_divideComplex, real part

Percentage Accurate: 61.9% → 79.2%

Time: 14.7s

Alternatives: 13

Speedup: 1.1×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 61.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 79.2% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(x.im + x.re \cdot \frac{y.re}{y.im}\right) \cdot \frac{--1}{y.im}\\ \mathbf{if}\;y.re \leq -3.6 \cdot 10^{-36}:\\ \;\;\;\;\frac{\frac{-x.im}{\frac{y.re}{y.im}} - x.re}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.re \leq 8 \cdot 10^{-49}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.re \leq 1.6 \cdot 10^{+21}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)}\\ \mathbf{elif}\;y.re \leq 1.3 \cdot 10^{+78}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re + \frac{x.im}{\frac{y.re}{y.im}}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (* (+ x.im (* x.re (/ y.re y.im))) (/ (- -1.0) y.im))))
   (if (<= y.re -3.6e-36)
     (/ (- (/ (- x.im) (/ y.re y.im)) x.re) (hypot y.re y.im))
     (if (<= y.re 8e-49)
       t_0
       (if (<= y.re 1.6e+21)
         (/ (fma x.re y.re (* x.im y.im)) (fma y.re y.re (* y.im y.im)))
         (if (<= y.re 1.3e+78)
           t_0
           (/ (+ x.re (/ x.im (/ y.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 = (x_46_im + (x_46_re * (y_46_re / y_46_im))) * (-(-1.0) / y_46_im);
	double tmp;
	if (y_46_re <= -3.6e-36) {
		tmp = ((-x_46_im / (y_46_re / y_46_im)) - x_46_re) / hypot(y_46_re, y_46_im);
	} else if (y_46_re <= 8e-49) {
		tmp = t_0;
	} else if (y_46_re <= 1.6e+21) {
		tmp = fma(x_46_re, y_46_re, (x_46_im * y_46_im)) / fma(y_46_re, y_46_re, (y_46_im * y_46_im));
	} else if (y_46_re <= 1.3e+78) {
		tmp = t_0;
	} else {
		tmp = (x_46_re + (x_46_im / (y_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(x_46_im + Float64(x_46_re * Float64(y_46_re / y_46_im))) * Float64(Float64(-(-1.0)) / y_46_im))
	tmp = 0.0
	if (y_46_re <= -3.6e-36)
		tmp = Float64(Float64(Float64(Float64(-x_46_im) / Float64(y_46_re / y_46_im)) - x_46_re) / hypot(y_46_re, y_46_im));
	elseif (y_46_re <= 8e-49)
		tmp = t_0;
	elseif (y_46_re <= 1.6e+21)
		tmp = Float64(fma(x_46_re, y_46_re, Float64(x_46_im * y_46_im)) / fma(y_46_re, y_46_re, Float64(y_46_im * y_46_im)));
	elseif (y_46_re <= 1.3e+78)
		tmp = t_0;
	else
		tmp = Float64(Float64(x_46_re + Float64(x_46_im / Float64(y_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[(x$46$im + N[(x$46$re * N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[((--1.0) / y$46$im), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$re, -3.6e-36], N[(N[(N[((-x$46$im) / N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision] - x$46$re), $MachinePrecision] / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 8e-49], t$95$0, If[LessEqual[y$46$re, 1.6e+21], N[(N[(x$46$re * y$46$re + N[(x$46$im * y$46$im), $MachinePrecision]), $MachinePrecision] / N[(y$46$re * y$46$re + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 1.3e+78], t$95$0, N[(N[(x$46$re + N[(x$46$im / N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if y.re < -3.60000000000000032e-36

   1. Initial program 49.4%

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

      1. *-un-lft-identity 49.4%

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

      2. associate-*r/ 49.4%

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

      3. fma-def 49.4%

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

      4. add-sqr-sqrt 49.4%

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

      5. times-frac 49.4%

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

      6. fma-def 49.4%

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

      7. hypot-def 49.4%

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

      8. fma-def 49.4%

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

      9. fma-def 49.4%

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

      10. hypot-def 59.7%

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

   4. Applied egg-rr 59.7%

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

      1. associate-*l/ 59.8%

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

      2. *-un-lft-identity 59.8%

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

   6. Applied egg-rr 59.8%

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

   7. Taylor expanded in y.re around -inf 73.2%

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

      1. neg-mul-1 73.2%

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

      2. +-commutative 73.2%

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

      3. unsub-neg 73.2%

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

      4. mul-1-neg 73.2%

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

      5. associate-/l* 76.4%

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

      6. distribute-neg-frac 76.4%

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

   9. Simplified 76.4%

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

   if -3.60000000000000032e-36 < y.re < 7.99999999999999949e-49 or 1.6e21 < y.re < 1.3e78

   1. Initial program 67.2%

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

      1. *-un-lft-identity 67.2%

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

      2. associate-*r/ 67.2%

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

      3. fma-def 67.2%

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

      4. add-sqr-sqrt 67.2%

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

      5. times-frac 67.2%

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

      6. fma-def 67.2%

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

      7. hypot-def 67.2%

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

      8. fma-def 67.2%

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

      9. fma-def 67.2%

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

      10. hypot-def 80.0%

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

   4. Applied egg-rr 80.0%

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

   5. Taylor expanded in y.im around -inf 51.5%

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

      1. neg-mul-1 51.5%

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

      2. mul-1-neg 51.5%

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

      3. unsub-neg 51.5%

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

      4. *-lft-identity 51.5%

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

      5. times-frac 51.5%

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

      6. /-rgt-identity 51.5%

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

   7. Simplified 51.5%

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

   8. Taylor expanded in y.im around -inf 88.3%

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

   if 7.99999999999999949e-49 < y.re < 1.6e21

   1. Initial program 90.2%

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

      1. fma-def 90.3%

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

      2. fma-def 90.3%

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

   3. Simplified 90.3%

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

   if 1.3e78 < y.re

   1. Initial program 47.8%

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

      1. *-un-lft-identity 47.8%

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

      2. associate-*r/ 47.8%

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

      3. fma-def 47.8%

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

      4. add-sqr-sqrt 47.8%

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

      5. times-frac 47.9%

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

      6. fma-def 47.9%

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

      7. hypot-def 47.9%

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

      8. fma-def 47.9%

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

      9. fma-def 47.9%

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

      10. hypot-def 70.2%

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

   4. Applied egg-rr 70.2%

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

      1. associate-*l/ 70.3%

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

      2. *-un-lft-identity 70.3%

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

   6. Applied egg-rr 70.3%

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

   7. Taylor expanded in y.re around inf 84.3%

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

      1. associate-/l* 89.9%

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

   9. Simplified 89.9%

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

4. Final simplification 85.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -3.6 \cdot 10^{-36}:\\ \;\;\;\;\frac{\frac{-x.im}{\frac{y.re}{y.im}} - x.re}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.re \leq 8 \cdot 10^{-49}:\\ \;\;\;\;\left(x.im + x.re \cdot \frac{y.re}{y.im}\right) \cdot \frac{--1}{y.im}\\ \mathbf{elif}\;y.re \leq 1.6 \cdot 10^{+21}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)}\\ \mathbf{elif}\;y.re \leq 1.3 \cdot 10^{+78}:\\ \;\;\;\;\left(x.im + x.re \cdot \frac{y.re}{y.im}\right) \cdot \frac{--1}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re + \frac{x.im}{\frac{y.re}{y.im}}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 84.9% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \leq 5 \cdot 10^{+299}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}{\mathsf{hypot}\left(y.re, y.im\right)}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{x.im}{y.re \cdot \frac{y.re}{y.im}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<=
      (/ (+ (* x.re y.re) (* x.im y.im)) (+ (* y.re y.re) (* y.im y.im)))
      5e+299)
   (/ (/ (fma x.re y.re (* x.im y.im)) (hypot y.re y.im)) (hypot y.re 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 ((((x_46_re * y_46_re) + (x_46_im * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im))) <= 5e+299) {
		tmp = (fma(x_46_re, y_46_re, (x_46_im * y_46_im)) / hypot(y_46_re, y_46_im)) / hypot(y_46_re, 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 (Float64(Float64(Float64(x_46_re * y_46_re) + Float64(x_46_im * y_46_im)) / Float64(Float64(y_46_re * y_46_re) + Float64(y_46_im * y_46_im))) <= 5e+299)
		tmp = Float64(Float64(fma(x_46_re, y_46_re, Float64(x_46_im * y_46_im)) / hypot(y_46_re, y_46_im)) / hypot(y_46_re, 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

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[N[(N[(N[(x$46$re * y$46$re), $MachinePrecision] + N[(x$46$im * y$46$im), $MachinePrecision]), $MachinePrecision] / N[(N[(y$46$re * y$46$re), $MachinePrecision] + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 5e+299], N[(N[(N[(x$46$re * y$46$re + N[(x$46$im * y$46$im), $MachinePrecision]), $MachinePrecision] / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision] / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $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 (/.f64 (+.f64 (*.f64 x.re y.re) (*.f64 x.im y.im)) (+.f64 (*.f64 y.re y.re) (*.f64 y.im y.im))) < 5.0000000000000003e299

