_divideComplex, imaginary part

Percentage Accurate: 62.2% → 80.2%

Time: 10.6s

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: 62.2% 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: 80.2% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -9 \cdot 10^{+75}:\\ \;\;\;\;\frac{y.re \cdot \frac{x.im}{y.im} - x.re}{y.im}\\ \mathbf{elif}\;y.im \leq -3.3 \cdot 10^{-64}:\\ \;\;\;\;\frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.im \leq 6.4 \cdot 10^{-128}:\\ \;\;\;\;\frac{x.im - \frac{y.im \cdot x.re}{y.re}}{y.re}\\ \mathbf{elif}\;y.im \leq 1.02 \cdot 10^{+43}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x.im, y.re, y.im \cdot \left(-x.re\right)\right)}{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)}\\ \mathbf{elif}\;y.im \leq 7.5 \cdot 10^{+125}:\\ \;\;\;\;y.im \cdot \left({\left(\mathsf{hypot}\left(y.im, y.re\right)\right)}^{-2} \cdot \left(-x.re\right)\right)\\ \mathbf{elif}\;y.im \leq 3.5 \cdot 10^{+170}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x.im, \frac{y.re}{y.im}, -x.re\right)}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.im -9e+75)
   (/ (- (* y.re (/ x.im y.im)) x.re) y.im)
   (if (<= y.im -3.3e-64)
     (/ (- (* y.re x.im) (* y.im x.re)) (+ (* y.re y.re) (* y.im y.im)))
     (if (<= y.im 6.4e-128)
       (/ (- x.im (/ (* y.im x.re) y.re)) y.re)
       (if (<= y.im 1.02e+43)
         (/ (fma x.im y.re (* y.im (- x.re))) (fma y.im y.im (* y.re y.re)))
         (if (<= y.im 7.5e+125)
           (* y.im (* (pow (hypot y.im y.re) -2.0) (- x.re)))
           (if (<= y.im 3.5e+170)
             (/ x.im y.re)
             (/ (fma 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_im <= -9e+75) {
		tmp = ((y_46_re * (x_46_im / y_46_im)) - x_46_re) / y_46_im;
	} else if (y_46_im <= -3.3e-64) {
		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 <= 6.4e-128) {
		tmp = (x_46_im - ((y_46_im * x_46_re) / y_46_re)) / y_46_re;
	} else if (y_46_im <= 1.02e+43) {
		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 <= 7.5e+125) {
		tmp = y_46_im * (pow(hypot(y_46_im, y_46_re), -2.0) * -x_46_re);
	} else if (y_46_im <= 3.5e+170) {
		tmp = x_46_im / y_46_re;
	} else {
		tmp = fma(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_im <= -9e+75)
		tmp = Float64(Float64(Float64(y_46_re * Float64(x_46_im / y_46_im)) - x_46_re) / y_46_im);
	elseif (y_46_im <= -3.3e-64)
		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 <= 6.4e-128)
		tmp = Float64(Float64(x_46_im - Float64(Float64(y_46_im * x_46_re) / y_46_re)) / y_46_re);
	elseif (y_46_im <= 1.02e+43)
		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 <= 7.5e+125)
		tmp = Float64(y_46_im * Float64((hypot(y_46_im, y_46_re) ^ -2.0) * Float64(-x_46_re)));
	elseif (y_46_im <= 3.5e+170)
		tmp = Float64(x_46_im / y_46_re);
	else
		tmp = Float64(fma(x_46_im, Float64(y_46_re / y_46_im), Float64(-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, -9e+75], 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, -3.3e-64], 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, 6.4e-128], 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, 1.02e+43], 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, 7.5e+125], N[(y$46$im * N[(N[Power[N[Sqrt[y$46$im ^ 2 + y$46$re ^ 2], $MachinePrecision], -2.0], $MachinePrecision] * (-x$46$re)), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, 3.5e+170], N[(x$46$im / y$46$re), $MachinePrecision], N[(N[(x$46$im * N[(y$46$re / y$46$im), $MachinePrecision] + (-x$46$re)), $MachinePrecision] / y$46$im), $MachinePrecision]]]]]]]

Derivation

1. Split input into 7 regimes

2. if y.im < -9.0000000000000007e75

   1. Initial program 32.7%

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

   3. Taylor expanded in y.re around 0 83.8%

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

      1. +-commutative 83.8%

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

      2. mul-1-neg 83.8%

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

      3. unsub-neg 83.8%

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

      4. unpow2 83.8%

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

      5. associate-/r* 85.8%

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

      6. div-sub 85.8%

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

      7. *-commutative 85.8%

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

      8. associate-/l* 89.7%

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

   5. Simplified 89.7%

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

   if -9.0000000000000007e75 < y.im < -3.2999999999999999e-64

   1. Initial program 86.0%

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

   if -3.2999999999999999e-64 < y.im < 6.3999999999999995e-128

   1. Initial program 66.6%

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

   3. Taylor expanded in y.re around inf 87.3%

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

      1. mul-1-neg 87.3%

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

      2. unsub-neg 87.3%

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

      3. unsub-neg 87.3%

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

      4. remove-double-neg 87.3%

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

      5. mul-1-neg 87.3%

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

      6. neg-mul-1 87.3%

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

      7. mul-1-neg 87.3%

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

      8. distribute-lft-in 87.3%

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

      9. distribute-lft-in 87.3%

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

      10. mul-1-neg 87.3%

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

      11. unsub-neg 87.3%

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

      12. neg-mul-1 87.3%

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

      13. mul-1-neg 87.3%

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

      14. remove-double-neg 87.3%

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

      15. associate-/l* 86.9%

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

   5. Simplified 86.9%

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

      1. add-sqr-sqrt 43.6%

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

      2. sqrt-unprod 54.7%

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

      3. sqr-neg 54.7%

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

      4. sqrt-unprod 37.1%

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

      5. add-sqr-sqrt 62.5%

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

      6. associate-/l* 62.6%

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

      7. *-commutative 62.6%

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

      8. add-sqr-sqrt 37.1%

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

      9. sqrt-unprod 55.0%

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

      10. sqr-neg 55.0%

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

      11. sqrt-unprod 43.8%

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

      12. add-sqr-sqrt 87.3%

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

   7. Applied egg-rr 87.3%

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

   if 6.3999999999999995e-128 < y.im < 1.02e43

   1. Initial program 85.6%

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

      1. fma-neg 85.6%

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

      2. distribute-rgt-neg-out 85.6%

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

      3. +-commutative 85.6%

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

      4. fma-define 85.6%

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

   3. Simplified 85.6%

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

   if 1.02e43 < y.im < 7.5000000000000006e125

   1. Initial program 56.6%

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

      1. fma-define 56.7%

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

      2. add-cube-cbrt 56.1%

         \[\leadsto \frac{x.im \cdot y.re - x.re \cdot y.im}{\color{blue}{\left(\sqrt[3]{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)} \cdot \sqrt[3]{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)}\right) \cdot \sqrt[3]{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)}}} \]

