_divideComplex, imaginary part

Percentage Accurate: 61.4% → 83.8%

Time: 12.9s

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: 61.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 83.8% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -4 \cdot 10^{+43}:\\ \;\;\;\;\frac{y.re \cdot \frac{x.im}{y.im} - x.re}{y.im}\\ \mathbf{elif}\;y.im \leq -3.8 \cdot 10^{-132}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x.im, y.re, y.im \cdot \left(-x.re\right)\right)}{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)}\\ \mathbf{elif}\;y.im \leq 1.16 \cdot 10^{-157}:\\ \;\;\;\;\frac{x.im - \frac{y.im \cdot x.re}{y.re}}{y.re}\\ \mathbf{elif}\;y.im \leq 1.8 \cdot 10^{+91}:\\ \;\;\;\;\frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\frac{\frac{y.im}{x.im}}{y.re}} - x.re}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.im -4e+43)
   (/ (- (* y.re (/ x.im y.im)) x.re) y.im)
   (if (<= y.im -3.8e-132)
     (/ (fma x.im y.re (* y.im (- x.re))) (fma y.im y.im (* y.re y.re)))
     (if (<= y.im 1.16e-157)
       (/ (- x.im (/ (* y.im x.re) y.re)) y.re)
       (if (<= y.im 1.8e+91)
         (/ (- (* y.re x.im) (* y.im x.re)) (+ (* y.re y.re) (* y.im y.im)))
         (/ (- (/ 1.0 (/ (/ y.im 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_im <= -4e+43) {
		tmp = ((y_46_re * (x_46_im / y_46_im)) - x_46_re) / y_46_im;
	} else if (y_46_im <= -3.8e-132) {
		tmp = fma(x_46_im, y_46_re, (y_46_im * -x_46_re)) / fma(y_46_im, y_46_im, (y_46_re * y_46_re));
	} else if (y_46_im <= 1.16e-157) {
		tmp = (x_46_im - ((y_46_im * x_46_re) / y_46_re)) / y_46_re;
	} else if (y_46_im <= 1.8e+91) {
		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 {
		tmp = ((1.0 / ((y_46_im / x_46_im) / y_46_re)) - x_46_re) / y_46_im;
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (y_46_im <= -4e+43)
		tmp = Float64(Float64(Float64(y_46_re * Float64(x_46_im / y_46_im)) - x_46_re) / y_46_im);
	elseif (y_46_im <= -3.8e-132)
		tmp = Float64(fma(x_46_im, y_46_re, Float64(y_46_im * Float64(-x_46_re))) / fma(y_46_im, y_46_im, Float64(y_46_re * y_46_re)));
	elseif (y_46_im <= 1.16e-157)
		tmp = Float64(Float64(x_46_im - Float64(Float64(y_46_im * x_46_re) / y_46_re)) / y_46_re);
	elseif (y_46_im <= 1.8e+91)
		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)));
	else
		tmp = Float64(Float64(Float64(1.0 / Float64(Float64(y_46_im / x_46_im) / y_46_re)) - x_46_re) / y_46_im);
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[y$46$im, -4e+43], N[(N[(N[(y$46$re * N[(x$46$im / y$46$im), $MachinePrecision]), $MachinePrecision] - x$46$re), $MachinePrecision] / y$46$im), $MachinePrecision], If[LessEqual[y$46$im, -3.8e-132], N[(N[(x$46$im * y$46$re + N[(y$46$im * (-x$46$re)), $MachinePrecision]), $MachinePrecision] / N[(y$46$im * y$46$im + N[(y$46$re * y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, 1.16e-157], N[(N[(x$46$im - N[(N[(y$46$im * x$46$re), $MachinePrecision] / y$46$re), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision], If[LessEqual[y$46$im, 1.8e+91], 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], N[(N[(N[(1.0 / N[(N[(y$46$im / x$46$im), $MachinePrecision] / y$46$re), $MachinePrecision]), $MachinePrecision] - x$46$re), $MachinePrecision] / y$46$im), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if y.im < -4.00000000000000006e43

