_divideComplex, real part

Percentage Accurate: 61.9% → 83.9%

Time: 9.1s

Alternatives: 12

Speedup: 2.1×

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: 61.9% 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: 83.9% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \leq 5 \cdot 10^{+265}:\\ \;\;\;\;\frac{1}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{\frac{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}{\mathsf{hypot}\left(y.re, y.im\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{\frac{y.im}{\frac{y.re}{x.im}}}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<=
      (/ (+ (* x.re y.re) (* x.im y.im)) (+ (* y.re y.re) (* y.im y.im)))
      5e+265)
   (/
    1.0
    (/ (hypot y.re y.im) (/ (fma x.re y.re (* x.im y.im)) (hypot y.re y.im))))
   (+ (/ x.re y.re) (/ (/ y.im (/ y.re x.im)) y.re))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((((x_46_re * y_46_re) + (x_46_im * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im))) <= 5e+265) {
		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_re / y_46_re) + ((y_46_im / (y_46_re / x_46_im)) / y_46_re);
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (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))) <= 5e+265)
		tmp = Float64(1.0 / Float64(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_re / y_46_re) + Float64(Float64(y_46_im / Float64(y_46_re / x_46_im)) / y_46_re));
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[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], 5e+265], N[(1.0 / N[(N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $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]), $MachinePrecision], N[(N[(x$46$re / y$46$re), $MachinePrecision] + N[(N[(y$46$im / N[(y$46$re / x$46$im), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 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))) < 5.0000000000000002e265

   1. Initial program 80.0%

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

      1. *-un-lft-identity 80.0%

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

      2. add-sqr-sqrt 80.0%

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

      3. times-frac 79.8%

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

      4. hypot-def 79.8%

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

      5. fma-def 79.8%

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

      6. hypot-def 96.4%

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

   3. Applied egg-rr 96.4%

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

      1. associate-*l/ 96.7%

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

      2. *-un-lft-identity 96.7%

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

      3. clear-num 96.5%

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

   5. Applied egg-rr 96.5%

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

   if 5.0000000000000002e265 < (/.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 8.3%

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

   2. Taylor expanded in y.re around inf 41.8%

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

      1. associate-/l* 46.8%

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

      2. associate-/r/ 46.8%

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

      3. unpow2 46.8%

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

   4. Simplified 46.8%

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

      1. frac-2neg 46.8%

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

      2. div-inv 46.8%

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

      3. distribute-rgt-neg-in 46.8%

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

   6. Applied egg-rr 46.8%

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

      1. associate-*r/ 46.8%

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

      2. *-commutative 46.8%

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

      3. times-frac 55.1%

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

      4. neg-mul-1 55.1%

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

      5. neg-mul-1 55.1%

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

      6. times-frac 55.1%

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

      7. metadata-eval 55.1%

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

      8. *-lft-identity 55.1%

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

      9. associate-*r/ 55.1%

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

      10. *-rgt-identity 55.1%

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

   8. Simplified 55.1%

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

      1. expm1-log1p-u 49.9%

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

      2. expm1-udef 48.4%

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

      3. associate-*l/ 48.5%

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

      4. *-commutative 48.5%

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

   10. Applied egg-rr 48.5%

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

       1. expm1-def 52.7%

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

       2. expm1-log1p 59.4%

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

       3. associate-*r/ 49.1%

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

       4. *-commutative 49.1%

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

       5. associate-/l* 59.4%

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

   12. Simplified 59.4%

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

4. Final simplification 86.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \leq 5 \cdot 10^{+265}:\\ \;\;\;\;\frac{1}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{\frac{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}{\mathsf{hypot}\left(y.re, y.im\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{\frac{y.im}{\frac{y.re}{x.im}}}{y.re}\\ \end{array} \]

Alternative 2: 84.5% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \leq 5 \cdot 10^{+265}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{\frac{y.im}{\frac{y.re}{x.im}}}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<=
      (/ (+ (* x.re y.re) (* x.im y.im)) (+ (* y.re y.re) (* y.im y.im)))
      5e+265)
   (*
    (/ (fma x.re y.re (* x.im y.im)) (hypot y.re y.im))
    (/ 1.0 (hypot y.re y.im)))
   (+ (/ x.re y.re) (/ (/ y.im (/ y.re x.im)) y.re))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((((x_46_re * y_46_re) + (x_46_im * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im))) <= 5e+265) {
		tmp = (fma(x_46_re, y_46_re, (x_46_im * y_46_im)) / hypot(y_46_re, y_46_im)) * (1.0 / hypot(y_46_re, y_46_im));
	} else {
		tmp = (x_46_re / y_46_re) + ((y_46_im / (y_46_re / x_46_im)) / y_46_re);
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (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))) <= 5e+265)
		tmp = Float64(Float64(fma(x_46_re, y_46_re, Float64(x_46_im * y_46_im)) / hypot(y_46_re, y_46_im)) * Float64(1.0 / hypot(y_46_re, y_46_im)));
	else
		tmp = Float64(Float64(x_46_re / y_46_re) + Float64(Float64(y_46_im / Float64(y_46_re / x_46_im)) / y_46_re));
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[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], 5e+265], N[(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] * N[(1.0 / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$re / y$46$re), $MachinePrecision] + N[(N[(y$46$im / N[(y$46$re / x$46$im), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 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))) < 5.0000000000000002e265

   1. Initial program 80.0%

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

      1. *-un-lft-identity 80.0%

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

      2. add-sqr-sqrt 80.0%

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

      3. times-frac 79.8%

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

      4. hypot-def 79.8%

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

      5. fma-def 79.8%

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

      6. hypot-def 96.4%

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

   3. Applied egg-rr 96.4%

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

   if 5.0000000000000002e265 < (/.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 8.3%

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

   2. Taylor expanded in y.re around inf 41.8%

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

      1. associate-/l* 46.8%

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

      2. associate-/r/ 46.8%

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

      3. unpow2 46.8%

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

   4. Simplified 46.8%

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

      1. frac-2neg 46.8%

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

      2. div-inv 46.8%

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

      3. distribute-rgt-neg-in 46.8%

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

   6. Applied egg-rr 46.8%

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

      1. associate-*r/ 46.8%

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

      2. *-commutative 46.8%

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

      3. times-frac 55.1%

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

      4. neg-mul-1 55.1%

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

      5. neg-mul-1 55.1%

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

      6. times-frac 55.1%

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

      7. metadata-eval 55.1%

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

      8. *-lft-identity 55.1%

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

      9. associate-*r/ 55.1%

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

      10. *-rgt-identity 55.1%

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

   8. Simplified 55.1%

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

      1. expm1-log1p-u 49.9%

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

      2. expm1-udef 48.4%

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

      3. associate-*l/ 48.5%

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

      4. *-commutative 48.5%

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

   10. Applied egg-rr 48.5%

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

       1. expm1-def 52.7%

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

       2. expm1-log1p 59.4%

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

       3. associate-*r/ 49.1%

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

       4. *-commutative 49.1%

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

       5. associate-/l* 59.4%

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

   12. Simplified 59.4%

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

4. Final simplification 86.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \leq 5 \cdot 10^{+265}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{\frac{y.im}{\frac{y.re}{x.im}}}{y.re}\\ \end{array} \]

