = (x_46im / y_46re) + (y_46im * ((x_46re / y_46re) * ((-1.0d0) / y_46re)))
    else if (y_46re <= 6.6d-35) then
        tmp = t_0
    else if (y_46re <= 4.8d+18) then
        tmp = (x_46im / y_46re) - (((y_46im * x_46re) / y_46re) / y_46re)
    else if (y_46re <= 9.2d+76) then
        tmp = t_0
    else
        tmp = (x_46im / y_46re) - (y_46im / (y_46re * (y_46re / x_46re)))
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = ((x_46_im * (y_46_re / y_46_im)) - x_46_re) / y_46_im;
	double tmp;
	if (y_46_re <= -8.2e-21) {
		tmp = (x_46_im / y_46_re) + (y_46_im * ((x_46_re / y_46_re) * (-1.0 / y_46_re)));
	} else if (y_46_re <= 6.6e-35) {
		tmp = t_0;
	} else if (y_46_re <= 4.8e+18) {
		tmp = (x_46_im / y_46_re) - (((y_46_im * x_46_re) / y_46_re) / y_46_re);
	} else if (y_46_re <= 9.2e+76) {
		tmp = t_0;
	} else {
		tmp = (x_46_im / y_46_re) - (y_46_im / (y_46_re * (y_46_re / x_46_re)));
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = ((x_46_im * (y_46_re / y_46_im)) - x_46_re) / y_46_im
	tmp = 0
	if y_46_re <= -8.2e-21:
		tmp = (x_46_im / y_46_re) + (y_46_im * ((x_46_re / y_46_re) * (-1.0 / y_46_re)))
	elif y_46_re <= 6.6e-35:
		tmp = t_0
	elif y_46_re <= 4.8e+18:
		tmp = (x_46_im / y_46_re) - (((y_46_im * x_46_re) / y_46_re) / y_46_re)
	elif y_46_re <= 9.2e+76:
		tmp = t_0
	else:
		tmp = (x_46_im / y_46_re) - (y_46_im / (y_46_re * (y_46_re / x_46_re)))
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(Float64(Float64(x_46_im * Float64(y_46_re / y_46_im)) - x_46_re) / y_46_im)
	tmp = 0.0
	if (y_46_re <= -8.2e-21)
		tmp = Float64(Float64(x_46_im / y_46_re) + Float64(y_46_im * Float64(Float64(x_46_re / y_46_re) * Float64(-1.0 / y_46_re))));
	elseif (y_46_re <= 6.6e-35)
		tmp = t_0;
	elseif (y_46_re <= 4.8e+18)
		tmp = Float64(Float64(x_46_im / y_46_re) - Float64(Float64(Float64(y_46_im * x_46_re) / y_46_re) / y_46_re));
	elseif (y_46_re <= 9.2e+76)
		tmp = t_0;
	else
		tmp = Float64(Float64(x_46_im / y_46_re) - Float64(y_46_im / Float64(y_46_re * Float64(y_46_re / x_46_re))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = ((x_46_im * (y_46_re / y_46_im)) - x_46_re) / y_46_im;
	tmp = 0.0;
	if (y_46_re <= -8.2e-21)
		tmp = (x_46_im / y_46_re) + (y_46_im * ((x_46_re / y_46_re) * (-1.0 / y_46_re)));
	elseif (y_46_re <= 6.6e-35)
		tmp = t_0;
	elseif (y_46_re <= 4.8e+18)
		tmp = (x_46_im / y_46_re) - (((y_46_im * x_46_re) / y_46_re) / y_46_re);
	elseif (y_46_re <= 9.2e+76)
		tmp = t_0;
	else
		tmp = (x_46_im / y_46_re) - (y_46_im / (y_46_re * (y_46_re / x_46_re)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[(N[(x$46$im * N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision] - x$46$re), $MachinePrecision] / y$46$im), $MachinePrecision]}, If[LessEqual[y$46$re, -8.2e-21], N[(N[(x$46$im / y$46$re), $MachinePrecision] + N[(y$46$im * N[(N[(x$46$re / y$46$re), $MachinePrecision] * N[(-1.0 / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 6.6e-35], t$95$0, If[LessEqual[y$46$re, 4.8e+18], N[(N[(x$46$im / y$46$re), $MachinePrecision] - N[(N[(N[(y$46$im * x$46$re), $MachinePrecision] / y$46$re), $MachinePrecision] / y$46$re), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 9.2e+76], t$95$0, N[(N[(x$46$im / y$46$re), $MachinePrecision] - N[(y$46$im / N[(y$46$re * N[(y$46$re / x$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if y.re < -8.19999999999999988e-21

