_divideComplex, real part

Percentage Accurate: 61.8% → 83.3%

Time: 10.4s

Alternatives: 11

Speedup: 1.3×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 61.8% 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: 83.3% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y.im}{\frac{y.im}{\frac{x.im}{y.im}} + y.re \cdot \frac{y.re}{x.im}} + \frac{x.re \cdot y.re}{y.re \cdot y.re + y.im \cdot y.im}\\ t_1 := \frac{y.im}{\frac{y.re}{x.im}}\\ \mathbf{if}\;y.re \leq -5.8 \cdot 10^{+110}:\\ \;\;\;\;\frac{\left(-x.re\right) - t_1}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.re \leq -5.5 \cdot 10^{-102}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq 2.2 \cdot 10^{-75}:\\ \;\;\;\;\frac{1}{y.im} \cdot \left(x.im + \frac{x.re}{\frac{y.im}{y.re}}\right)\\ \mathbf{elif}\;y.re \leq 6.6 \cdot 10^{+41}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq 4.4 \cdot 10^{+90}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re}{y.im} \cdot \frac{x.re}{y.im}\\ \mathbf{elif}\;y.re \leq 2.4 \cdot 10^{+157}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{t_1 + \mathsf{fma}\left(-0.5, \frac{x.re}{\frac{y.re}{y.im} \cdot \frac{y.re}{y.im}}, x.re\right)}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0
         (+
          (/ y.im (+ (/ y.im (/ x.im y.im)) (* y.re (/ y.re x.im))))
          (/ (* x.re y.re) (+ (* y.re y.re) (* y.im y.im)))))
        (t_1 (/ y.im (/ y.re x.im))))
   (if (<= y.re -5.8e+110)
     (/ (- (- x.re) t_1) (hypot y.re y.im))
     (if (<= y.re -5.5e-102)
       t_0
       (if (<= y.re 2.2e-75)
         (* (/ 1.0 y.im) (+ x.im (/ x.re (/ y.im y.re))))
         (if (<= y.re 6.6e+41)
           t_0
           (if (<= y.re 4.4e+90)
             (+ (/ x.im y.im) (* (/ y.re y.im) (/ x.re y.im)))
             (if (<= y.re 2.4e+157)
               t_0
               (/
                (+
                 t_1
                 (fma -0.5 (/ x.re (* (/ y.re y.im) (/ y.re y.im))) x.re))
                (hypot y.re y.im))))))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = (y_46_im / ((y_46_im / (x_46_im / y_46_im)) + (y_46_re * (y_46_re / x_46_im)))) + ((x_46_re * y_46_re) / ((y_46_re * y_46_re) + (y_46_im * y_46_im)));
	double t_1 = y_46_im / (y_46_re / x_46_im);
	double tmp;
	if (y_46_re <= -5.8e+110) {
		tmp = (-x_46_re - t_1) / hypot(y_46_re, y_46_im);
	} else if (y_46_re <= -5.5e-102) {
		tmp = t_0;
	} else if (y_46_re <= 2.2e-75) {
		tmp = (1.0 / y_46_im) * (x_46_im + (x_46_re / (y_46_im / y_46_re)));
	} else if (y_46_re <= 6.6e+41) {
		tmp = t_0;
	} else if (y_46_re <= 4.4e+90) {
