_divideComplex, real part

Percentage Accurate: 62.2% → 85.5%

Time: 12.2s

Alternatives: 12

Speedup: 1.3×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 62.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 85.5% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\frac{\mathsf{fma}\left(x.re, y.re, y.im \cdot x.im\right)}{\mathsf{hypot}\left(y.re, y.im\right)}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{if}\;y.im \leq -8.6 \cdot 10^{+122}:\\ \;\;\;\;\frac{\left(-x.im\right) - y.re \cdot \frac{x.re}{y.im}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.im \leq -6.4 \cdot 10^{-204}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq 7.8 \cdot 10^{-140}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{x.im}{y.re \cdot \frac{y.re}{y.im}}\\ \mathbf{elif}\;y.im \leq 7.6 \cdot 10^{+129}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im + \frac{y.re \cdot x.re}{y.im}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0
         (/
          (/ (fma x.re y.re (* y.im x.im)) (hypot y.re y.im))
          (hypot y.re y.im))))
   (if (<= y.im -8.6e+122)
     (/ (- (- x.im) (* y.re (/ x.re y.im))) (hypot y.re y.im))
     (if (<= y.im -6.4e-204)
       t_0
       (if (<= y.im 7.8e-140)
         (+ (/ x.re y.re) (/ x.im (* y.re (/ y.re y.im))))
         (if (<= y.im 7.6e+129)
           t_0
           (/ (+ x.im (/ (* y.re x.re) y.im)) (hypot y.re y.im))))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = (fma(x_46_re, y_46_re, (y_46_im * x_46_im)) / hypot(y_46_re, y_46_im)) / hypot(y_46_re, y_46_im);
	double tmp;
	if (y_46_im <= -8.6e+122) {
		tmp = (-x_46_im - (y_46_re * (x_46_re / y_46_im))) / hypot(y_46_re, y_46_im);
	} else if (y_46_im <= -6.4e-204) {
		tmp = t_0;
	} else if (y_46_im <= 7.8e-140) {
		tmp = (x_46_re / y_46_re) + (x_46_im / (y_46_re * (y_46_re / y_46_im)));
	} else if (y_46_im <= 7.6e+129) {
		tmp = t_0;
	} else {
		tmp = (x_46_im + ((y_46_re * x_46_re) / y_46_im)) / hypot(y_46_re, y_46_im);
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(Float64(fma(x_46_re, y_46_re, Float64(y_46_im * x_46_im)) / hypot(y_46_re, y_46_im)) / hypot(y_46_re, y_46_im))
	tmp = 0.0
	if (y_46_im <= -8.6e+122)
		tmp = Float64(Float64(Float64(-x_46_im) - Float64(y_46_re * Float64(x_46_re / y_46_im))) / hypot(y_46_re, y_46_im));
	elseif (y_46_im <= -6.4e-204)
		tmp = t_0;
	elseif (y_46_im <= 7.8e-140)
		tmp = Float64(Float64(x_46_re / y_46_re) + Float64(x_46_im / Float64(y_46_re * Float64(y_46_re / y_46_im))));
	elseif (y_46_im <= 7.6e+129)
		tmp = t_0;
	else
		tmp = Float64(Float64(x_46_im + Float64(Float64(y_46_re * x_46_re) / y_46_im)) / hypot(y_46_re, y_46_im));
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[(N[(x$46$re * y$46$re + N[(y$46$im * x$46$im), $MachinePrecision]), $MachinePrecision] / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision] / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$im, -8.6e+122], N[(N[((-x$46$im) - N[(y$46$re * N[(x$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, -6.4e-204], t$95$0, If[LessEqual[y$46$im, 7.8e-140], N[(N[(x$46$re / y$46$re), $MachinePrecision] + N[(x$46$im / N[(y$46$re * N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, 7.6e+129], t$95$0, N[(N[(x$46$im + N[(N[(y$46$re * x$46$re), $MachinePrecision] / y$46$im), $MachinePrecision]), $MachinePrecision] / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if y.im < -8.59999999999999943e122

   1. Initial program 32.9%

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

      1. *-un-lft-identity 32.9%

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

      2. +-commutative 32.9%

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

      3. fma-udef 32.9%

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

      4. add-sqr-sqrt 32.9%

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

      5. times-frac 32.9%

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

      6. fma-udef 32.9%

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

      7. +-commutative 32.9%

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

      8. hypot-def 32.9%

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

      9. fma-def 32.9%

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

      10. fma-udef 32.9%

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

      11. +-commutative 32.9%

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

      12. hypot-def 51.3%

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

   3. Applied egg-rr 51.3%

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

      1. associate-*l/ 51.4%

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

      2. *-un-lft-identity 51.4%

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

   5. Applied egg-rr 51.4%

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

      1. clear-num 51.3%

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

      2. inv-pow 51.3%

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

   7. Applied egg-rr 51.3%

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

      1. unpow-1 51.3%

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

      2. fma-udef 51.3%

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

      3. +-commutative 51.3%

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

      4. fma-def 51.3%

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

      5. *-commutative 51.3%

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

   9. Simplified 51.3%

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

   10. Taylor expanded in y.im around -inf 82.3%

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

       1. distribute-lft-out 82.3%

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

       2. *-commutative 82.3%

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

       3. associate-*r/ 90.3%

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

   12. Simplified 90.3%

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

   if -8.59999999999999943e122 < y.im < -6.4e-204 or 7.80000000000000038e-140 < y.im < 7.60000000000000011e129

   1. Initial program 82.9%

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

      1. *-un-lft-identity 82.9%

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

      2. +-commutative 82.9%

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

      3. fma-udef 82.9%

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

      4. add-sqr-sqrt 82.9%

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

      5. times-frac 82.9%

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

      6. fma-udef 82.9%

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

      7. +-commutative 82.9%

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

      8. hypot-def 82.9%

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

      9. fma-def 82.9%

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

      10. fma-udef 82.9%

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

      11. +-commutative 82.9%

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

      12. hypot-def 93.0%

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

   3. Applied egg-rr 93.0%

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

      1. associate-*l/ 93.1%

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

      2. *-un-lft-identity 93.1%

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

   5. Applied egg-rr 93.1%

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

   if -6.4e-204 < y.im < 7.80000000000000038e-140

   1. Initial program 65.4%

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

   2. Taylor expanded in y.re around inf 83.7%

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

      1. associate-/l* 90.0%

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

   4. Simplified 90.0%

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

      1. pow2 90.0%

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

      2. *-un-lft-identity 90.0%

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

      3. times-frac 92.1%

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

   6. Applied egg-rr 92.1%

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

   if 7.60000000000000011e129 < y.im

   1. Initial program 41.1%

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

      1. *-un-lft-identity 41.1%

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

      2. +-commutative 41.1%

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

      3. fma-udef 41.1%

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

      4. add-sqr-sqrt 41.1%

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

      5. times-frac 41.0%

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

      6. fma-udef 41.0%

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

      7. +-commutative 41.0%

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

      8. hypot-def 41.0%

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

      9. fma-def 41.0%

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

      10. fma-udef 41.0%

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

      11. +-commutative 41.0%

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

      12. hypot-def 58.9%

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

   3. Applied egg-rr 58.9%

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

      1. associate-*l/ 59.0%

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

      2. *-un-lft-identity 59.0%

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

   5. Applied egg-rr 59.0%

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

   6. Taylor expanded in y.re around 0 90.7%

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

4. Final simplification 92.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -8.6 \cdot 10^{+122}:\\ \;\;\;\;\frac{\left(-x.im\right) - y.re \cdot \frac{x.re}{y.im}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.im \leq -6.4 \cdot 10^{-204}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(x.re, y.re, y.im \cdot x.im\right)}{\mathsf{hypot}\left(y.re, y.im\right)}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.im \leq 7.8 \cdot 10^{-140}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{x.im}{y.re \cdot \frac{y.re}{y.im}}\\ \mathbf{elif}\;y.im \leq 7.6 \cdot 10^{+129}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(x.re, y.re, y.im \cdot x.im\right)}{\mathsf{hypot}\left(y.re, y.im\right)}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im + \frac{y.re \cdot x.re}{y.im}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \end{array} \]