   1. Initial program 49.6%

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

   3. Taylor expanded in y.re around inf 67.9%

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

      1. +-commutative 67.9%

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

      2. mul-1-neg 67.9%

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

      3. unsub-neg 67.9%

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

      4. associate-/l* 69.8%

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

      5. associate-/r/ 71.4%

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

   5. Simplified 71.4%

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

      1. *-un-lft-identity 71.4%

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

      2. pow2 71.4%

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

      3. times-frac 74.3%

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

   7. Applied egg-rr 74.3%

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

   if -8.19999999999999988e-21 < y.re < 6.6000000000000001e-35 or 4.8e18 < y.re < 9.20000000000000005e76

   1. Initial program 67.2%

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

   3. Taylor expanded in y.re around 0 80.2%

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

      1. +-commutative 80.2%

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

      2. mul-1-neg 80.2%

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

      3. unsub-neg 80.2%

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

      4. associate-/l* 81.2%

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

      5. associate-/r/ 80.3%

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

   5. Simplified 80.3%

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

      1. *-un-lft-identity 80.3%

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

      2. pow2 80.3%

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

      3. times-frac 83.4%

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

   7. Applied egg-rr 83.4%

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

      1. associate-*l* 87.8%

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

      2. fma-neg 87.8%

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

   9. Applied egg-rr 87.8%

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

       1. expm1-log1p-u 69.2%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{fma}\left(\frac{1}{y.im}, \frac{x.im}{y.im} \cdot y.re, -\frac{x.re}{y.im}\right)\right)\right)} \]

       2. expm1-udef 34.4%

          \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\mathsf{fma}\left(\frac{1}{y.im}, \frac{x.im}{y.im} \cdot y.re, -\frac{x.re}{y.im}\right)\right)} - 1} \]

   11. Applied egg-rr 34.4%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\frac{y.re}{\frac{y.im}{x.im}} - x.re}{y.im}\right)} - 1} \]
   12. Step-by-step derivation

       1. expm1-def 69.2%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{y.re}{\frac{y.im}{x.im}} - x.re}{y.im}\right)\right)} \]

