_divideComplex, real part

Percentage Accurate: 61.4% → 79.5%

Time: 9.5s

Alternatives: 12

Speedup: 2.1×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 61.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 79.5% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)\\ \mathbf{if}\;y.im \leq -1.12 \cdot 10^{+156}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re \cdot \frac{x.re}{y.im}}{y.im}\\ \mathbf{elif}\;y.im \leq -2.5 \cdot 10^{-136}:\\ \;\;\;\;\frac{x.re \cdot y.re + x.im \cdot y.im}{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)}\\ \mathbf{elif}\;y.im \leq 6.5 \cdot 10^{-111}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{x.im \cdot \frac{y.im}{y.re}}{y.re}\\ \mathbf{elif}\;y.im \leq 4.2 \cdot 10^{-8}:\\ \;\;\;\;\frac{y.im}{\frac{t_0}{x.im}} + \frac{x.re}{\frac{t_0}{y.re}}\\ \mathbf{elif}\;y.im \leq 1.92 \cdot 10^{+99}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{y.im}{y.re} \cdot \frac{x.im}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(x.im + \frac{x.re}{\frac{y.im}{y.re}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (fma y.im y.im (* y.re y.re))))
   (if (<= y.im -1.12e+156)
     (+ (/ x.im y.im) (/ (* y.re (/ x.re y.im)) y.im))
     (if (<= y.im -2.5e-136)
       (/ (+ (* x.re y.re) (* x.im y.im)) (fma y.re y.re (* y.im y.im)))
       (if (<= y.im 6.5e-111)
         (+ (/ x.re y.re) (/ (* x.im (/ y.im y.re)) y.re))
         (if (<= y.im 4.2e-8)
           (+ (/ y.im (/ t_0 x.im)) (/ x.re (/ t_0 y.re)))
           (if (<= y.im 1.92e+99)
             (+ (/ x.re y.re) (* (/ y.im y.re) (/ x.im y.re)))
             (*
              (/ 1.0 (hypot y.re y.im))
              (+ x.im (/ x.re (/ y.im y.re)))))))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = fma(y_46_im, y_46_im, (y_46_re * y_46_re));
	double tmp;
	if (y_46_im <= -1.12e+156) {
		tmp = (x_46_im / y_46_im) + ((y_46_re * (x_46_re / y_46_im)) / y_46_im);
	} else if (y_46_im <= -2.5e-136) {
		tmp = ((x_46_re * y_46_re) + (x_46_im * y_46_im)) / fma(y_46_re, y_46_re, (y_46_im * y_46_im));
	} else if (y_46_im <= 6.5e-111) {
		tmp = (x_46_re / y_46_re) + ((x_46_im * (y_46_im / y_46_re)) / y_46_re);
	} else if (y_46_im <= 4.2e-8) {
		tmp = (y_46_im / (t_0 / x_46_im)) + (x_46_re / (t_0 / y_46_re));
	} else if (y_46_im <= 1.92e+99) {
		tmp = (x_46_re / y_46_re) + ((y_46_im / y_46_re) * (x_46_im / y_46_re));
	} else {
		tmp = (1.0 / hypot(y_46_re, y_46_im)) * (x_46_im + (x_46_re / (y_46_im / y_46_re)));
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = fma(y_46_im, y_46_im, Float64(y_46_re * y_46_re))
	tmp = 0.0
	if (y_46_im <= -1.12e+156)
		tmp = Float64(Float64(x_46_im / y_46_im) + Float64(Float64(y_46_re * Float64(x_46_re / y_46_im)) / y_46_im));
	elseif (y_46_im <= -2.5e-136)
		tmp = Float64(Float64(Float64(x_46_re * y_46_re) + Float64(x_46_im * y_46_im)) / fma(y_46_re, y_46_re, Float64(y_46_im * y_46_im)));
	elseif (y_46_im <= 6.5e-111)
		tmp = Float64(Float64(x_46_re / y_46_re) + Float64(Float64(x_46_im * Float64(y_46_im / y_46_re)) / y_46_re));
	elseif (y_46_im <= 4.2e-8)
		tmp = Float64(Float64(y_46_im / Float64(t_0 / x_46_im)) + Float64(x_46_re / Float64(t_0 / y_46_re)));
	elseif (y_46_im <= 1.92e+99)
		tmp = Float64(Float64(x_46_re / y_46_re) + Float64(Float64(y_46_im / y_46_re) * Float64(x_46_im / y_46_re)));
	else
		tmp = Float64(Float64(1.0 / hypot(y_46_re, y_46_im)) * Float64(x_46_im + Float64(x_46_re / Float64(y_46_im / y_46_re))));
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(y$46$im * y$46$im + N[(y$46$re * y$46$re), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$im, -1.12e+156], N[(N[(x$46$im / y$46$im), $MachinePrecision] + N[(N[(y$46$re * N[(x$46$re / y$46$im), $MachinePrecision]), $MachinePrecision] / y$46$im), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, -2.5e-136], N[(N[(N[(x$46$re * y$46$re), $MachinePrecision] + N[(x$46$im * y$46$im), $MachinePrecision]), $MachinePrecision] / N[(y$46$re * y$46$re + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, 6.5e-111], N[(N[(x$46$re / y$46$re), $MachinePrecision] + N[(N[(x$46$im * N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, 4.2e-8], N[(N[(y$46$im / N[(t$95$0 / x$46$im), $MachinePrecision]), $MachinePrecision] + N[(x$46$re / N[(t$95$0 / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, 1.92e+99], N[(N[(x$46$re / y$46$re), $MachinePrecision] + N[(N[(y$46$im / y$46$re), $MachinePrecision] * N[(x$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision] * N[(x$46$im + N[(x$46$re / N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 6 regimes

2. if y.im < -1.12000000000000007e156

   1. Initial program 19.7%

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

      1. *-un-lft-identity 19.7%

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

      2. add-sqr-sqrt 19.7%

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

      3. times-frac 19.7%

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

      4. hypot-def 19.7%

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

      5. fma-def 19.7%

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

      6. hypot-def 48.4%

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

   3. Applied egg-rr 48.4%

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

   4. Taylor expanded in y.re around 0 77.4%

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

      1. +-commutative 77.4%

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

      2. *-commutative 77.4%

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

      3. unpow2 77.4%

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

      4. times-frac 92.4%

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

   6. Simplified 92.4%

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

      1. associate-*l/ 94.5%

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

   8. Applied egg-rr 94.5%

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

   if -1.12000000000000007e156 < y.im < -2.5000000000000001e-136

   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. Taylor expanded in y.re around 0 87.3%

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

      1. unpow2 87.3%

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

      2. unpow2 87.3%

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

      3. fma-def 87.3%

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

   4. Simplified 87.3%

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

   if -2.5000000000000001e-136 < y.im < 6.49999999999999974e-111

   1. Initial program 79.2%

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

   2. Taylor expanded in y.re around inf 88.6%

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

      1. unpow2 88.6%

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

      2. times-frac 93.2%

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

   4. Simplified 93.2%

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

      1. *-commutative 93.2%

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

      2. associate-*l/ 94.6%

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

   6. Applied egg-rr 94.6%

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

   if 6.49999999999999974e-111 < y.im < 4.19999999999999989e-8

   1. Initial program 84.0%

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

      1. *-un-lft-identity 84.0%

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

      2. add-sqr-sqrt 84.0%

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

      3. times-frac 84.1%

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

      4. hypot-def 84.2%

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

      5. fma-def 84.2%

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

      6. hypot-def 84.5%

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

   3. Applied egg-rr 84.5%

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

   4. Taylor expanded in x.re around 0 84.0%

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

      1. associate-/l* 84.1%

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

      2. unpow2 84.1%

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

      3. +-commutative 84.1%

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

      4. unpow2 84.1%

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

      5. fma-def 84.1%

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

      6. associate-/l* 89.6%

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

      7. unpow2 89.6%

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

      8. +-commutative 89.6%

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

      9. unpow2 89.6%

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

      10. fma-def 89.6%

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

   6. Simplified 89.6%

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

   if 4.19999999999999989e-8 < y.im < 1.9199999999999999e99

   1. Initial program 47.3%

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

   2. Taylor expanded in y.re around inf 75.6%

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

      1. unpow2 75.6%

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

      2. times-frac 87.9%

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

   4. Simplified 87.9%

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

   if 1.9199999999999999e99 < y.im

   1. Initial program 39.5%

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

      1. *-un-lft-identity 39.5%

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

      2. add-sqr-sqrt 39.5%

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

      3. times-frac 39.5%

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

      4. hypot-def 39.5%

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

      5. fma-def 39.6%

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

      6. hypot-def 57.4%

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

   3. Applied egg-rr 57.4%

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

   4. Taylor expanded in y.re around 0 78.6%

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

      1. +-commutative 78.6%

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

      2. associate-/l* 89.5%

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

   6. Simplified 89.5%

      \[\leadsto \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \color{blue}{\left(x.im + \frac{x.re}{\frac{y.im}{y.re}}\right)} \]
3. Recombined 6 regimes into one program.