Alternative 2: 81.5% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y.im \cdot x.im + y.re \cdot x.re\\ \mathbf{if}\;y.im \leq -3.7 \cdot 10^{+64}:\\ \;\;\;\;\frac{\left(-x.im\right) - y.re \cdot \frac{x.re}{y.im}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.im \leq -3.4 \cdot 10^{-145}:\\ \;\;\;\;\frac{t_0}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.im \leq 4.7 \cdot 10^{-139}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{x.im}{y.re \cdot \frac{y.re}{y.im}}\\ \mathbf{elif}\;y.im \leq 1.4 \cdot 10^{+37}:\\ \;\;\;\;\frac{t_0}{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im + \frac{y.re \cdot x.re}{y.im}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (+ (* y.im x.im) (* y.re x.re))))
   (if (<= y.im -3.7e+64)
     (/ (- (- x.im) (* y.re (/ x.re y.im))) (hypot y.re y.im))
     (if (<= y.im -3.4e-145)
       (/ t_0 (+ (* y.re y.re) (* y.im y.im)))
       (if (<= y.im 4.7e-139)
         (+ (/ x.re y.re) (/ x.im (* y.re (/ y.re y.im))))
         (if (<= y.im 1.4e+37)
           (/ t_0 (fma y.im y.im (* y.re y.re)))
           (/ (+ x.im (/ (* y.re x.re) y.im)) (hypot y.re y.im))))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = (y_46_im * x_46_im) + (y_46_re * x_46_re);
	double tmp;
	if (y_46_im <= -3.7e+64) {
		tmp = (-x_46_im - (y_46_re * (x_46_re / y_46_im))) / hypot(y_46_re, y_46_im);
	} else if (y_46_im <= -3.4e-145) {
		tmp = t_0 / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	} else if (y_46_im <= 4.7e-139) {
		tmp = (x_46_re / y_46_re) + (x_46_im / (y_46_re * (y_46_re / y_46_im)));
	} else if (y_46_im <= 1.4e+37) {
		tmp = t_0 / fma(y_46_im, y_46_im, (y_46_re * y_46_re));
	} else {
		tmp = (x_46_im + ((y_46_re * x_46_re) / y_46_im)) / hypot(y_46_re, y_46_im);
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(Float64(y_46_im * x_46_im) + Float64(y_46_re * x_46_re))
	tmp = 0.0
	if (y_46_im <= -3.7e+64)
		tmp = Float64(Float64(Float64(-x_46_im) - Float64(y_46_re * Float64(x_46_re / y_46_im))) / hypot(y_46_re, y_46_im));
	elseif (y_46_im <= -3.4e-145)
		tmp = Float64(t_0 / Float64(Float64(y_46_re * y_46_re) + Float64(y_46_im * y_46_im)));
	elseif (y_46_im <= 4.7e-139)
		tmp = Float64(Float64(x_46_re / y_46_re) + Float64(x_46_im / Float64(y_46_re * Float64(y_46_re / y_46_im))));
	elseif (y_46_im <= 1.4e+37)
		tmp = Float64(t_0 / fma(y_46_im, y_46_im, Float64(y_46_re * y_46_re)));
	else
		tmp = Float64(Float64(x_46_im + Float64(Float64(y_46_re * x_46_re) / y_46_im)) / hypot(y_46_re, y_46_im));
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[(y$46$im * x$46$im), $MachinePrecision] + N[(y$46$re * x$46$re), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$im, -3.7e+64], N[(N[((-x$46$im) - N[(y$46$re * N[(x$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, -3.4e-145], N[(t$95$0 / N[(N[(y$46$re * y$46$re), $MachinePrecision] + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, 4.7e-139], N[(N[(x$46$re / y$46$re), $MachinePrecision] + N[(x$46$im / N[(y$46$re * N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, 1.4e+37], N[(t$95$0 / N[(y$46$im * y$46$im + N[(y$46$re * y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$im + N[(N[(y$46$re * x$46$re), $MachinePrecision] / y$46$im), $MachinePrecision]), $MachinePrecision] / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if y.im < -3.69999999999999983e64

   1. Initial program 39.8%

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

      1. *-un-lft-identity 39.8%

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

      2. +-commutative 39.8%

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

      3. fma-udef 39.8%

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

      4. add-sqr-sqrt 39.8%

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

      5. times-frac 39.8%

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

      6. fma-udef 39.8%

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

      7. +-commutative 39.8%

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

      8. hypot-def 39.8%

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

      9. fma-def 39.8%

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

      10. fma-udef 39.8%

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

      11. +-commutative 39.8%

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

      12. hypot-def 57.8%

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

   3. Applied egg-rr 57.8%

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

      1. associate-*l/ 57.9%

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

      2. *-un-lft-identity 57.9%

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

   5. Applied egg-rr 57.9%

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

      1. clear-num 57.9%

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

      2. inv-pow 57.9%

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

   7. Applied egg-rr 57.9%

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

      1. unpow-1 57.9%

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

      2. fma-udef 57.9%

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

      3. +-commutative 57.9%

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

      4. fma-def 57.9%

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

      5. *-commutative 57.9%

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

   9. Simplified 57.9%

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

   10. Taylor expanded in y.im around -inf 82.6%

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

       1. distribute-lft-out 82.6%

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

       2. *-commutative 82.6%

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

       3. associate-*r/ 89.5%

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

   12. Simplified 89.5%

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

   if -3.69999999999999983e64 < y.im < -3.3999999999999999e-145

   1. Initial program 81.2%

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

   if -3.3999999999999999e-145 < y.im < 4.70000000000000027e-139

   1. Initial program 69.0%

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

   2. Taylor expanded in y.re around inf 84.4%

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

      1. associate-/l* 89.8%

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

   4. Simplified 89.8%

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

      1. pow2 89.8%

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

      2. *-un-lft-identity 89.8%

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

      3. times-frac 91.6%

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

   6. Applied egg-rr 91.6%

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

   if 4.70000000000000027e-139 < y.im < 1.3999999999999999e37

   1. Initial program 87.4%

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

      1. fma-def 87.4%

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

      2. +-commutative 87.4%

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

      3. fma-def 87.4%

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

   3. Simplified 87.4%

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

      1. fma-def 87.4%

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

   5. Applied egg-rr 87.4%

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

   if 1.3999999999999999e37 < y.im

   1. Initial program 48.4%

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

      1. *-un-lft-identity 48.4%

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

      2. +-commutative 48.4%

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

      3. fma-udef 48.4%

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

      4. add-sqr-sqrt 48.4%

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

      5. times-frac 48.4%

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

      6. fma-udef 48.4%

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

      7. +-commutative 48.4%

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

      8. hypot-def 48.4%

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

      9. fma-def 48.4%

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

      10. fma-udef 48.4%

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

      11. +-commutative 48.4%

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

      12. hypot-def 63.5%

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

   3. Applied egg-rr 63.5%

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

      1. associate-*l/ 63.7%

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

      2. *-un-lft-identity 63.7%

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

   5. Applied egg-rr 63.7%

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

   6. Taylor expanded in y.re around 0 85.5%

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

4. Final simplification 87.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -3.7 \cdot 10^{+64}:\\ \;\;\;\;\frac{\left(-x.im\right) - y.re \cdot \frac{x.re}{y.im}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.im \leq -3.4 \cdot 10^{-145}:\\ \;\;\;\;\frac{y.im \cdot x.im + y.re \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.im \leq 4.7 \cdot 10^{-139}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{x.im}{y.re \cdot \frac{y.re}{y.im}}\\ \mathbf{elif}\;y.im \leq 1.4 \cdot 10^{+37}:\\ \;\;\;\;\frac{y.im \cdot x.im + y.re \cdot x.re}{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im + \frac{y.re \cdot x.re}{y.im}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \end{array} \]

Alternative 3: 78.9% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y.im \cdot x.im + y.re \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{if}\;y.im \leq -2.25 \cdot 10^{+123}:\\ \;\;\;\;x.im \cdot \frac{-1}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.im \leq -2.35 \cdot 10^{-150}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq 7 \cdot 10^{-140}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{x.im}{y.re \cdot \frac{y.re}{y.im}}\\ \mathbf{elif}\;y.im \leq 1.4 \cdot 10^{+37}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{x.re}{\frac{{y.im}^{2}}{y.re}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0
         (/ (+ (* y.im x.im) (* y.re x.re)) (+ (* y.re y.re) (* y.im y.im)))))
   (if (<= y.im -2.25e+123)
     (* x.im (/ -1.0 (hypot y.re y.im)))
     (if (<= y.im -2.35e-150)
       t_0
       (if (<= y.im 7e-140)
         (+ (/ x.re y.re) (/ x.im (* y.re (/ y.re y.im))))
         (if (<= y.im 1.4e+37)
           t_0
           (+ (/ x.im y.im) (/ x.re (/ (pow y.im 2.0) y.re)))))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = ((y_46_im * x_46_im) + (y_46_re * x_46_re)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	double tmp;
	if (y_46_im <= -2.25e+123) {
		tmp = x_46_im * (-1.0 / hypot(y_46_re, y_46_im));
	} else if (y_46_im <= -2.35e-150) {
		tmp = t_0;
	} else if (y_46_im <= 7e-140) {
		tmp = (x_46_re / y_46_re) + (x_46_im / (y_46_re * (y_46_re / y_46_im)));
	} else if (y_46_im <= 1.4e+37) {
		tmp = t_0;
	} else {
		tmp = (x_46_im / y_46_im) + (x_46_re / (pow(y_46_im, 2.0) / 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 t_0 = ((y_46_im * x_46_im) + (y_46_re * x_46_re)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	double tmp;
	if (y_46_im <= -2.25e+123) {
		tmp = x_46_im * (-1.0 / Math.hypot(y_46_re, y_46_im));
	} else if (y_46_im <= -2.35e-150) {
		tmp = t_0;
	} else if (y_46_im <= 7e-140) {
		tmp = (x_46_re / y_46_re) + (x_46_im / (y_46_re * (y_46_re / y_46_im)));
	} else if (y_46_im <= 1.4e+37) {
		tmp = t_0;
	} else {
		tmp = (x_46_im / y_46_im) + (x_46_re / (Math.pow(y_46_im, 2.0) / y_46_re));
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = ((y_46_im * x_46_im) + (y_46_re * x_46_re)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im))
	tmp = 0
	if y_46_im <= -2.25e+123:
		tmp = x_46_im * (-1.0 / math.hypot(y_46_re, y_46_im))
	elif y_46_im <= -2.35e-150:
		tmp = t_0
	elif y_46_im <= 7e-140:
		tmp = (x_46_re / y_46_re) + (x_46_im / (y_46_re * (y_46_re / y_46_im)))
	elif y_46_im <= 1.4e+37:
		tmp = t_0
	else:
		tmp = (x_46_im / y_46_im) + (x_46_re / (math.pow(y_46_im, 2.0) / y_46_re))
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(Float64(Float64(y_46_im * x_46_im) + Float64(y_46_re * x_46_re)) / Float64(Float64(y_46_re * y_46_re) + Float64(y_46_im * y_46_im)))
	tmp = 0.0
	if (y_46_im <= -2.25e+123)
		tmp = Float64(x_46_im * Float64(-1.0 / hypot(y_46_re, y_46_im)));
	elseif (y_46_im <= -2.35e-150)
		tmp = t_0;
	elseif (y_46_im <= 7e-140)
		tmp = Float64(Float64(x_46_re / y_46_re) + Float64(x_46_im / Float64(y_46_re * Float64(y_46_re / y_46_im))));
	elseif (y_46_im <= 1.4e+37)
		tmp = t_0;
	else
		tmp = Float64(Float64(x_46_im / y_46_im) + Float64(x_46_re / Float64((y_46_im ^ 2.0) / 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 = ((y_46_im * x_46_im) + (y_46_re * x_46_re)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	tmp = 0.0;
	if (y_46_im <= -2.25e+123)
		tmp = x_46_im * (-1.0 / hypot(y_46_re, y_46_im));
	elseif (y_46_im <= -2.35e-150)
		tmp = t_0;
	elseif (y_46_im <= 7e-140)
		tmp = (x_46_re / y_46_re) + (x_46_im / (y_46_re * (y_46_re / y_46_im)));
	elseif (y_46_im <= 1.4e+37)
		tmp = t_0;
	else
		tmp = (x_46_im / y_46_im) + (x_46_re / ((y_46_im ^ 2.0) / 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[(N[(y$46$im * x$46$im), $MachinePrecision] + N[(y$46$re * x$46$re), $MachinePrecision]), $MachinePrecision] / N[(N[(y$46$re * y$46$re), $MachinePrecision] + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$im, -2.25e+123], N[(x$46$im * N[(-1.0 / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, -2.35e-150], t$95$0, If[LessEqual[y$46$im, 7e-140], N[(N[(x$46$re / y$46$re), $MachinePrecision] + N[(x$46$im / N[(y$46$re * N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, 1.4e+37], t$95$0, N[(N[(x$46$im / y$46$im), $MachinePrecision] + N[(x$46$re / N[(N[Power[y$46$im, 2.0], $MachinePrecision] / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if y.im < -2.24999999999999991e123