Alternative 3: 77.3% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x.re}{\frac{y.im}{y.re}}\\ \mathbf{if}\;y.im \leq -1.6 \cdot 10^{+154}:\\ \;\;\;\;\left(x.im + t_0\right) \cdot \frac{-1}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.im \leq -9 \cdot 10^{+49}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y.im, x.im, x.re \cdot y.re\right)}{{\left(\mathsf{hypot}\left(y.re, y.im\right)\right)}^{2}}\\ \mathbf{elif}\;y.im \leq -4 \cdot 10^{-13}:\\ \;\;\;\;\frac{1}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{\left(-x.im\right) - t_0}}\\ \mathbf{elif}\;y.im \leq 1.1 \cdot 10^{+19}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{x.im \cdot \frac{y.im}{y.re}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re}{y.im} \cdot \frac{x.re}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (/ x.re (/ y.im y.re))))
   (if (<= y.im -1.6e+154)
     (* (+ x.im t_0) (/ -1.0 (hypot y.re y.im)))
     (if (<= y.im -9e+49)
       (/ (fma y.im x.im (* x.re y.re)) (pow (hypot y.re y.im) 2.0))
       (if (<= y.im -4e-13)
         (/ 1.0 (/ (hypot y.re y.im) (- (- x.im) t_0)))
         (if (<= y.im 1.1e+19)
           (+ (/ x.re y.re) (/ (* x.im (/ y.im y.re)) y.re))
           (+ (/ x.im y.im) (* (/ y.re y.im) (/ x.re y.im)))))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = x_46_re / (y_46_im / y_46_re);
	double tmp;
	if (y_46_im <= -1.6e+154) {
		tmp = (x_46_im + t_0) * (-1.0 / hypot(y_46_re, y_46_im));
	} else if (y_46_im <= -9e+49) {
		tmp = fma(y_46_im, x_46_im, (x_46_re * y_46_re)) / pow(hypot(y_46_re, y_46_im), 2.0);
	} else if (y_46_im <= -4e-13) {
		tmp = 1.0 / (hypot(y_46_re, y_46_im) / (-x_46_im - t_0));
	} else if (y_46_im <= 1.1e+19) {
		tmp = (x_46_re / y_46_re) + ((x_46_im * (y_46_im / y_46_re)) / y_46_re);
	} else {
		tmp = (x_46_im / y_46_im) + ((y_46_re / y_46_im) * (x_46_re / y_46_im));
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(x_46_re / Float64(y_46_im / y_46_re))
	tmp = 0.0
	if (y_46_im <= -1.6e+154)
		tmp = Float64(Float64(x_46_im + t_0) * Float64(-1.0 / hypot(y_46_re, y_46_im)));
	elseif (y_46_im <= -9e+49)
		tmp = Float64(fma(y_46_im, x_46_im, Float64(x_46_re * y_46_re)) / (hypot(y_46_re, y_46_im) ^ 2.0));
	elseif (y_46_im <= -4e-13)
		tmp = Float64(1.0 / Float64(hypot(y_46_re, y_46_im) / Float64(Float64(-x_46_im) - t_0)));
	elseif (y_46_im <= 1.1e+19)
		tmp = Float64(Float64(x_46_re / y_46_re) + Float64(Float64(x_46_im * Float64(y_46_im / y_46_re)) / y_46_re));
	else
		tmp = Float64(Float64(x_46_im / y_46_im) + Float64(Float64(y_46_re / y_46_im) * Float64(x_46_re / y_46_im)));
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(x$46$re / N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$im, -1.6e+154], N[(N[(x$46$im + t$95$0), $MachinePrecision] * N[(-1.0 / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, -9e+49], N[(N[(y$46$im * x$46$im + N[(x$46$re * y$46$re), $MachinePrecision]), $MachinePrecision] / N[Power[N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, -4e-13], N[(1.0 / N[(N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision] / N[((-x$46$im) - t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, 1.1e+19], N[(N[(x$46$re / y$46$re), $MachinePrecision] + N[(N[(x$46$im * N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$im / y$46$im), $MachinePrecision] + N[(N[(y$46$re / y$46$im), $MachinePrecision] * N[(x$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if y.im < -1.6e154

   1. Initial program 27.1%

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

      1. *-un-lft-identity 27.1%

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

      2. add-sqr-sqrt 27.1%

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

      3. times-frac 27.1%

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

      4. hypot-def 27.1%

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

      5. fma-def 27.1%

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

      6. hypot-def 58.4%

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

   3. Applied egg-rr 58.4%

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

   4. Taylor expanded in y.im around -inf 80.7%

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

      1. +-commutative 80.7%

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

      2. mul-1-neg 80.7%

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

      3. unsub-neg 80.7%

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

      4. mul-1-neg 80.7%

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

      5. associate-/l* 85.7%

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

   6. Simplified 85.7%

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

   if -1.6e154 < y.im < -8.99999999999999965e49

   1. Initial program 75.1%

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

      1. add-log-exp 23.3%

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

      2. *-un-lft-identity 23.3%

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

      3. log-prod 23.3%

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

      4. metadata-eval 23.3%

         \[\leadsto \color{blue}{0} + \log \left(e^{\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}}\right) \]

      5. add-log-exp 75.1%

         \[\leadsto 0 + \color{blue}{\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}} \]

      6. add-sqr-sqrt 75.1%

         \[\leadsto 0 + \frac{x.re \cdot y.re + x.im \cdot y.im}{\color{blue}{\sqrt{y.re \cdot y.re + y.im \cdot y.im} \cdot \sqrt{y.re \cdot y.re + y.im \cdot y.im}}} \]

      7. fma-def 75.1%

         \[\leadsto 0 + \frac{\color{blue}{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}}{\sqrt{y.re \cdot y.re + y.im \cdot y.im} \cdot \sqrt{y.re \cdot y.re + y.im \cdot y.im}} \]