4. Final simplification 90.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -1.12 \cdot 10^{+156}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re \cdot \frac{x.re}{y.im}}{y.im}\\ \mathbf{elif}\;y.im \leq -2.5 \cdot 10^{-136}:\\ \;\;\;\;\frac{x.re \cdot y.re + x.im \cdot y.im}{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)}\\ \mathbf{elif}\;y.im \leq 6.5 \cdot 10^{-111}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{x.im \cdot \frac{y.im}{y.re}}{y.re}\\ \mathbf{elif}\;y.im \leq 4.2 \cdot 10^{-8}:\\ \;\;\;\;\frac{y.im}{\frac{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)}{x.im}} + \frac{x.re}{\frac{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)}{y.re}}\\ \mathbf{elif}\;y.im \leq 1.92 \cdot 10^{+99}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{y.im}{y.re} \cdot \frac{x.im}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(x.im + \frac{x.re}{\frac{y.im}{y.re}}\right)\\ \end{array} \]

Alternative 2: 88.2% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \leq 5 \cdot 10^{+172}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{x.re}{y.re + y.im \cdot \frac{y.im}{y.re}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<=
      (/ (+ (* x.re y.re) (* x.im y.im)) (+ (* y.re y.re) (* y.im y.im)))
      5e+172)
   (*
    (/ 1.0 (hypot y.re y.im))
    (/ (fma x.re y.re (* x.im y.im)) (hypot y.re y.im)))
   (+ (/ x.im y.im) (/ x.re (+ y.re (* y.im (/ y.im y.re)))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((((x_46_re * y_46_re) + (x_46_im * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im))) <= 5e+172) {
		tmp = (1.0 / hypot(y_46_re, y_46_im)) * (fma(x_46_re, y_46_re, (x_46_im * y_46_im)) / hypot(y_46_re, y_46_im));
	} else {
		tmp = (x_46_im / y_46_im) + (x_46_re / (y_46_re + (y_46_im * (y_46_im / y_46_re))));
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (Float64(Float64(Float64(x_46_re * y_46_re) + Float64(x_46_im * y_46_im)) / Float64(Float64(y_46_re * y_46_re) + Float64(y_46_im * y_46_im))) <= 5e+172)
		tmp = Float64(Float64(1.0 / hypot(y_46_re, y_46_im)) * Float64(fma(x_46_re, y_46_re, Float64(x_46_im * y_46_im)) / hypot(y_46_re, y_46_im)));
	else
		tmp = Float64(Float64(x_46_im / y_46_im) + Float64(x_46_re / Float64(y_46_re + Float64(y_46_im * Float64(y_46_im / y_46_re)))));
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[N[(N[(N[(x$46$re * y$46$re), $MachinePrecision] + N[(x$46$im * y$46$im), $MachinePrecision]), $MachinePrecision] / N[(N[(y$46$re * y$46$re), $MachinePrecision] + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 5e+172], 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[(x$46$im * y$46$im), $MachinePrecision]), $MachinePrecision] / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$im / y$46$im), $MachinePrecision] + N[(x$46$re / N[(y$46$re + N[(y$46$im * N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (*.f64 x.re y.re) (*.f64 x.im y.im)) (+.f64 (*.f64 y.re y.re) (*.f64 y.im y.im))) < 5.0000000000000001e172

   1. Initial program 82.8%

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

      1. *-un-lft-identity 82.8%

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

      2. add-sqr-sqrt 82.7%

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

      3. times-frac 82.7%

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

      4. hypot-def 82.7%

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

      5. fma-def 82.7%

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

      6. hypot-def 97.4%

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

   3. Applied egg-rr 97.4%

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

   if 5.0000000000000001e172 < (/.f64 (+.f64 (*.f64 x.re y.re) (*.f64 x.im y.im)) (+.f64 (*.f64 y.re y.re) (*.f64 y.im y.im)))

   1. Initial program 12.9%

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

      1. *-un-lft-identity 12.9%

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

      2. add-sqr-sqrt 12.9%

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

      3. times-frac 12.9%

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

      4. hypot-def 12.9%

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

      5. fma-def 13.0%

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

      6. hypot-def 17.8%

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

   3. Applied egg-rr 17.8%

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

   4. Taylor expanded in x.re around 0 8.6%

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

      1. associate-/l* 8.8%

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

      2. unpow2 8.8%

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

      3. +-commutative 8.8%

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

      4. unpow2 8.8%

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

      5. fma-def 8.8%

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

      6. associate-/l* 16.8%

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

      7. unpow2 16.8%

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

      8. +-commutative 16.8%

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

      9. unpow2 16.8%

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

      10. fma-def 16.8%

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

   6. Simplified 16.8%

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

   7. Taylor expanded in y.im around inf 58.3%

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

   8. Taylor expanded in y.im around 0 66.4%

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

      1. +-commutative 66.4%

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

      2. unpow2 66.4%

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

      3. associate-*r/ 74.2%

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

   10. Simplified 74.2%

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

4. Final simplification 90.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \leq 5 \cdot 10^{+172}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{x.re}{y.re + y.im \cdot \frac{y.im}{y.re}}\\ \end{array} \]

Alternative 3: 78.2% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -1.12 \cdot 10^{+156}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re \cdot \frac{x.re}{y.im}}{y.im}\\ \mathbf{elif}\;y.im \leq -1.8 \cdot 10^{-136}:\\ \;\;\;\;\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.im \leq 1.92 \cdot 10^{+99}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{x.im \cdot \frac{y.im}{y.re}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(x.im + \frac{x.re}{\frac{y.im}{y.re}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.im -1.12e+156)
   (+ (/ x.im y.im) (/ (* y.re (/ x.re y.im)) y.im))
   (if (<= y.im -1.8e-136)
     (/ (+ (* x.re y.re) (* x.im y.im)) (+ (* y.re y.re) (* y.im y.im)))
     (if (<= y.im 1.92e+99)
       (+ (/ x.re y.re) (/ (* x.im (/ y.im y.re)) y.re))
       (* (/ 1.0 (hypot y.re y.im)) (+ x.im (/ x.re (/ y.im y.re))))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_im <= -1.12e+156) {
		tmp = (x_46_im / y_46_im) + ((y_46_re * (x_46_re / y_46_im)) / y_46_im);
	} else if (y_46_im <= -1.8e-136) {
		tmp = ((x_46_re * y_46_re) + (x_46_im * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	} else if (y_46_im <= 1.92e+99) {
		tmp = (x_46_re / y_46_re) + ((x_46_im * (y_46_im / y_46_re)) / y_46_re);
	} else {
		tmp = (1.0 / hypot(y_46_re, y_46_im)) * (x_46_im + (x_46_re / (y_46_im / y_46_re)));
	}
	return tmp;
}