   1. Initial program 32.9%

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

      1. *-un-lft-identity 32.9%

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

      2. +-commutative 32.9%

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

      3. fma-udef 32.9%

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

      4. add-sqr-sqrt 32.9%

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

      5. times-frac 32.9%

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

      6. fma-udef 32.9%

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

      7. +-commutative 32.9%

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

      8. hypot-def 32.9%

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

      9. fma-def 32.9%

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

      10. fma-udef 32.9%

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

      11. +-commutative 32.9%

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

      12. hypot-def 51.3%

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

   3. Applied egg-rr 51.3%

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

   4. Taylor expanded in y.im around -inf 78.2%

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

      1. neg-mul-1 78.2%

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

   6. Simplified 78.2%

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

   if -2.24999999999999991e123 < y.im < -2.3499999999999999e-150 or 6.9999999999999996e-140 < y.im < 1.3999999999999999e37

   1. Initial program 84.0%

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

   if -2.3499999999999999e-150 < y.im < 6.9999999999999996e-140

   1. Initial program 69.0%

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

   2. Taylor expanded in y.re around inf 84.4%

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

      1. associate-/l* 89.8%

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

   4. Simplified 89.8%

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

      1. pow2 89.8%

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

      2. *-un-lft-identity 89.8%

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

      3. times-frac 91.6%

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

   6. Applied egg-rr 91.6%

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

   if 1.3999999999999999e37 < y.im

   1. Initial program 48.4%

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

   2. Taylor expanded in y.re around 0 81.4%

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

      1. associate-/l* 81.7%

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

   4. Simplified 81.7%

      \[\leadsto \color{blue}{\frac{x.im}{y.im} + \frac{x.re}{\frac{{y.im}^{2}}{y.re}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 84.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -2.25 \cdot 10^{+123}:\\ \;\;\;\;x.im \cdot \frac{-1}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.im \leq -2.35 \cdot 10^{-150}:\\ \;\;\;\;\frac{y.im \cdot x.im + y.re \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.im \leq 7 \cdot 10^{-140}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{x.im}{y.re \cdot \frac{y.re}{y.im}}\\ \mathbf{elif}\;y.im \leq 1.4 \cdot 10^{+37}:\\ \;\;\;\;\frac{y.im \cdot x.im + y.re \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{x.re}{\frac{{y.im}^{2}}{y.re}}\\ \end{array} \]

Alternative 4: 79.8% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y.im \cdot x.im + y.re \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{if}\;y.im \leq -1.95 \cdot 10^{+123}:\\ \;\;\;\;x.im \cdot \frac{-1}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.im \leq -1.1 \cdot 10^{-144}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq 5.4 \cdot 10^{-139}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{x.im}{y.re \cdot \frac{y.re}{y.im}}\\ \mathbf{elif}\;y.im \leq 1.4 \cdot 10^{+37}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im + \frac{y.re \cdot x.re}{y.im}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0
         (/ (+ (* y.im x.im) (* y.re x.re)) (+ (* y.re y.re) (* y.im y.im)))))
   (if (<= y.im -1.95e+123)
     (* x.im (/ -1.0 (hypot y.re y.im)))
     (if (<= y.im -1.1e-144)
       t_0
       (if (<= y.im 5.4e-139)
         (+ (/ x.re y.re) (/ x.im (* y.re (/ y.re y.im))))
         (if (<= y.im 1.4e+37)
           t_0
           (/ (+ x.im (/ (* y.re x.re) y.im)) (hypot y.re y.im))))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = ((y_46_im * x_46_im) + (y_46_re * x_46_re)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	double tmp;
	if (y_46_im <= -1.95e+123) {
		tmp = x_46_im * (-1.0 / hypot(y_46_re, y_46_im));
	} else if (y_46_im <= -1.1e-144) {
		tmp = t_0;
	} else if (y_46_im <= 5.4e-139) {
		tmp = (x_46_re / y_46_re) + (x_46_im / (y_46_re * (y_46_re / y_46_im)));
	} else if (y_46_im <= 1.4e+37) {
		tmp = t_0;
	} else {
		tmp = (x_46_im + ((y_46_re * x_46_re) / y_46_im)) / hypot(y_46_re, y_46_im);
	}
	return tmp;
}

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = ((y_46_im * x_46_im) + (y_46_re * x_46_re)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	double tmp;
	if (y_46_im <= -1.95e+123) {
		tmp = x_46_im * (-1.0 / Math.hypot(y_46_re, y_46_im));
	} else if (y_46_im <= -1.1e-144) {
		tmp = t_0;
	} else if (y_46_im <= 5.4e-139) {
		tmp = (x_46_re / y_46_re) + (x_46_im / (y_46_re * (y_46_re / y_46_im)));
	} else if (y_46_im <= 1.4e+37) {
		tmp = t_0;
	} else {
		tmp = (x_46_im + ((y_46_re * x_46_re) / y_46_im)) / Math.hypot(y_46_re, y_46_im);
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = ((y_46_im * x_46_im) + (y_46_re * x_46_re)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im))
	tmp = 0
	if y_46_im <= -1.95e+123:
		tmp = x_46_im * (-1.0 / math.hypot(y_46_re, y_46_im))
	elif y_46_im <= -1.1e-144:
		tmp = t_0
	elif y_46_im <= 5.4e-139:
		tmp = (x_46_re / y_46_re) + (x_46_im / (y_46_re * (y_46_re / y_46_im)))
	elif y_46_im <= 1.4e+37:
		tmp = t_0
	else:
		tmp = (x_46_im + ((y_46_re * x_46_re) / y_46_im)) / math.hypot(y_46_re, y_46_im)
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(Float64(Float64(y_46_im * x_46_im) + Float64(y_46_re * x_46_re)) / Float64(Float64(y_46_re * y_46_re) + Float64(y_46_im * y_46_im)))
	tmp = 0.0
	if (y_46_im <= -1.95e+123)
		tmp = Float64(x_46_im * Float64(-1.0 / hypot(y_46_re, y_46_im)));
	elseif (y_46_im <= -1.1e-144)
		tmp = t_0;
	elseif (y_46_im <= 5.4e-139)
		tmp = Float64(Float64(x_46_re / y_46_re) + Float64(x_46_im / Float64(y_46_re * Float64(y_46_re / y_46_im))));
	elseif (y_46_im <= 1.4e+37)
		tmp = t_0;
	else
		tmp = Float64(Float64(x_46_im + Float64(Float64(y_46_re * x_46_re) / y_46_im)) / hypot(y_46_re, y_46_im));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = ((y_46_im * x_46_im) + (y_46_re * x_46_re)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	tmp = 0.0;
	if (y_46_im <= -1.95e+123)
		tmp = x_46_im * (-1.0 / hypot(y_46_re, y_46_im));
	elseif (y_46_im <= -1.1e-144)
		tmp = t_0;
	elseif (y_46_im <= 5.4e-139)
		tmp = (x_46_re / y_46_re) + (x_46_im / (y_46_re * (y_46_re / y_46_im)));
	elseif (y_46_im <= 1.4e+37)
		tmp = t_0;
	else
		tmp = (x_46_im + ((y_46_re * x_46_re) / y_46_im)) / hypot(y_46_re, y_46_im);
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[(N[(y$46$im * x$46$im), $MachinePrecision] + N[(y$46$re * x$46$re), $MachinePrecision]), $MachinePrecision] / N[(N[(y$46$re * y$46$re), $MachinePrecision] + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$im, -1.95e+123], N[(x$46$im * N[(-1.0 / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, -1.1e-144], t$95$0, If[LessEqual[y$46$im, 5.4e-139], N[(N[(x$46$re / y$46$re), $MachinePrecision] + N[(x$46$im / N[(y$46$re * N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, 1.4e+37], t$95$0, N[(N[(x$46$im + N[(N[(y$46$re * x$46$re), $MachinePrecision] / y$46$im), $MachinePrecision]), $MachinePrecision] / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if y.im < -1.94999999999999996e123

