_divideComplex, real part

Percentage Accurate: 62.2% → 85.3%

Time: 12.4s

Alternatives: 17

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: 62.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 85.3% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)}\\ t_1 := \mathsf{fma}\left(x.re, y.re, y.im \cdot x.im\right)\\ \mathbf{if}\;y.re \leq -3.5 \cdot 10^{+76}:\\ \;\;\;\;\left(x.re + \frac{y.im}{\frac{y.re}{x.im}}\right) \cdot \frac{-1}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.re \leq -1.4 \cdot 10^{-144}:\\ \;\;\;\;\frac{t_0}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{t_1}}\\ \mathbf{elif}\;y.re \leq 5.8 \cdot 10^{-131}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re}{y.im} \cdot \frac{x.re}{y.im}\\ \mathbf{elif}\;y.re \leq 7.4 \cdot 10^{+131}:\\ \;\;\;\;t_0 \cdot \frac{t_1}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{1}{\frac{y.re}{\frac{x.im}{\frac{y.re}{y.im}}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (/ 1.0 (hypot y.re y.im))) (t_1 (fma x.re y.re (* y.im x.im))))
   (if (<= y.re -3.5e+76)
     (* (+ x.re (/ y.im (/ y.re x.im))) (/ -1.0 (hypot y.re y.im)))
     (if (<= y.re -1.4e-144)
       (/ t_0 (/ (hypot y.re y.im) t_1))
       (if (<= y.re 5.8e-131)
         (+ (/ x.im y.im) (* (/ y.re y.im) (/ x.re y.im)))
         (if (<= y.re 7.4e+131)
           (* t_0 (/ t_1 (hypot y.re y.im)))
           (+ (/ x.re y.re) (/ 1.0 (/ y.re (/ x.im (/ 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 / hypot(y_46_re, y_46_im);
	double t_1 = fma(x_46_re, y_46_re, (y_46_im * x_46_im));
	double tmp;
	if (y_46_re <= -3.5e+76) {
		tmp = (x_46_re + (y_46_im / (y_46_re / x_46_im))) * (-1.0 / hypot(y_46_re, y_46_im));
	} else if (y_46_re <= -1.4e-144) {
		tmp = t_0 / (hypot(y_46_re, y_46_im) / t_1);
	} else if (y_46_re <= 5.8e-131) {
		tmp = (x_46_im / y_46_im) + ((y_46_re / y_46_im) * (x_46_re / y_46_im));
	} else if (y_46_re <= 7.4e+131) {
		tmp = t_0 * (t_1 / hypot(y_46_re, y_46_im));
	} else {
		tmp = (x_46_re / y_46_re) + (1.0 / (y_46_re / (x_46_im / (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(1.0 / hypot(y_46_re, y_46_im))
	t_1 = fma(x_46_re, y_46_re, Float64(y_46_im * x_46_im))
	tmp = 0.0
	if (y_46_re <= -3.5e+76)
		tmp = Float64(Float64(x_46_re + Float64(y_46_im / Float64(y_46_re / x_46_im))) * Float64(-1.0 / hypot(y_46_re, y_46_im)));
	elseif (y_46_re <= -1.4e-144)
		tmp = Float64(t_0 / Float64(hypot(y_46_re, y_46_im) / t_1));
	elseif (y_46_re <= 5.8e-131)
		tmp = Float64(Float64(x_46_im / y_46_im) + Float64(Float64(y_46_re / y_46_im) * Float64(x_46_re / y_46_im)));
	elseif (y_46_re <= 7.4e+131)
		tmp = Float64(t_0 * Float64(t_1 / hypot(y_46_re, y_46_im)));
	else
		tmp = Float64(Float64(x_46_re / y_46_re) + Float64(1.0 / Float64(y_46_re / Float64(x_46_im / Float64(y_46_re / y_46_im)))));
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(1.0 / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x$46$re * y$46$re + N[(y$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$re, -3.5e+76], N[(N[(x$46$re + N[(y$46$im / N[(y$46$re / x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(-1.0 / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, -1.4e-144], N[(t$95$0 / N[(N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 5.8e-131], N[(N[(x$46$im / y$46$im), $MachinePrecision] + N[(N[(y$46$re / y$46$im), $MachinePrecision] * N[(x$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 7.4e+131], N[(t$95$0 * N[(t$95$1 / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$re / y$46$re), $MachinePrecision] + N[(1.0 / N[(y$46$re / N[(x$46$im / N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 5 regimes

2. if y.re < -3.5e76

   1. Initial program 32.0%

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

      1. *-un-lft-identity 32.0%

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

      2. add-sqr-sqrt 32.0%

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

      3. times-frac 32.1%

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

      4. hypot-def 32.1%

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

      5. fma-def 32.1%

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

      6. hypot-def 51.5%

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

   3. Applied egg-rr 51.5%

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

   4. Taylor expanded in y.re around -inf 82.9%

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

      1. distribute-lft-out 82.9%

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

      2. +-commutative 82.9%

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

      3. associate-/l* 89.5%

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

   6. Simplified 89.5%

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

   if -3.5e76 < y.re < -1.39999999999999999e-144

   1. Initial program 87.3%

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

      1. *-un-lft-identity 87.3%

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

      2. add-sqr-sqrt 87.2%

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

      3. times-frac 87.1%

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

      4. hypot-def 87.1%

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

      5. fma-def 87.1%

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

      6. hypot-def 95.2%

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

   3. Applied egg-rr 95.2%

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

      1. clear-num 95.3%

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

      2. un-div-inv 95.4%

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

   5. Applied egg-rr 95.4%

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

   if -1.39999999999999999e-144 < y.re < 5.8000000000000004e-131

   1. Initial program 61.6%

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

   2. Taylor expanded in y.re around 0 83.4%

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

      1. +-commutative 83.4%

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

      2. *-commutative 83.4%

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

      3. unpow2 83.4%

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

      4. times-frac 92.1%

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

   4. Simplified 92.1%

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

   if 5.8000000000000004e-131 < y.re < 7.3999999999999999e131

   1. Initial program 79.1%

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

      1. *-un-lft-identity 79.1%

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

      2. add-sqr-sqrt 79.1%

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

      3. times-frac 79.1%

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

      4. hypot-def 79.1%

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

      5. fma-def 79.1%

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

      6. hypot-def 90.2%

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

   3. Applied egg-rr 90.2%

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

   if 7.3999999999999999e131 < y.re

   1. Initial program 28.6%

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

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

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

      1. associate-/l* 74.0%

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

      2. unpow2 74.0%

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

   4. Simplified 74.0%

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

      1. clear-num 74.0%

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

      2. inv-pow 74.0%

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

      3. associate-/l* 85.9%

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

   6. Applied egg-rr 85.9%

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

      1. unpow-1 85.9%

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

      2. associate-/l/ 91.9%

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

      3. /-rgt-identity 91.9%

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

      4. times-frac 91.5%

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

      5. *-commutative 91.5%

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

      6. *-commutative 91.5%

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

      7. times-frac 91.9%

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

      8. /-rgt-identity 91.9%

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

      9. associate-/r/ 91.9%

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

   8. Simplified 91.9%

      \[\leadsto \frac{x.re}{y.re} + \color{blue}{\frac{1}{\frac{y.re}{\frac{x.im}{\frac{y.re}{y.im}}}}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 91.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -3.5 \cdot 10^{+76}:\\ \;\;\;\;\left(x.re + \frac{y.im}{\frac{y.re}{x.im}}\right) \cdot \frac{-1}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.re \leq -1.4 \cdot 10^{-144}:\\ \;\;\;\;\frac{\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)}}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{\mathsf{fma}\left(x.re, y.re, y.im \cdot x.im\right)}}\\ \mathbf{elif}\;y.re \leq 5.8 \cdot 10^{-131}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re}{y.im} \cdot \frac{x.re}{y.im}\\ \mathbf{elif}\;y.re \leq 7.4 \cdot 10^{+131}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{\mathsf{fma}\left(x.re, y.re, y.im \cdot x.im\right)}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{1}{\frac{y.re}{\frac{x.im}{\frac{y.re}{y.im}}}}\\ \end{array} \]

Alternative 2: 85.4% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{\mathsf{fma}\left(x.re, y.re, y.im \cdot x.im\right)}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{if}\;y.re \leq -1.45 \cdot 10^{+74}:\\ \;\;\;\;\left(x.re + \frac{y.im}{\frac{y.re}{x.im}}\right) \cdot \frac{-1}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.re \leq -1.4 \cdot 10^{-143}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq 4.7 \cdot 10^{-138}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re}{y.im} \cdot \frac{x.re}{y.im}\\ \mathbf{elif}\;y.re \leq 2.65 \cdot 10^{+131}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{1}{\frac{y.re}{\frac{x.im}{\frac{y.re}{y.im}}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0
         (*
          (/ 1.0 (hypot y.re y.im))
          (/ (fma x.re y.re (* y.im x.im)) (hypot y.re y.im)))))
   (if (<= y.re -1.45e+74)
     (* (+ x.re (/ y.im (/ y.re x.im))) (/ -1.0 (hypot y.re y.im)))
     (if (<= y.re -1.4e-143)
       t_0
       (if (<= y.re 4.7e-138)
         (+ (/ x.im y.im) (* (/ y.re y.im) (/ x.re y.im)))
         (if (<= y.re 2.65e+131)
           t_0
           (+ (/ x.re y.re) (/ 1.0 (/ y.re (/ x.im (/ 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 / hypot(y_46_re, y_46_im)) * (fma(x_46_re, y_46_re, (y_46_im * x_46_im)) / hypot(y_46_re, y_46_im));
	double tmp;
	if (y_46_re <= -1.45e+74) {
		tmp = (x_46_re + (y_46_im / (y_46_re / x_46_im))) * (-1.0 / hypot(y_46_re, y_46_im));
	} else if (y_46_re <= -1.4e-143) {
		tmp = t_0;
	} else if (y_46_re <= 4.7e-138) {
		tmp = (x_46_im / y_46_im) + ((y_46_re / y_46_im) * (x_46_re / y_46_im));
	} else if (y_46_re <= 2.65e+131) {
		tmp = t_0;
	} else {
		tmp = (x_46_re / y_46_re) + (1.0 / (y_46_re / (x_46_im / (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 / hypot(y_46_re, y_46_im)) * Float64(fma(x_46_re, y_46_re, Float64(y_46_im * x_46_im)) / hypot(y_46_re, y_46_im)))
	tmp = 0.0
	if (y_46_re <= -1.45e+74)
		tmp = Float64(Float64(x_46_re + Float64(y_46_im / Float64(y_46_re / x_46_im))) * Float64(-1.0 / hypot(y_46_re, y_46_im)));
	elseif (y_46_re <= -1.4e-143)
		tmp = t_0;
	elseif (y_46_re <= 4.7e-138)
		tmp = Float64(Float64(x_46_im / y_46_im) + Float64(Float64(y_46_re / y_46_im) * Float64(x_46_re / y_46_im)));
	elseif (y_46_re <= 2.65e+131)
		tmp = t_0;
	else
		tmp = Float64(Float64(x_46_re / y_46_re) + Float64(1.0 / Float64(y_46_re / Float64(x_46_im / Float64(y_46_re / y_46_im)))));
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[(1.0 / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision] * N[(N[(x$46$re * y$46$re + N[(y$46$im * x$46$im), $MachinePrecision]), $MachinePrecision] / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$re, -1.45e+74], N[(N[(x$46$re + N[(y$46$im / N[(y$46$re / x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(-1.0 / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, -1.4e-143], t$95$0, If[LessEqual[y$46$re, 4.7e-138], N[(N[(x$46$im / y$46$im), $MachinePrecision] + N[(N[(y$46$re / y$46$im), $MachinePrecision] * N[(x$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 2.65e+131], t$95$0, N[(N[(x$46$re / y$46$re), $MachinePrecision] + N[(1.0 / N[(y$46$re / N[(x$46$im / N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if y.re < -1.4500000000000001e74

   1. Initial program 33.4%

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

      1. *-un-lft-identity 33.4%

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

      2. add-sqr-sqrt 33.4%

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

      3. times-frac 33.5%

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

      4. hypot-def 33.5%

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

      5. fma-def 33.5%

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

      6. hypot-def 52.5%

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

   3. Applied egg-rr 52.5%

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

   4. Taylor expanded in y.re around -inf 83.2%

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

      1. distribute-lft-out 83.2%

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

      2. +-commutative 83.2%

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

      3. associate-/l* 89.7%

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

   6. Simplified 89.7%

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

   if -1.4500000000000001e74 < y.re < -1.3999999999999999e-143 or 4.7000000000000001e-138 < y.re < 2.6499999999999998e131

