_divideComplex, imaginary part

Percentage Accurate: 61.6% → 86.8%

Time: 13.8s

Alternatives: 13

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 61.6% 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: 86.8% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\frac{y.re}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{-x.re}{\frac{{\left(\mathsf{hypot}\left(y.re, y.im\right)\right)}^{2}}{y.im}}\right)\\ \mathbf{if}\;y.im \leq -1.25 \cdot 10^{+154}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(x.re - \frac{y.re \cdot x.im}{y.im}\right)\\ \mathbf{elif}\;y.im \leq -9.5 \cdot 10^{-280}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq 9 \cdot 10^{+42}:\\ \;\;\;\;\frac{\frac{y.re \cdot x.im - y.im \cdot x.re}{\mathsf{hypot}\left(y.re, y.im\right)}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.im \leq 3.9 \cdot 10^{+118}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{y.re}{\frac{1}{\frac{\frac{x.im}{y.im}}{y.im}}} - \frac{x.re}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0
         (fma
          (/ y.re (hypot y.re y.im))
          (/ x.im (hypot y.re y.im))
          (/ (- x.re) (/ (pow (hypot y.re y.im) 2.0) y.im)))))
   (if (<= y.im -1.25e+154)
     (* (/ 1.0 (hypot y.re y.im)) (- x.re (/ (* y.re x.im) y.im)))
     (if (<= y.im -9.5e-280)
       t_0
       (if (<= y.im 9e+42)
         (/
          (/ (- (* y.re x.im) (* y.im x.re)) (hypot y.re y.im))
          (hypot y.re y.im))
         (if (<= y.im 3.9e+118)
           t_0
           (- (/ y.re (/ 1.0 (/ (/ x.im y.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 t_0 = fma((y_46_re / hypot(y_46_re, y_46_im)), (x_46_im / hypot(y_46_re, y_46_im)), (-x_46_re / (pow(hypot(y_46_re, y_46_im), 2.0) / y_46_im)));
	double tmp;
	if (y_46_im <= -1.25e+154) {
		tmp = (1.0 / hypot(y_46_re, y_46_im)) * (x_46_re - ((y_46_re * x_46_im) / y_46_im));
	} else if (y_46_im <= -9.5e-280) {
		tmp = t_0;
	} else if (y_46_im <= 9e+42) {
		tmp = (((y_46_re * x_46_im) - (y_46_im * x_46_re)) / hypot(y_46_re, y_46_im)) / hypot(y_46_re, y_46_im);
	} else if (y_46_im <= 3.9e+118) {
		tmp = t_0;
	} else {
		tmp = (y_46_re / (1.0 / ((x_46_im / y_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)
	t_0 = fma(Float64(y_46_re / hypot(y_46_re, y_46_im)), Float64(x_46_im / hypot(y_46_re, y_46_im)), Float64(Float64(-x_46_re) / Float64((hypot(y_46_re, y_46_im) ^ 2.0) / y_46_im)))
	tmp = 0.0
	if (y_46_im <= -1.25e+154)
		tmp = Float64(Float64(1.0 / hypot(y_46_re, y_46_im)) * Float64(x_46_re - Float64(Float64(y_46_re * x_46_im) / y_46_im)));
	elseif (y_46_im <= -9.5e-280)
		tmp = t_0;
	elseif (y_46_im <= 9e+42)
		tmp = Float64(Float64(Float64(Float64(y_46_re * x_46_im) - Float64(y_46_im * x_46_re)) / hypot(y_46_re, y_46_im)) / hypot(y_46_re, y_46_im));
	elseif (y_46_im <= 3.9e+118)
		tmp = t_0;
	else
		tmp = Float64(Float64(y_46_re / Float64(1.0 / Float64(Float64(x_46_im / y_46_im) / 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_] := Block[{t$95$0 = N[(N[(y$46$re / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision] * N[(x$46$im / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision] + N[((-x$46$re) / N[(N[Power[N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision], 2.0], $MachinePrecision] / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$im, -1.25e+154], N[(N[(1.0 / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision] * N[(x$46$re - N[(N[(y$46$re * x$46$im), $MachinePrecision] / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, -9.5e-280], t$95$0, If[LessEqual[y$46$im, 9e+42], N[(N[(N[(N[(y$46$re * x$46$im), $MachinePrecision] - N[(y$46$im * x$46$re), $MachinePrecision]), $MachinePrecision] / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision] / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, 3.9e+118], t$95$0, N[(N[(y$46$re / N[(1.0 / N[(N[(x$46$im / y$46$im), $MachinePrecision] / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if y.im < -1.25000000000000001e154

   1. Initial program 19.9%

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

      1. *-un-lft-identity 19.9%

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

      2. add-sqr-sqrt 19.9%

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

      3. times-frac 19.9%

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

      4. hypot-def 19.9%

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

      5. hypot-def 45.3%

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

   3. Applied egg-rr 45.3%

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

   4. Taylor expanded in y.im around -inf 92.9%

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

      1. associate-*r/ 92.9%

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

      2. neg-mul-1 92.9%

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

      3. distribute-rgt-neg-in 92.9%

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

   6. Simplified 92.9%

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

   if -1.25000000000000001e154 < y.im < -9.50000000000000082e-280 or 9.00000000000000025e42 < y.im < 3.9e118

   1. Initial program 67.8%

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

      1. div-sub 63.9%

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

      2. sub-neg 63.9%

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

      3. *-commutative 63.9%

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

      4. add-sqr-sqrt 63.9%

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

      5. times-frac 71.0%

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

      6. fma-def 71.0%

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

      7. hypot-def 71.0%

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

      8. hypot-def 91.1%

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

      9. associate-/l* 93.8%

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

      10. add-sqr-sqrt 93.8%

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

      11. pow2 93.8%

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

      12. hypot-def 93.8%

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

   3. Applied egg-rr 93.8%

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

   if -9.50000000000000082e-280 < y.im < 9.00000000000000025e42

   1. Initial program 77.1%

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

      1. *-un-lft-identity 77.1%

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

      2. add-sqr-sqrt 77.1%

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

      3. times-frac 77.1%

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

      4. hypot-def 77.1%

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

      5. hypot-def 92.3%

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

   3. Applied egg-rr 92.3%

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

      1. associate-*l/ 92.5%

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

      2. *-un-lft-identity 92.5%

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

   5. Applied egg-rr 92.5%

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

   if 3.9e118 < y.im

   1. Initial program 19.2%

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

   2. Taylor expanded in y.re around 0 81.1%

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

      1. +-commutative 81.1%

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

      2. mul-1-neg 81.1%

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

      3. unsub-neg 81.1%

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

      4. *-commutative 81.1%

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

      5. associate-/l* 90.9%

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

   4. Simplified 90.9%

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

      1. pow2 90.9%

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

      2. add-sqr-sqrt 35.9%

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

      3. times-frac 41.9%

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

   6. Applied egg-rr 41.9%

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

      1. unpow2 41.9%

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

   8. Simplified 41.9%

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

      1. unpow2 41.9%

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

      2. clear-num 41.9%

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

      3. clear-num 41.9%

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

      4. frac-times 41.9%

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

      5. metadata-eval 41.9%

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

   10. Applied egg-rr 41.9%

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

       1. associate-*r/ 41.9%

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

       2. associate-*l/ 41.9%

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

       3. rem-square-sqrt 96.9%

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

   12. Simplified 96.9%

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

4. Final simplification 93.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -1.25 \cdot 10^{+154}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(x.re - \frac{y.re \cdot x.im}{y.im}\right)\\ \mathbf{elif}\;y.im \leq -9.5 \cdot 10^{-280}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y.re}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{-x.re}{\frac{{\left(\mathsf{hypot}\left(y.re, y.im\right)\right)}^{2}}{y.im}}\right)\\ \mathbf{elif}\;y.im \leq 9 \cdot 10^{+42}:\\ \;\;\;\;\frac{\frac{y.re \cdot x.im - y.im \cdot x.re}{\mathsf{hypot}\left(y.re, y.im\right)}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.im \leq 3.9 \cdot 10^{+118}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y.re}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{-x.re}{\frac{{\left(\mathsf{hypot}\left(y.re, y.im\right)\right)}^{2}}{y.im}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{y.re}{\frac{1}{\frac{\frac{x.im}{y.im}}{y.im}}} - \frac{x.re}{y.im}\\ \end{array} \]