   1. Initial program 32.9%

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

      1. *-un-lft-identity 32.9%

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

      2. +-commutative 32.9%

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

      3. fma-udef 32.9%

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

      4. add-sqr-sqrt 32.9%

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

      5. times-frac 32.9%

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

      6. fma-udef 32.9%

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

      7. +-commutative 32.9%

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

      8. hypot-def 32.9%

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

      9. fma-def 32.9%

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

      10. fma-udef 32.9%

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

      11. +-commutative 32.9%

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

      12. hypot-def 51.3%

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

   3. Applied egg-rr 51.3%

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

   4. Taylor expanded in y.im around -inf 78.2%

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

      1. neg-mul-1 78.2%

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

   6. Simplified 78.2%

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

   if -1.94999999999999996e123 < y.im < -1.10000000000000003e-144 or 5.3999999999999997e-139 < y.im < 1.3999999999999999e37

   1. Initial program 84.0%

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

   if -1.10000000000000003e-144 < y.im < 5.3999999999999997e-139

   1. Initial program 69.0%

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

   2. Taylor expanded in y.re around inf 84.4%

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

      1. associate-/l* 89.8%

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

   4. Simplified 89.8%

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

      1. pow2 89.8%

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

      2. *-un-lft-identity 89.8%

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

      3. times-frac 91.6%

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

   6. Applied egg-rr 91.6%

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

   if 1.3999999999999999e37 < y.im

   1. Initial program 48.4%

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

      1. *-un-lft-identity 48.4%

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

      2. +-commutative 48.4%

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

      3. fma-udef 48.4%

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

      4. add-sqr-sqrt 48.4%

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

      5. times-frac 48.4%

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

      6. fma-udef 48.4%

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

      7. +-commutative 48.4%

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

      8. hypot-def 48.4%

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

      9. fma-def 48.4%

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

      10. fma-udef 48.4%

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

      11. +-commutative 48.4%

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

      12. hypot-def 63.5%

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

   3. Applied egg-rr 63.5%

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

      1. associate-*l/ 63.7%

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

      2. *-un-lft-identity 63.7%

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

   5. Applied egg-rr 63.7%

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

   6. Taylor expanded in y.re around 0 85.5%

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

4. Final simplification 85.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -1.95 \cdot 10^{+123}:\\ \;\;\;\;x.im \cdot \frac{-1}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.im \leq -1.1 \cdot 10^{-144}:\\ \;\;\;\;\frac{y.im \cdot x.im + y.re \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.im \leq 5.4 \cdot 10^{-139}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{x.im}{y.re \cdot \frac{y.re}{y.im}}\\ \mathbf{elif}\;y.im \leq 1.4 \cdot 10^{+37}:\\ \;\;\;\;\frac{y.im \cdot x.im + y.re \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im + \frac{y.re \cdot x.re}{y.im}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \end{array} \]

Alternative 5: 81.3% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y.im \cdot x.im + y.re \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{if}\;y.im \leq -5.1 \cdot 10^{+67}:\\ \;\;\;\;\frac{\frac{-x.re}{\frac{y.im}{y.re}} - x.im}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.im \leq -7 \cdot 10^{-148}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq 4.8 \cdot 10^{-138}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{x.im}{y.re \cdot \frac{y.re}{y.im}}\\ \mathbf{elif}\;y.im \leq 1.4 \cdot 10^{+37}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im + \frac{y.re \cdot x.re}{y.im}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0
         (/ (+ (* y.im x.im) (* y.re x.re)) (+ (* y.re y.re) (* y.im y.im)))))
   (if (<= y.im -5.1e+67)
     (/ (- (/ (- x.re) (/ y.im y.re)) x.im) (hypot y.re y.im))
     (if (<= y.im -7e-148)
       t_0
       (if (<= y.im 4.8e-138)
         (+ (/ x.re y.re) (/ x.im (* y.re (/ y.re y.im))))
         (if (<= y.im 1.4e+37)
           t_0
           (/ (+ x.im (/ (* y.re x.re) y.im)) (hypot y.re y.im))))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = ((y_46_im * x_46_im) + (y_46_re * x_46_re)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	double tmp;
	if (y_46_im <= -5.1e+67) {
		tmp = ((-x_46_re / (y_46_im / y_46_re)) - x_46_im) / hypot(y_46_re, y_46_im);
	} else if (y_46_im <= -7e-148) {
		tmp = t_0;
	} else if (y_46_im <= 4.8e-138) {
		tmp = (x_46_re / y_46_re) + (x_46_im / (y_46_re * (y_46_re / y_46_im)));
	} else if (y_46_im <= 1.4e+37) {
		tmp = t_0;
	} else {
		tmp = (x_46_im + ((y_46_re * x_46_re) / y_46_im)) / hypot(y_46_re, y_46_im);
	}
	return tmp;
}

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = ((y_46_im * x_46_im) + (y_46_re * x_46_re)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	double tmp;
	if (y_46_im <= -5.1e+67) {
		tmp = ((-x_46_re / (y_46_im / y_46_re)) - x_46_im) / Math.hypot(y_46_re, y_46_im);
	} else if (y_46_im <= -7e-148) {
		tmp = t_0;
	} else if (y_46_im <= 4.8e-138) {
		tmp = (x_46_re / y_46_re) + (x_46_im / (y_46_re * (y_46_re / y_46_im)));
	} else if (y_46_im <= 1.4e+37) {
		tmp = t_0;
	} else {
		tmp = (x_46_im + ((y_46_re * x_46_re) / y_46_im)) / Math.hypot(y_46_re, y_46_im);
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = ((y_46_im * x_46_im) + (y_46_re * x_46_re)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im))
	tmp = 0
	if y_46_im <= -5.1e+67:
		tmp = ((-x_46_re / (y_46_im / y_46_re)) - x_46_im) / math.hypot(y_46_re, y_46_im)
	elif y_46_im <= -7e-148:
		tmp = t_0
	elif y_46_im <= 4.8e-138:
		tmp = (x_46_re / y_46_re) + (x_46_im / (y_46_re * (y_46_re / y_46_im)))
	elif y_46_im <= 1.4e+37:
		tmp = t_0
	else:
		tmp = (x_46_im + ((y_46_re * x_46_re) / y_46_im)) / math.hypot(y_46_re, y_46_im)
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(Float64(Float64(y_46_im * x_46_im) + Float64(y_46_re * x_46_re)) / Float64(Float64(y_46_re * y_46_re) + Float64(y_46_im * y_46_im)))
	tmp = 0.0
	if (y_46_im <= -5.1e+67)
		tmp = Float64(Float64(Float64(Float64(-x_46_re) / Float64(y_46_im / y_46_re)) - x_46_im) / hypot(y_46_re, y_46_im));
	elseif (y_46_im <= -7e-148)
		tmp = t_0;
	elseif (y_46_im <= 4.8e-138)
		tmp = Float64(Float64(x_46_re / y_46_re) + Float64(x_46_im / Float64(y_46_re * Float64(y_46_re / y_46_im))));
	elseif (y_46_im <= 1.4e+37)
		tmp = t_0;
	else
		tmp = Float64(Float64(x_46_im + Float64(Float64(y_46_re * x_46_re) / y_46_im)) / hypot(y_46_re, y_46_im));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = ((y_46_im * x_46_im) + (y_46_re * x_46_re)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	tmp = 0.0;
	if (y_46_im <= -5.1e+67)
		tmp = ((-x_46_re / (y_46_im / y_46_re)) - x_46_im) / hypot(y_46_re, y_46_im);
	elseif (y_46_im <= -7e-148)
		tmp = t_0;
	elseif (y_46_im <= 4.8e-138)
		tmp = (x_46_re / y_46_re) + (x_46_im / (y_46_re * (y_46_re / y_46_im)));
	elseif (y_46_im <= 1.4e+37)
		tmp = t_0;
	else
		tmp = (x_46_im + ((y_46_re * x_46_re) / y_46_im)) / hypot(y_46_re, y_46_im);
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[(N[(y$46$im * x$46$im), $MachinePrecision] + N[(y$46$re * x$46$re), $MachinePrecision]), $MachinePrecision] / N[(N[(y$46$re * y$46$re), $MachinePrecision] + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$im, -5.1e+67], N[(N[(N[((-x$46$re) / N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision] - x$46$im), $MachinePrecision] / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, -7e-148], t$95$0, If[LessEqual[y$46$im, 4.8e-138], N[(N[(x$46$re / y$46$re), $MachinePrecision] + N[(x$46$im / N[(y$46$re * N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, 1.4e+37], t$95$0, N[(N[(x$46$im + N[(N[(y$46$re * x$46$re), $MachinePrecision] / y$46$im), $MachinePrecision]), $MachinePrecision] / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if y.im < -5.1000000000000002e67