      8. pow2 75.1%

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

      9. hypot-def 75.1%

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

   3. Applied egg-rr 75.1%

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

      1. +-lft-identity 75.1%

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

      2. fma-udef 75.1%

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

      3. +-commutative 75.1%

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

      4. *-commutative 75.1%

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

      5. fma-def 75.1%

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

      6. *-commutative 75.1%

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

   5. Simplified 75.1%

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

   if -8.99999999999999965e49 < y.im < -4.0000000000000001e-13

   1. Initial program 80.6%

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

      1. *-un-lft-identity 80.6%

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

      2. add-sqr-sqrt 80.6%

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

      3. times-frac 80.5%

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

      4. hypot-def 80.5%

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

      5. fma-def 80.5%

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

      6. hypot-def 80.5%

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

   3. Applied egg-rr 80.5%

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

      1. associate-*l/ 80.8%

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

      2. *-un-lft-identity 80.8%

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

      3. clear-num 80.6%

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

   5. Applied egg-rr 80.6%

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

   6. Taylor expanded in y.im around -inf 90.3%

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

      1. +-commutative 90.0%

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

      2. mul-1-neg 90.0%

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

      3. unsub-neg 90.0%

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

      4. mul-1-neg 90.0%

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

      5. associate-/l* 90.0%

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

   8. Simplified 90.2%

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

   if -4.0000000000000001e-13 < y.im < 1.1e19

   1. Initial program 70.4%

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

   2. Taylor expanded in y.re around inf 81.7%

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

      1. associate-/l* 77.0%

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

      2. associate-/r/ 80.7%

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

      3. unpow2 80.7%

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

   4. Simplified 80.7%

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

      1. *-commutative 80.7%

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

      2. associate-/r* 84.2%

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

      3. associate-*r/ 86.7%

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

   6. Applied egg-rr 86.7%

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

   if 1.1e19 < y.im

   1. Initial program 52.9%

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

   2. Taylor expanded in y.re around 0 73.8%

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

      1. +-commutative 73.8%

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

      2. *-commutative 73.8%

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

      3. unpow2 73.8%

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

      4. times-frac 84.2%

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

   4. Simplified 84.2%

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

4. Final simplification 85.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -1.6 \cdot 10^{+154}:\\ \;\;\;\;\left(x.im + \frac{x.re}{\frac{y.im}{y.re}}\right) \cdot \frac{-1}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.im \leq -9 \cdot 10^{+49}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y.im, x.im, x.re \cdot y.re\right)}{{\left(\mathsf{hypot}\left(y.re, y.im\right)\right)}^{2}}\\ \mathbf{elif}\;y.im \leq -4 \cdot 10^{-13}:\\ \;\;\;\;\frac{1}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{\left(-x.im\right) - \frac{x.re}{\frac{y.im}{y.re}}}}\\ \mathbf{elif}\;y.im \leq 1.1 \cdot 10^{+19}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{x.im \cdot \frac{y.im}{y.re}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re}{y.im} \cdot \frac{x.re}{y.im}\\ \end{array} \]

Alternative 4: 77.4% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x.re}{\frac{y.im}{y.re}}\\ \mathbf{if}\;y.im \leq -1.6 \cdot 10^{+154}:\\ \;\;\;\;\left(x.im + t_0\right) \cdot \frac{-1}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.im \leq -1.8 \cdot 10^{+50}:\\ \;\;\;\;\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.im \leq -1 \cdot 10^{-14}:\\ \;\;\;\;\frac{1}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{\left(-x.im\right) - t_0}}\\ \mathbf{elif}\;y.im \leq 8.2 \cdot 10^{+18}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{x.im \cdot \frac{y.im}{y.re}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re}{y.im} \cdot \frac{x.re}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (/ x.re (/ y.im y.re))))
   (if (<= y.im -1.6e+154)
     (* (+ x.im t_0) (/ -1.0 (hypot y.re y.im)))
     (if (<= y.im -1.8e+50)
       (/ (+ (* x.re y.re) (* x.im y.im)) (+ (* y.re y.re) (* y.im y.im)))
       (if (<= y.im -1e-14)
         (/ 1.0 (/ (hypot y.re y.im) (- (- x.im) t_0)))
         (if (<= y.im 8.2e+18)
           (+ (/ x.re y.re) (/ (* x.im (/ y.im y.re)) y.re))
           (+ (/ x.im y.im) (* (/ y.re y.im) (/ x.re y.im)))))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = x_46_re / (y_46_im / y_46_re);
	double tmp;
	if (y_46_im <= -1.6e+154) {
		tmp = (x_46_im + t_0) * (-1.0 / hypot(y_46_re, y_46_im));
	} else if (y_46_im <= -1.8e+50) {
		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 <= -1e-14) {
		tmp = 1.0 / (hypot(y_46_re, y_46_im) / (-x_46_im - t_0));
	} else if (y_46_im <= 8.2e+18) {
		tmp = (x_46_re / y_46_re) + ((x_46_im * (y_46_im / y_46_re)) / y_46_re);
	} else {
		tmp = (x_46_im / y_46_im) + ((y_46_re / y_46_im) * (x_46_re / y_46_im));
	}
	return tmp;
}