   1. Initial program 82.6%

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

      1. *-un-lft-identity 82.6%

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

      2. add-sqr-sqrt 82.5%

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

      3. times-frac 82.5%

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

      4. hypot-def 82.5%

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

      5. fma-def 82.5%

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

      6. hypot-def 92.3%

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

   3. Applied egg-rr 92.3%

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

   if -1.3999999999999999e-143 < y.re < 4.7000000000000001e-138

   1. Initial program 61.6%

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

   2. Taylor expanded in y.re around 0 83.4%

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

      1. +-commutative 83.4%

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

      2. *-commutative 83.4%

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

      3. unpow2 83.4%

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

      4. times-frac 92.1%

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

   4. Simplified 92.1%

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

   if 2.6499999999999998e131 < y.re

   1. Initial program 28.6%

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

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

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

      1. associate-/l* 74.0%

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

      2. unpow2 74.0%

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

   4. Simplified 74.0%

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

      1. clear-num 74.0%

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

      2. inv-pow 74.0%

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

      3. associate-/l* 85.9%

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

   6. Applied egg-rr 85.9%

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

      1. unpow-1 85.9%

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

      2. associate-/l/ 91.9%

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

      3. /-rgt-identity 91.9%

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

      4. times-frac 91.5%

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

      5. *-commutative 91.5%

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

      6. *-commutative 91.5%

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

      7. times-frac 91.9%

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

      8. /-rgt-identity 91.9%

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

      9. associate-/r/ 91.9%

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

   8. Simplified 91.9%

      \[\leadsto \frac{x.re}{y.re} + \color{blue}{\frac{1}{\frac{y.re}{\frac{x.im}{\frac{y.re}{y.im}}}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 91.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -1.45 \cdot 10^{+74}:\\ \;\;\;\;\left(x.re + \frac{y.im}{\frac{y.re}{x.im}}\right) \cdot \frac{-1}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.re \leq -1.4 \cdot 10^{-143}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{\mathsf{fma}\left(x.re, y.re, y.im \cdot x.im\right)}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.re \leq 4.7 \cdot 10^{-138}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re}{y.im} \cdot \frac{x.re}{y.im}\\ \mathbf{elif}\;y.re \leq 2.65 \cdot 10^{+131}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{\mathsf{fma}\left(x.re, y.re, y.im \cdot x.im\right)}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{1}{\frac{y.re}{\frac{x.im}{\frac{y.re}{y.im}}}}\\ \end{array} \]

Alternative 3: 82.2% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y.im \cdot x.im + y.re \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{if}\;y.re \leq -6.5 \cdot 10^{+66}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y.im}{y.re}, \frac{x.im}{y.re}, \frac{x.re}{y.re}\right)\\ \mathbf{elif}\;y.re \leq -1.5 \cdot 10^{-144}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq 10^{-130}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re}{y.im} \cdot \frac{x.re}{y.im}\\ \mathbf{elif}\;y.re \leq 1.05 \cdot 10^{+27}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(x.re + \frac{y.im}{\frac{y.re}{x.im}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0
         (/ (+ (* y.im x.im) (* y.re x.re)) (+ (* y.re y.re) (* y.im y.im)))))
   (if (<= y.re -6.5e+66)
     (fma (/ y.im y.re) (/ x.im y.re) (/ x.re y.re))
     (if (<= y.re -1.5e-144)
       t_0
       (if (<= y.re 1e-130)
         (+ (/ x.im y.im) (* (/ y.re y.im) (/ x.re y.im)))
         (if (<= y.re 1.05e+27)
           t_0
           (* (/ 1.0 (hypot y.re y.im)) (+ x.re (/ y.im (/ y.re x.im))))))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = ((y_46_im * x_46_im) + (y_46_re * x_46_re)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	double tmp;
	if (y_46_re <= -6.5e+66) {
		tmp = fma((y_46_im / y_46_re), (x_46_im / y_46_re), (x_46_re / y_46_re));
	} else if (y_46_re <= -1.5e-144) {
		tmp = t_0;
	} else if (y_46_re <= 1e-130) {
		tmp = (x_46_im / y_46_im) + ((y_46_re / y_46_im) * (x_46_re / y_46_im));
	} else if (y_46_re <= 1.05e+27) {
		tmp = t_0;
	} else {
		tmp = (1.0 / hypot(y_46_re, y_46_im)) * (x_46_re + (y_46_im / (y_46_re / x_46_im)));
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(Float64(Float64(y_46_im * x_46_im) + Float64(y_46_re * x_46_re)) / Float64(Float64(y_46_re * y_46_re) + Float64(y_46_im * y_46_im)))
	tmp = 0.0
	if (y_46_re <= -6.5e+66)
		tmp = fma(Float64(y_46_im / y_46_re), Float64(x_46_im / y_46_re), Float64(x_46_re / y_46_re));
	elseif (y_46_re <= -1.5e-144)
		tmp = t_0;
	elseif (y_46_re <= 1e-130)
		tmp = Float64(Float64(x_46_im / y_46_im) + Float64(Float64(y_46_re / y_46_im) * Float64(x_46_re / y_46_im)));
	elseif (y_46_re <= 1.05e+27)
		tmp = t_0;
	else
		tmp = Float64(Float64(1.0 / hypot(y_46_re, y_46_im)) * Float64(x_46_re + Float64(y_46_im / Float64(y_46_re / x_46_im))));
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[(N[(y$46$im * x$46$im), $MachinePrecision] + N[(y$46$re * x$46$re), $MachinePrecision]), $MachinePrecision] / N[(N[(y$46$re * y$46$re), $MachinePrecision] + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$re, -6.5e+66], N[(N[(y$46$im / y$46$re), $MachinePrecision] * N[(x$46$im / y$46$re), $MachinePrecision] + N[(x$46$re / y$46$re), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, -1.5e-144], t$95$0, If[LessEqual[y$46$re, 1e-130], N[(N[(x$46$im / y$46$im), $MachinePrecision] + N[(N[(y$46$re / y$46$im), $MachinePrecision] * N[(x$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 1.05e+27], t$95$0, N[(N[(1.0 / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision] * N[(x$46$re + N[(y$46$im / N[(y$46$re / x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if y.re < -6.5000000000000001e66

   1. Initial program 37.3%

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

   2. Taylor expanded in y.re around inf 80.6%

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

      1. +-commutative 80.6%

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

      2. unpow2 80.6%

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

      3. times-frac 88.7%

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

      4. fma-def 88.7%

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

   4. Simplified 88.7%

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

   if -6.5000000000000001e66 < y.re < -1.4999999999999999e-144 or 1.0000000000000001e-130 < y.re < 1.04999999999999997e27

   1. Initial program 86.0%

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

   if -1.4999999999999999e-144 < y.re < 1.0000000000000001e-130

   1. Initial program 61.6%

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

   2. Taylor expanded in y.re around 0 83.4%

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

      1. +-commutative 83.4%

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

      2. *-commutative 83.4%

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

      3. unpow2 83.4%

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

      4. times-frac 92.1%

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

   4. Simplified 92.1%

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

   if 1.04999999999999997e27 < y.re

   1. Initial program 42.3%

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

      1. *-un-lft-identity 42.3%

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

      2. add-sqr-sqrt 42.3%

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

      3. times-frac 42.4%

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

      4. hypot-def 42.4%

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

      5. fma-def 42.4%

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

      6. hypot-def 66.8%

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

   3. Applied egg-rr 66.8%

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

   4. Taylor expanded in y.re around inf 83.1%

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

      1. associate-/l* 83.5%

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

   6. Simplified 83.5%

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

4. Final simplification 87.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -6.5 \cdot 10^{+66}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y.im}{y.re}, \frac{x.im}{y.re}, \frac{x.re}{y.re}\right)\\ \mathbf{elif}\;y.re \leq -1.5 \cdot 10^{-144}:\\ \;\;\;\;\frac{y.im \cdot x.im + y.re \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.re \leq 10^{-130}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re}{y.im} \cdot \frac{x.re}{y.im}\\ \mathbf{elif}\;y.re \leq 1.05 \cdot 10^{+27}:\\ \;\;\;\;\frac{y.im \cdot x.im + y.re \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(x.re + \frac{y.im}{\frac{y.re}{x.im}}\right)\\ \end{array} \]

Alternative 4: 82.5% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y.im \cdot x.im + y.re \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ t_1 := x.re + \frac{y.im}{\frac{y.re}{x.im}}\\ \mathbf{if}\;y.re \leq -3.5 \cdot 10^{+71}:\\ \;\;\;\;t_1 \cdot \frac{-1}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.re \leq -5.7 \cdot 10^{-143}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq 3 \cdot 10^{-132}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re}{y.im} \cdot \frac{x.re}{y.im}\\ \mathbf{elif}\;y.re \leq 1.05 \cdot 10^{+27}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0
         (/ (+ (* y.im x.im) (* y.re x.re)) (+ (* y.re y.re) (* y.im y.im))))
        (t_1 (+ x.re (/ y.im (/ y.re x.im)))))
   (if (<= y.re -3.5e+71)
     (* t_1 (/ -1.0 (hypot y.re y.im)))
     (if (<= y.re -5.7e-143)
       t_0
       (if (<= y.re 3e-132)
         (+ (/ x.im y.im) (* (/ y.re y.im) (/ x.re y.im)))
         (if (<= y.re 1.05e+27) t_0 (* (/ 1.0 (hypot y.re y.im)) t_1)))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = ((y_46_im * x_46_im) + (y_46_re * x_46_re)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	double t_1 = x_46_re + (y_46_im / (y_46_re / x_46_im));
	double tmp;
	if (y_46_re <= -3.5e+71) {
		tmp = t_1 * (-1.0 / hypot(y_46_re, y_46_im));
	} else if (y_46_re <= -5.7e-143) {
		tmp = t_0;
	} else if (y_46_re <= 3e-132) {
		tmp = (x_46_im / y_46_im) + ((y_46_re / y_46_im) * (x_46_re / y_46_im));
	} else if (y_46_re <= 1.05e+27) {
		tmp = t_0;
	} else {
		tmp = (1.0 / hypot(y_46_re, y_46_im)) * t_1;
	}
	return tmp;
}

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = ((y_46_im * x_46_im) + (y_46_re * x_46_re)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	double t_1 = x_46_re + (y_46_im / (y_46_re / x_46_im));
	double tmp;
	if (y_46_re <= -3.5e+71) {
		tmp = t_1 * (-1.0 / Math.hypot(y_46_re, y_46_im));
	} else if (y_46_re <= -5.7e-143) {
		tmp = t_0;
	} else if (y_46_re <= 3e-132) {
		tmp = (x_46_im / y_46_im) + ((y_46_re / y_46_im) * (x_46_re / y_46_im));
	} else if (y_46_re <= 1.05e+27) {
		tmp = t_0;
	} else {
		tmp = (1.0 / Math.hypot(y_46_re, y_46_im)) * t_1;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = ((y_46_im * x_46_im) + (y_46_re * x_46_re)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im))
	t_1 = x_46_re + (y_46_im / (y_46_re / x_46_im))
	tmp = 0
	if y_46_re <= -3.5e+71:
		tmp = t_1 * (-1.0 / math.hypot(y_46_re, y_46_im))
	elif y_46_re <= -5.7e-143:
		tmp = t_0
	elif y_46_re <= 3e-132:
		tmp = (x_46_im / y_46_im) + ((y_46_re / y_46_im) * (x_46_re / y_46_im))
	elif y_46_re <= 1.05e+27:
		tmp = t_0
	else:
		tmp = (1.0 / math.hypot(y_46_re, y_46_im)) * t_1
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(Float64(Float64(y_46_im * x_46_im) + Float64(y_46_re * x_46_re)) / Float64(Float64(y_46_re * y_46_re) + Float64(y_46_im * y_46_im)))
	t_1 = Float64(x_46_re + Float64(y_46_im / Float64(y_46_re / x_46_im)))
	tmp = 0.0
	if (y_46_re <= -3.5e+71)
		tmp = Float64(t_1 * Float64(-1.0 / hypot(y_46_re, y_46_im)));
	elseif (y_46_re <= -5.7e-143)
		tmp = t_0;
	elseif (y_46_re <= 3e-132)
		tmp = Float64(Float64(x_46_im / y_46_im) + Float64(Float64(y_46_re / y_46_im) * Float64(x_46_re / y_46_im)));
	elseif (y_46_re <= 1.05e+27)
		tmp = t_0;
	else
		tmp = Float64(Float64(1.0 / hypot(y_46_re, y_46_im)) * t_1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = ((y_46_im * x_46_im) + (y_46_re * x_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 / x_46_im));
	tmp = 0.0;
	if (y_46_re <= -3.5e+71)
		tmp = t_1 * (-1.0 / hypot(y_46_re, y_46_im));
	elseif (y_46_re <= -5.7e-143)
		tmp = t_0;
	elseif (y_46_re <= 3e-132)
		tmp = (x_46_im / y_46_im) + ((y_46_re / y_46_im) * (x_46_re / y_46_im));
	elseif (y_46_re <= 1.05e+27)
		tmp = t_0;
	else
		tmp = (1.0 / hypot(y_46_re, y_46_im)) * t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[(N[(y$46$im * x$46$im), $MachinePrecision] + N[(y$46$re * x$46$re), $MachinePrecision]), $MachinePrecision] / N[(N[(y$46$re * y$46$re), $MachinePrecision] + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x$46$re + N[(y$46$im / N[(y$46$re / x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$re, -3.5e+71], N[(t$95$1 * N[(-1.0 / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, -5.7e-143], t$95$0, If[LessEqual[y$46$re, 3e-132], N[(N[(x$46$im / y$46$im), $MachinePrecision] + N[(N[(y$46$re / y$46$im), $MachinePrecision] * N[(x$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 1.05e+27], t$95$0, N[(N[(1.0 / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if y.re < -3.4999999999999999e71

   1. Initial program 34.8%

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

      1. *-un-lft-identity 34.8%

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

      2. add-sqr-sqrt 34.8%

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

      3. times-frac 34.9%

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

      4. hypot-def 34.9%

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

      5. fma-def 34.9%

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

      6. hypot-def 53.5%

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

   3. Applied egg-rr 53.5%

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

   4. Taylor expanded in y.re around -inf 83.6%

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

      1. distribute-lft-out 83.6%

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

      2. +-commutative 83.6%

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

      3. associate-/l* 89.9%

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

   6. Simplified 89.9%

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

   if -3.4999999999999999e71 < y.re < -5.6999999999999999e-143 or 3e-132 < y.re < 1.04999999999999997e27