   1. Initial program 39.8%

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

      1. *-un-lft-identity 39.8%

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

      2. +-commutative 39.8%

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

      3. fma-udef 39.8%

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

      4. add-sqr-sqrt 39.8%

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

      5. times-frac 39.8%

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

      6. fma-udef 39.8%

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

      7. +-commutative 39.8%

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

      8. hypot-def 39.8%

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

      9. fma-def 39.8%

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

      10. fma-udef 39.8%

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

      11. +-commutative 39.8%

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

      12. hypot-def 57.8%

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

   3. Applied egg-rr 57.8%

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

      1. associate-*l/ 57.9%

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

      2. *-un-lft-identity 57.9%

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

   5. Applied egg-rr 57.9%

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

   6. Taylor expanded in y.im around -inf 82.6%

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

      1. neg-mul-1 82.6%

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

      2. +-commutative 82.6%

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

      3. unsub-neg 82.6%

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

      4. mul-1-neg 82.6%

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

      5. associate-/l* 87.5%

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

      6. distribute-neg-frac 87.5%

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

   8. Simplified 87.5%

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

   if -5.1000000000000002e67 < y.im < -7.0000000000000001e-148 or 4.7999999999999998e-138 < y.im < 1.3999999999999999e37

   1. Initial program 84.0%

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

   if -7.0000000000000001e-148 < y.im < 4.7999999999999998e-138

   1. Initial program 69.0%

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

   2. Taylor expanded in y.re around inf 84.4%

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

      1. associate-/l* 89.8%

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

   4. Simplified 89.8%

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

      1. pow2 89.8%

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

      2. *-un-lft-identity 89.8%

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

      3. times-frac 91.6%

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

   6. Applied egg-rr 91.6%

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

   if 1.3999999999999999e37 < y.im

   1. Initial program 48.4%

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

      1. *-un-lft-identity 48.4%

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

      2. +-commutative 48.4%

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

      3. fma-udef 48.4%

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

      4. add-sqr-sqrt 48.4%

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

      5. times-frac 48.4%

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

      6. fma-udef 48.4%

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

      7. +-commutative 48.4%

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

      8. hypot-def 48.4%

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

      9. fma-def 48.4%

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

      10. fma-udef 48.4%

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

      11. +-commutative 48.4%

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

      12. hypot-def 63.5%

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

   3. Applied egg-rr 63.5%

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

      1. associate-*l/ 63.7%

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

      2. *-un-lft-identity 63.7%

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

   5. Applied egg-rr 63.7%

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

   6. Taylor expanded in y.re around 0 85.5%

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

4. Final simplification 86.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -5.1 \cdot 10^{+67}:\\ \;\;\;\;\frac{\frac{-x.re}{\frac{y.im}{y.re}} - x.im}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.im \leq -7 \cdot 10^{-148}:\\ \;\;\;\;\frac{y.im \cdot x.im + y.re \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.im \leq 4.8 \cdot 10^{-138}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{x.im}{y.re \cdot \frac{y.re}{y.im}}\\ \mathbf{elif}\;y.im \leq 1.4 \cdot 10^{+37}:\\ \;\;\;\;\frac{y.im \cdot x.im + y.re \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im + \frac{y.re \cdot x.re}{y.im}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \end{array} \]

Alternative 6: 81.4% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y.im \cdot x.im + y.re \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{if}\;y.im \leq -3 \cdot 10^{+62}:\\ \;\;\;\;\frac{\left(-x.im\right) - y.re \cdot \frac{x.re}{y.im}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.im \leq -3.8 \cdot 10^{-155}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq 10^{-138}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{x.im}{y.re \cdot \frac{y.re}{y.im}}\\ \mathbf{elif}\;y.im \leq 6.4 \cdot 10^{+34}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im + \frac{y.re \cdot x.re}{y.im}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0
         (/ (+ (* y.im x.im) (* y.re x.re)) (+ (* y.re y.re) (* y.im y.im)))))
   (if (<= y.im -3e+62)
     (/ (- (- x.im) (* y.re (/ x.re y.im))) (hypot y.re y.im))
     (if (<= y.im -3.8e-155)
       t_0
       (if (<= y.im 1e-138)
         (+ (/ x.re y.re) (/ x.im (* y.re (/ y.re y.im))))
         (if (<= y.im 6.4e+34)
           t_0
           (/ (+ x.im (/ (* y.re x.re) y.im)) (hypot y.re y.im))))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = ((y_46_im * x_46_im) + (y_46_re * x_46_re)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	double tmp;
	if (y_46_im <= -3e+62) {
		tmp = (-x_46_im - (y_46_re * (x_46_re / y_46_im))) / hypot(y_46_re, y_46_im);
	} else if (y_46_im <= -3.8e-155) {
		tmp = t_0;
	} else if (y_46_im <= 1e-138) {
		tmp = (x_46_re / y_46_re) + (x_46_im / (y_46_re * (y_46_re / y_46_im)));
	} else if (y_46_im <= 6.4e+34) {
		tmp = t_0;
	} else {
		tmp = (x_46_im + ((y_46_re * x_46_re) / y_46_im)) / hypot(y_46_re, y_46_im);
	}
	return tmp;
}

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = ((y_46_im * x_46_im) + (y_46_re * x_46_re)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	double tmp;
	if (y_46_im <= -3e+62) {
		tmp = (-x_46_im - (y_46_re * (x_46_re / y_46_im))) / Math.hypot(y_46_re, y_46_im);
	} else if (y_46_im <= -3.8e-155) {
		tmp = t_0;
	} else if (y_46_im <= 1e-138) {
		tmp = (x_46_re / y_46_re) + (x_46_im / (y_46_re * (y_46_re / y_46_im)));
	} else if (y_46_im <= 6.4e+34) {
		tmp = t_0;
	} else {
		tmp = (x_46_im + ((y_46_re * x_46_re) / y_46_im)) / Math.hypot(y_46_re, y_46_im);
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = ((y_46_im * x_46_im) + (y_46_re * x_46_re)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im))
	tmp = 0
	if y_46_im <= -3e+62:
		tmp = (-x_46_im - (y_46_re * (x_46_re / y_46_im))) / math.hypot(y_46_re, y_46_im)
	elif y_46_im <= -3.8e-155:
		tmp = t_0
	elif y_46_im <= 1e-138:
		tmp = (x_46_re / y_46_re) + (x_46_im / (y_46_re * (y_46_re / y_46_im)))
	elif y_46_im <= 6.4e+34:
		tmp = t_0
	else:
		tmp = (x_46_im + ((y_46_re * x_46_re) / y_46_im)) / math.hypot(y_46_re, y_46_im)
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(Float64(Float64(y_46_im * x_46_im) + Float64(y_46_re * x_46_re)) / Float64(Float64(y_46_re * y_46_re) + Float64(y_46_im * y_46_im)))
	tmp = 0.0
	if (y_46_im <= -3e+62)
		tmp = Float64(Float64(Float64(-x_46_im) - Float64(y_46_re * Float64(x_46_re / y_46_im))) / hypot(y_46_re, y_46_im));
	elseif (y_46_im <= -3.8e-155)
		tmp = t_0;
	elseif (y_46_im <= 1e-138)
		tmp = Float64(Float64(x_46_re / y_46_re) + Float64(x_46_im / Float64(y_46_re * Float64(y_46_re / y_46_im))));
	elseif (y_46_im <= 6.4e+34)
		tmp = t_0;
	else
		tmp = Float64(Float64(x_46_im + Float64(Float64(y_46_re * x_46_re) / y_46_im)) / hypot(y_46_re, y_46_im));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = ((y_46_im * x_46_im) + (y_46_re * x_46_re)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	tmp = 0.0;
	if (y_46_im <= -3e+62)
		tmp = (-x_46_im - (y_46_re * (x_46_re / y_46_im))) / hypot(y_46_re, y_46_im);
	elseif (y_46_im <= -3.8e-155)
		tmp = t_0;
	elseif (y_46_im <= 1e-138)
		tmp = (x_46_re / y_46_re) + (x_46_im / (y_46_re * (y_46_re / y_46_im)));
	elseif (y_46_im <= 6.4e+34)
		tmp = t_0;
	else
		tmp = (x_46_im + ((y_46_re * x_46_re) / y_46_im)) / hypot(y_46_re, y_46_im);
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[(N[(y$46$im * x$46$im), $MachinePrecision] + N[(y$46$re * x$46$re), $MachinePrecision]), $MachinePrecision] / N[(N[(y$46$re * y$46$re), $MachinePrecision] + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$im, -3e+62], N[(N[((-x$46$im) - N[(y$46$re * N[(x$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, -3.8e-155], t$95$0, If[LessEqual[y$46$im, 1e-138], N[(N[(x$46$re / y$46$re), $MachinePrecision] + N[(x$46$im / N[(y$46$re * N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, 6.4e+34], t$95$0, N[(N[(x$46$im + N[(N[(y$46$re * x$46$re), $MachinePrecision] / y$46$im), $MachinePrecision]), $MachinePrecision] / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if y.im < -3e62