       2. expm1-log1p 88.7%

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

       3. associate-/r/ 89.3%

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

   13. Simplified 89.3%

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

   if 6.6000000000000001e-35 < y.re < 4.8e18

   1. Initial program 76.8%

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

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

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

      1. +-commutative 71.7%

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

      2. mul-1-neg 71.7%

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

      3. unsub-neg 71.7%

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

      4. associate-/l* 71.6%

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

      5. associate-/r/ 66.3%

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

   5. Simplified 66.3%

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

      1. associate-*l/ 71.7%

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

      2. pow2 71.7%

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

      3. associate-/r* 71.4%

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

      4. *-commutative 71.4%

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

   7. Applied egg-rr 71.4%

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

   if 9.20000000000000005e76 < y.re

   1. Initial program 57.7%

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

   3. Taylor expanded in y.re around inf 82.5%

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

      1. +-commutative 82.5%

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

      2. mul-1-neg 82.5%

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

      3. unsub-neg 82.5%

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

      4. associate-/l* 83.1%

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

      5. associate-/r/ 84.8%

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

   5. Simplified 84.8%

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

      1. *-commutative 84.8%

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

      2. clear-num 84.8%

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

      3. un-div-inv 84.8%

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

   7. Applied egg-rr 84.8%

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

      1. unpow2 84.8%

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

      2. *-un-lft-identity 84.8%

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

      3. times-frac 89.6%

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

   9. Applied egg-rr 89.6%

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

4. Final simplification 84.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -8.2 \cdot 10^{-21}:\\ \;\;\;\;\frac{x.im}{y.re} + y.im \cdot \left(\frac{x.re}{y.re} \cdot \frac{-1}{y.re}\right)\\ \mathbf{elif}\;y.re \leq 6.6 \cdot 10^{-35}:\\ \;\;\;\;\frac{x.im \cdot \frac{y.re}{y.im} - x.re}{y.im}\\ \mathbf{elif}\;y.re \leq 4.8 \cdot 10^{+18}:\\ \;\;\;\;\frac{x.im}{y.re} - \frac{\frac{y.im \cdot x.re}{y.re}}{y.re}\\ \mathbf{elif}\;y.re \leq 9.2 \cdot 10^{+76}:\\ \;\;\;\;\frac{x.im \cdot \frac{y.re}{y.im} - x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re} - \frac{y.im}{y.re \cdot \frac{y.re}{x.re}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 78.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -7.2 \cdot 10^{+25} \lor \neg \left(y.im \leq 5.5 \cdot 10^{-23}\right):\\ \;\;\;\;\frac{\frac{y.re}{\frac{y.im}{x.im}} - x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re} - \frac{\frac{y.im \cdot x.re}{y.re}}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.im -7.2e+25) (not (<= y.im 5.5e-23)))
   (/ (- (/ y.re (/ y.im x.im)) x.re) y.im)
   (- (/ x.im y.re) (/ (/ (* y.im x.re) y.re) y.re))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_im <= -7.2e+25) || !(y_46_im <= 5.5e-23)) {
		tmp = ((y_46_re / (y_46_im / x_46_im)) - x_46_re) / y_46_im;
	} else {
		tmp = (x_46_im / y_46_re) - (((y_46_im * x_46_re) / y_46_re) / y_46_re);
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if ((y_46im <= (-7.2d+25)) .or. (.not. (y_46im <= 5.5d-23))) then
        tmp = ((y_46re / (y_46im / x_46im)) - x_46re) / y_46im
    else
        tmp = (x_46im / y_46re) - (((y_46im * x_46re) / y_46re) / y_46re)
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_im <= -7.2e+25) || !(y_46_im <= 5.5e-23)) {
		tmp = ((y_46_re / (y_46_im / x_46_im)) - x_46_re) / y_46_im;
	} else {
		tmp = (x_46_im / y_46_re) - (((y_46_im * x_46_re) / y_46_re) / y_46_re);
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if (y_46_im <= -7.2e+25) or not (y_46_im <= 5.5e-23):
		tmp = ((y_46_re / (y_46_im / x_46_im)) - x_46_re) / y_46_im
	else:
		tmp = (x_46_im / y_46_re) - (((y_46_im * x_46_re) / y_46_re) / y_46_re)
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if ((y_46_im <= -7.2e+25) || !(y_46_im <= 5.5e-23))
		tmp = Float64(Float64(Float64(y_46_re / Float64(y_46_im / x_46_im)) - x_46_re) / y_46_im);
	else
		tmp = Float64(Float64(x_46_im / y_46_re) - Float64(Float64(Float64(y_46_im * x_46_re) / y_46_re) / y_46_re));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if ((y_46_im <= -7.2e+25) || ~((y_46_im <= 5.5e-23)))
		tmp = ((y_46_re / (y_46_im / x_46_im)) - x_46_re) / y_46_im;
	else
		tmp = (x_46_im / y_46_re) - (((y_46_im * x_46_re) / y_46_re) / y_46_re);
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[Or[LessEqual[y$46$im, -7.2e+25], N[Not[LessEqual[y$46$im, 5.5e-23]], $MachinePrecision]], N[(N[(N[(y$46$re / N[(y$46$im / x$46$im), $MachinePrecision]), $MachinePrecision] - x$46$re), $MachinePrecision] / y$46$im), $MachinePrecision], N[(N[(x$46$im / y$46$re), $MachinePrecision] - N[(N[(N[(y$46$im * x$46$re), $MachinePrecision] / y$46$re), $MachinePrecision] / y$46$re), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.im < -7.20000000000000031e25 or 5.5000000000000001e-23 < y.im