		tmp = (x_46_im / y_46_im) + ((y_46_re / y_46_im) * (x_46_re / y_46_im));
	} else if (y_46_re <= 2.4e+157) {
		tmp = t_0;
	} else {
		tmp = (t_1 + fma(-0.5, (x_46_re / ((y_46_re / y_46_im) * (y_46_re / y_46_im))), x_46_re)) / hypot(y_46_re, y_46_im);
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(Float64(y_46_im / Float64(Float64(y_46_im / Float64(x_46_im / y_46_im)) + Float64(y_46_re * Float64(y_46_re / x_46_im)))) + Float64(Float64(x_46_re * y_46_re) / Float64(Float64(y_46_re * y_46_re) + Float64(y_46_im * y_46_im))))
	t_1 = Float64(y_46_im / Float64(y_46_re / x_46_im))
	tmp = 0.0
	if (y_46_re <= -5.8e+110)
		tmp = Float64(Float64(Float64(-x_46_re) - t_1) / hypot(y_46_re, y_46_im));
	elseif (y_46_re <= -5.5e-102)
		tmp = t_0;
	elseif (y_46_re <= 2.2e-75)
		tmp = Float64(Float64(1.0 / y_46_im) * Float64(x_46_im + Float64(x_46_re / Float64(y_46_im / y_46_re))));
	elseif (y_46_re <= 6.6e+41)
		tmp = t_0;
	elseif (y_46_re <= 4.4e+90)
		tmp = Float64(Float64(x_46_im / y_46_im) + Float64(Float64(y_46_re / y_46_im) * Float64(x_46_re / y_46_im)));
	elseif (y_46_re <= 2.4e+157)
		tmp = t_0;
	else
		tmp = Float64(Float64(t_1 + fma(-0.5, Float64(x_46_re / Float64(Float64(y_46_re / y_46_im) * Float64(y_46_re / y_46_im))), x_46_re)) / hypot(y_46_re, y_46_im));
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[(y$46$im / N[(N[(y$46$im / N[(x$46$im / y$46$im), $MachinePrecision]), $MachinePrecision] + N[(y$46$re * N[(y$46$re / x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(x$46$re * y$46$re), $MachinePrecision] / N[(N[(y$46$re * y$46$re), $MachinePrecision] + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(y$46$im / N[(y$46$re / x$46$im), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$re, -5.8e+110], N[(N[((-x$46$re) - t$95$1), $MachinePrecision] / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, -5.5e-102], t$95$0, If[LessEqual[y$46$re, 2.2e-75], N[(N[(1.0 / y$46$im), $MachinePrecision] * N[(x$46$im + N[(x$46$re / N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 6.6e+41], t$95$0, If[LessEqual[y$46$re, 4.4e+90], N[(N[(x$46$im / y$46$im), $MachinePrecision] + N[(N[(y$46$re / y$46$im), $MachinePrecision] * N[(x$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 2.4e+157], t$95$0, N[(N[(t$95$1 + N[(-0.5 * N[(x$46$re / N[(N[(y$46$re / y$46$im), $MachinePrecision] * N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + x$46$re), $MachinePrecision]), $MachinePrecision] / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if y.re < -5.7999999999999999e110