   1. Initial program 39.8%

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

      1. *-un-lft-identity 39.8%

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

      2. +-commutative 39.8%

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

      3. fma-udef 39.8%

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

      4. add-sqr-sqrt 39.8%

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

      5. times-frac 39.8%

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

      6. fma-udef 39.8%

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

      7. +-commutative 39.8%

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

      8. hypot-def 39.8%

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

      9. fma-def 39.8%

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

      10. fma-udef 39.8%

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

      11. +-commutative 39.8%

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

      12. hypot-def 57.8%

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

   3. Applied egg-rr 57.8%

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

      1. associate-*l/ 57.9%

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

      2. *-un-lft-identity 57.9%

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

   5. Applied egg-rr 57.9%

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

      1. clear-num 57.9%

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

      2. inv-pow 57.9%

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

   7. Applied egg-rr 57.9%

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

      1. unpow-1 57.9%

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

      2. fma-udef 57.9%

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

      3. +-commutative 57.9%

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

      4. fma-def 57.9%

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

      5. *-commutative 57.9%

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

   9. Simplified 57.9%

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

   10. Taylor expanded in y.im around -inf 82.6%

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

       1. distribute-lft-out 82.6%

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

       2. *-commutative 82.6%

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

       3. associate-*r/ 89.5%

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

   12. Simplified 89.5%

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

   if -3e62 < y.im < -3.7999999999999998e-155 or 1.00000000000000007e-138 < y.im < 6.3999999999999997e34

   1. Initial program 84.0%

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

   if -3.7999999999999998e-155 < y.im < 1.00000000000000007e-138

   1. Initial program 69.0%

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

   2. Taylor expanded in y.re around inf 84.4%

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

      1. associate-/l* 89.8%

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

   4. Simplified 89.8%

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

      1. pow2 89.8%

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

      2. *-un-lft-identity 89.8%

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

      3. times-frac 91.6%

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

   6. Applied egg-rr 91.6%

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

   if 6.3999999999999997e34 < y.im

   1. Initial program 48.4%

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

      1. *-un-lft-identity 48.4%

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

      2. +-commutative 48.4%

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

      3. fma-udef 48.4%

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

      4. add-sqr-sqrt 48.4%

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

      5. times-frac 48.4%

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

      6. fma-udef 48.4%

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

      7. +-commutative 48.4%

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

      8. hypot-def 48.4%

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

      9. fma-def 48.4%

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

      10. fma-udef 48.4%

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

      11. +-commutative 48.4%

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

      12. hypot-def 63.5%

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

   3. Applied egg-rr 63.5%

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

      1. associate-*l/ 63.7%

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

      2. *-un-lft-identity 63.7%

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

   5. Applied egg-rr 63.7%

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

   6. Taylor expanded in y.re around 0 85.5%

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

4. Final simplification 86.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -3 \cdot 10^{+62}:\\ \;\;\;\;\frac{\left(-x.im\right) - y.re \cdot \frac{x.re}{y.im}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.im \leq -3.8 \cdot 10^{-155}:\\ \;\;\;\;\frac{y.im \cdot x.im + y.re \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.im \leq 10^{-138}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{x.im}{y.re \cdot \frac{y.re}{y.im}}\\ \mathbf{elif}\;y.im \leq 6.4 \cdot 10^{+34}:\\ \;\;\;\;\frac{y.im \cdot x.im + y.re \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im + \frac{y.re \cdot x.re}{y.im}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \end{array} \]

Alternative 7: 79.1% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y.im \cdot x.im + y.re \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{if}\;y.im \leq -1.9 \cdot 10^{+123}:\\ \;\;\;\;x.im \cdot \frac{-1}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.im \leq -3.5 \cdot 10^{-148}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq 1.6 \cdot 10^{-139}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{x.im}{y.re \cdot \frac{y.re}{y.im}}\\ \mathbf{elif}\;y.im \leq 3.2 \cdot 10^{+134}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0
         (/ (+ (* y.im x.im) (* y.re x.re)) (+ (* y.re y.re) (* y.im y.im)))))
   (if (<= y.im -1.9e+123)
     (* x.im (/ -1.0 (hypot y.re y.im)))
     (if (<= y.im -3.5e-148)
       t_0
       (if (<= y.im 1.6e-139)
         (+ (/ x.re y.re) (/ x.im (* y.re (/ y.re y.im))))
         (if (<= y.im 3.2e+134) t_0 (/ x.im y.im)))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = ((y_46_im * x_46_im) + (y_46_re * x_46_re)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	double tmp;
	if (y_46_im <= -1.9e+123) {
		tmp = x_46_im * (-1.0 / hypot(y_46_re, y_46_im));
	} else if (y_46_im <= -3.5e-148) {
		tmp = t_0;
	} else if (y_46_im <= 1.6e-139) {
		tmp = (x_46_re / y_46_re) + (x_46_im / (y_46_re * (y_46_re / y_46_im)));
	} else if (y_46_im <= 3.2e+134) {
		tmp = t_0;
	} else {
		tmp = x_46_im / y_46_im;
	}
	return tmp;
}

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = ((y_46_im * x_46_im) + (y_46_re * x_46_re)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	double tmp;
	if (y_46_im <= -1.9e+123) {
		tmp = x_46_im * (-1.0 / Math.hypot(y_46_re, y_46_im));
	} else if (y_46_im <= -3.5e-148) {
		tmp = t_0;
	} else if (y_46_im <= 1.6e-139) {
		tmp = (x_46_re / y_46_re) + (x_46_im / (y_46_re * (y_46_re / y_46_im)));
	} else if (y_46_im <= 3.2e+134) {
		tmp = t_0;
	} 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):
	t_0 = ((y_46_im * x_46_im) + (y_46_re * x_46_re)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im))
	tmp = 0
	if y_46_im <= -1.9e+123:
		tmp = x_46_im * (-1.0 / math.hypot(y_46_re, y_46_im))
	elif y_46_im <= -3.5e-148:
		tmp = t_0
	elif y_46_im <= 1.6e-139:
		tmp = (x_46_re / y_46_re) + (x_46_im / (y_46_re * (y_46_re / y_46_im)))
	elif y_46_im <= 3.2e+134:
		tmp = t_0
	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)
	t_0 = Float64(Float64(Float64(y_46_im * x_46_im) + Float64(y_46_re * x_46_re)) / Float64(Float64(y_46_re * y_46_re) + Float64(y_46_im * y_46_im)))
	tmp = 0.0
	if (y_46_im <= -1.9e+123)
		tmp = Float64(x_46_im * Float64(-1.0 / hypot(y_46_re, y_46_im)));
	elseif (y_46_im <= -3.5e-148)
		tmp = t_0;
	elseif (y_46_im <= 1.6e-139)
		tmp = Float64(Float64(x_46_re / y_46_re) + Float64(x_46_im / Float64(y_46_re * Float64(y_46_re / y_46_im))));
	elseif (y_46_im <= 3.2e+134)
		tmp = t_0;
	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)
	t_0 = ((y_46_im * x_46_im) + (y_46_re * x_46_re)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	tmp = 0.0;
	if (y_46_im <= -1.9e+123)
		tmp = x_46_im * (-1.0 / hypot(y_46_re, y_46_im));
	elseif (y_46_im <= -3.5e-148)
		tmp = t_0;
	elseif (y_46_im <= 1.6e-139)
		tmp = (x_46_re / y_46_re) + (x_46_im / (y_46_re * (y_46_re / y_46_im)));
	elseif (y_46_im <= 3.2e+134)
		tmp = t_0;
	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_] := Block[{t$95$0 = N[(N[(N[(y$46$im * x$46$im), $MachinePrecision] + N[(y$46$re * x$46$re), $MachinePrecision]), $MachinePrecision] / N[(N[(y$46$re * y$46$re), $MachinePrecision] + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$im, -1.9e+123], N[(x$46$im * N[(-1.0 / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, -3.5e-148], t$95$0, If[LessEqual[y$46$im, 1.6e-139], N[(N[(x$46$re / y$46$re), $MachinePrecision] + N[(x$46$im / N[(y$46$re * N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, 3.2e+134], t$95$0, N[(x$46$im / y$46$im), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if y.im < -1.89999999999999997e123

   1. Initial program 32.9%

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

      1. *-un-lft-identity 32.9%

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

      2. +-commutative 32.9%

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

      3. fma-udef 32.9%

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

      4. add-sqr-sqrt 32.9%

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

      5. times-frac 32.9%

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

      6. fma-udef 32.9%

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

      7. +-commutative 32.9%

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

      8. hypot-def 32.9%

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

      9. fma-def 32.9%

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

      10. fma-udef 32.9%

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

      11. +-commutative 32.9%

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

      12. hypot-def 51.3%

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

   3. Applied egg-rr 51.3%

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

   4. Taylor expanded in y.im around -inf 78.2%

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

      1. neg-mul-1 78.2%

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

   6. Simplified 78.2%

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

   if -1.89999999999999997e123 < y.im < -3.5e-148 or 1.6e-139 < y.im < 3.2000000000000001e134