Alternative 2: 84.8% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y.re \cdot x.im - y.im \cdot x.re\\ t_1 := \frac{t_0}{y.re \cdot y.re + y.im \cdot y.im}\\ t_2 := {\left(\mathsf{hypot}\left(y.re, y.im\right)\right)}^{2}\\ \mathbf{if}\;t_1 \leq 2 \cdot 10^{-137}:\\ \;\;\;\;\frac{\frac{t_0}{\mathsf{hypot}\left(y.re, y.im\right)}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;t_1 \leq \infty:\\ \;\;\;\;\frac{x.im}{\frac{t_2}{y.re}} - \frac{x.re}{\frac{t_2}{y.im}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y.re}{\frac{y.im}{\frac{x.im}{y.im}}} - \frac{x.re}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (- (* y.re x.im) (* y.im x.re)))
        (t_1 (/ t_0 (+ (* y.re y.re) (* y.im y.im))))
        (t_2 (pow (hypot y.re y.im) 2.0)))
   (if (<= t_1 2e-137)
     (/ (/ t_0 (hypot y.re y.im)) (hypot y.re y.im))
     (if (<= t_1 INFINITY)
       (- (/ x.im (/ t_2 y.re)) (/ x.re (/ t_2 y.im)))
       (- (/ y.re (/ y.im (/ 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 t_0 = (y_46_re * x_46_im) - (y_46_im * x_46_re);
	double t_1 = t_0 / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	double t_2 = pow(hypot(y_46_re, y_46_im), 2.0);
	double tmp;
	if (t_1 <= 2e-137) {
		tmp = (t_0 / hypot(y_46_re, y_46_im)) / hypot(y_46_re, y_46_im);
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = (x_46_im / (t_2 / y_46_re)) - (x_46_re / (t_2 / y_46_im));
	} else {
		tmp = (y_46_re / (y_46_im / (x_46_im / y_46_im))) - (x_46_re / y_46_im);
	}
	return tmp;
}

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = (y_46_re * x_46_im) - (y_46_im * x_46_re);
	double t_1 = t_0 / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	double t_2 = Math.pow(Math.hypot(y_46_re, y_46_im), 2.0);
	double tmp;
	if (t_1 <= 2e-137) {
		tmp = (t_0 / Math.hypot(y_46_re, y_46_im)) / Math.hypot(y_46_re, y_46_im);
	} else if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = (x_46_im / (t_2 / y_46_re)) - (x_46_re / (t_2 / y_46_im));
	} else {
		tmp = (y_46_re / (y_46_im / (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):
	t_0 = (y_46_re * x_46_im) - (y_46_im * x_46_re)
	t_1 = t_0 / ((y_46_re * y_46_re) + (y_46_im * y_46_im))
	t_2 = math.pow(math.hypot(y_46_re, y_46_im), 2.0)
	tmp = 0
	if t_1 <= 2e-137:
		tmp = (t_0 / math.hypot(y_46_re, y_46_im)) / math.hypot(y_46_re, y_46_im)
	elif t_1 <= math.inf:
		tmp = (x_46_im / (t_2 / y_46_re)) - (x_46_re / (t_2 / y_46_im))
	else:
		tmp = (y_46_re / (y_46_im / (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)
	t_0 = Float64(Float64(y_46_re * x_46_im) - Float64(y_46_im * x_46_re))
	t_1 = Float64(t_0 / Float64(Float64(y_46_re * y_46_re) + Float64(y_46_im * y_46_im)))
	t_2 = hypot(y_46_re, y_46_im) ^ 2.0
	tmp = 0.0
	if (t_1 <= 2e-137)
		tmp = Float64(Float64(t_0 / hypot(y_46_re, y_46_im)) / hypot(y_46_re, y_46_im));
	elseif (t_1 <= Inf)
		tmp = Float64(Float64(x_46_im / Float64(t_2 / y_46_re)) - Float64(x_46_re / Float64(t_2 / y_46_im)));
	else
		tmp = Float64(Float64(y_46_re / Float64(y_46_im / Float64(x_46_im / y_46_im))) - Float64(x_46_re / y_46_im));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = (y_46_re * x_46_im) - (y_46_im * x_46_re);
	t_1 = t_0 / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	t_2 = hypot(y_46_re, y_46_im) ^ 2.0;
	tmp = 0.0;
	if (t_1 <= 2e-137)
		tmp = (t_0 / hypot(y_46_re, y_46_im)) / hypot(y_46_re, y_46_im);
	elseif (t_1 <= Inf)
		tmp = (x_46_im / (t_2 / y_46_re)) - (x_46_re / (t_2 / y_46_im));
	else
		tmp = (y_46_re / (y_46_im / (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_] := Block[{t$95$0 = N[(N[(y$46$re * x$46$im), $MachinePrecision] - N[(y$46$im * x$46$re), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 / N[(N[(y$46$re * y$46$re), $MachinePrecision] + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Power[N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[t$95$1, 2e-137], N[(N[(t$95$0 / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision] / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[(N[(x$46$im / N[(t$95$2 / y$46$re), $MachinePrecision]), $MachinePrecision] - N[(x$46$re / N[(t$95$2 / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y$46$re / N[(y$46$im / N[(x$46$im / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 (*.f64 x.im y.re) (*.f64 x.re y.im)) (+.f64 (*.f64 y.re y.re) (*.f64 y.im y.im))) < 1.99999999999999996e-137

   1. Initial program 68.3%

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

      1. *-un-lft-identity 68.3%

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

      2. add-sqr-sqrt 68.3%

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

      3. times-frac 68.3%

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

      4. hypot-def 68.3%

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

      5. hypot-def 95.4%

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

   3. Applied egg-rr 95.4%

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

      1. associate-*l/ 95.5%

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

      2. *-un-lft-identity 95.5%

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

   5. Applied egg-rr 95.5%

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

   if 1.99999999999999996e-137 < (/.f64 (-.f64 (*.f64 x.im y.re) (*.f64 x.re y.im)) (+.f64 (*.f64 y.re y.re) (*.f64 y.im y.im))) < +inf.0

   1. Initial program 81.4%

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

      1. div-sub 75.5%

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

      2. associate-/l* 86.9%

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

      3. add-sqr-sqrt 86.9%

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

      4. pow2 86.9%

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

      5. hypot-def 86.9%

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

      6. associate-/l* 92.6%

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

      7. add-sqr-sqrt 92.6%

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

      8. pow2 92.6%

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

      9. hypot-def 92.6%

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

   3. Applied egg-rr 92.6%

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

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

   1. Initial program 0.0%

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

   2. Taylor expanded in y.re around 0 59.0%

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

      1. +-commutative 59.0%

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

      2. mul-1-neg 59.0%

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

      3. unsub-neg 59.0%

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

      4. *-commutative 59.0%

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

      5. associate-/l* 64.7%

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

   4. Simplified 64.7%

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

      1. pow2 64.7%

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

      2. add-sqr-sqrt 33.3%

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

      3. times-frac 33.5%

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

   6. Applied egg-rr 33.5%

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

      1. unpow2 33.5%

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

   8. Simplified 33.5%

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

      1. unpow2 33.5%

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

      2. clear-num 33.5%

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

      3. frac-times 33.5%

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

      4. *-un-lft-identity 33.5%

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

   10. Applied egg-rr 33.5%

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

       1. associate-*l/ 33.5%

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

       2. rem-square-sqrt 64.9%

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

   12. Simplified 64.9%

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

4. Final simplification 88.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im} \leq 2 \cdot 10^{-137}:\\ \;\;\;\;\frac{\frac{y.re \cdot x.im - y.im \cdot x.re}{\mathsf{hypot}\left(y.re, y.im\right)}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;\frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im} \leq \infty:\\ \;\;\;\;\frac{x.im}{\frac{{\left(\mathsf{hypot}\left(y.re, y.im\right)\right)}^{2}}{y.re}} - \frac{x.re}{\frac{{\left(\mathsf{hypot}\left(y.re, y.im\right)\right)}^{2}}{y.im}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y.re}{\frac{y.im}{\frac{x.im}{y.im}}} - \frac{x.re}{y.im}\\ \end{array} \]

