_divideComplex, real part

Percentage Accurate: 61.4% → 85.6%

Time: 11.7s

Alternatives: 11

Speedup: 1.3×

Specification

Math

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

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (/ (+ (* x.re y.re) (* x.im 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_re * y_46_re) + (x_46_im * 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_46re * y_46re) + (x_46im * 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_re * y_46_re) + (x_46_im * 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_re * y_46_re) + (x_46_im * 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_re * y_46_re) + Float64(x_46_im * 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_re * y_46_re) + (x_46_im * 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$46$re * y$46$re), $MachinePrecision] + N[(x$46$im * y$46$im), $MachinePrecision]), $MachinePrecision] / N[(N[(y$46$re * y$46$re), $MachinePrecision] + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 61.4% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (/ (+ (* x.re y.re) (* x.im 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_re * y_46_re) + (x_46_im * 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_46re * y_46re) + (x_46im * 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_re * y_46_re) + (x_46_im * 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_re * y_46_re) + (x_46_im * 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_re * y_46_re) + Float64(x_46_im * 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_re * y_46_re) + (x_46_im * 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$46$re * y$46$re), $MachinePrecision] + N[(x$46$im * 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: 85.6% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{if}\;t_0 \leq -\infty:\\ \;\;\;\;\frac{y.im + \frac{y.im}{x.im} \cdot \frac{x.re}{\frac{y.im}{y.re}}}{y.im \cdot \frac{y.im}{x.im}}\\ \mathbf{elif}\;t_0 \leq 10^{+294}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re \cdot \frac{x.re}{y.im}}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0
         (/ (+ (* x.re y.re) (* x.im y.im)) (+ (* y.re y.re) (* y.im y.im)))))
   (if (<= t_0 (- INFINITY))
     (/
      (+ y.im (* (/ y.im x.im) (/ x.re (/ y.im y.re))))
      (* y.im (/ y.im x.im)))
     (if (<= t_0 1e+294)
       (*
        (/ 1.0 (hypot y.re y.im))
        (/ (fma x.re y.re (* x.im y.im)) (hypot y.re y.im)))
       (+ (/ x.im y.im) (/ (* y.re (/ x.re y.im)) y.im))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = ((x_46_re * y_46_re) + (x_46_im * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = (y_46_im + ((y_46_im / x_46_im) * (x_46_re / (y_46_im / y_46_re)))) / (y_46_im * (y_46_im / x_46_im));
	} else if (t_0 <= 1e+294) {
		tmp = (1.0 / hypot(y_46_re, y_46_im)) * (fma(x_46_re, y_46_re, (x_46_im * y_46_im)) / hypot(y_46_re, y_46_im));
	} else {
		tmp = (x_46_im / y_46_im) + ((y_46_re * (x_46_re / y_46_im)) / y_46_im);
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(Float64(Float64(x_46_re * y_46_re) + Float64(x_46_im * y_46_im)) / Float64(Float64(y_46_re * y_46_re) + Float64(y_46_im * y_46_im)))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(Float64(y_46_im + Float64(Float64(y_46_im / x_46_im) * Float64(x_46_re / Float64(y_46_im / y_46_re)))) / Float64(y_46_im * Float64(y_46_im / x_46_im)));
	elseif (t_0 <= 1e+294)
		tmp = Float64(Float64(1.0 / hypot(y_46_re, y_46_im)) * Float64(fma(x_46_re, y_46_re, Float64(x_46_im * y_46_im)) / hypot(y_46_re, y_46_im)));
	else
		tmp = Float64(Float64(x_46_im / y_46_im) + Float64(Float64(y_46_re * Float64(x_46_re / y_46_im)) / 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[(N[(x$46$re * y$46$re), $MachinePrecision] + N[(x$46$im * y$46$im), $MachinePrecision]), $MachinePrecision] / N[(N[(y$46$re * y$46$re), $MachinePrecision] + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[(y$46$im + N[(N[(y$46$im / x$46$im), $MachinePrecision] * N[(x$46$re / N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(y$46$im * N[(y$46$im / x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1e+294], N[(N[(1.0 / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision] * N[(N[(x$46$re * y$46$re + N[(x$46$im * y$46$im), $MachinePrecision]), $MachinePrecision] / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$im / y$46$im), $MachinePrecision] + N[(N[(y$46$re * N[(x$46$re / y$46$im), $MachinePrecision]), $MachinePrecision] / y$46$im), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 49.5%

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

   2. Taylor expanded in y.re around 0 67.2%

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

      1. associate-/l* 73.7%

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

      2. associate-/r/ 73.7%

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

   4. Simplified 73.7%

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

      1. *-un-lft-identity 73.7%

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

      2. unpow2 73.7%

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

      3. times-frac 80.0%

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

   6. Applied egg-rr 80.0%

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

      1. associate-*l/ 80.0%

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

      2. *-un-lft-identity 80.0%

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

      3. associate-*l/ 80.0%

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

   8. Applied egg-rr 80.0%

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

      1. clear-num 79.8%

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

      2. frac-add 73.8%

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

      3. *-un-lft-identity 73.8%

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

      4. associate-*l/ 81.0%

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

      5. associate-/l* 87.1%

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

   10. Applied egg-rr 87.1%

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

   if -inf.0 < (/.f64 (+.f64 (*.f64 x.re y.re) (*.f64 x.im y.im)) (+.f64 (*.f64 y.re y.re) (*.f64 y.im y.im))) < 1.00000000000000007e294

   1. Initial program 79.2%

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

      1. *-un-lft-identity 79.2%

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

      2. +-commutative 79.2%

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

      3. fma-udef 79.2%

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

      4. add-sqr-sqrt 79.2%

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

      5. times-frac 79.2%

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

      6. fma-udef 79.2%

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

      7. +-commutative 79.2%

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

      8. hypot-def 79.2%

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

      9. fma-def 79.2%

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

      10. fma-udef 79.2%

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

      11. +-commutative 79.2%

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

      12. hypot-def 99.0%

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

   3. Applied egg-rr 99.0%

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

   if 1.00000000000000007e294 < (/.f64 (+.f64 (*.f64 x.re y.re) (*.f64 x.im y.im)) (+.f64 (*.f64 y.re y.re) (*.f64 y.im y.im)))

   1. Initial program 9.5%

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

   2. Taylor expanded in y.re around 0 40.8%

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

      1. associate-/l* 45.5%

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

      2. associate-/r/ 45.5%

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

   4. Simplified 45.5%

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

      1. *-un-lft-identity 45.5%

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

      2. unpow2 45.5%

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

      3. times-frac 54.6%

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

   6. Applied egg-rr 54.6%

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

      1. associate-*l/ 54.5%

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

      2. *-un-lft-identity 54.5%

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

      3. associate-*l/ 58.0%

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

   8. Applied egg-rr 58.0%

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

4. Final simplification 91.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \leq -\infty:\\ \;\;\;\;\frac{y.im + \frac{y.im}{x.im} \cdot \frac{x.re}{\frac{y.im}{y.re}}}{y.im \cdot \frac{y.im}{x.im}}\\ \mathbf{elif}\;\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \leq 10^{+294}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re \cdot \frac{x.re}{y.im}}{y.im}\\ \end{array} \]