   1. Initial program 82.5%

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

   if -3.5e-148 < y.im < 1.6e-139

   1. Initial program 69.0%

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

   2. Taylor expanded in y.re around inf 84.4%

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

      1. associate-/l* 89.8%

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

   4. Simplified 89.8%

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

      1. pow2 89.8%

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

      2. *-un-lft-identity 89.8%

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

      3. times-frac 91.6%

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

   6. Applied egg-rr 91.6%

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

   if 3.2000000000000001e134 < y.im

   1. Initial program 41.1%

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

   2. Taylor expanded in y.re around 0 85.5%

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

4. Final simplification 84.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -1.9 \cdot 10^{+123}:\\ \;\;\;\;x.im \cdot \frac{-1}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.im \leq -3.5 \cdot 10^{-148}:\\ \;\;\;\;\frac{y.im \cdot x.im + y.re \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.im \leq 1.6 \cdot 10^{-139}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{x.im}{y.re \cdot \frac{y.re}{y.im}}\\ \mathbf{elif}\;y.im \leq 3.2 \cdot 10^{+134}:\\ \;\;\;\;\frac{y.im \cdot x.im + y.re \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \end{array} \]

Alternative 8: 79.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y.im \cdot x.im + y.re \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{if}\;y.im \leq -1.16 \cdot 10^{+123}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{elif}\;y.im \leq -5.8 \cdot 10^{-145}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq 1.35 \cdot 10^{-139}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{x.im}{y.re \cdot \frac{y.re}{y.im}}\\ \mathbf{elif}\;y.im \leq 9.5 \cdot 10^{+134}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0
         (/ (+ (* y.im x.im) (* y.re x.re)) (+ (* y.re y.re) (* y.im y.im)))))
   (if (<= y.im -1.16e+123)
     (/ x.im y.im)
     (if (<= y.im -5.8e-145)
       t_0
       (if (<= y.im 1.35e-139)
         (+ (/ x.re y.re) (/ x.im (* y.re (/ y.re y.im))))
         (if (<= y.im 9.5e+134) t_0 (/ x.im y.im)))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = ((y_46_im * x_46_im) + (y_46_re * x_46_re)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	double tmp;
	if (y_46_im <= -1.16e+123) {
		tmp = x_46_im / y_46_im;
	} else if (y_46_im <= -5.8e-145) {
		tmp = t_0;
	} else if (y_46_im <= 1.35e-139) {
		tmp = (x_46_re / y_46_re) + (x_46_im / (y_46_re * (y_46_re / y_46_im)));
	} else if (y_46_im <= 9.5e+134) {
		tmp = t_0;
	} 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) :: t_0
    real(8) :: tmp
    t_0 = ((y_46im * x_46im) + (y_46re * x_46re)) / ((y_46re * y_46re) + (y_46im * y_46im))
    if (y_46im <= (-1.16d+123)) then
        tmp = x_46im / y_46im
    else if (y_46im <= (-5.8d-145)) then
        tmp = t_0
    else if (y_46im <= 1.35d-139) then
        tmp = (x_46re / y_46re) + (x_46im / (y_46re * (y_46re / y_46im)))
    else if (y_46im <= 9.5d+134) then
        tmp = t_0
    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 t_0 = ((y_46_im * x_46_im) + (y_46_re * x_46_re)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	double tmp;
	if (y_46_im <= -1.16e+123) {
		tmp = x_46_im / y_46_im;
	} else if (y_46_im <= -5.8e-145) {
		tmp = t_0;
	} else if (y_46_im <= 1.35e-139) {
		tmp = (x_46_re / y_46_re) + (x_46_im / (y_46_re * (y_46_re / y_46_im)));
	} else if (y_46_im <= 9.5e+134) {
		tmp = t_0;
	} 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):
	t_0 = ((y_46_im * x_46_im) + (y_46_re * x_46_re)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im))
	tmp = 0
	if y_46_im <= -1.16e+123:
		tmp = x_46_im / y_46_im
	elif y_46_im <= -5.8e-145:
		tmp = t_0
	elif y_46_im <= 1.35e-139:
		tmp = (x_46_re / y_46_re) + (x_46_im / (y_46_re * (y_46_re / y_46_im)))
	elif y_46_im <= 9.5e+134:
		tmp = t_0
	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)
	t_0 = Float64(Float64(Float64(y_46_im * x_46_im) + Float64(y_46_re * x_46_re)) / Float64(Float64(y_46_re * y_46_re) + Float64(y_46_im * y_46_im)))
	tmp = 0.0
	if (y_46_im <= -1.16e+123)
		tmp = Float64(x_46_im / y_46_im);
	elseif (y_46_im <= -5.8e-145)
		tmp = t_0;
	elseif (y_46_im <= 1.35e-139)
		tmp = Float64(Float64(x_46_re / y_46_re) + Float64(x_46_im / Float64(y_46_re * Float64(y_46_re / y_46_im))));
	elseif (y_46_im <= 9.5e+134)
		tmp = t_0;
	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)
	t_0 = ((y_46_im * x_46_im) + (y_46_re * x_46_re)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	tmp = 0.0;
	if (y_46_im <= -1.16e+123)
		tmp = x_46_im / y_46_im;
	elseif (y_46_im <= -5.8e-145)
		tmp = t_0;
	elseif (y_46_im <= 1.35e-139)
		tmp = (x_46_re / y_46_re) + (x_46_im / (y_46_re * (y_46_re / y_46_im)));
	elseif (y_46_im <= 9.5e+134)
		tmp = t_0;
	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_] := Block[{t$95$0 = N[(N[(N[(y$46$im * x$46$im), $MachinePrecision] + N[(y$46$re * x$46$re), $MachinePrecision]), $MachinePrecision] / N[(N[(y$46$re * y$46$re), $MachinePrecision] + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$im, -1.16e+123], N[(x$46$im / y$46$im), $MachinePrecision], If[LessEqual[y$46$im, -5.8e-145], t$95$0, If[LessEqual[y$46$im, 1.35e-139], N[(N[(x$46$re / y$46$re), $MachinePrecision] + N[(x$46$im / N[(y$46$re * N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, 9.5e+134], t$95$0, N[(x$46$im / y$46$im), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if y.im < -1.16e123 or 9.5000000000000004e134 < y.im

   1. Initial program 37.0%

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

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

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

   if -1.16e123 < y.im < -5.79999999999999968e-145 or 1.3499999999999999e-139 < y.im < 9.5000000000000004e134

   1. Initial program 82.5%

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

   if -5.79999999999999968e-145 < y.im < 1.3499999999999999e-139

   1. Initial program 69.0%

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

   2. Taylor expanded in y.re around inf 84.4%

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

      1. associate-/l* 89.8%

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

   4. Simplified 89.8%

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

      1. pow2 89.8%

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

      2. *-un-lft-identity 89.8%

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

      3. times-frac 91.6%

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

   6. Applied egg-rr 91.6%

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

4. Final simplification 84.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -1.16 \cdot 10^{+123}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{elif}\;y.im \leq -5.8 \cdot 10^{-145}:\\ \;\;\;\;\frac{y.im \cdot x.im + y.re \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.im \leq 1.35 \cdot 10^{-139}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{x.im}{y.re \cdot \frac{y.re}{y.im}}\\ \mathbf{elif}\;y.im \leq 9.5 \cdot 10^{+134}:\\ \;\;\;\;\frac{y.im \cdot x.im + y.re \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \end{array} \]

Alternative 9: 62.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y.im \cdot x.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{if}\;y.im \leq -1.85 \cdot 10^{+67}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{elif}\;y.im \leq -2.4 \cdot 10^{-232}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq 6 \cdot 10^{-82}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{elif}\;y.im \leq 1.1 \cdot 10^{+147}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (/ (* y.im x.im) (+ (* y.re y.re) (* y.im y.im)))))
   (if (<= y.im -1.85e+67)
     (/ x.im y.im)
     (if (<= y.im -2.4e-232)
       t_0
       (if (<= y.im 6e-82)
         (/ x.re y.re)
         (if (<= y.im 1.1e+147) t_0 (/ x.im y.im)))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = (y_46_im * x_46_im) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	double tmp;
	if (y_46_im <= -1.85e+67) {
		tmp = x_46_im / y_46_im;
	} else if (y_46_im <= -2.4e-232) {
		tmp = t_0;
	} else if (y_46_im <= 6e-82) {
		tmp = x_46_re / y_46_re;
	} else if (y_46_im <= 1.1e+147) {
		tmp = t_0;
	} 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) :: t_0
    real(8) :: tmp
    t_0 = (y_46im * x_46im) / ((y_46re * y_46re) + (y_46im * y_46im))
    if (y_46im <= (-1.85d+67)) then
        tmp = x_46im / y_46im
    else if (y_46im <= (-2.4d-232)) then
        tmp = t_0
    else if (y_46im <= 6d-82) then
        tmp = x_46re / y_46re
    else if (y_46im <= 1.1d+147) then
        tmp = t_0
    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 t_0 = (y_46_im * x_46_im) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	double tmp;
	if (y_46_im <= -1.85e+67) {
		tmp = x_46_im / y_46_im;
	} else if (y_46_im <= -2.4e-232) {
		tmp = t_0;
	} else if (y_46_im <= 6e-82) {
		tmp = x_46_re / y_46_re;
	} else if (y_46_im <= 1.1e+147) {
		tmp = t_0;
	} 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):
	t_0 = (y_46_im * x_46_im) / ((y_46_re * y_46_re) + (y_46_im * y_46_im))
	tmp = 0
	if y_46_im <= -1.85e+67:
		tmp = x_46_im / y_46_im
	elif y_46_im <= -2.4e-232:
		tmp = t_0
	elif y_46_im <= 6e-82:
		tmp = x_46_re / y_46_re
	elif y_46_im <= 1.1e+147:
		tmp = t_0
	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)
	t_0 = Float64(Float64(y_46_im * x_46_im) / Float64(Float64(y_46_re * y_46_re) + Float64(y_46_im * y_46_im)))
	tmp = 0.0
	if (y_46_im <= -1.85e+67)
		tmp = Float64(x_46_im / y_46_im);
	elseif (y_46_im <= -2.4e-232)
		tmp = t_0;
	elseif (y_46_im <= 6e-82)
		tmp = Float64(x_46_re / y_46_re);
	elseif (y_46_im <= 1.1e+147)
		tmp = t_0;
	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)
	t_0 = (y_46_im * x_46_im) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	tmp = 0.0;
	if (y_46_im <= -1.85e+67)
		tmp = x_46_im / y_46_im;
	elseif (y_46_im <= -2.4e-232)
		tmp = t_0;
	elseif (y_46_im <= 6e-82)
		tmp = x_46_re / y_46_re;
	elseif (y_46_im <= 1.1e+147)
		tmp = t_0;
	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_] := Block[{t$95$0 = N[(N[(y$46$im * x$46$im), $MachinePrecision] / N[(N[(y$46$re * y$46$re), $MachinePrecision] + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$im, -1.85e+67], N[(x$46$im / y$46$im), $MachinePrecision], If[LessEqual[y$46$im, -2.4e-232], t$95$0, If[LessEqual[y$46$im, 6e-82], N[(x$46$re / y$46$re), $MachinePrecision], If[LessEqual[y$46$im, 1.1e+147], t$95$0, N[(x$46$im / y$46$im), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if y.im < -1.8499999999999999e67 or 1.1000000000000001e147 < y.im