   1. Initial program 46.8%

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

   3. Taylor expanded in y.re around 0 67.5%

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

      1. +-commutative 67.5%

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

      2. mul-1-neg 67.5%

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

      3. unsub-neg 67.5%

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

      4. associate-/l* 70.8%

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

      5. associate-/r/ 71.5%

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

   5. Simplified 71.5%

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

      1. *-un-lft-identity 71.5%

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

      2. pow2 71.5%

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

      3. times-frac 76.6%

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

   7. Applied egg-rr 76.6%

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

      1. associate-*l* 79.3%

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

      2. fma-neg 79.3%

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

   9. Applied egg-rr 79.3%

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

       1. fma-udef 79.3%

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

       2. unsub-neg 79.3%

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

       3. associate-*r* 76.6%

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

       4. times-frac 71.5%

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

       5. *-un-lft-identity 71.5%

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

       6. associate-/l/ 76.6%

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

       7. associate-*l/ 79.3%

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

       8. sub-div 79.3%

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

       9. *-commutative 79.3%

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

       10. clear-num 79.3%

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

       11. un-div-inv 79.3%

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

   11. Applied egg-rr 79.3%

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

   if -7.20000000000000031e25 < y.im < 5.5000000000000001e-23

   1. Initial program 77.8%

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

   3. Taylor expanded in y.re around inf 75.7%

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

      1. +-commutative 75.7%

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

      2. mul-1-neg 75.7%

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

      3. unsub-neg 75.7%

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

      4. associate-/l* 75.0%

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

      5. associate-/r/ 75.1%

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

   5. Simplified 75.1%

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

      1. associate-*l/ 75.7%

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

      2. pow2 75.7%

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

      3. associate-/r* 82.3%

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

      4. *-commutative 82.3%

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

   7. Applied egg-rr 82.3%

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

4. Final simplification 80.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -7.2 \cdot 10^{+25} \lor \neg \left(y.im \leq 5.5 \cdot 10^{-23}\right):\\ \;\;\;\;\frac{\frac{y.re}{\frac{y.im}{x.im}} - x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re} - \frac{\frac{y.im \cdot x.re}{y.re}}{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 78.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y.re}{\frac{y.im}{x.im}}\\ \mathbf{if}\;y.im \leq -1.55 \cdot 10^{+28}:\\ \;\;\;\;\frac{t_0}{y.im} - \frac{x.re}{y.im}\\ \mathbf{elif}\;y.im \leq 5.5 \cdot 10^{-23}:\\ \;\;\;\;\frac{x.im}{y.re} - \frac{\frac{y.im \cdot x.re}{y.re}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0 - x.re}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (/ y.re (/ y.im x.im))))
   (if (<= y.im -1.55e+28)
     (- (/ t_0 y.im) (/ x.re y.im))
     (if (<= y.im 5.5e-23)
       (- (/ x.im y.re) (/ (/ (* y.im x.re) y.re) y.re))
       (/ (- t_0 x.re) y.im)))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = y_46_re / (y_46_im / x_46_im);
	double tmp;
	if (y_46_im <= -1.55e+28) {
		tmp = (t_0 / y_46_im) - (x_46_re / y_46_im);
	} else if (y_46_im <= 5.5e-23) {
		tmp = (x_46_im / y_46_re) - (((y_46_im * x_46_re) / y_46_re) / y_46_re);
	} else {
		tmp = (t_0 - 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_46re / (y_46im / x_46im)
    if (y_46im <= (-1.55d+28)) then
        tmp = (t_0 / y_46im) - (x_46re / y_46im)
    else if (y_46im <= 5.5d-23) then
        tmp = (x_46im / y_46re) - (((y_46im * x_46re) / y_46re) / y_46re)
    else
        tmp = (t_0 - 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_re / (y_46_im / x_46_im);
	double tmp;
	if (y_46_im <= -1.55e+28) {
		tmp = (t_0 / y_46_im) - (x_46_re / y_46_im);
	} else if (y_46_im <= 5.5e-23) {
		tmp = (x_46_im / y_46_re) - (((y_46_im * x_46_re) / y_46_re) / y_46_re);
	} else {
		tmp = (t_0 - x_46_re) / y_46_im;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = y_46_re / (y_46_im / x_46_im)
	tmp = 0
	if y_46_im <= -1.55e+28:
		tmp = (t_0 / y_46_im) - (x_46_re / y_46_im)
	elif y_46_im <= 5.5e-23:
		tmp = (x_46_im / y_46_re) - (((y_46_im * x_46_re) / y_46_re) / y_46_re)
	else:
		tmp = (t_0 - 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(y_46_re / Float64(y_46_im / x_46_im))
	tmp = 0.0
	if (y_46_im <= -1.55e+28)
		tmp = Float64(Float64(t_0 / y_46_im) - Float64(x_46_re / y_46_im));
	elseif (y_46_im <= 5.5e-23)
		tmp = Float64(Float64(x_46_im / y_46_re) - Float64(Float64(Float64(y_46_im * x_46_re) / y_46_re) / y_46_re));
	else
		tmp = Float64(Float64(t_0 - x_46_re) / y_46_im);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = y_46_re / (y_46_im / x_46_im);
	tmp = 0.0;
	if (y_46_im <= -1.55e+28)
		tmp = (t_0 / y_46_im) - (x_46_re / y_46_im);
	elseif (y_46_im <= 5.5e-23)
		tmp = (x_46_im / y_46_re) - (((y_46_im * x_46_re) / y_46_re) / y_46_re);
	else
		tmp = (t_0 - 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[(y$46$re / N[(y$46$im / x$46$im), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$im, -1.55e+28], N[(N[(t$95$0 / y$46$im), $MachinePrecision] - N[(x$46$re / y$46$im), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, 5.5e-23], N[(N[(x$46$im / y$46$re), $MachinePrecision] - N[(N[(N[(y$46$im * x$46$re), $MachinePrecision] / y$46$re), $MachinePrecision] / y$46$re), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$0 - x$46$re), $MachinePrecision] / y$46$im), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y.im < -1.55e28