   1. Initial program 86.3%

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

   if -5.6999999999999999e-143 < y.re < 3e-132

   1. Initial program 61.6%

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

   2. Taylor expanded in y.re around 0 83.4%

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

      1. +-commutative 83.4%

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

      2. *-commutative 83.4%

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

      3. unpow2 83.4%

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

      4. times-frac 92.1%

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

   4. Simplified 92.1%

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

   if 1.04999999999999997e27 < y.re

   1. Initial program 42.3%

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

      1. *-un-lft-identity 42.3%

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

      2. add-sqr-sqrt 42.3%

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

      3. times-frac 42.4%

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

      4. hypot-def 42.4%

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

      5. fma-def 42.4%

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

      6. hypot-def 66.8%

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

   3. Applied egg-rr 66.8%

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

   4. Taylor expanded in y.re around inf 83.1%

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

      1. associate-/l* 83.5%

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

   6. Simplified 83.5%

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

4. Final simplification 88.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -3.5 \cdot 10^{+71}:\\ \;\;\;\;\left(x.re + \frac{y.im}{\frac{y.re}{x.im}}\right) \cdot \frac{-1}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.re \leq -5.7 \cdot 10^{-143}:\\ \;\;\;\;\frac{y.im \cdot x.im + y.re \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.re \leq 3 \cdot 10^{-132}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re}{y.im} \cdot \frac{x.re}{y.im}\\ \mathbf{elif}\;y.re \leq 1.05 \cdot 10^{+27}:\\ \;\;\;\;\frac{y.im \cdot x.im + y.re \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(x.re + \frac{y.im}{\frac{y.re}{x.im}}\right)\\ \end{array} \]

Alternative 5: 82.5% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y.im \cdot x.im + y.re \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{if}\;y.re \leq -2.2 \cdot 10^{+67}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y.im}{y.re}, \frac{x.im}{y.re}, \frac{x.re}{y.re}\right)\\ \mathbf{elif}\;y.re \leq -5.6 \cdot 10^{-144}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq 2.4 \cdot 10^{-135}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re}{y.im} \cdot \frac{x.re}{y.im}\\ \mathbf{elif}\;y.re \leq 2.4 \cdot 10^{+74}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{1}{\frac{y.re}{\frac{x.im}{\frac{y.re}{y.im}}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0
         (/ (+ (* y.im x.im) (* y.re x.re)) (+ (* y.re y.re) (* y.im y.im)))))
   (if (<= y.re -2.2e+67)
     (fma (/ y.im y.re) (/ x.im y.re) (/ x.re y.re))
     (if (<= y.re -5.6e-144)
       t_0
       (if (<= y.re 2.4e-135)
         (+ (/ x.im y.im) (* (/ y.re y.im) (/ x.re y.im)))
         (if (<= y.re 2.4e+74)
           t_0
           (+ (/ x.re y.re) (/ 1.0 (/ y.re (/ x.im (/ y.re y.im)))))))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = ((y_46_im * x_46_im) + (y_46_re * x_46_re)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	double tmp;
	if (y_46_re <= -2.2e+67) {
		tmp = fma((y_46_im / y_46_re), (x_46_im / y_46_re), (x_46_re / y_46_re));
	} else if (y_46_re <= -5.6e-144) {
		tmp = t_0;
	} else if (y_46_re <= 2.4e-135) {
		tmp = (x_46_im / y_46_im) + ((y_46_re / y_46_im) * (x_46_re / y_46_im));
	} else if (y_46_re <= 2.4e+74) {
		tmp = t_0;
	} else {
		tmp = (x_46_re / y_46_re) + (1.0 / (y_46_re / (x_46_im / (y_46_re / y_46_im))));
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(Float64(Float64(y_46_im * x_46_im) + Float64(y_46_re * x_46_re)) / Float64(Float64(y_46_re * y_46_re) + Float64(y_46_im * y_46_im)))
	tmp = 0.0
	if (y_46_re <= -2.2e+67)
		tmp = fma(Float64(y_46_im / y_46_re), Float64(x_46_im / y_46_re), Float64(x_46_re / y_46_re));
	elseif (y_46_re <= -5.6e-144)
		tmp = t_0;
	elseif (y_46_re <= 2.4e-135)
		tmp = Float64(Float64(x_46_im / y_46_im) + Float64(Float64(y_46_re / y_46_im) * Float64(x_46_re / y_46_im)));
	elseif (y_46_re <= 2.4e+74)
		tmp = t_0;
	else
		tmp = Float64(Float64(x_46_re / y_46_re) + Float64(1.0 / Float64(y_46_re / Float64(x_46_im / Float64(y_46_re / y_46_im)))));
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[(N[(y$46$im * x$46$im), $MachinePrecision] + N[(y$46$re * x$46$re), $MachinePrecision]), $MachinePrecision] / N[(N[(y$46$re * y$46$re), $MachinePrecision] + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$re, -2.2e+67], N[(N[(y$46$im / y$46$re), $MachinePrecision] * N[(x$46$im / y$46$re), $MachinePrecision] + N[(x$46$re / y$46$re), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, -5.6e-144], t$95$0, If[LessEqual[y$46$re, 2.4e-135], N[(N[(x$46$im / y$46$im), $MachinePrecision] + N[(N[(y$46$re / y$46$im), $MachinePrecision] * N[(x$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 2.4e+74], t$95$0, N[(N[(x$46$re / y$46$re), $MachinePrecision] + N[(1.0 / N[(y$46$re / N[(x$46$im / N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if y.re < -2.2e67

   1. Initial program 37.3%

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

   2. Taylor expanded in y.re around inf 80.6%

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

      1. +-commutative 80.6%

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

      2. unpow2 80.6%

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

      3. times-frac 88.7%

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

      4. fma-def 88.7%

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

   4. Simplified 88.7%

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

   if -2.2e67 < y.re < -5.59999999999999995e-144 or 2.3999999999999999e-135 < y.re < 2.40000000000000008e74

   1. Initial program 82.3%

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

   if -5.59999999999999995e-144 < y.re < 2.3999999999999999e-135

   1. Initial program 61.6%

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

   2. Taylor expanded in y.re around 0 83.4%

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

      1. +-commutative 83.4%

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

      2. *-commutative 83.4%

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

      3. unpow2 83.4%

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

      4. times-frac 92.1%

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

   4. Simplified 92.1%

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

   if 2.40000000000000008e74 < y.re

   1. Initial program 41.2%

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

   2. Taylor expanded in y.re around inf 75.5%

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

      1. associate-/l* 75.7%

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

      2. unpow2 75.7%

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

   4. Simplified 75.7%

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

      1. clear-num 75.7%

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

      2. inv-pow 75.7%

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

      3. associate-/l* 84.7%

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

   6. Applied egg-rr 84.7%

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

      1. unpow-1 84.7%

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

      2. associate-/l/ 89.2%

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

      3. /-rgt-identity 89.2%

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

      4. times-frac 88.8%

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

      5. *-commutative 88.8%

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

      6. *-commutative 88.8%

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

      7. times-frac 89.2%

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

      8. /-rgt-identity 89.2%

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

      9. associate-/r/ 89.2%

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

   8. Simplified 89.2%

      \[\leadsto \frac{x.re}{y.re} + \color{blue}{\frac{1}{\frac{y.re}{\frac{x.im}{\frac{y.re}{y.im}}}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 87.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -2.2 \cdot 10^{+67}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y.im}{y.re}, \frac{x.im}{y.re}, \frac{x.re}{y.re}\right)\\ \mathbf{elif}\;y.re \leq -5.6 \cdot 10^{-144}:\\ \;\;\;\;\frac{y.im \cdot x.im + y.re \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.re \leq 2.4 \cdot 10^{-135}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re}{y.im} \cdot \frac{x.re}{y.im}\\ \mathbf{elif}\;y.re \leq 2.4 \cdot 10^{+74}:\\ \;\;\;\;\frac{y.im \cdot x.im + y.re \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{1}{\frac{y.re}{\frac{x.im}{\frac{y.re}{y.im}}}}\\ \end{array} \]

Alternative 6: 82.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y.im \cdot x.im + y.re \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ t_1 := \frac{x.re}{y.re} + \frac{1}{\frac{y.re}{\frac{x.im}{\frac{y.re}{y.im}}}}\\ \mathbf{if}\;y.re \leq -1.32 \cdot 10^{+67}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.re \leq -2.35 \cdot 10^{-144}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq 1.02 \cdot 10^{-134}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re}{y.im} \cdot \frac{x.re}{y.im}\\ \mathbf{elif}\;y.re \leq 7.1 \cdot 10^{+75}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0
         (/ (+ (* y.im x.im) (* y.re x.re)) (+ (* y.re y.re) (* y.im y.im))))
        (t_1 (+ (/ x.re y.re) (/ 1.0 (/ y.re (/ x.im (/ y.re y.im)))))))
   (if (<= y.re -1.32e+67)
     t_1
     (if (<= y.re -2.35e-144)
       t_0
       (if (<= y.re 1.02e-134)
         (+ (/ x.im y.im) (* (/ y.re y.im) (/ x.re y.im)))
         (if (<= y.re 7.1e+75) t_0 t_1))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = ((y_46_im * x_46_im) + (y_46_re * x_46_re)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	double t_1 = (x_46_re / y_46_re) + (1.0 / (y_46_re / (x_46_im / (y_46_re / y_46_im))));
	double tmp;
	if (y_46_re <= -1.32e+67) {
		tmp = t_1;
	} else if (y_46_re <= -2.35e-144) {
		tmp = t_0;
	} else if (y_46_re <= 1.02e-134) {
		tmp = (x_46_im / y_46_im) + ((y_46_re / y_46_im) * (x_46_re / y_46_im));
	} else if (y_46_re <= 7.1e+75) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = ((y_46im * x_46im) + (y_46re * x_46re)) / ((y_46re * y_46re) + (y_46im * y_46im))
    t_1 = (x_46re / y_46re) + (1.0d0 / (y_46re / (x_46im / (y_46re / y_46im))))
    if (y_46re <= (-1.32d+67)) then
        tmp = t_1
    else if (y_46re <= (-2.35d-144)) then
        tmp = t_0
    else if (y_46re <= 1.02d-134) then
        tmp = (x_46im / y_46im) + ((y_46re / y_46im) * (x_46re / y_46im))
    else if (y_46re <= 7.1d+75) then
        tmp = t_0
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = ((y_46_im * x_46_im) + (y_46_re * x_46_re)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	double t_1 = (x_46_re / y_46_re) + (1.0 / (y_46_re / (x_46_im / (y_46_re / y_46_im))));
	double tmp;
	if (y_46_re <= -1.32e+67) {
		tmp = t_1;
	} else if (y_46_re <= -2.35e-144) {
		tmp = t_0;
	} else if (y_46_re <= 1.02e-134) {
		tmp = (x_46_im / y_46_im) + ((y_46_re / y_46_im) * (x_46_re / y_46_im));
	} else if (y_46_re <= 7.1e+75) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = ((y_46_im * x_46_im) + (y_46_re * x_46_re)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im))
	t_1 = (x_46_re / y_46_re) + (1.0 / (y_46_re / (x_46_im / (y_46_re / y_46_im))))
	tmp = 0
	if y_46_re <= -1.32e+67:
		tmp = t_1
	elif y_46_re <= -2.35e-144:
		tmp = t_0
	elif y_46_re <= 1.02e-134:
		tmp = (x_46_im / y_46_im) + ((y_46_re / y_46_im) * (x_46_re / y_46_im))
	elif y_46_re <= 7.1e+75:
		tmp = t_0
	else:
		tmp = t_1
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(Float64(Float64(y_46_im * x_46_im) + Float64(y_46_re * x_46_re)) / Float64(Float64(y_46_re * y_46_re) + Float64(y_46_im * y_46_im)))
	t_1 = Float64(Float64(x_46_re / y_46_re) + Float64(1.0 / Float64(y_46_re / Float64(x_46_im / Float64(y_46_re / y_46_im)))))
	tmp = 0.0
	if (y_46_re <= -1.32e+67)
		tmp = t_1;
	elseif (y_46_re <= -2.35e-144)
		tmp = t_0;
	elseif (y_46_re <= 1.02e-134)
		tmp = Float64(Float64(x_46_im / y_46_im) + Float64(Float64(y_46_re / y_46_im) * Float64(x_46_re / y_46_im)));
	elseif (y_46_re <= 7.1e+75)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = ((y_46_im * x_46_im) + (y_46_re * x_46_re)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	t_1 = (x_46_re / y_46_re) + (1.0 / (y_46_re / (x_46_im / (y_46_re / y_46_im))));
	tmp = 0.0;
	if (y_46_re <= -1.32e+67)
		tmp = t_1;
	elseif (y_46_re <= -2.35e-144)
		tmp = t_0;
	elseif (y_46_re <= 1.02e-134)
		tmp = (x_46_im / y_46_im) + ((y_46_re / y_46_im) * (x_46_re / y_46_im));
	elseif (y_46_re <= 7.1e+75)
		tmp = t_0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[(N[(y$46$im * x$46$im), $MachinePrecision] + N[(y$46$re * x$46$re), $MachinePrecision]), $MachinePrecision] / N[(N[(y$46$re * y$46$re), $MachinePrecision] + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x$46$re / y$46$re), $MachinePrecision] + N[(1.0 / N[(y$46$re / N[(x$46$im / N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$re, -1.32e+67], t$95$1, If[LessEqual[y$46$re, -2.35e-144], t$95$0, If[LessEqual[y$46$re, 1.02e-134], N[(N[(x$46$im / y$46$im), $MachinePrecision] + N[(N[(y$46$re / y$46$im), $MachinePrecision] * N[(x$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 7.1e+75], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -1.3200000000000001e67 or 7.09999999999999982e75 < y.re