      3. pow3 56.2%

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

      4. add-sqr-sqrt 56.2%

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

      5. pow2 56.2%

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

      6. fma-define 56.2%

         \[\leadsto \frac{x.im \cdot y.re - x.re \cdot y.im}{{\left(\sqrt[3]{{\left(\sqrt{\color{blue}{y.re \cdot y.re + y.im \cdot y.im}}\right)}^{2}}\right)}^{3}} \]

      7. hypot-define 56.2%

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

   4. Applied egg-rr 56.2%

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

   5. Taylor expanded in x.im around 0 48.0%

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

      1. mul-1-neg 48.0%

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

      2. associate-/l* 77.9%

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

      3. distribute-lft-neg-in 77.9%

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

      4. rem-square-sqrt 77.9%

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

      5. +-commutative 77.9%

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

      6. unpow2 77.9%

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

      7. unpow2 77.9%

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

      8. hypot-undefine 77.9%

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

      9. +-commutative 77.9%

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

      10. unpow2 77.9%

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

      11. unpow2 77.9%

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

      12. hypot-undefine 78.0%

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

      13. unpow2 78.0%

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

      14. hypot-undefine 77.9%

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

      15. unpow2 77.9%

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

      16. unpow2 77.9%

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

      17. +-commutative 77.9%

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

      18. unpow2 77.9%

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

      19. unpow2 77.9%

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

      20. hypot-define 78.0%

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

   7. Simplified 78.0%

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

      1. distribute-lft-neg-out 78.0%

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

      2. neg-sub0 78.0%

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

      3. *-commutative 78.0%

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

      4. add-sqr-sqrt 27.6%

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

      5. sqrt-unprod 20.4%

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

      6. sqr-neg 20.4%

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

      7. sqrt-unprod 5.6%

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

      8. add-sqr-sqrt 6.6%

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

      9. div-inv 6.6%

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

      10. associate-*l* 6.6%

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

      11. pow-flip 6.6%

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

      12. metadata-eval 6.6%

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

      13. add-sqr-sqrt 5.6%

          \[\leadsto 0 - y.im \cdot \left({\left(\mathsf{hypot}\left(y.im, y.re\right)\right)}^{-2} \cdot \color{blue}{\left(\sqrt{-x.re} \cdot \sqrt{-x.re}\right)}\right) \]

      14. sqrt-unprod 20.5%

          \[\leadsto 0 - y.im \cdot \left({\left(\mathsf{hypot}\left(y.im, y.re\right)\right)}^{-2} \cdot \color{blue}{\sqrt{\left(-x.re\right) \cdot \left(-x.re\right)}}\right) \]

      15. sqr-neg 20.5%

          \[\leadsto 0 - y.im \cdot \left({\left(\mathsf{hypot}\left(y.im, y.re\right)\right)}^{-2} \cdot \sqrt{\color{blue}{x.re \cdot x.re}}\right) \]

      16. sqrt-unprod 27.5%

          \[\leadsto 0 - y.im \cdot \left({\left(\mathsf{hypot}\left(y.im, y.re\right)\right)}^{-2} \cdot \color{blue}{\left(\sqrt{x.re} \cdot \sqrt{x.re}\right)}\right) \]

      17. add-sqr-sqrt 78.1%

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

   9. Applied egg-rr 78.1%

      \[\leadsto \color{blue}{0 - y.im \cdot \left({\left(\mathsf{hypot}\left(y.im, y.re\right)\right)}^{-2} \cdot x.re\right)} \]
   10. Step-by-step derivation

       1. neg-sub0 78.1%

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

       2. distribute-lft-neg-in 78.1%

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

       3. *-commutative 78.1%

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

   11. Simplified 78.1%

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

   if 7.5000000000000006e125 < y.im < 3.50000000000000005e170

   1. Initial program 0.0%

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

   3. Taylor expanded in y.re around inf 100.0%

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

   if 3.50000000000000005e170 < y.im

   1. Initial program 41.7%

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

   3. Taylor expanded in y.re around 0 81.6%

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

      1. +-commutative 81.6%

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

      2. mul-1-neg 81.6%

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

      3. unsub-neg 81.6%

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

      4. unpow2 81.6%

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

      5. associate-/r* 81.8%

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

      6. div-sub 81.8%

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

      7. associate-/l* 89.2%

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

      8. fma-neg 89.2%

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

   5. Simplified 89.2%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(x.im, \frac{y.re}{y.im}, -x.re\right)}{y.im}} \]
3. Recombined 7 regimes into one program.