Alternative 2: 80.5% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -1.16 \cdot 10^{+63}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{x.re \cdot \frac{y.re}{y.im}}{y.im}\\ \mathbf{elif}\;y.im \leq -1.4 \cdot 10^{-83}:\\ \;\;\;\;\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.im \leq 4.2 \cdot 10^{-96}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{x.im}{\frac{{y.re}^{2}}{y.im}}\\ \mathbf{elif}\;y.im \leq 2.25 \cdot 10^{+63}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(x.im + y.re \cdot \frac{x.re}{y.im}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.im -1.16e+63)
   (+ (/ x.im y.im) (/ (* x.re (/ y.re y.im)) y.im))
   (if (<= y.im -1.4e-83)
     (/ (+ (* x.re y.re) (* x.im y.im)) (+ (* y.re y.re) (* y.im y.im)))
     (if (<= y.im 4.2e-96)
       (+ (/ x.re y.re) (/ x.im (/ (pow y.re 2.0) y.im)))
       (if (<= y.im 2.25e+63)
         (/ (fma x.re y.re (* x.im y.im)) (fma y.im y.im (* y.re y.re)))
         (* (/ 1.0 (hypot y.re y.im)) (+ x.im (* y.re (/ x.re y.im)))))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_im <= -1.16e+63) {
		tmp = (x_46_im / y_46_im) + ((x_46_re * (y_46_re / y_46_im)) / y_46_im);
	} else if (y_46_im <= -1.4e-83) {
		tmp = ((x_46_re * y_46_re) + (x_46_im * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	} else if (y_46_im <= 4.2e-96) {
		tmp = (x_46_re / y_46_re) + (x_46_im / (pow(y_46_re, 2.0) / y_46_im));
	} else if (y_46_im <= 2.25e+63) {
		tmp = fma(x_46_re, y_46_re, (x_46_im * y_46_im)) / fma(y_46_im, y_46_im, (y_46_re * y_46_re));
	} else {
		tmp = (1.0 / hypot(y_46_re, y_46_im)) * (x_46_im + (y_46_re * (x_46_re / y_46_im)));
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (y_46_im <= -1.16e+63)
		tmp = Float64(Float64(x_46_im / y_46_im) + Float64(Float64(x_46_re * Float64(y_46_re / y_46_im)) / y_46_im));
	elseif (y_46_im <= -1.4e-83)
		tmp = Float64(Float64(Float64(x_46_re * y_46_re) + Float64(x_46_im * y_46_im)) / Float64(Float64(y_46_re * y_46_re) + Float64(y_46_im * y_46_im)));
	elseif (y_46_im <= 4.2e-96)
		tmp = Float64(Float64(x_46_re / y_46_re) + Float64(x_46_im / Float64((y_46_re ^ 2.0) / y_46_im)));
	elseif (y_46_im <= 2.25e+63)
		tmp = Float64(fma(x_46_re, y_46_re, Float64(x_46_im * y_46_im)) / fma(y_46_im, y_46_im, Float64(y_46_re * y_46_re)));
	else
		tmp = Float64(Float64(1.0 / hypot(y_46_re, y_46_im)) * Float64(x_46_im + Float64(y_46_re * Float64(x_46_re / y_46_im))));
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[y$46$im, -1.16e+63], N[(N[(x$46$im / y$46$im), $MachinePrecision] + N[(N[(x$46$re * N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision] / y$46$im), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, -1.4e-83], N[(N[(N[(x$46$re * y$46$re), $MachinePrecision] + N[(x$46$im * y$46$im), $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, 4.2e-96], N[(N[(x$46$re / y$46$re), $MachinePrecision] + N[(x$46$im / N[(N[Power[y$46$re, 2.0], $MachinePrecision] / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, 2.25e+63], N[(N[(x$46$re * y$46$re + N[(x$46$im * y$46$im), $MachinePrecision]), $MachinePrecision] / N[(y$46$im * y$46$im + N[(y$46$re * y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision] * N[(x$46$im + N[(y$46$re * N[(x$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if y.im < -1.15999999999999994e63

   1. Initial program 37.8%

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

   2. Taylor expanded in y.re around 0 81.8%

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

      1. associate-/l* 80.6%

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

      2. associate-/r/ 83.7%

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

   4. Simplified 83.7%

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

      1. *-un-lft-identity 83.7%

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

      2. unpow2 83.7%

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

      3. times-frac 89.8%

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

   6. Applied egg-rr 89.8%

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

      1. associate-*l/ 89.8%

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

      2. *-un-lft-identity 89.8%

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

      3. associate-*l/ 92.1%

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

   8. Applied egg-rr 92.1%

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

   9. Taylor expanded in x.re around 0 87.3%

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

       1. *-commutative 87.3%

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

       2. associate-*l/ 92.1%

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

       3. *-commutative 92.1%

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

   11. Simplified 92.1%

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

   if -1.15999999999999994e63 < y.im < -1.4e-83

   1. Initial program 83.7%

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

   if -1.4e-83 < y.im < 4.20000000000000002e-96

   1. Initial program 78.8%

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

   2. Taylor expanded in y.re around inf 89.9%

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

      1. associate-/l* 90.1%

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

   4. Simplified 90.1%

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

   if 4.20000000000000002e-96 < y.im < 2.25000000000000008e63

   1. Initial program 76.6%

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

      1. fma-def 76.6%

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

      2. +-commutative 76.6%

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

      3. fma-def 76.6%

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

   3. Simplified 76.6%

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

   if 2.25000000000000008e63 < y.im

   1. Initial program 55.5%

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

      1. *-un-lft-identity 55.5%

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

      2. +-commutative 55.5%

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

      3. fma-udef 55.5%

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

      4. add-sqr-sqrt 55.5%

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

      5. times-frac 55.4%

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

      6. fma-udef 55.4%

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

      7. +-commutative 55.4%

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

      8. hypot-def 55.4%

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

      9. fma-def 55.4%

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

      10. fma-udef 55.4%

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

      11. +-commutative 55.4%

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

      12. hypot-def 71.6%

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

   3. Applied egg-rr 71.6%

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

   4. Taylor expanded in y.re around 0 80.0%

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

      1. associate-/l* 82.0%

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

      2. associate-/r/ 84.1%

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

   6. Simplified 84.1%

      \[\leadsto \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \color{blue}{\left(x.im + \frac{x.re}{y.im} \cdot y.re\right)} \]
3. Recombined 5 regimes into one program.

4. Final simplification 86.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -1.16 \cdot 10^{+63}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{x.re \cdot \frac{y.re}{y.im}}{y.im}\\ \mathbf{elif}\;y.im \leq -1.4 \cdot 10^{-83}:\\ \;\;\;\;\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.im \leq 4.2 \cdot 10^{-96}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{x.im}{\frac{{y.re}^{2}}{y.im}}\\ \mathbf{elif}\;y.im \leq 2.25 \cdot 10^{+63}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(x.im + y.re \cdot \frac{x.re}{y.im}\right)\\ \end{array} \]