   1. Initial program 79.3%

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

      1. *-un-lft-identity 79.3%

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

      2. associate-*r/ 79.3%

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

      3. fma-def 79.3%

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

      4. add-sqr-sqrt 79.3%

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

      5. times-frac 79.3%

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

      6. fma-def 79.3%

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

      7. hypot-def 79.3%

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

      8. fma-def 79.3%

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

      9. fma-def 79.3%

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

      10. hypot-def 95.2%

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

   4. Applied egg-rr 95.2%

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

      1. associate-*l/ 95.4%

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

      2. *-un-lft-identity 95.4%

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

   6. Applied egg-rr 95.4%

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

   if 5.0000000000000003e299 < (/.f64 (+.f64 (*.f64 x.re y.re) (*.f64 x.im y.im)) (+.f64 (*.f64 y.re y.re) (*.f64 y.im y.im)))

   1. Initial program 3.8%

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

   3. Taylor expanded in y.re around inf 38.9%

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

      1. associate-/l* 44.4%

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

   5. Simplified 44.4%

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

      1. pow2 44.4%

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

      2. *-un-lft-identity 44.4%

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

      3. times-frac 55.0%

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

   7. Applied egg-rr 55.0%

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

4. Final simplification 85.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \leq 5 \cdot 10^{+299}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}{\mathsf{hypot}\left(y.re, y.im\right)}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{x.im}{y.re \cdot \frac{y.re}{y.im}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 82.7% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{if}\;y.im \leq -7 \cdot 10^{+61}:\\ \;\;\;\;\left(x.im + x.re \cdot \frac{y.re}{y.im}\right) \cdot \frac{--1}{y.im}\\ \mathbf{elif}\;y.im \leq -2.35 \cdot 10^{-82}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.im \leq 1.3 \cdot 10^{-113}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{x.im}{y.re \cdot \frac{y.re}{y.im}}\\ \mathbf{elif}\;y.im \leq 1.85 \cdot 10^{+79}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im + \frac{x.re}{\frac{y.im}{y.re}}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0
         (/ (+ (* x.re y.re) (* x.im y.im)) (+ (* y.re y.re) (* y.im y.im)))))
   (if (<= y.im -7e+61)
     (* (+ x.im (* x.re (/ y.re y.im))) (/ (- -1.0) y.im))
     (if (<= y.im -2.35e-82)
       t_0
       (if (<= y.im 1.3e-113)
         (+ (/ x.re y.re) (/ x.im (* y.re (/ y.re y.im))))
         (if (<= y.im 1.85e+79)
           t_0
           (/ (+ x.im (/ x.re (/ y.im y.re))) (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 = ((x_46_re * y_46_re) + (x_46_im * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	double tmp;
	if (y_46_im <= -7e+61) {
		tmp = (x_46_im + (x_46_re * (y_46_re / y_46_im))) * (-(-1.0) / y_46_im);
	} else if (y_46_im <= -2.35e-82) {
		tmp = t_0;
	} else if (y_46_im <= 1.3e-113) {
		tmp = (x_46_re / y_46_re) + (x_46_im / (y_46_re * (y_46_re / y_46_im)));
	} else if (y_46_im <= 1.85e+79) {
		tmp = t_0;
	} else {
		tmp = (x_46_im + (x_46_re / (y_46_im / y_46_re))) / 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 = ((x_46_re * y_46_re) + (x_46_im * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	double tmp;
	if (y_46_im <= -7e+61) {
		tmp = (x_46_im + (x_46_re * (y_46_re / y_46_im))) * (-(-1.0) / y_46_im);
	} else if (y_46_im <= -2.35e-82) {
		tmp = t_0;
	} else if (y_46_im <= 1.3e-113) {
		tmp = (x_46_re / y_46_re) + (x_46_im / (y_46_re * (y_46_re / y_46_im)));
	} else if (y_46_im <= 1.85e+79) {
		tmp = t_0;
	} else {
		tmp = (x_46_im + (x_46_re / (y_46_im / y_46_re))) / 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 = ((x_46_re * y_46_re) + (x_46_im * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im))
	tmp = 0
	if y_46_im <= -7e+61:
		tmp = (x_46_im + (x_46_re * (y_46_re / y_46_im))) * (-(-1.0) / y_46_im)
	elif y_46_im <= -2.35e-82:
		tmp = t_0
	elif y_46_im <= 1.3e-113:
		tmp = (x_46_re / y_46_re) + (x_46_im / (y_46_re * (y_46_re / y_46_im)))
	elif y_46_im <= 1.85e+79:
		tmp = t_0
	else:
		tmp = (x_46_im + (x_46_re / (y_46_im / y_46_re))) / 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(x_46_re * y_46_re) + Float64(x_46_im * y_46_im)) / Float64(Float64(y_46_re * y_46_re) + Float64(y_46_im * y_46_im)))
	tmp = 0.0
	if (y_46_im <= -7e+61)
		tmp = Float64(Float64(x_46_im + Float64(x_46_re * Float64(y_46_re / y_46_im))) * Float64(Float64(-(-1.0)) / y_46_im));
	elseif (y_46_im <= -2.35e-82)
		tmp = t_0;
	elseif (y_46_im <= 1.3e-113)
		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 <= 1.85e+79)
		tmp = t_0;
	else
		tmp = Float64(Float64(x_46_im + Float64(x_46_re / Float64(y_46_im / y_46_re))) / 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 = ((x_46_re * y_46_re) + (x_46_im * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	tmp = 0.0;
	if (y_46_im <= -7e+61)
		tmp = (x_46_im + (x_46_re * (y_46_re / y_46_im))) * (-(-1.0) / y_46_im);
	elseif (y_46_im <= -2.35e-82)
		tmp = t_0;
	elseif (y_46_im <= 1.3e-113)
		tmp = (x_46_re / y_46_re) + (x_46_im / (y_46_re * (y_46_re / y_46_im)));
	elseif (y_46_im <= 1.85e+79)
		tmp = t_0;
	else
		tmp = (x_46_im + (x_46_re / (y_46_im / y_46_re))) / 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[(x$46$re * y$46$re), $MachinePrecision] + N[(x$46$im * y$46$im), $MachinePrecision]), $MachinePrecision] / N[(N[(y$46$re * y$46$re), $MachinePrecision] + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$im, -7e+61], N[(N[(x$46$im + N[(x$46$re * N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[((--1.0) / y$46$im), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, -2.35e-82], t$95$0, If[LessEqual[y$46$im, 1.3e-113], 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, 1.85e+79], t$95$0, N[(N[(x$46$im + N[(x$46$re / N[(y$46$im / y$46$re), $MachinePrecision]), $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 < -7.00000000000000036e61

   1. Initial program 47.4%

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

      1. *-un-lft-identity 47.4%

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

      2. associate-*r/ 47.4%

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

      3. fma-def 47.4%

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

      4. add-sqr-sqrt 47.4%

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

      5. times-frac 47.3%

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

      6. fma-def 47.3%

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

      7. hypot-def 47.3%

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

      8. fma-def 47.3%

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

      9. fma-def 47.3%

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

      10. hypot-def 63.9%

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

   4. Applied egg-rr 63.9%

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

   5. Taylor expanded in y.im around -inf 82.2%

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

      1. neg-mul-1 82.2%

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

      2. mul-1-neg 82.2%

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

      3. unsub-neg 82.2%

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

      4. *-lft-identity 82.2%

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

      5. times-frac 87.7%

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

      6. /-rgt-identity 87.7%

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

   7. Simplified 87.7%

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

   8. Taylor expanded in y.im around -inf 87.5%

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

   if -7.00000000000000036e61 < y.im < -2.35e-82 or 1.3e-113 < y.im < 1.85000000000000005e79