   1. Initial program 57.1%

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

      1. fma-neg 57.1%

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

      2. distribute-rgt-neg-out 57.1%

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

      3. +-commutative 57.1%

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

      4. fma-define 57.1%

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

   3. Simplified 57.1%

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

   5. Taylor expanded in y.re around 0 89.9%

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

      1. +-commutative 89.9%

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

      2. mul-1-neg 89.9%

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

      3. unsub-neg 89.9%

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

      4. unpow2 89.9%

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

      5. associate-/r* 89.0%

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

      6. div-sub 89.0%

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

      7. *-commutative 89.0%

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

      8. associate-/l* 94.8%

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

   7. Simplified 94.8%

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

   if -4.00000000000000006e43 < y.im < -3.7999999999999997e-132

   1. Initial program 84.7%

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

      1. fma-neg 84.7%

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

      2. distribute-rgt-neg-out 84.7%

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

      3. +-commutative 84.7%

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

      4. fma-define 84.7%

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

   3. Simplified 84.7%

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

   if -3.7999999999999997e-132 < y.im < 1.15999999999999992e-157

   1. Initial program 61.8%

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

      1. fma-neg 61.8%

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

      2. distribute-rgt-neg-out 61.8%

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

      3. +-commutative 61.8%

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

      4. fma-define 61.8%

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

   3. Simplified 61.8%

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

   5. Taylor expanded in y.re around inf 90.6%

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

      1. mul-1-neg 90.6%

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

      2. unsub-neg 90.6%

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

      3. *-commutative 90.6%

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

      4. associate-/l* 89.6%

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

   7. Simplified 89.6%

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

      1. associate-*r/ 90.6%

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

   9. Applied egg-rr 90.6%

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

   if 1.15999999999999992e-157 < y.im < 1.8e91

   1. Initial program 79.8%

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

   if 1.8e91 < y.im

   1. Initial program 39.3%

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

      1. fma-neg 39.3%

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

      2. distribute-rgt-neg-out 39.3%

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

      3. +-commutative 39.3%

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

      4. fma-define 39.3%

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

   3. Simplified 39.3%

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

   5. Taylor expanded in y.im around inf 77.6%

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

      1. clear-num 77.6%

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

      2. inv-pow 77.6%

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

   7. Applied egg-rr 77.6%

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

      1. unpow-1 77.6%

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

      2. associate-/r* 77.8%

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

   9. Simplified 77.8%

      \[\leadsto \frac{-1 \cdot x.re + \color{blue}{\frac{1}{\frac{\frac{y.im}{x.im}}{y.re}}}}{y.im} \]
3. Recombined 5 regimes into one program.