Alternative 3: 80.5% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{if}\;y.im \leq -6.6 \cdot 10^{+64}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{x.re \cdot \frac{y.re}{y.im}}{y.im}\\ \mathbf{elif}\;y.im \leq -4.2 \cdot 10^{-83}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq 3.55 \cdot 10^{-97}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{x.im}{\frac{{y.re}^{2}}{y.im}}\\ \mathbf{elif}\;y.im \leq 5.8 \cdot 10^{+65}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(x.im + y.re \cdot \frac{x.re}{y.im}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0
         (/ (+ (* x.re y.re) (* x.im y.im)) (+ (* y.re y.re) (* y.im y.im)))))
   (if (<= y.im -6.6e+64)
     (+ (/ x.im y.im) (/ (* x.re (/ y.re y.im)) y.im))
     (if (<= y.im -4.2e-83)
       t_0
       (if (<= y.im 3.55e-97)
         (+ (/ x.re y.re) (/ x.im (/ (pow y.re 2.0) y.im)))
         (if (<= y.im 5.8e+65)
           t_0
           (* (/ 1.0 (hypot y.re y.im)) (+ x.im (* y.re (/ x.re y.im))))))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = ((x_46_re * y_46_re) + (x_46_im * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	double tmp;
	if (y_46_im <= -6.6e+64) {
		tmp = (x_46_im / y_46_im) + ((x_46_re * (y_46_re / y_46_im)) / y_46_im);
	} else if (y_46_im <= -4.2e-83) {
		tmp = t_0;
	} else if (y_46_im <= 3.55e-97) {
		tmp = (x_46_re / y_46_re) + (x_46_im / (pow(y_46_re, 2.0) / y_46_im));
	} else if (y_46_im <= 5.8e+65) {
		tmp = t_0;
	} else {
		tmp = (1.0 / hypot(y_46_re, y_46_im)) * (x_46_im + (y_46_re * (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 = ((x_46_re * y_46_re) + (x_46_im * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	double tmp;
	if (y_46_im <= -6.6e+64) {
		tmp = (x_46_im / y_46_im) + ((x_46_re * (y_46_re / y_46_im)) / y_46_im);
	} else if (y_46_im <= -4.2e-83) {
		tmp = t_0;
	} else if (y_46_im <= 3.55e-97) {
		tmp = (x_46_re / y_46_re) + (x_46_im / (Math.pow(y_46_re, 2.0) / y_46_im));
	} else if (y_46_im <= 5.8e+65) {
		tmp = t_0;
	} else {
		tmp = (1.0 / Math.hypot(y_46_re, y_46_im)) * (x_46_im + (y_46_re * (x_46_re / y_46_im)));
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = ((x_46_re * y_46_re) + (x_46_im * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im))
	tmp = 0
	if y_46_im <= -6.6e+64:
		tmp = (x_46_im / y_46_im) + ((x_46_re * (y_46_re / y_46_im)) / y_46_im)
	elif y_46_im <= -4.2e-83:
		tmp = t_0
	elif y_46_im <= 3.55e-97:
		tmp = (x_46_re / y_46_re) + (x_46_im / (math.pow(y_46_re, 2.0) / y_46_im))
	elif y_46_im <= 5.8e+65:
		tmp = t_0
	else:
		tmp = (1.0 / math.hypot(y_46_re, y_46_im)) * (x_46_im + (y_46_re * (x_46_re / y_46_im)))
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(Float64(Float64(x_46_re * y_46_re) + Float64(x_46_im * y_46_im)) / Float64(Float64(y_46_re * y_46_re) + Float64(y_46_im * y_46_im)))
	tmp = 0.0
	if (y_46_im <= -6.6e+64)
		tmp = Float64(Float64(x_46_im / y_46_im) + Float64(Float64(x_46_re * Float64(y_46_re / y_46_im)) / y_46_im));
	elseif (y_46_im <= -4.2e-83)
		tmp = t_0;
	elseif (y_46_im <= 3.55e-97)
		tmp = Float64(Float64(x_46_re / y_46_re) + Float64(x_46_im / Float64((y_46_re ^ 2.0) / y_46_im)));
	elseif (y_46_im <= 5.8e+65)
		tmp = t_0;
	else
		tmp = Float64(Float64(1.0 / hypot(y_46_re, y_46_im)) * Float64(x_46_im + Float64(y_46_re * 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 = ((x_46_re * y_46_re) + (x_46_im * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	tmp = 0.0;
	if (y_46_im <= -6.6e+64)
		tmp = (x_46_im / y_46_im) + ((x_46_re * (y_46_re / y_46_im)) / y_46_im);
	elseif (y_46_im <= -4.2e-83)
		tmp = t_0;
	elseif (y_46_im <= 3.55e-97)
		tmp = (x_46_re / y_46_re) + (x_46_im / ((y_46_re ^ 2.0) / y_46_im));
	elseif (y_46_im <= 5.8e+65)
		tmp = t_0;
	else
		tmp = (1.0 / hypot(y_46_re, y_46_im)) * (x_46_im + (y_46_re * (x_46_re / y_46_im)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[(N[(x$46$re * y$46$re), $MachinePrecision] + N[(x$46$im * y$46$im), $MachinePrecision]), $MachinePrecision] / N[(N[(y$46$re * y$46$re), $MachinePrecision] + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$im, -6.6e+64], N[(N[(x$46$im / y$46$im), $MachinePrecision] + N[(N[(x$46$re * N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision] / y$46$im), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, -4.2e-83], t$95$0, If[LessEqual[y$46$im, 3.55e-97], N[(N[(x$46$re / y$46$re), $MachinePrecision] + N[(x$46$im / N[(N[Power[y$46$re, 2.0], $MachinePrecision] / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, 5.8e+65], t$95$0, N[(N[(1.0 / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision] * N[(x$46$im + N[(y$46$re * N[(x$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if y.im < -6.59999999999999976e64

   1. Initial program 37.8%

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

   2. Taylor expanded in y.re around 0 81.8%

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

      1. associate-/l* 80.6%

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

      2. associate-/r/ 83.7%

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

   4. Simplified 83.7%

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

      1. *-un-lft-identity 83.7%

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

      2. unpow2 83.7%

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

      3. times-frac 89.8%

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

   6. Applied egg-rr 89.8%

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

      1. associate-*l/ 89.8%

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

      2. *-un-lft-identity 89.8%

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

      3. associate-*l/ 92.1%

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

   8. Applied egg-rr 92.1%

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

   9. Taylor expanded in x.re around 0 87.3%

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

       1. *-commutative 87.3%

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

       2. associate-*l/ 92.1%

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

       3. *-commutative 92.1%

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

   11. Simplified 92.1%

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

   if -6.59999999999999976e64 < y.im < -4.1999999999999998e-83 or 3.55000000000000008e-97 < y.im < 5.8000000000000001e65

   1. Initial program 79.7%

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

   if -4.1999999999999998e-83 < y.im < 3.55000000000000008e-97

   1. Initial program 78.8%

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

   2. Taylor expanded in y.re around inf 89.9%

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

      1. associate-/l* 90.1%

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

   4. Simplified 90.1%

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

   if 5.8000000000000001e65 < y.im

   1. Initial program 55.5%

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

      1. *-un-lft-identity 55.5%

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

      2. +-commutative 55.5%

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

      3. fma-udef 55.5%

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

      4. add-sqr-sqrt 55.5%

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

      5. times-frac 55.4%

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

      6. fma-udef 55.4%

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

      7. +-commutative 55.4%

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

      8. hypot-def 55.4%

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

      9. fma-def 55.4%

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

      10. fma-udef 55.4%

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

      11. +-commutative 55.4%

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

      12. hypot-def 71.6%

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

   3. Applied egg-rr 71.6%

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

   4. Taylor expanded in y.re around 0 80.0%

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

      1. associate-/l* 82.0%

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

      2. associate-/r/ 84.1%

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

   6. Simplified 84.1%

      \[\leadsto \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \color{blue}{\left(x.im + \frac{x.re}{y.im} \cdot y.re\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 86.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -6.6 \cdot 10^{+64}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{x.re \cdot \frac{y.re}{y.im}}{y.im}\\ \mathbf{elif}\;y.im \leq -4.2 \cdot 10^{-83}:\\ \;\;\;\;\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.im \leq 3.55 \cdot 10^{-97}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{x.im}{\frac{{y.re}^{2}}{y.im}}\\ \mathbf{elif}\;y.im \leq 5.8 \cdot 10^{+65}:\\ \;\;\;\;\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(x.im + y.re \cdot \frac{x.re}{y.im}\right)\\ \end{array} \]