   1. Initial program 75.7%

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

   if -2.35e-82 < y.im < 1.3e-113

   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. Add Preprocessing

   3. Taylor expanded in y.re around inf 85.0%

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

      1. associate-/l* 82.4%

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

   5. Simplified 82.4%

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

      1. pow2 82.4%

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

      2. *-un-lft-identity 82.4%

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

      3. times-frac 91.9%

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

   7. Applied egg-rr 91.9%

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

   if 1.85000000000000005e79 < y.im

   1. Initial program 42.2%

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

      1. *-un-lft-identity 42.2%

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

      2. associate-*r/ 42.2%

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

      3. fma-def 42.2%

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

      4. add-sqr-sqrt 42.2%

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

      5. times-frac 42.2%

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

      6. fma-def 42.2%

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

      7. hypot-def 42.2%

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

      8. fma-def 42.2%

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

      9. fma-def 42.2%

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

      10. hypot-def 59.9%

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

   4. Applied egg-rr 59.9%

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

      1. associate-*l/ 60.0%

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

      2. *-un-lft-identity 60.0%

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

   6. Applied egg-rr 60.0%

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

   7. Taylor expanded in y.re around 0 73.5%

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

      1. associate-/l* 80.7%

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

   9. Simplified 80.7%

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

4. Final simplification 83.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -7 \cdot 10^{+61}:\\ \;\;\;\;\left(x.im + x.re \cdot \frac{y.re}{y.im}\right) \cdot \frac{--1}{y.im}\\ \mathbf{elif}\;y.im \leq -2.35 \cdot 10^{-82}:\\ \;\;\;\;\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.im \leq 1.3 \cdot 10^{-113}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{x.im}{y.re \cdot \frac{y.re}{y.im}}\\ \mathbf{elif}\;y.im \leq 1.85 \cdot 10^{+79}:\\ \;\;\;\;\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im + \frac{x.re}{\frac{y.im}{y.re}}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 79.6% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(x.im + x.re \cdot \frac{y.re}{y.im}\right) \cdot \frac{--1}{y.im}\\ \mathbf{if}\;y.re \leq -1.45 \cdot 10^{-32}:\\ \;\;\;\;\frac{\frac{-x.im}{\frac{y.re}{y.im}} - x.re}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.re \leq 6.5 \cdot 10^{-53}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.re \leq 2.05 \cdot 10^{+24}:\\ \;\;\;\;\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.re \leq 1.4 \cdot 10^{+76}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re + \frac{x.im}{\frac{y.re}{y.im}}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (* (+ x.im (* x.re (/ y.re y.im))) (/ (- -1.0) y.im))))
   (if (<= y.re -1.45e-32)
     (/ (- (/ (- x.im) (/ y.re y.im)) x.re) (hypot y.re y.im))
     (if (<= y.re 6.5e-53)
       t_0
       (if (<= y.re 2.05e+24)
         (/ (+ (* x.re y.re) (* x.im y.im)) (+ (* y.re y.re) (* y.im y.im)))
         (if (<= y.re 1.4e+76)
           t_0
           (/ (+ x.re (/ x.im (/ y.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 = (x_46_im + (x_46_re * (y_46_re / y_46_im))) * (-(-1.0) / y_46_im);
	double tmp;
	if (y_46_re <= -1.45e-32) {
		tmp = ((-x_46_im / (y_46_re / y_46_im)) - x_46_re) / hypot(y_46_re, y_46_im);
	} else if (y_46_re <= 6.5e-53) {
		tmp = t_0;
	} else if (y_46_re <= 2.05e+24) {
		tmp = ((x_46_re * y_46_re) + (x_46_im * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	} else if (y_46_re <= 1.4e+76) {
		tmp = t_0;
	} else {
		tmp = (x_46_re + (x_46_im / (y_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 = (x_46_im + (x_46_re * (y_46_re / y_46_im))) * (-(-1.0) / y_46_im);
	double tmp;
	if (y_46_re <= -1.45e-32) {
		tmp = ((-x_46_im / (y_46_re / y_46_im)) - x_46_re) / Math.hypot(y_46_re, y_46_im);
	} else if (y_46_re <= 6.5e-53) {
		tmp = t_0;
	} else if (y_46_re <= 2.05e+24) {
		tmp = ((x_46_re * y_46_re) + (x_46_im * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	} else if (y_46_re <= 1.4e+76) {
		tmp = t_0;
	} else {
		tmp = (x_46_re + (x_46_im / (y_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 = (x_46_im + (x_46_re * (y_46_re / y_46_im))) * (-(-1.0) / y_46_im)
	tmp = 0
	if y_46_re <= -1.45e-32:
		tmp = ((-x_46_im / (y_46_re / y_46_im)) - x_46_re) / math.hypot(y_46_re, y_46_im)
	elif y_46_re <= 6.5e-53:
		tmp = t_0
	elif y_46_re <= 2.05e+24:
		tmp = ((x_46_re * y_46_re) + (x_46_im * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im))
	elif y_46_re <= 1.4e+76:
		tmp = t_0
	else:
		tmp = (x_46_re + (x_46_im / (y_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(x_46_im + Float64(x_46_re * Float64(y_46_re / y_46_im))) * Float64(Float64(-(-1.0)) / y_46_im))
	tmp = 0.0
	if (y_46_re <= -1.45e-32)
		tmp = Float64(Float64(Float64(Float64(-x_46_im) / Float64(y_46_re / y_46_im)) - x_46_re) / hypot(y_46_re, y_46_im));
	elseif (y_46_re <= 6.5e-53)
		tmp = t_0;
	elseif (y_46_re <= 2.05e+24)
		tmp = Float64(Float64(Float64(x_46_re * y_46_re) + Float64(x_46_im * y_46_im)) / Float64(Float64(y_46_re * y_46_re) + Float64(y_46_im * y_46_im)));
	elseif (y_46_re <= 1.4e+76)
		tmp = t_0;
	else
		tmp = Float64(Float64(x_46_re + Float64(x_46_im / Float64(y_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 = (x_46_im + (x_46_re * (y_46_re / y_46_im))) * (-(-1.0) / y_46_im);
	tmp = 0.0;
	if (y_46_re <= -1.45e-32)
		tmp = ((-x_46_im / (y_46_re / y_46_im)) - x_46_re) / hypot(y_46_re, y_46_im);
	elseif (y_46_re <= 6.5e-53)
		tmp = t_0;
	elseif (y_46_re <= 2.05e+24)
		tmp = ((x_46_re * y_46_re) + (x_46_im * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	elseif (y_46_re <= 1.4e+76)
		tmp = t_0;
	else
		tmp = (x_46_re + (x_46_im / (y_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[(x$46$im + N[(x$46$re * N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[((--1.0) / y$46$im), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$re, -1.45e-32], N[(N[(N[((-x$46$im) / N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision] - x$46$re), $MachinePrecision] / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 6.5e-53], t$95$0, If[LessEqual[y$46$re, 2.05e+24], N[(N[(N[(x$46$re * y$46$re), $MachinePrecision] + N[(x$46$im * y$46$im), $MachinePrecision]), $MachinePrecision] / N[(N[(y$46$re * y$46$re), $MachinePrecision] + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 1.4e+76], t$95$0, N[(N[(x$46$re + N[(x$46$im / N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if y.re < -1.44999999999999998e-32

   1. Initial program 49.4%

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

      1. *-un-lft-identity 49.4%

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

      2. associate-*r/ 49.4%

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

      3. fma-def 49.4%

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

      4. add-sqr-sqrt 49.4%

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

      5. times-frac 49.4%

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

      6. fma-def 49.4%

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

      7. hypot-def 49.4%

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

      8. fma-def 49.4%

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

      9. fma-def 49.4%

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

      10. hypot-def 59.7%

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

   4. Applied egg-rr 59.7%

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

      1. associate-*l/ 59.8%

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

      2. *-un-lft-identity 59.8%

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

   6. Applied egg-rr 59.8%

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

   7. Taylor expanded in y.re around -inf 73.2%

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

      1. neg-mul-1 73.2%

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

      2. +-commutative 73.2%

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

      3. unsub-neg 73.2%

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

      4. mul-1-neg 73.2%

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

      5. associate-/l* 76.4%

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

      6. distribute-neg-frac 76.4%

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

   9. Simplified 76.4%

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

   if -1.44999999999999998e-32 < y.re < 6.4999999999999997e-53 or 2.05e24 < y.re < 1.3999999999999999e76