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = x_46_re / (y_46_im / y_46_re);
	double tmp;
	if (y_46_im <= -1.6e+154) {
		tmp = (x_46_im + t_0) * (-1.0 / Math.hypot(y_46_re, y_46_im));
	} else if (y_46_im <= -1.8e+50) {
		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 <= -1e-14) {
		tmp = 1.0 / (Math.hypot(y_46_re, y_46_im) / (-x_46_im - t_0));
	} else if (y_46_im <= 8.2e+18) {
		tmp = (x_46_re / y_46_re) + ((x_46_im * (y_46_im / y_46_re)) / y_46_re);
	} else {
		tmp = (x_46_im / y_46_im) + ((y_46_re / y_46_im) * (x_46_re / y_46_im));
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = x_46_re / (y_46_im / y_46_re)
	tmp = 0
	if y_46_im <= -1.6e+154:
		tmp = (x_46_im + t_0) * (-1.0 / math.hypot(y_46_re, y_46_im))
	elif y_46_im <= -1.8e+50:
		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))
	elif y_46_im <= -1e-14:
		tmp = 1.0 / (math.hypot(y_46_re, y_46_im) / (-x_46_im - t_0))
	elif y_46_im <= 8.2e+18:
		tmp = (x_46_re / y_46_re) + ((x_46_im * (y_46_im / y_46_re)) / y_46_re)
	else:
		tmp = (x_46_im / y_46_im) + ((y_46_re / y_46_im) * (x_46_re / y_46_im))
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(x_46_re / Float64(y_46_im / y_46_re))
	tmp = 0.0
	if (y_46_im <= -1.6e+154)
		tmp = Float64(Float64(x_46_im + t_0) * Float64(-1.0 / hypot(y_46_re, y_46_im)));
	elseif (y_46_im <= -1.8e+50)
		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 <= -1e-14)
		tmp = Float64(1.0 / Float64(hypot(y_46_re, y_46_im) / Float64(Float64(-x_46_im) - t_0)));
	elseif (y_46_im <= 8.2e+18)
		tmp = Float64(Float64(x_46_re / y_46_re) + Float64(Float64(x_46_im * Float64(y_46_im / y_46_re)) / y_46_re));
	else
		tmp = Float64(Float64(x_46_im / y_46_im) + Float64(Float64(y_46_re / y_46_im) * Float64(x_46_re / y_46_im)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = x_46_re / (y_46_im / y_46_re);
	tmp = 0.0;
	if (y_46_im <= -1.6e+154)
		tmp = (x_46_im + t_0) * (-1.0 / hypot(y_46_re, y_46_im));
	elseif (y_46_im <= -1.8e+50)
		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));
	elseif (y_46_im <= -1e-14)
		tmp = 1.0 / (hypot(y_46_re, y_46_im) / (-x_46_im - t_0));
	elseif (y_46_im <= 8.2e+18)
		tmp = (x_46_re / y_46_re) + ((x_46_im * (y_46_im / y_46_re)) / y_46_re);
	else
		tmp = (x_46_im / y_46_im) + ((y_46_re / y_46_im) * (x_46_re / y_46_im));
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(x$46$re / N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$im, -1.6e+154], N[(N[(x$46$im + t$95$0), $MachinePrecision] * N[(-1.0 / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, -1.8e+50], 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, -1e-14], N[(1.0 / N[(N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision] / N[((-x$46$im) - t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, 8.2e+18], N[(N[(x$46$re / y$46$re), $MachinePrecision] + N[(N[(x$46$im * N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$im / y$46$im), $MachinePrecision] + N[(N[(y$46$re / y$46$im), $MachinePrecision] * N[(x$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if y.im < -1.6e154

   1. Initial program 27.1%

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

      1. *-un-lft-identity 27.1%

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

      2. add-sqr-sqrt 27.1%

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

      3. times-frac 27.1%

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

      4. hypot-def 27.1%

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

      5. fma-def 27.1%

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

      6. hypot-def 58.4%

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

   3. Applied egg-rr 58.4%

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

   4. Taylor expanded in y.im around -inf 80.7%

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

      1. +-commutative 80.7%

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

      2. mul-1-neg 80.7%

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

      3. unsub-neg 80.7%

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

      4. mul-1-neg 80.7%

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

      5. associate-/l* 85.7%

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

   6. Simplified 85.7%

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

   if -1.6e154 < y.im < -1.79999999999999993e50

   1. Initial program 75.1%

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

   if -1.79999999999999993e50 < y.im < -9.99999999999999999e-15

   1. Initial program 80.6%

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

      1. *-un-lft-identity 80.6%

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

      2. add-sqr-sqrt 80.6%

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

      3. times-frac 80.5%

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

      4. hypot-def 80.5%

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

      5. fma-def 80.5%

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

      6. hypot-def 80.5%

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

   3. Applied egg-rr 80.5%

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

      1. associate-*l/ 80.8%

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

      2. *-un-lft-identity 80.8%

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

      3. clear-num 80.6%

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

   5. Applied egg-rr 80.6%

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

   6. Taylor expanded in y.im around -inf 90.3%

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

      1. +-commutative 90.0%

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

      2. mul-1-neg 90.0%

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

      3. unsub-neg 90.0%

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

      4. mul-1-neg 90.0%

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

      5. associate-/l* 90.0%

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

   8. Simplified 90.2%

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

   if -9.99999999999999999e-15 < y.im < 8.2e18

   1. Initial program 70.4%

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

   2. Taylor expanded in y.re around inf 81.7%

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

      1. associate-/l* 77.0%

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

      2. associate-/r/ 80.7%

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

      3. unpow2 80.7%

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

   4. Simplified 80.7%

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

      1. *-commutative 80.7%

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

      2. associate-/r* 84.2%

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

      3. associate-*r/ 86.7%

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

   6. Applied egg-rr 86.7%

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

   if 8.2e18 < y.im

   1. Initial program 52.9%

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

   2. Taylor expanded in y.re around 0 73.8%

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

      1. +-commutative 73.8%

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

      2. *-commutative 73.8%

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

      3. unpow2 73.8%

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

      4. times-frac 84.2%

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

   4. Simplified 84.2%

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

4. Final simplification 85.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -1.6 \cdot 10^{+154}:\\ \;\;\;\;\left(x.im + \frac{x.re}{\frac{y.im}{y.re}}\right) \cdot \frac{-1}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.im \leq -1.8 \cdot 10^{+50}:\\ \;\;\;\;\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.im \leq -1 \cdot 10^{-14}:\\ \;\;\;\;\frac{1}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{\left(-x.im\right) - \frac{x.re}{\frac{y.im}{y.re}}}}\\ \mathbf{elif}\;y.im \leq 8.2 \cdot 10^{+18}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{x.im \cdot \frac{y.im}{y.re}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re}{y.im} \cdot \frac{x.re}{y.im}\\ \end{array} \]

Alternative 5: 77.1% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -1.6 \cdot 10^{+154}:\\ \;\;\;\;\left(x.im + \frac{x.re}{\frac{y.im}{y.re}}\right) \cdot \frac{-1}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.im \leq -9 \cdot 10^{+49}:\\ \;\;\;\;\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.im \leq -9.5 \cdot 10^{-22} \lor \neg \left(y.im \leq 7.2 \cdot 10^{+18}\right):\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re}{y.im} \cdot \frac{x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{x.im \cdot \frac{y.im}{y.re}}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.im -1.6e+154)
   (* (+ x.im (/ x.re (/ y.im y.re))) (/ -1.0 (hypot y.re y.im)))
   (if (<= y.im -9e+49)
     (/ (+ (* x.re y.re) (* x.im y.im)) (+ (* y.re y.re) (* y.im y.im)))
     (if (or (<= y.im -9.5e-22) (not (<= y.im 7.2e+18)))
       (+ (/ x.im y.im) (* (/ y.re y.im) (/ x.re y.im)))
       (+ (/ x.re y.re) (/ (* x.im (/ y.im y.re)) y.re))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_im <= -1.6e+154) {
		tmp = (x_46_im + (x_46_re / (y_46_im / y_46_re))) * (-1.0 / hypot(y_46_re, y_46_im));
	} else if (y_46_im <= -9e+49) {
		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 <= -9.5e-22) || !(y_46_im <= 7.2e+18)) {
		tmp = (x_46_im / y_46_im) + ((y_46_re / y_46_im) * (x_46_re / y_46_im));
	} else {
		tmp = (x_46_re / y_46_re) + ((x_46_im * (y_46_im / y_46_re)) / y_46_re);
	}
	return tmp;
}