Alternative 4: 80.4% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{if}\;y.im \leq -2.05 \cdot 10^{+64}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{x.re \cdot \frac{y.re}{y.im}}{y.im}\\ \mathbf{elif}\;y.im \leq -4.4 \cdot 10^{-83}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq 1.9 \cdot 10^{-97}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{x.im}{\frac{{y.re}^{2}}{y.im}}\\ \mathbf{elif}\;y.im \leq 3.3 \cdot 10^{+60}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re \cdot \frac{x.re}{y.im}}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0
         (/ (+ (* x.re y.re) (* x.im y.im)) (+ (* y.re y.re) (* y.im y.im)))))
   (if (<= y.im -2.05e+64)
     (+ (/ x.im y.im) (/ (* x.re (/ y.re y.im)) y.im))
     (if (<= y.im -4.4e-83)
       t_0
       (if (<= y.im 1.9e-97)
         (+ (/ x.re y.re) (/ x.im (/ (pow y.re 2.0) y.im)))
         (if (<= y.im 3.3e+60)
           t_0
           (+ (/ x.im y.im) (/ (* y.re (/ x.re y.im)) y.im))))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = ((x_46_re * y_46_re) + (x_46_im * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	double tmp;
	if (y_46_im <= -2.05e+64) {
		tmp = (x_46_im / y_46_im) + ((x_46_re * (y_46_re / y_46_im)) / y_46_im);
	} else if (y_46_im <= -4.4e-83) {
		tmp = t_0;
	} else if (y_46_im <= 1.9e-97) {
		tmp = (x_46_re / y_46_re) + (x_46_im / (pow(y_46_re, 2.0) / y_46_im));
	} else if (y_46_im <= 3.3e+60) {
		tmp = t_0;
	} else {
		tmp = (x_46_im / y_46_im) + ((y_46_re * (x_46_re / y_46_im)) / y_46_im);
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ((x_46re * y_46re) + (x_46im * y_46im)) / ((y_46re * y_46re) + (y_46im * y_46im))
    if (y_46im <= (-2.05d+64)) then
        tmp = (x_46im / y_46im) + ((x_46re * (y_46re / y_46im)) / y_46im)
    else if (y_46im <= (-4.4d-83)) then
        tmp = t_0
    else if (y_46im <= 1.9d-97) then
        tmp = (x_46re / y_46re) + (x_46im / ((y_46re ** 2.0d0) / y_46im))
    else if (y_46im <= 3.3d+60) then
        tmp = t_0
    else
        tmp = (x_46im / y_46im) + ((y_46re * (x_46re / y_46im)) / y_46im)
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = ((x_46_re * y_46_re) + (x_46_im * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	double tmp;
	if (y_46_im <= -2.05e+64) {
		tmp = (x_46_im / y_46_im) + ((x_46_re * (y_46_re / y_46_im)) / y_46_im);
	} else if (y_46_im <= -4.4e-83) {
		tmp = t_0;
	} else if (y_46_im <= 1.9e-97) {
		tmp = (x_46_re / y_46_re) + (x_46_im / (Math.pow(y_46_re, 2.0) / y_46_im));
	} else if (y_46_im <= 3.3e+60) {
		tmp = t_0;
	} else {
		tmp = (x_46_im / y_46_im) + ((y_46_re * (x_46_re / y_46_im)) / y_46_im);
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = ((x_46_re * y_46_re) + (x_46_im * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im))
	tmp = 0
	if y_46_im <= -2.05e+64:
		tmp = (x_46_im / y_46_im) + ((x_46_re * (y_46_re / y_46_im)) / y_46_im)
	elif y_46_im <= -4.4e-83:
		tmp = t_0
	elif y_46_im <= 1.9e-97:
		tmp = (x_46_re / y_46_re) + (x_46_im / (math.pow(y_46_re, 2.0) / y_46_im))
	elif y_46_im <= 3.3e+60:
		tmp = t_0
	else:
		tmp = (x_46_im / y_46_im) + ((y_46_re * (x_46_re / y_46_im)) / y_46_im)
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(Float64(Float64(x_46_re * y_46_re) + Float64(x_46_im * y_46_im)) / Float64(Float64(y_46_re * y_46_re) + Float64(y_46_im * y_46_im)))
	tmp = 0.0
	if (y_46_im <= -2.05e+64)
		tmp = Float64(Float64(x_46_im / y_46_im) + Float64(Float64(x_46_re * Float64(y_46_re / y_46_im)) / y_46_im));
	elseif (y_46_im <= -4.4e-83)
		tmp = t_0;
	elseif (y_46_im <= 1.9e-97)
		tmp = Float64(Float64(x_46_re / y_46_re) + Float64(x_46_im / Float64((y_46_re ^ 2.0) / y_46_im)));
	elseif (y_46_im <= 3.3e+60)
		tmp = t_0;
	else
		tmp = Float64(Float64(x_46_im / y_46_im) + Float64(Float64(y_46_re * Float64(x_46_re / y_46_im)) / y_46_im));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = ((x_46_re * y_46_re) + (x_46_im * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	tmp = 0.0;
	if (y_46_im <= -2.05e+64)
		tmp = (x_46_im / y_46_im) + ((x_46_re * (y_46_re / y_46_im)) / y_46_im);
	elseif (y_46_im <= -4.4e-83)
		tmp = t_0;
	elseif (y_46_im <= 1.9e-97)
		tmp = (x_46_re / y_46_re) + (x_46_im / ((y_46_re ^ 2.0) / y_46_im));
	elseif (y_46_im <= 3.3e+60)
		tmp = t_0;
	else
		tmp = (x_46_im / y_46_im) + ((y_46_re * (x_46_re / y_46_im)) / y_46_im);
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[(N[(x$46$re * y$46$re), $MachinePrecision] + N[(x$46$im * y$46$im), $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, -2.05e+64], N[(N[(x$46$im / y$46$im), $MachinePrecision] + N[(N[(x$46$re * N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision] / y$46$im), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, -4.4e-83], t$95$0, If[LessEqual[y$46$im, 1.9e-97], N[(N[(x$46$re / y$46$re), $MachinePrecision] + N[(x$46$im / N[(N[Power[y$46$re, 2.0], $MachinePrecision] / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, 3.3e+60], t$95$0, N[(N[(x$46$im / y$46$im), $MachinePrecision] + N[(N[(y$46$re * N[(x$46$re / y$46$im), $MachinePrecision]), $MachinePrecision] / y$46$im), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if y.im < -2.04999999999999989e64

   1. Initial program 37.8%

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

   2. Taylor expanded in y.re around 0 81.8%

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

      1. associate-/l* 80.6%

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

      2. associate-/r/ 83.7%

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

   4. Simplified 83.7%

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

      1. *-un-lft-identity 83.7%

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

      2. unpow2 83.7%

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

      3. times-frac 89.8%

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

   6. Applied egg-rr 89.8%

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

      1. associate-*l/ 89.8%

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

      2. *-un-lft-identity 89.8%

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

      3. associate-*l/ 92.1%

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

   8. Applied egg-rr 92.1%

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

   9. Taylor expanded in x.re around 0 87.3%

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

       1. *-commutative 87.3%

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

       2. associate-*l/ 92.1%

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

       3. *-commutative 92.1%

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

   11. Simplified 92.1%

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

   if -2.04999999999999989e64 < y.im < -4.40000000000000015e-83 or 1.9e-97 < y.im < 3.2999999999999998e60

   1. Initial program 79.7%

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

   if -4.40000000000000015e-83 < y.im < 1.9e-97

   1. Initial program 78.8%

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

   2. Taylor expanded in y.re around inf 89.9%

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

      1. associate-/l* 90.1%

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

   4. Simplified 90.1%

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

   if 3.2999999999999998e60 < y.im

   1. Initial program 55.5%

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

   2. Taylor expanded in y.re around 0 78.3%

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

      1. associate-/l* 78.4%

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

      2. associate-/r/ 80.4%

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

   4. Simplified 80.4%

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

      1. *-un-lft-identity 80.4%

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

      2. unpow2 80.4%

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

      3. times-frac 81.2%

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

   6. Applied egg-rr 81.2%

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

      1. associate-*l/ 81.1%

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

      2. *-un-lft-identity 81.1%

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

      3. associate-*l/ 82.4%

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

   8. Applied egg-rr 82.4%

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

4. Final simplification 86.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -2.05 \cdot 10^{+64}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{x.re \cdot \frac{y.re}{y.im}}{y.im}\\ \mathbf{elif}\;y.im \leq -4.4 \cdot 10^{-83}:\\ \;\;\;\;\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.im \leq 1.9 \cdot 10^{-97}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{x.im}{\frac{{y.re}^{2}}{y.im}}\\ \mathbf{elif}\;y.im \leq 3.3 \cdot 10^{+60}:\\ \;\;\;\;\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re \cdot \frac{x.re}{y.im}}{y.im}\\ \end{array} \]

Alternative 5: 67.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -1.05 \cdot 10^{-57} \lor \neg \left(y.im \leq 4 \cdot 10^{-88} \lor \neg \left(y.im \leq 3.2\right) \land y.im \leq 4 \cdot 10^{+39}\right):\\ \;\;\;\;\frac{x.im}{y.im} + y.re \cdot \frac{x.re}{y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.im -1.05e-57)
         (not (or (<= y.im 4e-88) (and (not (<= y.im 3.2)) (<= y.im 4e+39)))))
   (+ (/ x.im y.im) (* y.re (/ x.re (* y.im y.im))))
   (/ x.re y.re)))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_im <= -1.05e-57) || !((y_46_im <= 4e-88) || (!(y_46_im <= 3.2) && (y_46_im <= 4e+39)))) {
		tmp = (x_46_im / y_46_im) + (y_46_re * (x_46_re / (y_46_im * y_46_im)));
	} else {
		tmp = x_46_re / y_46_re;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if ((y_46im <= (-1.05d-57)) .or. (.not. (y_46im <= 4d-88) .or. (.not. (y_46im <= 3.2d0)) .and. (y_46im <= 4d+39))) then
        tmp = (x_46im / y_46im) + (y_46re * (x_46re / (y_46im * y_46im)))
    else
        tmp = x_46re / y_46re
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_im <= -1.05e-57) || !((y_46_im <= 4e-88) || (!(y_46_im <= 3.2) && (y_46_im <= 4e+39)))) {
		tmp = (x_46_im / y_46_im) + (y_46_re * (x_46_re / (y_46_im * y_46_im)));
	} else {
		tmp = x_46_re / y_46_re;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if (y_46_im <= -1.05e-57) or not ((y_46_im <= 4e-88) or (not (y_46_im <= 3.2) and (y_46_im <= 4e+39))):
		tmp = (x_46_im / y_46_im) + (y_46_re * (x_46_re / (y_46_im * y_46_im)))
	else:
		tmp = x_46_re / y_46_re
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if ((y_46_im <= -1.05e-57) || !((y_46_im <= 4e-88) || (!(y_46_im <= 3.2) && (y_46_im <= 4e+39))))
		tmp = Float64(Float64(x_46_im / y_46_im) + Float64(y_46_re * Float64(x_46_re / Float64(y_46_im * y_46_im))));
	else
		tmp = Float64(x_46_re / y_46_re);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if ((y_46_im <= -1.05e-57) || ~(((y_46_im <= 4e-88) || (~((y_46_im <= 3.2)) && (y_46_im <= 4e+39)))))
		tmp = (x_46_im / y_46_im) + (y_46_re * (x_46_re / (y_46_im * y_46_im)));
	else
		tmp = x_46_re / y_46_re;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[Or[LessEqual[y$46$im, -1.05e-57], N[Not[Or[LessEqual[y$46$im, 4e-88], And[N[Not[LessEqual[y$46$im, 3.2]], $MachinePrecision], LessEqual[y$46$im, 4e+39]]]], $MachinePrecision]], N[(N[(x$46$im / y$46$im), $MachinePrecision] + N[(y$46$re * N[(x$46$re / N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x$46$re / y$46$re), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.im < -1.05e-57 or 3.99999999999999974e-88 < y.im < 3.2000000000000002 or 3.99999999999999976e39 < y.im