4. Final simplification 87.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -9 \cdot 10^{+75}:\\ \;\;\;\;\frac{y.re \cdot \frac{x.im}{y.im} - x.re}{y.im}\\ \mathbf{elif}\;y.im \leq -3.3 \cdot 10^{-64}:\\ \;\;\;\;\frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.im \leq 6.4 \cdot 10^{-128}:\\ \;\;\;\;\frac{x.im - \frac{y.im \cdot x.re}{y.re}}{y.re}\\ \mathbf{elif}\;y.im \leq 1.02 \cdot 10^{+43}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x.im, y.re, y.im \cdot \left(-x.re\right)\right)}{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)}\\ \mathbf{elif}\;y.im \leq 7.5 \cdot 10^{+125}:\\ \;\;\;\;y.im \cdot \left({\left(\mathsf{hypot}\left(y.im, y.re\right)\right)}^{-2} \cdot \left(-x.re\right)\right)\\ \mathbf{elif}\;y.im \leq 3.5 \cdot 10^{+170}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x.im, \frac{y.re}{y.im}, -x.re\right)}{y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 82.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ t_1 := \frac{y.re \cdot \frac{x.im}{y.im} - x.re}{y.im}\\ \mathbf{if}\;y.im \leq -3.1 \cdot 10^{+76}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y.im \leq -9 \cdot 10^{-66}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.im \leq 2.15 \cdot 10^{-127}:\\ \;\;\;\;\frac{x.im - \frac{y.im \cdot x.re}{y.re}}{y.re}\\ \mathbf{elif}\;y.im \leq 9 \cdot 10^{+42}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0
         (/ (- (* y.re x.im) (* y.im x.re)) (+ (* y.re y.re) (* y.im y.im))))
        (t_1 (/ (- (* y.re (/ x.im y.im)) x.re) y.im)))
   (if (<= y.im -3.1e+76)
     t_1
     (if (<= y.im -9e-66)
       t_0
       (if (<= y.im 2.15e-127)
         (/ (- x.im (/ (* y.im x.re) y.re)) y.re)
         (if (<= y.im 9e+42) t_0 t_1))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = ((y_46_re * x_46_im) - (y_46_im * x_46_re)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	double t_1 = ((y_46_re * (x_46_im / y_46_im)) - x_46_re) / y_46_im;
	double tmp;
	if (y_46_im <= -3.1e+76) {
		tmp = t_1;
	} else if (y_46_im <= -9e-66) {
		tmp = t_0;
	} else if (y_46_im <= 2.15e-127) {
		tmp = (x_46_im - ((y_46_im * x_46_re) / y_46_re)) / y_46_re;
	} else if (y_46_im <= 9e+42) {
		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 = ((y_46re * x_46im) - (y_46im * x_46re)) / ((y_46re * y_46re) + (y_46im * y_46im))
    t_1 = ((y_46re * (x_46im / y_46im)) - x_46re) / y_46im
    if (y_46im <= (-3.1d+76)) then
        tmp = t_1
    else if (y_46im <= (-9d-66)) then
        tmp = t_0
    else if (y_46im <= 2.15d-127) then
        tmp = (x_46im - ((y_46im * x_46re) / y_46re)) / y_46re
    else if (y_46im <= 9d+42) 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 = ((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 * (x_46_im / y_46_im)) - x_46_re) / y_46_im;
	double tmp;
	if (y_46_im <= -3.1e+76) {
		tmp = t_1;
	} else if (y_46_im <= -9e-66) {
		tmp = t_0;
	} else if (y_46_im <= 2.15e-127) {
		tmp = (x_46_im - ((y_46_im * x_46_re) / y_46_re)) / y_46_re;
	} else if (y_46_im <= 9e+42) {
		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 = ((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 * (x_46_im / y_46_im)) - x_46_re) / y_46_im
	tmp = 0
	if y_46_im <= -3.1e+76:
		tmp = t_1
	elif y_46_im <= -9e-66:
		tmp = t_0
	elif y_46_im <= 2.15e-127:
		tmp = (x_46_im - ((y_46_im * x_46_re) / y_46_re)) / y_46_re
	elif y_46_im <= 9e+42:
		tmp = t_0
	else:
		tmp = t_1
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(Float64(Float64(y_46_re * x_46_im) - Float64(y_46_im * x_46_re)) / Float64(Float64(y_46_re * y_46_re) + Float64(y_46_im * y_46_im)))
	t_1 = Float64(Float64(Float64(y_46_re * Float64(x_46_im / y_46_im)) - x_46_re) / y_46_im)
	tmp = 0.0
	if (y_46_im <= -3.1e+76)
		tmp = t_1;
	elseif (y_46_im <= -9e-66)
		tmp = t_0;
	elseif (y_46_im <= 2.15e-127)
		tmp = Float64(Float64(x_46_im - Float64(Float64(y_46_im * x_46_re) / y_46_re)) / y_46_re);
	elseif (y_46_im <= 9e+42)
		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 = ((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 * (x_46_im / y_46_im)) - x_46_re) / y_46_im;
	tmp = 0.0;
	if (y_46_im <= -3.1e+76)
		tmp = t_1;
	elseif (y_46_im <= -9e-66)
		tmp = t_0;
	elseif (y_46_im <= 2.15e-127)
		tmp = (x_46_im - ((y_46_im * x_46_re) / y_46_re)) / y_46_re;
	elseif (y_46_im <= 9e+42)
		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[(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[(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, -3.1e+76], t$95$1, If[LessEqual[y$46$im, -9e-66], t$95$0, If[LessEqual[y$46$im, 2.15e-127], 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, 9e+42], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if y.im < -3.10000000000000011e76 or 9.00000000000000025e42 < y.im

   1. Initial program 38.4%

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

   3. Taylor expanded in y.re around 0 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. unpow2 77.1%

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

      5. associate-/r* 78.2%

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

      6. div-sub 78.2%

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

      7. *-commutative 78.2%

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

      8. associate-/l* 82.0%

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

   5. Simplified 82.0%

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

   if -3.10000000000000011e76 < y.im < -8.9999999999999995e-66 or 2.14999999999999992e-127 < y.im < 9.00000000000000025e42