   1. Initial program 29.3%

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

      1. *-un-lft-identity 29.3%

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

      2. add-sqr-sqrt 29.3%

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

      3. times-frac 29.3%

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

      4. hypot-def 29.3%

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

      5. fma-def 29.3%

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

      6. hypot-def 51.4%

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

   3. Applied egg-rr 51.4%

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

      1. associate-*l/ 51.5%

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

      2. *-un-lft-identity 51.5%

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

   5. Applied egg-rr 51.5%

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

   6. Taylor expanded in y.re around -inf 80.1%

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

      1. +-commutative 80.1%

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

      2. mul-1-neg 80.1%

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

      3. unsub-neg 80.1%

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

      4. mul-1-neg 80.1%

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

      5. associate-/l* 86.4%

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

   8. Simplified 86.4%

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

   if -5.7999999999999999e110 < y.re < -5.4999999999999997e-102 or 2.20000000000000005e-75 < y.re < 6.6000000000000001e41 or 4.39999999999999981e90 < y.re < 2.4e157

   1. Initial program 73.3%

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

      1. *-un-lft-identity 73.3%

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

      2. add-sqr-sqrt 73.3%

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

      3. times-frac 73.3%

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

      4. hypot-def 73.3%

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

      5. fma-def 73.3%

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

      6. hypot-def 79.7%

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

   3. Applied egg-rr 79.7%

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

   4. Taylor expanded in x.re around 0 73.4%

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

      1. associate-/l* 78.7%

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

      2. unpow2 78.7%

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

      3. unpow2 78.7%

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

      4. *-commutative 78.7%

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

      5. unpow2 78.7%

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

      6. unpow2 78.7%

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

   6. Simplified 78.7%

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

   7. Taylor expanded in y.re around 0 78.7%

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

      1. unpow2 78.7%

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

      2. associate-/l* 89.8%

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

      3. unpow2 89.8%

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

      4. associate-/l* 91.1%

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

      5. associate-/r/ 91.1%

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

   9. Simplified 91.1%

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

   if -5.4999999999999997e-102 < y.re < 2.20000000000000005e-75

   1. Initial program 66.9%

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

      1. *-un-lft-identity 66.9%

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

      2. add-sqr-sqrt 66.9%

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

      3. times-frac 67.0%

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

      4. hypot-def 67.0%

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

      5. fma-def 67.0%

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

      6. hypot-def 82.3%

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

   3. Applied egg-rr 82.3%

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

   4. Taylor expanded in y.re around 0 55.2%

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

      1. +-commutative 55.2%

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

      2. associate-/l* 56.1%

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

   6. Simplified 56.1%

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

   7. Taylor expanded in y.re around 0 91.6%

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

   if 6.6000000000000001e41 < y.re < 4.39999999999999981e90

   1. Initial program 27.2%

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

   2. Taylor expanded in y.re around 0 43.8%

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

      1. +-commutative 43.8%

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

      2. *-commutative 43.8%

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

      3. unpow2 43.8%

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

      4. times-frac 75.5%

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

   4. Simplified 75.5%

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

   if 2.4e157 < y.re

   1. Initial program 34.2%

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

      1. *-un-lft-identity 34.2%

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

      2. add-sqr-sqrt 34.2%

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

      3. times-frac 34.2%

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

      4. hypot-def 34.2%

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

      5. fma-def 34.2%

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

      6. hypot-def 68.7%

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

   3. Applied egg-rr 68.7%

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

      1. associate-*l/ 68.8%

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

      2. *-un-lft-identity 68.8%

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

   5. Applied egg-rr 68.8%

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

   6. Taylor expanded in y.re around inf 67.9%

      \[\leadsto \frac{\color{blue}{-0.5 \cdot \frac{x.re \cdot {y.im}^{2}}{{y.re}^{2}} + \left(x.re + \frac{y.im \cdot x.im}{y.re}\right)}}{\mathsf{hypot}\left(y.re, y.im\right)} \]
   7. Step-by-step derivation

      1. associate-+r+ 67.9%

         \[\leadsto \frac{\color{blue}{\left(-0.5 \cdot \frac{x.re \cdot {y.im}^{2}}{{y.re}^{2}} + x.re\right) + \frac{y.im \cdot x.im}{y.re}}}{\mathsf{hypot}\left(y.re, y.im\right)} \]

      2. fma-def 67.9%

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(-0.5, \frac{x.re \cdot {y.im}^{2}}{{y.re}^{2}}, x.re\right)} + \frac{y.im \cdot x.im}{y.re}}{\mathsf{hypot}\left(y.re, y.im\right)} \]

      3. unpow2 67.9%

         \[\leadsto \frac{\mathsf{fma}\left(-0.5, \frac{x.re \cdot \color{blue}{\left(y.im \cdot y.im\right)}}{{y.re}^{2}}, x.re\right) + \frac{y.im \cdot x.im}{y.re}}{\mathsf{hypot}\left(y.re, y.im\right)} \]

      4. associate-/l* 67.9%

         \[\leadsto \frac{\mathsf{fma}\left(-0.5, \color{blue}{\frac{x.re}{\frac{{y.re}^{2}}{y.im \cdot y.im}}}, x.re\right) + \frac{y.im \cdot x.im}{y.re}}{\mathsf{hypot}\left(y.re, y.im\right)} \]

      5. unpow2 67.9%

         \[\leadsto \frac{\mathsf{fma}\left(-0.5, \frac{x.re}{\frac{\color{blue}{y.re \cdot y.re}}{y.im \cdot y.im}}, x.re\right) + \frac{y.im \cdot x.im}{y.re}}{\mathsf{hypot}\left(y.re, y.im\right)} \]

