_divideComplex, real part

Percentage Accurate: 62.0% → 85.1%

Time: 12.5s

Alternatives: 13

Speedup: 2.1×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 62.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 85.1% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(y.im, y.im, {y.re}^{2}\right)\\ t_1 := \frac{x.re}{\frac{y.im}{y.re}}\\ \mathbf{if}\;y.im \leq -3.7 \cdot 10^{+151}:\\ \;\;\;\;\frac{\left(-x.im\right) - t_1}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.im \leq -7.5 \cdot 10^{-185}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(x.re, y.re, y.im \cdot x.im\right)}{\mathsf{hypot}\left(y.re, y.im\right)}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.im \leq 1.55 \cdot 10^{-133}:\\ \;\;\;\;\frac{-1}{y.re} \cdot \left(\left(-x.re\right) - \frac{y.im \cdot x.im}{y.re}\right)\\ \mathbf{elif}\;y.im \leq 1.65 \cdot 10^{+101}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x.im}{t_0}, y.im, \frac{x.re}{\frac{t_0}{y.re}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im + t_1}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (fma y.im y.im (pow y.re 2.0))) (t_1 (/ x.re (/ y.im y.re))))
   (if (<= y.im -3.7e+151)
     (/ (- (- x.im) t_1) (hypot y.re y.im))
     (if (<= y.im -7.5e-185)
       (/
        (/ (fma x.re y.re (* y.im x.im)) (hypot y.re y.im))
        (hypot y.re y.im))
       (if (<= y.im 1.55e-133)
         (* (/ -1.0 y.re) (- (- x.re) (/ (* y.im x.im) y.re)))
         (if (<= y.im 1.65e+101)
           (fma (/ x.im t_0) y.im (/ x.re (/ t_0 y.re)))
           (/ (+ x.im t_1) (hypot y.re y.im))))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = fma(y_46_im, y_46_im, pow(y_46_re, 2.0));
	double t_1 = x_46_re / (y_46_im / y_46_re);
	double tmp;
	if (y_46_im <= -3.7e+151) {
		tmp = (-x_46_im - t_1) / hypot(y_46_re, y_46_im);
	} else if (y_46_im <= -7.5e-185) {
		tmp = (fma(x_46_re, y_46_re, (y_46_im * x_46_im)) / hypot(y_46_re, y_46_im)) / hypot(y_46_re, y_46_im);
	} else if (y_46_im <= 1.55e-133) {
		tmp = (-1.0 / y_46_re) * (-x_46_re - ((y_46_im * x_46_im) / y_46_re));
	} else if (y_46_im <= 1.65e+101) {
		tmp = fma((x_46_im / t_0), y_46_im, (x_46_re / (t_0 / y_46_re)));
	} else {
		tmp = (x_46_im + t_1) / 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 = fma(y_46_im, y_46_im, (y_46_re ^ 2.0))
	t_1 = Float64(x_46_re / Float64(y_46_im / y_46_re))
	tmp = 0.0
	if (y_46_im <= -3.7e+151)
		tmp = Float64(Float64(Float64(-x_46_im) - t_1) / hypot(y_46_re, y_46_im));
	elseif (y_46_im <= -7.5e-185)
		tmp = Float64(Float64(fma(x_46_re, y_46_re, Float64(y_46_im * x_46_im)) / hypot(y_46_re, y_46_im)) / hypot(y_46_re, y_46_im));
	elseif (y_46_im <= 1.55e-133)
		tmp = Float64(Float64(-1.0 / y_46_re) * Float64(Float64(-x_46_re) - Float64(Float64(y_46_im * x_46_im) / y_46_re)));
	elseif (y_46_im <= 1.65e+101)
		tmp = fma(Float64(x_46_im / t_0), y_46_im, Float64(x_46_re / Float64(t_0 / y_46_re)));
	else
		tmp = Float64(Float64(x_46_im + t_1) / 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[(y$46$im * y$46$im + N[Power[y$46$re, 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x$46$re / N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$im, -3.7e+151], N[(N[((-x$46$im) - t$95$1), $MachinePrecision] / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, -7.5e-185], N[(N[(N[(x$46$re * y$46$re + N[(y$46$im * x$46$im), $MachinePrecision]), $MachinePrecision] / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision] / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, 1.55e-133], N[(N[(-1.0 / y$46$re), $MachinePrecision] * N[((-x$46$re) - N[(N[(y$46$im * x$46$im), $MachinePrecision] / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, 1.65e+101], N[(N[(x$46$im / t$95$0), $MachinePrecision] * y$46$im + N[(x$46$re / N[(t$95$0 / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$im + t$95$1), $MachinePrecision] / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 5 regimes

2. if y.im < -3.6999999999999997e151

   1. Initial program 21.5%

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

      1. *-un-lft-identity 21.5%

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

      2. add-sqr-sqrt 21.5%

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

      3. times-frac 21.5%

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

      4. hypot-def 21.5%

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

      5. fma-def 21.5%

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

      6. hypot-def 50.0%

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

   3. Applied egg-rr 50.0%

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

      1. associate-*l/ 50.0%

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

      2. *-un-lft-identity 50.0%

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

   5. Applied egg-rr 50.0%

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

   6. Taylor expanded in y.im around -inf 78.9%

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

      1. mul-1-neg 78.9%

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

      2. unsub-neg 78.9%

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

      3. mul-1-neg 78.9%

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

      4. associate-/l* 93.2%

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

   8. Simplified 93.2%

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

   if -3.6999999999999997e151 < y.im < -7.49999999999999978e-185

   1. Initial program 69.0%

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

      1. *-un-lft-identity 69.0%

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

      2. add-sqr-sqrt 68.9%

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

      3. times-frac 69.0%

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

      4. hypot-def 69.1%

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

      5. fma-def 69.1%

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

      6. hypot-def 83.2%

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

   3. Applied egg-rr 83.2%

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

      1. associate-*l/ 83.4%

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

      2. *-un-lft-identity 83.4%

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

   5. Applied egg-rr 83.4%

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

   if -7.49999999999999978e-185 < y.im < 1.55000000000000008e-133

   1. Initial program 72.7%

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

      1. *-un-lft-identity 72.7%

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

      2. add-sqr-sqrt 72.7%

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

      3. times-frac 72.8%

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

      4. hypot-def 72.9%

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

      5. fma-def 72.9%

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

      6. hypot-def 87.0%

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

   3. Applied egg-rr 87.0%

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

   4. Taylor expanded in y.re around -inf 46.5%

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

   5. Taylor expanded in y.re around -inf 98.1%

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

   if 1.55000000000000008e-133 < y.im < 1.65000000000000006e101

   1. Initial program 74.6%

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

   2. Taylor expanded in x.re around 0 74.6%

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

      1. associate-/l* 79.6%

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

      2. associate-/r/ 79.5%

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

      3. fma-def 79.5%

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

      4. unpow2 79.5%

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

      5. fma-def 79.5%

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

      6. associate-/l* 81.8%

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

      7. unpow2 81.8%

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

      8. fma-def 81.8%

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

   4. Simplified 81.8%

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

   if 1.65000000000000006e101 < y.im

   1. Initial program 35.4%

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

      1. *-un-lft-identity 35.4%

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

      2. add-sqr-sqrt 35.4%

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

      3. times-frac 35.3%

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

      4. hypot-def 35.3%

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

      5. fma-def 35.3%

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

      6. hypot-def 44.3%

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

   3. Applied egg-rr 44.3%

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

      1. associate-*l/ 44.5%

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

      2. *-un-lft-identity 44.5%

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

   5. Applied egg-rr 44.5%

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

   6. Taylor expanded in y.re around 0 86.2%

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

      1. associate-/l* 97.3%

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

   8. Simplified 97.3%

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

4. Final simplification 90.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -3.7 \cdot 10^{+151}:\\ \;\;\;\;\frac{\left(-x.im\right) - \frac{x.re}{\frac{y.im}{y.re}}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.im \leq -7.5 \cdot 10^{-185}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(x.re, y.re, y.im \cdot x.im\right)}{\mathsf{hypot}\left(y.re, y.im\right)}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.im \leq 1.55 \cdot 10^{-133}:\\ \;\;\;\;\frac{-1}{y.re} \cdot \left(\left(-x.re\right) - \frac{y.im \cdot x.im}{y.re}\right)\\ \mathbf{elif}\;y.im \leq 1.65 \cdot 10^{+101}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x.im}{\mathsf{fma}\left(y.im, y.im, {y.re}^{2}\right)}, y.im, \frac{x.re}{\frac{\mathsf{fma}\left(y.im, y.im, {y.re}^{2}\right)}{y.re}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im + \frac{x.re}{\frac{y.im}{y.re}}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \end{array} \]

Alternative 2: 85.0% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{y.im \cdot x.im + x.re \cdot y.re}{y.re \cdot y.re + y.im \cdot y.im} \leq 5 \cdot 10^{+293}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(x.re, y.re, y.im \cdot x.im\right)}{\mathsf{hypot}\left(y.re, y.im\right)}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{x.re}{y.im \cdot \frac{y.im}{y.re}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<=
      (/ (+ (* y.im x.im) (* x.re y.re)) (+ (* y.re y.re) (* y.im y.im)))
      5e+293)
   (/ (/ (fma x.re y.re (* y.im x.im)) (hypot y.re y.im)) (hypot y.re y.im))
   (+ (/ x.im y.im) (/ x.re (* y.im (/ y.im y.re))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((((y_46_im * x_46_im) + (x_46_re * y_46_re)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im))) <= 5e+293) {
		tmp = (fma(x_46_re, y_46_re, (y_46_im * x_46_im)) / hypot(y_46_re, y_46_im)) / hypot(y_46_re, y_46_im);
	} else {
		tmp = (x_46_im / y_46_im) + (x_46_re / (y_46_im * (y_46_im / y_46_re)));
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (Float64(Float64(Float64(y_46_im * x_46_im) + Float64(x_46_re * y_46_re)) / Float64(Float64(y_46_re * y_46_re) + Float64(y_46_im * y_46_im))) <= 5e+293)
		tmp = Float64(Float64(fma(x_46_re, y_46_re, Float64(y_46_im * x_46_im)) / hypot(y_46_re, y_46_im)) / hypot(y_46_re, y_46_im));
	else
		tmp = Float64(Float64(x_46_im / y_46_im) + Float64(x_46_re / Float64(y_46_im * Float64(y_46_im / y_46_re))));
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[N[(N[(N[(y$46$im * x$46$im), $MachinePrecision] + N[(x$46$re * y$46$re), $MachinePrecision]), $MachinePrecision] / N[(N[(y$46$re * y$46$re), $MachinePrecision] + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 5e+293], N[(N[(N[(x$46$re * y$46$re + N[(y$46$im * x$46$im), $MachinePrecision]), $MachinePrecision] / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision] / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision], N[(N[(x$46$im / y$46$im), $MachinePrecision] + N[(x$46$re / N[(y$46$im * N[(y$46$im / y$46$re), $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))) < 5.00000000000000033e293