4. Final simplification 86.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -4 \cdot 10^{+43}:\\ \;\;\;\;\frac{y.re \cdot \frac{x.im}{y.im} - x.re}{y.im}\\ \mathbf{elif}\;y.im \leq -3.8 \cdot 10^{-132}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x.im, y.re, y.im \cdot \left(-x.re\right)\right)}{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)}\\ \mathbf{elif}\;y.im \leq 1.16 \cdot 10^{-157}:\\ \;\;\;\;\frac{x.im - \frac{y.im \cdot x.re}{y.re}}{y.re}\\ \mathbf{elif}\;y.im \leq 1.8 \cdot 10^{+91}:\\ \;\;\;\;\frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\frac{\frac{y.im}{x.im}}{y.re}} - x.re}{y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 83.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{if}\;y.im \leq -2.9 \cdot 10^{+43}:\\ \;\;\;\;\frac{y.re \cdot \frac{x.im}{y.im} - x.re}{y.im}\\ \mathbf{elif}\;y.im \leq -9.8 \cdot 10^{-131}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.im \leq 1.05 \cdot 10^{-157}:\\ \;\;\;\;\frac{x.im - \frac{y.im \cdot x.re}{y.re}}{y.re}\\ \mathbf{elif}\;y.im \leq 2.5 \cdot 10^{+91}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\frac{\frac{y.im}{x.im}}{y.re}} - x.re}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0
         (/ (- (* y.re x.im) (* y.im x.re)) (+ (* y.re y.re) (* y.im y.im)))))
   (if (<= y.im -2.9e+43)
     (/ (- (* y.re (/ x.im y.im)) x.re) y.im)
     (if (<= y.im -9.8e-131)
       t_0
       (if (<= y.im 1.05e-157)
         (/ (- x.im (/ (* y.im x.re) y.re)) y.re)
         (if (<= y.im 2.5e+91)
           t_0
           (/ (- (/ 1.0 (/ (/ y.im 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 t_0 = ((y_46_re * x_46_im) - (y_46_im * x_46_re)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	double tmp;
	if (y_46_im <= -2.9e+43) {
		tmp = ((y_46_re * (x_46_im / y_46_im)) - x_46_re) / y_46_im;
	} else if (y_46_im <= -9.8e-131) {
		tmp = t_0;
	} else if (y_46_im <= 1.05e-157) {
		tmp = (x_46_im - ((y_46_im * x_46_re) / y_46_re)) / y_46_re;
	} else if (y_46_im <= 2.5e+91) {
		tmp = t_0;
	} else {
		tmp = ((1.0 / ((y_46_im / x_46_im) / y_46_re)) - 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 * x_46im) - (y_46im * x_46re)) / ((y_46re * y_46re) + (y_46im * y_46im))
    if (y_46im <= (-2.9d+43)) then
        tmp = ((y_46re * (x_46im / y_46im)) - x_46re) / y_46im
    else if (y_46im <= (-9.8d-131)) then
        tmp = t_0
    else if (y_46im <= 1.05d-157) then
        tmp = (x_46im - ((y_46im * x_46re) / y_46re)) / y_46re
    else if (y_46im <= 2.5d+91) then
        tmp = t_0
    else
        tmp = ((1.0d0 / ((y_46im / x_46im) / y_46re)) - 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 tmp;
	if (y_46_im <= -2.9e+43) {
		tmp = ((y_46_re * (x_46_im / y_46_im)) - x_46_re) / y_46_im;
	} else if (y_46_im <= -9.8e-131) {
		tmp = t_0;
	} else if (y_46_im <= 1.05e-157) {
		tmp = (x_46_im - ((y_46_im * x_46_re) / y_46_re)) / y_46_re;
	} else if (y_46_im <= 2.5e+91) {
		tmp = t_0;
	} else {
		tmp = ((1.0 / ((y_46_im / x_46_im) / y_46_re)) - x_46_re) / y_46_im;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = ((y_46_re * x_46_im) - (y_46_im * x_46_re)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im))
	tmp = 0
	if y_46_im <= -2.9e+43:
		tmp = ((y_46_re * (x_46_im / y_46_im)) - x_46_re) / y_46_im
	elif y_46_im <= -9.8e-131:
		tmp = t_0
	elif y_46_im <= 1.05e-157:
		tmp = (x_46_im - ((y_46_im * x_46_re) / y_46_re)) / y_46_re
	elif y_46_im <= 2.5e+91:
		tmp = t_0
	else:
		tmp = ((1.0 / ((y_46_im / x_46_im) / y_46_re)) - x_46_re) / y_46_im
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(Float64(Float64(y_46_re * x_46_im) - Float64(y_46_im * x_46_re)) / Float64(Float64(y_46_re * y_46_re) + Float64(y_46_im * y_46_im)))
	tmp = 0.0
	if (y_46_im <= -2.9e+43)
		tmp = Float64(Float64(Float64(y_46_re * Float64(x_46_im / y_46_im)) - x_46_re) / y_46_im);
	elseif (y_46_im <= -9.8e-131)
		tmp = t_0;
	elseif (y_46_im <= 1.05e-157)
		tmp = Float64(Float64(x_46_im - Float64(Float64(y_46_im * x_46_re) / y_46_re)) / y_46_re);
	elseif (y_46_im <= 2.5e+91)
		tmp = t_0;
	else
		tmp = Float64(Float64(Float64(1.0 / Float64(Float64(y_46_im / x_46_im) / y_46_re)) - x_46_re) / y_46_im);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = ((y_46_re * x_46_im) - (y_46_im * x_46_re)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	tmp = 0.0;
	if (y_46_im <= -2.9e+43)
		tmp = ((y_46_re * (x_46_im / y_46_im)) - x_46_re) / y_46_im;
	elseif (y_46_im <= -9.8e-131)
		tmp = t_0;
	elseif (y_46_im <= 1.05e-157)
		tmp = (x_46_im - ((y_46_im * x_46_re) / y_46_re)) / y_46_re;
	elseif (y_46_im <= 2.5e+91)
		tmp = t_0;
	else
		tmp = ((1.0 / ((y_46_im / x_46_im) / y_46_re)) - x_46_re) / y_46_im;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[(N[(y$46$re * x$46$im), $MachinePrecision] - N[(y$46$im * x$46$re), $MachinePrecision]), $MachinePrecision] / N[(N[(y$46$re * y$46$re), $MachinePrecision] + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$im, -2.9e+43], N[(N[(N[(y$46$re * N[(x$46$im / y$46$im), $MachinePrecision]), $MachinePrecision] - x$46$re), $MachinePrecision] / y$46$im), $MachinePrecision], If[LessEqual[y$46$im, -9.8e-131], t$95$0, If[LessEqual[y$46$im, 1.05e-157], N[(N[(x$46$im - N[(N[(y$46$im * x$46$re), $MachinePrecision] / y$46$re), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision], If[LessEqual[y$46$im, 2.5e+91], t$95$0, N[(N[(N[(1.0 / N[(N[(y$46$im / x$46$im), $MachinePrecision] / y$46$re), $MachinePrecision]), $MachinePrecision] - x$46$re), $MachinePrecision] / y$46$im), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if y.im < -2.9000000000000002e43

   1. Initial program 57.1%

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

      1. fma-neg 57.1%

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

      2. distribute-rgt-neg-out 57.1%

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

      3. +-commutative 57.1%

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

      4. fma-define 57.1%

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

   3. Simplified 57.1%

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

   5. Taylor expanded in y.re around 0 89.9%

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

      1. +-commutative 89.9%

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

      2. mul-1-neg 89.9%

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

      3. unsub-neg 89.9%

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

      4. unpow2 89.9%

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

      5. associate-/r* 89.0%

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

      6. div-sub 89.0%

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

      7. *-commutative 89.0%

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

      8. associate-/l* 94.8%

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

   7. Simplified 94.8%

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

   if -2.9000000000000002e43 < y.im < -9.8000000000000004e-131 or 1.05e-157 < y.im < 2.5000000000000001e91