   1. Initial program 85.8%

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

   if -8.9999999999999995e-66 < y.im < 2.14999999999999992e-127

   1. Initial program 66.6%

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

   3. Taylor expanded in y.re around inf 87.3%

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

      1. mul-1-neg 87.3%

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

      2. unsub-neg 87.3%

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

      3. unsub-neg 87.3%

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

      4. remove-double-neg 87.3%

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

      5. mul-1-neg 87.3%

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

      6. neg-mul-1 87.3%

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

      7. mul-1-neg 87.3%

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

      8. distribute-lft-in 87.3%

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

      9. distribute-lft-in 87.3%

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

      10. mul-1-neg 87.3%

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

      11. unsub-neg 87.3%

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

      12. neg-mul-1 87.3%

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

      13. mul-1-neg 87.3%

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

      14. remove-double-neg 87.3%

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

      15. associate-/l* 86.9%

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

   5. Simplified 86.9%

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

      1. add-sqr-sqrt 43.6%

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

      2. sqrt-unprod 54.7%

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

      3. sqr-neg 54.7%

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

      4. sqrt-unprod 37.1%

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

      5. add-sqr-sqrt 62.5%

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

      6. associate-/l* 62.6%

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

      7. *-commutative 62.6%

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

      8. add-sqr-sqrt 37.1%

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

      9. sqrt-unprod 55.0%

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

      10. sqr-neg 55.0%

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

      11. sqrt-unprod 43.8%

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

      12. add-sqr-sqrt 87.3%

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

   7. Applied egg-rr 87.3%

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

4. Final simplification 84.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -3.1 \cdot 10^{+76}:\\ \;\;\;\;\frac{y.re \cdot \frac{x.im}{y.im} - x.re}{y.im}\\ \mathbf{elif}\;y.im \leq -9 \cdot 10^{-66}:\\ \;\;\;\;\frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.im \leq 2.15 \cdot 10^{-127}:\\ \;\;\;\;\frac{x.im - \frac{y.im \cdot x.re}{y.re}}{y.re}\\ \mathbf{elif}\;y.im \leq 9 \cdot 10^{+42}:\\ \;\;\;\;\frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{y.re \cdot \frac{x.im}{y.im} - x.re}{y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 71.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x.re}{-y.im}\\ \mathbf{if}\;y.im \leq -600000000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.im \leq 1.6 \cdot 10^{+41}:\\ \;\;\;\;\frac{x.im - \frac{y.im \cdot x.re}{y.re}}{y.re}\\ \mathbf{elif}\;y.im \leq 7.5 \cdot 10^{+125} \lor \neg \left(y.im \leq 3.5 \cdot 10^{+170}\right):\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (/ x.re (- y.im))))
   (if (<= y.im -600000000000.0)
     t_0
     (if (<= y.im 1.6e+41)
       (/ (- x.im (/ (* y.im x.re) y.re)) y.re)
       (if (or (<= y.im 7.5e+125) (not (<= y.im 3.5e+170)))
         t_0
         (/ x.im y.re))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = x_46_re / -y_46_im;
	double tmp;
	if (y_46_im <= -600000000000.0) {
		tmp = t_0;
	} else if (y_46_im <= 1.6e+41) {
		tmp = (x_46_im - ((y_46_im * x_46_re) / y_46_re)) / y_46_re;
	} else if ((y_46_im <= 7.5e+125) || !(y_46_im <= 3.5e+170)) {
		tmp = t_0;
	} else {
		tmp = x_46_im / y_46_re;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x_46re / -y_46im
    if (y_46im <= (-600000000000.0d0)) then
        tmp = t_0
    else if (y_46im <= 1.6d+41) then
        tmp = (x_46im - ((y_46im * x_46re) / y_46re)) / y_46re
    else if ((y_46im <= 7.5d+125) .or. (.not. (y_46im <= 3.5d+170))) then
        tmp = t_0
    else
        tmp = x_46im / y_46re
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = x_46_re / -y_46_im;
	double tmp;
	if (y_46_im <= -600000000000.0) {
		tmp = t_0;
	} else if (y_46_im <= 1.6e+41) {
		tmp = (x_46_im - ((y_46_im * x_46_re) / y_46_re)) / y_46_re;
	} else if ((y_46_im <= 7.5e+125) || !(y_46_im <= 3.5e+170)) {
		tmp = t_0;
	} else {
		tmp = x_46_im / y_46_re;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = x_46_re / -y_46_im
	tmp = 0
	if y_46_im <= -600000000000.0:
		tmp = t_0
	elif y_46_im <= 1.6e+41:
		tmp = (x_46_im - ((y_46_im * x_46_re) / y_46_re)) / y_46_re
	elif (y_46_im <= 7.5e+125) or not (y_46_im <= 3.5e+170):
		tmp = t_0
	else:
		tmp = x_46_im / y_46_re
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(x_46_re / Float64(-y_46_im))
	tmp = 0.0
	if (y_46_im <= -600000000000.0)
		tmp = t_0;
	elseif (y_46_im <= 1.6e+41)
		tmp = Float64(Float64(x_46_im - Float64(Float64(y_46_im * x_46_re) / y_46_re)) / y_46_re);
	elseif ((y_46_im <= 7.5e+125) || !(y_46_im <= 3.5e+170))
		tmp = t_0;
	else
		tmp = Float64(x_46_im / y_46_re);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = x_46_re / -y_46_im;
	tmp = 0.0;
	if (y_46_im <= -600000000000.0)
		tmp = t_0;
	elseif (y_46_im <= 1.6e+41)
		tmp = (x_46_im - ((y_46_im * x_46_re) / y_46_re)) / y_46_re;
	elseif ((y_46_im <= 7.5e+125) || ~((y_46_im <= 3.5e+170)))
		tmp = t_0;
	else
		tmp = x_46_im / y_46_re;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(x$46$re / (-y$46$im)), $MachinePrecision]}, If[LessEqual[y$46$im, -600000000000.0], t$95$0, If[LessEqual[y$46$im, 1.6e+41], N[(N[(x$46$im - N[(N[(y$46$im * x$46$re), $MachinePrecision] / y$46$re), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision], If[Or[LessEqual[y$46$im, 7.5e+125], N[Not[LessEqual[y$46$im, 3.5e+170]], $MachinePrecision]], t$95$0, N[(x$46$im / y$46$re), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if y.im < -6e11 or 1.60000000000000005e41 < y.im < 7.5000000000000006e125 or 3.50000000000000005e170 < 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 78.1%