      6. times-frac 90.1%

         \[\leadsto \frac{\mathsf{fma}\left(-0.5, \frac{x.re}{\color{blue}{\frac{y.re}{y.im} \cdot \frac{y.re}{y.im}}}, x.re\right) + \frac{y.im \cdot x.im}{y.re}}{\mathsf{hypot}\left(y.re, y.im\right)} \]

      7. associate-/l* 90.3%

         \[\leadsto \frac{\mathsf{fma}\left(-0.5, \frac{x.re}{\frac{y.re}{y.im} \cdot \frac{y.re}{y.im}}, x.re\right) + \color{blue}{\frac{y.im}{\frac{y.re}{x.im}}}}{\mathsf{hypot}\left(y.re, y.im\right)} \]

   8. Simplified 90.3%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(-0.5, \frac{x.re}{\frac{y.re}{y.im} \cdot \frac{y.re}{y.im}}, x.re\right) + \frac{y.im}{\frac{y.re}{x.im}}}}{\mathsf{hypot}\left(y.re, y.im\right)} \]
3. Recombined 5 regimes into one program.

4. Final simplification 89.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -5.8 \cdot 10^{+110}:\\ \;\;\;\;\frac{\left(-x.re\right) - \frac{y.im}{\frac{y.re}{x.im}}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.re \leq -5.5 \cdot 10^{-102}:\\ \;\;\;\;\frac{y.im}{\frac{y.im}{\frac{x.im}{y.im}} + y.re \cdot \frac{y.re}{x.im}} + \frac{x.re \cdot y.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.re \leq 2.2 \cdot 10^{-75}:\\ \;\;\;\;\frac{1}{y.im} \cdot \left(x.im + \frac{x.re}{\frac{y.im}{y.re}}\right)\\ \mathbf{elif}\;y.re \leq 6.6 \cdot 10^{+41}:\\ \;\;\;\;\frac{y.im}{\frac{y.im}{\frac{x.im}{y.im}} + y.re \cdot \frac{y.re}{x.im}} + \frac{x.re \cdot y.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.re \leq 4.4 \cdot 10^{+90}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re}{y.im} \cdot \frac{x.re}{y.im}\\ \mathbf{elif}\;y.re \leq 2.4 \cdot 10^{+157}:\\ \;\;\;\;\frac{y.im}{\frac{y.im}{\frac{x.im}{y.im}} + y.re \cdot \frac{y.re}{x.im}} + \frac{x.re \cdot y.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y.im}{\frac{y.re}{x.im}} + \mathsf{fma}\left(-0.5, \frac{x.re}{\frac{y.re}{y.im} \cdot \frac{y.re}{y.im}}, x.re\right)}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \end{array} \]

Alternative 2: 85.6% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{if}\;t_0 \leq -2 \cdot 10^{+281} \lor \neg \left(t_0 \leq 10^{+299}\right):\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re}{y.im} \cdot \frac{x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}{\mathsf{hypot}\left(y.re, y.im\right)}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0
         (/ (+ (* x.re y.re) (* x.im y.im)) (+ (* y.re y.re) (* y.im y.im)))))
   (if (or (<= t_0 -2e+281) (not (<= t_0 1e+299)))
     (+ (/ x.im y.im) (* (/ y.re y.im) (/ x.re y.im)))
     (/
      (/ (fma x.re y.re (* x.im y.im)) (hypot y.re y.im))
      (hypot y.re y.im)))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = ((x_46_re * y_46_re) + (x_46_im * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	double tmp;
	if ((t_0 <= -2e+281) || !(t_0 <= 1e+299)) {
		tmp = (x_46_im / y_46_im) + ((y_46_re / y_46_im) * (x_46_re / y_46_im));
	} else {
		tmp = (fma(x_46_re, y_46_re, (x_46_im * y_46_im)) / hypot(y_46_re, y_46_im)) / hypot(y_46_re, y_46_im);
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(Float64(Float64(x_46_re * y_46_re) + Float64(x_46_im * y_46_im)) / Float64(Float64(y_46_re * y_46_re) + Float64(y_46_im * y_46_im)))
	tmp = 0.0
	if ((t_0 <= -2e+281) || !(t_0 <= 1e+299))
		tmp = Float64(Float64(x_46_im / y_46_im) + Float64(Float64(y_46_re / y_46_im) * Float64(x_46_re / y_46_im)));
	else
		tmp = Float64(Float64(fma(x_46_re, y_46_re, Float64(x_46_im * y_46_im)) / hypot(y_46_re, y_46_im)) / hypot(y_46_re, y_46_im));
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[(N[(x$46$re * y$46$re), $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[Or[LessEqual[t$95$0, -2e+281], N[Not[LessEqual[t$95$0, 1e+299]], $MachinePrecision]], N[(N[(x$46$im / y$46$im), $MachinePrecision] + N[(N[(y$46$re / y$46$im), $MachinePrecision] * N[(x$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x$46$re * y$46$re + N[(x$46$im * y$46$im), $MachinePrecision]), $MachinePrecision] / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision] / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision]]]