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_im <= -1.12e+156) {
		tmp = (x_46_im / y_46_im) + ((y_46_re * (x_46_re / y_46_im)) / y_46_im);
	} else if (y_46_im <= -1.8e-136) {
		tmp = ((x_46_re * y_46_re) + (x_46_im * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	} else if (y_46_im <= 1.92e+99) {
		tmp = (x_46_re / y_46_re) + ((x_46_im * (y_46_im / y_46_re)) / y_46_re);
	} else {
		tmp = (1.0 / Math.hypot(y_46_re, y_46_im)) * (x_46_im + (x_46_re / (y_46_im / y_46_re)));
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if y_46_im <= -1.12e+156:
		tmp = (x_46_im / y_46_im) + ((y_46_re * (x_46_re / y_46_im)) / y_46_im)
	elif y_46_im <= -1.8e-136:
		tmp = ((x_46_re * y_46_re) + (x_46_im * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im))
	elif y_46_im <= 1.92e+99:
		tmp = (x_46_re / y_46_re) + ((x_46_im * (y_46_im / y_46_re)) / y_46_re)
	else:
		tmp = (1.0 / math.hypot(y_46_re, y_46_im)) * (x_46_im + (x_46_re / (y_46_im / y_46_re)))
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (y_46_im <= -1.12e+156)
		tmp = Float64(Float64(x_46_im / y_46_im) + Float64(Float64(y_46_re * Float64(x_46_re / y_46_im)) / y_46_im));
	elseif (y_46_im <= -1.8e-136)
		tmp = Float64(Float64(Float64(x_46_re * y_46_re) + Float64(x_46_im * y_46_im)) / Float64(Float64(y_46_re * y_46_re) + Float64(y_46_im * y_46_im)));
	elseif (y_46_im <= 1.92e+99)
		tmp = Float64(Float64(x_46_re / y_46_re) + Float64(Float64(x_46_im * Float64(y_46_im / y_46_re)) / y_46_re));
	else
		tmp = Float64(Float64(1.0 / hypot(y_46_re, y_46_im)) * Float64(x_46_im + Float64(x_46_re / Float64(y_46_im / y_46_re))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if (y_46_im <= -1.12e+156)
		tmp = (x_46_im / y_46_im) + ((y_46_re * (x_46_re / y_46_im)) / y_46_im);
	elseif (y_46_im <= -1.8e-136)
		tmp = ((x_46_re * y_46_re) + (x_46_im * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	elseif (y_46_im <= 1.92e+99)
		tmp = (x_46_re / y_46_re) + ((x_46_im * (y_46_im / y_46_re)) / y_46_re);
	else
		tmp = (1.0 / hypot(y_46_re, y_46_im)) * (x_46_im + (x_46_re / (y_46_im / y_46_re)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[y$46$im, -1.12e+156], N[(N[(x$46$im / y$46$im), $MachinePrecision] + N[(N[(y$46$re * N[(x$46$re / y$46$im), $MachinePrecision]), $MachinePrecision] / y$46$im), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, -1.8e-136], N[(N[(N[(x$46$re * y$46$re), $MachinePrecision] + N[(x$46$im * y$46$im), $MachinePrecision]), $MachinePrecision] / N[(N[(y$46$re * y$46$re), $MachinePrecision] + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, 1.92e+99], N[(N[(x$46$re / y$46$re), $MachinePrecision] + N[(N[(x$46$im * N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision] * N[(x$46$im + N[(x$46$re / N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if y.im < -1.12000000000000007e156

   1. Initial program 19.7%

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

      1. *-un-lft-identity 19.7%

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

      2. add-sqr-sqrt 19.7%

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

      3. times-frac 19.7%

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

      4. hypot-def 19.7%

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

      5. fma-def 19.7%

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

      6. hypot-def 48.4%

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

   3. Applied egg-rr 48.4%

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

   4. Taylor expanded in y.re around 0 77.4%

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

      1. +-commutative 77.4%

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

      2. *-commutative 77.4%

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

      3. unpow2 77.4%

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

      4. times-frac 92.4%

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

   6. Simplified 92.4%

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

      1. associate-*l/ 94.5%

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

   8. Applied egg-rr 94.5%

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

   if -1.12000000000000007e156 < y.im < -1.7999999999999999e-136

   1. Initial program 87.3%

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

   if -1.7999999999999999e-136 < y.im < 1.9199999999999999e99

   1. Initial program 75.2%

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

   2. Taylor expanded in y.re around inf 82.2%

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

      1. unpow2 82.2%

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

      2. times-frac 87.1%

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

   4. Simplified 87.1%

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

      1. *-commutative 87.1%

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

      2. associate-*l/ 88.1%

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

   6. Applied egg-rr 88.1%

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

   if 1.9199999999999999e99 < y.im

   1. Initial program 39.5%

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

      1. *-un-lft-identity 39.5%

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

      2. add-sqr-sqrt 39.5%

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

      3. times-frac 39.5%

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

      4. hypot-def 39.5%

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

      5. fma-def 39.6%

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

      6. hypot-def 57.4%

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

   3. Applied egg-rr 57.4%

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

   4. Taylor expanded in y.re around 0 78.6%

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

      1. +-commutative 78.6%

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

      2. associate-/l* 89.5%

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

   6. Simplified 89.5%

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

4. Final simplification 89.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -1.12 \cdot 10^{+156}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re \cdot \frac{x.re}{y.im}}{y.im}\\ \mathbf{elif}\;y.im \leq -1.8 \cdot 10^{-136}:\\ \;\;\;\;\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.im \leq 1.92 \cdot 10^{+99}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{x.im \cdot \frac{y.im}{y.re}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(x.im + \frac{x.re}{\frac{y.im}{y.re}}\right)\\ \end{array} \]

Alternative 4: 78.2% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -1.12 \cdot 10^{+156}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re \cdot \frac{x.re}{y.im}}{y.im}\\ \mathbf{elif}\;y.im \leq -2.9 \cdot 10^{-136}:\\ \;\;\;\;\frac{x.re \cdot y.re + x.im \cdot y.im}{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)}\\ \mathbf{elif}\;y.im \leq 2 \cdot 10^{+99}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{x.im \cdot \frac{y.im}{y.re}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(x.im + \frac{x.re}{\frac{y.im}{y.re}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.im -1.12e+156)
   (+ (/ x.im y.im) (/ (* y.re (/ x.re y.im)) y.im))
   (if (<= y.im -2.9e-136)
     (/ (+ (* x.re y.re) (* x.im y.im)) (fma y.re y.re (* y.im y.im)))
     (if (<= y.im 2e+99)
       (+ (/ x.re y.re) (/ (* x.im (/ y.im y.re)) y.re))
       (* (/ 1.0 (hypot y.re y.im)) (+ x.im (/ x.re (/ y.im y.re))))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_im <= -1.12e+156) {
		tmp = (x_46_im / y_46_im) + ((y_46_re * (x_46_re / y_46_im)) / y_46_im);
	} else if (y_46_im <= -2.9e-136) {
		tmp = ((x_46_re * y_46_re) + (x_46_im * y_46_im)) / fma(y_46_re, y_46_re, (y_46_im * y_46_im));
	} else if (y_46_im <= 2e+99) {
		tmp = (x_46_re / y_46_re) + ((x_46_im * (y_46_im / y_46_re)) / y_46_re);
	} else {
		tmp = (1.0 / hypot(y_46_re, y_46_im)) * (x_46_im + (x_46_re / (y_46_im / y_46_re)));
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (y_46_im <= -1.12e+156)
		tmp = Float64(Float64(x_46_im / y_46_im) + Float64(Float64(y_46_re * Float64(x_46_re / y_46_im)) / y_46_im));
	elseif (y_46_im <= -2.9e-136)
		tmp = Float64(Float64(Float64(x_46_re * y_46_re) + Float64(x_46_im * y_46_im)) / fma(y_46_re, y_46_re, Float64(y_46_im * y_46_im)));
	elseif (y_46_im <= 2e+99)
		tmp = Float64(Float64(x_46_re / y_46_re) + Float64(Float64(x_46_im * Float64(y_46_im / y_46_re)) / y_46_re));
	else
		tmp = Float64(Float64(1.0 / hypot(y_46_re, y_46_im)) * Float64(x_46_im + Float64(x_46_re / Float64(y_46_im / y_46_re))));
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[y$46$im, -1.12e+156], N[(N[(x$46$im / y$46$im), $MachinePrecision] + N[(N[(y$46$re * N[(x$46$re / y$46$im), $MachinePrecision]), $MachinePrecision] / y$46$im), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, -2.9e-136], N[(N[(N[(x$46$re * y$46$re), $MachinePrecision] + N[(x$46$im * y$46$im), $MachinePrecision]), $MachinePrecision] / N[(y$46$re * y$46$re + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, 2e+99], N[(N[(x$46$re / y$46$re), $MachinePrecision] + N[(N[(x$46$im * N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision] * N[(x$46$im + N[(x$46$re / N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if y.im < -1.12000000000000007e156