Alternative 3: 84.4% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y.re \cdot x.im - y.im \cdot x.re\\ \mathbf{if}\;\frac{t_0}{y.re \cdot y.re + y.im \cdot y.im} \leq 2 \cdot 10^{+297}:\\ \;\;\;\;\frac{\frac{t_0}{\mathsf{hypot}\left(y.re, y.im\right)}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{y.re}{\frac{y.im}{\frac{x.im}{y.im}}} - \frac{x.re}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (- (* y.re x.im) (* y.im x.re))))
   (if (<= (/ t_0 (+ (* y.re y.re) (* y.im y.im))) 2e+297)
     (/ (/ t_0 (hypot y.re y.im)) (hypot y.re y.im))
     (- (/ y.re (/ y.im (/ 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 t_0 = (y_46_re * x_46_im) - (y_46_im * x_46_re);
	double tmp;
	if ((t_0 / ((y_46_re * y_46_re) + (y_46_im * y_46_im))) <= 2e+297) {
		tmp = (t_0 / hypot(y_46_re, y_46_im)) / hypot(y_46_re, y_46_im);
	} else {
		tmp = (y_46_re / (y_46_im / (x_46_im / y_46_im))) - (x_46_re / y_46_im);
	}
	return tmp;
}

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = (y_46_re * x_46_im) - (y_46_im * x_46_re);
	double tmp;
	if ((t_0 / ((y_46_re * y_46_re) + (y_46_im * y_46_im))) <= 2e+297) {
		tmp = (t_0 / Math.hypot(y_46_re, y_46_im)) / Math.hypot(y_46_re, y_46_im);
	} else {
		tmp = (y_46_re / (y_46_im / (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):
	t_0 = (y_46_re * x_46_im) - (y_46_im * x_46_re)
	tmp = 0
	if (t_0 / ((y_46_re * y_46_re) + (y_46_im * y_46_im))) <= 2e+297:
		tmp = (t_0 / math.hypot(y_46_re, y_46_im)) / math.hypot(y_46_re, y_46_im)
	else:
		tmp = (y_46_re / (y_46_im / (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)
	t_0 = Float64(Float64(y_46_re * x_46_im) - Float64(y_46_im * x_46_re))
	tmp = 0.0
	if (Float64(t_0 / Float64(Float64(y_46_re * y_46_re) + Float64(y_46_im * y_46_im))) <= 2e+297)
		tmp = Float64(Float64(t_0 / hypot(y_46_re, y_46_im)) / hypot(y_46_re, y_46_im));
	else
		tmp = Float64(Float64(y_46_re / Float64(y_46_im / Float64(x_46_im / y_46_im))) - Float64(x_46_re / y_46_im));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = (y_46_re * x_46_im) - (y_46_im * x_46_re);
	tmp = 0.0;
	if ((t_0 / ((y_46_re * y_46_re) + (y_46_im * y_46_im))) <= 2e+297)
		tmp = (t_0 / hypot(y_46_re, y_46_im)) / hypot(y_46_re, y_46_im);
	else
		tmp = (y_46_re / (y_46_im / (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_] := Block[{t$95$0 = N[(N[(y$46$re * x$46$im), $MachinePrecision] - N[(y$46$im * x$46$re), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t$95$0 / N[(N[(y$46$re * y$46$re), $MachinePrecision] + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2e+297], N[(N[(t$95$0 / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision] / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision], N[(N[(y$46$re / N[(y$46$im / N[(x$46$im / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 (*.f64 x.im y.re) (*.f64 x.re y.im)) (+.f64 (*.f64 y.re y.re) (*.f64 y.im y.im))) < 2e297

   1. Initial program 76.6%

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

      1. *-un-lft-identity 76.6%

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

      2. add-sqr-sqrt 76.6%

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

      3. times-frac 76.5%

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

      4. hypot-def 76.5%

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

      5. hypot-def 96.4%

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

   3. Applied egg-rr 96.4%

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

      1. associate-*l/ 96.6%

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

      2. *-un-lft-identity 96.6%

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

   5. Applied egg-rr 96.6%

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

   if 2e297 < (/.f64 (-.f64 (*.f64 x.im y.re) (*.f64 x.re y.im)) (+.f64 (*.f64 y.re y.re) (*.f64 y.im y.im)))

   1. Initial program 10.1%

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

   2. Taylor expanded in y.re around 0 54.5%

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

      1. +-commutative 54.5%

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

      2. mul-1-neg 54.5%

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

      3. unsub-neg 54.5%

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

      4. *-commutative 54.5%

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

      5. associate-/l* 61.8%

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

   4. Simplified 61.8%

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

      1. pow2 61.8%

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

      2. add-sqr-sqrt 31.4%

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

      3. times-frac 31.6%

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

   6. Applied egg-rr 31.6%

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

      1. unpow2 31.6%

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

   8. Simplified 31.6%

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

      1. unpow2 31.6%

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

      2. clear-num 31.6%

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

      3. frac-times 31.6%

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

      4. *-un-lft-identity 31.6%

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

   10. Applied egg-rr 31.6%

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

       1. associate-*l/ 31.6%

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

       2. rem-square-sqrt 62.1%

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

   12. Simplified 62.1%

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

4. Final simplification 86.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im} \leq 2 \cdot 10^{+297}:\\ \;\;\;\;\frac{\frac{y.re \cdot x.im - y.im \cdot x.re}{\mathsf{hypot}\left(y.re, y.im\right)}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{y.re}{\frac{y.im}{\frac{x.im}{y.im}}} - \frac{x.re}{y.im}\\ \end{array} \]

Alternative 4: 78.7% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{if}\;y.im \leq -2.5 \cdot 10^{+95}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(x.re - \frac{y.re \cdot x.im}{y.im}\right)\\ \mathbf{elif}\;y.im \leq -3.8 \cdot 10^{-63}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq 3.9 \cdot 10^{-82}:\\ \;\;\;\;\frac{x.im}{y.re} - y.im \cdot \frac{\frac{x.re}{y.re}}{y.re}\\ \mathbf{elif}\;y.im \leq 2 \cdot 10^{+118}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{y.re}{\frac{1}{\frac{\frac{x.im}{y.im}}{y.im}}} - \frac{x.re}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0
         (/ (- (* y.re x.im) (* y.im x.re)) (+ (* y.re y.re) (* y.im y.im)))))
   (if (<= y.im -2.5e+95)
     (* (/ 1.0 (hypot y.re y.im)) (- x.re (/ (* y.re x.im) y.im)))
     (if (<= y.im -3.8e-63)
       t_0
       (if (<= y.im 3.9e-82)
         (- (/ x.im y.re) (* y.im (/ (/ x.re y.re) y.re)))
         (if (<= y.im 2e+118)
           t_0
           (- (/ y.re (/ 1.0 (/ (/ x.im y.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 t_0 = ((y_46_re * x_46_im) - (y_46_im * x_46_re)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	double tmp;
	if (y_46_im <= -2.5e+95) {
		tmp = (1.0 / hypot(y_46_re, y_46_im)) * (x_46_re - ((y_46_re * x_46_im) / y_46_im));
	} else if (y_46_im <= -3.8e-63) {
		tmp = t_0;
	} else if (y_46_im <= 3.9e-82) {
		tmp = (x_46_im / y_46_re) - (y_46_im * ((x_46_re / y_46_re) / y_46_re));
	} else if (y_46_im <= 2e+118) {
		tmp = t_0;
	} else {
		tmp = (y_46_re / (1.0 / ((x_46_im / y_46_im) / y_46_im))) - (x_46_re / y_46_im);
	}
	return tmp;
}