   1. Initial program 78.3%

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

      1. *-un-lft-identity 78.3%

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

      2. add-sqr-sqrt 78.3%

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

      3. times-frac 78.3%

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

      4. hypot-def 78.3%

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

      5. fma-def 78.3%

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

      6. hypot-def 95.4%

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

   3. Applied egg-rr 95.4%

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

      1. associate-*l/ 95.7%

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

      2. *-un-lft-identity 95.7%

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

   5. Applied egg-rr 95.7%

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

   if 5.00000000000000033e293 < (/.f64 (+.f64 (*.f64 x.re y.re) (*.f64 x.im y.im)) (+.f64 (*.f64 y.re y.re) (*.f64 y.im y.im)))

   1. Initial program 9.1%

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

   2. Taylor expanded in y.re around 0 49.8%

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

      1. associate-/l* 52.1%

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

   4. Simplified 52.1%

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

      1. unpow2 52.1%

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

      2. *-un-lft-identity 52.1%

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

      3. times-frac 58.7%

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

   6. Applied egg-rr 58.7%

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

4. Final simplification 85.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y.im \cdot x.im + x.re \cdot y.re}{y.re \cdot y.re + y.im \cdot y.im} \leq 5 \cdot 10^{+293}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(x.re, y.re, y.im \cdot x.im\right)}{\mathsf{hypot}\left(y.re, y.im\right)}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{x.re}{y.im \cdot \frac{y.im}{y.re}}\\ \end{array} \]

Alternative 3: 83.2% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x.re}{\frac{y.im}{y.re}}\\ \mathbf{if}\;y.im \leq -3.2 \cdot 10^{+29}:\\ \;\;\;\;\frac{\left(-x.im\right) - t_0}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.im \leq -1.7 \cdot 10^{-97}:\\ \;\;\;\;\frac{y.im \cdot x.im + x.re \cdot y.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.im \leq 9.2 \cdot 10^{-88}:\\ \;\;\;\;\frac{-1}{y.re} \cdot \left(\left(-x.re\right) - \frac{y.im \cdot x.im}{y.re}\right)\\ \mathbf{elif}\;y.im \leq 1.25 \cdot 10^{+100}:\\ \;\;\;\;\mathsf{fma}\left(x.re, y.re, y.im \cdot x.im\right) \cdot \frac{1}{{\left(\mathsf{hypot}\left(y.re, y.im\right)\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im + t_0}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (/ x.re (/ y.im y.re))))
   (if (<= y.im -3.2e+29)
     (/ (- (- x.im) t_0) (hypot y.re y.im))
     (if (<= y.im -1.7e-97)
       (/ (+ (* y.im x.im) (* x.re y.re)) (+ (* y.re y.re) (* y.im y.im)))
       (if (<= y.im 9.2e-88)
         (* (/ -1.0 y.re) (- (- x.re) (/ (* y.im x.im) y.re)))
         (if (<= y.im 1.25e+100)
           (*
            (fma x.re y.re (* y.im x.im))
            (/ 1.0 (pow (hypot y.re y.im) 2.0)))
           (/ (+ x.im t_0) (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_im / y_46_re);
	double tmp;
	if (y_46_im <= -3.2e+29) {
		tmp = (-x_46_im - t_0) / hypot(y_46_re, y_46_im);
	} else if (y_46_im <= -1.7e-97) {
		tmp = ((y_46_im * x_46_im) + (x_46_re * y_46_re)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	} else if (y_46_im <= 9.2e-88) {
		tmp = (-1.0 / y_46_re) * (-x_46_re - ((y_46_im * x_46_im) / y_46_re));
	} else if (y_46_im <= 1.25e+100) {
		tmp = fma(x_46_re, y_46_re, (y_46_im * x_46_im)) * (1.0 / pow(hypot(y_46_re, y_46_im), 2.0));
	} else {
		tmp = (x_46_im + t_0) / 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(x_46_re / Float64(y_46_im / y_46_re))
	tmp = 0.0
	if (y_46_im <= -3.2e+29)
		tmp = Float64(Float64(Float64(-x_46_im) - t_0) / hypot(y_46_re, y_46_im));
	elseif (y_46_im <= -1.7e-97)
		tmp = Float64(Float64(Float64(y_46_im * x_46_im) + 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_im <= 9.2e-88)
		tmp = Float64(Float64(-1.0 / y_46_re) * Float64(Float64(-x_46_re) - Float64(Float64(y_46_im * x_46_im) / y_46_re)));
	elseif (y_46_im <= 1.25e+100)
		tmp = Float64(fma(x_46_re, y_46_re, Float64(y_46_im * x_46_im)) * Float64(1.0 / (hypot(y_46_re, y_46_im) ^ 2.0)));
	else
		tmp = Float64(Float64(x_46_im + t_0) / 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[(x$46$re / N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$im, -3.2e+29], N[(N[((-x$46$im) - t$95$0), $MachinePrecision] / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, -1.7e-97], N[(N[(N[(y$46$im * x$46$im), $MachinePrecision] + N[(x$46$re * y$46$re), $MachinePrecision]), $MachinePrecision] / N[(N[(y$46$re * y$46$re), $MachinePrecision] + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, 9.2e-88], N[(N[(-1.0 / y$46$re), $MachinePrecision] * N[((-x$46$re) - N[(N[(y$46$im * x$46$im), $MachinePrecision] / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, 1.25e+100], N[(N[(x$46$re * y$46$re + N[(y$46$im * x$46$im), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[Power[N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$im + t$95$0), $MachinePrecision] / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if y.im < -3.19999999999999987e29

   1. Initial program 30.3%

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

      1. *-un-lft-identity 30.3%

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

      2. add-sqr-sqrt 30.3%

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

      3. times-frac 30.3%

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

      4. hypot-def 30.3%

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

      5. fma-def 30.3%

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

      6. hypot-def 54.4%

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

   3. Applied egg-rr 54.4%

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

      1. associate-*l/ 54.4%

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

      2. *-un-lft-identity 54.4%

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

   5. Applied egg-rr 54.4%

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

   6. Taylor expanded in y.im around -inf 68.7%

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

      1. mul-1-neg 68.7%

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

      2. unsub-neg 68.7%

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

      3. mul-1-neg 68.7%

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

      4. associate-/l* 79.4%

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

   8. Simplified 79.4%

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

   if -3.19999999999999987e29 < y.im < -1.6999999999999999e-97

   1. Initial program 86.1%

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

   if -1.6999999999999999e-97 < y.im < 9.19999999999999945e-88

   1. Initial program 73.5%

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

      1. *-un-lft-identity 73.5%

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

      2. add-sqr-sqrt 73.5%

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

      3. times-frac 73.6%

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

      4. hypot-def 73.6%

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

      5. fma-def 73.6%

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

      6. hypot-def 87.0%

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

   3. Applied egg-rr 87.0%

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

   4. Taylor expanded in y.re around -inf 49.4%

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

   5. Taylor expanded in y.re around -inf 94.0%

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

   if 9.19999999999999945e-88 < y.im < 1.25e100

   1. Initial program 75.0%

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

      1. div-inv 75.1%

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

      2. fma-def 75.1%

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

      3. add-sqr-sqrt 75.1%

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

      4. pow2 75.1%

         \[\leadsto \mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right) \cdot \frac{1}{\color{blue}{{\left(\sqrt{y.re \cdot y.re + y.im \cdot y.im}\right)}^{2}}} \]

      5. hypot-def 75.1%

         \[\leadsto \mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right) \cdot \frac{1}{{\color{blue}{\left(\mathsf{hypot}\left(y.re, y.im\right)\right)}}^{2}} \]

   3. Applied egg-rr 75.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right) \cdot \frac{1}{{\left(\mathsf{hypot}\left(y.re, y.im\right)\right)}^{2}}} \]

   if 1.25e100 < y.im

   1. Initial program 35.4%

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

      1. *-un-lft-identity 35.4%

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

      2. add-sqr-sqrt 35.4%

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

      3. times-frac 35.3%

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

      4. hypot-def 35.3%

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

      5. fma-def 35.3%

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

      6. hypot-def 44.3%

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

   3. Applied egg-rr 44.3%

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

      1. associate-*l/ 44.5%

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

      2. *-un-lft-identity 44.5%

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

   5. Applied egg-rr 44.5%

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

   6. Taylor expanded in y.re around 0 86.2%

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

      1. associate-/l* 97.3%

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

   8. Simplified 97.3%

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

4. Final simplification 86.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -3.2 \cdot 10^{+29}:\\ \;\;\;\;\frac{\left(-x.im\right) - \frac{x.re}{\frac{y.im}{y.re}}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.im \leq -1.7 \cdot 10^{-97}:\\ \;\;\;\;\frac{y.im \cdot x.im + x.re \cdot y.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.im \leq 9.2 \cdot 10^{-88}:\\ \;\;\;\;\frac{-1}{y.re} \cdot \left(\left(-x.re\right) - \frac{y.im \cdot x.im}{y.re}\right)\\ \mathbf{elif}\;y.im \leq 1.25 \cdot 10^{+100}:\\ \;\;\;\;\mathsf{fma}\left(x.re, y.re, y.im \cdot x.im\right) \cdot \frac{1}{{\left(\mathsf{hypot}\left(y.re, y.im\right)\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im + \frac{x.re}{\frac{y.im}{y.re}}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \end{array} \]