   1. Initial program 82.1%

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

   if -9.8000000000000004e-131 < y.im < 1.05e-157

   1. Initial program 61.8%

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

      1. fma-neg 61.8%

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

      2. distribute-rgt-neg-out 61.8%

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

      3. +-commutative 61.8%

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

      4. fma-define 61.8%

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

   3. Simplified 61.8%

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

   5. Taylor expanded in y.re around inf 90.6%

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

      1. mul-1-neg 90.6%

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

      2. unsub-neg 90.6%

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

      3. *-commutative 90.6%

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

      4. associate-/l* 89.6%

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

   7. Simplified 89.6%

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

      1. associate-*r/ 90.6%

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

   9. Applied egg-rr 90.6%

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

   if 2.5000000000000001e91 < y.im

   1. Initial program 39.3%

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

      1. fma-neg 39.3%

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

      2. distribute-rgt-neg-out 39.3%

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

      3. +-commutative 39.3%

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

      4. fma-define 39.3%

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

   3. Simplified 39.3%

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

   5. Taylor expanded in y.im around inf 77.6%

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

      1. clear-num 77.6%

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

      2. inv-pow 77.6%

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

   7. Applied egg-rr 77.6%

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

      1. unpow-1 77.6%

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

      2. associate-/r* 77.8%

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

   9. Simplified 77.8%

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

4. Final simplification 86.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -2.9 \cdot 10^{+43}:\\ \;\;\;\;\frac{y.re \cdot \frac{x.im}{y.im} - x.re}{y.im}\\ \mathbf{elif}\;y.im \leq -9.8 \cdot 10^{-131}:\\ \;\;\;\;\frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.im \leq 1.05 \cdot 10^{-157}:\\ \;\;\;\;\frac{x.im - \frac{y.im \cdot x.re}{y.re}}{y.re}\\ \mathbf{elif}\;y.im \leq 2.5 \cdot 10^{+91}:\\ \;\;\;\;\frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\frac{\frac{y.im}{x.im}}{y.re}} - x.re}{y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 79.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -1.05 \cdot 10^{+40} \lor \neg \left(y.im \leq 2600000000\right):\\ \;\;\;\;\frac{y.re \cdot \frac{x.im}{y.im} - x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im - \frac{y.im \cdot x.re}{y.re}}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.im -1.05e+40) (not (<= y.im 2600000000.0)))
   (/ (- (* y.re (/ x.im y.im)) x.re) y.im)
   (/ (- x.im (/ (* 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 <= -1.05e+40) || !(y_46_im <= 2600000000.0)) {
		tmp = ((y_46_re * (x_46_im / y_46_im)) - x_46_re) / y_46_im;
	} else {
		tmp = (x_46_im - ((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 <= (-1.05d+40)) .or. (.not. (y_46im <= 2600000000.0d0))) then
        tmp = ((y_46re * (x_46im / y_46im)) - x_46re) / y_46im
    else
        tmp = (x_46im - ((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 <= -1.05e+40) || !(y_46_im <= 2600000000.0)) {
		tmp = ((y_46_re * (x_46_im / y_46_im)) - x_46_re) / y_46_im;
	} else {
		tmp = (x_46_im - ((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 <= -1.05e+40) or not (y_46_im <= 2600000000.0):
		tmp = ((y_46_re * (x_46_im / y_46_im)) - x_46_re) / y_46_im
	else:
		tmp = (x_46_im - ((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 <= -1.05e+40) || !(y_46_im <= 2600000000.0))
		tmp = Float64(Float64(Float64(y_46_re * Float64(x_46_im / y_46_im)) - x_46_re) / y_46_im);
	else
		tmp = Float64(Float64(x_46_im - Float64(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 <= -1.05e+40) || ~((y_46_im <= 2600000000.0)))
		tmp = ((y_46_re * (x_46_im / y_46_im)) - x_46_re) / y_46_im;
	else
		tmp = (x_46_im - ((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, -1.05e+40], N[Not[LessEqual[y$46$im, 2600000000.0]], $MachinePrecision]], N[(N[(N[(y$46$re * N[(x$46$im / y$46$im), $MachinePrecision]), $MachinePrecision] - x$46$re), $MachinePrecision] / y$46$im), $MachinePrecision], N[(N[(x$46$im - N[(N[(y$46$im * x$46$re), $MachinePrecision] / y$46$re), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.im < -1.05000000000000005e40 or 2.6e9 < y.im