   1. Initial program 43.1%

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

   3. Taylor expanded in y.re around 0 77.1%

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

      1. +-commutative 77.1%

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

      2. mul-1-neg 77.1%

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

      3. unsub-neg 77.1%

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

      4. associate-/l* 79.0%

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

      5. associate-/r/ 80.5%

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

   5. Simplified 80.5%

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

      1. *-un-lft-identity 80.5%

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

      2. pow2 80.5%

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

      3. times-frac 85.0%

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

   7. Applied egg-rr 85.0%

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

      1. associate-*l/ 85.1%

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

      2. *-un-lft-identity 85.1%

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

   9. Applied egg-rr 85.1%

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

       1. associate-*l/ 87.4%

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

       2. *-commutative 87.4%

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

       3. clear-num 87.4%

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

       4. un-div-inv 87.4%

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

   11. Applied egg-rr 87.4%

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

   if -1.55e28 < y.im < 5.5000000000000001e-23

   1. Initial program 77.8%

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

   3. Taylor expanded in y.re around inf 75.7%

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

      1. +-commutative 75.7%

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

      2. mul-1-neg 75.7%

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

      3. unsub-neg 75.7%

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

      4. associate-/l* 75.0%

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

      5. associate-/r/ 75.1%

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

   5. Simplified 75.1%

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

      1. associate-*l/ 75.7%

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

      2. pow2 75.7%

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

      3. associate-/r* 82.3%

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

      4. *-commutative 82.3%

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

   7. Applied egg-rr 82.3%

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

   if 5.5000000000000001e-23 < y.im

   1. Initial program 49.8%

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

   3. Taylor expanded in y.re around 0 59.9%

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

      1. +-commutative 59.9%

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

      2. mul-1-neg 59.9%

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

      3. unsub-neg 59.9%

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

      4. associate-/l* 64.3%

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

      5. associate-/r/ 64.4%

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

   5. Simplified 64.4%

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

      1. *-un-lft-identity 64.4%

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

      2. pow2 64.4%

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

      3. times-frac 70.0%

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

   7. Applied egg-rr 70.0%

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

      1. associate-*l* 72.9%

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

      2. fma-neg 72.9%

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

   9. Applied egg-rr 72.9%

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

       1. fma-udef 72.9%

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

       2. unsub-neg 72.9%

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

       3. associate-*r* 70.0%

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

       4. times-frac 64.4%

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

       5. *-un-lft-identity 64.4%

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

       6. associate-/l/ 70.0%