   1. Initial program 19.7%

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

      1. *-un-lft-identity 19.7%

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

      2. add-sqr-sqrt 19.7%

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

      3. times-frac 19.7%

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

      4. hypot-def 19.7%

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

      5. fma-def 19.7%

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

      6. hypot-def 48.4%

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

   3. Applied egg-rr 48.4%

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

   4. Taylor expanded in y.re around 0 77.4%

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

      1. +-commutative 77.4%

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

      2. *-commutative 77.4%

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

      3. unpow2 77.4%

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

      4. times-frac 92.4%

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

   6. Simplified 92.4%

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

      1. associate-*l/ 94.5%

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

   8. Applied egg-rr 94.5%

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

   if -1.12000000000000007e156 < y.im < -2.89999999999999995e-136

   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. Taylor expanded in y.re around 0 87.3%

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

      1. unpow2 87.3%

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

      2. unpow2 87.3%

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

      3. fma-def 87.3%

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

   4. Simplified 87.3%

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

   if -2.89999999999999995e-136 < y.im < 1.9999999999999999e99

   1. Initial program 75.2%

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

   2. Taylor expanded in y.re around inf 82.2%

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

      1. unpow2 82.2%

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

      2. times-frac 87.1%

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

   4. Simplified 87.1%

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

      1. *-commutative 87.1%

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

      2. associate-*l/ 88.1%

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

   6. Applied egg-rr 88.1%

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

   if 1.9999999999999999e99 < y.im

   1. Initial program 39.5%

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

      1. *-un-lft-identity 39.5%

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

      2. add-sqr-sqrt 39.5%

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

      3. times-frac 39.5%

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

      4. hypot-def 39.5%

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

      5. fma-def 39.6%

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

      6. hypot-def 57.4%

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

   3. Applied egg-rr 57.4%

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

   4. Taylor expanded in y.re around 0 78.6%

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

      1. +-commutative 78.6%

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

      2. associate-/l* 89.5%

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

   6. Simplified 89.5%

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

4. Final simplification 89.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -1.12 \cdot 10^{+156}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re \cdot \frac{x.re}{y.im}}{y.im}\\ \mathbf{elif}\;y.im \leq -2.9 \cdot 10^{-136}:\\ \;\;\;\;\frac{x.re \cdot y.re + x.im \cdot y.im}{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)}\\ \mathbf{elif}\;y.im \leq 2 \cdot 10^{+99}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{x.im \cdot \frac{y.im}{y.re}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(x.im + \frac{x.re}{\frac{y.im}{y.re}}\right)\\ \end{array} \]

Alternative 5: 78.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -1.12 \cdot 10^{+156}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re \cdot \frac{x.re}{y.im}}{y.im}\\ \mathbf{elif}\;y.im \leq -4.2 \cdot 10^{-135}:\\ \;\;\;\;\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.im \leq 1.92 \cdot 10^{+99}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{x.im \cdot \frac{y.im}{y.re}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{x.re}{y.re + y.im \cdot \frac{y.im}{y.re}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.im -1.12e+156)
   (+ (/ x.im y.im) (/ (* y.re (/ x.re y.im)) y.im))
   (if (<= y.im -4.2e-135)
     (/ (+ (* x.re y.re) (* x.im y.im)) (+ (* y.re y.re) (* y.im y.im)))
     (if (<= y.im 1.92e+99)
       (+ (/ x.re y.re) (/ (* x.im (/ y.im y.re)) y.re))
       (+ (/ x.im y.im) (/ x.re (+ y.re (* y.im (/ y.im y.re)))))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_im <= -1.12e+156) {
		tmp = (x_46_im / y_46_im) + ((y_46_re * (x_46_re / y_46_im)) / y_46_im);
	} else if (y_46_im <= -4.2e-135) {
		tmp = ((x_46_re * y_46_re) + (x_46_im * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	} else if (y_46_im <= 1.92e+99) {
		tmp = (x_46_re / y_46_re) + ((x_46_im * (y_46_im / y_46_re)) / y_46_re);
	} else {
		tmp = (x_46_im / y_46_im) + (x_46_re / (y_46_re + (y_46_im * (y_46_im / y_46_re))));
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if (y_46im <= (-1.12d+156)) then
        tmp = (x_46im / y_46im) + ((y_46re * (x_46re / y_46im)) / y_46im)
    else if (y_46im <= (-4.2d-135)) then
        tmp = ((x_46re * y_46re) + (x_46im * y_46im)) / ((y_46re * y_46re) + (y_46im * y_46im))
    else if (y_46im <= 1.92d+99) then
        tmp = (x_46re / y_46re) + ((x_46im * (y_46im / y_46re)) / y_46re)
    else
        tmp = (x_46im / y_46im) + (x_46re / (y_46re + (y_46im * (y_46im / y_46re))))
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_im <= -1.12e+156) {
		tmp = (x_46_im / y_46_im) + ((y_46_re * (x_46_re / y_46_im)) / y_46_im);
	} else if (y_46_im <= -4.2e-135) {
		tmp = ((x_46_re * y_46_re) + (x_46_im * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	} else if (y_46_im <= 1.92e+99) {
		tmp = (x_46_re / y_46_re) + ((x_46_im * (y_46_im / y_46_re)) / y_46_re);
	} else {
		tmp = (x_46_im / y_46_im) + (x_46_re / (y_46_re + (y_46_im * (y_46_im / y_46_re))));
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if y_46_im <= -1.12e+156:
		tmp = (x_46_im / y_46_im) + ((y_46_re * (x_46_re / y_46_im)) / y_46_im)
	elif y_46_im <= -4.2e-135:
		tmp = ((x_46_re * y_46_re) + (x_46_im * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im))
	elif y_46_im <= 1.92e+99:
		tmp = (x_46_re / y_46_re) + ((x_46_im * (y_46_im / y_46_re)) / y_46_re)
	else:
		tmp = (x_46_im / y_46_im) + (x_46_re / (y_46_re + (y_46_im * (y_46_im / y_46_re))))
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (y_46_im <= -1.12e+156)
		tmp = Float64(Float64(x_46_im / y_46_im) + Float64(Float64(y_46_re * Float64(x_46_re / y_46_im)) / y_46_im));
	elseif (y_46_im <= -4.2e-135)
		tmp = Float64(Float64(Float64(x_46_re * y_46_re) + Float64(x_46_im * y_46_im)) / Float64(Float64(y_46_re * y_46_re) + Float64(y_46_im * y_46_im)));
	elseif (y_46_im <= 1.92e+99)
		tmp = Float64(Float64(x_46_re / y_46_re) + Float64(Float64(x_46_im * Float64(y_46_im / y_46_re)) / y_46_re));
	else
		tmp = Float64(Float64(x_46_im / y_46_im) + Float64(x_46_re / Float64(y_46_re + Float64(y_46_im * Float64(y_46_im / y_46_re)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if (y_46_im <= -1.12e+156)
		tmp = (x_46_im / y_46_im) + ((y_46_re * (x_46_re / y_46_im)) / y_46_im);
	elseif (y_46_im <= -4.2e-135)
		tmp = ((x_46_re * y_46_re) + (x_46_im * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	elseif (y_46_im <= 1.92e+99)
		tmp = (x_46_re / y_46_re) + ((x_46_im * (y_46_im / y_46_re)) / y_46_re);
	else
		tmp = (x_46_im / y_46_im) + (x_46_re / (y_46_re + (y_46_im * (y_46_im / y_46_re))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[y$46$im, -1.12e+156], N[(N[(x$46$im / y$46$im), $MachinePrecision] + N[(N[(y$46$re * N[(x$46$re / y$46$im), $MachinePrecision]), $MachinePrecision] / y$46$im), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, -4.2e-135], N[(N[(N[(x$46$re * y$46$re), $MachinePrecision] + N[(x$46$im * y$46$im), $MachinePrecision]), $MachinePrecision] / N[(N[(y$46$re * y$46$re), $MachinePrecision] + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, 1.92e+99], N[(N[(x$46$re / y$46$re), $MachinePrecision] + N[(N[(x$46$im * N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$im / y$46$im), $MachinePrecision] + N[(x$46$re / N[(y$46$re + N[(y$46$im * N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if y.im < -1.12000000000000007e156