   1. Initial program 53.7%

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

      1. fma-neg 53.7%

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

      2. distribute-rgt-neg-out 53.7%

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

      3. +-commutative 53.7%

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

      4. fma-define 53.7%

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

   3. Simplified 53.7%

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

   5. Taylor expanded in y.re around 0 78.1%

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

      1. +-commutative 78.1%

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

      2. mul-1-neg 78.1%

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

      3. unsub-neg 78.1%

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

      4. unpow2 78.1%

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

      5. associate-/r* 79.3%

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

      6. div-sub 79.3%

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

      7. *-commutative 79.3%

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

      8. associate-/l* 82.8%

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

   7. Simplified 82.8%

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

   if -1.05000000000000005e40 < y.im < 2.6e9

   1. Initial program 71.7%

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

      1. fma-neg 71.7%

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

      2. distribute-rgt-neg-out 71.7%

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

      3. +-commutative 71.7%

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

      4. fma-define 71.7%

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

   3. Simplified 71.7%

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

   5. Taylor expanded in y.re around inf 81.5%

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

      1. mul-1-neg 81.5%

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

      2. unsub-neg 81.5%

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

      3. *-commutative 81.5%

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

      4. associate-/l* 80.3%

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

   7. Simplified 80.3%

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

      1. associate-*r/ 81.5%

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

   9. Applied egg-rr 81.5%

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

4. Final simplification 82.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -1.05 \cdot 10^{+40} \lor \neg \left(y.im \leq 2600000000\right):\\ \;\;\;\;\frac{y.re \cdot \frac{x.im}{y.im} - x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im - \frac{y.im \cdot x.re}{y.re}}{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 78.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -4.8 \cdot 10^{+39} \lor \neg \left(y.im \leq 1800000000\right):\\ \;\;\;\;\frac{x.im \cdot \frac{y.re}{y.im} - x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im - \frac{y.im \cdot x.re}{y.re}}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.im -4.8e+39) (not (<= y.im 1800000000.0)))
   (/ (- (* x.im (/ y.re y.im)) x.re) y.im)
   (/ (- x.im (/ (* 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 <= -4.8e+39) || !(y_46_im <= 1800000000.0)) {
		tmp = ((x_46_im * (y_46_re / y_46_im)) - x_46_re) / y_46_im;
	} else {
		tmp = (x_46_im - ((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 <= (-4.8d+39)) .or. (.not. (y_46im <= 1800000000.0d0))) then
        tmp = ((x_46im * (y_46re / y_46im)) - x_46re) / y_46im
    else
        tmp = (x_46im - ((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 <= -4.8e+39) || !(y_46_im <= 1800000000.0)) {
		tmp = ((x_46_im * (y_46_re / y_46_im)) - x_46_re) / y_46_im;
	} else {
		tmp = (x_46_im - ((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 <= -4.8e+39) or not (y_46_im <= 1800000000.0):
		tmp = ((x_46_im * (y_46_re / y_46_im)) - x_46_re) / y_46_im
	else:
		tmp = (x_46_im - ((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 <= -4.8e+39) || !(y_46_im <= 1800000000.0))
		tmp = Float64(Float64(Float64(x_46_im * Float64(y_46_re / y_46_im)) - x_46_re) / y_46_im);
	else
		tmp = Float64(Float64(x_46_im - Float64(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 <= -4.8e+39) || ~((y_46_im <= 1800000000.0)))
		tmp = ((x_46_im * (y_46_re / y_46_im)) - x_46_re) / y_46_im;
	else
		tmp = (x_46_im - ((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, -4.8e+39], N[Not[LessEqual[y$46$im, 1800000000.0]], $MachinePrecision]], N[(N[(N[(x$46$im * N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision] - x$46$re), $MachinePrecision] / y$46$im), $MachinePrecision], N[(N[(x$46$im - N[(N[(y$46$im * x$46$re), $MachinePrecision] / y$46$re), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.im < -4.8000000000000002e39 or 1.8e9 < y.im

   1. Initial program 53.7%

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

      1. fma-neg 53.7%

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

      2. distribute-rgt-neg-out 53.7%

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

      3. +-commutative 53.7%

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

      4. fma-define 53.7%

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

   3. Simplified 53.7%

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

   5. Taylor expanded in y.im around inf 79.3%

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

      1. neg-mul-1 79.3%

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

      2. +-commutative 79.3%

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

      3. unsub-neg 79.3%

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

      4. associate-/l* 81.7%

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

   7. Applied egg-rr 81.7%

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

   if -4.8000000000000002e39 < y.im < 1.8e9

   1. Initial program 71.7%

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

      1. fma-neg 71.7%

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

      2. distribute-rgt-neg-out 71.7%

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

      3. +-commutative 71.7%

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

      4. fma-define 71.7%

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

   3. Simplified 71.7%

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

   5. Taylor expanded in y.re around inf 81.5%

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

      1. mul-1-neg 81.5%

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

      2. unsub-neg 81.5%

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

      3. *-commutative 81.5%

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

      4. associate-/l* 80.3%

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

   7. Simplified 80.3%

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

      1. associate-*r/ 81.5%

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

   9. Applied egg-rr 81.5%

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

4. Final simplification 81.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -4.8 \cdot 10^{+39} \lor \neg \left(y.im \leq 1800000000\right):\\ \;\;\;\;\frac{x.im \cdot \frac{y.re}{y.im} - x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im - \frac{y.im \cdot x.re}{y.re}}{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 73.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -2.7 \cdot 10^{+39} \lor \neg \left(y.im \leq 9 \cdot 10^{+113}\right):\\ \;\;\;\;\frac{-x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im - \frac{y.im \cdot x.re}{y.re}}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.im -2.7e+39) (not (<= y.im 9e+113)))
   (/ (- x.re) y.im)
   (/ (- x.im (/ (* 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 <= -2.7e+39) || !(y_46_im <= 9e+113)) {
		tmp = -x_46_re / y_46_im;
	} else {
		tmp = (x_46_im - ((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 <= (-2.7d+39)) .or. (.not. (y_46im <= 9d+113))) then
        tmp = -x_46re / y_46im
    else
        tmp = (x_46im - ((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 <= -2.7e+39) || !(y_46_im <= 9e+113)) {
		tmp = -x_46_re / y_46_im;
	} else {
		tmp = (x_46_im - ((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 <= -2.7e+39) or not (y_46_im <= 9e+113):
		tmp = -x_46_re / y_46_im
	else:
		tmp = (x_46_im - ((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 <= -2.7e+39) || !(y_46_im <= 9e+113))
		tmp = Float64(Float64(-x_46_re) / y_46_im);
	else
		tmp = Float64(Float64(x_46_im - 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 <= -2.7e+39) || ~((y_46_im <= 9e+113)))
		tmp = -x_46_re / y_46_im;
	else
		tmp = (x_46_im - ((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, -2.7e+39], N[Not[LessEqual[y$46$im, 9e+113]], $MachinePrecision]], N[((-x$46$re) / y$46$im), $MachinePrecision], N[(N[(x$46$im - N[(N[(y$46$im * x$46$re), $MachinePrecision] / y$46$re), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.im < -2.70000000000000003e39 or 9.0000000000000001e113 < y.im