Alternative 4: 80.3% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-1}{y.re} \cdot \left(\left(-x.re\right) - \frac{y.im \cdot x.im}{y.re}\right)\\ t_1 := \frac{y.im \cdot x.im + x.re \cdot y.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{if}\;y.im \leq -5.2 \cdot 10^{+143}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{x.re}{y.im \cdot \frac{y.im}{y.re}}\\ \mathbf{elif}\;y.im \leq -8.8 \cdot 10^{+80}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq -3.8 \cdot 10^{+29}:\\ \;\;\;\;\frac{x.im}{y.im} + y.re \cdot \left(x.re \cdot {y.im}^{-2}\right)\\ \mathbf{elif}\;y.im \leq -1.72 \cdot 10^{-97}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.im \leq 9.6 \cdot 10^{-88}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq 1.6 \cdot 10^{+100}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im + \frac{x.re}{\frac{y.im}{y.re}}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (* (/ -1.0 y.re) (- (- x.re) (/ (* y.im x.im) y.re))))
        (t_1
         (/ (+ (* y.im x.im) (* x.re y.re)) (+ (* y.re y.re) (* y.im y.im)))))
   (if (<= y.im -5.2e+143)
     (+ (/ x.im y.im) (/ x.re (* y.im (/ y.im y.re))))
     (if (<= y.im -8.8e+80)
       t_0
       (if (<= y.im -3.8e+29)
         (+ (/ x.im y.im) (* y.re (* x.re (pow y.im -2.0))))
         (if (<= y.im -1.72e-97)
           t_1
           (if (<= y.im 9.6e-88)
             t_0
             (if (<= y.im 1.6e+100)
               t_1
               (/ (+ 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 = (-1.0 / y_46_re) * (-x_46_re - ((y_46_im * x_46_im) / y_46_re));
	double t_1 = ((y_46_im * x_46_im) + (x_46_re * y_46_re)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	double tmp;
	if (y_46_im <= -5.2e+143) {
		tmp = (x_46_im / y_46_im) + (x_46_re / (y_46_im * (y_46_im / y_46_re)));
	} else if (y_46_im <= -8.8e+80) {
		tmp = t_0;
	} else if (y_46_im <= -3.8e+29) {
		tmp = (x_46_im / y_46_im) + (y_46_re * (x_46_re * pow(y_46_im, -2.0)));
	} else if (y_46_im <= -1.72e-97) {
		tmp = t_1;
	} else if (y_46_im <= 9.6e-88) {
		tmp = t_0;
	} else if (y_46_im <= 1.6e+100) {
		tmp = t_1;
	} 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 = (-1.0 / y_46_re) * (-x_46_re - ((y_46_im * x_46_im) / y_46_re));
	double t_1 = ((y_46_im * x_46_im) + (x_46_re * y_46_re)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	double tmp;
	if (y_46_im <= -5.2e+143) {
		tmp = (x_46_im / y_46_im) + (x_46_re / (y_46_im * (y_46_im / y_46_re)));
	} else if (y_46_im <= -8.8e+80) {
		tmp = t_0;
	} else if (y_46_im <= -3.8e+29) {
		tmp = (x_46_im / y_46_im) + (y_46_re * (x_46_re * Math.pow(y_46_im, -2.0)));
	} else if (y_46_im <= -1.72e-97) {
		tmp = t_1;
	} else if (y_46_im <= 9.6e-88) {
		tmp = t_0;
	} else if (y_46_im <= 1.6e+100) {
		tmp = t_1;
	} 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 = (-1.0 / y_46_re) * (-x_46_re - ((y_46_im * x_46_im) / y_46_re))
	t_1 = ((y_46_im * x_46_im) + (x_46_re * y_46_re)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im))
	tmp = 0
	if y_46_im <= -5.2e+143:
		tmp = (x_46_im / y_46_im) + (x_46_re / (y_46_im * (y_46_im / y_46_re)))
	elif y_46_im <= -8.8e+80:
		tmp = t_0
	elif y_46_im <= -3.8e+29:
		tmp = (x_46_im / y_46_im) + (y_46_re * (x_46_re * math.pow(y_46_im, -2.0)))
	elif y_46_im <= -1.72e-97:
		tmp = t_1
	elif y_46_im <= 9.6e-88:
		tmp = t_0
	elif y_46_im <= 1.6e+100:
		tmp = t_1
	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(-1.0 / y_46_re) * Float64(Float64(-x_46_re) - Float64(Float64(y_46_im * x_46_im) / y_46_re)))
	t_1 = Float64(Float64(Float64(y_46_im * x_46_im) + 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_im <= -5.2e+143)
		tmp = Float64(Float64(x_46_im / y_46_im) + Float64(x_46_re / Float64(y_46_im * Float64(y_46_im / y_46_re))));
	elseif (y_46_im <= -8.8e+80)
		tmp = t_0;
	elseif (y_46_im <= -3.8e+29)
		tmp = Float64(Float64(x_46_im / y_46_im) + Float64(y_46_re * Float64(x_46_re * (y_46_im ^ -2.0))));
	elseif (y_46_im <= -1.72e-97)
		tmp = t_1;
	elseif (y_46_im <= 9.6e-88)
		tmp = t_0;
	elseif (y_46_im <= 1.6e+100)
		tmp = t_1;
	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 = (-1.0 / y_46_re) * (-x_46_re - ((y_46_im * x_46_im) / y_46_re));
	t_1 = ((y_46_im * x_46_im) + (x_46_re * y_46_re)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	tmp = 0.0;
	if (y_46_im <= -5.2e+143)
		tmp = (x_46_im / y_46_im) + (x_46_re / (y_46_im * (y_46_im / y_46_re)));
	elseif (y_46_im <= -8.8e+80)
		tmp = t_0;
	elseif (y_46_im <= -3.8e+29)
		tmp = (x_46_im / y_46_im) + (y_46_re * (x_46_re * (y_46_im ^ -2.0)));
	elseif (y_46_im <= -1.72e-97)
		tmp = t_1;
	elseif (y_46_im <= 9.6e-88)
		tmp = t_0;
	elseif (y_46_im <= 1.6e+100)
		tmp = t_1;
	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[(-1.0 / y$46$re), $MachinePrecision] * N[((-x$46$re) - N[(N[(y$46$im * x$46$im), $MachinePrecision] / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(y$46$im * x$46$im), $MachinePrecision] + N[(x$46$re * y$46$re), $MachinePrecision]), $MachinePrecision] / N[(N[(y$46$re * y$46$re), $MachinePrecision] + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$im, -5.2e+143], N[(N[(x$46$im / y$46$im), $MachinePrecision] + N[(x$46$re / N[(y$46$im * N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, -8.8e+80], t$95$0, If[LessEqual[y$46$im, -3.8e+29], N[(N[(x$46$im / y$46$im), $MachinePrecision] + N[(y$46$re * N[(x$46$re * N[Power[y$46$im, -2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, -1.72e-97], t$95$1, If[LessEqual[y$46$im, 9.6e-88], t$95$0, If[LessEqual[y$46$im, 1.6e+100], t$95$1, 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 5 regimes

2. if y.im < -5.1999999999999998e143

   1. Initial program 24.8%

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

   2. Taylor expanded in y.re around 0 69.4%

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

      1. associate-/l* 70.2%

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

   4. Simplified 70.2%

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

      1. unpow2 70.2%

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

      2. *-un-lft-identity 70.2%

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

      3. times-frac 80.9%

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

   6. Applied egg-rr 80.9%

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

   if -5.1999999999999998e143 < y.im < -8.80000000000000011e80 or -1.71999999999999995e-97 < y.im < 9.5999999999999998e-88

   1. Initial program 67.7%

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

      1. *-un-lft-identity 67.7%

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

      2. add-sqr-sqrt 67.7%

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

      3. times-frac 67.8%

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

      4. hypot-def 67.8%

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

      5. fma-def 67.8%

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

      6. hypot-def 82.4%

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

   3. Applied egg-rr 82.4%

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

   4. Taylor expanded in y.re around -inf 47.2%

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

   5. Taylor expanded in y.re around -inf 90.7%

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

   if -8.80000000000000011e80 < y.im < -3.79999999999999971e29

   1. Initial program 66.7%

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

   2. Taylor expanded in y.re around 0 78.2%

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

      1. associate-/l* 78.2%

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

      2. associate-/r/ 78.2%

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

   4. Simplified 78.2%

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

      1. expm1-log1p-u 55.9%

         \[\leadsto \frac{x.im}{y.im} + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{x.re}{{y.im}^{2}}\right)\right)} \cdot y.re \]

      2. expm1-udef 55.9%

         \[\leadsto \frac{x.im}{y.im} + \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{x.re}{{y.im}^{2}}\right)} - 1\right)} \cdot y.re \]

      3. pow2 55.9%

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

      4. div-inv 55.9%

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

      5. pow2 55.9%

         \[\leadsto \frac{x.im}{y.im} + \left(e^{\mathsf{log1p}\left(x.re \cdot \frac{1}{\color{blue}{{y.im}^{2}}}\right)} - 1\right) \cdot y.re \]

      6. pow-flip 55.9%

         \[\leadsto \frac{x.im}{y.im} + \left(e^{\mathsf{log1p}\left(x.re \cdot \color{blue}{{y.im}^{\left(-2\right)}}\right)} - 1\right) \cdot y.re \]

      7. metadata-eval 55.9%

         \[\leadsto \frac{x.im}{y.im} + \left(e^{\mathsf{log1p}\left(x.re \cdot {y.im}^{\color{blue}{-2}}\right)} - 1\right) \cdot y.re \]

   6. Applied egg-rr 55.9%

      \[\leadsto \frac{x.im}{y.im} + \color{blue}{\left(e^{\mathsf{log1p}\left(x.re \cdot {y.im}^{-2}\right)} - 1\right)} \cdot y.re \]
   7. Step-by-step derivation

      1. expm1-def 55.9%

         \[\leadsto \frac{x.im}{y.im} + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(x.re \cdot {y.im}^{-2}\right)\right)} \cdot y.re \]

      2. expm1-log1p 78.4%

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

   8. Simplified 78.4%

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

   if -3.79999999999999971e29 < y.im < -1.71999999999999995e-97 or 9.5999999999999998e-88 < y.im < 1.5999999999999999e100

   1. Initial program 78.3%

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

   if 1.5999999999999999e100 < y.im

   1. Initial program 35.4%

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

      1. *-un-lft-identity 35.4%

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

      2. add-sqr-sqrt 35.4%

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

      3. times-frac 35.3%

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

      4. hypot-def 35.3%

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

      5. fma-def 35.3%

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

      6. hypot-def 44.3%

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

   3. Applied egg-rr 44.3%

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

      1. associate-*l/ 44.5%

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

      2. *-un-lft-identity 44.5%

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

   5. Applied egg-rr 44.5%

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

   6. Taylor expanded in y.re around 0 86.2%

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

      1. associate-/l* 97.3%

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

   8. Simplified 97.3%

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

4. Final simplification 86.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -5.2 \cdot 10^{+143}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{x.re}{y.im \cdot \frac{y.im}{y.re}}\\ \mathbf{elif}\;y.im \leq -8.8 \cdot 10^{+80}:\\ \;\;\;\;\frac{-1}{y.re} \cdot \left(\left(-x.re\right) - \frac{y.im \cdot x.im}{y.re}\right)\\ \mathbf{elif}\;y.im \leq -3.8 \cdot 10^{+29}:\\ \;\;\;\;\frac{x.im}{y.im} + y.re \cdot \left(x.re \cdot {y.im}^{-2}\right)\\ \mathbf{elif}\;y.im \leq -1.72 \cdot 10^{-97}:\\ \;\;\;\;\frac{y.im \cdot x.im + x.re \cdot y.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.im \leq 9.6 \cdot 10^{-88}:\\ \;\;\;\;\frac{-1}{y.re} \cdot \left(\left(-x.re\right) - \frac{y.im \cdot x.im}{y.re}\right)\\ \mathbf{elif}\;y.im \leq 1.6 \cdot 10^{+100}:\\ \;\;\;\;\frac{y.im \cdot x.im + x.re \cdot y.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im + \frac{x.re}{\frac{y.im}{y.re}}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \end{array} \]