   1. Initial program 67.2%

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

      1. *-un-lft-identity 67.2%

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

      2. associate-*r/ 67.2%

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

      3. fma-def 67.2%

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

      4. add-sqr-sqrt 67.2%

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

      5. times-frac 67.2%

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

      6. fma-def 67.2%

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

      7. hypot-def 67.2%

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

      8. fma-def 67.2%

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

      9. fma-def 67.2%

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

      10. hypot-def 80.0%

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

   4. Applied egg-rr 80.0%

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

   5. Taylor expanded in y.im around -inf 51.5%

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

      1. neg-mul-1 51.5%

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

      2. mul-1-neg 51.5%

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

      3. unsub-neg 51.5%

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

      4. *-lft-identity 51.5%

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

      5. times-frac 51.5%

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

      6. /-rgt-identity 51.5%

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

   7. Simplified 51.5%

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

   8. Taylor expanded in y.im around -inf 88.3%

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

   if 6.4999999999999997e-53 < y.re < 2.05e24

   1. Initial program 90.2%

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

   if 1.3999999999999999e76 < y.re

   1. Initial program 47.8%

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

      1. *-un-lft-identity 47.8%

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

      2. associate-*r/ 47.8%

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

      3. fma-def 47.8%

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

      4. add-sqr-sqrt 47.8%

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

      5. times-frac 47.9%

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

      6. fma-def 47.9%

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

      7. hypot-def 47.9%

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

      8. fma-def 47.9%

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

      9. fma-def 47.9%

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

      10. hypot-def 70.2%

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

   4. Applied egg-rr 70.2%

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

      1. associate-*l/ 70.3%

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

      2. *-un-lft-identity 70.3%

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

   6. Applied egg-rr 70.3%

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

   7. Taylor expanded in y.re around inf 84.3%

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

      1. associate-/l* 89.9%

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

   9. Simplified 89.9%

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

4. Final simplification 85.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -1.45 \cdot 10^{-32}:\\ \;\;\;\;\frac{\frac{-x.im}{\frac{y.re}{y.im}} - x.re}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.re \leq 6.5 \cdot 10^{-53}:\\ \;\;\;\;\left(x.im + x.re \cdot \frac{y.re}{y.im}\right) \cdot \frac{--1}{y.im}\\ \mathbf{elif}\;y.re \leq 2.05 \cdot 10^{+24}:\\ \;\;\;\;\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.re \leq 1.4 \cdot 10^{+76}:\\ \;\;\;\;\left(x.im + x.re \cdot \frac{y.re}{y.im}\right) \cdot \frac{--1}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re + \frac{x.im}{\frac{y.re}{y.im}}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 64.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -2.8 \cdot 10^{+29}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{elif}\;y.im \leq -1.4 \cdot 10^{-274}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{elif}\;y.im \leq -8.2 \cdot 10^{-303}:\\ \;\;\;\;x.im \cdot \left(\frac{y.im}{y.re} \cdot \frac{1}{y.re}\right)\\ \mathbf{elif}\;y.im \leq 9 \cdot 10^{-23}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{elif}\;y.im \leq 2.6 \cdot 10^{+78}:\\ \;\;\;\;\frac{x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.im -2.8e+29)
   (/ x.im y.im)
   (if (<= y.im -1.4e-274)
     (/ x.re y.re)
     (if (<= y.im -8.2e-303)
       (* x.im (* (/ y.im y.re) (/ 1.0 y.re)))
       (if (<= y.im 9e-23)
         (/ x.re y.re)
         (if (<= y.im 2.6e+78)
           (/ (* x.im y.im) (+ (* y.re y.re) (* y.im y.im)))
           (/ x.im y.im)))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_im <= -2.8e+29) {
		tmp = x_46_im / y_46_im;
	} else if (y_46_im <= -1.4e-274) {
		tmp = x_46_re / y_46_re;
	} else if (y_46_im <= -8.2e-303) {
		tmp = x_46_im * ((y_46_im / y_46_re) * (1.0 / y_46_re));
	} else if (y_46_im <= 9e-23) {
		tmp = x_46_re / y_46_re;
	} else if (y_46_im <= 2.6e+78) {
		tmp = (x_46_im * y_46_im) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	} else {
		tmp = x_46_im / y_46_im;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if (y_46im <= (-2.8d+29)) then
        tmp = x_46im / y_46im
    else if (y_46im <= (-1.4d-274)) then
        tmp = x_46re / y_46re
    else if (y_46im <= (-8.2d-303)) then
        tmp = x_46im * ((y_46im / y_46re) * (1.0d0 / y_46re))
    else if (y_46im <= 9d-23) then
        tmp = x_46re / y_46re
    else if (y_46im <= 2.6d+78) then
        tmp = (x_46im * y_46im) / ((y_46re * y_46re) + (y_46im * y_46im))
    else
        tmp = x_46im / y_46im
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_im <= -2.8e+29) {
		tmp = x_46_im / y_46_im;
	} else if (y_46_im <= -1.4e-274) {
		tmp = x_46_re / y_46_re;
	} else if (y_46_im <= -8.2e-303) {
		tmp = x_46_im * ((y_46_im / y_46_re) * (1.0 / y_46_re));
	} else if (y_46_im <= 9e-23) {
		tmp = x_46_re / y_46_re;
	} else if (y_46_im <= 2.6e+78) {
		tmp = (x_46_im * y_46_im) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	} else {
		tmp = x_46_im / y_46_im;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if y_46_im <= -2.8e+29:
		tmp = x_46_im / y_46_im
	elif y_46_im <= -1.4e-274:
		tmp = x_46_re / y_46_re
	elif y_46_im <= -8.2e-303:
		tmp = x_46_im * ((y_46_im / y_46_re) * (1.0 / y_46_re))
	elif y_46_im <= 9e-23:
		tmp = x_46_re / y_46_re
	elif y_46_im <= 2.6e+78:
		tmp = (x_46_im * y_46_im) / ((y_46_re * y_46_re) + (y_46_im * y_46_im))
	else:
		tmp = x_46_im / y_46_im
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (y_46_im <= -2.8e+29)
		tmp = Float64(x_46_im / y_46_im);
	elseif (y_46_im <= -1.4e-274)
		tmp = Float64(x_46_re / y_46_re);
	elseif (y_46_im <= -8.2e-303)
		tmp = Float64(x_46_im * Float64(Float64(y_46_im / y_46_re) * Float64(1.0 / y_46_re)));
	elseif (y_46_im <= 9e-23)
		tmp = Float64(x_46_re / y_46_re);
	elseif (y_46_im <= 2.6e+78)
		tmp = Float64(Float64(x_46_im * y_46_im) / Float64(Float64(y_46_re * y_46_re) + Float64(y_46_im * y_46_im)));
	else
		tmp = Float64(x_46_im / y_46_im);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if (y_46_im <= -2.8e+29)
		tmp = x_46_im / y_46_im;
	elseif (y_46_im <= -1.4e-274)
		tmp = x_46_re / y_46_re;
	elseif (y_46_im <= -8.2e-303)
		tmp = x_46_im * ((y_46_im / y_46_re) * (1.0 / y_46_re));
	elseif (y_46_im <= 9e-23)
		tmp = x_46_re / y_46_re;
	elseif (y_46_im <= 2.6e+78)
		tmp = (x_46_im * y_46_im) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	else
		tmp = x_46_im / y_46_im;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[y$46$im, -2.8e+29], N[(x$46$im / y$46$im), $MachinePrecision], If[LessEqual[y$46$im, -1.4e-274], N[(x$46$re / y$46$re), $MachinePrecision], If[LessEqual[y$46$im, -8.2e-303], N[(x$46$im * N[(N[(y$46$im / y$46$re), $MachinePrecision] * N[(1.0 / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, 9e-23], N[(x$46$re / y$46$re), $MachinePrecision], If[LessEqual[y$46$im, 2.6e+78], N[(N[(x$46$im * y$46$im), $MachinePrecision] / N[(N[(y$46$re * y$46$re), $MachinePrecision] + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x$46$im / y$46$im), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if y.im < -2.8e29 or 2.6e78 < y.im

   1. Initial program 46.8%

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

   3. Taylor expanded in y.re around 0 67.9%

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

   if -2.8e29 < y.im < -1.39999999999999988e-274 or -8.20000000000000037e-303 < y.im < 8.9999999999999995e-23

   1. Initial program 71.2%

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

   3. Taylor expanded in y.re around inf 69.1%

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

   if -1.39999999999999988e-274 < y.im < -8.20000000000000037e-303

   1. Initial program 45.3%

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

   3. Taylor expanded in x.re around 0 45.7%

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

   4. Taylor expanded in y.im around 0 45.7%

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

      1. associate-*r/ 45.7%

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

   6. Simplified 45.7%

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

      1. *-un-lft-identity 45.7%

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

      2. unpow2 45.7%

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

      3. times-frac 85.7%

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

   8. Applied egg-rr 85.7%

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

   if 8.9999999999999995e-23 < y.im < 2.6e78

   1. Initial program 84.3%

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

   3. Taylor expanded in x.re around 0 64.5%

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

4. Final simplification 68.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -2.8 \cdot 10^{+29}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{elif}\;y.im \leq -1.4 \cdot 10^{-274}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{elif}\;y.im \leq -8.2 \cdot 10^{-303}:\\ \;\;\;\;x.im \cdot \left(\frac{y.im}{y.re} \cdot \frac{1}{y.re}\right)\\ \mathbf{elif}\;y.im \leq 9 \cdot 10^{-23}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{elif}\;y.im \leq 2.6 \cdot 10^{+78}:\\ \;\;\;\;\frac{x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 82.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ t_1 := \left(x.im + x.re \cdot \frac{y.re}{y.im}\right) \cdot \frac{--1}{y.im}\\ \mathbf{if}\;y.im \leq -1.4 \cdot 10^{+58}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y.im \leq -1.85 \cdot 10^{-77}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.im \leq 2.3 \cdot 10^{-113}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{x.im}{y.re \cdot \frac{y.re}{y.im}}\\ \mathbf{elif}\;y.im \leq 7 \cdot 10^{+139}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0
         (/ (+ (* x.re y.re) (* x.im y.im)) (+ (* y.re y.re) (* y.im y.im))))
        (t_1 (* (+ x.im (* x.re (/ y.re y.im))) (/ (- -1.0) y.im))))
   (if (<= y.im -1.4e+58)
     t_1
     (if (<= y.im -1.85e-77)
       t_0
       (if (<= y.im 2.3e-113)
         (+ (/ x.re y.re) (/ x.im (* y.re (/ y.re y.im))))
         (if (<= y.im 7e+139) t_0 t_1))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = ((x_46_re * y_46_re) + (x_46_im * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	double t_1 = (x_46_im + (x_46_re * (y_46_re / y_46_im))) * (-(-1.0) / y_46_im);
	double tmp;
	if (y_46_im <= -1.4e+58) {
		tmp = t_1;
	} else if (y_46_im <= -1.85e-77) {
		tmp = t_0;
	} else if (y_46_im <= 2.3e-113) {
		tmp = (x_46_re / y_46_re) + (x_46_im / (y_46_re * (y_46_re / y_46_im)));
	} else if (y_46_im <= 7e+139) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = ((x_46re * y_46re) + (x_46im * y_46im)) / ((y_46re * y_46re) + (y_46im * y_46im))
    t_1 = (x_46im + (x_46re * (y_46re / y_46im))) * (-(-1.0d0) / y_46im)
    if (y_46im <= (-1.4d+58)) then
        tmp = t_1
    else if (y_46im <= (-1.85d-77)) then
        tmp = t_0
    else if (y_46im <= 2.3d-113) then
        tmp = (x_46re / y_46re) + (x_46im / (y_46re * (y_46re / y_46im)))
    else if (y_46im <= 7d+139) then
        tmp = t_0
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = ((x_46_re * y_46_re) + (x_46_im * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	double t_1 = (x_46_im + (x_46_re * (y_46_re / y_46_im))) * (-(-1.0) / y_46_im);
	double tmp;
	if (y_46_im <= -1.4e+58) {
		tmp = t_1;
	} else if (y_46_im <= -1.85e-77) {
		tmp = t_0;
	} else if (y_46_im <= 2.3e-113) {
		tmp = (x_46_re / y_46_re) + (x_46_im / (y_46_re * (y_46_re / y_46_im)));
	} else if (y_46_im <= 7e+139) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = ((x_46_re * y_46_re) + (x_46_im * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im))
	t_1 = (x_46_im + (x_46_re * (y_46_re / y_46_im))) * (-(-1.0) / y_46_im)
	tmp = 0
	if y_46_im <= -1.4e+58:
		tmp = t_1
	elif y_46_im <= -1.85e-77:
		tmp = t_0
	elif y_46_im <= 2.3e-113:
		tmp = (x_46_re / y_46_re) + (x_46_im / (y_46_re * (y_46_re / y_46_im)))
	elif y_46_im <= 7e+139:
		tmp = t_0
	else:
		tmp = t_1
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(Float64(Float64(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)))
	t_1 = Float64(Float64(x_46_im + Float64(x_46_re * Float64(y_46_re / y_46_im))) * Float64(Float64(-(-1.0)) / y_46_im))
	tmp = 0.0
	if (y_46_im <= -1.4e+58)
		tmp = t_1;
	elseif (y_46_im <= -1.85e-77)
		tmp = t_0;
	elseif (y_46_im <= 2.3e-113)
		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 <= 7e+139)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = ((x_46_re * y_46_re) + (x_46_im * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	t_1 = (x_46_im + (x_46_re * (y_46_re / y_46_im))) * (-(-1.0) / y_46_im);
	tmp = 0.0;
	if (y_46_im <= -1.4e+58)
		tmp = t_1;
	elseif (y_46_im <= -1.85e-77)
		tmp = t_0;
	elseif (y_46_im <= 2.3e-113)
		tmp = (x_46_re / y_46_re) + (x_46_im / (y_46_re * (y_46_re / y_46_im)));
	elseif (y_46_im <= 7e+139)
		tmp = t_0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[(N[(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]}, Block[{t$95$1 = N[(N[(x$46$im + N[(x$46$re * N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[((--1.0) / y$46$im), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$im, -1.4e+58], t$95$1, If[LessEqual[y$46$im, -1.85e-77], t$95$0, If[LessEqual[y$46$im, 2.3e-113], 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, 7e+139], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if y.im < -1.3999999999999999e58 or 6.99999999999999957e139 < y.im