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 <= -1.6e+154) {
		tmp = (x_46_im + (x_46_re / (y_46_im / y_46_re))) * (-1.0 / Math.hypot(y_46_re, y_46_im));
	} else if (y_46_im <= -9e+49) {
		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 <= -9.5e-22) || !(y_46_im <= 7.2e+18)) {
		tmp = (x_46_im / y_46_im) + ((y_46_re / y_46_im) * (x_46_re / y_46_im));
	} else {
		tmp = (x_46_re / y_46_re) + ((x_46_im * (y_46_im / y_46_re)) / y_46_re);
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if y_46_im <= -1.6e+154:
		tmp = (x_46_im + (x_46_re / (y_46_im / y_46_re))) * (-1.0 / math.hypot(y_46_re, y_46_im))
	elif y_46_im <= -9e+49:
		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))
	elif (y_46_im <= -9.5e-22) or not (y_46_im <= 7.2e+18):
		tmp = (x_46_im / y_46_im) + ((y_46_re / y_46_im) * (x_46_re / y_46_im))
	else:
		tmp = (x_46_re / y_46_re) + ((x_46_im * (y_46_im / y_46_re)) / y_46_re)
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (y_46_im <= -1.6e+154)
		tmp = Float64(Float64(x_46_im + Float64(x_46_re / Float64(y_46_im / y_46_re))) * Float64(-1.0 / hypot(y_46_re, y_46_im)));
	elseif (y_46_im <= -9e+49)
		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 <= -9.5e-22) || !(y_46_im <= 7.2e+18))
		tmp = Float64(Float64(x_46_im / y_46_im) + Float64(Float64(y_46_re / y_46_im) * Float64(x_46_re / y_46_im)));
	else
		tmp = Float64(Float64(x_46_re / y_46_re) + Float64(Float64(x_46_im * Float64(y_46_im / y_46_re)) / y_46_re));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if (y_46_im <= -1.6e+154)
		tmp = (x_46_im + (x_46_re / (y_46_im / y_46_re))) * (-1.0 / hypot(y_46_re, y_46_im));
	elseif (y_46_im <= -9e+49)
		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));
	elseif ((y_46_im <= -9.5e-22) || ~((y_46_im <= 7.2e+18)))
		tmp = (x_46_im / y_46_im) + ((y_46_re / y_46_im) * (x_46_re / y_46_im));
	else
		tmp = (x_46_re / y_46_re) + ((x_46_im * (y_46_im / y_46_re)) / y_46_re);
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[y$46$im, -1.6e+154], N[(N[(x$46$im + N[(x$46$re / N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(-1.0 / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, -9e+49], 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[Or[LessEqual[y$46$im, -9.5e-22], N[Not[LessEqual[y$46$im, 7.2e+18]], $MachinePrecision]], N[(N[(x$46$im / y$46$im), $MachinePrecision] + N[(N[(y$46$re / y$46$im), $MachinePrecision] * N[(x$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$re / y$46$re), $MachinePrecision] + N[(N[(x$46$im * N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if y.im < -1.6e154

   1. Initial program 27.1%

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

      1. *-un-lft-identity 27.1%

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

      2. add-sqr-sqrt 27.1%

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

      3. times-frac 27.1%

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

      4. hypot-def 27.1%

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

      5. fma-def 27.1%

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

      6. hypot-def 58.4%

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

   3. Applied egg-rr 58.4%

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

   4. Taylor expanded in y.im around -inf 80.7%

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

      1. +-commutative 80.7%

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

      2. mul-1-neg 80.7%

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

      3. unsub-neg 80.7%

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

      4. mul-1-neg 80.7%

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

      5. associate-/l* 85.7%

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

   6. Simplified 85.7%

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

   if -1.6e154 < y.im < -8.99999999999999965e49

   1. Initial program 75.1%

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

   if -8.99999999999999965e49 < y.im < -9.4999999999999994e-22 or 7.2e18 < y.im

   1. Initial program 57.7%

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

   2. Taylor expanded in y.re around 0 76.7%

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

      1. +-commutative 76.7%

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

      2. *-commutative 76.7%

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

      3. unpow2 76.7%

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

      4. times-frac 85.2%

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

   4. Simplified 85.2%

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

   if -9.4999999999999994e-22 < y.im < 7.2e18

   1. Initial program 70.4%

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

   2. Taylor expanded in y.re around inf 81.7%

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

      1. associate-/l* 77.0%

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

      2. associate-/r/ 80.7%

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

      3. unpow2 80.7%

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

   4. Simplified 80.7%

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

      1. *-commutative 80.7%

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

      2. associate-/r* 84.2%

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

      3. associate-*r/ 86.7%

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

   6. Applied egg-rr 86.7%

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

4. Final simplification 85.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -1.6 \cdot 10^{+154}:\\ \;\;\;\;\left(x.im + \frac{x.re}{\frac{y.im}{y.re}}\right) \cdot \frac{-1}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.im \leq -9 \cdot 10^{+49}:\\ \;\;\;\;\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.im \leq -9.5 \cdot 10^{-22} \lor \neg \left(y.im \leq 7.2 \cdot 10^{+18}\right):\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re}{y.im} \cdot \frac{x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{x.im \cdot \frac{y.im}{y.re}}{y.re}\\ \end{array} \]