   1. Initial program 19.7%

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

      1. *-un-lft-identity 19.7%

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

      2. add-sqr-sqrt 19.7%

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

      3. times-frac 19.7%

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

      4. hypot-def 19.7%

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

      5. fma-def 19.7%

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

      6. hypot-def 48.4%

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

   3. Applied egg-rr 48.4%

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

   4. Taylor expanded in y.re around 0 77.4%

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

      1. +-commutative 77.4%

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

      2. *-commutative 77.4%

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

      3. unpow2 77.4%

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

      4. times-frac 92.4%

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

   6. Simplified 92.4%

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

      1. associate-*l/ 94.5%

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

   8. Applied egg-rr 94.5%

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

   if -1.12000000000000007e156 < y.im < -4.2e-135

   1. Initial program 87.3%

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

   if -4.2e-135 < y.im < 1.9199999999999999e99

   1. Initial program 75.2%

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

   2. Taylor expanded in y.re around inf 82.2%

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

      1. unpow2 82.2%

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

      2. times-frac 87.1%

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

   4. Simplified 87.1%

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

      1. *-commutative 87.1%

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

      2. associate-*l/ 88.1%

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

   6. Applied egg-rr 88.1%

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

   if 1.9199999999999999e99 < y.im

   1. Initial program 39.5%

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

      1. *-un-lft-identity 39.5%

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

      2. add-sqr-sqrt 39.5%

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

      3. times-frac 39.5%

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

      4. hypot-def 39.5%

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

      5. fma-def 39.6%

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

      6. hypot-def 57.4%

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

   3. Applied egg-rr 57.4%

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

   4. Taylor expanded in x.re around 0 39.5%

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

      1. associate-/l* 33.7%

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

      2. unpow2 33.7%

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

      3. +-commutative 33.7%

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

      4. unpow2 33.7%

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

      5. fma-def 33.7%

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

      6. associate-/l* 37.7%

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

      7. unpow2 37.7%

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

      8. +-commutative 37.7%

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

      9. unpow2 37.7%

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

      10. fma-def 37.7%

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

   6. Simplified 37.7%

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

   7. Taylor expanded in y.im around inf 81.0%

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

   8. Taylor expanded in y.im around 0 81.0%

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

      1. +-commutative 81.0%

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

      2. unpow2 81.0%

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

      3. associate-*r/ 89.4%

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

   10. Simplified 89.4%

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

4. Final simplification 89.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -1.12 \cdot 10^{+156}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re \cdot \frac{x.re}{y.im}}{y.im}\\ \mathbf{elif}\;y.im \leq -4.2 \cdot 10^{-135}:\\ \;\;\;\;\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.im \leq 1.92 \cdot 10^{+99}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{x.im \cdot \frac{y.im}{y.re}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{x.re}{y.re + y.im \cdot \frac{y.im}{y.re}}\\ \end{array} \]

Alternative 6: 76.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -1.45 \cdot 10^{-25}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re \cdot \frac{x.re}{y.im}}{y.im}\\ \mathbf{elif}\;y.im \leq 1.92 \cdot 10^{+99}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{x.im \cdot \frac{y.im}{y.re}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{x.re}{y.re + y.im \cdot \frac{y.im}{y.re}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.im -1.45e-25)
   (+ (/ x.im y.im) (/ (* y.re (/ x.re y.im)) y.im))
   (if (<= y.im 1.92e+99)
     (+ (/ x.re y.re) (/ (* x.im (/ y.im y.re)) y.re))
     (+ (/ x.im y.im) (/ x.re (+ y.re (* y.im (/ y.im y.re))))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_im <= -1.45e-25) {
		tmp = (x_46_im / y_46_im) + ((y_46_re * (x_46_re / y_46_im)) / y_46_im);
	} else if (y_46_im <= 1.92e+99) {
		tmp = (x_46_re / y_46_re) + ((x_46_im * (y_46_im / y_46_re)) / y_46_re);
	} else {
		tmp = (x_46_im / y_46_im) + (x_46_re / (y_46_re + (y_46_im * (y_46_im / y_46_re))));
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if (y_46im <= (-1.45d-25)) then
        tmp = (x_46im / y_46im) + ((y_46re * (x_46re / y_46im)) / y_46im)
    else if (y_46im <= 1.92d+99) then
        tmp = (x_46re / y_46re) + ((x_46im * (y_46im / y_46re)) / y_46re)
    else
        tmp = (x_46im / y_46im) + (x_46re / (y_46re + (y_46im * (y_46im / y_46re))))
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_im <= -1.45e-25) {
		tmp = (x_46_im / y_46_im) + ((y_46_re * (x_46_re / y_46_im)) / y_46_im);
	} else if (y_46_im <= 1.92e+99) {
		tmp = (x_46_re / y_46_re) + ((x_46_im * (y_46_im / y_46_re)) / y_46_re);
	} else {
		tmp = (x_46_im / y_46_im) + (x_46_re / (y_46_re + (y_46_im * (y_46_im / y_46_re))));
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if y_46_im <= -1.45e-25:
		tmp = (x_46_im / y_46_im) + ((y_46_re * (x_46_re / y_46_im)) / y_46_im)
	elif y_46_im <= 1.92e+99:
		tmp = (x_46_re / y_46_re) + ((x_46_im * (y_46_im / y_46_re)) / y_46_re)
	else:
		tmp = (x_46_im / y_46_im) + (x_46_re / (y_46_re + (y_46_im * (y_46_im / y_46_re))))
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (y_46_im <= -1.45e-25)
		tmp = Float64(Float64(x_46_im / y_46_im) + Float64(Float64(y_46_re * Float64(x_46_re / y_46_im)) / y_46_im));
	elseif (y_46_im <= 1.92e+99)
		tmp = Float64(Float64(x_46_re / y_46_re) + Float64(Float64(x_46_im * Float64(y_46_im / y_46_re)) / y_46_re));
	else
		tmp = Float64(Float64(x_46_im / y_46_im) + Float64(x_46_re / Float64(y_46_re + Float64(y_46_im * Float64(y_46_im / y_46_re)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if (y_46_im <= -1.45e-25)
		tmp = (x_46_im / y_46_im) + ((y_46_re * (x_46_re / y_46_im)) / y_46_im);
	elseif (y_46_im <= 1.92e+99)
		tmp = (x_46_re / y_46_re) + ((x_46_im * (y_46_im / y_46_re)) / y_46_re);
	else
		tmp = (x_46_im / y_46_im) + (x_46_re / (y_46_re + (y_46_im * (y_46_im / y_46_re))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[y$46$im, -1.45e-25], N[(N[(x$46$im / y$46$im), $MachinePrecision] + N[(N[(y$46$re * N[(x$46$re / y$46$im), $MachinePrecision]), $MachinePrecision] / y$46$im), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, 1.92e+99], N[(N[(x$46$re / y$46$re), $MachinePrecision] + N[(N[(x$46$im * N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$im / y$46$im), $MachinePrecision] + N[(x$46$re / N[(y$46$re + N[(y$46$im * N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y.im < -1.45e-25