   1. Initial program 41.3%

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

      1. *-un-lft-identity 41.3%

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

      2. associate-*r/ 41.3%

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

      3. fma-def 41.3%

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

      4. add-sqr-sqrt 41.3%

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

      5. times-frac 41.3%

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

      6. fma-def 41.3%

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

      7. hypot-def 41.3%

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

      8. fma-def 41.3%

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

      9. fma-def 41.3%

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

      10. hypot-def 57.5%

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

   4. Applied egg-rr 57.5%

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

   5. Taylor expanded in y.im around -inf 60.6%

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

      1. neg-mul-1 60.6%

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

      2. mul-1-neg 60.6%

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

      3. unsub-neg 60.6%

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

      4. *-lft-identity 60.6%

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

      5. times-frac 63.8%

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

      6. /-rgt-identity 63.8%

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

   7. Simplified 63.8%

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

   8. Taylor expanded in y.im around -inf 87.2%

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

   if -1.3999999999999999e58 < y.im < -1.84999999999999998e-77 or 2.30000000000000008e-113 < y.im < 6.99999999999999957e139

   1. Initial program 73.8%

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

   if -1.84999999999999998e-77 < y.im < 2.30000000000000008e-113

   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. Add Preprocessing

   3. Taylor expanded in y.re around inf 85.0%

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

      1. associate-/l* 82.4%

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

   5. Simplified 82.4%

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

      1. pow2 82.4%

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

      2. *-un-lft-identity 82.4%

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

      3. times-frac 91.9%

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

   7. Applied egg-rr 91.9%

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

4. Final simplification 83.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -1.4 \cdot 10^{+58}:\\ \;\;\;\;\left(x.im + x.re \cdot \frac{y.re}{y.im}\right) \cdot \frac{--1}{y.im}\\ \mathbf{elif}\;y.im \leq -1.85 \cdot 10^{-77}:\\ \;\;\;\;\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.im \leq 2.3 \cdot 10^{-113}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{x.im}{y.re \cdot \frac{y.re}{y.im}}\\ \mathbf{elif}\;y.im \leq 7 \cdot 10^{+139}:\\ \;\;\;\;\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\left(x.im + x.re \cdot \frac{y.re}{y.im}\right) \cdot \frac{--1}{y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 73.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(x.im + x.re \cdot \frac{y.re}{y.im}\right) \cdot \frac{--1}{y.im}\\ \mathbf{if}\;y.re \leq -3.2 \cdot 10^{-19}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{elif}\;y.re \leq 1.75 \cdot 10^{-35}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.re \leq 3.8 \cdot 10^{+20}:\\ \;\;\;\;\frac{x.re \cdot y.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.re \leq 5.8 \cdot 10^{+77}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (* (+ x.im (* x.re (/ y.re y.im))) (/ (- -1.0) y.im))))
   (if (<= y.re -3.2e-19)
     (/ x.re y.re)
     (if (<= y.re 1.75e-35)
       t_0
       (if (<= y.re 3.8e+20)
         (/ (* x.re y.re) (+ (* y.re y.re) (* y.im y.im)))
         (if (<= y.re 5.8e+77) t_0 (/ x.re y.re)))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = (x_46_im + (x_46_re * (y_46_re / y_46_im))) * (-(-1.0) / y_46_im);
	double tmp;
	if (y_46_re <= -3.2e-19) {
		tmp = x_46_re / y_46_re;
	} else if (y_46_re <= 1.75e-35) {
		tmp = t_0;
	} else if (y_46_re <= 3.8e+20) {
		tmp = (x_46_re * y_46_re) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	} else if (y_46_re <= 5.8e+77) {
		tmp = t_0;
	} 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) :: t_0
    real(8) :: tmp
    t_0 = (x_46im + (x_46re * (y_46re / y_46im))) * (-(-1.0d0) / y_46im)
    if (y_46re <= (-3.2d-19)) then
        tmp = x_46re / y_46re
    else if (y_46re <= 1.75d-35) then
        tmp = t_0
    else if (y_46re <= 3.8d+20) then
        tmp = (x_46re * y_46re) / ((y_46re * y_46re) + (y_46im * y_46im))
    else if (y_46re <= 5.8d+77) then
        tmp = t_0
    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 t_0 = (x_46_im + (x_46_re * (y_46_re / y_46_im))) * (-(-1.0) / y_46_im);
	double tmp;
	if (y_46_re <= -3.2e-19) {
		tmp = x_46_re / y_46_re;
	} else if (y_46_re <= 1.75e-35) {
		tmp = t_0;
	} else if (y_46_re <= 3.8e+20) {
		tmp = (x_46_re * y_46_re) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	} else if (y_46_re <= 5.8e+77) {
		tmp = t_0;
	} 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):
	t_0 = (x_46_im + (x_46_re * (y_46_re / y_46_im))) * (-(-1.0) / y_46_im)
	tmp = 0
	if y_46_re <= -3.2e-19:
		tmp = x_46_re / y_46_re
	elif y_46_re <= 1.75e-35:
		tmp = t_0
	elif y_46_re <= 3.8e+20:
		tmp = (x_46_re * y_46_re) / ((y_46_re * y_46_re) + (y_46_im * y_46_im))
	elif y_46_re <= 5.8e+77:
		tmp = t_0
	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)
	t_0 = Float64(Float64(x_46_im + Float64(x_46_re * Float64(y_46_re / y_46_im))) * Float64(Float64(-(-1.0)) / y_46_im))
	tmp = 0.0
	if (y_46_re <= -3.2e-19)
		tmp = Float64(x_46_re / y_46_re);
	elseif (y_46_re <= 1.75e-35)
		tmp = t_0;
	elseif (y_46_re <= 3.8e+20)
		tmp = Float64(Float64(x_46_re * y_46_re) / Float64(Float64(y_46_re * y_46_re) + Float64(y_46_im * y_46_im)));
	elseif (y_46_re <= 5.8e+77)
		tmp = t_0;
	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)
	t_0 = (x_46_im + (x_46_re * (y_46_re / y_46_im))) * (-(-1.0) / y_46_im);
	tmp = 0.0;
	if (y_46_re <= -3.2e-19)
		tmp = x_46_re / y_46_re;
	elseif (y_46_re <= 1.75e-35)
		tmp = t_0;
	elseif (y_46_re <= 3.8e+20)
		tmp = (x_46_re * y_46_re) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	elseif (y_46_re <= 5.8e+77)
		tmp = t_0;
	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_] := Block[{t$95$0 = N[(N[(x$46$im + N[(x$46$re * N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[((--1.0) / y$46$im), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$re, -3.2e-19], N[(x$46$re / y$46$re), $MachinePrecision], If[LessEqual[y$46$re, 1.75e-35], t$95$0, If[LessEqual[y$46$re, 3.8e+20], N[(N[(x$46$re * y$46$re), $MachinePrecision] / N[(N[(y$46$re * y$46$re), $MachinePrecision] + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 5.8e+77], t$95$0, N[(x$46$re / y$46$re), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -3.19999999999999982e-19 or 5.8000000000000003e77 < y.re