Alternative 6: 77.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x.im}{y.im} + \frac{y.re}{y.im} \cdot \frac{x.re}{y.im}\\ \mathbf{if}\;y.im \leq -1.6 \cdot 10^{+154}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq -1 \cdot 10^{+50}:\\ \;\;\;\;\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.im \leq -1.45 \cdot 10^{-14} \lor \neg \left(y.im \leq 3.1 \cdot 10^{+19}\right):\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{x.im \cdot \frac{y.im}{y.re}}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (+ (/ x.im y.im) (* (/ y.re y.im) (/ x.re y.im)))))
   (if (<= y.im -1.6e+154)
     t_0
     (if (<= y.im -1e+50)
       (/ (+ (* x.re y.re) (* x.im y.im)) (+ (* y.re y.re) (* y.im y.im)))
       (if (or (<= y.im -1.45e-14) (not (<= y.im 3.1e+19)))
         t_0
         (+ (/ x.re y.re) (/ (* x.im (/ y.im y.re)) y.re)))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = (x_46_im / y_46_im) + ((y_46_re / y_46_im) * (x_46_re / y_46_im));
	double tmp;
	if (y_46_im <= -1.6e+154) {
		tmp = t_0;
	} else if (y_46_im <= -1e+50) {
		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 <= -1.45e-14) || !(y_46_im <= 3.1e+19)) {
		tmp = t_0;
	} else {
		tmp = (x_46_re / y_46_re) + ((x_46_im * (y_46_im / y_46_re)) / y_46_re);
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (x_46im / y_46im) + ((y_46re / y_46im) * (x_46re / y_46im))
    if (y_46im <= (-1.6d+154)) then
        tmp = t_0
    else if (y_46im <= (-1d+50)) then
        tmp = ((x_46re * y_46re) + (x_46im * y_46im)) / ((y_46re * y_46re) + (y_46im * y_46im))
    else if ((y_46im <= (-1.45d-14)) .or. (.not. (y_46im <= 3.1d+19))) then
        tmp = t_0
    else
        tmp = (x_46re / y_46re) + ((x_46im * (y_46im / y_46re)) / y_46re)
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = (x_46_im / y_46_im) + ((y_46_re / y_46_im) * (x_46_re / y_46_im));
	double tmp;
	if (y_46_im <= -1.6e+154) {
		tmp = t_0;
	} else if (y_46_im <= -1e+50) {
		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 <= -1.45e-14) || !(y_46_im <= 3.1e+19)) {
		tmp = t_0;
	} else {
		tmp = (x_46_re / y_46_re) + ((x_46_im * (y_46_im / y_46_re)) / y_46_re);
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = (x_46_im / y_46_im) + ((y_46_re / y_46_im) * (x_46_re / y_46_im))
	tmp = 0
	if y_46_im <= -1.6e+154:
		tmp = t_0
	elif y_46_im <= -1e+50:
		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))
	elif (y_46_im <= -1.45e-14) or not (y_46_im <= 3.1e+19):
		tmp = t_0
	else:
		tmp = (x_46_re / y_46_re) + ((x_46_im * (y_46_im / y_46_re)) / y_46_re)
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(Float64(x_46_im / y_46_im) + Float64(Float64(y_46_re / y_46_im) * Float64(x_46_re / y_46_im)))
	tmp = 0.0
	if (y_46_im <= -1.6e+154)
		tmp = t_0;
	elseif (y_46_im <= -1e+50)
		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 <= -1.45e-14) || !(y_46_im <= 3.1e+19))
		tmp = t_0;
	else
		tmp = Float64(Float64(x_46_re / y_46_re) + Float64(Float64(x_46_im * Float64(y_46_im / y_46_re)) / y_46_re));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = (x_46_im / y_46_im) + ((y_46_re / y_46_im) * (x_46_re / y_46_im));
	tmp = 0.0;
	if (y_46_im <= -1.6e+154)
		tmp = t_0;
	elseif (y_46_im <= -1e+50)
		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));
	elseif ((y_46_im <= -1.45e-14) || ~((y_46_im <= 3.1e+19)))
		tmp = t_0;
	else
		tmp = (x_46_re / y_46_re) + ((x_46_im * (y_46_im / y_46_re)) / y_46_re);
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[(x$46$im / y$46$im), $MachinePrecision] + N[(N[(y$46$re / y$46$im), $MachinePrecision] * N[(x$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$im, -1.6e+154], t$95$0, If[LessEqual[y$46$im, -1e+50], 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[Or[LessEqual[y$46$im, -1.45e-14], N[Not[LessEqual[y$46$im, 3.1e+19]], $MachinePrecision]], t$95$0, N[(N[(x$46$re / y$46$re), $MachinePrecision] + N[(N[(x$46$im * N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if y.im < -1.6e154 or -1.0000000000000001e50 < y.im < -1.4500000000000001e-14 or 3.1e19 < y.im

   1. Initial program 45.2%

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

   2. Taylor expanded in y.re around 0 73.6%

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

      1. +-commutative 73.6%

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

      2. *-commutative 73.6%

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

      3. unpow2 73.6%

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

      4. times-frac 85.2%

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

   4. Simplified 85.2%

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

   if -1.6e154 < y.im < -1.0000000000000001e50

   1. Initial program 75.1%

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

   if -1.4500000000000001e-14 < y.im < 3.1e19

   1. Initial program 70.4%

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

   2. Taylor expanded in y.re around inf 81.7%

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

      1. associate-/l* 77.0%

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

      2. associate-/r/ 80.7%

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

      3. unpow2 80.7%

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

   4. Simplified 80.7%

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

      1. *-commutative 80.7%

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

      2. associate-/r* 84.2%

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

      3. associate-*r/ 86.7%

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

   6. Applied egg-rr 86.7%

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

4. Final simplification 85.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -1.6 \cdot 10^{+154}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re}{y.im} \cdot \frac{x.re}{y.im}\\ \mathbf{elif}\;y.im \leq -1 \cdot 10^{+50}:\\ \;\;\;\;\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.im \leq -1.45 \cdot 10^{-14} \lor \neg \left(y.im \leq 3.1 \cdot 10^{+19}\right):\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re}{y.im} \cdot \frac{x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{x.im \cdot \frac{y.im}{y.re}}{y.re}\\ \end{array} \]