   1. Initial program 39.1%

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

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

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

      1. associate-/l* 79.7%

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

      2. unpow2 79.7%

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

   4. Simplified 79.7%

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

      1. clear-num 79.7%

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

      2. inv-pow 79.7%

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

      3. associate-/l* 84.8%

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

   6. Applied egg-rr 84.8%

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

      1. unpow-1 84.8%

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

      2. associate-/l/ 89.9%

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

      3. /-rgt-identity 89.9%

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

      4. times-frac 86.4%

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

      5. *-commutative 86.4%

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

      6. *-commutative 86.4%

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

      7. times-frac 89.9%

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

      8. /-rgt-identity 89.9%

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

      9. associate-/r/ 88.9%

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

   8. Simplified 88.9%

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

   if -1.3200000000000001e67 < y.re < -2.3500000000000001e-144 or 1.02e-134 < y.re < 7.09999999999999982e75

   1. Initial program 82.3%

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

   if -2.3500000000000001e-144 < y.re < 1.02e-134

   1. Initial program 61.6%

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

   2. Taylor expanded in y.re around 0 83.4%

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

      1. +-commutative 83.4%

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

      2. *-commutative 83.4%

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

      3. unpow2 83.4%

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

      4. times-frac 92.1%

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

   4. Simplified 92.1%

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

4. Final simplification 87.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -1.32 \cdot 10^{+67}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{1}{\frac{y.re}{\frac{x.im}{\frac{y.re}{y.im}}}}\\ \mathbf{elif}\;y.re \leq -2.35 \cdot 10^{-144}:\\ \;\;\;\;\frac{y.im \cdot x.im + y.re \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.re \leq 1.02 \cdot 10^{-134}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re}{y.im} \cdot \frac{x.re}{y.im}\\ \mathbf{elif}\;y.re \leq 7.1 \cdot 10^{+75}:\\ \;\;\;\;\frac{y.im \cdot x.im + y.re \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{1}{\frac{y.re}{\frac{x.im}{\frac{y.re}{y.im}}}}\\ \end{array} \]

Alternative 7: 75.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x.im}{y.im} + \frac{1}{\frac{y.im}{y.re} \cdot \frac{y.im}{x.re}}\\ t_1 := \frac{-1}{y.re} \cdot \left(\left(-x.re\right) - x.im \cdot \frac{y.im}{y.re}\right)\\ \mathbf{if}\;y.re \leq -3 \cdot 10^{+49}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.re \leq -980:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq -1.8 \cdot 10^{-117}:\\ \;\;\;\;\frac{y.re \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.re \leq 4.15 \cdot 10^{+61}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (+ (/ x.im y.im) (/ 1.0 (* (/ y.im y.re) (/ y.im x.re)))))
        (t_1 (* (/ -1.0 y.re) (- (- x.re) (* x.im (/ y.im y.re))))))
   (if (<= y.re -3e+49)
     t_1
     (if (<= y.re -980.0)
       t_0
       (if (<= y.re -1.8e-117)
         (/ (* y.re x.re) (+ (* y.re y.re) (* y.im y.im)))
         (if (<= y.re 4.15e+61) t_0 t_1))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = (x_46_im / y_46_im) + (1.0 / ((y_46_im / y_46_re) * (y_46_im / x_46_re)));
	double t_1 = (-1.0 / y_46_re) * (-x_46_re - (x_46_im * (y_46_im / y_46_re)));
	double tmp;
	if (y_46_re <= -3e+49) {
		tmp = t_1;
	} else if (y_46_re <= -980.0) {
		tmp = t_0;
	} else if (y_46_re <= -1.8e-117) {
		tmp = (y_46_re * x_46_re) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	} else if (y_46_re <= 4.15e+61) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (x_46im / y_46im) + (1.0d0 / ((y_46im / y_46re) * (y_46im / x_46re)))
    t_1 = ((-1.0d0) / y_46re) * (-x_46re - (x_46im * (y_46im / y_46re)))
    if (y_46re <= (-3d+49)) then
        tmp = t_1
    else if (y_46re <= (-980.0d0)) then
        tmp = t_0
    else if (y_46re <= (-1.8d-117)) then
        tmp = (y_46re * x_46re) / ((y_46re * y_46re) + (y_46im * y_46im))
    else if (y_46re <= 4.15d+61) then
        tmp = t_0
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = (x_46_im / y_46_im) + (1.0 / ((y_46_im / y_46_re) * (y_46_im / x_46_re)));
	double t_1 = (-1.0 / y_46_re) * (-x_46_re - (x_46_im * (y_46_im / y_46_re)));
	double tmp;
	if (y_46_re <= -3e+49) {
		tmp = t_1;
	} else if (y_46_re <= -980.0) {
		tmp = t_0;
	} else if (y_46_re <= -1.8e-117) {
		tmp = (y_46_re * x_46_re) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	} else if (y_46_re <= 4.15e+61) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = (x_46_im / y_46_im) + (1.0 / ((y_46_im / y_46_re) * (y_46_im / x_46_re)))
	t_1 = (-1.0 / y_46_re) * (-x_46_re - (x_46_im * (y_46_im / y_46_re)))
	tmp = 0
	if y_46_re <= -3e+49:
		tmp = t_1
	elif y_46_re <= -980.0:
		tmp = t_0
	elif y_46_re <= -1.8e-117:
		tmp = (y_46_re * x_46_re) / ((y_46_re * y_46_re) + (y_46_im * y_46_im))
	elif y_46_re <= 4.15e+61:
		tmp = t_0
	else:
		tmp = t_1
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(Float64(x_46_im / y_46_im) + Float64(1.0 / Float64(Float64(y_46_im / y_46_re) * Float64(y_46_im / x_46_re))))
	t_1 = Float64(Float64(-1.0 / y_46_re) * Float64(Float64(-x_46_re) - Float64(x_46_im * Float64(y_46_im / y_46_re))))
	tmp = 0.0
	if (y_46_re <= -3e+49)
		tmp = t_1;
	elseif (y_46_re <= -980.0)
		tmp = t_0;
	elseif (y_46_re <= -1.8e-117)
		tmp = Float64(Float64(y_46_re * x_46_re) / Float64(Float64(y_46_re * y_46_re) + Float64(y_46_im * y_46_im)));
	elseif (y_46_re <= 4.15e+61)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = (x_46_im / y_46_im) + (1.0 / ((y_46_im / y_46_re) * (y_46_im / x_46_re)));
	t_1 = (-1.0 / y_46_re) * (-x_46_re - (x_46_im * (y_46_im / y_46_re)));
	tmp = 0.0;
	if (y_46_re <= -3e+49)
		tmp = t_1;
	elseif (y_46_re <= -980.0)
		tmp = t_0;
	elseif (y_46_re <= -1.8e-117)
		tmp = (y_46_re * x_46_re) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	elseif (y_46_re <= 4.15e+61)
		tmp = t_0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[(x$46$im / y$46$im), $MachinePrecision] + N[(1.0 / N[(N[(y$46$im / y$46$re), $MachinePrecision] * N[(y$46$im / x$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(-1.0 / y$46$re), $MachinePrecision] * N[((-x$46$re) - N[(x$46$im * N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$re, -3e+49], t$95$1, If[LessEqual[y$46$re, -980.0], t$95$0, If[LessEqual[y$46$re, -1.8e-117], N[(N[(y$46$re * x$46$re), $MachinePrecision] / N[(N[(y$46$re * y$46$re), $MachinePrecision] + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 4.15e+61], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -3.0000000000000002e49 or 4.15000000000000025e61 < y.re

   1. Initial program 42.2%

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

      1. *-un-lft-identity 42.2%

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

      2. add-sqr-sqrt 42.2%

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

      3. times-frac 42.3%

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

      4. hypot-def 42.3%

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

      5. fma-def 42.3%

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

      6. hypot-def 62.4%

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

   3. Applied egg-rr 62.4%

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

   4. Taylor expanded in y.re around -inf 54.8%

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

      1. +-commutative 54.8%

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

      2. mul-1-neg 54.8%

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

      3. unsub-neg 54.8%

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

      4. neg-mul-1 54.8%

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

      5. associate-*l/ 57.0%

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

   6. Simplified 57.0%

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

   7. Taylor expanded in y.re around -inf 87.3%

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

   if -3.0000000000000002e49 < y.re < -980 or -1.8e-117 < y.re < 4.15000000000000025e61

   1. Initial program 69.0%

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

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

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

      1. +-commutative 72.2%

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

      2. *-commutative 72.2%

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

      3. unpow2 72.2%

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

   4. Simplified 72.2%

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

      1. clear-num 72.2%

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

      2. inv-pow 72.2%

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

      3. *-commutative 72.2%

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

      4. times-frac 80.5%

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

   6. Applied egg-rr 80.5%

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

      1. unpow-1 80.5%

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

   8. Simplified 80.5%

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

   if -980 < y.re < -1.8e-117

   1. Initial program 95.3%

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

   2. Taylor expanded in x.re around inf 86.1%

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

      1. *-commutative 86.1%

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

      2. unpow2 86.1%

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

      3. unpow2 86.1%

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

   4. Simplified 86.1%

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

4. Final simplification 83.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -3 \cdot 10^{+49}:\\ \;\;\;\;\frac{-1}{y.re} \cdot \left(\left(-x.re\right) - x.im \cdot \frac{y.im}{y.re}\right)\\ \mathbf{elif}\;y.re \leq -980:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{1}{\frac{y.im}{y.re} \cdot \frac{y.im}{x.re}}\\ \mathbf{elif}\;y.re \leq -1.8 \cdot 10^{-117}:\\ \;\;\;\;\frac{y.re \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.re \leq 4.15 \cdot 10^{+61}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{1}{\frac{y.im}{y.re} \cdot \frac{y.im}{x.re}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{y.re} \cdot \left(\left(-x.re\right) - x.im \cdot \frac{y.im}{y.re}\right)\\ \end{array} \]

Alternative 8: 75.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x.im}{y.im} + \frac{1}{\frac{y.im}{y.re} \cdot \frac{y.im}{x.re}}\\ t_1 := \frac{x.re}{y.re} + \frac{1}{\frac{y.re}{\frac{x.im}{\frac{y.re}{y.im}}}}\\ \mathbf{if}\;y.re \leq -2.3 \cdot 10^{+49}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.re \leq -0.0033:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq -1.8 \cdot 10^{-117}:\\ \;\;\;\;\frac{y.re \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.re \leq 7.5 \cdot 10^{+58}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (+ (/ x.im y.im) (/ 1.0 (* (/ y.im y.re) (/ y.im x.re)))))
        (t_1 (+ (/ x.re y.re) (/ 1.0 (/ y.re (/ x.im (/ y.re y.im)))))))
   (if (<= y.re -2.3e+49)
     t_1
     (if (<= y.re -0.0033)
       t_0
       (if (<= y.re -1.8e-117)
         (/ (* y.re x.re) (+ (* y.re y.re) (* y.im y.im)))
         (if (<= y.re 7.5e+58) t_0 t_1))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = (x_46_im / y_46_im) + (1.0 / ((y_46_im / y_46_re) * (y_46_im / x_46_re)));
	double t_1 = (x_46_re / y_46_re) + (1.0 / (y_46_re / (x_46_im / (y_46_re / y_46_im))));
	double tmp;
	if (y_46_re <= -2.3e+49) {
		tmp = t_1;
	} else if (y_46_re <= -0.0033) {
		tmp = t_0;
	} else if (y_46_re <= -1.8e-117) {
		tmp = (y_46_re * x_46_re) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	} else if (y_46_re <= 7.5e+58) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (x_46im / y_46im) + (1.0d0 / ((y_46im / y_46re) * (y_46im / x_46re)))
    t_1 = (x_46re / y_46re) + (1.0d0 / (y_46re / (x_46im / (y_46re / y_46im))))
    if (y_46re <= (-2.3d+49)) then
        tmp = t_1
    else if (y_46re <= (-0.0033d0)) then
        tmp = t_0
    else if (y_46re <= (-1.8d-117)) then
        tmp = (y_46re * x_46re) / ((y_46re * y_46re) + (y_46im * y_46im))
    else if (y_46re <= 7.5d+58) then
        tmp = t_0
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = (x_46_im / y_46_im) + (1.0 / ((y_46_im / y_46_re) * (y_46_im / x_46_re)));
	double t_1 = (x_46_re / y_46_re) + (1.0 / (y_46_re / (x_46_im / (y_46_re / y_46_im))));
	double tmp;
	if (y_46_re <= -2.3e+49) {
		tmp = t_1;
	} else if (y_46_re <= -0.0033) {
		tmp = t_0;
	} else if (y_46_re <= -1.8e-117) {
		tmp = (y_46_re * x_46_re) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	} else if (y_46_re <= 7.5e+58) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = (x_46_im / y_46_im) + (1.0 / ((y_46_im / y_46_re) * (y_46_im / x_46_re)))
	t_1 = (x_46_re / y_46_re) + (1.0 / (y_46_re / (x_46_im / (y_46_re / y_46_im))))
	tmp = 0
	if y_46_re <= -2.3e+49:
		tmp = t_1
	elif y_46_re <= -0.0033:
		tmp = t_0
	elif y_46_re <= -1.8e-117:
		tmp = (y_46_re * x_46_re) / ((y_46_re * y_46_re) + (y_46_im * y_46_im))
	elif y_46_re <= 7.5e+58:
		tmp = t_0
	else:
		tmp = t_1
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(Float64(x_46_im / y_46_im) + Float64(1.0 / Float64(Float64(y_46_im / y_46_re) * Float64(y_46_im / x_46_re))))
	t_1 = Float64(Float64(x_46_re / y_46_re) + Float64(1.0 / Float64(y_46_re / Float64(x_46_im / Float64(y_46_re / y_46_im)))))
	tmp = 0.0
	if (y_46_re <= -2.3e+49)
		tmp = t_1;
	elseif (y_46_re <= -0.0033)
		tmp = t_0;
	elseif (y_46_re <= -1.8e-117)
		tmp = Float64(Float64(y_46_re * x_46_re) / Float64(Float64(y_46_re * y_46_re) + Float64(y_46_im * y_46_im)));
	elseif (y_46_re <= 7.5e+58)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = (x_46_im / y_46_im) + (1.0 / ((y_46_im / y_46_re) * (y_46_im / x_46_re)));
	t_1 = (x_46_re / y_46_re) + (1.0 / (y_46_re / (x_46_im / (y_46_re / y_46_im))));
	tmp = 0.0;
	if (y_46_re <= -2.3e+49)
		tmp = t_1;
	elseif (y_46_re <= -0.0033)
		tmp = t_0;
	elseif (y_46_re <= -1.8e-117)
		tmp = (y_46_re * x_46_re) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	elseif (y_46_re <= 7.5e+58)
		tmp = t_0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[(x$46$im / y$46$im), $MachinePrecision] + N[(1.0 / N[(N[(y$46$im / y$46$re), $MachinePrecision] * N[(y$46$im / x$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x$46$re / y$46$re), $MachinePrecision] + N[(1.0 / N[(y$46$re / N[(x$46$im / N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$re, -2.3e+49], t$95$1, If[LessEqual[y$46$re, -0.0033], t$95$0, If[LessEqual[y$46$re, -1.8e-117], N[(N[(y$46$re * x$46$re), $MachinePrecision] / N[(N[(y$46$re * y$46$re), $MachinePrecision] + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 7.5e+58], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -2.30000000000000002e49 or 7.5000000000000001e58 < y.re