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = ((y_46_re * x_46_im) - (y_46_im * x_46_re)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	double tmp;
	if (y_46_im <= -2.5e+95) {
		tmp = (1.0 / Math.hypot(y_46_re, y_46_im)) * (x_46_re - ((y_46_re * x_46_im) / y_46_im));
	} else if (y_46_im <= -3.8e-63) {
		tmp = t_0;
	} else if (y_46_im <= 3.9e-82) {
		tmp = (x_46_im / y_46_re) - (y_46_im * ((x_46_re / y_46_re) / y_46_re));
	} else if (y_46_im <= 2e+118) {
		tmp = t_0;
	} else {
		tmp = (y_46_re / (1.0 / ((x_46_im / y_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):
	t_0 = ((y_46_re * x_46_im) - (y_46_im * x_46_re)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im))
	tmp = 0
	if y_46_im <= -2.5e+95:
		tmp = (1.0 / math.hypot(y_46_re, y_46_im)) * (x_46_re - ((y_46_re * x_46_im) / y_46_im))
	elif y_46_im <= -3.8e-63:
		tmp = t_0
	elif y_46_im <= 3.9e-82:
		tmp = (x_46_im / y_46_re) - (y_46_im * ((x_46_re / y_46_re) / y_46_re))
	elif y_46_im <= 2e+118:
		tmp = t_0
	else:
		tmp = (y_46_re / (1.0 / ((x_46_im / y_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)
	t_0 = Float64(Float64(Float64(y_46_re * x_46_im) - Float64(y_46_im * x_46_re)) / Float64(Float64(y_46_re * y_46_re) + Float64(y_46_im * y_46_im)))
	tmp = 0.0
	if (y_46_im <= -2.5e+95)
		tmp = Float64(Float64(1.0 / hypot(y_46_re, y_46_im)) * Float64(x_46_re - Float64(Float64(y_46_re * x_46_im) / y_46_im)));
	elseif (y_46_im <= -3.8e-63)
		tmp = t_0;
	elseif (y_46_im <= 3.9e-82)
		tmp = Float64(Float64(x_46_im / y_46_re) - Float64(y_46_im * Float64(Float64(x_46_re / y_46_re) / y_46_re)));
	elseif (y_46_im <= 2e+118)
		tmp = t_0;
	else
		tmp = Float64(Float64(y_46_re / Float64(1.0 / Float64(Float64(x_46_im / y_46_im) / y_46_im))) - Float64(x_46_re / y_46_im));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = ((y_46_re * x_46_im) - (y_46_im * x_46_re)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	tmp = 0.0;
	if (y_46_im <= -2.5e+95)
		tmp = (1.0 / hypot(y_46_re, y_46_im)) * (x_46_re - ((y_46_re * x_46_im) / y_46_im));
	elseif (y_46_im <= -3.8e-63)
		tmp = t_0;
	elseif (y_46_im <= 3.9e-82)
		tmp = (x_46_im / y_46_re) - (y_46_im * ((x_46_re / y_46_re) / y_46_re));
	elseif (y_46_im <= 2e+118)
		tmp = t_0;
	else
		tmp = (y_46_re / (1.0 / ((x_46_im / y_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_] := Block[{t$95$0 = N[(N[(N[(y$46$re * x$46$im), $MachinePrecision] - N[(y$46$im * x$46$re), $MachinePrecision]), $MachinePrecision] / N[(N[(y$46$re * y$46$re), $MachinePrecision] + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$im, -2.5e+95], N[(N[(1.0 / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision] * N[(x$46$re - N[(N[(y$46$re * x$46$im), $MachinePrecision] / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, -3.8e-63], t$95$0, If[LessEqual[y$46$im, 3.9e-82], N[(N[(x$46$im / y$46$re), $MachinePrecision] - N[(y$46$im * N[(N[(x$46$re / y$46$re), $MachinePrecision] / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, 2e+118], t$95$0, N[(N[(y$46$re / N[(1.0 / N[(N[(x$46$im / y$46$im), $MachinePrecision] / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if y.im < -2.50000000000000012e95

   1. Initial program 20.7%

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

      1. *-un-lft-identity 20.7%

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

      2. add-sqr-sqrt 20.7%

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

      3. times-frac 20.7%

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

      4. hypot-def 20.7%

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

      5. hypot-def 47.6%

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

   3. Applied egg-rr 47.6%

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

   4. Taylor expanded in y.im around -inf 81.7%

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

      1. associate-*r/ 81.7%

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

      2. neg-mul-1 81.7%

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

      3. distribute-rgt-neg-in 81.7%

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

   6. Simplified 81.7%

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

   if -2.50000000000000012e95 < y.im < -3.80000000000000017e-63 or 3.89999999999999973e-82 < y.im < 1.99999999999999993e118

   1. Initial program 77.4%

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

   if -3.80000000000000017e-63 < y.im < 3.89999999999999973e-82

   1. Initial program 72.7%

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

   2. Taylor expanded in y.re around inf 84.8%

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

      1. +-commutative 84.8%

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

      2. mul-1-neg 84.8%

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

      3. unsub-neg 84.8%

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

      4. associate-/l* 81.9%

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

      5. associate-/r/ 80.0%

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

   4. Simplified 80.0%

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

      1. *-un-lft-identity 80.0%

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

      2. pow2 80.0%

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

      3. times-frac 84.1%

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

   6. Applied egg-rr 84.1%

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

      1. associate-*l/ 84.1%

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

      2. *-lft-identity 84.1%

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

   8. Simplified 84.1%

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

   if 1.99999999999999993e118 < y.im

   1. Initial program 19.2%

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

   2. Taylor expanded in y.re around 0 81.1%

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

      1. +-commutative 81.1%

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

      2. mul-1-neg 81.1%

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

      3. unsub-neg 81.1%

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

      4. *-commutative 81.1%

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

      5. associate-/l* 90.9%

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

   4. Simplified 90.9%

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

      1. pow2 90.9%

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

      2. add-sqr-sqrt 35.9%

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

      3. times-frac 41.9%

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

   6. Applied egg-rr 41.9%

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

      1. unpow2 41.9%

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

   8. Simplified 41.9%

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

      1. unpow2 41.9%

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

      2. clear-num 41.9%

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

      3. clear-num 41.9%

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

      4. frac-times 41.9%

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

      5. metadata-eval 41.9%

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

   10. Applied egg-rr 41.9%

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

       1. associate-*r/ 41.9%

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

       2. associate-*l/ 41.9%

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

       3. rem-square-sqrt 96.9%

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

   12. Simplified 96.9%

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

4. Final simplification 83.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -2.5 \cdot 10^{+95}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(x.re - \frac{y.re \cdot x.im}{y.im}\right)\\ \mathbf{elif}\;y.im \leq -3.8 \cdot 10^{-63}:\\ \;\;\;\;\frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.im \leq 3.9 \cdot 10^{-82}:\\ \;\;\;\;\frac{x.im}{y.re} - y.im \cdot \frac{\frac{x.re}{y.re}}{y.re}\\ \mathbf{elif}\;y.im \leq 2 \cdot 10^{+118}:\\ \;\;\;\;\frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{y.re}{\frac{1}{\frac{\frac{x.im}{y.im}}{y.im}}} - \frac{x.re}{y.im}\\ \end{array} \]