Alternative 5: 83.3% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y.im \cdot x.im + x.re \cdot y.re}{y.re \cdot y.re + y.im \cdot y.im}\\ t_1 := \frac{x.re}{\frac{y.im}{y.re}}\\ \mathbf{if}\;y.im \leq -3.8 \cdot 10^{+29}:\\ \;\;\;\;\frac{\left(-x.im\right) - t_1}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.im \leq -1.7 \cdot 10^{-97}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq 1.02 \cdot 10^{-87}:\\ \;\;\;\;\frac{-1}{y.re} \cdot \left(\left(-x.re\right) - \frac{y.im \cdot x.im}{y.re}\right)\\ \mathbf{elif}\;y.im \leq 3.8 \cdot 10^{+101}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im + t_1}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0
         (/ (+ (* y.im x.im) (* x.re y.re)) (+ (* y.re y.re) (* y.im y.im))))
        (t_1 (/ x.re (/ y.im y.re))))
   (if (<= y.im -3.8e+29)
     (/ (- (- x.im) t_1) (hypot y.re y.im))
     (if (<= y.im -1.7e-97)
       t_0
       (if (<= y.im 1.02e-87)
         (* (/ -1.0 y.re) (- (- x.re) (/ (* y.im x.im) y.re)))
         (if (<= y.im 3.8e+101) t_0 (/ (+ x.im t_1) (hypot y.re y.im))))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = ((y_46_im * x_46_im) + (x_46_re * y_46_re)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	double t_1 = x_46_re / (y_46_im / y_46_re);
	double tmp;
	if (y_46_im <= -3.8e+29) {
		tmp = (-x_46_im - t_1) / hypot(y_46_re, y_46_im);
	} else if (y_46_im <= -1.7e-97) {
		tmp = t_0;
	} else if (y_46_im <= 1.02e-87) {
		tmp = (-1.0 / y_46_re) * (-x_46_re - ((y_46_im * x_46_im) / y_46_re));
	} else if (y_46_im <= 3.8e+101) {
		tmp = t_0;
	} else {
		tmp = (x_46_im + t_1) / hypot(y_46_re, y_46_im);
	}
	return tmp;
}

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = ((y_46_im * x_46_im) + (x_46_re * y_46_re)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	double t_1 = x_46_re / (y_46_im / y_46_re);
	double tmp;
	if (y_46_im <= -3.8e+29) {
		tmp = (-x_46_im - t_1) / Math.hypot(y_46_re, y_46_im);
	} else if (y_46_im <= -1.7e-97) {
		tmp = t_0;
	} else if (y_46_im <= 1.02e-87) {
		tmp = (-1.0 / y_46_re) * (-x_46_re - ((y_46_im * x_46_im) / y_46_re));
	} else if (y_46_im <= 3.8e+101) {
		tmp = t_0;
	} else {
		tmp = (x_46_im + t_1) / Math.hypot(y_46_re, y_46_im);
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = ((y_46_im * x_46_im) + (x_46_re * y_46_re)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im))
	t_1 = x_46_re / (y_46_im / y_46_re)
	tmp = 0
	if y_46_im <= -3.8e+29:
		tmp = (-x_46_im - t_1) / math.hypot(y_46_re, y_46_im)
	elif y_46_im <= -1.7e-97:
		tmp = t_0
	elif y_46_im <= 1.02e-87:
		tmp = (-1.0 / y_46_re) * (-x_46_re - ((y_46_im * x_46_im) / y_46_re))
	elif y_46_im <= 3.8e+101:
		tmp = t_0
	else:
		tmp = (x_46_im + t_1) / math.hypot(y_46_re, y_46_im)
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(Float64(Float64(y_46_im * x_46_im) + Float64(x_46_re * y_46_re)) / Float64(Float64(y_46_re * y_46_re) + Float64(y_46_im * y_46_im)))
	t_1 = Float64(x_46_re / Float64(y_46_im / y_46_re))
	tmp = 0.0
	if (y_46_im <= -3.8e+29)
		tmp = Float64(Float64(Float64(-x_46_im) - t_1) / hypot(y_46_re, y_46_im));
	elseif (y_46_im <= -1.7e-97)
		tmp = t_0;
	elseif (y_46_im <= 1.02e-87)
		tmp = Float64(Float64(-1.0 / y_46_re) * Float64(Float64(-x_46_re) - Float64(Float64(y_46_im * x_46_im) / y_46_re)));
	elseif (y_46_im <= 3.8e+101)
		tmp = t_0;
	else
		tmp = Float64(Float64(x_46_im + t_1) / hypot(y_46_re, y_46_im));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = ((y_46_im * x_46_im) + (x_46_re * y_46_re)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	t_1 = x_46_re / (y_46_im / y_46_re);
	tmp = 0.0;
	if (y_46_im <= -3.8e+29)
		tmp = (-x_46_im - t_1) / hypot(y_46_re, y_46_im);
	elseif (y_46_im <= -1.7e-97)
		tmp = t_0;
	elseif (y_46_im <= 1.02e-87)
		tmp = (-1.0 / y_46_re) * (-x_46_re - ((y_46_im * x_46_im) / y_46_re));
	elseif (y_46_im <= 3.8e+101)
		tmp = t_0;
	else
		tmp = (x_46_im + t_1) / hypot(y_46_re, y_46_im);
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[(N[(y$46$im * x$46$im), $MachinePrecision] + N[(x$46$re * y$46$re), $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[(x$46$re / N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$im, -3.8e+29], N[(N[((-x$46$im) - t$95$1), $MachinePrecision] / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, -1.7e-97], t$95$0, If[LessEqual[y$46$im, 1.02e-87], N[(N[(-1.0 / y$46$re), $MachinePrecision] * N[((-x$46$re) - N[(N[(y$46$im * x$46$im), $MachinePrecision] / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, 3.8e+101], t$95$0, N[(N[(x$46$im + t$95$1), $MachinePrecision] / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if y.im < -3.79999999999999971e29

   1. Initial program 30.3%

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

      1. *-un-lft-identity 30.3%

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

      2. add-sqr-sqrt 30.3%

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

      3. times-frac 30.3%

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

      4. hypot-def 30.3%

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

      5. fma-def 30.3%

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

      6. hypot-def 54.4%

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

   3. Applied egg-rr 54.4%

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

      1. associate-*l/ 54.4%

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

      2. *-un-lft-identity 54.4%

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

   5. Applied egg-rr 54.4%

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

   6. Taylor expanded in y.im around -inf 68.7%

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

      1. mul-1-neg 68.7%

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

      2. unsub-neg 68.7%

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

      3. mul-1-neg 68.7%

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

      4. associate-/l* 79.4%

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

   8. Simplified 79.4%

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

   if -3.79999999999999971e29 < y.im < -1.6999999999999999e-97 or 1.02000000000000009e-87 < y.im < 3.7999999999999998e101

   1. Initial program 78.3%

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

   if -1.6999999999999999e-97 < y.im < 1.02000000000000009e-87

   1. Initial program 73.5%

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

      1. *-un-lft-identity 73.5%

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

      2. add-sqr-sqrt 73.5%

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

      3. times-frac 73.6%

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

      4. hypot-def 73.6%

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

      5. fma-def 73.6%

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

      6. hypot-def 87.0%

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

   3. Applied egg-rr 87.0%

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

   4. Taylor expanded in y.re around -inf 49.4%

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

   5. Taylor expanded in y.re around -inf 94.0%

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

   if 3.7999999999999998e101 < y.im

   1. Initial program 35.4%

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

      1. *-un-lft-identity 35.4%

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

      2. add-sqr-sqrt 35.4%

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

      3. times-frac 35.3%

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

      4. hypot-def 35.3%

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

      5. fma-def 35.3%

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

      6. hypot-def 44.3%

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

   3. Applied egg-rr 44.3%

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

      1. associate-*l/ 44.5%

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

      2. *-un-lft-identity 44.5%

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

   5. Applied egg-rr 44.5%

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

   6. Taylor expanded in y.re around 0 86.2%

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

      1. associate-/l* 97.3%

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

   8. Simplified 97.3%

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

4. Final simplification 86.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -3.8 \cdot 10^{+29}:\\ \;\;\;\;\frac{\left(-x.im\right) - \frac{x.re}{\frac{y.im}{y.re}}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.im \leq -1.7 \cdot 10^{-97}:\\ \;\;\;\;\frac{y.im \cdot x.im + x.re \cdot y.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.im \leq 1.02 \cdot 10^{-87}:\\ \;\;\;\;\frac{-1}{y.re} \cdot \left(\left(-x.re\right) - \frac{y.im \cdot x.im}{y.re}\right)\\ \mathbf{elif}\;y.im \leq 3.8 \cdot 10^{+101}:\\ \;\;\;\;\frac{y.im \cdot x.im + x.re \cdot y.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im + \frac{x.re}{\frac{y.im}{y.re}}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \end{array} \]