   1. Initial program 38.6%

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

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

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

   if -1.8499999999999999e67 < y.im < -2.39999999999999999e-232 or 5.9999999999999998e-82 < y.im < 1.1000000000000001e147

   1. Initial program 82.9%

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

   2. Taylor expanded in x.re around 0 54.9%

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

   if -2.39999999999999999e-232 < y.im < 5.9999999999999998e-82

   1. Initial program 68.6%

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

   2. Taylor expanded in y.re around inf 77.2%

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

4. Final simplification 68.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -1.85 \cdot 10^{+67}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{elif}\;y.im \leq -2.4 \cdot 10^{-232}:\\ \;\;\;\;\frac{y.im \cdot x.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.im \leq 6 \cdot 10^{-82}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{elif}\;y.im \leq 1.1 \cdot 10^{+147}:\\ \;\;\;\;\frac{y.im \cdot x.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \end{array} \]

Alternative 10: 71.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -29000000000 \lor \neg \left(y.im \leq 950000\right):\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{x.im}{y.re \cdot \frac{y.re}{y.im}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.im -29000000000.0) (not (<= y.im 950000.0)))
   (/ x.im y.im)
   (+ (/ x.re y.re) (/ x.im (* y.re (/ y.re y.im))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_im <= -29000000000.0) || !(y_46_im <= 950000.0)) {
		tmp = x_46_im / y_46_im;
	} else {
		tmp = (x_46_re / y_46_re) + (x_46_im / (y_46_re * (y_46_re / y_46_im)));
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if ((y_46im <= (-29000000000.0d0)) .or. (.not. (y_46im <= 950000.0d0))) then
        tmp = x_46im / y_46im
    else
        tmp = (x_46re / y_46re) + (x_46im / (y_46re * (y_46re / y_46im)))
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_im <= -29000000000.0) || !(y_46_im <= 950000.0)) {
		tmp = x_46_im / y_46_im;
	} else {
		tmp = (x_46_re / y_46_re) + (x_46_im / (y_46_re * (y_46_re / y_46_im)));
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if (y_46_im <= -29000000000.0) or not (y_46_im <= 950000.0):
		tmp = x_46_im / y_46_im
	else:
		tmp = (x_46_re / y_46_re) + (x_46_im / (y_46_re * (y_46_re / y_46_im)))
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if ((y_46_im <= -29000000000.0) || !(y_46_im <= 950000.0))
		tmp = Float64(x_46_im / y_46_im);
	else
		tmp = Float64(Float64(x_46_re / y_46_re) + Float64(x_46_im / Float64(y_46_re * Float64(y_46_re / y_46_im))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if ((y_46_im <= -29000000000.0) || ~((y_46_im <= 950000.0)))
		tmp = x_46_im / y_46_im;
	else
		tmp = (x_46_re / y_46_re) + (x_46_im / (y_46_re * (y_46_re / y_46_im)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[Or[LessEqual[y$46$im, -29000000000.0], N[Not[LessEqual[y$46$im, 950000.0]], $MachinePrecision]], N[(x$46$im / y$46$im), $MachinePrecision], N[(N[(x$46$re / y$46$re), $MachinePrecision] + N[(x$46$im / N[(y$46$re * N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.im < -2.9e10 or 9.5e5 < y.im

   1. Initial program 51.6%

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

   2. Taylor expanded in y.re around 0 72.9%

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

   if -2.9e10 < y.im < 9.5e5

   1. Initial program 77.6%

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

   2. Taylor expanded in y.re around inf 69.9%

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

      1. associate-/l* 71.5%

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

   4. Simplified 71.5%

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

      1. pow2 71.5%

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

      2. *-un-lft-identity 71.5%

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

      3. times-frac 72.3%

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

   6. Applied egg-rr 72.3%

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

4. Final simplification 72.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -29000000000 \lor \neg \left(y.im \leq 950000\right):\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{x.im}{y.re \cdot \frac{y.re}{y.im}}\\ \end{array} \]

Alternative 11: 63.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -1.5 \cdot 10^{+20} \lor \neg \left(y.im \leq -2.2 \cdot 10^{-94} \lor \neg \left(y.im \leq -4.3 \cdot 10^{-107}\right) \land y.im \leq 4.3 \cdot 10^{-41}\right):\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.im -1.5e+20)
         (not
          (or (<= y.im -2.2e-94)
              (and (not (<= y.im -4.3e-107)) (<= y.im 4.3e-41)))))
   (/ x.im y.im)
   (/ x.re y.re)))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_im <= -1.5e+20) || !((y_46_im <= -2.2e-94) || (!(y_46_im <= -4.3e-107) && (y_46_im <= 4.3e-41)))) {
		tmp = x_46_im / y_46_im;
	} else {
		tmp = x_46_re / y_46_re;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if ((y_46im <= (-1.5d+20)) .or. (.not. (y_46im <= (-2.2d-94)) .or. (.not. (y_46im <= (-4.3d-107))) .and. (y_46im <= 4.3d-41))) then
        tmp = x_46im / y_46im
    else
        tmp = x_46re / y_46re
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_im <= -1.5e+20) || !((y_46_im <= -2.2e-94) || (!(y_46_im <= -4.3e-107) && (y_46_im <= 4.3e-41)))) {
		tmp = x_46_im / y_46_im;
	} else {
		tmp = x_46_re / y_46_re;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if (y_46_im <= -1.5e+20) or not ((y_46_im <= -2.2e-94) or (not (y_46_im <= -4.3e-107) and (y_46_im <= 4.3e-41))):
		tmp = x_46_im / y_46_im
	else:
		tmp = x_46_re / y_46_re
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if ((y_46_im <= -1.5e+20) || !((y_46_im <= -2.2e-94) || (!(y_46_im <= -4.3e-107) && (y_46_im <= 4.3e-41))))
		tmp = Float64(x_46_im / y_46_im);
	else
		tmp = Float64(x_46_re / y_46_re);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if ((y_46_im <= -1.5e+20) || ~(((y_46_im <= -2.2e-94) || (~((y_46_im <= -4.3e-107)) && (y_46_im <= 4.3e-41)))))
		tmp = x_46_im / y_46_im;
	else
		tmp = x_46_re / y_46_re;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[Or[LessEqual[y$46$im, -1.5e+20], N[Not[Or[LessEqual[y$46$im, -2.2e-94], And[N[Not[LessEqual[y$46$im, -4.3e-107]], $MachinePrecision], LessEqual[y$46$im, 4.3e-41]]]], $MachinePrecision]], N[(x$46$im / y$46$im), $MachinePrecision], N[(x$46$re / y$46$re), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.im < -1.5e20 or -2.20000000000000001e-94 < y.im < -4.2999999999999997e-107 or 4.2999999999999999e-41 < y.im

   1. Initial program 57.0%

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

   2. Taylor expanded in y.re around 0 69.9%

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

   if -1.5e20 < y.im < -2.20000000000000001e-94 or -4.2999999999999997e-107 < y.im < 4.2999999999999999e-41

   1. Initial program 75.3%

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

   2. Taylor expanded in y.re around inf 63.1%

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

4. Final simplification 66.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -1.5 \cdot 10^{+20} \lor \neg \left(y.im \leq -2.2 \cdot 10^{-94} \lor \neg \left(y.im \leq -4.3 \cdot 10^{-107}\right) \land y.im \leq 4.3 \cdot 10^{-41}\right):\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \end{array} \]

Alternative 12: 43.2% 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 65.9%

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

2. Taylor expanded in y.re around 0 43.9%

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

3. Final simplification 43.9%

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

Reproduce

herbie shell --seed 2023298 
(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))))