Alternative 5: 78.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{if}\;y.im \leq -1.78 \cdot 10^{+109}:\\ \;\;\;\;\left(x.re - \frac{y.re \cdot x.im}{y.im}\right) \cdot \frac{-1}{y.im}\\ \mathbf{elif}\;y.im \leq -3.6 \cdot 10^{-61}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq 3.5 \cdot 10^{-82}:\\ \;\;\;\;\frac{x.im}{y.re} - y.im \cdot \frac{\frac{x.re}{y.re}}{y.re}\\ \mathbf{elif}\;y.im \leq 3.6 \cdot 10^{+118}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{y.re}{\frac{1}{\frac{\frac{x.im}{y.im}}{y.im}}} - \frac{x.re}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0
         (/ (- (* y.re x.im) (* y.im x.re)) (+ (* y.re y.re) (* y.im y.im)))))
   (if (<= y.im -1.78e+109)
     (* (- x.re (/ (* y.re x.im) y.im)) (/ -1.0 y.im))
     (if (<= y.im -3.6e-61)
       t_0
       (if (<= y.im 3.5e-82)
         (- (/ x.im y.re) (* y.im (/ (/ x.re y.re) y.re)))
         (if (<= y.im 3.6e+118)
           t_0
           (- (/ y.re (/ 1.0 (/ (/ x.im y.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 t_0 = ((y_46_re * x_46_im) - (y_46_im * x_46_re)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	double tmp;
	if (y_46_im <= -1.78e+109) {
		tmp = (x_46_re - ((y_46_re * x_46_im) / y_46_im)) * (-1.0 / y_46_im);
	} else if (y_46_im <= -3.6e-61) {
		tmp = t_0;
	} else if (y_46_im <= 3.5e-82) {
		tmp = (x_46_im / y_46_re) - (y_46_im * ((x_46_re / y_46_re) / y_46_re));
	} else if (y_46_im <= 3.6e+118) {
		tmp = t_0;
	} else {
		tmp = (y_46_re / (1.0 / ((x_46_im / y_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) :: t_0
    real(8) :: tmp
    t_0 = ((y_46re * x_46im) - (y_46im * x_46re)) / ((y_46re * y_46re) + (y_46im * y_46im))
    if (y_46im <= (-1.78d+109)) then
        tmp = (x_46re - ((y_46re * x_46im) / y_46im)) * ((-1.0d0) / y_46im)
    else if (y_46im <= (-3.6d-61)) then
        tmp = t_0
    else if (y_46im <= 3.5d-82) then
        tmp = (x_46im / y_46re) - (y_46im * ((x_46re / y_46re) / y_46re))
    else if (y_46im <= 3.6d+118) then
        tmp = t_0
    else
        tmp = (y_46re / (1.0d0 / ((x_46im / y_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 t_0 = ((y_46_re * x_46_im) - (y_46_im * x_46_re)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	double tmp;
	if (y_46_im <= -1.78e+109) {
		tmp = (x_46_re - ((y_46_re * x_46_im) / y_46_im)) * (-1.0 / y_46_im);
	} else if (y_46_im <= -3.6e-61) {
		tmp = t_0;
	} else if (y_46_im <= 3.5e-82) {
		tmp = (x_46_im / y_46_re) - (y_46_im * ((x_46_re / y_46_re) / y_46_re));
	} else if (y_46_im <= 3.6e+118) {
		tmp = t_0;
	} else {
		tmp = (y_46_re / (1.0 / ((x_46_im / y_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):
	t_0 = ((y_46_re * x_46_im) - (y_46_im * x_46_re)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im))
	tmp = 0
	if y_46_im <= -1.78e+109:
		tmp = (x_46_re - ((y_46_re * x_46_im) / y_46_im)) * (-1.0 / y_46_im)
	elif y_46_im <= -3.6e-61:
		tmp = t_0
	elif y_46_im <= 3.5e-82:
		tmp = (x_46_im / y_46_re) - (y_46_im * ((x_46_re / y_46_re) / y_46_re))
	elif y_46_im <= 3.6e+118:
		tmp = t_0
	else:
		tmp = (y_46_re / (1.0 / ((x_46_im / y_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)
	t_0 = Float64(Float64(Float64(y_46_re * x_46_im) - Float64(y_46_im * x_46_re)) / Float64(Float64(y_46_re * y_46_re) + Float64(y_46_im * y_46_im)))
	tmp = 0.0
	if (y_46_im <= -1.78e+109)
		tmp = Float64(Float64(x_46_re - Float64(Float64(y_46_re * x_46_im) / y_46_im)) * Float64(-1.0 / y_46_im));
	elseif (y_46_im <= -3.6e-61)
		tmp = t_0;
	elseif (y_46_im <= 3.5e-82)
		tmp = Float64(Float64(x_46_im / y_46_re) - Float64(y_46_im * Float64(Float64(x_46_re / y_46_re) / y_46_re)));
	elseif (y_46_im <= 3.6e+118)
		tmp = t_0;
	else
		tmp = Float64(Float64(y_46_re / Float64(1.0 / Float64(Float64(x_46_im / y_46_im) / y_46_im))) - Float64(x_46_re / y_46_im));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = ((y_46_re * x_46_im) - (y_46_im * x_46_re)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	tmp = 0.0;
	if (y_46_im <= -1.78e+109)
		tmp = (x_46_re - ((y_46_re * x_46_im) / y_46_im)) * (-1.0 / y_46_im);
	elseif (y_46_im <= -3.6e-61)
		tmp = t_0;
	elseif (y_46_im <= 3.5e-82)
		tmp = (x_46_im / y_46_re) - (y_46_im * ((x_46_re / y_46_re) / y_46_re));
	elseif (y_46_im <= 3.6e+118)
		tmp = t_0;
	else
		tmp = (y_46_re / (1.0 / ((x_46_im / y_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_] := Block[{t$95$0 = N[(N[(N[(y$46$re * x$46$im), $MachinePrecision] - N[(y$46$im * x$46$re), $MachinePrecision]), $MachinePrecision] / N[(N[(y$46$re * y$46$re), $MachinePrecision] + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$im, -1.78e+109], N[(N[(x$46$re - N[(N[(y$46$re * x$46$im), $MachinePrecision] / y$46$im), $MachinePrecision]), $MachinePrecision] * N[(-1.0 / y$46$im), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, -3.6e-61], t$95$0, If[LessEqual[y$46$im, 3.5e-82], N[(N[(x$46$im / y$46$re), $MachinePrecision] - N[(y$46$im * N[(N[(x$46$re / y$46$re), $MachinePrecision] / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, 3.6e+118], t$95$0, N[(N[(y$46$re / N[(1.0 / N[(N[(x$46$im / y$46$im), $MachinePrecision] / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if y.im < -1.7800000000000001e109

   1. Initial program 21.3%

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

      1. *-un-lft-identity 21.3%

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

      2. add-sqr-sqrt 21.3%

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

      3. times-frac 21.3%

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

      4. hypot-def 21.3%

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

      5. hypot-def 47.4%

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

   3. Applied egg-rr 47.4%

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

   4. Taylor expanded in y.im around -inf 84.7%

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

      1. associate-*r/ 84.7%

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

      2. neg-mul-1 84.7%

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

      3. distribute-rgt-neg-in 84.7%

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

   6. Simplified 84.7%

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

   7. Taylor expanded in y.im around -inf 84.6%

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

   if -1.7800000000000001e109 < y.im < -3.60000000000000014e-61 or 3.4999999999999999e-82 < y.im < 3.6e118

   1. Initial program 75.6%

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

   if -3.60000000000000014e-61 < y.im < 3.4999999999999999e-82

   1. Initial program 72.7%

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

   2. Taylor expanded in y.re around inf 84.8%

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

      1. +-commutative 84.8%

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

      2. mul-1-neg 84.8%

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

      3. unsub-neg 84.8%

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

      4. associate-/l* 81.9%

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

      5. associate-/r/ 80.0%

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

   4. Simplified 80.0%

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

      1. *-un-lft-identity 80.0%

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

      2. pow2 80.0%

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

      3. times-frac 84.1%

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

   6. Applied egg-rr 84.1%

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

      1. associate-*l/ 84.1%

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

      2. *-lft-identity 84.1%

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

   8. Simplified 84.1%

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

   if 3.6e118 < y.im

   1. Initial program 19.2%

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

   2. Taylor expanded in y.re around 0 81.1%

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

      1. +-commutative 81.1%

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

      2. mul-1-neg 81.1%

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

      3. unsub-neg 81.1%

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

      4. *-commutative 81.1%

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

      5. associate-/l* 90.9%

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

   4. Simplified 90.9%

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

      1. pow2 90.9%

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

      2. add-sqr-sqrt 35.9%

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

      3. times-frac 41.9%

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

   6. Applied egg-rr 41.9%

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

      1. unpow2 41.9%

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

   8. Simplified 41.9%

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

      1. unpow2 41.9%

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

      2. clear-num 41.9%

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

      3. clear-num 41.9%

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

      4. frac-times 41.9%

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

      5. metadata-eval 41.9%

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

   10. Applied egg-rr 41.9%

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

       1. associate-*r/ 41.9%

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

       2. associate-*l/ 41.9%

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

       3. rem-square-sqrt 96.9%

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

   12. Simplified 96.9%

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

4. Final simplification 83.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -1.78 \cdot 10^{+109}:\\ \;\;\;\;\left(x.re - \frac{y.re \cdot x.im}{y.im}\right) \cdot \frac{-1}{y.im}\\ \mathbf{elif}\;y.im \leq -3.6 \cdot 10^{-61}:\\ \;\;\;\;\frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.im \leq 3.5 \cdot 10^{-82}:\\ \;\;\;\;\frac{x.im}{y.re} - y.im \cdot \frac{\frac{x.re}{y.re}}{y.re}\\ \mathbf{elif}\;y.im \leq 3.6 \cdot 10^{+118}:\\ \;\;\;\;\frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{y.re}{\frac{1}{\frac{\frac{x.im}{y.im}}{y.im}}} - \frac{x.re}{y.im}\\ \end{array} \]