Alternative 6: 79.3% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-1}{y.re} \cdot \left(\left(-x.re\right) - \frac{y.im \cdot x.im}{y.re}\right)\\ t_1 := \frac{y.im \cdot x.im + x.re \cdot y.re}{y.re \cdot y.re + y.im \cdot y.im}\\ t_2 := \frac{x.im}{y.im} + \frac{x.re}{y.im \cdot \frac{y.im}{y.re}}\\ \mathbf{if}\;y.im \leq -5.2 \cdot 10^{+143}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y.im \leq -1.9 \cdot 10^{+81}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq -3.8 \cdot 10^{+29}:\\ \;\;\;\;\frac{x.im}{y.im} + y.re \cdot \left(x.re \cdot {y.im}^{-2}\right)\\ \mathbf{elif}\;y.im \leq -1.75 \cdot 10^{-97}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.im \leq 8 \cdot 10^{-88}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq 1.02 \cdot 10^{+124}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (* (/ -1.0 y.re) (- (- x.re) (/ (* y.im x.im) y.re))))
        (t_1
         (/ (+ (* y.im x.im) (* x.re y.re)) (+ (* y.re y.re) (* y.im y.im))))
        (t_2 (+ (/ x.im y.im) (/ x.re (* y.im (/ y.im y.re))))))
   (if (<= y.im -5.2e+143)
     t_2
     (if (<= y.im -1.9e+81)
       t_0
       (if (<= y.im -3.8e+29)
         (+ (/ x.im y.im) (* y.re (* x.re (pow y.im -2.0))))
         (if (<= y.im -1.75e-97)
           t_1
           (if (<= y.im 8e-88) t_0 (if (<= y.im 1.02e+124) t_1 t_2))))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = (-1.0 / y_46_re) * (-x_46_re - ((y_46_im * x_46_im) / y_46_re));
	double t_1 = ((y_46_im * x_46_im) + (x_46_re * y_46_re)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	double t_2 = (x_46_im / y_46_im) + (x_46_re / (y_46_im * (y_46_im / y_46_re)));
	double tmp;
	if (y_46_im <= -5.2e+143) {
		tmp = t_2;
	} else if (y_46_im <= -1.9e+81) {
		tmp = t_0;
	} else if (y_46_im <= -3.8e+29) {
		tmp = (x_46_im / y_46_im) + (y_46_re * (x_46_re * pow(y_46_im, -2.0)));
	} else if (y_46_im <= -1.75e-97) {
		tmp = t_1;
	} else if (y_46_im <= 8e-88) {
		tmp = t_0;
	} else if (y_46_im <= 1.02e+124) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	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) :: t_2
    real(8) :: tmp
    t_0 = ((-1.0d0) / y_46re) * (-x_46re - ((y_46im * x_46im) / y_46re))
    t_1 = ((y_46im * x_46im) + (x_46re * y_46re)) / ((y_46re * y_46re) + (y_46im * y_46im))
    t_2 = (x_46im / y_46im) + (x_46re / (y_46im * (y_46im / y_46re)))
    if (y_46im <= (-5.2d+143)) then
        tmp = t_2
    else if (y_46im <= (-1.9d+81)) then
        tmp = t_0
    else if (y_46im <= (-3.8d+29)) then
        tmp = (x_46im / y_46im) + (y_46re * (x_46re * (y_46im ** (-2.0d0))))
    else if (y_46im <= (-1.75d-97)) then
        tmp = t_1
    else if (y_46im <= 8d-88) then
        tmp = t_0
    else if (y_46im <= 1.02d+124) then
        tmp = t_1
    else
        tmp = t_2
    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 = (-1.0 / y_46_re) * (-x_46_re - ((y_46_im * x_46_im) / y_46_re));
	double t_1 = ((y_46_im * x_46_im) + (x_46_re * y_46_re)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	double t_2 = (x_46_im / y_46_im) + (x_46_re / (y_46_im * (y_46_im / y_46_re)));
	double tmp;
	if (y_46_im <= -5.2e+143) {
		tmp = t_2;
	} else if (y_46_im <= -1.9e+81) {
		tmp = t_0;
	} else if (y_46_im <= -3.8e+29) {
		tmp = (x_46_im / y_46_im) + (y_46_re * (x_46_re * Math.pow(y_46_im, -2.0)));
	} else if (y_46_im <= -1.75e-97) {
		tmp = t_1;
	} else if (y_46_im <= 8e-88) {
		tmp = t_0;
	} else if (y_46_im <= 1.02e+124) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = (-1.0 / y_46_re) * (-x_46_re - ((y_46_im * x_46_im) / y_46_re))
	t_1 = ((y_46_im * x_46_im) + (x_46_re * y_46_re)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im))
	t_2 = (x_46_im / y_46_im) + (x_46_re / (y_46_im * (y_46_im / y_46_re)))
	tmp = 0
	if y_46_im <= -5.2e+143:
		tmp = t_2
	elif y_46_im <= -1.9e+81:
		tmp = t_0
	elif y_46_im <= -3.8e+29:
		tmp = (x_46_im / y_46_im) + (y_46_re * (x_46_re * math.pow(y_46_im, -2.0)))
	elif y_46_im <= -1.75e-97:
		tmp = t_1
	elif y_46_im <= 8e-88:
		tmp = t_0
	elif y_46_im <= 1.02e+124:
		tmp = t_1
	else:
		tmp = t_2
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(Float64(-1.0 / y_46_re) * Float64(Float64(-x_46_re) - Float64(Float64(y_46_im * x_46_im) / y_46_re)))
	t_1 = Float64(Float64(Float64(y_46_im * x_46_im) + Float64(x_46_re * y_46_re)) / Float64(Float64(y_46_re * y_46_re) + Float64(y_46_im * y_46_im)))
	t_2 = Float64(Float64(x_46_im / y_46_im) + Float64(x_46_re / Float64(y_46_im * Float64(y_46_im / y_46_re))))
	tmp = 0.0
	if (y_46_im <= -5.2e+143)
		tmp = t_2;
	elseif (y_46_im <= -1.9e+81)
		tmp = t_0;
	elseif (y_46_im <= -3.8e+29)
		tmp = Float64(Float64(x_46_im / y_46_im) + Float64(y_46_re * Float64(x_46_re * (y_46_im ^ -2.0))));
	elseif (y_46_im <= -1.75e-97)
		tmp = t_1;
	elseif (y_46_im <= 8e-88)
		tmp = t_0;
	elseif (y_46_im <= 1.02e+124)
		tmp = t_1;
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = (-1.0 / y_46_re) * (-x_46_re - ((y_46_im * x_46_im) / y_46_re));
	t_1 = ((y_46_im * x_46_im) + (x_46_re * y_46_re)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	t_2 = (x_46_im / y_46_im) + (x_46_re / (y_46_im * (y_46_im / y_46_re)));
	tmp = 0.0;
	if (y_46_im <= -5.2e+143)
		tmp = t_2;
	elseif (y_46_im <= -1.9e+81)
		tmp = t_0;
	elseif (y_46_im <= -3.8e+29)
		tmp = (x_46_im / y_46_im) + (y_46_re * (x_46_re * (y_46_im ^ -2.0)));
	elseif (y_46_im <= -1.75e-97)
		tmp = t_1;
	elseif (y_46_im <= 8e-88)
		tmp = t_0;
	elseif (y_46_im <= 1.02e+124)
		tmp = t_1;
	else
		tmp = t_2;
	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[(-1.0 / y$46$re), $MachinePrecision] * N[((-x$46$re) - N[(N[(y$46$im * x$46$im), $MachinePrecision] / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(y$46$im * x$46$im), $MachinePrecision] + N[(x$46$re * y$46$re), $MachinePrecision]), $MachinePrecision] / N[(N[(y$46$re * y$46$re), $MachinePrecision] + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x$46$im / y$46$im), $MachinePrecision] + N[(x$46$re / N[(y$46$im * N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$im, -5.2e+143], t$95$2, If[LessEqual[y$46$im, -1.9e+81], t$95$0, If[LessEqual[y$46$im, -3.8e+29], N[(N[(x$46$im / y$46$im), $MachinePrecision] + N[(y$46$re * N[(x$46$re * N[Power[y$46$im, -2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, -1.75e-97], t$95$1, If[LessEqual[y$46$im, 8e-88], t$95$0, If[LessEqual[y$46$im, 1.02e+124], t$95$1, t$95$2]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if y.im < -5.1999999999999998e143 or 1.01999999999999994e124 < y.im

   1. Initial program 23.9%

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

   2. Taylor expanded in y.re around 0 73.2%

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

      1. associate-/l* 74.0%

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

   4. Simplified 74.0%

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

      1. unpow2 74.0%

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

      2. *-un-lft-identity 74.0%

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

      3. times-frac 83.2%

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

   6. Applied egg-rr 83.2%

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

   if -5.1999999999999998e143 < y.im < -1.9e81 or -1.7500000000000001e-97 < y.im < 7.99999999999999947e-88

   1. Initial program 67.7%

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

      1. *-un-lft-identity 67.7%

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

      2. add-sqr-sqrt 67.7%

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

      3. times-frac 67.8%

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

      4. hypot-def 67.8%

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

      5. fma-def 67.8%

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

      6. hypot-def 82.4%

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

   3. Applied egg-rr 82.4%

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

   4. Taylor expanded in y.re around -inf 47.2%

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

   5. Taylor expanded in y.re around -inf 90.7%

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

   if -1.9e81 < y.im < -3.79999999999999971e29

   1. Initial program 66.7%

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

   2. Taylor expanded in y.re around 0 78.2%

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

      1. associate-/l* 78.2%

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

      2. associate-/r/ 78.2%

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

   4. Simplified 78.2%

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

      1. expm1-log1p-u 55.9%

         \[\leadsto \frac{x.im}{y.im} + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{x.re}{{y.im}^{2}}\right)\right)} \cdot y.re \]

      2. expm1-udef 55.9%

         \[\leadsto \frac{x.im}{y.im} + \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{x.re}{{y.im}^{2}}\right)} - 1\right)} \cdot y.re \]

      3. pow2 55.9%

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

      4. div-inv 55.9%

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

      5. pow2 55.9%

         \[\leadsto \frac{x.im}{y.im} + \left(e^{\mathsf{log1p}\left(x.re \cdot \frac{1}{\color{blue}{{y.im}^{2}}}\right)} - 1\right) \cdot y.re \]

      6. pow-flip 55.9%

         \[\leadsto \frac{x.im}{y.im} + \left(e^{\mathsf{log1p}\left(x.re \cdot \color{blue}{{y.im}^{\left(-2\right)}}\right)} - 1\right) \cdot y.re \]

      7. metadata-eval 55.9%

         \[\leadsto \frac{x.im}{y.im} + \left(e^{\mathsf{log1p}\left(x.re \cdot {y.im}^{\color{blue}{-2}}\right)} - 1\right) \cdot y.re \]

   6. Applied egg-rr 55.9%

      \[\leadsto \frac{x.im}{y.im} + \color{blue}{\left(e^{\mathsf{log1p}\left(x.re \cdot {y.im}^{-2}\right)} - 1\right)} \cdot y.re \]
   7. Step-by-step derivation

      1. expm1-def 55.9%

         \[\leadsto \frac{x.im}{y.im} + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(x.re \cdot {y.im}^{-2}\right)\right)} \cdot y.re \]

      2. expm1-log1p 78.4%

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

   8. Simplified 78.4%

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

   if -3.79999999999999971e29 < y.im < -1.7500000000000001e-97 or 7.99999999999999947e-88 < y.im < 1.01999999999999994e124

   1. Initial program 79.0%

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

4. Final simplification 84.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -5.2 \cdot 10^{+143}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{x.re}{y.im \cdot \frac{y.im}{y.re}}\\ \mathbf{elif}\;y.im \leq -1.9 \cdot 10^{+81}:\\ \;\;\;\;\frac{-1}{y.re} \cdot \left(\left(-x.re\right) - \frac{y.im \cdot x.im}{y.re}\right)\\ \mathbf{elif}\;y.im \leq -3.8 \cdot 10^{+29}:\\ \;\;\;\;\frac{x.im}{y.im} + y.re \cdot \left(x.re \cdot {y.im}^{-2}\right)\\ \mathbf{elif}\;y.im \leq -1.75 \cdot 10^{-97}:\\ \;\;\;\;\frac{y.im \cdot x.im + x.re \cdot y.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.im \leq 8 \cdot 10^{-88}:\\ \;\;\;\;\frac{-1}{y.re} \cdot \left(\left(-x.re\right) - \frac{y.im \cdot x.im}{y.re}\right)\\ \mathbf{elif}\;y.im \leq 1.02 \cdot 10^{+124}:\\ \;\;\;\;\frac{y.im \cdot x.im + x.re \cdot y.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{x.re}{y.im \cdot \frac{y.im}{y.re}}\\ \end{array} \]