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

      1. associate-*r/ 78.1%

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

      2. neg-mul-1 78.1%

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

   5. Simplified 78.1%

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

   if -6e11 < y.im < 1.60000000000000005e41

   1. Initial program 73.5%

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

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

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

      1. mul-1-neg 73.9%

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

      2. unsub-neg 73.9%

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

      3. unsub-neg 73.9%

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

      4. remove-double-neg 73.9%

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

      5. mul-1-neg 73.9%

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

      6. neg-mul-1 73.9%

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

      7. mul-1-neg 73.9%

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

      8. distribute-lft-in 73.9%

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

      9. distribute-lft-in 73.9%

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

      10. mul-1-neg 73.9%

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

      11. unsub-neg 73.9%

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

      12. neg-mul-1 73.9%

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

      13. mul-1-neg 73.9%

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

      14. remove-double-neg 73.9%

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

      15. associate-/l* 73.6%

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

   5. Simplified 73.6%

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

      1. add-sqr-sqrt 36.3%

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

      2. sqrt-unprod 47.6%

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

      3. sqr-neg 47.6%

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

      4. sqrt-unprod 31.1%

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

      5. add-sqr-sqrt 54.7%

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

      6. associate-/l* 54.7%

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

      7. *-commutative 54.7%

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

      8. add-sqr-sqrt 31.1%

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

      9. sqrt-unprod 47.8%

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

      10. sqr-neg 47.8%

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

      11. sqrt-unprod 36.5%

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

      12. add-sqr-sqrt 73.9%

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

   7. Applied egg-rr 73.9%

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

   if 7.5000000000000006e125 < y.im < 3.50000000000000005e170

   1. Initial program 0.0%

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

   3. Taylor expanded in y.re around inf 100.0%

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

4. Final simplification 76.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -600000000000:\\ \;\;\;\;\frac{x.re}{-y.im}\\ \mathbf{elif}\;y.im \leq 1.6 \cdot 10^{+41}:\\ \;\;\;\;\frac{x.im - \frac{y.im \cdot x.re}{y.re}}{y.re}\\ \mathbf{elif}\;y.im \leq 7.5 \cdot 10^{+125} \lor \neg \left(y.im \leq 3.5 \cdot 10^{+170}\right):\\ \;\;\;\;\frac{x.re}{-y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 71.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x.re}{-y.im}\\ \mathbf{if}\;y.im \leq -220000000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.im \leq 1.9 \cdot 10^{+40}:\\ \;\;\;\;\frac{x.im - \frac{x.re}{\frac{y.re}{y.im}}}{y.re}\\ \mathbf{elif}\;y.im \leq 7.5 \cdot 10^{+125} \lor \neg \left(y.im \leq 3.5 \cdot 10^{+170}\right):\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (/ x.re (- y.im))))
   (if (<= y.im -220000000000.0)
     t_0
     (if (<= y.im 1.9e+40)
       (/ (- x.im (/ x.re (/ y.re y.im))) y.re)
       (if (or (<= y.im 7.5e+125) (not (<= y.im 3.5e+170)))
         t_0
         (/ x.im y.re))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = x_46_re / -y_46_im;
	double tmp;
	if (y_46_im <= -220000000000.0) {
		tmp = t_0;
	} else if (y_46_im <= 1.9e+40) {
		tmp = (x_46_im - (x_46_re / (y_46_re / y_46_im))) / y_46_re;
	} else if ((y_46_im <= 7.5e+125) || !(y_46_im <= 3.5e+170)) {
		tmp = t_0;
	} else {
		tmp = x_46_im / y_46_re;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x_46re / -y_46im
    if (y_46im <= (-220000000000.0d0)) then
        tmp = t_0
    else if (y_46im <= 1.9d+40) then
        tmp = (x_46im - (x_46re / (y_46re / y_46im))) / y_46re
    else if ((y_46im <= 7.5d+125) .or. (.not. (y_46im <= 3.5d+170))) then
        tmp = t_0
    else
        tmp = x_46im / y_46re
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = x_46_re / -y_46_im;
	double tmp;
	if (y_46_im <= -220000000000.0) {
		tmp = t_0;
	} else if (y_46_im <= 1.9e+40) {
		tmp = (x_46_im - (x_46_re / (y_46_re / y_46_im))) / y_46_re;
	} else if ((y_46_im <= 7.5e+125) || !(y_46_im <= 3.5e+170)) {
		tmp = t_0;
	} else {
		tmp = x_46_im / y_46_re;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = x_46_re / -y_46_im
	tmp = 0
	if y_46_im <= -220000000000.0:
		tmp = t_0
	elif y_46_im <= 1.9e+40:
		tmp = (x_46_im - (x_46_re / (y_46_re / y_46_im))) / y_46_re
	elif (y_46_im <= 7.5e+125) or not (y_46_im <= 3.5e+170):
		tmp = t_0
	else:
		tmp = x_46_im / y_46_re
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(x_46_re / Float64(-y_46_im))
	tmp = 0.0
	if (y_46_im <= -220000000000.0)
		tmp = t_0;
	elseif (y_46_im <= 1.9e+40)
		tmp = Float64(Float64(x_46_im - Float64(x_46_re / Float64(y_46_re / y_46_im))) / y_46_re);
	elseif ((y_46_im <= 7.5e+125) || !(y_46_im <= 3.5e+170))
		tmp = t_0;
	else
		tmp = Float64(x_46_im / y_46_re);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = x_46_re / -y_46_im;
	tmp = 0.0;
	if (y_46_im <= -220000000000.0)
		tmp = t_0;
	elseif (y_46_im <= 1.9e+40)
		tmp = (x_46_im - (x_46_re / (y_46_re / y_46_im))) / y_46_re;
	elseif ((y_46_im <= 7.5e+125) || ~((y_46_im <= 3.5e+170)))
		tmp = t_0;
	else
		tmp = x_46_im / y_46_re;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(x$46$re / (-y$46$im)), $MachinePrecision]}, If[LessEqual[y$46$im, -220000000000.0], t$95$0, If[LessEqual[y$46$im, 1.9e+40], N[(N[(x$46$im - N[(x$46$re / N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision], If[Or[LessEqual[y$46$im, 7.5e+125], N[Not[LessEqual[y$46$im, 3.5e+170]], $MachinePrecision]], t$95$0, N[(x$46$im / y$46$re), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if y.im < -2.2e11 or 1.90000000000000002e40 < y.im < 7.5000000000000006e125 or 3.50000000000000005e170 < 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 78.1%