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))) < -2.0000000000000001e281 or 1.0000000000000001e299 < (/.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 15.2%

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

   2. Taylor expanded in y.re around 0 52.8%

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

      1. +-commutative 52.8%

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

      2. *-commutative 52.8%

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

      3. unpow2 52.8%

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

      4. times-frac 65.4%

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

   4. Simplified 65.4%

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

   if -2.0000000000000001e281 < (/.f64 (+.f64 (*.f64 x.re y.re) (*.f64 x.im y.im)) (+.f64 (*.f64 y.re y.re) (*.f64 y.im y.im))) < 1.0000000000000001e299

   1. Initial program 79.5%

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

      1. *-un-lft-identity 79.5%

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

      2. add-sqr-sqrt 79.5%

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

      3. times-frac 79.5%

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

      4. hypot-def 79.5%

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

      5. fma-def 79.5%

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

      6. hypot-def 98.5%

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

   3. Applied egg-rr 98.5%

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

      1. associate-*l/ 98.7%

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

      2. *-un-lft-identity 98.7%

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

   5. Applied egg-rr 98.7%

      \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}{\mathsf{hypot}\left(y.re, y.im\right)}}{\mathsf{hypot}\left(y.re, y.im\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 87.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \leq -2 \cdot 10^{+281} \lor \neg \left(\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \leq 10^{+299}\right):\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re}{y.im} \cdot \frac{x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}{\mathsf{hypot}\left(y.re, y.im\right)}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \end{array} \]