   1. Initial program 42.2%

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

   2. Taylor expanded in y.re around inf 77.5%

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

      1. associate-/l* 78.9%

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

      2. unpow2 78.9%

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

   4. Simplified 78.9%

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

      1. clear-num 78.9%

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

      2. inv-pow 78.9%

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

      3. associate-/l* 83.7%

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

   6. Applied egg-rr 83.7%

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

      1. unpow-1 83.7%

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

      2. associate-/l/ 88.5%

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

      3. /-rgt-identity 88.5%

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

      4. times-frac 85.2%

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

      5. *-commutative 85.2%

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

      6. *-commutative 85.2%

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

      7. times-frac 88.5%

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

      8. /-rgt-identity 88.5%

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

      9. associate-/r/ 87.6%

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

   8. Simplified 87.6%

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

   if -2.30000000000000002e49 < y.re < -0.0033 or -1.8e-117 < y.re < 7.5000000000000001e58

   1. Initial program 69.0%

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

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

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

      1. +-commutative 72.2%

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

      2. *-commutative 72.2%

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

      3. unpow2 72.2%

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

   4. Simplified 72.2%

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

      1. clear-num 72.2%

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

      2. inv-pow 72.2%

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

      3. *-commutative 72.2%

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

      4. times-frac 80.5%

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

   6. Applied egg-rr 80.5%

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

      1. unpow-1 80.5%

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

   8. Simplified 80.5%

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

   if -0.0033 < y.re < -1.8e-117

   1. Initial program 95.3%

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

   2. Taylor expanded in x.re around inf 86.1%

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

      1. *-commutative 86.1%

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

      2. unpow2 86.1%

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

      3. unpow2 86.1%

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

   4. Simplified 86.1%

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

4. Final simplification 83.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -2.3 \cdot 10^{+49}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{1}{\frac{y.re}{\frac{x.im}{\frac{y.re}{y.im}}}}\\ \mathbf{elif}\;y.re \leq -0.0033:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{1}{\frac{y.im}{y.re} \cdot \frac{y.im}{x.re}}\\ \mathbf{elif}\;y.re \leq -1.8 \cdot 10^{-117}:\\ \;\;\;\;\frac{y.re \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.re \leq 7.5 \cdot 10^{+58}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{1}{\frac{y.im}{y.re} \cdot \frac{y.im}{x.re}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{1}{\frac{y.re}{\frac{x.im}{\frac{y.re}{y.im}}}}\\ \end{array} \]

Alternative 9: 75.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x.im}{y.im} + \frac{y.re}{y.im} \cdot \frac{x.re}{y.im}\\ t_1 := \frac{-1}{y.re} \cdot \left(\left(-x.re\right) - x.im \cdot \frac{y.im}{y.re}\right)\\ \mathbf{if}\;y.re \leq -8 \cdot 10^{+48}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.re \leq -2.12 \cdot 10^{-7}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq -1.7 \cdot 10^{-117}:\\ \;\;\;\;\frac{y.re \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.re \leq 6.3 \cdot 10^{+61}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (+ (/ x.im y.im) (* (/ y.re y.im) (/ x.re y.im))))
        (t_1 (* (/ -1.0 y.re) (- (- x.re) (* x.im (/ y.im y.re))))))
   (if (<= y.re -8e+48)
     t_1
     (if (<= y.re -2.12e-7)
       t_0
       (if (<= y.re -1.7e-117)
         (/ (* y.re x.re) (+ (* y.re y.re) (* y.im y.im)))
         (if (<= y.re 6.3e+61) t_0 t_1))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = (x_46_im / y_46_im) + ((y_46_re / y_46_im) * (x_46_re / y_46_im));
	double t_1 = (-1.0 / y_46_re) * (-x_46_re - (x_46_im * (y_46_im / y_46_re)));
	double tmp;
	if (y_46_re <= -8e+48) {
		tmp = t_1;
	} else if (y_46_re <= -2.12e-7) {
		tmp = t_0;
	} else if (y_46_re <= -1.7e-117) {
		tmp = (y_46_re * x_46_re) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	} else if (y_46_re <= 6.3e+61) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (x_46im / y_46im) + ((y_46re / y_46im) * (x_46re / y_46im))
    t_1 = ((-1.0d0) / y_46re) * (-x_46re - (x_46im * (y_46im / y_46re)))
    if (y_46re <= (-8d+48)) then
        tmp = t_1
    else if (y_46re <= (-2.12d-7)) then
        tmp = t_0
    else if (y_46re <= (-1.7d-117)) then
        tmp = (y_46re * x_46re) / ((y_46re * y_46re) + (y_46im * y_46im))
    else if (y_46re <= 6.3d+61) then
        tmp = t_0
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = (x_46_im / y_46_im) + ((y_46_re / y_46_im) * (x_46_re / y_46_im));
	double t_1 = (-1.0 / y_46_re) * (-x_46_re - (x_46_im * (y_46_im / y_46_re)));
	double tmp;
	if (y_46_re <= -8e+48) {
		tmp = t_1;
	} else if (y_46_re <= -2.12e-7) {
		tmp = t_0;
	} else if (y_46_re <= -1.7e-117) {
		tmp = (y_46_re * x_46_re) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	} else if (y_46_re <= 6.3e+61) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = (x_46_im / y_46_im) + ((y_46_re / y_46_im) * (x_46_re / y_46_im))
	t_1 = (-1.0 / y_46_re) * (-x_46_re - (x_46_im * (y_46_im / y_46_re)))
	tmp = 0
	if y_46_re <= -8e+48:
		tmp = t_1
	elif y_46_re <= -2.12e-7:
		tmp = t_0
	elif y_46_re <= -1.7e-117:
		tmp = (y_46_re * x_46_re) / ((y_46_re * y_46_re) + (y_46_im * y_46_im))
	elif y_46_re <= 6.3e+61:
		tmp = t_0
	else:
		tmp = t_1
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(Float64(x_46_im / y_46_im) + Float64(Float64(y_46_re / y_46_im) * Float64(x_46_re / y_46_im)))
	t_1 = Float64(Float64(-1.0 / y_46_re) * Float64(Float64(-x_46_re) - Float64(x_46_im * Float64(y_46_im / y_46_re))))
	tmp = 0.0
	if (y_46_re <= -8e+48)
		tmp = t_1;
	elseif (y_46_re <= -2.12e-7)
		tmp = t_0;
	elseif (y_46_re <= -1.7e-117)
		tmp = Float64(Float64(y_46_re * x_46_re) / Float64(Float64(y_46_re * y_46_re) + Float64(y_46_im * y_46_im)));
	elseif (y_46_re <= 6.3e+61)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = (x_46_im / y_46_im) + ((y_46_re / y_46_im) * (x_46_re / y_46_im));
	t_1 = (-1.0 / y_46_re) * (-x_46_re - (x_46_im * (y_46_im / y_46_re)));
	tmp = 0.0;
	if (y_46_re <= -8e+48)
		tmp = t_1;
	elseif (y_46_re <= -2.12e-7)
		tmp = t_0;
	elseif (y_46_re <= -1.7e-117)
		tmp = (y_46_re * x_46_re) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	elseif (y_46_re <= 6.3e+61)
		tmp = t_0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[(x$46$im / y$46$im), $MachinePrecision] + N[(N[(y$46$re / y$46$im), $MachinePrecision] * N[(x$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(-1.0 / y$46$re), $MachinePrecision] * N[((-x$46$re) - N[(x$46$im * N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$re, -8e+48], t$95$1, If[LessEqual[y$46$re, -2.12e-7], t$95$0, If[LessEqual[y$46$re, -1.7e-117], N[(N[(y$46$re * x$46$re), $MachinePrecision] / N[(N[(y$46$re * y$46$re), $MachinePrecision] + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 6.3e+61], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -8.00000000000000035e48 or 6.29999999999999975e61 < y.re

   1. Initial program 42.2%

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

      1. *-un-lft-identity 42.2%

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

      2. add-sqr-sqrt 42.2%

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

      3. times-frac 42.3%

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

      4. hypot-def 42.3%

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

      5. fma-def 42.3%

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

      6. hypot-def 62.4%

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

   3. Applied egg-rr 62.4%

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

   4. Taylor expanded in y.re around -inf 54.8%

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

      1. +-commutative 54.8%

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

      2. mul-1-neg 54.8%

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

      3. unsub-neg 54.8%

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

      4. neg-mul-1 54.8%

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

      5. associate-*l/ 57.0%

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

   6. Simplified 57.0%

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

   7. Taylor expanded in y.re around -inf 87.3%

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

   if -8.00000000000000035e48 < y.re < -2.1199999999999999e-7 or -1.70000000000000017e-117 < y.re < 6.29999999999999975e61

   1. Initial program 69.0%

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

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

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

      1. +-commutative 72.2%

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

      2. *-commutative 72.2%

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

      3. unpow2 72.2%

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

      4. times-frac 80.3%

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

   4. Simplified 80.3%

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

   if -2.1199999999999999e-7 < y.re < -1.70000000000000017e-117

   1. Initial program 95.3%

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

   2. Taylor expanded in x.re around inf 86.1%

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

      1. *-commutative 86.1%

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

      2. unpow2 86.1%

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

      3. unpow2 86.1%

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

   4. Simplified 86.1%

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

4. Final simplification 83.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -8 \cdot 10^{+48}:\\ \;\;\;\;\frac{-1}{y.re} \cdot \left(\left(-x.re\right) - x.im \cdot \frac{y.im}{y.re}\right)\\ \mathbf{elif}\;y.re \leq -2.12 \cdot 10^{-7}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re}{y.im} \cdot \frac{x.re}{y.im}\\ \mathbf{elif}\;y.re \leq -1.7 \cdot 10^{-117}:\\ \;\;\;\;\frac{y.re \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.re \leq 6.3 \cdot 10^{+61}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re}{y.im} \cdot \frac{x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{y.re} \cdot \left(\left(-x.re\right) - x.im \cdot \frac{y.im}{y.re}\right)\\ \end{array} \]