   1. Initial program 50.3%

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

      1. fma-neg 50.3%

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

      2. distribute-rgt-neg-out 50.3%

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

      3. +-commutative 50.3%

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

      4. fma-define 50.3%

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

   3. Simplified 50.3%

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

   5. Taylor expanded in y.re around 0 79.0%

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

      1. associate-*r/ 79.0%

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

      2. neg-mul-1 79.0%

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

   7. Simplified 79.0%

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

   if -2.70000000000000003e39 < y.im < 9.0000000000000001e113

   1. Initial program 70.7%

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

      1. fma-neg 70.7%

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

      2. distribute-rgt-neg-out 70.7%

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

      3. +-commutative 70.7%

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

      4. fma-define 70.7%

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

   3. Simplified 70.7%

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

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

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

      1. mul-1-neg 76.4%

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

      2. unsub-neg 76.4%

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

      3. *-commutative 76.4%

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

      4. associate-/l* 75.9%

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

   7. Simplified 75.9%

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

      1. associate-*r/ 76.4%

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

   9. Applied egg-rr 76.4%

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

4. Final simplification 77.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -2.7 \cdot 10^{+39} \lor \neg \left(y.im \leq 9 \cdot 10^{+113}\right):\\ \;\;\;\;\frac{-x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im - \frac{y.im \cdot x.re}{y.re}}{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 72.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -5.5 \cdot 10^{+42} \lor \neg \left(y.im \leq 2 \cdot 10^{+114}\right):\\ \;\;\;\;\frac{-x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im - y.im \cdot \frac{x.re}{y.re}}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.im -5.5e+42) (not (<= y.im 2e+114)))
   (/ (- x.re) y.im)
   (/ (- x.im (* 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 <= -5.5e+42) || !(y_46_im <= 2e+114)) {
		tmp = -x_46_re / y_46_im;
	} else {
		tmp = (x_46_im - (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 <= (-5.5d+42)) .or. (.not. (y_46im <= 2d+114))) then
        tmp = -x_46re / y_46im
    else
        tmp = (x_46im - (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 <= -5.5e+42) || !(y_46_im <= 2e+114)) {
		tmp = -x_46_re / y_46_im;
	} else {
		tmp = (x_46_im - (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 <= -5.5e+42) or not (y_46_im <= 2e+114):
		tmp = -x_46_re / y_46_im
	else:
		tmp = (x_46_im - (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 <= -5.5e+42) || !(y_46_im <= 2e+114))
		tmp = Float64(Float64(-x_46_re) / y_46_im);
	else
		tmp = Float64(Float64(x_46_im - Float64(y_46_im * Float64(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 <= -5.5e+42) || ~((y_46_im <= 2e+114)))
		tmp = -x_46_re / y_46_im;
	else
		tmp = (x_46_im - (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, -5.5e+42], N[Not[LessEqual[y$46$im, 2e+114]], $MachinePrecision]], N[((-x$46$re) / y$46$im), $MachinePrecision], N[(N[(x$46$im - N[(y$46$im * N[(x$46$re / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.im < -5.50000000000000001e42 or 2e114 < y.im