Alternative 7: 79.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y.im \cdot x.im + x.re \cdot y.re}{y.re \cdot y.re + y.im \cdot y.im}\\ t_1 := \frac{-1}{y.re} \cdot \left(\left(-x.re\right) - \frac{y.im \cdot x.im}{y.re}\right)\\ t_2 := \frac{x.im}{y.im} + \frac{x.re}{y.im \cdot \frac{y.im}{y.re}}\\ \mathbf{if}\;y.im \leq -5.2 \cdot 10^{+143}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y.im \leq -1.95 \cdot 10^{+81}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.im \leq -2.05 \cdot 10^{+51}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{elif}\;y.im \leq -1.8 \cdot 10^{-97}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq 8.8 \cdot 10^{-88}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.im \leq 1.65 \cdot 10^{+123}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0
         (/ (+ (* y.im x.im) (* x.re y.re)) (+ (* y.re y.re) (* y.im y.im))))
        (t_1 (* (/ -1.0 y.re) (- (- x.re) (/ (* y.im x.im) y.re))))
        (t_2 (+ (/ x.im y.im) (/ x.re (* y.im (/ y.im y.re))))))
   (if (<= y.im -5.2e+143)
     t_2
     (if (<= y.im -1.95e+81)
       t_1
       (if (<= y.im -2.05e+51)
         (/ x.im y.im)
         (if (<= y.im -1.8e-97)
           t_0
           (if (<= y.im 8.8e-88) t_1 (if (<= y.im 1.65e+123) t_0 t_2))))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = ((y_46_im * x_46_im) + (x_46_re * y_46_re)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	double t_1 = (-1.0 / y_46_re) * (-x_46_re - ((y_46_im * x_46_im) / y_46_re));
	double t_2 = (x_46_im / y_46_im) + (x_46_re / (y_46_im * (y_46_im / y_46_re)));
	double tmp;
	if (y_46_im <= -5.2e+143) {
		tmp = t_2;
	} else if (y_46_im <= -1.95e+81) {
		tmp = t_1;
	} else if (y_46_im <= -2.05e+51) {
		tmp = x_46_im / y_46_im;
	} else if (y_46_im <= -1.8e-97) {
		tmp = t_0;
	} else if (y_46_im <= 8.8e-88) {
		tmp = t_1;
	} else if (y_46_im <= 1.65e+123) {
		tmp = t_0;
	} else {
		tmp = t_2;
	}
	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) :: t_2
    real(8) :: tmp
    t_0 = ((y_46im * x_46im) + (x_46re * y_46re)) / ((y_46re * y_46re) + (y_46im * y_46im))
    t_1 = ((-1.0d0) / y_46re) * (-x_46re - ((y_46im * x_46im) / y_46re))
    t_2 = (x_46im / y_46im) + (x_46re / (y_46im * (y_46im / y_46re)))
    if (y_46im <= (-5.2d+143)) then
        tmp = t_2
    else if (y_46im <= (-1.95d+81)) then
        tmp = t_1
    else if (y_46im <= (-2.05d+51)) then
        tmp = x_46im / y_46im
    else if (y_46im <= (-1.8d-97)) then
        tmp = t_0
    else if (y_46im <= 8.8d-88) then
        tmp = t_1
    else if (y_46im <= 1.65d+123) then
        tmp = t_0
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = ((y_46_im * x_46_im) + (x_46_re * y_46_re)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	double t_1 = (-1.0 / y_46_re) * (-x_46_re - ((y_46_im * x_46_im) / y_46_re));
	double t_2 = (x_46_im / y_46_im) + (x_46_re / (y_46_im * (y_46_im / y_46_re)));
	double tmp;
	if (y_46_im <= -5.2e+143) {
		tmp = t_2;
	} else if (y_46_im <= -1.95e+81) {
		tmp = t_1;
	} else if (y_46_im <= -2.05e+51) {
		tmp = x_46_im / y_46_im;
	} else if (y_46_im <= -1.8e-97) {
		tmp = t_0;
	} else if (y_46_im <= 8.8e-88) {
		tmp = t_1;
	} else if (y_46_im <= 1.65e+123) {
		tmp = t_0;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = ((y_46_im * x_46_im) + (x_46_re * y_46_re)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im))
	t_1 = (-1.0 / y_46_re) * (-x_46_re - ((y_46_im * x_46_im) / y_46_re))
	t_2 = (x_46_im / y_46_im) + (x_46_re / (y_46_im * (y_46_im / y_46_re)))
	tmp = 0
	if y_46_im <= -5.2e+143:
		tmp = t_2
	elif y_46_im <= -1.95e+81:
		tmp = t_1
	elif y_46_im <= -2.05e+51:
		tmp = x_46_im / y_46_im
	elif y_46_im <= -1.8e-97:
		tmp = t_0
	elif y_46_im <= 8.8e-88:
		tmp = t_1
	elif y_46_im <= 1.65e+123:
		tmp = t_0
	else:
		tmp = t_2
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(Float64(Float64(y_46_im * x_46_im) + Float64(x_46_re * y_46_re)) / Float64(Float64(y_46_re * y_46_re) + Float64(y_46_im * y_46_im)))
	t_1 = Float64(Float64(-1.0 / y_46_re) * Float64(Float64(-x_46_re) - Float64(Float64(y_46_im * x_46_im) / y_46_re)))
	t_2 = Float64(Float64(x_46_im / y_46_im) + Float64(x_46_re / Float64(y_46_im * Float64(y_46_im / y_46_re))))
	tmp = 0.0
	if (y_46_im <= -5.2e+143)
		tmp = t_2;
	elseif (y_46_im <= -1.95e+81)
		tmp = t_1;
	elseif (y_46_im <= -2.05e+51)
		tmp = Float64(x_46_im / y_46_im);
	elseif (y_46_im <= -1.8e-97)
		tmp = t_0;
	elseif (y_46_im <= 8.8e-88)
		tmp = t_1;
	elseif (y_46_im <= 1.65e+123)
		tmp = t_0;
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = ((y_46_im * x_46_im) + (x_46_re * y_46_re)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	t_1 = (-1.0 / y_46_re) * (-x_46_re - ((y_46_im * x_46_im) / y_46_re));
	t_2 = (x_46_im / y_46_im) + (x_46_re / (y_46_im * (y_46_im / y_46_re)));
	tmp = 0.0;
	if (y_46_im <= -5.2e+143)
		tmp = t_2;
	elseif (y_46_im <= -1.95e+81)
		tmp = t_1;
	elseif (y_46_im <= -2.05e+51)
		tmp = x_46_im / y_46_im;
	elseif (y_46_im <= -1.8e-97)
		tmp = t_0;
	elseif (y_46_im <= 8.8e-88)
		tmp = t_1;
	elseif (y_46_im <= 1.65e+123)
		tmp = t_0;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[(N[(y$46$im * x$46$im), $MachinePrecision] + N[(x$46$re * y$46$re), $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[(-1.0 / y$46$re), $MachinePrecision] * N[((-x$46$re) - N[(N[(y$46$im * x$46$im), $MachinePrecision] / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x$46$im / y$46$im), $MachinePrecision] + N[(x$46$re / N[(y$46$im * N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$im, -5.2e+143], t$95$2, If[LessEqual[y$46$im, -1.95e+81], t$95$1, If[LessEqual[y$46$im, -2.05e+51], N[(x$46$im / y$46$im), $MachinePrecision], If[LessEqual[y$46$im, -1.8e-97], t$95$0, If[LessEqual[y$46$im, 8.8e-88], t$95$1, If[LessEqual[y$46$im, 1.65e+123], t$95$0, t$95$2]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if y.im < -5.1999999999999998e143 or 1.65000000000000001e123 < y.im

   1. Initial program 23.9%

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

   2. Taylor expanded in y.re around 0 73.2%

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

      1. associate-/l* 74.0%

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

   4. Simplified 74.0%

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

      1. unpow2 74.0%

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

      2. *-un-lft-identity 74.0%

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

      3. times-frac 83.2%

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

   6. Applied egg-rr 83.2%

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

   if -5.1999999999999998e143 < y.im < -1.95e81 or -1.79999999999999999e-97 < y.im < 8.8000000000000002e-88

   1. Initial program 67.7%

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

      1. *-un-lft-identity 67.7%

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

      2. add-sqr-sqrt 67.7%

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

      3. times-frac 67.8%

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

      4. hypot-def 67.8%

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

      5. fma-def 67.8%

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

      6. hypot-def 82.4%

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

   3. Applied egg-rr 82.4%

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

   4. Taylor expanded in y.re around -inf 47.2%

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

   5. Taylor expanded in y.re around -inf 90.7%

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

   if -1.95e81 < y.im < -2.05000000000000005e51

   1. Initial program 67.0%

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

   2. Taylor expanded in y.re around 0 84.4%

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

   if -2.05000000000000005e51 < y.im < -1.79999999999999999e-97 or 8.8000000000000002e-88 < y.im < 1.65000000000000001e123

   1. Initial program 78.5%

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

4. Final simplification 84.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -5.2 \cdot 10^{+143}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{x.re}{y.im \cdot \frac{y.im}{y.re}}\\ \mathbf{elif}\;y.im \leq -1.95 \cdot 10^{+81}:\\ \;\;\;\;\frac{-1}{y.re} \cdot \left(\left(-x.re\right) - \frac{y.im \cdot x.im}{y.re}\right)\\ \mathbf{elif}\;y.im \leq -2.05 \cdot 10^{+51}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{elif}\;y.im \leq -1.8 \cdot 10^{-97}:\\ \;\;\;\;\frac{y.im \cdot x.im + x.re \cdot y.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.im \leq 8.8 \cdot 10^{-88}:\\ \;\;\;\;\frac{-1}{y.re} \cdot \left(\left(-x.re\right) - \frac{y.im \cdot x.im}{y.re}\right)\\ \mathbf{elif}\;y.im \leq 1.65 \cdot 10^{+123}:\\ \;\;\;\;\frac{y.im \cdot x.im + x.re \cdot y.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{x.re}{y.im \cdot \frac{y.im}{y.re}}\\ \end{array} \]