   1. Initial program 57.6%

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

   2. Taylor expanded in y.re around 0 73.4%

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

      1. associate-/l* 73.5%

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

      2. associate-/r/ 74.7%

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

   4. Simplified 74.7%

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

      1. unpow2 74.7%

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

   6. Applied egg-rr 74.7%

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

   if -1.05e-57 < y.im < 3.99999999999999974e-88 or 3.2000000000000002 < y.im < 3.99999999999999976e39

   1. Initial program 77.2%

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

   2. Taylor expanded in y.re around inf 72.5%

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

4. Final simplification 73.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -1.05 \cdot 10^{-57} \lor \neg \left(y.im \leq 4 \cdot 10^{-88} \lor \neg \left(y.im \leq 3.2\right) \land y.im \leq 4 \cdot 10^{+39}\right):\\ \;\;\;\;\frac{x.im}{y.im} + y.re \cdot \frac{x.re}{y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \end{array} \]

Alternative 6: 69.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x.im}{y.im} + y.re \cdot \frac{\frac{x.re}{y.im}}{y.im}\\ \mathbf{if}\;y.im \leq -1.2 \cdot 10^{-57}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq 5.6 \cdot 10^{-89}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{elif}\;y.im \leq 4:\\ \;\;\;\;\frac{x.im}{y.im} + y.re \cdot \frac{x.re}{y.im \cdot y.im}\\ \mathbf{elif}\;y.im \leq 4.1 \cdot 10^{+40}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (+ (/ x.im y.im) (* y.re (/ (/ x.re y.im) y.im)))))
   (if (<= y.im -1.2e-57)
     t_0
     (if (<= y.im 5.6e-89)
       (/ x.re y.re)
       (if (<= y.im 4.0)
         (+ (/ x.im y.im) (* y.re (/ x.re (* y.im y.im))))
         (if (<= y.im 4.1e+40) (/ x.re y.re) t_0))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = (x_46_im / y_46_im) + (y_46_re * ((x_46_re / y_46_im) / y_46_im));
	double tmp;
	if (y_46_im <= -1.2e-57) {
		tmp = t_0;
	} else if (y_46_im <= 5.6e-89) {
		tmp = x_46_re / y_46_re;
	} else if (y_46_im <= 4.0) {
		tmp = (x_46_im / y_46_im) + (y_46_re * (x_46_re / (y_46_im * y_46_im)));
	} else if (y_46_im <= 4.1e+40) {
		tmp = x_46_re / y_46_re;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (x_46im / y_46im) + (y_46re * ((x_46re / y_46im) / y_46im))
    if (y_46im <= (-1.2d-57)) then
        tmp = t_0
    else if (y_46im <= 5.6d-89) then
        tmp = x_46re / y_46re
    else if (y_46im <= 4.0d0) then
        tmp = (x_46im / y_46im) + (y_46re * (x_46re / (y_46im * y_46im)))
    else if (y_46im <= 4.1d+40) then
        tmp = x_46re / y_46re
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = (x_46_im / y_46_im) + (y_46_re * ((x_46_re / y_46_im) / y_46_im));
	double tmp;
	if (y_46_im <= -1.2e-57) {
		tmp = t_0;
	} else if (y_46_im <= 5.6e-89) {
		tmp = x_46_re / y_46_re;
	} else if (y_46_im <= 4.0) {
		tmp = (x_46_im / y_46_im) + (y_46_re * (x_46_re / (y_46_im * y_46_im)));
	} else if (y_46_im <= 4.1e+40) {
		tmp = x_46_re / y_46_re;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = (x_46_im / y_46_im) + (y_46_re * ((x_46_re / y_46_im) / y_46_im))
	tmp = 0
	if y_46_im <= -1.2e-57:
		tmp = t_0
	elif y_46_im <= 5.6e-89:
		tmp = x_46_re / y_46_re
	elif y_46_im <= 4.0:
		tmp = (x_46_im / y_46_im) + (y_46_re * (x_46_re / (y_46_im * y_46_im)))
	elif y_46_im <= 4.1e+40:
		tmp = x_46_re / 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_im / y_46_im) + Float64(y_46_re * Float64(Float64(x_46_re / y_46_im) / y_46_im)))
	tmp = 0.0
	if (y_46_im <= -1.2e-57)
		tmp = t_0;
	elseif (y_46_im <= 5.6e-89)
		tmp = Float64(x_46_re / y_46_re);
	elseif (y_46_im <= 4.0)
		tmp = Float64(Float64(x_46_im / y_46_im) + Float64(y_46_re * Float64(x_46_re / Float64(y_46_im * y_46_im))));
	elseif (y_46_im <= 4.1e+40)
		tmp = Float64(x_46_re / y_46_re);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = (x_46_im / y_46_im) + (y_46_re * ((x_46_re / y_46_im) / y_46_im));
	tmp = 0.0;
	if (y_46_im <= -1.2e-57)
		tmp = t_0;
	elseif (y_46_im <= 5.6e-89)
		tmp = x_46_re / y_46_re;
	elseif (y_46_im <= 4.0)
		tmp = (x_46_im / y_46_im) + (y_46_re * (x_46_re / (y_46_im * y_46_im)));
	elseif (y_46_im <= 4.1e+40)
		tmp = x_46_re / y_46_re;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[(x$46$im / y$46$im), $MachinePrecision] + N[(y$46$re * N[(N[(x$46$re / y$46$im), $MachinePrecision] / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$im, -1.2e-57], t$95$0, If[LessEqual[y$46$im, 5.6e-89], N[(x$46$re / y$46$re), $MachinePrecision], If[LessEqual[y$46$im, 4.0], N[(N[(x$46$im / y$46$im), $MachinePrecision] + N[(y$46$re * N[(x$46$re / N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, 4.1e+40], N[(x$46$re / y$46$re), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 3 regimes

2. if y.im < -1.20000000000000003e-57 or 4.1000000000000002e40 < y.im

   1. Initial program 54.5%

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

   2. Taylor expanded in y.re around 0 74.0%

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

      1. associate-/l* 74.2%

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

      2. associate-/r/ 76.3%

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

   4. Simplified 76.3%

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

      1. *-un-lft-identity 76.3%

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

      2. unpow2 76.3%

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

      3. times-frac 79.2%

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

   6. Applied egg-rr 79.2%

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

      1. associate-*l/ 79.2%

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

      2. *-un-lft-identity 79.2%

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

   8. Applied egg-rr 79.2%

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

   if -1.20000000000000003e-57 < y.im < 5.5999999999999998e-89 or 4 < y.im < 4.1000000000000002e40