Alternative 6: 60.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-x.re}{y.im}\\ \mathbf{if}\;y.im \leq -7.6 \cdot 10^{+128}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq -115:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{elif}\;y.im \leq -5.8 \cdot 10^{-58}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq 4.6 \cdot 10^{+15}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{elif}\;y.im \leq 4.5 \cdot 10^{+48}:\\ \;\;\;\;\frac{x.im \cdot \frac{y.re}{y.im}}{y.im}\\ \mathbf{elif}\;y.im \leq 8.2 \cdot 10^{+99}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (/ (- x.re) y.im)))
   (if (<= y.im -7.6e+128)
     t_0
     (if (<= y.im -115.0)
       (/ x.im y.re)
       (if (<= y.im -5.8e-58)
         t_0
         (if (<= y.im 4.6e+15)
           (/ x.im y.re)
           (if (<= y.im 4.5e+48)
             (/ (* x.im (/ y.re y.im)) y.im)
             (if (<= y.im 8.2e+99) (/ x.im y.re) t_0))))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = -x_46_re / y_46_im;
	double tmp;
	if (y_46_im <= -7.6e+128) {
		tmp = t_0;
	} else if (y_46_im <= -115.0) {
		tmp = x_46_im / y_46_re;
	} else if (y_46_im <= -5.8e-58) {
		tmp = t_0;
	} else if (y_46_im <= 4.6e+15) {
		tmp = x_46_im / y_46_re;
	} else if (y_46_im <= 4.5e+48) {
		tmp = (x_46_im * (y_46_re / y_46_im)) / y_46_im;
	} else if (y_46_im <= 8.2e+99) {
		tmp = x_46_im / y_46_re;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = -x_46re / y_46im
    if (y_46im <= (-7.6d+128)) then
        tmp = t_0
    else if (y_46im <= (-115.0d0)) then
        tmp = x_46im / y_46re
    else if (y_46im <= (-5.8d-58)) then
        tmp = t_0
    else if (y_46im <= 4.6d+15) then
        tmp = x_46im / y_46re
    else if (y_46im <= 4.5d+48) then
        tmp = (x_46im * (y_46re / y_46im)) / y_46im
    else if (y_46im <= 8.2d+99) then
        tmp = x_46im / y_46re
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = -x_46_re / y_46_im;
	double tmp;
	if (y_46_im <= -7.6e+128) {
		tmp = t_0;
	} else if (y_46_im <= -115.0) {
		tmp = x_46_im / y_46_re;
	} else if (y_46_im <= -5.8e-58) {
		tmp = t_0;
	} else if (y_46_im <= 4.6e+15) {
		tmp = x_46_im / y_46_re;
	} else if (y_46_im <= 4.5e+48) {
		tmp = (x_46_im * (y_46_re / y_46_im)) / y_46_im;
	} else if (y_46_im <= 8.2e+99) {
		tmp = x_46_im / y_46_re;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = -x_46_re / y_46_im
	tmp = 0
	if y_46_im <= -7.6e+128:
		tmp = t_0
	elif y_46_im <= -115.0:
		tmp = x_46_im / y_46_re
	elif y_46_im <= -5.8e-58:
		tmp = t_0
	elif y_46_im <= 4.6e+15:
		tmp = x_46_im / y_46_re
	elif y_46_im <= 4.5e+48:
		tmp = (x_46_im * (y_46_re / y_46_im)) / y_46_im
	elif y_46_im <= 8.2e+99:
		tmp = x_46_im / y_46_re
	else:
		tmp = t_0
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(Float64(-x_46_re) / y_46_im)
	tmp = 0.0
	if (y_46_im <= -7.6e+128)
		tmp = t_0;
	elseif (y_46_im <= -115.0)
		tmp = Float64(x_46_im / y_46_re);
	elseif (y_46_im <= -5.8e-58)
		tmp = t_0;
	elseif (y_46_im <= 4.6e+15)
		tmp = Float64(x_46_im / y_46_re);
	elseif (y_46_im <= 4.5e+48)
		tmp = Float64(Float64(x_46_im * Float64(y_46_re / y_46_im)) / y_46_im);
	elseif (y_46_im <= 8.2e+99)
		tmp = Float64(x_46_im / y_46_re);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = -x_46_re / y_46_im;
	tmp = 0.0;
	if (y_46_im <= -7.6e+128)
		tmp = t_0;
	elseif (y_46_im <= -115.0)
		tmp = x_46_im / y_46_re;
	elseif (y_46_im <= -5.8e-58)
		tmp = t_0;
	elseif (y_46_im <= 4.6e+15)
		tmp = x_46_im / y_46_re;
	elseif (y_46_im <= 4.5e+48)
		tmp = (x_46_im * (y_46_re / y_46_im)) / y_46_im;
	elseif (y_46_im <= 8.2e+99)
		tmp = x_46_im / y_46_re;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[((-x$46$re) / y$46$im), $MachinePrecision]}, If[LessEqual[y$46$im, -7.6e+128], t$95$0, If[LessEqual[y$46$im, -115.0], N[(x$46$im / y$46$re), $MachinePrecision], If[LessEqual[y$46$im, -5.8e-58], t$95$0, If[LessEqual[y$46$im, 4.6e+15], N[(x$46$im / y$46$re), $MachinePrecision], If[LessEqual[y$46$im, 4.5e+48], N[(N[(x$46$im * N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision] / y$46$im), $MachinePrecision], If[LessEqual[y$46$im, 8.2e+99], N[(x$46$im / y$46$re), $MachinePrecision], t$95$0]]]]]]]

Derivation

1. Split input into 3 regimes

2. if y.im < -7.5999999999999998e128 or -115 < y.im < -5.7999999999999998e-58 or 8.19999999999999959e99 < y.im

   1. Initial program 31.7%

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

   2. Taylor expanded in y.re around 0 81.9%

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

      1. associate-*r/ 81.9%

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

      2. neg-mul-1 81.9%

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

   4. Simplified 81.9%

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

   if -7.5999999999999998e128 < y.im < -115 or -5.7999999999999998e-58 < y.im < 4.6e15 or 4.49999999999999995e48 < y.im < 8.19999999999999959e99

   1. Initial program 71.8%

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

   2. Taylor expanded in y.re around inf 61.7%

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

   if 4.6e15 < y.im < 4.49999999999999995e48

   1. Initial program 68.5%

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

   2. Taylor expanded in x.im around inf 57.5%

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

      1. *-commutative 57.5%

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

   4. Simplified 57.5%

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

   5. Taylor expanded in y.re around 0 56.2%

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

      1. *-commutative 56.2%

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

      2. associate-*l/ 45.4%

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

      3. *-commutative 45.4%

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

   7. Simplified 45.4%

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

      1. *-un-lft-identity 45.4%

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

      2. unpow2 45.4%

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

      3. times-frac 45.4%

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

   9. Applied egg-rr 45.4%

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

       1. associate-*l/ 45.5%

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

       2. *-lft-identity 45.5%

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

   11. Simplified 45.5%

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

       1. associate-*r/ 54.7%

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

   13. Applied egg-rr 54.7%

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

4. Final simplification 68.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -7.6 \cdot 10^{+128}:\\ \;\;\;\;\frac{-x.re}{y.im}\\ \mathbf{elif}\;y.im \leq -115:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{elif}\;y.im \leq -5.8 \cdot 10^{-58}:\\ \;\;\;\;\frac{-x.re}{y.im}\\ \mathbf{elif}\;y.im \leq 4.6 \cdot 10^{+15}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{elif}\;y.im \leq 4.5 \cdot 10^{+48}:\\ \;\;\;\;\frac{x.im \cdot \frac{y.re}{y.im}}{y.im}\\ \mathbf{elif}\;y.im \leq 8.2 \cdot 10^{+99}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{-x.re}{y.im}\\ \end{array} \]

Alternative 7: 68.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -7.6 \cdot 10^{+128} \lor \neg \left(y.im \leq 8.5 \cdot 10^{+99}\right):\\ \;\;\;\;\frac{-x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re} - y.im \cdot \frac{\frac{x.re}{y.re}}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.im -7.6e+128) (not (<= y.im 8.5e+99)))
   (/ (- x.re) y.im)
   (- (/ x.im y.re) (* y.im (/ (/ x.re y.re) y.re)))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_im <= -7.6e+128) || !(y_46_im <= 8.5e+99)) {
		tmp = -x_46_re / y_46_im;
	} else {
		tmp = (x_46_im / y_46_re) - (y_46_im * ((x_46_re / y_46_re) / y_46_re));
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if ((y_46im <= (-7.6d+128)) .or. (.not. (y_46im <= 8.5d+99))) then
        tmp = -x_46re / y_46im
    else
        tmp = (x_46im / y_46re) - (y_46im * ((x_46re / y_46re) / y_46re))
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_im <= -7.6e+128) || !(y_46_im <= 8.5e+99)) {
		tmp = -x_46_re / y_46_im;
	} else {
		tmp = (x_46_im / y_46_re) - (y_46_im * ((x_46_re / y_46_re) / y_46_re));
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if (y_46_im <= -7.6e+128) or not (y_46_im <= 8.5e+99):
		tmp = -x_46_re / y_46_im
	else:
		tmp = (x_46_im / y_46_re) - (y_46_im * ((x_46_re / y_46_re) / y_46_re))
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if ((y_46_im <= -7.6e+128) || !(y_46_im <= 8.5e+99))
		tmp = Float64(Float64(-x_46_re) / y_46_im);
	else
		tmp = Float64(Float64(x_46_im / y_46_re) - Float64(y_46_im * Float64(Float64(x_46_re / y_46_re) / y_46_re)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if ((y_46_im <= -7.6e+128) || ~((y_46_im <= 8.5e+99)))
		tmp = -x_46_re / y_46_im;
	else
		tmp = (x_46_im / y_46_re) - (y_46_im * ((x_46_re / y_46_re) / y_46_re));
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[Or[LessEqual[y$46$im, -7.6e+128], N[Not[LessEqual[y$46$im, 8.5e+99]], $MachinePrecision]], N[((-x$46$re) / y$46$im), $MachinePrecision], N[(N[(x$46$im / y$46$re), $MachinePrecision] - N[(y$46$im * N[(N[(x$46$re / y$46$re), $MachinePrecision] / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.im < -7.5999999999999998e128 or 8.49999999999999984e99 < y.im