Alternative 10: 72.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -7.5 \cdot 10^{+48}:\\ \;\;\;\;\frac{x.re}{y.re} + y.im \cdot \frac{x.im}{y.re \cdot y.re}\\ \mathbf{elif}\;y.re \leq -2.3 \cdot 10^{-77} \lor \neg \left(y.re \leq -6.2 \cdot 10^{-143}\right) \land y.re \leq 4 \cdot 10^{+59}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re}{y.im} \cdot \frac{x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re} + x.im \cdot \frac{y.im}{y.re \cdot y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.re -7.5e+48)
   (+ (/ x.re y.re) (* y.im (/ x.im (* y.re y.re))))
   (if (or (<= y.re -2.3e-77) (and (not (<= y.re -6.2e-143)) (<= y.re 4e+59)))
     (+ (/ x.im y.im) (* (/ y.re y.im) (/ x.re y.im)))
     (+ (/ x.re y.re) (* x.im (/ y.im (* y.re y.re)))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_re <= -7.5e+48) {
		tmp = (x_46_re / y_46_re) + (y_46_im * (x_46_im / (y_46_re * y_46_re)));
	} else if ((y_46_re <= -2.3e-77) || (!(y_46_re <= -6.2e-143) && (y_46_re <= 4e+59))) {
		tmp = (x_46_im / y_46_im) + ((y_46_re / y_46_im) * (x_46_re / y_46_im));
	} else {
		tmp = (x_46_re / y_46_re) + (x_46_im * (y_46_im / (y_46_re * y_46_re)));
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if (y_46re <= (-7.5d+48)) then
        tmp = (x_46re / y_46re) + (y_46im * (x_46im / (y_46re * y_46re)))
    else if ((y_46re <= (-2.3d-77)) .or. (.not. (y_46re <= (-6.2d-143))) .and. (y_46re <= 4d+59)) then
        tmp = (x_46im / y_46im) + ((y_46re / y_46im) * (x_46re / y_46im))
    else
        tmp = (x_46re / y_46re) + (x_46im * (y_46im / (y_46re * y_46re)))
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_re <= -7.5e+48) {
		tmp = (x_46_re / y_46_re) + (y_46_im * (x_46_im / (y_46_re * y_46_re)));
	} else if ((y_46_re <= -2.3e-77) || (!(y_46_re <= -6.2e-143) && (y_46_re <= 4e+59))) {
		tmp = (x_46_im / y_46_im) + ((y_46_re / y_46_im) * (x_46_re / y_46_im));
	} else {
		tmp = (x_46_re / y_46_re) + (x_46_im * (y_46_im / (y_46_re * y_46_re)));
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if y_46_re <= -7.5e+48:
		tmp = (x_46_re / y_46_re) + (y_46_im * (x_46_im / (y_46_re * y_46_re)))
	elif (y_46_re <= -2.3e-77) or (not (y_46_re <= -6.2e-143) and (y_46_re <= 4e+59)):
		tmp = (x_46_im / y_46_im) + ((y_46_re / y_46_im) * (x_46_re / y_46_im))
	else:
		tmp = (x_46_re / y_46_re) + (x_46_im * (y_46_im / (y_46_re * y_46_re)))
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (y_46_re <= -7.5e+48)
		tmp = Float64(Float64(x_46_re / y_46_re) + Float64(y_46_im * Float64(x_46_im / Float64(y_46_re * y_46_re))));
	elseif ((y_46_re <= -2.3e-77) || (!(y_46_re <= -6.2e-143) && (y_46_re <= 4e+59)))
		tmp = Float64(Float64(x_46_im / y_46_im) + Float64(Float64(y_46_re / y_46_im) * Float64(x_46_re / y_46_im)));
	else
		tmp = Float64(Float64(x_46_re / y_46_re) + Float64(x_46_im * Float64(y_46_im / Float64(y_46_re * y_46_re))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if (y_46_re <= -7.5e+48)
		tmp = (x_46_re / y_46_re) + (y_46_im * (x_46_im / (y_46_re * y_46_re)));
	elseif ((y_46_re <= -2.3e-77) || (~((y_46_re <= -6.2e-143)) && (y_46_re <= 4e+59)))
		tmp = (x_46_im / y_46_im) + ((y_46_re / y_46_im) * (x_46_re / y_46_im));
	else
		tmp = (x_46_re / y_46_re) + (x_46_im * (y_46_im / (y_46_re * y_46_re)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[y$46$re, -7.5e+48], N[(N[(x$46$re / y$46$re), $MachinePrecision] + N[(y$46$im * N[(x$46$im / N[(y$46$re * y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[y$46$re, -2.3e-77], And[N[Not[LessEqual[y$46$re, -6.2e-143]], $MachinePrecision], LessEqual[y$46$re, 4e+59]]], N[(N[(x$46$im / y$46$im), $MachinePrecision] + N[(N[(y$46$re / y$46$im), $MachinePrecision] * N[(x$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$re / y$46$re), $MachinePrecision] + N[(x$46$im * N[(y$46$im / N[(y$46$re * y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -7.5000000000000006e48

   1. Initial program 41.9%

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

   2. Taylor expanded in y.re around inf 78.7%

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

      1. associate-/l* 80.9%

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

      2. unpow2 80.9%

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

   4. Simplified 80.9%

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

      1. div-inv 80.9%

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

      2. clear-num 80.9%

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

   6. Applied egg-rr 80.9%

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

   if -7.5000000000000006e48 < y.re < -2.29999999999999999e-77 or -6.20000000000000015e-143 < y.re < 3.99999999999999989e59

   1. Initial program 69.6%

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

   2. Taylor expanded in y.re around 0 73.3%

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

      1. +-commutative 73.3%

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

      2. *-commutative 73.3%

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

      3. unpow2 73.3%

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

      4. times-frac 81.0%

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

   4. Simplified 81.0%

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

   if -2.29999999999999999e-77 < y.re < -6.20000000000000015e-143 or 3.99999999999999989e59 < y.re

   1. Initial program 57.7%

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

   2. Taylor expanded in y.re around inf 74.1%

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

      1. associate-/l* 71.0%

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

      2. associate-/r/ 74.2%

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

      3. unpow2 74.2%

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

   4. Simplified 74.2%

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

4. Final simplification 79.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -7.5 \cdot 10^{+48}:\\ \;\;\;\;\frac{x.re}{y.re} + y.im \cdot \frac{x.im}{y.re \cdot y.re}\\ \mathbf{elif}\;y.re \leq -2.3 \cdot 10^{-77} \lor \neg \left(y.re \leq -6.2 \cdot 10^{-143}\right) \land y.re \leq 4 \cdot 10^{+59}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re}{y.im} \cdot \frac{x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re} + x.im \cdot \frac{y.im}{y.re \cdot y.re}\\ \end{array} \]

Alternative 11: 72.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x.im}{y.im} + \frac{y.re}{y.im} \cdot \frac{x.re}{y.im}\\ \mathbf{if}\;y.re \leq -9.5 \cdot 10^{+48}:\\ \;\;\;\;\frac{x.re}{y.re} + y.im \cdot \frac{x.im}{y.re \cdot y.re}\\ \mathbf{elif}\;y.re \leq -3.1 \cdot 10^{-77}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq -6.2 \cdot 10^{-143}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{y.im \cdot x.im}{y.re \cdot y.re}\\ \mathbf{elif}\;y.re \leq 3 \cdot 10^{+64}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re} + x.im \cdot \frac{y.im}{y.re \cdot y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (+ (/ x.im y.im) (* (/ y.re y.im) (/ x.re y.im)))))
   (if (<= y.re -9.5e+48)
     (+ (/ x.re y.re) (* y.im (/ x.im (* y.re y.re))))
     (if (<= y.re -3.1e-77)
       t_0
       (if (<= y.re -6.2e-143)
         (+ (/ x.re y.re) (/ (* y.im x.im) (* y.re y.re)))
         (if (<= y.re 3e+64)
           t_0
           (+ (/ x.re y.re) (* x.im (/ y.im (* y.re y.re))))))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = (x_46_im / y_46_im) + ((y_46_re / y_46_im) * (x_46_re / y_46_im));
	double tmp;
	if (y_46_re <= -9.5e+48) {
		tmp = (x_46_re / y_46_re) + (y_46_im * (x_46_im / (y_46_re * y_46_re)));
	} else if (y_46_re <= -3.1e-77) {
		tmp = t_0;
	} else if (y_46_re <= -6.2e-143) {
		tmp = (x_46_re / y_46_re) + ((y_46_im * x_46_im) / (y_46_re * y_46_re));
	} else if (y_46_re <= 3e+64) {
		tmp = t_0;
	} else {
		tmp = (x_46_re / y_46_re) + (x_46_im * (y_46_im / (y_46_re * y_46_re)));
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (x_46im / y_46im) + ((y_46re / y_46im) * (x_46re / y_46im))
    if (y_46re <= (-9.5d+48)) then
        tmp = (x_46re / y_46re) + (y_46im * (x_46im / (y_46re * y_46re)))
    else if (y_46re <= (-3.1d-77)) then
        tmp = t_0
    else if (y_46re <= (-6.2d-143)) then
        tmp = (x_46re / y_46re) + ((y_46im * x_46im) / (y_46re * y_46re))
    else if (y_46re <= 3d+64) then
        tmp = t_0
    else
        tmp = (x_46re / y_46re) + (x_46im * (y_46im / (y_46re * y_46re)))
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = (x_46_im / y_46_im) + ((y_46_re / y_46_im) * (x_46_re / y_46_im));
	double tmp;
	if (y_46_re <= -9.5e+48) {
		tmp = (x_46_re / y_46_re) + (y_46_im * (x_46_im / (y_46_re * y_46_re)));
	} else if (y_46_re <= -3.1e-77) {
		tmp = t_0;
	} else if (y_46_re <= -6.2e-143) {
		tmp = (x_46_re / y_46_re) + ((y_46_im * x_46_im) / (y_46_re * y_46_re));
	} else if (y_46_re <= 3e+64) {
		tmp = t_0;
	} else {
		tmp = (x_46_re / y_46_re) + (x_46_im * (y_46_im / (y_46_re * y_46_re)));
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = (x_46_im / y_46_im) + ((y_46_re / y_46_im) * (x_46_re / y_46_im))
	tmp = 0
	if y_46_re <= -9.5e+48:
		tmp = (x_46_re / y_46_re) + (y_46_im * (x_46_im / (y_46_re * y_46_re)))
	elif y_46_re <= -3.1e-77:
		tmp = t_0
	elif y_46_re <= -6.2e-143:
		tmp = (x_46_re / y_46_re) + ((y_46_im * x_46_im) / (y_46_re * y_46_re))
	elif y_46_re <= 3e+64:
		tmp = t_0
	else:
		tmp = (x_46_re / y_46_re) + (x_46_im * (y_46_im / (y_46_re * y_46_re)))
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(Float64(x_46_im / y_46_im) + Float64(Float64(y_46_re / y_46_im) * Float64(x_46_re / y_46_im)))
	tmp = 0.0
	if (y_46_re <= -9.5e+48)
		tmp = Float64(Float64(x_46_re / y_46_re) + Float64(y_46_im * Float64(x_46_im / Float64(y_46_re * y_46_re))));
	elseif (y_46_re <= -3.1e-77)
		tmp = t_0;
	elseif (y_46_re <= -6.2e-143)
		tmp = Float64(Float64(x_46_re / y_46_re) + Float64(Float64(y_46_im * x_46_im) / Float64(y_46_re * y_46_re)));
	elseif (y_46_re <= 3e+64)
		tmp = t_0;
	else
		tmp = Float64(Float64(x_46_re / y_46_re) + Float64(x_46_im * Float64(y_46_im / Float64(y_46_re * y_46_re))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = (x_46_im / y_46_im) + ((y_46_re / y_46_im) * (x_46_re / y_46_im));
	tmp = 0.0;
	if (y_46_re <= -9.5e+48)
		tmp = (x_46_re / y_46_re) + (y_46_im * (x_46_im / (y_46_re * y_46_re)));
	elseif (y_46_re <= -3.1e-77)
		tmp = t_0;
	elseif (y_46_re <= -6.2e-143)
		tmp = (x_46_re / y_46_re) + ((y_46_im * x_46_im) / (y_46_re * y_46_re));
	elseif (y_46_re <= 3e+64)
		tmp = t_0;
	else
		tmp = (x_46_re / y_46_re) + (x_46_im * (y_46_im / (y_46_re * y_46_re)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[(x$46$im / y$46$im), $MachinePrecision] + N[(N[(y$46$re / y$46$im), $MachinePrecision] * N[(x$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$re, -9.5e+48], N[(N[(x$46$re / y$46$re), $MachinePrecision] + N[(y$46$im * N[(x$46$im / N[(y$46$re * y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, -3.1e-77], t$95$0, If[LessEqual[y$46$re, -6.2e-143], N[(N[(x$46$re / y$46$re), $MachinePrecision] + N[(N[(y$46$im * x$46$im), $MachinePrecision] / N[(y$46$re * y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 3e+64], t$95$0, N[(N[(x$46$re / y$46$re), $MachinePrecision] + N[(x$46$im * N[(y$46$im / N[(y$46$re * y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if y.re < -9.4999999999999997e48

   1. Initial program 41.9%

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

   2. Taylor expanded in y.re around inf 78.7%

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

      1. associate-/l* 80.9%

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

      2. unpow2 80.9%

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

   4. Simplified 80.9%

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

      1. div-inv 80.9%

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

      2. clear-num 80.9%

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

   6. Applied egg-rr 80.9%

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

   if -9.4999999999999997e48 < y.re < -3.10000000000000008e-77 or -6.20000000000000015e-143 < y.re < 3.0000000000000002e64

   1. Initial program 69.6%

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

   2. Taylor expanded in y.re around 0 73.3%

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

      1. +-commutative 73.3%

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

      2. *-commutative 73.3%

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

      3. unpow2 73.3%

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

      4. times-frac 81.0%

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

   4. Simplified 81.0%

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

   if -3.10000000000000008e-77 < y.re < -6.20000000000000015e-143

   1. Initial program 99.7%

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

   2. Taylor expanded in y.re around inf 68.7%

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

      1. +-commutative 68.7%

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

      2. unpow2 68.7%

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

      3. times-frac 62.5%

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

      4. fma-def 62.5%

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

   4. Simplified 62.5%

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

   5. Taylor expanded in y.im around 0 68.7%

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

      1. unpow2 68.7%

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

   7. Simplified 68.7%

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

   if 3.0000000000000002e64 < y.re

   1. Initial program 42.6%

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

   2. Taylor expanded in y.re around inf 76.0%

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

      1. associate-/l* 76.2%

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

      2. associate-/r/ 76.2%

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

      3. unpow2 76.2%

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

   4. Simplified 76.2%

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

4. Final simplification 79.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -9.5 \cdot 10^{+48}:\\ \;\;\;\;\frac{x.re}{y.re} + y.im \cdot \frac{x.im}{y.re \cdot y.re}\\ \mathbf{elif}\;y.re \leq -3.1 \cdot 10^{-77}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re}{y.im} \cdot \frac{x.re}{y.im}\\ \mathbf{elif}\;y.re \leq -6.2 \cdot 10^{-143}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{y.im \cdot x.im}{y.re \cdot y.re}\\ \mathbf{elif}\;y.re \leq 3 \cdot 10^{+64}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re}{y.im} \cdot \frac{x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re} + x.im \cdot \frac{y.im}{y.re \cdot y.re}\\ \end{array} \]