   1. Initial program 77.2%

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

   2. Taylor expanded in y.re around inf 72.5%

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

   if 5.5999999999999998e-89 < y.im < 4

   1. Initial program 77.6%

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

   2. Taylor expanded in y.re around 0 69.2%

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

      1. associate-/l* 69.3%

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

      2. associate-/r/ 65.0%

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

   4. Simplified 65.0%

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

      1. unpow2 65.0%

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

   6. Applied egg-rr 65.0%

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

4. Final simplification 75.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -1.2 \cdot 10^{-57}:\\ \;\;\;\;\frac{x.im}{y.im} + y.re \cdot \frac{\frac{x.re}{y.im}}{y.im}\\ \mathbf{elif}\;y.im \leq 5.6 \cdot 10^{-89}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{elif}\;y.im \leq 4:\\ \;\;\;\;\frac{x.im}{y.im} + y.re \cdot \frac{x.re}{y.im \cdot y.im}\\ \mathbf{elif}\;y.im \leq 4.1 \cdot 10^{+40}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im} + y.re \cdot \frac{\frac{x.re}{y.im}}{y.im}\\ \end{array} \]

Alternative 7: 69.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x.im}{y.im} + \frac{x.re \cdot \frac{y.re}{y.im}}{y.im}\\ \mathbf{if}\;y.im \leq -1.16 \cdot 10^{-57}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq 1.15 \cdot 10^{-102}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{elif}\;y.im \leq 4:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq 4 \cdot 10^{+39}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im} + y.re \cdot \frac{\frac{x.re}{y.im}}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (+ (/ x.im y.im) (/ (* x.re (/ y.re y.im)) y.im))))
   (if (<= y.im -1.16e-57)
     t_0
     (if (<= y.im 1.15e-102)
       (/ x.re y.re)
       (if (<= y.im 4.0)
         t_0
         (if (<= y.im 4e+39)
           (/ x.re y.re)
           (+ (/ x.im y.im) (* y.re (/ (/ x.re y.im) y.im)))))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = (x_46_im / y_46_im) + ((x_46_re * (y_46_re / y_46_im)) / y_46_im);
	double tmp;
	if (y_46_im <= -1.16e-57) {
		tmp = t_0;
	} else if (y_46_im <= 1.15e-102) {
		tmp = x_46_re / y_46_re;
	} else if (y_46_im <= 4.0) {
		tmp = t_0;
	} else if (y_46_im <= 4e+39) {
		tmp = x_46_re / y_46_re;
	} else {
		tmp = (x_46_im / y_46_im) + (y_46_re * ((x_46_re / y_46_im) / y_46_im));
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (x_46im / y_46im) + ((x_46re * (y_46re / y_46im)) / y_46im)
    if (y_46im <= (-1.16d-57)) then
        tmp = t_0
    else if (y_46im <= 1.15d-102) then
        tmp = x_46re / y_46re
    else if (y_46im <= 4.0d0) then
        tmp = t_0
    else if (y_46im <= 4d+39) then
        tmp = x_46re / y_46re
    else
        tmp = (x_46im / y_46im) + (y_46re * ((x_46re / y_46im) / y_46im))
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = (x_46_im / y_46_im) + ((x_46_re * (y_46_re / y_46_im)) / y_46_im);
	double tmp;
	if (y_46_im <= -1.16e-57) {
		tmp = t_0;
	} else if (y_46_im <= 1.15e-102) {
		tmp = x_46_re / y_46_re;
	} else if (y_46_im <= 4.0) {
		tmp = t_0;
	} else if (y_46_im <= 4e+39) {
		tmp = x_46_re / y_46_re;
	} else {
		tmp = (x_46_im / y_46_im) + (y_46_re * ((x_46_re / y_46_im) / y_46_im));
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = (x_46_im / y_46_im) + ((x_46_re * (y_46_re / y_46_im)) / y_46_im)
	tmp = 0
	if y_46_im <= -1.16e-57:
		tmp = t_0
	elif y_46_im <= 1.15e-102:
		tmp = x_46_re / y_46_re
	elif y_46_im <= 4.0:
		tmp = t_0
	elif y_46_im <= 4e+39:
		tmp = x_46_re / y_46_re
	else:
		tmp = (x_46_im / y_46_im) + (y_46_re * ((x_46_re / y_46_im) / y_46_im))
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(Float64(x_46_im / y_46_im) + Float64(Float64(x_46_re * Float64(y_46_re / y_46_im)) / y_46_im))
	tmp = 0.0
	if (y_46_im <= -1.16e-57)
		tmp = t_0;
	elseif (y_46_im <= 1.15e-102)
		tmp = Float64(x_46_re / y_46_re);
	elseif (y_46_im <= 4.0)
		tmp = t_0;
	elseif (y_46_im <= 4e+39)
		tmp = Float64(x_46_re / y_46_re);
	else
		tmp = Float64(Float64(x_46_im / y_46_im) + Float64(y_46_re * Float64(Float64(x_46_re / y_46_im) / y_46_im)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = (x_46_im / y_46_im) + ((x_46_re * (y_46_re / y_46_im)) / y_46_im);
	tmp = 0.0;
	if (y_46_im <= -1.16e-57)
		tmp = t_0;
	elseif (y_46_im <= 1.15e-102)
		tmp = x_46_re / y_46_re;
	elseif (y_46_im <= 4.0)
		tmp = t_0;
	elseif (y_46_im <= 4e+39)
		tmp = x_46_re / y_46_re;
	else
		tmp = (x_46_im / y_46_im) + (y_46_re * ((x_46_re / y_46_im) / 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[(x$46$im / y$46$im), $MachinePrecision] + N[(N[(x$46$re * N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision] / y$46$im), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$im, -1.16e-57], t$95$0, If[LessEqual[y$46$im, 1.15e-102], N[(x$46$re / y$46$re), $MachinePrecision], If[LessEqual[y$46$im, 4.0], t$95$0, If[LessEqual[y$46$im, 4e+39], N[(x$46$re / y$46$re), $MachinePrecision], N[(N[(x$46$im / y$46$im), $MachinePrecision] + N[(y$46$re * N[(N[(x$46$re / y$46$im), $MachinePrecision] / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if y.im < -1.15999999999999996e-57 or 1.14999999999999993e-102 < y.im < 4

   1. Initial program 56.1%

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

   2. Taylor expanded in y.re around 0 71.7%

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

      1. associate-/l* 72.0%

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

      2. associate-/r/ 72.0%

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

   4. Simplified 72.0%

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

      1. *-un-lft-identity 72.0%

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

      2. unpow2 72.0%

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

      3. times-frac 75.4%

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

   6. Applied egg-rr 75.4%

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

      1. associate-*l/ 75.4%

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

      2. *-un-lft-identity 75.4%

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

      3. associate-*l/ 77.5%

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

   8. Applied egg-rr 77.5%

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

   9. Taylor expanded in x.re around 0 74.7%

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

       1. *-commutative 74.7%

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

       2. associate-*l/ 78.4%

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

       3. *-commutative 78.4%

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

   11. Simplified 78.4%

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

   if -1.15999999999999996e-57 < y.im < 1.14999999999999993e-102 or 4 < y.im < 3.99999999999999976e39

   1. Initial program 78.4%

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

   2. Taylor expanded in y.re around inf 74.5%

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

   if 3.99999999999999976e39 < y.im

   1. Initial program 60.3%

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

   2. Taylor expanded in y.re around 0 74.9%

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

      1. associate-/l* 75.0%

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

      2. associate-/r/ 76.8%

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

   4. Simplified 76.8%

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

      1. *-un-lft-identity 76.8%

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

      2. unpow2 76.8%

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

      3. times-frac 77.4%

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

   6. Applied egg-rr 77.4%

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

      1. associate-*l/ 77.4%

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

      2. *-un-lft-identity 77.4%

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

   8. Applied egg-rr 77.4%

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

4. Final simplification 76.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -1.16 \cdot 10^{-57}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{x.re \cdot \frac{y.re}{y.im}}{y.im}\\ \mathbf{elif}\;y.im \leq 1.15 \cdot 10^{-102}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{elif}\;y.im \leq 4:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{x.re \cdot \frac{y.re}{y.im}}{y.im}\\ \mathbf{elif}\;y.im \leq 4 \cdot 10^{+39}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im} + y.re \cdot \frac{\frac{x.re}{y.im}}{y.im}\\ \end{array} \]