   1. Initial program 24.8%

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

   2. Taylor expanded in y.re around 0 84.3%

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

      1. associate-*r/ 84.3%

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

      2. neg-mul-1 84.3%

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

   4. Simplified 84.3%

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

   if -7.5999999999999998e128 < y.im < 8.49999999999999984e99

   1. Initial program 72.6%

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

   2. Taylor expanded in y.re around inf 66.3%

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

      1. +-commutative 66.3%

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

      2. mul-1-neg 66.3%

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

      3. unsub-neg 66.3%

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

      4. associate-/l* 65.2%

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

      5. associate-/r/ 63.7%

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

   4. Simplified 63.7%

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

      1. *-un-lft-identity 63.7%

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

      2. pow2 63.7%

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

      3. times-frac 67.6%

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

   6. Applied egg-rr 67.6%

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

      1. associate-*l/ 67.5%

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

      2. *-lft-identity 67.5%

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

   8. Simplified 67.5%

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

4. Final simplification 73.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -7.6 \cdot 10^{+128} \lor \neg \left(y.im \leq 8.5 \cdot 10^{+99}\right):\\ \;\;\;\;\frac{-x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re} - y.im \cdot \frac{\frac{x.re}{y.re}}{y.re}\\ \end{array} \]

Alternative 8: 74.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -1.2 \cdot 10^{+32} \lor \neg \left(y.im \leq 8.5 \cdot 10^{-35}\right):\\ \;\;\;\;\frac{y.re}{\frac{y.im}{\frac{x.im}{y.im}}} - \frac{x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re} - y.im \cdot \frac{\frac{x.re}{y.re}}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.im -1.2e+32) (not (<= y.im 8.5e-35)))
   (- (/ y.re (/ y.im (/ x.im y.im))) (/ x.re y.im))
   (- (/ x.im y.re) (* y.im (/ (/ x.re y.re) y.re)))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_im <= -1.2e+32) || !(y_46_im <= 8.5e-35)) {
		tmp = (y_46_re / (y_46_im / (x_46_im / y_46_im))) - (x_46_re / y_46_im);
	} else {
		tmp = (x_46_im / y_46_re) - (y_46_im * ((x_46_re / y_46_re) / y_46_re));
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if ((y_46im <= (-1.2d+32)) .or. (.not. (y_46im <= 8.5d-35))) then
        tmp = (y_46re / (y_46im / (x_46im / y_46im))) - (x_46re / y_46im)
    else
        tmp = (x_46im / y_46re) - (y_46im * ((x_46re / y_46re) / y_46re))
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_im <= -1.2e+32) || !(y_46_im <= 8.5e-35)) {
		tmp = (y_46_re / (y_46_im / (x_46_im / y_46_im))) - (x_46_re / y_46_im);
	} else {
		tmp = (x_46_im / y_46_re) - (y_46_im * ((x_46_re / y_46_re) / y_46_re));
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if (y_46_im <= -1.2e+32) or not (y_46_im <= 8.5e-35):
		tmp = (y_46_re / (y_46_im / (x_46_im / y_46_im))) - (x_46_re / y_46_im)
	else:
		tmp = (x_46_im / y_46_re) - (y_46_im * ((x_46_re / y_46_re) / y_46_re))
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if ((y_46_im <= -1.2e+32) || !(y_46_im <= 8.5e-35))
		tmp = Float64(Float64(y_46_re / Float64(y_46_im / Float64(x_46_im / y_46_im))) - Float64(x_46_re / y_46_im));
	else
		tmp = Float64(Float64(x_46_im / y_46_re) - Float64(y_46_im * Float64(Float64(x_46_re / y_46_re) / y_46_re)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if ((y_46_im <= -1.2e+32) || ~((y_46_im <= 8.5e-35)))
		tmp = (y_46_re / (y_46_im / (x_46_im / y_46_im))) - (x_46_re / y_46_im);
	else
		tmp = (x_46_im / y_46_re) - (y_46_im * ((x_46_re / y_46_re) / y_46_re));
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[Or[LessEqual[y$46$im, -1.2e+32], N[Not[LessEqual[y$46$im, 8.5e-35]], $MachinePrecision]], N[(N[(y$46$re / N[(y$46$im / N[(x$46$im / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x$46$re / y$46$im), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$im / y$46$re), $MachinePrecision] - N[(y$46$im * N[(N[(x$46$re / y$46$re), $MachinePrecision] / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.im < -1.19999999999999996e32 or 8.5000000000000001e-35 < y.im

   1. Initial program 41.6%

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

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

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

      1. +-commutative 70.9%

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

      2. mul-1-neg 70.9%

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

      3. unsub-neg 70.9%

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

      4. *-commutative 70.9%

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

      5. associate-/l* 72.8%

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

   4. Simplified 72.8%

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

      1. pow2 72.8%

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

      2. add-sqr-sqrt 36.9%

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

      3. times-frac 39.0%

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

   6. Applied egg-rr 39.0%

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

      1. unpow2 39.0%

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

   8. Simplified 39.0%

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

      1. unpow2 39.0%

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

      2. clear-num 39.0%

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

      3. frac-times 39.0%

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

      4. *-un-lft-identity 39.0%

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

   10. Applied egg-rr 39.0%

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

       1. associate-*l/ 39.0%

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

       2. rem-square-sqrt 76.2%

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

   12. Simplified 76.2%

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

   if -1.19999999999999996e32 < y.im < 8.5000000000000001e-35

   1. Initial program 74.4%

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

   2. Taylor expanded in y.re around inf 77.7%

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

      1. +-commutative 77.7%

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

      2. mul-1-neg 77.7%

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

      3. unsub-neg 77.7%

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

      4. associate-/l* 75.5%

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

      5. associate-/r/ 74.0%

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

   4. Simplified 74.0%

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

      1. *-un-lft-identity 74.0%

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

      2. pow2 74.0%

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

      3. times-frac 77.2%

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

   6. Applied egg-rr 77.2%

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

      1. associate-*l/ 77.2%

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

      2. *-lft-identity 77.2%

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

   8. Simplified 77.2%

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

4. Final simplification 76.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -1.2 \cdot 10^{+32} \lor \neg \left(y.im \leq 8.5 \cdot 10^{-35}\right):\\ \;\;\;\;\frac{y.re}{\frac{y.im}{\frac{x.im}{y.im}}} - \frac{x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re} - y.im \cdot \frac{\frac{x.re}{y.re}}{y.re}\\ \end{array} \]