   1. Initial program 54.7%

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

      1. *-un-lft-identity 54.7%

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

      2. add-sqr-sqrt 54.7%

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

      3. times-frac 54.6%

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

      4. hypot-def 54.6%

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

      5. fma-def 54.6%

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

      6. hypot-def 71.5%

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

   3. Applied egg-rr 71.5%

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

   4. Taylor expanded in y.re around 0 78.9%

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

      1. +-commutative 78.9%

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

      2. *-commutative 78.9%

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

      3. unpow2 78.9%

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

      4. times-frac 87.6%

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

   6. Simplified 87.6%

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

      1. associate-*l/ 88.7%

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

   8. Applied egg-rr 88.7%

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

   if -1.45e-25 < y.im < 1.9199999999999999e99

   1. Initial program 77.4%

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

   2. Taylor expanded in y.re around inf 79.4%

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

      1. unpow2 79.4%

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

      2. times-frac 82.7%

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

   4. Simplified 82.7%

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

      1. *-commutative 82.7%

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

      2. associate-*l/ 84.1%

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

   6. Applied egg-rr 84.1%

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

   if 1.9199999999999999e99 < y.im

   1. Initial program 39.5%

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

      1. *-un-lft-identity 39.5%

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

      2. add-sqr-sqrt 39.5%

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

      3. times-frac 39.5%

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

      4. hypot-def 39.5%

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

      5. fma-def 39.6%

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

      6. hypot-def 57.4%

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

   3. Applied egg-rr 57.4%

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

   4. Taylor expanded in x.re around 0 39.5%

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

      1. associate-/l* 33.7%

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

      2. unpow2 33.7%

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

      3. +-commutative 33.7%

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

      4. unpow2 33.7%

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

      5. fma-def 33.7%

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

      6. associate-/l* 37.7%

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

      7. unpow2 37.7%

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

      8. +-commutative 37.7%

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

      9. unpow2 37.7%

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

      10. fma-def 37.7%

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

   6. Simplified 37.7%

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

   7. Taylor expanded in y.im around inf 81.0%

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

   8. Taylor expanded in y.im around 0 81.0%

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

      1. +-commutative 81.0%

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

      2. unpow2 81.0%

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

      3. associate-*r/ 89.4%

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

   10. Simplified 89.4%

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

4. Final simplification 86.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -1.45 \cdot 10^{-25}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re \cdot \frac{x.re}{y.im}}{y.im}\\ \mathbf{elif}\;y.im \leq 1.92 \cdot 10^{+99}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{x.im \cdot \frac{y.im}{y.re}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{x.re}{y.re + y.im \cdot \frac{y.im}{y.re}}\\ \end{array} \]

Alternative 7: 67.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -6.1 \cdot 10^{-87} \lor \neg \left(y.im \leq 1.92 \cdot 10^{+99}\right):\\ \;\;\;\;\frac{x.im}{y.im} + \frac{x.re}{y.im} \cdot \frac{y.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.im -6.1e-87) (not (<= y.im 1.92e+99)))
   (+ (/ x.im y.im) (* (/ x.re y.im) (/ y.re y.im)))
   (/ x.re y.re)))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_im <= -6.1e-87) || !(y_46_im <= 1.92e+99)) {
		tmp = (x_46_im / y_46_im) + ((x_46_re / y_46_im) * (y_46_re / y_46_im));
	} else {
		tmp = x_46_re / y_46_re;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if ((y_46im <= (-6.1d-87)) .or. (.not. (y_46im <= 1.92d+99))) then
        tmp = (x_46im / y_46im) + ((x_46re / y_46im) * (y_46re / y_46im))
    else
        tmp = x_46re / y_46re
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_im <= -6.1e-87) || !(y_46_im <= 1.92e+99)) {
		tmp = (x_46_im / y_46_im) + ((x_46_re / y_46_im) * (y_46_re / y_46_im));
	} else {
		tmp = x_46_re / y_46_re;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if (y_46_im <= -6.1e-87) or not (y_46_im <= 1.92e+99):
		tmp = (x_46_im / y_46_im) + ((x_46_re / y_46_im) * (y_46_re / y_46_im))
	else:
		tmp = x_46_re / y_46_re
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if ((y_46_im <= -6.1e-87) || !(y_46_im <= 1.92e+99))
		tmp = Float64(Float64(x_46_im / y_46_im) + Float64(Float64(x_46_re / y_46_im) * Float64(y_46_re / y_46_im)));
	else
		tmp = Float64(x_46_re / y_46_re);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if ((y_46_im <= -6.1e-87) || ~((y_46_im <= 1.92e+99)))
		tmp = (x_46_im / y_46_im) + ((x_46_re / y_46_im) * (y_46_re / y_46_im));
	else
		tmp = x_46_re / y_46_re;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[Or[LessEqual[y$46$im, -6.1e-87], N[Not[LessEqual[y$46$im, 1.92e+99]], $MachinePrecision]], N[(N[(x$46$im / y$46$im), $MachinePrecision] + N[(N[(x$46$re / y$46$im), $MachinePrecision] * N[(y$46$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.im < -6.10000000000000014e-87 or 1.9199999999999999e99 < y.im

   1. Initial program 52.5%

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

      1. *-un-lft-identity 52.5%

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

      2. add-sqr-sqrt 52.5%

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

      3. times-frac 52.4%

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

      4. hypot-def 52.4%

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

      5. fma-def 52.4%

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

      6. hypot-def 68.0%

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

   3. Applied egg-rr 68.0%

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

   4. Taylor expanded in y.re around 0 74.2%

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

      1. +-commutative 74.2%

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

      2. *-commutative 74.2%

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

      3. unpow2 74.2%

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

      4. times-frac 84.3%

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

   6. Simplified 84.3%

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

   if -6.10000000000000014e-87 < y.im < 1.9199999999999999e99

   1. Initial program 75.6%

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

   2. Taylor expanded in y.re around inf 70.5%

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

4. Final simplification 77.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -6.1 \cdot 10^{-87} \lor \neg \left(y.im \leq 1.92 \cdot 10^{+99}\right):\\ \;\;\;\;\frac{x.im}{y.im} + \frac{x.re}{y.im} \cdot \frac{y.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \end{array} \]

Alternative 8: 67.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -6.2 \cdot 10^{-87}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re \cdot \frac{x.re}{y.im}}{y.im}\\ \mathbf{elif}\;y.im \leq 1.92 \cdot 10^{+99}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{x.re}{y.im} \cdot \frac{y.re}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.im -6.2e-87)
   (+ (/ x.im y.im) (/ (* y.re (/ x.re y.im)) y.im))
   (if (<= y.im 1.92e+99)
     (/ x.re y.re)
     (+ (/ x.im y.im) (* (/ x.re y.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 tmp;
	if (y_46_im <= -6.2e-87) {
		tmp = (x_46_im / y_46_im) + ((y_46_re * (x_46_re / y_46_im)) / y_46_im);
	} else if (y_46_im <= 1.92e+99) {
		tmp = x_46_re / y_46_re;
	} else {
		tmp = (x_46_im / y_46_im) + ((x_46_re / y_46_im) * (y_46_re / y_46_im));
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if (y_46im <= (-6.2d-87)) then
        tmp = (x_46im / y_46im) + ((y_46re * (x_46re / y_46im)) / y_46im)
    else if (y_46im <= 1.92d+99) then
        tmp = x_46re / y_46re
    else
        tmp = (x_46im / y_46im) + ((x_46re / y_46im) * (y_46re / y_46im))
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_im <= -6.2e-87) {
		tmp = (x_46_im / y_46_im) + ((y_46_re * (x_46_re / y_46_im)) / y_46_im);
	} else if (y_46_im <= 1.92e+99) {
		tmp = x_46_re / y_46_re;
	} else {
		tmp = (x_46_im / y_46_im) + ((x_46_re / y_46_im) * (y_46_re / y_46_im));
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if y_46_im <= -6.2e-87:
		tmp = (x_46_im / y_46_im) + ((y_46_re * (x_46_re / y_46_im)) / y_46_im)
	elif y_46_im <= 1.92e+99:
		tmp = x_46_re / y_46_re
	else:
		tmp = (x_46_im / y_46_im) + ((x_46_re / y_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)
	tmp = 0.0
	if (y_46_im <= -6.2e-87)
		tmp = Float64(Float64(x_46_im / y_46_im) + Float64(Float64(y_46_re * Float64(x_46_re / y_46_im)) / y_46_im));
	elseif (y_46_im <= 1.92e+99)
		tmp = Float64(x_46_re / y_46_re);
	else
		tmp = Float64(Float64(x_46_im / y_46_im) + Float64(Float64(x_46_re / y_46_im) * Float64(y_46_re / y_46_im)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if (y_46_im <= -6.2e-87)
		tmp = (x_46_im / y_46_im) + ((y_46_re * (x_46_re / y_46_im)) / y_46_im);
	elseif (y_46_im <= 1.92e+99)
		tmp = x_46_re / y_46_re;
	else
		tmp = (x_46_im / y_46_im) + ((x_46_re / y_46_im) * (y_46_re / y_46_im));
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[y$46$im, -6.2e-87], N[(N[(x$46$im / y$46$im), $MachinePrecision] + N[(N[(y$46$re * N[(x$46$re / y$46$im), $MachinePrecision]), $MachinePrecision] / y$46$im), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, 1.92e+99], N[(x$46$re / y$46$re), $MachinePrecision], N[(N[(x$46$im / y$46$im), $MachinePrecision] + N[(N[(x$46$re / y$46$im), $MachinePrecision] * N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y.im < -6.19999999999999995e-87