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

      1. associate-*r/ 78.1%

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

      2. neg-mul-1 78.1%

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

   5. Simplified 78.1%

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

   if -2.2e11 < y.im < 1.90000000000000002e40

   1. Initial program 73.5%

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

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

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

      1. mul-1-neg 73.9%

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

      2. unsub-neg 73.9%

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

      3. unsub-neg 73.9%

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

      4. remove-double-neg 73.9%

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

      5. mul-1-neg 73.9%

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

      6. neg-mul-1 73.9%

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

      7. mul-1-neg 73.9%

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

      8. distribute-lft-in 73.9%

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

      9. distribute-lft-in 73.9%

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

      10. mul-1-neg 73.9%

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

      11. unsub-neg 73.9%

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

      12. neg-mul-1 73.9%

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

      13. mul-1-neg 73.9%

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

      14. remove-double-neg 73.9%

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

      15. associate-/l* 73.6%

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

   5. Simplified 73.6%

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

      1. clear-num 73.6%

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

      2. un-div-inv 73.7%

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

   7. Applied egg-rr 73.7%

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

   if 7.5000000000000006e125 < y.im < 3.50000000000000005e170

   1. Initial program 0.0%

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

   3. Taylor expanded in y.re around inf 100.0%

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

4. Final simplification 76.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -220000000000:\\ \;\;\;\;\frac{x.re}{-y.im}\\ \mathbf{elif}\;y.im \leq 1.9 \cdot 10^{+40}:\\ \;\;\;\;\frac{x.im - \frac{x.re}{\frac{y.re}{y.im}}}{y.re}\\ \mathbf{elif}\;y.im \leq 7.5 \cdot 10^{+125} \lor \neg \left(y.im \leq 3.5 \cdot 10^{+170}\right):\\ \;\;\;\;\frac{x.re}{-y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 71.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x.re}{-y.im}\\ \mathbf{if}\;y.im \leq -750000000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.im \leq 2.3 \cdot 10^{+41}:\\ \;\;\;\;\frac{x.im - x.re \cdot \frac{y.im}{y.re}}{y.re}\\ \mathbf{elif}\;y.im \leq 7.5 \cdot 10^{+125} \lor \neg \left(y.im \leq 3.5 \cdot 10^{+170}\right):\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (/ x.re (- y.im))))
   (if (<= y.im -750000000000.0)
     t_0
     (if (<= y.im 2.3e+41)
       (/ (- x.im (* x.re (/ y.im y.re))) y.re)
       (if (or (<= y.im 7.5e+125) (not (<= y.im 3.5e+170)))
         t_0
         (/ x.im y.re))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = x_46_re / -y_46_im;
	double tmp;
	if (y_46_im <= -750000000000.0) {
		tmp = t_0;
	} else if (y_46_im <= 2.3e+41) {
		tmp = (x_46_im - (x_46_re * (y_46_im / y_46_re))) / y_46_re;
	} else if ((y_46_im <= 7.5e+125) || !(y_46_im <= 3.5e+170)) {
		tmp = t_0;
	} else {
		tmp = x_46_im / y_46_re;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x_46re / -y_46im
    if (y_46im <= (-750000000000.0d0)) then
        tmp = t_0
    else if (y_46im <= 2.3d+41) then
        tmp = (x_46im - (x_46re * (y_46im / y_46re))) / y_46re
    else if ((y_46im <= 7.5d+125) .or. (.not. (y_46im <= 3.5d+170))) then
        tmp = t_0
    else
        tmp = x_46im / y_46re
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = x_46_re / -y_46_im;
	double tmp;
	if (y_46_im <= -750000000000.0) {
		tmp = t_0;
	} else if (y_46_im <= 2.3e+41) {
		tmp = (x_46_im - (x_46_re * (y_46_im / y_46_re))) / y_46_re;
	} else if ((y_46_im <= 7.5e+125) || !(y_46_im <= 3.5e+170)) {
		tmp = t_0;
	} else {
		tmp = x_46_im / y_46_re;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = x_46_re / -y_46_im
	tmp = 0
	if y_46_im <= -750000000000.0:
		tmp = t_0
	elif y_46_im <= 2.3e+41:
		tmp = (x_46_im - (x_46_re * (y_46_im / y_46_re))) / y_46_re
	elif (y_46_im <= 7.5e+125) or not (y_46_im <= 3.5e+170):
		tmp = t_0
	else:
		tmp = x_46_im / y_46_re
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(x_46_re / Float64(-y_46_im))
	tmp = 0.0
	if (y_46_im <= -750000000000.0)
		tmp = t_0;
	elseif (y_46_im <= 2.3e+41)
		tmp = Float64(Float64(x_46_im - Float64(x_46_re * Float64(y_46_im / y_46_re))) / y_46_re);
	elseif ((y_46_im <= 7.5e+125) || !(y_46_im <= 3.5e+170))
		tmp = t_0;
	else
		tmp = Float64(x_46_im / y_46_re);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = x_46_re / -y_46_im;
	tmp = 0.0;
	if (y_46_im <= -750000000000.0)
		tmp = t_0;
	elseif (y_46_im <= 2.3e+41)
		tmp = (x_46_im - (x_46_re * (y_46_im / y_46_re))) / y_46_re;
	elseif ((y_46_im <= 7.5e+125) || ~((y_46_im <= 3.5e+170)))
		tmp = t_0;
	else
		tmp = x_46_im / y_46_re;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(x$46$re / (-y$46$im)), $MachinePrecision]}, If[LessEqual[y$46$im, -750000000000.0], t$95$0, If[LessEqual[y$46$im, 2.3e+41], N[(N[(x$46$im - N[(x$46$re * N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision], If[Or[LessEqual[y$46$im, 7.5e+125], N[Not[LessEqual[y$46$im, 3.5e+170]], $MachinePrecision]], t$95$0, N[(x$46$im / y$46$re), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if y.im < -7.5e11 or 2.2999999999999998e41 < y.im < 7.5000000000000006e125 or 3.50000000000000005e170 < 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 78.1%