   1. Initial program 50.3%

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

      1. fma-neg 50.3%

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

      2. distribute-rgt-neg-out 50.3%

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

      3. +-commutative 50.3%

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

      4. fma-define 50.3%

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

   3. Simplified 50.3%

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

   5. Taylor expanded in y.re around 0 79.0%

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

      1. associate-*r/ 79.0%

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

      2. neg-mul-1 79.0%

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

   7. Simplified 79.0%

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

   if -5.50000000000000001e42 < y.im < 2e114

   1. Initial program 70.7%

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

      1. fma-neg 70.7%

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

      2. distribute-rgt-neg-out 70.7%

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

      3. +-commutative 70.7%

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

      4. fma-define 70.7%

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

   3. Simplified 70.7%

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

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

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

      1. mul-1-neg 76.4%

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

      2. unsub-neg 76.4%

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

      3. *-commutative 76.4%

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

      4. associate-/l* 75.9%

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

   7. Simplified 75.9%

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

4. Final simplification 76.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -5.5 \cdot 10^{+42} \lor \neg \left(y.im \leq 2 \cdot 10^{+114}\right):\\ \;\;\;\;\frac{-x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im - y.im \cdot \frac{x.re}{y.re}}{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 73.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -4.1 \cdot 10^{+40} \lor \neg \left(y.im \leq 4.4 \cdot 10^{+114}\right):\\ \;\;\;\;\frac{-x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im - x.re \cdot \frac{y.im}{y.re}}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.im -4.1e+40) (not (<= y.im 4.4e+114)))
   (/ (- x.re) y.im)
   (/ (- x.im (* x.re (/ y.im y.re))) y.re)))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_im <= -4.1e+40) || !(y_46_im <= 4.4e+114)) {
		tmp = -x_46_re / y_46_im;
	} else {
		tmp = (x_46_im - (x_46_re * (y_46_im / y_46_re))) / y_46_re;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if ((y_46im <= (-4.1d+40)) .or. (.not. (y_46im <= 4.4d+114))) then
        tmp = -x_46re / y_46im
    else
        tmp = (x_46im - (x_46re * (y_46im / y_46re))) / y_46re
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_im <= -4.1e+40) || !(y_46_im <= 4.4e+114)) {
		tmp = -x_46_re / y_46_im;
	} else {
		tmp = (x_46_im - (x_46_re * (y_46_im / y_46_re))) / y_46_re;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if (y_46_im <= -4.1e+40) or not (y_46_im <= 4.4e+114):
		tmp = -x_46_re / y_46_im
	else:
		tmp = (x_46_im - (x_46_re * (y_46_im / y_46_re))) / y_46_re
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if ((y_46_im <= -4.1e+40) || !(y_46_im <= 4.4e+114))
		tmp = Float64(Float64(-x_46_re) / y_46_im);
	else
		tmp = Float64(Float64(x_46_im - Float64(x_46_re * Float64(y_46_im / y_46_re))) / y_46_re);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if ((y_46_im <= -4.1e+40) || ~((y_46_im <= 4.4e+114)))
		tmp = -x_46_re / y_46_im;
	else
		tmp = (x_46_im - (x_46_re * (y_46_im / y_46_re))) / y_46_re;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[Or[LessEqual[y$46$im, -4.1e+40], N[Not[LessEqual[y$46$im, 4.4e+114]], $MachinePrecision]], N[((-x$46$re) / y$46$im), $MachinePrecision], N[(N[(x$46$im - N[(x$46$re * N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.im < -4.1000000000000002e40 or 4.4000000000000001e114 < y.im

   1. Initial program 50.3%

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

      1. fma-neg 50.3%

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

      2. distribute-rgt-neg-out 50.3%

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

      3. +-commutative 50.3%

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

      4. fma-define 50.3%

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

   3. Simplified 50.3%

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

   5. Taylor expanded in y.re around 0 79.0%

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

      1. associate-*r/ 79.0%

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

      2. neg-mul-1 79.0%

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

   7. Simplified 79.0%

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

   if -4.1000000000000002e40 < y.im < 4.4000000000000001e114

   1. Initial program 70.7%

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

      1. fma-neg 70.7%

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

      2. distribute-rgt-neg-out 70.7%

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

      3. +-commutative 70.7%

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

      4. fma-define 70.7%

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

   3. Simplified 70.7%

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

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

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

      1. mul-1-neg 76.4%

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

      2. unsub-neg 76.4%

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

      3. *-commutative 76.4%

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

      4. associate-/l* 75.9%

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

   7. Simplified 75.9%

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

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

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

      1. mul-1-neg 76.4%

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

      2. sub-neg 76.4%

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

      3. associate-*r/ 75.6%

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

   10. Simplified 75.6%

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

4. Final simplification 76.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -4.1 \cdot 10^{+40} \lor \neg \left(y.im \leq 4.4 \cdot 10^{+114}\right):\\ \;\;\;\;\frac{-x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im - x.re \cdot \frac{y.im}{y.re}}{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 63.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -2.2 \cdot 10^{+39} \lor \neg \left(y.im \leq 9 \cdot 10^{+113}\right):\\ \;\;\;\;\frac{-x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.im -2.2e+39) (not (<= y.im 9e+113)))
   (/ (- x.re) y.im)
   (/ x.im y.re)))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_im <= -2.2e+39) || !(y_46_im <= 9e+113)) {
		tmp = -x_46_re / y_46_im;
	} else {
		tmp = x_46_im / y_46_re;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if ((y_46im <= (-2.2d+39)) .or. (.not. (y_46im <= 9d+113))) then
        tmp = -x_46re / y_46im
    else
        tmp = x_46im / y_46re
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_im <= -2.2e+39) || !(y_46_im <= 9e+113)) {
		tmp = -x_46_re / y_46_im;
	} else {
		tmp = x_46_im / y_46_re;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if (y_46_im <= -2.2e+39) or not (y_46_im <= 9e+113):
		tmp = -x_46_re / y_46_im
	else:
		tmp = x_46_im / y_46_re
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if ((y_46_im <= -2.2e+39) || !(y_46_im <= 9e+113))
		tmp = Float64(Float64(-x_46_re) / y_46_im);
	else
		tmp = Float64(x_46_im / y_46_re);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if ((y_46_im <= -2.2e+39) || ~((y_46_im <= 9e+113)))
		tmp = -x_46_re / y_46_im;
	else
		tmp = x_46_im / y_46_re;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[Or[LessEqual[y$46$im, -2.2e+39], N[Not[LessEqual[y$46$im, 9e+113]], $MachinePrecision]], N[((-x$46$re) / y$46$im), $MachinePrecision], N[(x$46$im / y$46$re), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.im < -2.2000000000000001e39 or 9.0000000000000001e113 < y.im