   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. Step-by-step derivation

      1. *-un-lft-identity 61.1%

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

      2. add-sqr-sqrt 61.1%

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

      3. times-frac 61.0%

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

      4. hypot-def 61.0%

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

      5. fma-def 61.0%

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

      6. hypot-def 75.1%

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

   3. Applied egg-rr 75.1%

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

   4. Taylor expanded in y.re around 0 74.6%

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

      1. +-commutative 74.6%

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

      2. *-commutative 74.6%

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

      3. unpow2 74.6%

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

      4. times-frac 81.9%

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

   6. Simplified 81.9%

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

      1. associate-*l/ 82.8%

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

   8. Applied egg-rr 82.8%

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

   if -6.19999999999999995e-87 < y.im < 1.9199999999999999e99

   1. Initial program 75.6%

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

   2. Taylor expanded in y.re around inf 70.5%

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

   if 1.9199999999999999e99 < y.im

   1. Initial program 39.5%

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

      1. *-un-lft-identity 39.5%

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

      2. add-sqr-sqrt 39.5%

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

      3. times-frac 39.5%

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

      4. hypot-def 39.5%

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

      5. fma-def 39.6%

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

      6. hypot-def 57.4%

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

   3. Applied egg-rr 57.4%

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

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

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

      1. +-commutative 73.5%

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

      2. *-commutative 73.5%

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

      3. unpow2 73.5%

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

      4. times-frac 87.9%

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

   6. Simplified 87.9%

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

4. Final simplification 78.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -6.2 \cdot 10^{-87}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re \cdot \frac{x.re}{y.im}}{y.im}\\ \mathbf{elif}\;y.im \leq 1.92 \cdot 10^{+99}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{x.re}{y.im} \cdot \frac{y.re}{y.im}\\ \end{array} \]

Alternative 9: 75.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -1.05 \cdot 10^{-26}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re \cdot \frac{x.re}{y.im}}{y.im}\\ \mathbf{elif}\;y.im \leq 2.8 \cdot 10^{+99}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{y.im}{y.re} \cdot \frac{x.im}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{x.re}{y.im} \cdot \frac{y.re}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.im -1.05e-26)
   (+ (/ x.im y.im) (/ (* y.re (/ x.re y.im)) y.im))
   (if (<= y.im 2.8e+99)
     (+ (/ x.re y.re) (* (/ y.im y.re) (/ x.im y.re)))
     (+ (/ x.im y.im) (* (/ x.re y.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 tmp;
	if (y_46_im <= -1.05e-26) {
		tmp = (x_46_im / y_46_im) + ((y_46_re * (x_46_re / y_46_im)) / y_46_im);
	} else if (y_46_im <= 2.8e+99) {
		tmp = (x_46_re / y_46_re) + ((y_46_im / y_46_re) * (x_46_im / y_46_re));
	} else {
		tmp = (x_46_im / y_46_im) + ((x_46_re / y_46_im) * (y_46_re / y_46_im));
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if (y_46im <= (-1.05d-26)) then
        tmp = (x_46im / y_46im) + ((y_46re * (x_46re / y_46im)) / y_46im)
    else if (y_46im <= 2.8d+99) then
        tmp = (x_46re / y_46re) + ((y_46im / y_46re) * (x_46im / y_46re))
    else
        tmp = (x_46im / y_46im) + ((x_46re / y_46im) * (y_46re / y_46im))
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_im <= -1.05e-26) {
		tmp = (x_46_im / y_46_im) + ((y_46_re * (x_46_re / y_46_im)) / y_46_im);
	} else if (y_46_im <= 2.8e+99) {
		tmp = (x_46_re / y_46_re) + ((y_46_im / y_46_re) * (x_46_im / y_46_re));
	} else {
		tmp = (x_46_im / y_46_im) + ((x_46_re / y_46_im) * (y_46_re / y_46_im));
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if y_46_im <= -1.05e-26:
		tmp = (x_46_im / y_46_im) + ((y_46_re * (x_46_re / y_46_im)) / y_46_im)
	elif y_46_im <= 2.8e+99:
		tmp = (x_46_re / y_46_re) + ((y_46_im / y_46_re) * (x_46_im / y_46_re))
	else:
		tmp = (x_46_im / y_46_im) + ((x_46_re / y_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)
	tmp = 0.0
	if (y_46_im <= -1.05e-26)
		tmp = Float64(Float64(x_46_im / y_46_im) + Float64(Float64(y_46_re * Float64(x_46_re / y_46_im)) / y_46_im));
	elseif (y_46_im <= 2.8e+99)
		tmp = Float64(Float64(x_46_re / y_46_re) + Float64(Float64(y_46_im / y_46_re) * Float64(x_46_im / y_46_re)));
	else
		tmp = Float64(Float64(x_46_im / y_46_im) + Float64(Float64(x_46_re / y_46_im) * Float64(y_46_re / y_46_im)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if (y_46_im <= -1.05e-26)
		tmp = (x_46_im / y_46_im) + ((y_46_re * (x_46_re / y_46_im)) / y_46_im);
	elseif (y_46_im <= 2.8e+99)
		tmp = (x_46_re / y_46_re) + ((y_46_im / y_46_re) * (x_46_im / y_46_re));
	else
		tmp = (x_46_im / y_46_im) + ((x_46_re / y_46_im) * (y_46_re / y_46_im));
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[y$46$im, -1.05e-26], N[(N[(x$46$im / y$46$im), $MachinePrecision] + N[(N[(y$46$re * N[(x$46$re / y$46$im), $MachinePrecision]), $MachinePrecision] / y$46$im), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, 2.8e+99], N[(N[(x$46$re / y$46$re), $MachinePrecision] + N[(N[(y$46$im / y$46$re), $MachinePrecision] * N[(x$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$im / y$46$im), $MachinePrecision] + N[(N[(x$46$re / y$46$im), $MachinePrecision] * N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y.im < -1.05000000000000004e-26

   1. Initial program 54.7%

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

      1. *-un-lft-identity 54.7%

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

      2. add-sqr-sqrt 54.7%

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

      3. times-frac 54.6%

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

      4. hypot-def 54.6%

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

      5. fma-def 54.6%

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

      6. hypot-def 71.5%

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

   3. Applied egg-rr 71.5%

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

   4. Taylor expanded in y.re around 0 78.9%

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

      1. +-commutative 78.9%

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

      2. *-commutative 78.9%

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

      3. unpow2 78.9%

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

      4. times-frac 87.6%

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

   6. Simplified 87.6%

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

      1. associate-*l/ 88.7%

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

   8. Applied egg-rr 88.7%

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

   if -1.05000000000000004e-26 < y.im < 2.8e99

   1. Initial program 77.4%

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

   2. Taylor expanded in y.re around inf 79.4%

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

      1. unpow2 79.4%

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

      2. times-frac 82.7%

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

   4. Simplified 82.7%

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

   if 2.8e99 < y.im

   1. Initial program 39.5%

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

      1. *-un-lft-identity 39.5%

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

      2. add-sqr-sqrt 39.5%

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

      3. times-frac 39.5%

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

      4. hypot-def 39.5%

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

      5. fma-def 39.6%

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

      6. hypot-def 57.4%

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

   3. Applied egg-rr 57.4%

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

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

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

      1. +-commutative 73.5%

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

      2. *-commutative 73.5%

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

      3. unpow2 73.5%

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

      4. times-frac 87.9%

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

   6. Simplified 87.9%

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

4. Final simplification 85.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -1.05 \cdot 10^{-26}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re \cdot \frac{x.re}{y.im}}{y.im}\\ \mathbf{elif}\;y.im \leq 2.8 \cdot 10^{+99}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{y.im}{y.re} \cdot \frac{x.im}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{x.re}{y.im} \cdot \frac{y.re}{y.im}\\ \end{array} \]