   1. Initial program 47.8%

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

   3. Taylor expanded in y.re around inf 66.8%

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

   if -3.19999999999999982e-19 < y.re < 1.74999999999999998e-35 or 3.8e20 < y.re < 5.8000000000000003e77

   1. Initial program 68.2%

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

      1. *-un-lft-identity 68.2%

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

      2. associate-*r/ 68.2%

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

      3. fma-def 68.2%

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

      4. add-sqr-sqrt 68.2%

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

      5. times-frac 68.2%

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

      6. fma-def 68.2%

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

      7. hypot-def 68.2%

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

      8. fma-def 68.2%

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

      9. fma-def 68.2%

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

      10. hypot-def 80.4%

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

   4. Applied egg-rr 80.4%

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

   5. Taylor expanded in y.im around -inf 51.1%

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

      1. neg-mul-1 51.1%

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

      2. mul-1-neg 51.1%

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

      3. unsub-neg 51.1%

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

      4. *-lft-identity 51.1%

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

      5. times-frac 51.1%

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

      6. /-rgt-identity 51.1%

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

   7. Simplified 51.1%

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

   8. Taylor expanded in y.im around -inf 85.8%

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

   if 1.74999999999999998e-35 < y.re < 3.8e20

   1. Initial program 93.5%

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

   3. Taylor expanded in x.re around inf 64.1%

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

      1. *-commutative 64.1%

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

   5. Simplified 64.1%

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

4. Final simplification 75.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -3.2 \cdot 10^{-19}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{elif}\;y.re \leq 1.75 \cdot 10^{-35}:\\ \;\;\;\;\left(x.im + x.re \cdot \frac{y.re}{y.im}\right) \cdot \frac{--1}{y.im}\\ \mathbf{elif}\;y.re \leq 3.8 \cdot 10^{+20}:\\ \;\;\;\;\frac{x.re \cdot y.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.re \leq 5.8 \cdot 10^{+77}:\\ \;\;\;\;\left(x.im + x.re \cdot \frac{y.re}{y.im}\right) \cdot \frac{--1}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 76.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x.re}{y.re} + \frac{x.im}{y.re \cdot \frac{y.re}{y.im}}\\ t_1 := \left(x.im + x.re \cdot \frac{y.re}{y.im}\right) \cdot \frac{--1}{y.im}\\ \mathbf{if}\;y.im \leq -6 \cdot 10^{+25}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y.im \leq 6.5 \cdot 10^{-21}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.im \leq 2.05 \cdot 10^{+27}:\\ \;\;\;\;\frac{x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.im \leq 2.9 \cdot 10^{+78}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (+ (/ x.re y.re) (/ x.im (* y.re (/ y.re y.im)))))
        (t_1 (* (+ x.im (* x.re (/ y.re y.im))) (/ (- -1.0) y.im))))
   (if (<= y.im -6e+25)
     t_1
     (if (<= y.im 6.5e-21)
       t_0
       (if (<= y.im 2.05e+27)
         (/ (* x.im y.im) (+ (* y.re y.re) (* y.im y.im)))
         (if (<= y.im 2.9e+78) t_0 t_1))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = (x_46_re / y_46_re) + (x_46_im / (y_46_re * (y_46_re / y_46_im)));
	double t_1 = (x_46_im + (x_46_re * (y_46_re / y_46_im))) * (-(-1.0) / y_46_im);
	double tmp;
	if (y_46_im <= -6e+25) {
		tmp = t_1;
	} else if (y_46_im <= 6.5e-21) {
		tmp = t_0;
	} else if (y_46_im <= 2.05e+27) {
		tmp = (x_46_im * y_46_im) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	} else if (y_46_im <= 2.9e+78) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (x_46re / y_46re) + (x_46im / (y_46re * (y_46re / y_46im)))
    t_1 = (x_46im + (x_46re * (y_46re / y_46im))) * (-(-1.0d0) / y_46im)
    if (y_46im <= (-6d+25)) then
        tmp = t_1
    else if (y_46im <= 6.5d-21) then
        tmp = t_0
    else if (y_46im <= 2.05d+27) then
        tmp = (x_46im * y_46im) / ((y_46re * y_46re) + (y_46im * y_46im))
    else if (y_46im <= 2.9d+78) then
        tmp = t_0
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = (x_46_re / y_46_re) + (x_46_im / (y_46_re * (y_46_re / y_46_im)));
	double t_1 = (x_46_im + (x_46_re * (y_46_re / y_46_im))) * (-(-1.0) / y_46_im);
	double tmp;
	if (y_46_im <= -6e+25) {
		tmp = t_1;
	} else if (y_46_im <= 6.5e-21) {
		tmp = t_0;
	} else if (y_46_im <= 2.05e+27) {
		tmp = (x_46_im * y_46_im) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	} else if (y_46_im <= 2.9e+78) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = (x_46_re / y_46_re) + (x_46_im / (y_46_re * (y_46_re / y_46_im)))
	t_1 = (x_46_im + (x_46_re * (y_46_re / y_46_im))) * (-(-1.0) / y_46_im)
	tmp = 0
	if y_46_im <= -6e+25:
		tmp = t_1
	elif y_46_im <= 6.5e-21:
		tmp = t_0
	elif y_46_im <= 2.05e+27:
		tmp = (x_46_im * y_46_im) / ((y_46_re * y_46_re) + (y_46_im * y_46_im))
	elif y_46_im <= 2.9e+78:
		tmp = t_0
	else:
		tmp = t_1
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(Float64(x_46_re / y_46_re) + Float64(x_46_im / Float64(y_46_re * Float64(y_46_re / y_46_im))))
	t_1 = Float64(Float64(x_46_im + Float64(x_46_re * Float64(y_46_re / y_46_im))) * Float64(Float64(-(-1.0)) / y_46_im))
	tmp = 0.0
	if (y_46_im <= -6e+25)
		tmp = t_1;
	elseif (y_46_im <= 6.5e-21)
		tmp = t_0;
	elseif (y_46_im <= 2.05e+27)
		tmp = Float64(Float64(x_46_im * y_46_im) / Float64(Float64(y_46_re * y_46_re) + Float64(y_46_im * y_46_im)));
	elseif (y_46_im <= 2.9e+78)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = (x_46_re / y_46_re) + (x_46_im / (y_46_re * (y_46_re / y_46_im)));
	t_1 = (x_46_im + (x_46_re * (y_46_re / y_46_im))) * (-(-1.0) / y_46_im);
	tmp = 0.0;
	if (y_46_im <= -6e+25)
		tmp = t_1;
	elseif (y_46_im <= 6.5e-21)
		tmp = t_0;
	elseif (y_46_im <= 2.05e+27)
		tmp = (x_46_im * y_46_im) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	elseif (y_46_im <= 2.9e+78)
		tmp = t_0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[(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]}, Block[{t$95$1 = N[(N[(x$46$im + N[(x$46$re * N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[((--1.0) / y$46$im), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$im, -6e+25], t$95$1, If[LessEqual[y$46$im, 6.5e-21], t$95$0, If[LessEqual[y$46$im, 2.05e+27], N[(N[(x$46$im * y$46$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, 2.9e+78], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if y.im < -6.00000000000000011e25 or 2.90000000000000017e78 < y.im

   1. Initial program 46.8%

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

      1. *-un-lft-identity 46.8%

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

      2. associate-*r/ 46.8%

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

      3. fma-def 46.8%

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

      4. add-sqr-sqrt 46.8%

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

      5. times-frac 46.7%

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

      6. fma-def 46.7%

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

      7. hypot-def 46.7%

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

      8. fma-def 46.8%

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

      9. fma-def 46.8%

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

      10. hypot-def 63.0%

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

   4. Applied egg-rr 63.0%

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

   5. Taylor expanded in y.im around -inf 54.0%

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

      1. neg-mul-1 54.0%

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

      2. mul-1-neg 54.0%

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

      3. unsub-neg 54.0%

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

      4. *-lft-identity 54.0%

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

      5. times-frac 57.4%

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

      6. /-rgt-identity 57.4%

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

   7. Simplified 57.4%

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

   8. Taylor expanded in y.im around -inf 83.8%

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

   if -6.00000000000000011e25 < y.im < 6.49999999999999987e-21 or 2.0500000000000001e27 < y.im < 2.90000000000000017e78