   1. Initial program 50.3%

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

      1. fma-neg 50.3%

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

      2. distribute-rgt-neg-out 50.3%

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

      3. +-commutative 50.3%

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

      4. fma-define 50.3%

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

   3. Simplified 50.3%

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

   5. Taylor expanded in y.re around 0 79.0%

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

      1. associate-*r/ 79.0%

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

      2. neg-mul-1 79.0%

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

   7. Simplified 79.0%

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

   if -2.2000000000000001e39 < y.im < 9.0000000000000001e113

   1. Initial program 70.7%

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

      1. fma-neg 70.7%

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

      2. distribute-rgt-neg-out 70.7%

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

      3. +-commutative 70.7%

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

      4. fma-define 70.7%

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

   3. Simplified 70.7%

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

   5. Taylor expanded in y.re around inf 59.1%

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

4. Final simplification 65.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -2.2 \cdot 10^{+39} \lor \neg \left(y.im \leq 9 \cdot 10^{+113}\right):\\ \;\;\;\;\frac{-x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 46.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -2.95 \cdot 10^{+113} \lor \neg \left(y.im \leq 7.6 \cdot 10^{+203}\right):\\ \;\;\;\;\frac{x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.im -2.95e+113) (not (<= y.im 7.6e+203)))
   (/ x.re y.im)
   (/ x.im y.re)))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_im <= -2.95e+113) || !(y_46_im <= 7.6e+203)) {
		tmp = x_46_re / y_46_im;
	} else {
		tmp = x_46_im / y_46_re;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if ((y_46im <= (-2.95d+113)) .or. (.not. (y_46im <= 7.6d+203))) then
        tmp = x_46re / y_46im
    else
        tmp = x_46im / y_46re
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_im <= -2.95e+113) || !(y_46_im <= 7.6e+203)) {
		tmp = x_46_re / y_46_im;
	} else {
		tmp = x_46_im / y_46_re;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if (y_46_im <= -2.95e+113) or not (y_46_im <= 7.6e+203):
		tmp = x_46_re / y_46_im
	else:
		tmp = x_46_im / y_46_re
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if ((y_46_im <= -2.95e+113) || !(y_46_im <= 7.6e+203))
		tmp = Float64(x_46_re / y_46_im);
	else
		tmp = Float64(x_46_im / y_46_re);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if ((y_46_im <= -2.95e+113) || ~((y_46_im <= 7.6e+203)))
		tmp = x_46_re / y_46_im;
	else
		tmp = x_46_im / y_46_re;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[Or[LessEqual[y$46$im, -2.95e+113], N[Not[LessEqual[y$46$im, 7.6e+203]], $MachinePrecision]], N[(x$46$re / y$46$im), $MachinePrecision], N[(x$46$im / y$46$re), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.im < -2.95000000000000011e113 or 7.60000000000000047e203 < y.im

   1. Initial program 44.3%

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

      1. fma-neg 44.3%

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

      2. distribute-rgt-neg-out 44.3%

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

      3. +-commutative 44.3%

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

      4. fma-define 44.3%

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

   3. Simplified 44.3%

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

   5. Taylor expanded in y.re around 0 87.4%

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

      1. associate-*r/ 87.4%

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

      2. neg-mul-1 87.4%

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

   7. Simplified 87.4%

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

      1. add-sqr-sqrt 21.1%

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

      2. sqrt-unprod 39.6%

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

      3. sqr-neg 39.6%

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

      4. sqrt-unprod 20.4%

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

      5. add-sqr-sqrt 34.2%

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

      6. div-inv 34.2%

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

   9. Applied egg-rr 34.2%

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

       1. associate-*r/ 34.2%

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

       2. *-rgt-identity 34.2%

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

   11. Simplified 34.2%

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

   if -2.95000000000000011e113 < y.im < 7.60000000000000047e203

   1. Initial program 69.1%

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

      1. fma-neg 69.1%

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

      2. distribute-rgt-neg-out 69.1%

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

      3. +-commutative 69.1%

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

      4. fma-define 69.1%

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

   3. Simplified 69.1%

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

   5. Taylor expanded in y.re around inf 52.6%

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

4. Final simplification 48.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -2.95 \cdot 10^{+113} \lor \neg \left(y.im \leq 7.6 \cdot 10^{+203}\right):\\ \;\;\;\;\frac{x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 42.9% accurate, 5.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 64.0%

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

   1. fma-neg 64.0%

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

   2. distribute-rgt-neg-out 64.0%

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

   3. +-commutative 64.0%

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

   4. fma-define 64.0%

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

3. Simplified 64.0%

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

5. Taylor expanded in y.re around inf 42.9%

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

Reproduce

herbie shell --seed 2024136 
(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))))