Alternative 8: 69.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x.im}{y.im} + \frac{x.re \cdot \frac{y.re}{y.im}}{y.im}\\ \mathbf{if}\;y.im \leq -1.2 \cdot 10^{-57}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq 2.05 \cdot 10^{-103}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{elif}\;y.im \leq 4.2:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq 5.1 \cdot 10^{+39}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re \cdot \frac{x.re}{y.im}}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (+ (/ x.im y.im) (/ (* x.re (/ y.re y.im)) y.im))))
   (if (<= y.im -1.2e-57)
     t_0
     (if (<= y.im 2.05e-103)
       (/ x.re y.re)
       (if (<= y.im 4.2)
         t_0
         (if (<= y.im 5.1e+39)
           (/ x.re y.re)
           (+ (/ x.im y.im) (/ (* y.re (/ x.re y.im)) y.im))))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = (x_46_im / y_46_im) + ((x_46_re * (y_46_re / y_46_im)) / y_46_im);
	double tmp;
	if (y_46_im <= -1.2e-57) {
		tmp = t_0;
	} else if (y_46_im <= 2.05e-103) {
		tmp = x_46_re / y_46_re;
	} else if (y_46_im <= 4.2) {
		tmp = t_0;
	} else if (y_46_im <= 5.1e+39) {
		tmp = x_46_re / y_46_re;
	} else {
		tmp = (x_46_im / y_46_im) + ((y_46_re * (x_46_re / y_46_im)) / y_46_im);
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (x_46im / y_46im) + ((x_46re * (y_46re / y_46im)) / y_46im)
    if (y_46im <= (-1.2d-57)) then
        tmp = t_0
    else if (y_46im <= 2.05d-103) then
        tmp = x_46re / y_46re
    else if (y_46im <= 4.2d0) then
        tmp = t_0
    else if (y_46im <= 5.1d+39) then
        tmp = x_46re / y_46re
    else
        tmp = (x_46im / y_46im) + ((y_46re * (x_46re / y_46im)) / y_46im)
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = (x_46_im / y_46_im) + ((x_46_re * (y_46_re / y_46_im)) / y_46_im);
	double tmp;
	if (y_46_im <= -1.2e-57) {
		tmp = t_0;
	} else if (y_46_im <= 2.05e-103) {
		tmp = x_46_re / y_46_re;
	} else if (y_46_im <= 4.2) {
		tmp = t_0;
	} else if (y_46_im <= 5.1e+39) {
		tmp = x_46_re / y_46_re;
	} else {
		tmp = (x_46_im / y_46_im) + ((y_46_re * (x_46_re / y_46_im)) / y_46_im);
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = (x_46_im / y_46_im) + ((x_46_re * (y_46_re / y_46_im)) / y_46_im)
	tmp = 0
	if y_46_im <= -1.2e-57:
		tmp = t_0
	elif y_46_im <= 2.05e-103:
		tmp = x_46_re / y_46_re
	elif y_46_im <= 4.2:
		tmp = t_0
	elif y_46_im <= 5.1e+39:
		tmp = x_46_re / y_46_re
	else:
		tmp = (x_46_im / y_46_im) + ((y_46_re * (x_46_re / y_46_im)) / y_46_im)
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(Float64(x_46_im / y_46_im) + Float64(Float64(x_46_re * Float64(y_46_re / y_46_im)) / y_46_im))
	tmp = 0.0
	if (y_46_im <= -1.2e-57)
		tmp = t_0;
	elseif (y_46_im <= 2.05e-103)
		tmp = Float64(x_46_re / y_46_re);
	elseif (y_46_im <= 4.2)
		tmp = t_0;
	elseif (y_46_im <= 5.1e+39)
		tmp = Float64(x_46_re / y_46_re);
	else
		tmp = Float64(Float64(x_46_im / y_46_im) + Float64(Float64(y_46_re * Float64(x_46_re / y_46_im)) / y_46_im));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = (x_46_im / y_46_im) + ((x_46_re * (y_46_re / y_46_im)) / y_46_im);
	tmp = 0.0;
	if (y_46_im <= -1.2e-57)
		tmp = t_0;
	elseif (y_46_im <= 2.05e-103)
		tmp = x_46_re / y_46_re;
	elseif (y_46_im <= 4.2)
		tmp = t_0;
	elseif (y_46_im <= 5.1e+39)
		tmp = x_46_re / y_46_re;
	else
		tmp = (x_46_im / y_46_im) + ((y_46_re * (x_46_re / y_46_im)) / 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[(x$46$im / y$46$im), $MachinePrecision] + N[(N[(x$46$re * N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision] / y$46$im), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$im, -1.2e-57], t$95$0, If[LessEqual[y$46$im, 2.05e-103], N[(x$46$re / y$46$re), $MachinePrecision], If[LessEqual[y$46$im, 4.2], t$95$0, If[LessEqual[y$46$im, 5.1e+39], N[(x$46$re / y$46$re), $MachinePrecision], N[(N[(x$46$im / y$46$im), $MachinePrecision] + N[(N[(y$46$re * N[(x$46$re / y$46$im), $MachinePrecision]), $MachinePrecision] / y$46$im), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if y.im < -1.20000000000000003e-57 or 2.04999999999999998e-103 < y.im < 4.20000000000000018

   1. Initial program 56.1%

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

   2. Taylor expanded in y.re around 0 71.7%

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

      1. associate-/l* 72.0%

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

      2. associate-/r/ 72.0%

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

   4. Simplified 72.0%

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

      1. *-un-lft-identity 72.0%

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

      2. unpow2 72.0%

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

      3. times-frac 75.4%

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

   6. Applied egg-rr 75.4%

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

      1. associate-*l/ 75.4%

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

      2. *-un-lft-identity 75.4%

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

      3. associate-*l/ 77.5%

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

   8. Applied egg-rr 77.5%

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

   9. Taylor expanded in x.re around 0 74.7%

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

       1. *-commutative 74.7%

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

       2. associate-*l/ 78.4%

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

       3. *-commutative 78.4%

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

   11. Simplified 78.4%

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

   if -1.20000000000000003e-57 < y.im < 2.04999999999999998e-103 or 4.20000000000000018 < y.im < 5.0999999999999998e39

   1. Initial program 78.4%

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

   2. Taylor expanded in y.re around inf 74.5%

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

   if 5.0999999999999998e39 < y.im

   1. Initial program 60.3%

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

   2. Taylor expanded in y.re around 0 74.9%

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

      1. associate-/l* 75.0%

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

      2. associate-/r/ 76.8%

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

   4. Simplified 76.8%

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

      1. *-un-lft-identity 76.8%

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

      2. unpow2 76.8%

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

      3. times-frac 77.4%

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

   6. Applied egg-rr 77.4%

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

      1. associate-*l/ 77.4%

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

      2. *-un-lft-identity 77.4%

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

      3. associate-*l/ 78.5%

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

   8. Applied egg-rr 78.5%

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

4. Final simplification 77.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -1.2 \cdot 10^{-57}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{x.re \cdot \frac{y.re}{y.im}}{y.im}\\ \mathbf{elif}\;y.im \leq 2.05 \cdot 10^{-103}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{elif}\;y.im \leq 4.2:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{x.re \cdot \frac{y.re}{y.im}}{y.im}\\ \mathbf{elif}\;y.im \leq 5.1 \cdot 10^{+39}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re \cdot \frac{x.re}{y.im}}{y.im}\\ \end{array} \]