Alternative 8: 74.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-1}{y.re} \cdot \left(\left(-x.re\right) - \frac{y.im \cdot x.im}{y.re}\right)\\ t_1 := \frac{x.im}{y.im} + \frac{x.re}{y.im \cdot \frac{y.im}{y.re}}\\ \mathbf{if}\;y.im \leq -5.2 \cdot 10^{+143}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.im \leq -1.75 \cdot 10^{+81}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq -4.4 \cdot 10^{+29}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.im \leq 1.7 \cdot 10^{-71}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im + y.re \cdot \frac{y.re}{y.im}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (* (/ -1.0 y.re) (- (- x.re) (/ (* y.im x.im) y.re))))
        (t_1 (+ (/ x.im y.im) (/ x.re (* y.im (/ y.im y.re))))))
   (if (<= y.im -5.2e+143)
     t_1
     (if (<= y.im -1.75e+81)
       t_0
       (if (<= y.im -4.4e+29)
         t_1
         (if (<= y.im 1.7e-71)
           t_0
           (/ x.im (+ y.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 t_0 = (-1.0 / y_46_re) * (-x_46_re - ((y_46_im * x_46_im) / y_46_re));
	double t_1 = (x_46_im / y_46_im) + (x_46_re / (y_46_im * (y_46_im / y_46_re)));
	double tmp;
	if (y_46_im <= -5.2e+143) {
		tmp = t_1;
	} else if (y_46_im <= -1.75e+81) {
		tmp = t_0;
	} else if (y_46_im <= -4.4e+29) {
		tmp = t_1;
	} else if (y_46_im <= 1.7e-71) {
		tmp = t_0;
	} else {
		tmp = x_46_im / (y_46_im + (y_46_re * (y_46_re / y_46_im)));
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = ((-1.0d0) / y_46re) * (-x_46re - ((y_46im * x_46im) / y_46re))
    t_1 = (x_46im / y_46im) + (x_46re / (y_46im * (y_46im / y_46re)))
    if (y_46im <= (-5.2d+143)) then
        tmp = t_1
    else if (y_46im <= (-1.75d+81)) then
        tmp = t_0
    else if (y_46im <= (-4.4d+29)) then
        tmp = t_1
    else if (y_46im <= 1.7d-71) then
        tmp = t_0
    else
        tmp = x_46im / (y_46im + (y_46re * (y_46re / y_46im)))
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = (-1.0 / y_46_re) * (-x_46_re - ((y_46_im * x_46_im) / y_46_re));
	double t_1 = (x_46_im / y_46_im) + (x_46_re / (y_46_im * (y_46_im / y_46_re)));
	double tmp;
	if (y_46_im <= -5.2e+143) {
		tmp = t_1;
	} else if (y_46_im <= -1.75e+81) {
		tmp = t_0;
	} else if (y_46_im <= -4.4e+29) {
		tmp = t_1;
	} else if (y_46_im <= 1.7e-71) {
		tmp = t_0;
	} else {
		tmp = x_46_im / (y_46_im + (y_46_re * (y_46_re / y_46_im)));
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = (-1.0 / y_46_re) * (-x_46_re - ((y_46_im * x_46_im) / y_46_re))
	t_1 = (x_46_im / y_46_im) + (x_46_re / (y_46_im * (y_46_im / y_46_re)))
	tmp = 0
	if y_46_im <= -5.2e+143:
		tmp = t_1
	elif y_46_im <= -1.75e+81:
		tmp = t_0
	elif y_46_im <= -4.4e+29:
		tmp = t_1
	elif y_46_im <= 1.7e-71:
		tmp = t_0
	else:
		tmp = x_46_im / (y_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)
	t_0 = Float64(Float64(-1.0 / y_46_re) * Float64(Float64(-x_46_re) - Float64(Float64(y_46_im * x_46_im) / y_46_re)))
	t_1 = Float64(Float64(x_46_im / y_46_im) + Float64(x_46_re / Float64(y_46_im * Float64(y_46_im / y_46_re))))
	tmp = 0.0
	if (y_46_im <= -5.2e+143)
		tmp = t_1;
	elseif (y_46_im <= -1.75e+81)
		tmp = t_0;
	elseif (y_46_im <= -4.4e+29)
		tmp = t_1;
	elseif (y_46_im <= 1.7e-71)
		tmp = t_0;
	else
		tmp = Float64(x_46_im / Float64(y_46_im + Float64(y_46_re * Float64(y_46_re / y_46_im))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = (-1.0 / y_46_re) * (-x_46_re - ((y_46_im * x_46_im) / y_46_re));
	t_1 = (x_46_im / y_46_im) + (x_46_re / (y_46_im * (y_46_im / y_46_re)));
	tmp = 0.0;
	if (y_46_im <= -5.2e+143)
		tmp = t_1;
	elseif (y_46_im <= -1.75e+81)
		tmp = t_0;
	elseif (y_46_im <= -4.4e+29)
		tmp = t_1;
	elseif (y_46_im <= 1.7e-71)
		tmp = t_0;
	else
		tmp = x_46_im / (y_46_im + (y_46_re * (y_46_re / y_46_im)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[(-1.0 / y$46$re), $MachinePrecision] * N[((-x$46$re) - N[(N[(y$46$im * x$46$im), $MachinePrecision] / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x$46$im / y$46$im), $MachinePrecision] + N[(x$46$re / N[(y$46$im * N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$im, -5.2e+143], t$95$1, If[LessEqual[y$46$im, -1.75e+81], t$95$0, If[LessEqual[y$46$im, -4.4e+29], t$95$1, If[LessEqual[y$46$im, 1.7e-71], t$95$0, N[(x$46$im / N[(y$46$im + N[(y$46$re * N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if y.im < -5.1999999999999998e143 or -1.75e81 < y.im < -4.4000000000000003e29

   1. Initial program 32.5%

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

   2. Taylor expanded in y.re around 0 72.2%

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

      1. associate-/l* 72.9%

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

   4. Simplified 72.9%

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

      1. unpow2 72.9%

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

      2. *-un-lft-identity 72.9%

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

      3. times-frac 82.0%

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

   6. Applied egg-rr 82.0%

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

   if -5.1999999999999998e143 < y.im < -1.75e81 or -4.4000000000000003e29 < y.im < 1.70000000000000002e-71

   1. Initial program 71.1%

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

      1. *-un-lft-identity 71.1%

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

      2. add-sqr-sqrt 71.1%

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

      3. times-frac 71.1%

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

      4. hypot-def 71.2%

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

      5. fma-def 71.2%

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

      6. hypot-def 85.2%

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

   3. Applied egg-rr 85.2%

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

   4. Taylor expanded in y.re around -inf 44.6%

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

   5. Taylor expanded in y.re around -inf 85.3%

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

   if 1.70000000000000002e-71 < y.im

   1. Initial program 57.2%

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

   2. Taylor expanded in x.re around 0 46.1%

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

      1. associate-/l* 54.6%

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

   4. Simplified 54.6%

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

   5. Taylor expanded in y.im around 0 69.8%

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

      1. unpow2 69.8%

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

      2. *-un-lft-identity 69.8%

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

      3. times-frac 72.2%

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

   7. Applied egg-rr 72.2%

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

4. Final simplification 80.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -5.2 \cdot 10^{+143}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{x.re}{y.im \cdot \frac{y.im}{y.re}}\\ \mathbf{elif}\;y.im \leq -1.75 \cdot 10^{+81}:\\ \;\;\;\;\frac{-1}{y.re} \cdot \left(\left(-x.re\right) - \frac{y.im \cdot x.im}{y.re}\right)\\ \mathbf{elif}\;y.im \leq -4.4 \cdot 10^{+29}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{x.re}{y.im \cdot \frac{y.im}{y.re}}\\ \mathbf{elif}\;y.im \leq 1.7 \cdot 10^{-71}:\\ \;\;\;\;\frac{-1}{y.re} \cdot \left(\left(-x.re\right) - \frac{y.im \cdot x.im}{y.re}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im + y.re \cdot \frac{y.re}{y.im}}\\ \end{array} \]

Alternative 9: 69.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x.im}{y.im + y.re \cdot \frac{y.re}{y.im}}\\ \mathbf{if}\;y.im \leq -9.5 \cdot 10^{-15}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq -1.7 \cdot 10^{-55}:\\ \;\;\;\;\frac{x.re \cdot y.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.im \leq -8.2 \cdot 10^{-70} \lor \neg \left(y.im \leq 4 \cdot 10^{-71}\right):\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (/ x.im (+ y.im (* y.re (/ y.re y.im))))))
   (if (<= y.im -9.5e-15)
     t_0
     (if (<= y.im -1.7e-55)
       (/ (* x.re y.re) (+ (* y.re y.re) (* y.im y.im)))
       (if (or (<= y.im -8.2e-70) (not (<= y.im 4e-71))) 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 / (y_46_im + (y_46_re * (y_46_re / y_46_im)));
	double tmp;
	if (y_46_im <= -9.5e-15) {
		tmp = t_0;
	} else if (y_46_im <= -1.7e-55) {
		tmp = (x_46_re * y_46_re) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	} else if ((y_46_im <= -8.2e-70) || !(y_46_im <= 4e-71)) {
		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 / (y_46im + (y_46re * (y_46re / y_46im)))
    if (y_46im <= (-9.5d-15)) then
        tmp = t_0
    else if (y_46im <= (-1.7d-55)) then
        tmp = (x_46re * y_46re) / ((y_46re * y_46re) + (y_46im * y_46im))
    else if ((y_46im <= (-8.2d-70)) .or. (.not. (y_46im <= 4d-71))) 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 / (y_46_im + (y_46_re * (y_46_re / y_46_im)));
	double tmp;
	if (y_46_im <= -9.5e-15) {
		tmp = t_0;
	} else if (y_46_im <= -1.7e-55) {
		tmp = (x_46_re * y_46_re) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	} else if ((y_46_im <= -8.2e-70) || !(y_46_im <= 4e-71)) {
		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 / (y_46_im + (y_46_re * (y_46_re / y_46_im)))
	tmp = 0
	if y_46_im <= -9.5e-15:
		tmp = t_0
	elif y_46_im <= -1.7e-55:
		tmp = (x_46_re * y_46_re) / ((y_46_re * y_46_re) + (y_46_im * y_46_im))
	elif (y_46_im <= -8.2e-70) or not (y_46_im <= 4e-71):
		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(x_46_im / Float64(y_46_im + Float64(y_46_re * Float64(y_46_re / y_46_im))))
	tmp = 0.0
	if (y_46_im <= -9.5e-15)
		tmp = t_0;
	elseif (y_46_im <= -1.7e-55)
		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_im <= -8.2e-70) || !(y_46_im <= 4e-71))
		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 / (y_46_im + (y_46_re * (y_46_re / y_46_im)));
	tmp = 0.0;
	if (y_46_im <= -9.5e-15)
		tmp = t_0;
	elseif (y_46_im <= -1.7e-55)
		tmp = (x_46_re * y_46_re) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	elseif ((y_46_im <= -8.2e-70) || ~((y_46_im <= 4e-71)))
		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[(x$46$im / N[(y$46$im + N[(y$46$re * N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$im, -9.5e-15], t$95$0, If[LessEqual[y$46$im, -1.7e-55], 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[Or[LessEqual[y$46$im, -8.2e-70], N[Not[LessEqual[y$46$im, 4e-71]], $MachinePrecision]], t$95$0, N[(x$46$re / y$46$re), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if y.im < -9.5000000000000005e-15 or -1.69999999999999986e-55 < y.im < -8.19999999999999955e-70 or 3.9999999999999997e-71 < y.im