Alternative 12: 72.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x.im}{y.im} + \frac{y.re}{y.im} \cdot \frac{x.re}{y.im}\\ \mathbf{if}\;y.re \leq -1.06 \cdot 10^{+49}:\\ \;\;\;\;\frac{x.re}{y.re} + y.im \cdot \frac{x.im}{y.re \cdot y.re}\\ \mathbf{elif}\;y.re \leq -1650:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq -1.8 \cdot 10^{-117}:\\ \;\;\;\;\frac{y.re}{\frac{y.re \cdot y.re + y.im \cdot y.im}{x.re}}\\ \mathbf{elif}\;y.re \leq 7.6 \cdot 10^{+61}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re} + x.im \cdot \frac{y.im}{y.re \cdot y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (+ (/ x.im y.im) (* (/ y.re y.im) (/ x.re y.im)))))
   (if (<= y.re -1.06e+49)
     (+ (/ x.re y.re) (* y.im (/ x.im (* y.re y.re))))
     (if (<= y.re -1650.0)
       t_0
       (if (<= y.re -1.8e-117)
         (/ y.re (/ (+ (* y.re y.re) (* y.im y.im)) x.re))
         (if (<= y.re 7.6e+61)
           t_0
           (+ (/ x.re y.re) (* x.im (/ y.im (* y.re y.re))))))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = (x_46_im / y_46_im) + ((y_46_re / y_46_im) * (x_46_re / y_46_im));
	double tmp;
	if (y_46_re <= -1.06e+49) {
		tmp = (x_46_re / y_46_re) + (y_46_im * (x_46_im / (y_46_re * y_46_re)));
	} else if (y_46_re <= -1650.0) {
		tmp = t_0;
	} else if (y_46_re <= -1.8e-117) {
		tmp = y_46_re / (((y_46_re * y_46_re) + (y_46_im * y_46_im)) / x_46_re);
	} else if (y_46_re <= 7.6e+61) {
		tmp = t_0;
	} else {
		tmp = (x_46_re / y_46_re) + (x_46_im * (y_46_im / (y_46_re * y_46_re)));
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (x_46im / y_46im) + ((y_46re / y_46im) * (x_46re / y_46im))
    if (y_46re <= (-1.06d+49)) then
        tmp = (x_46re / y_46re) + (y_46im * (x_46im / (y_46re * y_46re)))
    else if (y_46re <= (-1650.0d0)) then
        tmp = t_0
    else if (y_46re <= (-1.8d-117)) then
        tmp = y_46re / (((y_46re * y_46re) + (y_46im * y_46im)) / x_46re)
    else if (y_46re <= 7.6d+61) then
        tmp = t_0
    else
        tmp = (x_46re / y_46re) + (x_46im * (y_46im / (y_46re * y_46re)))
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = (x_46_im / y_46_im) + ((y_46_re / y_46_im) * (x_46_re / y_46_im));
	double tmp;
	if (y_46_re <= -1.06e+49) {
		tmp = (x_46_re / y_46_re) + (y_46_im * (x_46_im / (y_46_re * y_46_re)));
	} else if (y_46_re <= -1650.0) {
		tmp = t_0;
	} else if (y_46_re <= -1.8e-117) {
		tmp = y_46_re / (((y_46_re * y_46_re) + (y_46_im * y_46_im)) / x_46_re);
	} else if (y_46_re <= 7.6e+61) {
		tmp = t_0;
	} else {
		tmp = (x_46_re / y_46_re) + (x_46_im * (y_46_im / (y_46_re * y_46_re)));
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = (x_46_im / y_46_im) + ((y_46_re / y_46_im) * (x_46_re / y_46_im))
	tmp = 0
	if y_46_re <= -1.06e+49:
		tmp = (x_46_re / y_46_re) + (y_46_im * (x_46_im / (y_46_re * y_46_re)))
	elif y_46_re <= -1650.0:
		tmp = t_0
	elif y_46_re <= -1.8e-117:
		tmp = y_46_re / (((y_46_re * y_46_re) + (y_46_im * y_46_im)) / x_46_re)
	elif y_46_re <= 7.6e+61:
		tmp = t_0
	else:
		tmp = (x_46_re / y_46_re) + (x_46_im * (y_46_im / (y_46_re * y_46_re)))
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(Float64(x_46_im / y_46_im) + Float64(Float64(y_46_re / y_46_im) * Float64(x_46_re / y_46_im)))
	tmp = 0.0
	if (y_46_re <= -1.06e+49)
		tmp = Float64(Float64(x_46_re / y_46_re) + Float64(y_46_im * Float64(x_46_im / Float64(y_46_re * y_46_re))));
	elseif (y_46_re <= -1650.0)
		tmp = t_0;
	elseif (y_46_re <= -1.8e-117)
		tmp = Float64(y_46_re / Float64(Float64(Float64(y_46_re * y_46_re) + Float64(y_46_im * y_46_im)) / x_46_re));
	elseif (y_46_re <= 7.6e+61)
		tmp = t_0;
	else
		tmp = Float64(Float64(x_46_re / y_46_re) + Float64(x_46_im * Float64(y_46_im / Float64(y_46_re * y_46_re))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = (x_46_im / y_46_im) + ((y_46_re / y_46_im) * (x_46_re / y_46_im));
	tmp = 0.0;
	if (y_46_re <= -1.06e+49)
		tmp = (x_46_re / y_46_re) + (y_46_im * (x_46_im / (y_46_re * y_46_re)));
	elseif (y_46_re <= -1650.0)
		tmp = t_0;
	elseif (y_46_re <= -1.8e-117)
		tmp = y_46_re / (((y_46_re * y_46_re) + (y_46_im * y_46_im)) / x_46_re);
	elseif (y_46_re <= 7.6e+61)
		tmp = t_0;
	else
		tmp = (x_46_re / y_46_re) + (x_46_im * (y_46_im / (y_46_re * y_46_re)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[(x$46$im / y$46$im), $MachinePrecision] + N[(N[(y$46$re / y$46$im), $MachinePrecision] * N[(x$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$re, -1.06e+49], N[(N[(x$46$re / y$46$re), $MachinePrecision] + N[(y$46$im * N[(x$46$im / N[(y$46$re * y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, -1650.0], t$95$0, If[LessEqual[y$46$re, -1.8e-117], N[(y$46$re / N[(N[(N[(y$46$re * y$46$re), $MachinePrecision] + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision] / x$46$re), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 7.6e+61], t$95$0, N[(N[(x$46$re / y$46$re), $MachinePrecision] + N[(x$46$im * N[(y$46$im / N[(y$46$re * y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if y.re < -1.06e49

   1. Initial program 41.9%

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

   2. Taylor expanded in y.re around inf 78.7%

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

      1. associate-/l* 80.9%

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

      2. unpow2 80.9%

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

   4. Simplified 80.9%

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

      1. div-inv 80.9%

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

      2. clear-num 80.9%

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

   6. Applied egg-rr 80.9%

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

   if -1.06e49 < y.re < -1650 or -1.8e-117 < y.re < 7.5999999999999999e61

   1. Initial program 69.0%

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

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

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

      1. +-commutative 72.2%

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

      2. *-commutative 72.2%

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

      3. unpow2 72.2%

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

      4. times-frac 80.3%

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

   4. Simplified 80.3%

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

   if -1650 < y.re < -1.8e-117

   1. Initial program 95.3%

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

   2. Taylor expanded in x.re around inf 86.1%

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

      1. *-commutative 86.1%

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

      2. associate-/l* 82.6%

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

      3. +-commutative 82.6%

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

      4. unpow2 82.6%

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

      5. fma-def 82.6%

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

      6. unpow2 82.6%

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

   4. Simplified 82.6%

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

   5. Taylor expanded in x.re around 0 82.6%

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

      1. unpow2 82.6%

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

      2. unpow2 82.6%

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

   7. Simplified 82.6%

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

   if 7.5999999999999999e61 < y.re

   1. Initial program 42.6%

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

   2. Taylor expanded in y.re around inf 76.0%

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

      1. associate-/l* 76.2%

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

      2. associate-/r/ 76.2%

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

      3. unpow2 76.2%

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

   4. Simplified 76.2%

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

4. Final simplification 80.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -1.06 \cdot 10^{+49}:\\ \;\;\;\;\frac{x.re}{y.re} + y.im \cdot \frac{x.im}{y.re \cdot y.re}\\ \mathbf{elif}\;y.re \leq -1650:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re}{y.im} \cdot \frac{x.re}{y.im}\\ \mathbf{elif}\;y.re \leq -1.8 \cdot 10^{-117}:\\ \;\;\;\;\frac{y.re}{\frac{y.re \cdot y.re + y.im \cdot y.im}{x.re}}\\ \mathbf{elif}\;y.re \leq 7.6 \cdot 10^{+61}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re}{y.im} \cdot \frac{x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re} + x.im \cdot \frac{y.im}{y.re \cdot y.re}\\ \end{array} \]

Alternative 13: 72.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x.im}{y.im} + \frac{y.re}{y.im} \cdot \frac{x.re}{y.im}\\ \mathbf{if}\;y.re \leq -1 \cdot 10^{+49}:\\ \;\;\;\;\frac{x.re}{y.re} + y.im \cdot \frac{x.im}{y.re \cdot y.re}\\ \mathbf{elif}\;y.re \leq -1220:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq -1.6 \cdot 10^{-117}:\\ \;\;\;\;\frac{y.re \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.re \leq 3.9 \cdot 10^{+68}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re} + x.im \cdot \frac{y.im}{y.re \cdot y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (+ (/ x.im y.im) (* (/ y.re y.im) (/ x.re y.im)))))
   (if (<= y.re -1e+49)
     (+ (/ x.re y.re) (* y.im (/ x.im (* y.re y.re))))
     (if (<= y.re -1220.0)
       t_0
       (if (<= y.re -1.6e-117)
         (/ (* y.re x.re) (+ (* y.re y.re) (* y.im y.im)))
         (if (<= y.re 3.9e+68)
           t_0
           (+ (/ x.re y.re) (* x.im (/ y.im (* y.re y.re))))))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = (x_46_im / y_46_im) + ((y_46_re / y_46_im) * (x_46_re / y_46_im));
	double tmp;
	if (y_46_re <= -1e+49) {
		tmp = (x_46_re / y_46_re) + (y_46_im * (x_46_im / (y_46_re * y_46_re)));
	} else if (y_46_re <= -1220.0) {
		tmp = t_0;
	} else if (y_46_re <= -1.6e-117) {
		tmp = (y_46_re * x_46_re) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	} else if (y_46_re <= 3.9e+68) {
		tmp = t_0;
	} else {
		tmp = (x_46_re / y_46_re) + (x_46_im * (y_46_im / (y_46_re * y_46_re)));
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (x_46im / y_46im) + ((y_46re / y_46im) * (x_46re / y_46im))
    if (y_46re <= (-1d+49)) then
        tmp = (x_46re / y_46re) + (y_46im * (x_46im / (y_46re * y_46re)))
    else if (y_46re <= (-1220.0d0)) then
        tmp = t_0
    else if (y_46re <= (-1.6d-117)) then
        tmp = (y_46re * x_46re) / ((y_46re * y_46re) + (y_46im * y_46im))
    else if (y_46re <= 3.9d+68) then
        tmp = t_0
    else
        tmp = (x_46re / y_46re) + (x_46im * (y_46im / (y_46re * y_46re)))
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = (x_46_im / y_46_im) + ((y_46_re / y_46_im) * (x_46_re / y_46_im));
	double tmp;
	if (y_46_re <= -1e+49) {
		tmp = (x_46_re / y_46_re) + (y_46_im * (x_46_im / (y_46_re * y_46_re)));
	} else if (y_46_re <= -1220.0) {
		tmp = t_0;
	} else if (y_46_re <= -1.6e-117) {
		tmp = (y_46_re * x_46_re) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	} else if (y_46_re <= 3.9e+68) {
		tmp = t_0;
	} else {
		tmp = (x_46_re / y_46_re) + (x_46_im * (y_46_im / (y_46_re * y_46_re)));
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = (x_46_im / y_46_im) + ((y_46_re / y_46_im) * (x_46_re / y_46_im))
	tmp = 0
	if y_46_re <= -1e+49:
		tmp = (x_46_re / y_46_re) + (y_46_im * (x_46_im / (y_46_re * y_46_re)))
	elif y_46_re <= -1220.0:
		tmp = t_0
	elif y_46_re <= -1.6e-117:
		tmp = (y_46_re * x_46_re) / ((y_46_re * y_46_re) + (y_46_im * y_46_im))
	elif y_46_re <= 3.9e+68:
		tmp = t_0
	else:
		tmp = (x_46_re / y_46_re) + (x_46_im * (y_46_im / (y_46_re * y_46_re)))
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(Float64(x_46_im / y_46_im) + Float64(Float64(y_46_re / y_46_im) * Float64(x_46_re / y_46_im)))
	tmp = 0.0
	if (y_46_re <= -1e+49)
		tmp = Float64(Float64(x_46_re / y_46_re) + Float64(y_46_im * Float64(x_46_im / Float64(y_46_re * y_46_re))));
	elseif (y_46_re <= -1220.0)
		tmp = t_0;
	elseif (y_46_re <= -1.6e-117)
		tmp = Float64(Float64(y_46_re * x_46_re) / Float64(Float64(y_46_re * y_46_re) + Float64(y_46_im * y_46_im)));
	elseif (y_46_re <= 3.9e+68)
		tmp = t_0;
	else
		tmp = Float64(Float64(x_46_re / y_46_re) + Float64(x_46_im * Float64(y_46_im / Float64(y_46_re * y_46_re))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = (x_46_im / y_46_im) + ((y_46_re / y_46_im) * (x_46_re / y_46_im));
	tmp = 0.0;
	if (y_46_re <= -1e+49)
		tmp = (x_46_re / y_46_re) + (y_46_im * (x_46_im / (y_46_re * y_46_re)));
	elseif (y_46_re <= -1220.0)
		tmp = t_0;
	elseif (y_46_re <= -1.6e-117)
		tmp = (y_46_re * x_46_re) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	elseif (y_46_re <= 3.9e+68)
		tmp = t_0;
	else
		tmp = (x_46_re / y_46_re) + (x_46_im * (y_46_im / (y_46_re * y_46_re)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[(x$46$im / y$46$im), $MachinePrecision] + N[(N[(y$46$re / y$46$im), $MachinePrecision] * N[(x$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$re, -1e+49], N[(N[(x$46$re / y$46$re), $MachinePrecision] + N[(y$46$im * N[(x$46$im / N[(y$46$re * y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, -1220.0], t$95$0, If[LessEqual[y$46$re, -1.6e-117], N[(N[(y$46$re * x$46$re), $MachinePrecision] / N[(N[(y$46$re * y$46$re), $MachinePrecision] + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 3.9e+68], t$95$0, N[(N[(x$46$re / y$46$re), $MachinePrecision] + N[(x$46$im * N[(y$46$im / N[(y$46$re * y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if y.re < -9.99999999999999946e48