Alternative 9: 77.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -6.8 \cdot 10^{+65}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{x.re \cdot \frac{y.re}{y.im}}{y.im}\\ \mathbf{elif}\;y.im \leq 1.28 \cdot 10^{+61}:\\ \;\;\;\;\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re \cdot \frac{x.re}{y.im}}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.im -6.8e+65)
   (+ (/ x.im y.im) (/ (* x.re (/ y.re y.im)) y.im))
   (if (<= y.im 1.28e+61)
     (/ (+ (* x.re y.re) (* x.im y.im)) (+ (* y.re y.re) (* y.im y.im)))
     (+ (/ x.im y.im) (/ (* y.re (/ x.re y.im)) y.im)))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_im <= -6.8e+65) {
		tmp = (x_46_im / y_46_im) + ((x_46_re * (y_46_re / y_46_im)) / y_46_im);
	} else if (y_46_im <= 1.28e+61) {
		tmp = ((x_46_re * y_46_re) + (x_46_im * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	} else {
		tmp = (x_46_im / y_46_im) + ((y_46_re * (x_46_re / y_46_im)) / y_46_im);
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if (y_46im <= (-6.8d+65)) then
        tmp = (x_46im / y_46im) + ((x_46re * (y_46re / y_46im)) / y_46im)
    else if (y_46im <= 1.28d+61) then
        tmp = ((x_46re * y_46re) + (x_46im * y_46im)) / ((y_46re * y_46re) + (y_46im * y_46im))
    else
        tmp = (x_46im / y_46im) + ((y_46re * (x_46re / y_46im)) / y_46im)
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_im <= -6.8e+65) {
		tmp = (x_46_im / y_46_im) + ((x_46_re * (y_46_re / y_46_im)) / y_46_im);
	} else if (y_46_im <= 1.28e+61) {
		tmp = ((x_46_re * y_46_re) + (x_46_im * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	} else {
		tmp = (x_46_im / y_46_im) + ((y_46_re * (x_46_re / y_46_im)) / y_46_im);
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if y_46_im <= -6.8e+65:
		tmp = (x_46_im / y_46_im) + ((x_46_re * (y_46_re / y_46_im)) / y_46_im)
	elif y_46_im <= 1.28e+61:
		tmp = ((x_46_re * y_46_re) + (x_46_im * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im))
	else:
		tmp = (x_46_im / y_46_im) + ((y_46_re * (x_46_re / y_46_im)) / y_46_im)
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (y_46_im <= -6.8e+65)
		tmp = Float64(Float64(x_46_im / y_46_im) + Float64(Float64(x_46_re * Float64(y_46_re / y_46_im)) / y_46_im));
	elseif (y_46_im <= 1.28e+61)
		tmp = Float64(Float64(Float64(x_46_re * y_46_re) + Float64(x_46_im * y_46_im)) / Float64(Float64(y_46_re * y_46_re) + Float64(y_46_im * y_46_im)));
	else
		tmp = Float64(Float64(x_46_im / y_46_im) + Float64(Float64(y_46_re * Float64(x_46_re / y_46_im)) / y_46_im));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if (y_46_im <= -6.8e+65)
		tmp = (x_46_im / y_46_im) + ((x_46_re * (y_46_re / y_46_im)) / y_46_im);
	elseif (y_46_im <= 1.28e+61)
		tmp = ((x_46_re * y_46_re) + (x_46_im * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	else
		tmp = (x_46_im / y_46_im) + ((y_46_re * (x_46_re / y_46_im)) / y_46_im);
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[y$46$im, -6.8e+65], N[(N[(x$46$im / y$46$im), $MachinePrecision] + N[(N[(x$46$re * N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision] / y$46$im), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, 1.28e+61], N[(N[(N[(x$46$re * y$46$re), $MachinePrecision] + N[(x$46$im * y$46$im), $MachinePrecision]), $MachinePrecision] / N[(N[(y$46$re * y$46$re), $MachinePrecision] + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$im / y$46$im), $MachinePrecision] + N[(N[(y$46$re * N[(x$46$re / y$46$im), $MachinePrecision]), $MachinePrecision] / y$46$im), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y.im < -6.7999999999999999e65

   1. Initial program 37.8%

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

   2. Taylor expanded in y.re around 0 81.8%

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

      1. associate-/l* 80.6%

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

      2. associate-/r/ 83.7%

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

   4. Simplified 83.7%

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

      1. *-un-lft-identity 83.7%

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

      2. unpow2 83.7%

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

      3. times-frac 89.8%

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

   6. Applied egg-rr 89.8%

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

      1. associate-*l/ 89.8%

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

      2. *-un-lft-identity 89.8%

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

      3. associate-*l/ 92.1%

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

   8. Applied egg-rr 92.1%

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

   9. Taylor expanded in x.re around 0 87.3%

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

       1. *-commutative 87.3%

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

       2. associate-*l/ 92.1%

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

       3. *-commutative 92.1%

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

   11. Simplified 92.1%

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

   if -6.7999999999999999e65 < y.im < 1.27999999999999996e61

   1. Initial program 79.2%

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

   if 1.27999999999999996e61 < y.im

   1. Initial program 55.5%

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

   2. Taylor expanded in y.re around 0 78.3%

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

      1. associate-/l* 78.4%

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

      2. associate-/r/ 80.4%

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

   4. Simplified 80.4%

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

      1. *-un-lft-identity 80.4%

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

      2. unpow2 80.4%

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

      3. times-frac 81.2%

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

   6. Applied egg-rr 81.2%

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

      1. associate-*l/ 81.1%

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

      2. *-un-lft-identity 81.1%

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

      3. associate-*l/ 82.4%

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

   8. Applied egg-rr 82.4%

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

4. Final simplification 82.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -6.8 \cdot 10^{+65}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{x.re \cdot \frac{y.re}{y.im}}{y.im}\\ \mathbf{elif}\;y.im \leq 1.28 \cdot 10^{+61}:\\ \;\;\;\;\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re \cdot \frac{x.re}{y.im}}{y.im}\\ \end{array} \]

Alternative 10: 63.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -2.55 \cdot 10^{-27} \lor \neg \left(y.im \leq 1.5 \cdot 10^{-88} \lor \neg \left(y.im \leq 3.3\right) \land y.im \leq 9.5 \cdot 10^{+31}\right):\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.im -2.55e-27)
         (not
          (or (<= y.im 1.5e-88) (and (not (<= y.im 3.3)) (<= y.im 9.5e+31)))))
   (/ x.im y.im)
   (/ x.re y.re)))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_im <= -2.55e-27) || !((y_46_im <= 1.5e-88) || (!(y_46_im <= 3.3) && (y_46_im <= 9.5e+31)))) {
		tmp = x_46_im / y_46_im;
	} else {
		tmp = x_46_re / y_46_re;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if ((y_46im <= (-2.55d-27)) .or. (.not. (y_46im <= 1.5d-88) .or. (.not. (y_46im <= 3.3d0)) .and. (y_46im <= 9.5d+31))) then
        tmp = x_46im / y_46im
    else
        tmp = x_46re / y_46re
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_im <= -2.55e-27) || !((y_46_im <= 1.5e-88) || (!(y_46_im <= 3.3) && (y_46_im <= 9.5e+31)))) {
		tmp = x_46_im / y_46_im;
	} else {
		tmp = x_46_re / y_46_re;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if (y_46_im <= -2.55e-27) or not ((y_46_im <= 1.5e-88) or (not (y_46_im <= 3.3) and (y_46_im <= 9.5e+31))):
		tmp = x_46_im / y_46_im
	else:
		tmp = x_46_re / y_46_re
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if ((y_46_im <= -2.55e-27) || !((y_46_im <= 1.5e-88) || (!(y_46_im <= 3.3) && (y_46_im <= 9.5e+31))))
		tmp = Float64(x_46_im / y_46_im);
	else
		tmp = Float64(x_46_re / y_46_re);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if ((y_46_im <= -2.55e-27) || ~(((y_46_im <= 1.5e-88) || (~((y_46_im <= 3.3)) && (y_46_im <= 9.5e+31)))))
		tmp = x_46_im / y_46_im;
	else
		tmp = x_46_re / y_46_re;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[Or[LessEqual[y$46$im, -2.55e-27], N[Not[Or[LessEqual[y$46$im, 1.5e-88], And[N[Not[LessEqual[y$46$im, 3.3]], $MachinePrecision], LessEqual[y$46$im, 9.5e+31]]]], $MachinePrecision]], N[(x$46$im / y$46$im), $MachinePrecision], N[(x$46$re / y$46$re), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.im < -2.55e-27 or 1.5e-88 < y.im < 3.2999999999999998 or 9.5000000000000008e31 < y.im

   1. Initial program 57.1%

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

   2. Taylor expanded in y.re around 0 65.0%

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

   if -2.55e-27 < y.im < 1.5e-88 or 3.2999999999999998 < y.im < 9.5000000000000008e31

   1. Initial program 78.2%

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

   2. Taylor expanded in y.re around inf 72.5%

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

4. Final simplification 67.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -2.55 \cdot 10^{-27} \lor \neg \left(y.im \leq 1.5 \cdot 10^{-88} \lor \neg \left(y.im \leq 3.3\right) \land y.im \leq 9.5 \cdot 10^{+31}\right):\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \end{array} \]

Alternative 11: 43.7% accurate, 5.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 64.9%

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

2. Taylor expanded in y.re around 0 45.9%

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

3. Final simplification 45.9%

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

Reproduce

herbie shell --seed 2023326 
(FPCore (x.re x.im y.re y.im)
  :name "_divideComplex, real part"
  :precision binary64
  (/ (+ (* x.re y.re) (* x.im y.im)) (+ (* y.re y.re) (* y.im y.im))))