   1. Initial program 47.9%

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

   2. Taylor expanded in x.re around 0 40.7%

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

      1. associate-/l* 46.9%

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

   4. Simplified 46.9%

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

   5. Taylor expanded in y.im around 0 66.7%

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

      1. unpow2 66.7%

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

      2. *-un-lft-identity 66.7%

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

      3. times-frac 70.3%

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

   7. Applied egg-rr 70.3%

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

   if -9.5000000000000005e-15 < y.im < -1.69999999999999986e-55

   1. Initial program 83.6%

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

   2. Taylor expanded in x.re around inf 83.6%

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

      1. *-commutative 83.6%

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

   4. Simplified 83.6%

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

   if -8.19999999999999955e-70 < y.im < 3.9999999999999997e-71

   1. Initial program 75.0%

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

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

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

4. Final simplification 70.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -9.5 \cdot 10^{-15}:\\ \;\;\;\;\frac{x.im}{y.im + y.re \cdot \frac{y.re}{y.im}}\\ \mathbf{elif}\;y.im \leq -1.7 \cdot 10^{-55}:\\ \;\;\;\;\frac{x.re \cdot y.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.im \leq -8.2 \cdot 10^{-70} \lor \neg \left(y.im \leq 4 \cdot 10^{-71}\right):\\ \;\;\;\;\frac{x.im}{y.im + y.re \cdot \frac{y.re}{y.im}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \end{array} \]

Alternative 10: 69.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -2.9 \cdot 10^{-13} \lor \neg \left(y.im \leq 3.5 \cdot 10^{-73}\right):\\ \;\;\;\;\frac{x.im}{y.im + y.re \cdot \frac{y.re}{y.im}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.im -2.9e-13) (not (<= y.im 3.5e-73)))
   (/ x.im (+ y.im (* y.re (/ y.re y.im))))
   (/ x.re y.re)))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_im <= -2.9e-13) || !(y_46_im <= 3.5e-73)) {
		tmp = x_46_im / (y_46_im + (y_46_re * (y_46_re / y_46_im)));
	} else {
		tmp = x_46_re / y_46_re;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if ((y_46im <= (-2.9d-13)) .or. (.not. (y_46im <= 3.5d-73))) then
        tmp = x_46im / (y_46im + (y_46re * (y_46re / y_46im)))
    else
        tmp = x_46re / y_46re
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_im <= -2.9e-13) || !(y_46_im <= 3.5e-73)) {
		tmp = x_46_im / (y_46_im + (y_46_re * (y_46_re / y_46_im)));
	} else {
		tmp = x_46_re / y_46_re;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if (y_46_im <= -2.9e-13) or not (y_46_im <= 3.5e-73):
		tmp = x_46_im / (y_46_im + (y_46_re * (y_46_re / y_46_im)))
	else:
		tmp = x_46_re / y_46_re
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if ((y_46_im <= -2.9e-13) || !(y_46_im <= 3.5e-73))
		tmp = Float64(x_46_im / Float64(y_46_im + Float64(y_46_re * Float64(y_46_re / y_46_im))));
	else
		tmp = Float64(x_46_re / y_46_re);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if ((y_46_im <= -2.9e-13) || ~((y_46_im <= 3.5e-73)))
		tmp = x_46_im / (y_46_im + (y_46_re * (y_46_re / y_46_im)));
	else
		tmp = x_46_re / y_46_re;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[Or[LessEqual[y$46$im, -2.9e-13], N[Not[LessEqual[y$46$im, 3.5e-73]], $MachinePrecision]], N[(x$46$im / N[(y$46$im + N[(y$46$re * N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x$46$re / y$46$re), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.im < -2.8999999999999998e-13 or 3.4999999999999998e-73 < y.im

   1. Initial program 47.2%

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

   2. Taylor expanded in x.re around 0 40.0%

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

      1. associate-/l* 46.2%

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

   4. Simplified 46.2%

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

   5. Taylor expanded in y.im around 0 66.2%

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

      1. unpow2 66.2%

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

      2. *-un-lft-identity 66.2%

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

      3. times-frac 69.9%

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

   7. Applied egg-rr 69.9%

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

   if -2.8999999999999998e-13 < y.im < 3.4999999999999998e-73

   1. Initial program 76.0%

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

   2. Taylor expanded in y.re around inf 66.1%

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

4. Final simplification 68.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -2.9 \cdot 10^{-13} \lor \neg \left(y.im \leq 3.5 \cdot 10^{-73}\right):\\ \;\;\;\;\frac{x.im}{y.im + y.re \cdot \frac{y.re}{y.im}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \end{array} \]

Alternative 11: 69.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -3.1 \cdot 10^{-58}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{x.re}{y.im \cdot \frac{y.im}{y.re}}\\ \mathbf{elif}\;y.im \leq 3.2 \cdot 10^{-71}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im + y.re \cdot \frac{y.re}{y.im}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.im -3.1e-58)
   (+ (/ x.im y.im) (/ x.re (* y.im (/ y.im y.re))))
   (if (<= y.im 3.2e-71)
     (/ x.re y.re)
     (/ x.im (+ y.im (* y.re (/ y.re y.im)))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_im <= -3.1e-58) {
		tmp = (x_46_im / y_46_im) + (x_46_re / (y_46_im * (y_46_im / y_46_re)));
	} else if (y_46_im <= 3.2e-71) {
		tmp = x_46_re / y_46_re;
	} else {
		tmp = x_46_im / (y_46_im + (y_46_re * (y_46_re / y_46_im)));
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if (y_46im <= (-3.1d-58)) then
        tmp = (x_46im / y_46im) + (x_46re / (y_46im * (y_46im / y_46re)))
    else if (y_46im <= 3.2d-71) then
        tmp = x_46re / y_46re
    else
        tmp = x_46im / (y_46im + (y_46re * (y_46re / y_46im)))
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_im <= -3.1e-58) {
		tmp = (x_46_im / y_46_im) + (x_46_re / (y_46_im * (y_46_im / y_46_re)));
	} else if (y_46_im <= 3.2e-71) {
		tmp = x_46_re / y_46_re;
	} else {
		tmp = x_46_im / (y_46_im + (y_46_re * (y_46_re / y_46_im)));
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if y_46_im <= -3.1e-58:
		tmp = (x_46_im / y_46_im) + (x_46_re / (y_46_im * (y_46_im / y_46_re)))
	elif y_46_im <= 3.2e-71:
		tmp = x_46_re / y_46_re
	else:
		tmp = x_46_im / (y_46_im + (y_46_re * (y_46_re / y_46_im)))
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (y_46_im <= -3.1e-58)
		tmp = Float64(Float64(x_46_im / y_46_im) + Float64(x_46_re / Float64(y_46_im * Float64(y_46_im / y_46_re))));
	elseif (y_46_im <= 3.2e-71)
		tmp = Float64(x_46_re / y_46_re);
	else
		tmp = Float64(x_46_im / Float64(y_46_im + Float64(y_46_re * Float64(y_46_re / y_46_im))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if (y_46_im <= -3.1e-58)
		tmp = (x_46_im / y_46_im) + (x_46_re / (y_46_im * (y_46_im / y_46_re)));
	elseif (y_46_im <= 3.2e-71)
		tmp = x_46_re / y_46_re;
	else
		tmp = x_46_im / (y_46_im + (y_46_re * (y_46_re / y_46_im)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[y$46$im, -3.1e-58], N[(N[(x$46$im / y$46$im), $MachinePrecision] + N[(x$46$re / N[(y$46$im * N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, 3.2e-71], N[(x$46$re / y$46$re), $MachinePrecision], N[(x$46$im / N[(y$46$im + N[(y$46$re * N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y.im < -3.0999999999999999e-58

   1. Initial program 40.2%

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

   2. Taylor expanded in y.re around 0 60.6%

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

      1. associate-/l* 61.8%

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

   4. Simplified 61.8%

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

      1. unpow2 61.8%

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

      2. *-un-lft-identity 61.8%

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

      3. times-frac 67.8%

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

   6. Applied egg-rr 67.8%

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

   if -3.0999999999999999e-58 < y.im < 3.1999999999999999e-71

   1. Initial program 75.3%

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

   2. Taylor expanded in y.re around inf 68.6%

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

   if 3.1999999999999999e-71 < y.im

   1. Initial program 57.2%

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

   2. Taylor expanded in x.re around 0 46.1%

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

      1. associate-/l* 54.6%

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

   4. Simplified 54.6%

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

   5. Taylor expanded in y.im around 0 69.8%

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

      1. unpow2 69.8%

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

      2. *-un-lft-identity 69.8%

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

      3. times-frac 72.2%

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

   7. Applied egg-rr 72.2%

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

4. Final simplification 69.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -3.1 \cdot 10^{-58}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{x.re}{y.im \cdot \frac{y.im}{y.re}}\\ \mathbf{elif}\;y.im \leq 3.2 \cdot 10^{-71}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im + y.re \cdot \frac{y.re}{y.im}}\\ \end{array} \]

Alternative 12: 64.5% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -1.56 \cdot 10^{+38} \lor \neg \left(y.im \leq 4.5 \cdot 10^{-71}\right):\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.im -1.56e+38) (not (<= y.im 4.5e-71)))
   (/ x.im y.im)
   (/ x.re y.re)))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_im <= -1.56e+38) || !(y_46_im <= 4.5e-71)) {
		tmp = x_46_im / y_46_im;
	} else {
		tmp = x_46_re / y_46_re;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if ((y_46im <= (-1.56d+38)) .or. (.not. (y_46im <= 4.5d-71))) then
        tmp = x_46im / y_46im
    else
        tmp = x_46re / y_46re
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_im <= -1.56e+38) || !(y_46_im <= 4.5e-71)) {
		tmp = x_46_im / y_46_im;
	} else {
		tmp = x_46_re / y_46_re;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if (y_46_im <= -1.56e+38) or not (y_46_im <= 4.5e-71):
		tmp = x_46_im / y_46_im
	else:
		tmp = x_46_re / y_46_re
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if ((y_46_im <= -1.56e+38) || !(y_46_im <= 4.5e-71))
		tmp = Float64(x_46_im / y_46_im);
	else
		tmp = Float64(x_46_re / y_46_re);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if ((y_46_im <= -1.56e+38) || ~((y_46_im <= 4.5e-71)))
		tmp = x_46_im / y_46_im;
	else
		tmp = x_46_re / y_46_re;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[Or[LessEqual[y$46$im, -1.56e+38], N[Not[LessEqual[y$46$im, 4.5e-71]], $MachinePrecision]], N[(x$46$im / y$46$im), $MachinePrecision], N[(x$46$re / y$46$re), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.im < -1.5599999999999999e38 or 4.5000000000000002e-71 < y.im

   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. Taylor expanded in y.re around 0 61.6%

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

   if -1.5599999999999999e38 < y.im < 4.5000000000000002e-71

   1. Initial program 76.1%

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

   2. Taylor expanded in y.re around inf 64.2%

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

4. Final simplification 62.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -1.56 \cdot 10^{+38} \lor \neg \left(y.im \leq 4.5 \cdot 10^{-71}\right):\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \end{array} \]

Alternative 13: 43.0% 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 58.8%

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

2. Taylor expanded in y.re around 0 40.3%

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

3. Final simplification 40.3%

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

Reproduce

herbie shell --seed 2023333 
(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))))