   1. Initial program 69.4%

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

   3. Taylor expanded in y.re around inf 74.0%

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

      1. associate-/l* 71.8%

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

   5. Simplified 71.8%

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

      1. pow2 71.8%

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

      2. *-un-lft-identity 71.8%

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

      3. times-frac 77.1%

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

   7. Applied egg-rr 77.1%

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

   if 6.49999999999999987e-21 < y.im < 2.0500000000000001e27

   1. Initial program 99.3%

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

   3. Taylor expanded in x.re around 0 90.4%

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

4. Final simplification 80.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -6 \cdot 10^{+25}:\\ \;\;\;\;\left(x.im + x.re \cdot \frac{y.re}{y.im}\right) \cdot \frac{--1}{y.im}\\ \mathbf{elif}\;y.im \leq 6.5 \cdot 10^{-21}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{x.im}{y.re \cdot \frac{y.re}{y.im}}\\ \mathbf{elif}\;y.im \leq 2.05 \cdot 10^{+27}:\\ \;\;\;\;\frac{x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.im \leq 2.9 \cdot 10^{+78}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{x.im}{y.re \cdot \frac{y.re}{y.im}}\\ \mathbf{else}:\\ \;\;\;\;\left(x.im + x.re \cdot \frac{y.re}{y.im}\right) \cdot \frac{--1}{y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 65.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x.re \cdot y.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{if}\;y.re \leq -1.35 \cdot 10^{+56}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{elif}\;y.re \leq -1.25 \cdot 10^{-127}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.re \leq 1.75 \cdot 10^{-35}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{elif}\;y.re \leq 10^{+44}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.re \leq 3.5 \cdot 10^{+76}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (/ (* x.re y.re) (+ (* y.re y.re) (* y.im y.im)))))
   (if (<= y.re -1.35e+56)
     (/ x.re y.re)
     (if (<= y.re -1.25e-127)
       t_0
       (if (<= y.re 1.75e-35)
         (/ x.im y.im)
         (if (<= y.re 1e+44)
           t_0
           (if (<= y.re 3.5e+76) (/ 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 t_0 = (x_46_re * y_46_re) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	double tmp;
	if (y_46_re <= -1.35e+56) {
		tmp = x_46_re / y_46_re;
	} else if (y_46_re <= -1.25e-127) {
		tmp = t_0;
	} else if (y_46_re <= 1.75e-35) {
		tmp = x_46_im / y_46_im;
	} else if (y_46_re <= 1e+44) {
		tmp = t_0;
	} else if (y_46_re <= 3.5e+76) {
		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) :: t_0
    real(8) :: tmp
    t_0 = (x_46re * y_46re) / ((y_46re * y_46re) + (y_46im * y_46im))
    if (y_46re <= (-1.35d+56)) then
        tmp = x_46re / y_46re
    else if (y_46re <= (-1.25d-127)) then
        tmp = t_0
    else if (y_46re <= 1.75d-35) then
        tmp = x_46im / y_46im
    else if (y_46re <= 1d+44) then
        tmp = t_0
    else if (y_46re <= 3.5d+76) 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 t_0 = (x_46_re * y_46_re) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	double tmp;
	if (y_46_re <= -1.35e+56) {
		tmp = x_46_re / y_46_re;
	} else if (y_46_re <= -1.25e-127) {
		tmp = t_0;
	} else if (y_46_re <= 1.75e-35) {
		tmp = x_46_im / y_46_im;
	} else if (y_46_re <= 1e+44) {
		tmp = t_0;
	} else if (y_46_re <= 3.5e+76) {
		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):
	t_0 = (x_46_re * y_46_re) / ((y_46_re * y_46_re) + (y_46_im * y_46_im))
	tmp = 0
	if y_46_re <= -1.35e+56:
		tmp = x_46_re / y_46_re
	elif y_46_re <= -1.25e-127:
		tmp = t_0
	elif y_46_re <= 1.75e-35:
		tmp = x_46_im / y_46_im
	elif y_46_re <= 1e+44:
		tmp = t_0
	elif y_46_re <= 3.5e+76:
		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)
	t_0 = Float64(Float64(x_46_re * y_46_re) / Float64(Float64(y_46_re * y_46_re) + Float64(y_46_im * y_46_im)))
	tmp = 0.0
	if (y_46_re <= -1.35e+56)
		tmp = Float64(x_46_re / y_46_re);
	elseif (y_46_re <= -1.25e-127)
		tmp = t_0;
	elseif (y_46_re <= 1.75e-35)
		tmp = Float64(x_46_im / y_46_im);
	elseif (y_46_re <= 1e+44)
		tmp = t_0;
	elseif (y_46_re <= 3.5e+76)
		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)
	t_0 = (x_46_re * y_46_re) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	tmp = 0.0;
	if (y_46_re <= -1.35e+56)
		tmp = x_46_re / y_46_re;
	elseif (y_46_re <= -1.25e-127)
		tmp = t_0;
	elseif (y_46_re <= 1.75e-35)
		tmp = x_46_im / y_46_im;
	elseif (y_46_re <= 1e+44)
		tmp = t_0;
	elseif (y_46_re <= 3.5e+76)
		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_] := Block[{t$95$0 = N[(N[(x$46$re * y$46$re), $MachinePrecision] / N[(N[(y$46$re * y$46$re), $MachinePrecision] + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$re, -1.35e+56], N[(x$46$re / y$46$re), $MachinePrecision], If[LessEqual[y$46$re, -1.25e-127], t$95$0, If[LessEqual[y$46$re, 1.75e-35], N[(x$46$im / y$46$im), $MachinePrecision], If[LessEqual[y$46$re, 1e+44], t$95$0, If[LessEqual[y$46$re, 3.5e+76], N[(x$46$im / y$46$im), $MachinePrecision], N[(x$46$re / y$46$re), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -1.35000000000000005e56 or 3.5e76 < y.re

   1. Initial program 43.8%

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

   3. Taylor expanded in y.re around inf 68.4%

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

   if -1.35000000000000005e56 < y.re < -1.2499999999999999e-127 or 1.74999999999999998e-35 < y.re < 1.0000000000000001e44

   1. Initial program 85.4%

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

   3. Taylor expanded in x.re around inf 65.6%

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

      1. *-commutative 65.6%

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

   5. Simplified 65.6%

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

   if -1.2499999999999999e-127 < y.re < 1.74999999999999998e-35 or 1.0000000000000001e44 < y.re < 3.5e76

   1. Initial program 65.0%

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

   3. Taylor expanded in y.re around 0 75.3%

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

4. Final simplification 70.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -1.35 \cdot 10^{+56}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{elif}\;y.re \leq -1.25 \cdot 10^{-127}:\\ \;\;\;\;\frac{x.re \cdot y.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.re \leq 1.75 \cdot 10^{-35}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{elif}\;y.re \leq 10^{+44}:\\ \;\;\;\;\frac{x.re \cdot y.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.re \leq 3.5 \cdot 10^{+76}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 63.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -2.5 \cdot 10^{+29}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{elif}\;y.im \leq -1.4 \cdot 10^{-274}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{elif}\;y.im \leq -8.2 \cdot 10^{-303}:\\ \;\;\;\;x.im \cdot \left(\frac{y.im}{y.re} \cdot \frac{1}{y.re}\right)\\ \mathbf{elif}\;y.im \leq 6.2 \cdot 10^{-19}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.im -2.5e+29)
   (/ x.im y.im)
   (if (<= y.im -1.4e-274)
     (/ x.re y.re)
     (if (<= y.im -8.2e-303)
       (* x.im (* (/ y.im y.re) (/ 1.0 y.re)))
       (if (<= y.im 6.2e-19) (/ x.re y.re) (/ x.im y.im))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_im <= -2.5e+29) {
		tmp = x_46_im / y_46_im;
	} else if (y_46_im <= -1.4e-274) {
		tmp = x_46_re / y_46_re;
	} else if (y_46_im <= -8.2e-303) {
		tmp = x_46_im * ((y_46_im / y_46_re) * (1.0 / y_46_re));
	} else if (y_46_im <= 6.2e-19) {
		tmp = x_46_re / y_46_re;
	} else {
		tmp = x_46_im / y_46_im;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if (y_46im <= (-2.5d+29)) then
        tmp = x_46im / y_46im
    else if (y_46im <= (-1.4d-274)) then
        tmp = x_46re / y_46re
    else if (y_46im <= (-8.2d-303)) then
        tmp = x_46im * ((y_46im / y_46re) * (1.0d0 / y_46re))
    else if (y_46im <= 6.2d-19) then
        tmp = x_46re / y_46re
    else
        tmp = x_46im / y_46im
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_im <= -2.5e+29) {
		tmp = x_46_im / y_46_im;
	} else if (y_46_im <= -1.4e-274) {
		tmp = x_46_re / y_46_re;
	} else if (y_46_im <= -8.2e-303) {
		tmp = x_46_im * ((y_46_im / y_46_re) * (1.0 / y_46_re));
	} else if (y_46_im <= 6.2e-19) {
		tmp = x_46_re / y_46_re;
	} else {
		tmp = x_46_im / y_46_im;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if y_46_im <= -2.5e+29:
		tmp = x_46_im / y_46_im
	elif y_46_im <= -1.4e-274:
		tmp = x_46_re / y_46_re
	elif y_46_im <= -8.2e-303:
		tmp = x_46_im * ((y_46_im / y_46_re) * (1.0 / y_46_re))
	elif y_46_im <= 6.2e-19:
		tmp = x_46_re / y_46_re
	else:
		tmp = x_46_im / y_46_im
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (y_46_im <= -2.5e+29)
		tmp = Float64(x_46_im / y_46_im);
	elseif (y_46_im <= -1.4e-274)
		tmp = Float64(x_46_re / y_46_re);
	elseif (y_46_im <= -8.2e-303)
		tmp = Float64(x_46_im * Float64(Float64(y_46_im / y_46_re) * Float64(1.0 / y_46_re)));
	elseif (y_46_im <= 6.2e-19)
		tmp = Float64(x_46_re / y_46_re);
	else
		tmp = Float64(x_46_im / y_46_im);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if (y_46_im <= -2.5e+29)
		tmp = x_46_im / y_46_im;
	elseif (y_46_im <= -1.4e-274)
		tmp = x_46_re / y_46_re;
	elseif (y_46_im <= -8.2e-303)
		tmp = x_46_im * ((y_46_im / y_46_re) * (1.0 / y_46_re));
	elseif (y_46_im <= 6.2e-19)
		tmp = x_46_re / y_46_re;
	else
		tmp = x_46_im / y_46_im;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[y$46$im, -2.5e+29], N[(x$46$im / y$46$im), $MachinePrecision], If[LessEqual[y$46$im, -1.4e-274], N[(x$46$re / y$46$re), $MachinePrecision], If[LessEqual[y$46$im, -8.2e-303], N[(x$46$im * N[(N[(y$46$im / y$46$re), $MachinePrecision] * N[(1.0 / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, 6.2e-19], N[(x$46$re / y$46$re), $MachinePrecision], N[(x$46$im / y$46$im), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if y.im < -2.5e29 or 6.1999999999999998e-19 < y.im