   1. Initial program 41.9%

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

   2. Taylor expanded in y.re around inf 78.7%

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

      1. associate-/l* 80.9%

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

      2. unpow2 80.9%

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

   4. Simplified 80.9%

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

      1. div-inv 80.9%

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

      2. clear-num 80.9%

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

   6. Applied egg-rr 80.9%

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

   if -9.99999999999999946e48 < y.re < -1220 or -1.59999999999999998e-117 < y.re < 3.90000000000000019e68

   1. Initial program 69.0%

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

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

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

      1. +-commutative 72.2%

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

      2. *-commutative 72.2%

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

      3. unpow2 72.2%

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

      4. times-frac 80.3%

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

   4. Simplified 80.3%

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

   if -1220 < y.re < -1.59999999999999998e-117

   1. Initial program 95.3%

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

   2. Taylor expanded in x.re around inf 86.1%

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

      1. *-commutative 86.1%

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

      2. unpow2 86.1%

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

      3. unpow2 86.1%

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

   4. Simplified 86.1%

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

   if 3.90000000000000019e68 < y.re

   1. Initial program 42.6%

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

   2. Taylor expanded in y.re around inf 76.0%

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

      1. associate-/l* 76.2%

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

      2. associate-/r/ 76.2%

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

      3. unpow2 76.2%

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

   4. Simplified 76.2%

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

4. Final simplification 80.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -1 \cdot 10^{+49}:\\ \;\;\;\;\frac{x.re}{y.re} + y.im \cdot \frac{x.im}{y.re \cdot y.re}\\ \mathbf{elif}\;y.re \leq -1220:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re}{y.im} \cdot \frac{x.re}{y.im}\\ \mathbf{elif}\;y.re \leq -1.6 \cdot 10^{-117}:\\ \;\;\;\;\frac{y.re \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.re \leq 3.9 \cdot 10^{+68}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re}{y.im} \cdot \frac{x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re} + x.im \cdot \frac{y.im}{y.re \cdot y.re}\\ \end{array} \]

Alternative 14: 74.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -3.3 \cdot 10^{+49} \lor \neg \left(y.re \leq 3.1 \cdot 10^{+62}\right):\\ \;\;\;\;\frac{x.re}{y.re} + y.im \cdot \frac{x.im}{y.re \cdot y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re}{y.im} \cdot \frac{x.re}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.re -3.3e+49) (not (<= y.re 3.1e+62)))
   (+ (/ x.re y.re) (* y.im (/ x.im (* y.re y.re))))
   (+ (/ x.im y.im) (* (/ y.re y.im) (/ x.re y.im)))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_re <= -3.3e+49) || !(y_46_re <= 3.1e+62)) {
		tmp = (x_46_re / y_46_re) + (y_46_im * (x_46_im / (y_46_re * y_46_re)));
	} else {
		tmp = (x_46_im / y_46_im) + ((y_46_re / y_46_im) * (x_46_re / y_46_im));
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if ((y_46re <= (-3.3d+49)) .or. (.not. (y_46re <= 3.1d+62))) then
        tmp = (x_46re / y_46re) + (y_46im * (x_46im / (y_46re * y_46re)))
    else
        tmp = (x_46im / y_46im) + ((y_46re / y_46im) * (x_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_re <= -3.3e+49) || !(y_46_re <= 3.1e+62)) {
		tmp = (x_46_re / y_46_re) + (y_46_im * (x_46_im / (y_46_re * y_46_re)));
	} else {
		tmp = (x_46_im / y_46_im) + ((y_46_re / y_46_im) * (x_46_re / y_46_im));
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if (y_46_re <= -3.3e+49) or not (y_46_re <= 3.1e+62):
		tmp = (x_46_re / y_46_re) + (y_46_im * (x_46_im / (y_46_re * y_46_re)))
	else:
		tmp = (x_46_im / y_46_im) + ((y_46_re / y_46_im) * (x_46_re / y_46_im))
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if ((y_46_re <= -3.3e+49) || !(y_46_re <= 3.1e+62))
		tmp = Float64(Float64(x_46_re / y_46_re) + Float64(y_46_im * Float64(x_46_im / Float64(y_46_re * y_46_re))));
	else
		tmp = Float64(Float64(x_46_im / y_46_im) + Float64(Float64(y_46_re / y_46_im) * Float64(x_46_re / y_46_im)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if ((y_46_re <= -3.3e+49) || ~((y_46_re <= 3.1e+62)))
		tmp = (x_46_re / y_46_re) + (y_46_im * (x_46_im / (y_46_re * y_46_re)));
	else
		tmp = (x_46_im / y_46_im) + ((y_46_re / y_46_im) * (x_46_re / y_46_im));
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[Or[LessEqual[y$46$re, -3.3e+49], N[Not[LessEqual[y$46$re, 3.1e+62]], $MachinePrecision]], N[(N[(x$46$re / y$46$re), $MachinePrecision] + N[(y$46$im * N[(x$46$im / N[(y$46$re * y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$im / y$46$im), $MachinePrecision] + N[(N[(y$46$re / y$46$im), $MachinePrecision] * N[(x$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.re < -3.2999999999999998e49 or 3.10000000000000014e62 < y.re

   1. Initial program 42.2%

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

   2. Taylor expanded in y.re around inf 77.5%

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

      1. associate-/l* 78.9%

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

      2. unpow2 78.9%

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

   4. Simplified 78.9%

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

      1. div-inv 78.9%

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

      2. clear-num 78.9%

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

   6. Applied egg-rr 78.9%

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

   if -3.2999999999999998e49 < y.re < 3.10000000000000014e62

   1. Initial program 72.4%

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

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

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

      1. +-commutative 69.9%

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

      2. *-commutative 69.9%

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

      3. unpow2 69.9%

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

      4. times-frac 76.9%

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

   4. Simplified 76.9%

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

4. Final simplification 77.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -3.3 \cdot 10^{+49} \lor \neg \left(y.re \leq 3.1 \cdot 10^{+62}\right):\\ \;\;\;\;\frac{x.re}{y.re} + y.im \cdot \frac{x.im}{y.re \cdot y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re}{y.im} \cdot \frac{x.re}{y.im}\\ \end{array} \]

Alternative 15: 72.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -9 \cdot 10^{+49}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{elif}\;y.re \leq 3 \cdot 10^{+64}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re}{y.im} \cdot \frac{x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.re -9e+49)
   (/ x.re y.re)
   (if (<= y.re 3e+64)
     (+ (/ x.im y.im) (* (/ y.re y.im) (/ x.re y.im)))
     (/ x.re y.re))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_re <= -9e+49) {
		tmp = x_46_re / y_46_re;
	} else if (y_46_re <= 3e+64) {
		tmp = (x_46_im / y_46_im) + ((y_46_re / y_46_im) * (x_46_re / y_46_im));
	} else {
		tmp = x_46_re / y_46_re;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if (y_46re <= (-9d+49)) then
        tmp = x_46re / y_46re
    else if (y_46re <= 3d+64) then
        tmp = (x_46im / y_46im) + ((y_46re / y_46im) * (x_46re / y_46im))
    else
        tmp = x_46re / y_46re
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_re <= -9e+49) {
		tmp = x_46_re / y_46_re;
	} else if (y_46_re <= 3e+64) {
		tmp = (x_46_im / y_46_im) + ((y_46_re / y_46_im) * (x_46_re / y_46_im));
	} else {
		tmp = x_46_re / y_46_re;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if y_46_re <= -9e+49:
		tmp = x_46_re / y_46_re
	elif y_46_re <= 3e+64:
		tmp = (x_46_im / y_46_im) + ((y_46_re / y_46_im) * (x_46_re / y_46_im))
	else:
		tmp = x_46_re / y_46_re
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (y_46_re <= -9e+49)
		tmp = Float64(x_46_re / y_46_re);
	elseif (y_46_re <= 3e+64)
		tmp = Float64(Float64(x_46_im / y_46_im) + Float64(Float64(y_46_re / y_46_im) * Float64(x_46_re / y_46_im)));
	else
		tmp = Float64(x_46_re / y_46_re);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if (y_46_re <= -9e+49)
		tmp = x_46_re / y_46_re;
	elseif (y_46_re <= 3e+64)
		tmp = (x_46_im / y_46_im) + ((y_46_re / y_46_im) * (x_46_re / y_46_im));
	else
		tmp = x_46_re / y_46_re;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[y$46$re, -9e+49], N[(x$46$re / y$46$re), $MachinePrecision], If[LessEqual[y$46$re, 3e+64], N[(N[(x$46$im / y$46$im), $MachinePrecision] + N[(N[(y$46$re / y$46$im), $MachinePrecision] * N[(x$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x$46$re / y$46$re), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y.re < -8.99999999999999965e49 or 3.0000000000000002e64 < y.re

   1. Initial program 42.2%

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

   2. Taylor expanded in y.re around inf 74.8%

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

   if -8.99999999999999965e49 < y.re < 3.0000000000000002e64

   1. Initial program 72.4%

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

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

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

      1. +-commutative 69.9%

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

      2. *-commutative 69.9%

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

      3. unpow2 69.9%

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

      4. times-frac 76.9%

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

   4. Simplified 76.9%

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

4. Final simplification 76.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -9 \cdot 10^{+49}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{elif}\;y.re \leq 3 \cdot 10^{+64}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re}{y.im} \cdot \frac{x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \end{array} \]

Alternative 16: 63.9% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -4.7 \cdot 10^{+56}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{elif}\;y.im \leq 1.9 \cdot 10^{-7}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.im -4.7e+56)
   (/ x.im y.im)
   (if (<= y.im 1.9e-7) (/ x.re y.re) (/ x.im y.im))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_im <= -4.7e+56) {
		tmp = x_46_im / y_46_im;
	} else if (y_46_im <= 1.9e-7) {
		tmp = x_46_re / y_46_re;
	} else {
		tmp = x_46_im / y_46_im;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if (y_46im <= (-4.7d+56)) then
        tmp = x_46im / y_46im
    else if (y_46im <= 1.9d-7) then
        tmp = x_46re / y_46re
    else
        tmp = x_46im / y_46im
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_im <= -4.7e+56) {
		tmp = x_46_im / y_46_im;
	} else if (y_46_im <= 1.9e-7) {
		tmp = x_46_re / y_46_re;
	} else {
		tmp = x_46_im / y_46_im;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if y_46_im <= -4.7e+56:
		tmp = x_46_im / y_46_im
	elif y_46_im <= 1.9e-7:
		tmp = x_46_re / y_46_re
	else:
		tmp = x_46_im / y_46_im
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (y_46_im <= -4.7e+56)
		tmp = Float64(x_46_im / y_46_im);
	elseif (y_46_im <= 1.9e-7)
		tmp = Float64(x_46_re / y_46_re);
	else
		tmp = Float64(x_46_im / y_46_im);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if (y_46_im <= -4.7e+56)
		tmp = x_46_im / y_46_im;
	elseif (y_46_im <= 1.9e-7)
		tmp = x_46_re / y_46_re;
	else
		tmp = x_46_im / y_46_im;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[y$46$im, -4.7e+56], N[(x$46$im / y$46$im), $MachinePrecision], If[LessEqual[y$46$im, 1.9e-7], N[(x$46$re / y$46$re), $MachinePrecision], N[(x$46$im / y$46$im), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y.im < -4.7e56 or 1.90000000000000007e-7 < y.im

   1. Initial program 52.0%

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

   2. Taylor expanded in y.re around 0 65.7%

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

   if -4.7e56 < y.im < 1.90000000000000007e-7

   1. Initial program 68.6%

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

   2. Taylor expanded in y.re around inf 61.3%

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

4. Final simplification 63.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -4.7 \cdot 10^{+56}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{elif}\;y.im \leq 1.9 \cdot 10^{-7}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \end{array} \]

Alternative 17: 42.9% accurate, 5.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 61.1%

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

2. Taylor expanded in y.re around 0 39.9%

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

3. Final simplification 39.9%

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

Reproduce

herbie shell --seed 2023174 
(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))))