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

      1. associate-*r/ 78.1%

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

      2. neg-mul-1 78.1%

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

   5. Simplified 78.1%

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

   if -7.5e11 < y.im < 2.2999999999999998e41

   1. Initial program 73.5%

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

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

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

      1. mul-1-neg 73.9%

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

      2. unsub-neg 73.9%

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

      3. unsub-neg 73.9%

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

      4. remove-double-neg 73.9%

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

      5. mul-1-neg 73.9%

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

      6. neg-mul-1 73.9%

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

      7. mul-1-neg 73.9%

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

      8. distribute-lft-in 73.9%

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

      9. distribute-lft-in 73.9%

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

      10. mul-1-neg 73.9%

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

      11. unsub-neg 73.9%

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

      12. neg-mul-1 73.9%

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

      13. mul-1-neg 73.9%

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

      14. remove-double-neg 73.9%

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

      15. associate-/l* 73.6%

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

   5. Simplified 73.6%

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

   if 7.5000000000000006e125 < y.im < 3.50000000000000005e170

   1. Initial program 0.0%

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

   3. Taylor expanded in y.re around inf 100.0%

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

4. Final simplification 76.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -750000000000:\\ \;\;\;\;\frac{x.re}{-y.im}\\ \mathbf{elif}\;y.im \leq 2.3 \cdot 10^{+41}:\\ \;\;\;\;\frac{x.im - x.re \cdot \frac{y.im}{y.re}}{y.re}\\ \mathbf{elif}\;y.im \leq 7.5 \cdot 10^{+125} \lor \neg \left(y.im \leq 3.5 \cdot 10^{+170}\right):\\ \;\;\;\;\frac{x.re}{-y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 77.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -4.8 \cdot 10^{-20} \lor \neg \left(y.re \leq 1.45 \cdot 10^{+40}\right):\\ \;\;\;\;\frac{x.im - x.re \cdot \frac{y.im}{y.re}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y.re \cdot x.im}{y.im} - x.re}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.re -4.8e-20) (not (<= y.re 1.45e+40)))
   (/ (- x.im (* x.re (/ y.im y.re))) y.re)
   (/ (- (/ (* y.re x.im) 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 <= -4.8e-20) || !(y_46_re <= 1.45e+40)) {
		tmp = (x_46_im - (x_46_re * (y_46_im / y_46_re))) / y_46_re;
	} else {
		tmp = (((y_46_re * x_46_im) / 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 <= (-4.8d-20)) .or. (.not. (y_46re <= 1.45d+40))) then
        tmp = (x_46im - (x_46re * (y_46im / y_46re))) / y_46re
    else
        tmp = (((y_46re * x_46im) / 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 <= -4.8e-20) || !(y_46_re <= 1.45e+40)) {
		tmp = (x_46_im - (x_46_re * (y_46_im / y_46_re))) / y_46_re;
	} else {
		tmp = (((y_46_re * x_46_im) / 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 <= -4.8e-20) or not (y_46_re <= 1.45e+40):
		tmp = (x_46_im - (x_46_re * (y_46_im / y_46_re))) / y_46_re
	else:
		tmp = (((y_46_re * x_46_im) / 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 <= -4.8e-20) || !(y_46_re <= 1.45e+40))
		tmp = Float64(Float64(x_46_im - Float64(x_46_re * Float64(y_46_im / y_46_re))) / y_46_re);
	else
		tmp = Float64(Float64(Float64(Float64(y_46_re * x_46_im) / 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 <= -4.8e-20) || ~((y_46_re <= 1.45e+40)))
		tmp = (x_46_im - (x_46_re * (y_46_im / y_46_re))) / y_46_re;
	else
		tmp = (((y_46_re * x_46_im) / 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, -4.8e-20], N[Not[LessEqual[y$46$re, 1.45e+40]], $MachinePrecision]], N[(N[(x$46$im - N[(x$46$re * N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision], N[(N[(N[(N[(y$46$re * x$46$im), $MachinePrecision] / y$46$im), $MachinePrecision] - x$46$re), $MachinePrecision] / y$46$im), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.re < -4.79999999999999986e-20 or 1.45000000000000009e40 < y.re

   1. Initial program 51.5%

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

   3. Taylor expanded in y.re around inf 74.8%

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

      1. mul-1-neg 74.8%

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

      2. unsub-neg 74.8%

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

      3. unsub-neg 74.8%

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

      4. remove-double-neg 74.8%

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

      5. mul-1-neg 74.8%

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

      6. neg-mul-1 74.8%

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

      7. mul-1-neg 74.8%

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

      8. distribute-lft-in 74.8%

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

      9. distribute-lft-in 74.8%

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

      10. mul-1-neg 74.8%

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

      11. unsub-neg 74.8%

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

      12. neg-mul-1 74.8%

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

      13. mul-1-neg 74.8%

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

      14. remove-double-neg 74.8%

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

      15. associate-/l* 77.2%

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

   5. Simplified 77.2%

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

   if -4.79999999999999986e-20 < y.re < 1.45000000000000009e40

   1. Initial program 68.0%

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

   3. Taylor expanded in y.re around 0 77.7%

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

      1. +-commutative 77.7%

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

      2. mul-1-neg 77.7%

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

      3. unsub-neg 77.7%

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

      4. unpow2 77.7%

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

      5. associate-/r* 84.7%

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

      6. div-sub 84.7%

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

      7. *-commutative 84.7%

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

      8. associate-/l* 80.5%

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

   5. Simplified 80.5%

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

      1. associate-*r/ 84.7%

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

   7. Applied egg-rr 84.7%

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

4. Final simplification 81.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -4.8 \cdot 10^{-20} \lor \neg \left(y.re \leq 1.45 \cdot 10^{+40}\right):\\ \;\;\;\;\frac{x.im - x.re \cdot \frac{y.im}{y.re}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y.re \cdot x.im}{y.im} - x.re}{y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 77.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -2.3 \cdot 10^{-20} \lor \neg \left(y.re \leq 1.95 \cdot 10^{+40}\right):\\ \;\;\;\;\frac{x.im - x.re \cdot \frac{y.im}{y.re}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{y.re \cdot \frac{x.im}{y.im} - x.re}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.re -2.3e-20) (not (<= y.re 1.95e+40)))
   (/ (- x.im (* x.re (/ y.im y.re))) y.re)
   (/ (- (* y.re (/ x.im 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 <= -2.3e-20) || !(y_46_re <= 1.95e+40)) {
		tmp = (x_46_im - (x_46_re * (y_46_im / y_46_re))) / y_46_re;
	} else {
		tmp = ((y_46_re * (x_46_im / 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 <= (-2.3d-20)) .or. (.not. (y_46re <= 1.95d+40))) then
        tmp = (x_46im - (x_46re * (y_46im / y_46re))) / y_46re
    else
        tmp = ((y_46re * (x_46im / 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 <= -2.3e-20) || !(y_46_re <= 1.95e+40)) {
		tmp = (x_46_im - (x_46_re * (y_46_im / y_46_re))) / y_46_re;
	} else {
		tmp = ((y_46_re * (x_46_im / 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 <= -2.3e-20) or not (y_46_re <= 1.95e+40):
		tmp = (x_46_im - (x_46_re * (y_46_im / y_46_re))) / y_46_re
	else:
		tmp = ((y_46_re * (x_46_im / 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 <= -2.3e-20) || !(y_46_re <= 1.95e+40))
		tmp = Float64(Float64(x_46_im - Float64(x_46_re * Float64(y_46_im / y_46_re))) / y_46_re);
	else
		tmp = Float64(Float64(Float64(y_46_re * Float64(x_46_im / 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 <= -2.3e-20) || ~((y_46_re <= 1.95e+40)))
		tmp = (x_46_im - (x_46_re * (y_46_im / y_46_re))) / y_46_re;
	else
		tmp = ((y_46_re * (x_46_im / 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, -2.3e-20], N[Not[LessEqual[y$46$re, 1.95e+40]], $MachinePrecision]], N[(N[(x$46$im - N[(x$46$re * N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision], N[(N[(N[(y$46$re * N[(x$46$im / y$46$im), $MachinePrecision]), $MachinePrecision] - x$46$re), $MachinePrecision] / y$46$im), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.re < -2.2999999999999999e-20 or 1.95e40 < y.re