Alternative 10: 76.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -1.95 \cdot 10^{-30}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re \cdot \frac{x.re}{y.im}}{y.im}\\ \mathbf{elif}\;y.im \leq 1.92 \cdot 10^{+99}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{x.im \cdot \frac{y.im}{y.re}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{x.re}{y.im} \cdot \frac{y.re}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.im -1.95e-30)
   (+ (/ x.im y.im) (/ (* y.re (/ x.re y.im)) y.im))
   (if (<= y.im 1.92e+99)
     (+ (/ x.re y.re) (/ (* x.im (/ y.im y.re)) y.re))
     (+ (/ x.im y.im) (* (/ x.re y.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 tmp;
	if (y_46_im <= -1.95e-30) {
		tmp = (x_46_im / y_46_im) + ((y_46_re * (x_46_re / y_46_im)) / y_46_im);
	} else if (y_46_im <= 1.92e+99) {
		tmp = (x_46_re / y_46_re) + ((x_46_im * (y_46_im / y_46_re)) / y_46_re);
	} else {
		tmp = (x_46_im / y_46_im) + ((x_46_re / y_46_im) * (y_46_re / y_46_im));
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if (y_46im <= (-1.95d-30)) then
        tmp = (x_46im / y_46im) + ((y_46re * (x_46re / y_46im)) / y_46im)
    else if (y_46im <= 1.92d+99) then
        tmp = (x_46re / y_46re) + ((x_46im * (y_46im / y_46re)) / y_46re)
    else
        tmp = (x_46im / y_46im) + ((x_46re / y_46im) * (y_46re / y_46im))
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_im <= -1.95e-30) {
		tmp = (x_46_im / y_46_im) + ((y_46_re * (x_46_re / y_46_im)) / y_46_im);
	} else if (y_46_im <= 1.92e+99) {
		tmp = (x_46_re / y_46_re) + ((x_46_im * (y_46_im / y_46_re)) / y_46_re);
	} else {
		tmp = (x_46_im / y_46_im) + ((x_46_re / y_46_im) * (y_46_re / y_46_im));
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if y_46_im <= -1.95e-30:
		tmp = (x_46_im / y_46_im) + ((y_46_re * (x_46_re / y_46_im)) / y_46_im)
	elif y_46_im <= 1.92e+99:
		tmp = (x_46_re / y_46_re) + ((x_46_im * (y_46_im / y_46_re)) / y_46_re)
	else:
		tmp = (x_46_im / y_46_im) + ((x_46_re / y_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)
	tmp = 0.0
	if (y_46_im <= -1.95e-30)
		tmp = Float64(Float64(x_46_im / y_46_im) + Float64(Float64(y_46_re * Float64(x_46_re / y_46_im)) / y_46_im));
	elseif (y_46_im <= 1.92e+99)
		tmp = Float64(Float64(x_46_re / y_46_re) + Float64(Float64(x_46_im * Float64(y_46_im / y_46_re)) / y_46_re));
	else
		tmp = Float64(Float64(x_46_im / y_46_im) + Float64(Float64(x_46_re / y_46_im) * Float64(y_46_re / y_46_im)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if (y_46_im <= -1.95e-30)
		tmp = (x_46_im / y_46_im) + ((y_46_re * (x_46_re / y_46_im)) / y_46_im);
	elseif (y_46_im <= 1.92e+99)
		tmp = (x_46_re / y_46_re) + ((x_46_im * (y_46_im / y_46_re)) / y_46_re);
	else
		tmp = (x_46_im / y_46_im) + ((x_46_re / y_46_im) * (y_46_re / y_46_im));
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[y$46$im, -1.95e-30], N[(N[(x$46$im / y$46$im), $MachinePrecision] + N[(N[(y$46$re * N[(x$46$re / y$46$im), $MachinePrecision]), $MachinePrecision] / y$46$im), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, 1.92e+99], N[(N[(x$46$re / y$46$re), $MachinePrecision] + N[(N[(x$46$im * N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$im / y$46$im), $MachinePrecision] + N[(N[(x$46$re / y$46$im), $MachinePrecision] * N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y.im < -1.9500000000000002e-30

   1. Initial program 54.7%

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

      1. *-un-lft-identity 54.7%

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

      2. add-sqr-sqrt 54.7%

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

      3. times-frac 54.6%

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

      4. hypot-def 54.6%

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

      5. fma-def 54.6%

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

      6. hypot-def 71.5%

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

   3. Applied egg-rr 71.5%

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

   4. Taylor expanded in y.re around 0 78.9%

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

      1. +-commutative 78.9%

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

      2. *-commutative 78.9%

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

      3. unpow2 78.9%

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

      4. times-frac 87.6%

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

   6. Simplified 87.6%

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

      1. associate-*l/ 88.7%

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

   8. Applied egg-rr 88.7%

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

   if -1.9500000000000002e-30 < y.im < 1.9199999999999999e99

   1. Initial program 77.4%

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

   2. Taylor expanded in y.re around inf 79.4%

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

      1. unpow2 79.4%

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

      2. times-frac 82.7%

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

   4. Simplified 82.7%

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

      1. *-commutative 82.7%

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

      2. associate-*l/ 84.1%

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

   6. Applied egg-rr 84.1%

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

   if 1.9199999999999999e99 < y.im

   1. Initial program 39.5%

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

      1. *-un-lft-identity 39.5%

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

      2. add-sqr-sqrt 39.5%

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

      3. times-frac 39.5%

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

      4. hypot-def 39.5%

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

      5. fma-def 39.6%

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

      6. hypot-def 57.4%

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

   3. Applied egg-rr 57.4%

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

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

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

      1. +-commutative 73.5%

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

      2. *-commutative 73.5%

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

      3. unpow2 73.5%

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

      4. times-frac 87.9%

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

   6. Simplified 87.9%

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

4. Final simplification 86.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -1.95 \cdot 10^{-30}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re \cdot \frac{x.re}{y.im}}{y.im}\\ \mathbf{elif}\;y.im \leq 1.92 \cdot 10^{+99}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{x.im \cdot \frac{y.im}{y.re}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{x.re}{y.im} \cdot \frac{y.re}{y.im}\\ \end{array} \]

Alternative 11: 62.8% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -4.8 \cdot 10^{-87}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{elif}\;y.im \leq 1.92 \cdot 10^{+99}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.im -4.8e-87)
   (/ x.im y.im)
   (if (<= y.im 1.92e+99) (/ 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 <= -4.8e-87) {
		tmp = x_46_im / y_46_im;
	} else if (y_46_im <= 1.92e+99) {
		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 <= (-4.8d-87)) then
        tmp = x_46im / y_46im
    else if (y_46im <= 1.92d+99) 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 <= -4.8e-87) {
		tmp = x_46_im / y_46_im;
	} else if (y_46_im <= 1.92e+99) {
		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 <= -4.8e-87:
		tmp = x_46_im / y_46_im
	elif y_46_im <= 1.92e+99:
		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 <= -4.8e-87)
		tmp = Float64(x_46_im / y_46_im);
	elseif (y_46_im <= 1.92e+99)
		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 <= -4.8e-87)
		tmp = x_46_im / y_46_im;
	elseif (y_46_im <= 1.92e+99)
		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, -4.8e-87], N[(x$46$im / y$46$im), $MachinePrecision], If[LessEqual[y$46$im, 1.92e+99], 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 < -4.7999999999999999e-87 or 1.9199999999999999e99 < y.im

   1. Initial program 52.5%

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

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

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

   if -4.7999999999999999e-87 < y.im < 1.9199999999999999e99

   1. Initial program 75.6%

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

   2. Taylor expanded in y.re around inf 70.5%

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

4. Final simplification 69.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -4.8 \cdot 10^{-87}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{elif}\;y.im \leq 1.92 \cdot 10^{+99}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \end{array} \]

Alternative 12: 44.1% accurate, 5.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 63.1%

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

2. Taylor expanded in y.re around 0 47.1%

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

3. Final simplification 47.1%

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

Reproduce

herbie shell --seed 2023257 
(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))))