Alternative 7: 76.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -6.2 \cdot 10^{-27} \lor \neg \left(y.im \leq 8.5 \cdot 10^{+18}\right):\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re}{y.im} \cdot \frac{x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{y.im}{y.re} \cdot \frac{x.im}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.im -6.2e-27) (not (<= y.im 8.5e+18)))
   (+ (/ x.im y.im) (* (/ y.re y.im) (/ x.re y.im)))
   (+ (/ x.re y.re) (* (/ y.im y.re) (/ x.im y.re)))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_im <= -6.2e-27) || !(y_46_im <= 8.5e+18)) {
		tmp = (x_46_im / y_46_im) + ((y_46_re / y_46_im) * (x_46_re / y_46_im));
	} else {
		tmp = (x_46_re / y_46_re) + ((y_46_im / y_46_re) * (x_46_im / y_46_re));
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if ((y_46im <= (-6.2d-27)) .or. (.not. (y_46im <= 8.5d+18))) then
        tmp = (x_46im / y_46im) + ((y_46re / y_46im) * (x_46re / y_46im))
    else
        tmp = (x_46re / y_46re) + ((y_46im / y_46re) * (x_46im / y_46re))
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_im <= -6.2e-27) || !(y_46_im <= 8.5e+18)) {
		tmp = (x_46_im / y_46_im) + ((y_46_re / y_46_im) * (x_46_re / y_46_im));
	} else {
		tmp = (x_46_re / y_46_re) + ((y_46_im / y_46_re) * (x_46_im / y_46_re));
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if (y_46_im <= -6.2e-27) or not (y_46_im <= 8.5e+18):
		tmp = (x_46_im / y_46_im) + ((y_46_re / y_46_im) * (x_46_re / y_46_im))
	else:
		tmp = (x_46_re / y_46_re) + ((y_46_im / y_46_re) * (x_46_im / y_46_re))
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if ((y_46_im <= -6.2e-27) || !(y_46_im <= 8.5e+18))
		tmp = Float64(Float64(x_46_im / y_46_im) + Float64(Float64(y_46_re / y_46_im) * Float64(x_46_re / y_46_im)));
	else
		tmp = Float64(Float64(x_46_re / y_46_re) + Float64(Float64(y_46_im / y_46_re) * Float64(x_46_im / y_46_re)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if ((y_46_im <= -6.2e-27) || ~((y_46_im <= 8.5e+18)))
		tmp = (x_46_im / y_46_im) + ((y_46_re / y_46_im) * (x_46_re / y_46_im));
	else
		tmp = (x_46_re / y_46_re) + ((y_46_im / y_46_re) * (x_46_im / y_46_re));
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[Or[LessEqual[y$46$im, -6.2e-27], N[Not[LessEqual[y$46$im, 8.5e+18]], $MachinePrecision]], N[(N[(x$46$im / y$46$im), $MachinePrecision] + N[(N[(y$46$re / y$46$im), $MachinePrecision] * N[(x$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$re / y$46$re), $MachinePrecision] + N[(N[(y$46$im / y$46$re), $MachinePrecision] * N[(x$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.im < -6.1999999999999997e-27 or 8.5e18 < y.im

   1. Initial program 51.1%

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

   2. Taylor expanded in y.re around 0 69.1%

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

      1. +-commutative 69.1%

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

      2. *-commutative 69.1%

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

      3. unpow2 69.1%

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

      4. times-frac 79.3%

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

   4. Simplified 79.3%

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

   if -6.1999999999999997e-27 < y.im < 8.5e18

   1. Initial program 70.4%

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

   2. Taylor expanded in y.re around inf 81.7%

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

      1. associate-/l* 77.0%

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

      2. associate-/r/ 80.7%

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

      3. unpow2 80.7%

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

   4. Simplified 80.7%

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

   5. Taylor expanded in y.im around 0 81.7%

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

      1. unpow2 81.7%

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

      2. times-frac 85.3%

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

   7. Simplified 85.3%

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

4. Final simplification 82.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -6.2 \cdot 10^{-27} \lor \neg \left(y.im \leq 8.5 \cdot 10^{+18}\right):\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re}{y.im} \cdot \frac{x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{y.im}{y.re} \cdot \frac{x.im}{y.re}\\ \end{array} \]

Alternative 8: 77.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -5.2 \cdot 10^{-29} \lor \neg \left(y.im \leq 1.25 \cdot 10^{+19}\right):\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re}{y.im} \cdot \frac{x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{x.im \cdot \frac{y.im}{y.re}}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.im -5.2e-29) (not (<= y.im 1.25e+19)))
   (+ (/ x.im y.im) (* (/ y.re y.im) (/ x.re y.im)))
   (+ (/ x.re y.re) (/ (* x.im (/ y.im y.re)) y.re))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_im <= -5.2e-29) || !(y_46_im <= 1.25e+19)) {
		tmp = (x_46_im / y_46_im) + ((y_46_re / y_46_im) * (x_46_re / y_46_im));
	} else {
		tmp = (x_46_re / y_46_re) + ((x_46_im * (y_46_im / y_46_re)) / y_46_re);
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if ((y_46im <= (-5.2d-29)) .or. (.not. (y_46im <= 1.25d+19))) then
        tmp = (x_46im / y_46im) + ((y_46re / y_46im) * (x_46re / y_46im))
    else
        tmp = (x_46re / y_46re) + ((x_46im * (y_46im / y_46re)) / y_46re)
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_im <= -5.2e-29) || !(y_46_im <= 1.25e+19)) {
		tmp = (x_46_im / y_46_im) + ((y_46_re / y_46_im) * (x_46_re / y_46_im));
	} else {
		tmp = (x_46_re / y_46_re) + ((x_46_im * (y_46_im / y_46_re)) / y_46_re);
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if (y_46_im <= -5.2e-29) or not (y_46_im <= 1.25e+19):
		tmp = (x_46_im / y_46_im) + ((y_46_re / y_46_im) * (x_46_re / y_46_im))
	else:
		tmp = (x_46_re / y_46_re) + ((x_46_im * (y_46_im / y_46_re)) / y_46_re)
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if ((y_46_im <= -5.2e-29) || !(y_46_im <= 1.25e+19))
		tmp = Float64(Float64(x_46_im / y_46_im) + Float64(Float64(y_46_re / y_46_im) * Float64(x_46_re / y_46_im)));
	else
		tmp = Float64(Float64(x_46_re / y_46_re) + Float64(Float64(x_46_im * Float64(y_46_im / y_46_re)) / y_46_re));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if ((y_46_im <= -5.2e-29) || ~((y_46_im <= 1.25e+19)))
		tmp = (x_46_im / y_46_im) + ((y_46_re / y_46_im) * (x_46_re / y_46_im));
	else
		tmp = (x_46_re / y_46_re) + ((x_46_im * (y_46_im / y_46_re)) / y_46_re);
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[Or[LessEqual[y$46$im, -5.2e-29], N[Not[LessEqual[y$46$im, 1.25e+19]], $MachinePrecision]], N[(N[(x$46$im / y$46$im), $MachinePrecision] + N[(N[(y$46$re / y$46$im), $MachinePrecision] * N[(x$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$re / y$46$re), $MachinePrecision] + N[(N[(x$46$im * N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.im < -5.2000000000000004e-29 or 1.25e19 < y.im

   1. Initial program 51.1%

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

   2. Taylor expanded in y.re around 0 69.1%

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

      1. +-commutative 69.1%

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

      2. *-commutative 69.1%

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

      3. unpow2 69.1%

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

      4. times-frac 79.3%

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

   4. Simplified 79.3%

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

   if -5.2000000000000004e-29 < y.im < 1.25e19

   1. Initial program 70.4%

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

   2. Taylor expanded in y.re around inf 81.7%

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

      1. associate-/l* 77.0%

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

      2. associate-/r/ 80.7%

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

      3. unpow2 80.7%

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

   4. Simplified 80.7%

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

      1. *-commutative 80.7%

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

      2. associate-/r* 84.2%

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

      3. associate-*r/ 86.7%

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

   6. Applied egg-rr 86.7%

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

4. Final simplification 83.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -5.2 \cdot 10^{-29} \lor \neg \left(y.im \leq 1.25 \cdot 10^{+19}\right):\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re}{y.im} \cdot \frac{x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{x.im \cdot \frac{y.im}{y.re}}{y.re}\\ \end{array} \]