   1. Initial program 52.1%

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

   3. Taylor expanded in y.re around 0 64.0%

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

   if -2.5e29 < y.im < -1.39999999999999988e-274 or -8.20000000000000037e-303 < y.im < 6.1999999999999998e-19

   1. Initial program 71.2%

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

   3. Taylor expanded in y.re around inf 69.1%

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

   if -1.39999999999999988e-274 < y.im < -8.20000000000000037e-303

   1. Initial program 45.3%

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

   3. Taylor expanded in x.re around 0 45.7%

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

   4. Taylor expanded in y.im around 0 45.7%

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

      1. associate-*r/ 45.7%

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

   6. Simplified 45.7%

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

      1. *-un-lft-identity 45.7%

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

      2. unpow2 45.7%

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

      3. times-frac 85.7%

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

   8. Applied egg-rr 85.7%

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

4. Final simplification 66.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -2.5 \cdot 10^{+29}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{elif}\;y.im \leq -1.4 \cdot 10^{-274}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{elif}\;y.im \leq -8.2 \cdot 10^{-303}:\\ \;\;\;\;x.im \cdot \left(\frac{y.im}{y.re} \cdot \frac{1}{y.re}\right)\\ \mathbf{elif}\;y.im \leq 6.2 \cdot 10^{-19}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 63.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -8.6 \cdot 10^{+23}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{elif}\;y.im \leq -1.4 \cdot 10^{-274}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{elif}\;y.im \leq -8.2 \cdot 10^{-303}:\\ \;\;\;\;\frac{\frac{x.im \cdot y.im}{y.re}}{y.re}\\ \mathbf{elif}\;y.im \leq 8 \cdot 10^{-19}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.im -8.6e+23)
   (/ x.im y.im)
   (if (<= y.im -1.4e-274)
     (/ x.re y.re)
     (if (<= y.im -8.2e-303)
       (/ (/ (* x.im y.im) y.re) y.re)
       (if (<= y.im 8e-19) (/ x.re y.re) (/ x.im y.im))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_im <= -8.6e+23) {
		tmp = x_46_im / y_46_im;
	} else if (y_46_im <= -1.4e-274) {
		tmp = x_46_re / y_46_re;
	} else if (y_46_im <= -8.2e-303) {
		tmp = ((x_46_im * y_46_im) / y_46_re) / y_46_re;
	} else if (y_46_im <= 8e-19) {
		tmp = x_46_re / y_46_re;
	} else {
		tmp = x_46_im / y_46_im;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if (y_46im <= (-8.6d+23)) then
        tmp = x_46im / y_46im
    else if (y_46im <= (-1.4d-274)) then
        tmp = x_46re / y_46re
    else if (y_46im <= (-8.2d-303)) then
        tmp = ((x_46im * y_46im) / y_46re) / y_46re
    else if (y_46im <= 8d-19) then
        tmp = x_46re / y_46re
    else
        tmp = x_46im / y_46im
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_im <= -8.6e+23) {
		tmp = x_46_im / y_46_im;
	} else if (y_46_im <= -1.4e-274) {
		tmp = x_46_re / y_46_re;
	} else if (y_46_im <= -8.2e-303) {
		tmp = ((x_46_im * y_46_im) / y_46_re) / y_46_re;
	} else if (y_46_im <= 8e-19) {
		tmp = x_46_re / y_46_re;
	} else {
		tmp = x_46_im / y_46_im;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if y_46_im <= -8.6e+23:
		tmp = x_46_im / y_46_im
	elif y_46_im <= -1.4e-274:
		tmp = x_46_re / y_46_re
	elif y_46_im <= -8.2e-303:
		tmp = ((x_46_im * y_46_im) / y_46_re) / y_46_re
	elif y_46_im <= 8e-19:
		tmp = x_46_re / y_46_re
	else:
		tmp = x_46_im / y_46_im
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (y_46_im <= -8.6e+23)
		tmp = Float64(x_46_im / y_46_im);
	elseif (y_46_im <= -1.4e-274)
		tmp = Float64(x_46_re / y_46_re);
	elseif (y_46_im <= -8.2e-303)
		tmp = Float64(Float64(Float64(x_46_im * y_46_im) / y_46_re) / y_46_re);
	elseif (y_46_im <= 8e-19)
		tmp = Float64(x_46_re / y_46_re);
	else
		tmp = Float64(x_46_im / y_46_im);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if (y_46_im <= -8.6e+23)
		tmp = x_46_im / y_46_im;
	elseif (y_46_im <= -1.4e-274)
		tmp = x_46_re / y_46_re;
	elseif (y_46_im <= -8.2e-303)
		tmp = ((x_46_im * y_46_im) / y_46_re) / y_46_re;
	elseif (y_46_im <= 8e-19)
		tmp = x_46_re / y_46_re;
	else
		tmp = x_46_im / y_46_im;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[y$46$im, -8.6e+23], N[(x$46$im / y$46$im), $MachinePrecision], If[LessEqual[y$46$im, -1.4e-274], N[(x$46$re / y$46$re), $MachinePrecision], If[LessEqual[y$46$im, -8.2e-303], N[(N[(N[(x$46$im * y$46$im), $MachinePrecision] / y$46$re), $MachinePrecision] / y$46$re), $MachinePrecision], If[LessEqual[y$46$im, 8e-19], N[(x$46$re / y$46$re), $MachinePrecision], N[(x$46$im / y$46$im), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if y.im < -8.5999999999999997e23 or 7.9999999999999998e-19 < y.im

   1. Initial program 52.1%

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

   3. Taylor expanded in y.re around 0 64.0%

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

   if -8.5999999999999997e23 < y.im < -1.39999999999999988e-274 or -8.20000000000000037e-303 < y.im < 7.9999999999999998e-19

   1. Initial program 71.2%

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

   3. Taylor expanded in y.re around inf 69.1%

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

   if -1.39999999999999988e-274 < y.im < -8.20000000000000037e-303

   1. Initial program 45.3%

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

   3. Taylor expanded in x.re around 0 45.7%

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

   4. Taylor expanded in y.im around 0 45.7%

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

      1. associate-*r/ 45.7%

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

   6. Simplified 45.7%

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

      1. associate-*r/ 45.7%

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

      2. unpow2 45.7%

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

      3. associate-/r* 85.5%

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

   8. Applied egg-rr 85.5%

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

4. Final simplification 66.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -8.6 \cdot 10^{+23}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{elif}\;y.im \leq -1.4 \cdot 10^{-274}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{elif}\;y.im \leq -8.2 \cdot 10^{-303}:\\ \;\;\;\;\frac{\frac{x.im \cdot y.im}{y.re}}{y.re}\\ \mathbf{elif}\;y.im \leq 8 \cdot 10^{-19}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 64.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -6.2 \cdot 10^{+31} \lor \neg \left(y.im \leq 7.5 \cdot 10^{-20}\right):\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.im -6.2e+31) (not (<= y.im 7.5e-20)))
   (/ 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 <= -6.2e+31) || !(y_46_im <= 7.5e-20)) {
		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 <= (-6.2d+31)) .or. (.not. (y_46im <= 7.5d-20))) 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 <= -6.2e+31) || !(y_46_im <= 7.5e-20)) {
		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 <= -6.2e+31) or not (y_46_im <= 7.5e-20):
		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 <= -6.2e+31) || !(y_46_im <= 7.5e-20))
		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 <= -6.2e+31) || ~((y_46_im <= 7.5e-20)))
		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, -6.2e+31], N[Not[LessEqual[y$46$im, 7.5e-20]], $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 < -6.2000000000000004e31 or 7.49999999999999981e-20 < y.im

   1. Initial program 52.1%

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

   3. Taylor expanded in y.re around 0 64.0%

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

   if -6.2000000000000004e31 < y.im < 7.49999999999999981e-20

   1. Initial program 69.7%

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

   3. Taylor expanded in y.re around inf 66.2%

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

4. Final simplification 65.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -6.2 \cdot 10^{+31} \lor \neg \left(y.im \leq 7.5 \cdot 10^{-20}\right):\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 42.9% 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 60.4%

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

3. Taylor expanded in y.re around 0 41.7%

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

4. Final simplification 41.7%

   \[\leadsto \frac{x.im}{y.im} \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024026 
(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))))