   1. Initial program 51.5%

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

   3. Taylor expanded in y.re around inf 74.8%

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

      1. mul-1-neg 74.8%

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

      2. unsub-neg 74.8%

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

      3. unsub-neg 74.8%

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

      4. remove-double-neg 74.8%

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

      5. mul-1-neg 74.8%

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

      6. neg-mul-1 74.8%

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

      7. mul-1-neg 74.8%

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

      8. distribute-lft-in 74.8%

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

      9. distribute-lft-in 74.8%

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

      10. mul-1-neg 74.8%

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

      11. unsub-neg 74.8%

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

      12. neg-mul-1 74.8%

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

      13. mul-1-neg 74.8%

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

      14. remove-double-neg 74.8%

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

      15. associate-/l* 77.2%

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

   5. Simplified 77.2%

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

   if -2.2999999999999999e-20 < y.re < 1.95e40

   1. Initial program 68.0%

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

   3. Taylor expanded in y.re around 0 77.7%

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

      1. +-commutative 77.7%

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

      2. mul-1-neg 77.7%

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

      3. unsub-neg 77.7%

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

      4. unpow2 77.7%

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

      5. associate-/r* 84.7%

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

      6. div-sub 84.7%

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

      7. *-commutative 84.7%

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

      8. associate-/l* 80.5%

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

   5. Simplified 80.5%

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

4. Final simplification 79.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -2.3 \cdot 10^{-20} \lor \neg \left(y.re \leq 1.95 \cdot 10^{+40}\right):\\ \;\;\;\;\frac{x.im - x.re \cdot \frac{y.im}{y.re}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{y.re \cdot \frac{x.im}{y.im} - x.re}{y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 63.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -1.5 \cdot 10^{-31} \lor \neg \left(y.re \leq 6.2 \cdot 10^{+75}\right):\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{-y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.re -1.5e-31) (not (<= y.re 6.2e+75)))
   (/ 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 <= -1.5e-31) || !(y_46_re <= 6.2e+75)) {
		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 <= (-1.5d-31)) .or. (.not. (y_46re <= 6.2d+75))) 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 <= -1.5e-31) || !(y_46_re <= 6.2e+75)) {
		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 <= -1.5e-31) or not (y_46_re <= 6.2e+75):
		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 <= -1.5e-31) || !(y_46_re <= 6.2e+75))
		tmp = Float64(x_46_im / y_46_re);
	else
		tmp = Float64(x_46_re / Float64(-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 <= -1.5e-31) || ~((y_46_re <= 6.2e+75)))
		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, -1.5e-31], N[Not[LessEqual[y$46$re, 6.2e+75]], $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 < -1.49999999999999991e-31 or 6.2000000000000002e75 < y.re

   1. Initial program 48.9%

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

   3. Taylor expanded in y.re around inf 68.5%

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

   if -1.49999999999999991e-31 < y.re < 6.2000000000000002e75

   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. Add Preprocessing

   3. Taylor expanded in y.re around 0 63.8%

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

      1. associate-*r/ 63.8%

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

      2. neg-mul-1 63.8%

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

   5. Simplified 63.8%

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

4. Final simplification 65.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -1.5 \cdot 10^{-31} \lor \neg \left(y.re \leq 6.2 \cdot 10^{+75}\right):\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{-y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 47.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -2.3 \cdot 10^{+195} \lor \neg \left(y.im \leq 8 \cdot 10^{+172}\right):\\ \;\;\;\;\frac{x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.im -2.3e+195) (not (<= y.im 8e+172)))
   (/ 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 <= -2.3e+195) || !(y_46_im <= 8e+172)) {
		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 <= (-2.3d+195)) .or. (.not. (y_46im <= 8d+172))) 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 <= -2.3e+195) || !(y_46_im <= 8e+172)) {
		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 <= -2.3e+195) or not (y_46_im <= 8e+172):
		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 <= -2.3e+195) || !(y_46_im <= 8e+172))
		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 <= -2.3e+195) || ~((y_46_im <= 8e+172)))
		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, -2.3e+195], N[Not[LessEqual[y$46$im, 8e+172]], $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 < -2.3000000000000001e195 or 8.0000000000000007e172 < y.im

   1. Initial program 41.1%

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

      1. fma-define 41.1%

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

      2. add-cube-cbrt 41.1%

         \[\leadsto \frac{x.im \cdot y.re - x.re \cdot y.im}{\color{blue}{\left(\sqrt[3]{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)} \cdot \sqrt[3]{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)}\right) \cdot \sqrt[3]{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)}}} \]

      3. pow3 41.1%

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

      4. add-sqr-sqrt 41.1%

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

      5. pow2 41.1%

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

      6. fma-define 41.1%

         \[\leadsto \frac{x.im \cdot y.re - x.re \cdot y.im}{{\left(\sqrt[3]{{\left(\sqrt{\color{blue}{y.re \cdot y.re + y.im \cdot y.im}}\right)}^{2}}\right)}^{3}} \]

      7. hypot-define 41.1%

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

   4. Applied egg-rr 41.1%

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

   5. Taylor expanded in x.im around 0 41.3%

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

      1. mul-1-neg 41.3%

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

      2. associate-/l* 42.7%

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

      3. distribute-lft-neg-in 42.7%

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

      4. rem-square-sqrt 42.7%

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

      5. +-commutative 42.7%

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

      6. unpow2 42.7%

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

      7. unpow2 42.7%

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

      8. hypot-undefine 42.7%

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

      9. +-commutative 42.7%

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

      10. unpow2 42.7%

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

      11. unpow2 42.7%

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

      12. hypot-undefine 42.7%

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

      13. unpow2 42.7%

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

      14. hypot-undefine 42.7%

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

      15. unpow2 42.7%

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

      16. unpow2 42.7%

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

      17. +-commutative 42.7%

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

      18. unpow2 42.7%

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

      19. unpow2 42.7%

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

      20. hypot-define 42.7%

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

   7. Simplified 42.7%

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

   8. Taylor expanded in y.im around inf 89.7%

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

      1. un-div-inv 90.0%

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

      2. add-sqr-sqrt 43.1%

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

      3. sqrt-unprod 51.4%

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

      4. sqr-neg 51.4%

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

      5. sqrt-unprod 17.0%

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

      6. add-sqr-sqrt 38.7%

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

   10. Applied egg-rr 38.7%

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

   if -2.3000000000000001e195 < y.im < 8.0000000000000007e172

   1. Initial program 65.9%

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

   3. Taylor expanded in y.re around inf 47.3%

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

4. Final simplification 45.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -2.3 \cdot 10^{+195} \lor \neg \left(y.im \leq 8 \cdot 10^{+172}\right):\\ \;\;\;\;\frac{x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \end{array} \]
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 60.5%

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

3. Taylor expanded in y.re around inf 38.8%

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

Reproduce

herbie shell --seed 2024101 
(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))))