Alternative 9: 77.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -820000000 \lor \neg \left(y.re \leq 96000\right):\\ \;\;\;\;\frac{x.im}{y.re} - y.im \cdot \frac{\frac{x.re}{y.re}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\left(x.re - \frac{y.re \cdot x.im}{y.im}\right) \cdot \frac{-1}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.re -820000000.0) (not (<= y.re 96000.0)))
   (- (/ x.im y.re) (* y.im (/ (/ x.re y.re) y.re)))
   (* (- x.re (/ (* y.re x.im) y.im)) (/ -1.0 y.im))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_re <= -820000000.0) || !(y_46_re <= 96000.0)) {
		tmp = (x_46_im / y_46_re) - (y_46_im * ((x_46_re / y_46_re) / y_46_re));
	} else {
		tmp = (x_46_re - ((y_46_re * x_46_im) / y_46_im)) * (-1.0 / y_46_im);
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if ((y_46re <= (-820000000.0d0)) .or. (.not. (y_46re <= 96000.0d0))) then
        tmp = (x_46im / y_46re) - (y_46im * ((x_46re / y_46re) / y_46re))
    else
        tmp = (x_46re - ((y_46re * x_46im) / y_46im)) * ((-1.0d0) / y_46im)
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_re <= -820000000.0) || !(y_46_re <= 96000.0)) {
		tmp = (x_46_im / y_46_re) - (y_46_im * ((x_46_re / y_46_re) / y_46_re));
	} else {
		tmp = (x_46_re - ((y_46_re * x_46_im) / y_46_im)) * (-1.0 / y_46_im);
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if (y_46_re <= -820000000.0) or not (y_46_re <= 96000.0):
		tmp = (x_46_im / y_46_re) - (y_46_im * ((x_46_re / y_46_re) / y_46_re))
	else:
		tmp = (x_46_re - ((y_46_re * x_46_im) / y_46_im)) * (-1.0 / y_46_im)
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if ((y_46_re <= -820000000.0) || !(y_46_re <= 96000.0))
		tmp = Float64(Float64(x_46_im / y_46_re) - Float64(y_46_im * Float64(Float64(x_46_re / y_46_re) / y_46_re)));
	else
		tmp = Float64(Float64(x_46_re - Float64(Float64(y_46_re * x_46_im) / y_46_im)) * Float64(-1.0 / y_46_im));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if ((y_46_re <= -820000000.0) || ~((y_46_re <= 96000.0)))
		tmp = (x_46_im / y_46_re) - (y_46_im * ((x_46_re / y_46_re) / y_46_re));
	else
		tmp = (x_46_re - ((y_46_re * x_46_im) / y_46_im)) * (-1.0 / 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, -820000000.0], N[Not[LessEqual[y$46$re, 96000.0]], $MachinePrecision]], N[(N[(x$46$im / y$46$re), $MachinePrecision] - N[(y$46$im * N[(N[(x$46$re / y$46$re), $MachinePrecision] / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$re - N[(N[(y$46$re * x$46$im), $MachinePrecision] / y$46$im), $MachinePrecision]), $MachinePrecision] * N[(-1.0 / y$46$im), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.re < -8.2e8 or 96000 < y.re

   1. Initial program 43.0%

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

   2. Taylor expanded in y.re around inf 68.8%

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

      1. +-commutative 68.8%

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

      2. mul-1-neg 68.8%

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

      3. unsub-neg 68.8%

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

      4. associate-/l* 68.4%

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

      5. associate-/r/ 70.1%

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

   4. Simplified 70.1%

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

      1. *-un-lft-identity 70.1%

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

      2. pow2 70.1%

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

      3. times-frac 73.3%

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

   6. Applied egg-rr 73.3%

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

      1. associate-*l/ 73.3%

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

      2. *-lft-identity 73.3%

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

   8. Simplified 73.3%

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

   if -8.2e8 < y.re < 96000

   1. Initial program 69.8%

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

      1. *-un-lft-identity 69.8%

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

      2. add-sqr-sqrt 69.8%

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

      3. times-frac 69.7%

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

      4. hypot-def 69.7%

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

      5. hypot-def 82.4%

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

   3. Applied egg-rr 82.4%

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

   4. Taylor expanded in y.im around -inf 43.8%

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

      1. associate-*r/ 43.8%

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

      2. neg-mul-1 43.8%

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

      3. distribute-rgt-neg-in 43.8%

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

   6. Simplified 43.8%

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

   7. Taylor expanded in y.im around -inf 79.9%

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

4. Final simplification 76.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -820000000 \lor \neg \left(y.re \leq 96000\right):\\ \;\;\;\;\frac{x.im}{y.re} - y.im \cdot \frac{\frac{x.re}{y.re}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\left(x.re - \frac{y.re \cdot x.im}{y.im}\right) \cdot \frac{-1}{y.im}\\ \end{array} \]

Alternative 10: 61.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -7.6 \cdot 10^{+128} \lor \neg \left(y.im \leq -900 \lor \neg \left(y.im \leq -5.5 \cdot 10^{-56}\right) \land y.im \leq 1.75 \cdot 10^{+100}\right):\\ \;\;\;\;\frac{-x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.im -7.6e+128)
         (not
          (or (<= y.im -900.0)
              (and (not (<= y.im -5.5e-56)) (<= y.im 1.75e+100)))))
   (/ (- 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 <= -7.6e+128) || !((y_46_im <= -900.0) || (!(y_46_im <= -5.5e-56) && (y_46_im <= 1.75e+100)))) {
		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 <= (-7.6d+128)) .or. (.not. (y_46im <= (-900.0d0)) .or. (.not. (y_46im <= (-5.5d-56))) .and. (y_46im <= 1.75d+100))) 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 <= -7.6e+128) || !((y_46_im <= -900.0) || (!(y_46_im <= -5.5e-56) && (y_46_im <= 1.75e+100)))) {
		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 <= -7.6e+128) or not ((y_46_im <= -900.0) or (not (y_46_im <= -5.5e-56) and (y_46_im <= 1.75e+100))):
		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 <= -7.6e+128) || !((y_46_im <= -900.0) || (!(y_46_im <= -5.5e-56) && (y_46_im <= 1.75e+100))))
		tmp = Float64(Float64(-x_46_re) / y_46_im);
	else
		tmp = Float64(x_46_im / y_46_re);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if ((y_46_im <= -7.6e+128) || ~(((y_46_im <= -900.0) || (~((y_46_im <= -5.5e-56)) && (y_46_im <= 1.75e+100)))))
		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, -7.6e+128], N[Not[Or[LessEqual[y$46$im, -900.0], And[N[Not[LessEqual[y$46$im, -5.5e-56]], $MachinePrecision], LessEqual[y$46$im, 1.75e+100]]]], $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 < -7.5999999999999998e128 or -900 < y.im < -5.4999999999999999e-56 or 1.74999999999999988e100 < y.im

   1. Initial program 31.7%

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

   2. Taylor expanded in y.re around 0 81.9%

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

      1. associate-*r/ 81.9%

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

      2. neg-mul-1 81.9%

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

   4. Simplified 81.9%

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

   if -7.5999999999999998e128 < y.im < -900 or -5.4999999999999999e-56 < y.im < 1.74999999999999988e100

   1. Initial program 71.6%

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

   2. Taylor expanded in y.re around inf 58.5%

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

4. Final simplification 67.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -7.6 \cdot 10^{+128} \lor \neg \left(y.im \leq -900 \lor \neg \left(y.im \leq -5.5 \cdot 10^{-56}\right) \land y.im \leq 1.75 \cdot 10^{+100}\right):\\ \;\;\;\;\frac{-x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \end{array} \]

Alternative 11: 44.6% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -2.4 \cdot 10^{+150}:\\ \;\;\;\;\frac{x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.im -2.4e+150) (/ 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 <= -2.4e+150) {
		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 <= (-2.4d+150)) 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 <= -2.4e+150) {
		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 <= -2.4e+150:
		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 <= -2.4e+150)
		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 <= -2.4e+150)
		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[LessEqual[y$46$im, -2.4e+150], 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 < -2.40000000000000003e150

   1. Initial program 21.8%

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

      1. *-un-lft-identity 21.8%

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

      2. add-sqr-sqrt 21.8%

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

      3. times-frac 21.8%

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

      4. hypot-def 21.8%

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

      5. hypot-def 46.6%

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

   3. Applied egg-rr 46.6%

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

   4. Taylor expanded in y.im around -inf 93.0%

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

      1. associate-*r/ 93.0%

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

      2. neg-mul-1 93.0%

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

      3. distribute-rgt-neg-in 93.0%

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

   6. Simplified 93.0%

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

   7. Taylor expanded in y.re around 0 22.8%

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

   if -2.40000000000000003e150 < y.im

   1. Initial program 64.0%

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

   2. 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 43.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -2.4 \cdot 10^{+150}:\\ \;\;\;\;\frac{x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \end{array} \]

Alternative 12: 9.6% accurate, 5.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 57.1%

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

   1. *-un-lft-identity 57.1%

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

   2. add-sqr-sqrt 57.1%

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

   3. times-frac 57.1%

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

   4. hypot-def 57.1%

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

   5. hypot-def 72.8%

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

3. Applied egg-rr 72.8%

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

4. Taylor expanded in y.im around -inf 32.0%

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

   1. associate-*r/ 32.0%

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

   2. neg-mul-1 32.0%

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

   3. distribute-rgt-neg-in 32.0%

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

6. Simplified 32.0%

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

7. Taylor expanded in y.re around -inf 7.9%

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

8. Final simplification 7.9%

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

Alternative 13: 42.3% accurate, 5.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 57.1%

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

2. Taylor expanded in y.re around inf 41.1%

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

3. Final simplification 41.1%

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

Reproduce

herbie shell --seed 2023333 
(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))))