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

       7. associate-*l/ 72.9%

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

       8. sub-div 72.9%

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

       9. *-commutative 72.9%

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

       10. clear-num 72.9%

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

       11. un-div-inv 72.9%

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

   11. Applied egg-rr 72.9%

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

4. Final simplification 80.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -1.55 \cdot 10^{+28}:\\ \;\;\;\;\frac{\frac{y.re}{\frac{y.im}{x.im}}}{y.im} - \frac{x.re}{y.im}\\ \mathbf{elif}\;y.im \leq 5.5 \cdot 10^{-23}:\\ \;\;\;\;\frac{x.im}{y.re} - \frac{\frac{y.im \cdot x.re}{y.re}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y.re}{\frac{y.im}{x.im}} - x.re}{y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 73.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -1.75 \cdot 10^{-20} \lor \neg \left(y.re \leq 7.4 \cdot 10^{+78}\right):\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im \cdot \frac{y.re}{y.im} - x.re}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.re -1.75e-20) (not (<= y.re 7.4e+78)))
   (/ x.im y.re)
   (/ (- (* x.im (/ y.re y.im)) x.re) y.im)))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_re <= -1.75e-20) || !(y_46_re <= 7.4e+78)) {
		tmp = x_46_im / y_46_re;
	} else {
		tmp = ((x_46_im * (y_46_re / y_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_46re <= (-1.75d-20)) .or. (.not. (y_46re <= 7.4d+78))) then
        tmp = x_46im / y_46re
    else
        tmp = ((x_46im * (y_46re / y_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_re <= -1.75e-20) || !(y_46_re <= 7.4e+78)) {
		tmp = x_46_im / y_46_re;
	} else {
		tmp = ((x_46_im * (y_46_re / y_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_re <= -1.75e-20) or not (y_46_re <= 7.4e+78):
		tmp = x_46_im / y_46_re
	else:
		tmp = ((x_46_im * (y_46_re / y_46_im)) - x_46_re) / y_46_im
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if ((y_46_re <= -1.75e-20) || !(y_46_re <= 7.4e+78))
		tmp = Float64(x_46_im / y_46_re);
	else
		tmp = Float64(Float64(Float64(x_46_im * Float64(y_46_re / y_46_im)) - x_46_re) / y_46_im);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if ((y_46_re <= -1.75e-20) || ~((y_46_re <= 7.4e+78)))
		tmp = x_46_im / y_46_re;
	else
		tmp = ((x_46_im * (y_46_re / y_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[Or[LessEqual[y$46$re, -1.75e-20], N[Not[LessEqual[y$46$re, 7.4e+78]], $MachinePrecision]], N[(x$46$im / y$46$re), $MachinePrecision], N[(N[(N[(x$46$im * N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision] - x$46$re), $MachinePrecision] / y$46$im), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.re < -1.75000000000000002e-20 or 7.39999999999999969e78 < y.re

   1. Initial program 53.0%

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

   3. Taylor expanded in y.re around inf 67.7%

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

   if -1.75000000000000002e-20 < y.re < 7.39999999999999969e78

   1. Initial program 68.6%

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

   3. Taylor expanded in y.re around 0 73.9%

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

      1. +-commutative 73.9%

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

      2. mul-1-neg 73.9%

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

      3. unsub-neg 73.9%

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

      4. associate-/l* 74.7%

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

      5. associate-/r/ 73.9%

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

   5. Simplified 73.9%

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

      1. *-un-lft-identity 73.9%

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

      2. pow2 73.9%

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

      3. times-frac 76.6%

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

   7. Applied egg-rr 76.6%

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

      1. associate-*l* 80.5%

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

      2. fma-neg 80.5%

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

   9. Applied egg-rr 80.5%

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

       1. expm1-log1p-u 63.2%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{fma}\left(\frac{1}{y.im}, \frac{x.im}{y.im} \cdot y.re, -\frac{x.re}{y.im}\right)\right)\right)} \]