Alternative 9: 71.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -5.7 \cdot 10^{+38}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{elif}\;y.re \leq 63000000000000:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re}{y.im} \cdot \frac{x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.re -5.7e+38)
   (/ x.re y.re)
   (if (<= y.re 63000000000000.0)
     (+ (/ x.im y.im) (* (/ y.re y.im) (/ x.re 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_re <= -5.7e+38) {
		tmp = x_46_re / y_46_re;
	} else if (y_46_re <= 63000000000000.0) {
		tmp = (x_46_im / y_46_im) + ((y_46_re / y_46_im) * (x_46_re / 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_46re <= (-5.7d+38)) then
        tmp = x_46re / y_46re
    else if (y_46re <= 63000000000000.0d0) then
        tmp = (x_46im / y_46im) + ((y_46re / y_46im) * (x_46re / 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_re <= -5.7e+38) {
		tmp = x_46_re / y_46_re;
	} else if (y_46_re <= 63000000000000.0) {
		tmp = (x_46_im / y_46_im) + ((y_46_re / y_46_im) * (x_46_re / 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_re <= -5.7e+38:
		tmp = x_46_re / y_46_re
	elif y_46_re <= 63000000000000.0:
		tmp = (x_46_im / y_46_im) + ((y_46_re / y_46_im) * (x_46_re / 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_re <= -5.7e+38)
		tmp = Float64(x_46_re / y_46_re);
	elseif (y_46_re <= 63000000000000.0)
		tmp = Float64(Float64(x_46_im / y_46_im) + Float64(Float64(y_46_re / y_46_im) * Float64(x_46_re / 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_re <= -5.7e+38)
		tmp = x_46_re / y_46_re;
	elseif (y_46_re <= 63000000000000.0)
		tmp = (x_46_im / y_46_im) + ((y_46_re / y_46_im) * (x_46_re / 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[LessEqual[y$46$re, -5.7e+38], N[(x$46$re / y$46$re), $MachinePrecision], If[LessEqual[y$46$re, 63000000000000.0], N[(N[(x$46$im / y$46$im), $MachinePrecision] + N[(N[(y$46$re / y$46$im), $MachinePrecision] * N[(x$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x$46$re / y$46$re), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y.re < -5.6999999999999997e38 or 6.3e13 < y.re

   1. Initial program 50.2%

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

   2. Taylor expanded in y.re around inf 70.6%

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

   if -5.6999999999999997e38 < y.re < 6.3e13

   1. Initial program 71.9%

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

   2. Taylor expanded in y.re around 0 69.3%

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

      1. +-commutative 69.3%

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

      2. *-commutative 69.3%

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

      3. unpow2 69.3%

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

      4. times-frac 74.7%

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

   4. Simplified 74.7%

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

4. Final simplification 72.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -5.7 \cdot 10^{+38}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{elif}\;y.re \leq 63000000000000:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re}{y.im} \cdot \frac{x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \end{array} \]

Alternative 10: 63.8% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -1.12 \cdot 10^{-38}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{elif}\;y.re \leq 160000000:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.re -1.12e-38)
   (/ x.re y.re)
   (if (<= y.re 160000000.0) (/ 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_re <= -1.12e-38) {
		tmp = x_46_re / y_46_re;
	} else if (y_46_re <= 160000000.0) {
		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_46re <= (-1.12d-38)) then
        tmp = x_46re / y_46re
    else if (y_46re <= 160000000.0d0) 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_re <= -1.12e-38) {
		tmp = x_46_re / y_46_re;
	} else if (y_46_re <= 160000000.0) {
		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_re <= -1.12e-38:
		tmp = x_46_re / y_46_re
	elif y_46_re <= 160000000.0:
		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_re <= -1.12e-38)
		tmp = Float64(x_46_re / y_46_re);
	elseif (y_46_re <= 160000000.0)
		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_re <= -1.12e-38)
		tmp = x_46_re / y_46_re;
	elseif (y_46_re <= 160000000.0)
		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[LessEqual[y$46$re, -1.12e-38], N[(x$46$re / y$46$re), $MachinePrecision], If[LessEqual[y$46$re, 160000000.0], 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.re < -1.1200000000000001e-38 or 1.6e8 < y.re

   1. Initial program 51.5%

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

   2. Taylor expanded in y.re around inf 66.0%

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

   if -1.1200000000000001e-38 < y.re < 1.6e8

   1. Initial program 72.8%

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

   2. Taylor expanded in y.re around 0 70.3%

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

4. Final simplification 67.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -1.12 \cdot 10^{-38}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{elif}\;y.re \leq 160000000:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \end{array} \]

Alternative 11: 43.1% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -3 \cdot 10^{+115}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.re -3e+115) (/ x.im y.re) (/ x.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_re <= -3e+115) {
		tmp = x_46_im / y_46_re;
	} else {
		tmp = x_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_46re <= (-3d+115)) then
        tmp = x_46im / y_46re
    else
        tmp = x_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_re <= -3e+115) {
		tmp = x_46_im / y_46_re;
	} else {
		tmp = x_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_re <= -3e+115:
		tmp = x_46_im / y_46_re
	else:
		tmp = x_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_re <= -3e+115)
		tmp = Float64(x_46_im / y_46_re);
	else
		tmp = Float64(x_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_re <= -3e+115)
		tmp = x_46_im / y_46_re;
	else
		tmp = x_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$46$re, -3e+115], N[(x$46$im / y$46$re), $MachinePrecision], N[(x$46$im / y$46$im), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.re < -3e115

   1. Initial program 44.8%

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

      1. *-un-lft-identity 44.8%

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

      2. add-sqr-sqrt 44.8%

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

      3. times-frac 44.9%

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

      4. hypot-def 44.9%

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

      5. fma-def 44.9%

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

      6. hypot-def 59.2%

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

   3. Applied egg-rr 59.2%

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

   4. Taylor expanded in y.re around 0 20.1%

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

   5. Taylor expanded in y.re around inf 20.5%

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

   if -3e115 < y.re

   1. Initial program 64.5%

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

   2. Taylor expanded in y.re around 0 46.2%

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

4. Final simplification 41.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -3 \cdot 10^{+115}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \end{array} \]

Alternative 12: 43.1% accurate, 5.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 61.2%

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

2. Taylor expanded in y.re around 0 39.8%

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

3. Final simplification 39.8%

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

Reproduce

herbie shell --seed 2023242 
(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))))