       2. expm1-udef 32.2%

          \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\mathsf{fma}\left(\frac{1}{y.im}, \frac{x.im}{y.im} \cdot y.re, -\frac{x.re}{y.im}\right)\right)} - 1} \]

   11. Applied egg-rr 32.3%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\frac{y.re}{\frac{y.im}{x.im}} - x.re}{y.im}\right)} - 1} \]
   12. Step-by-step derivation

       1. expm1-def 63.2%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{y.re}{\frac{y.im}{x.im}} - x.re}{y.im}\right)\right)} \]

       2. expm1-log1p 81.2%

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

       3. associate-/r/ 81.7%

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

   13. Simplified 81.7%

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

4. Final simplification 75.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -1.75 \cdot 10^{-20} \lor \neg \left(y.re \leq 7.4 \cdot 10^{+78}\right):\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im \cdot \frac{y.re}{y.im} - x.re}{y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 64.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -2.1 \cdot 10^{-33} \lor \neg \left(y.re \leq 5.4 \cdot 10^{+25}\right):\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{-x.re}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.re -2.1e-33) (not (<= y.re 5.4e+25)))
   (/ x.im y.re)
   (/ (- x.re) y.im)))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_re <= -2.1e-33) || !(y_46_re <= 5.4e+25)) {
		tmp = x_46_im / y_46_re;
	} else {
		tmp = -x_46_re / y_46_im;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if ((y_46re <= (-2.1d-33)) .or. (.not. (y_46re <= 5.4d+25))) then
        tmp = x_46im / y_46re
    else
        tmp = -x_46re / y_46im
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_re <= -2.1e-33) || !(y_46_re <= 5.4e+25)) {
		tmp = x_46_im / y_46_re;
	} else {
		tmp = -x_46_re / y_46_im;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if (y_46_re <= -2.1e-33) or not (y_46_re <= 5.4e+25):
		tmp = x_46_im / y_46_re
	else:
		tmp = -x_46_re / y_46_im
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if ((y_46_re <= -2.1e-33) || !(y_46_re <= 5.4e+25))
		tmp = Float64(x_46_im / y_46_re);
	else
		tmp = Float64(Float64(-x_46_re) / y_46_im);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if ((y_46_re <= -2.1e-33) || ~((y_46_re <= 5.4e+25)))
		tmp = x_46_im / y_46_re;
	else
		tmp = -x_46_re / y_46_im;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[Or[LessEqual[y$46$re, -2.1e-33], N[Not[LessEqual[y$46$re, 5.4e+25]], $MachinePrecision]], N[(x$46$im / y$46$re), $MachinePrecision], N[((-x$46$re) / y$46$im), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.re < -2.1e-33 or 5.4e25 < y.re

   1. Initial program 53.3%

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

   3. Taylor expanded in y.re around inf 65.1%

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

   if -2.1e-33 < y.re < 5.4e25

   1. Initial program 69.9%

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

   3. Taylor expanded in y.re around 0 72.6%

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

      1. associate-*r/ 72.6%

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

      2. neg-mul-1 72.6%

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

   5. Simplified 72.6%

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

4. Final simplification 68.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -2.1 \cdot 10^{-33} \lor \neg \left(y.re \leq 5.4 \cdot 10^{+25}\right):\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{-x.re}{y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 9.7% accurate, 5.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 61.3%

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

3. Taylor expanded in x.im around inf 36.5%

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

   1. *-commutative 36.5%

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

5. Simplified 36.5%

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

   1. pow2 36.5%

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

   2. add-sqr-sqrt 36.5%

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

   3. times-frac 38.9%

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

   4. +-commutative 38.9%

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

   5. pow2 38.9%

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

   6. hypot-def 38.9%

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

   7. +-commutative 38.9%

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

   8. pow2 38.9%

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

   9. hypot-def 53.0%

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

7. Applied egg-rr 53.0%

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

8. Taylor expanded in y.im around inf 15.5%

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

9. Taylor expanded in y.re around inf 8.8%

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

10. Final simplification 8.8%

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

Alternative 10: 43.3% 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 61.3%

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

3. Taylor expanded in y.re around inf 40.4%

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

4. Final simplification 40.4%

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

Reproduce

herbie shell --seed 2024026 
(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))))