Alternative 3: 83.4% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y.im}{\frac{y.im}{\frac{x.im}{y.im}} + y.re \cdot \frac{y.re}{x.im}} + \frac{x.re \cdot y.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{if}\;y.re \leq -3.1 \cdot 10^{+109}:\\ \;\;\;\;\frac{\left(-x.re\right) - \frac{y.im}{\frac{y.re}{x.im}}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.re \leq -4.4 \cdot 10^{-102}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq 5 \cdot 10^{-71}:\\ \;\;\;\;\frac{1}{y.im} \cdot \left(x.im + \frac{x.re}{\frac{y.im}{y.re}}\right)\\ \mathbf{elif}\;y.re \leq 6.6 \cdot 10^{+41}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq 3.3 \cdot 10^{+89}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re}{y.im} \cdot \frac{x.re}{y.im}\\ \mathbf{elif}\;y.re \leq 2.4 \cdot 10^{+157}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{y.im}{y.re} \cdot \frac{x.im}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0
         (+
          (/ y.im (+ (/ y.im (/ x.im y.im)) (* y.re (/ y.re x.im))))
          (/ (* x.re y.re) (+ (* y.re y.re) (* y.im y.im))))))
   (if (<= y.re -3.1e+109)
     (/ (- (- x.re) (/ y.im (/ y.re x.im))) (hypot y.re y.im))
     (if (<= y.re -4.4e-102)
       t_0
       (if (<= y.re 5e-71)
         (* (/ 1.0 y.im) (+ x.im (/ x.re (/ y.im y.re))))
         (if (<= y.re 6.6e+41)
           t_0
           (if (<= y.re 3.3e+89)
             (+ (/ x.im y.im) (* (/ y.re y.im) (/ x.re y.im)))
             (if (<= y.re 2.4e+157)
               t_0
               (+ (/ x.re y.re) (* (/ y.im y.re) (/ x.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 = (y_46_im / ((y_46_im / (x_46_im / y_46_im)) + (y_46_re * (y_46_re / x_46_im)))) + ((x_46_re * y_46_re) / ((y_46_re * y_46_re) + (y_46_im * y_46_im)));
	double tmp;
	if (y_46_re <= -3.1e+109) {
		tmp = (-x_46_re - (y_46_im / (y_46_re / x_46_im))) / hypot(y_46_re, y_46_im);
	} else if (y_46_re <= -4.4e-102) {
		tmp = t_0;
	} else if (y_46_re <= 5e-71) {
		tmp = (1.0 / y_46_im) * (x_46_im + (x_46_re / (y_46_im / y_46_re)));
	} else if (y_46_re <= 6.6e+41) {
		tmp = t_0;
	} else if (y_46_re <= 3.3e+89) {
		tmp = (x_46_im / y_46_im) + ((y_46_re / y_46_im) * (x_46_re / y_46_im));
	} else if (y_46_re <= 2.4e+157) {
		tmp = t_0;
	} else {
		tmp = (x_46_re / y_46_re) + ((y_46_im / y_46_re) * (x_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 t_0 = (y_46_im / ((y_46_im / (x_46_im / y_46_im)) + (y_46_re * (y_46_re / x_46_im)))) + ((x_46_re * y_46_re) / ((y_46_re * y_46_re) + (y_46_im * y_46_im)));
	double tmp;
	if (y_46_re <= -3.1e+109) {
		tmp = (-x_46_re - (y_46_im / (y_46_re / x_46_im))) / Math.hypot(y_46_re, y_46_im);
	} else if (y_46_re <= -4.4e-102) {
		tmp = t_0;
	} else if (y_46_re <= 5e-71) {
		tmp = (1.0 / y_46_im) * (x_46_im + (x_46_re / (y_46_im / y_46_re)));
	} else if (y_46_re <= 6.6e+41) {
		tmp = t_0;
	} else if (y_46_re <= 3.3e+89) {
		tmp = (x_46_im / y_46_im) + ((y_46_re / y_46_im) * (x_46_re / y_46_im));
	} else if (y_46_re <= 2.4e+157) {
		tmp = t_0;
	} else {
		tmp = (x_46_re / y_46_re) + ((y_46_im / y_46_re) * (x_46_im / y_46_re));
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = (y_46_im / ((y_46_im / (x_46_im / y_46_im)) + (y_46_re * (y_46_re / x_46_im)))) + ((x_46_re * y_46_re) / ((y_46_re * y_46_re) + (y_46_im * y_46_im)))
	tmp = 0
	if y_46_re <= -3.1e+109:
		tmp = (-x_46_re - (y_46_im / (y_46_re / x_46_im))) / math.hypot(y_46_re, y_46_im)
	elif y_46_re <= -4.4e-102:
		tmp = t_0
	elif y_46_re <= 5e-71:
		tmp = (1.0 / y_46_im) * (x_46_im + (x_46_re / (y_46_im / y_46_re)))
	elif y_46_re <= 6.6e+41:
		tmp = t_0
	elif y_46_re <= 3.3e+89:
		tmp = (x_46_im / y_46_im) + ((y_46_re / y_46_im) * (x_46_re / y_46_im))
	elif y_46_re <= 2.4e+157:
		tmp = t_0
	else:
		tmp = (x_46_re / y_46_re) + ((y_46_im / y_46_re) * (x_46_im / y_46_re))
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(Float64(y_46_im / Float64(Float64(y_46_im / Float64(x_46_im / y_46_im)) + Float64(y_46_re * Float64(y_46_re / x_46_im)))) + Float64(Float64(x_46_re * y_46_re) / Float64(Float64(y_46_re * y_46_re) + Float64(y_46_im * y_46_im))))
	tmp = 0.0
	if (y_46_re <= -3.1e+109)
		tmp = Float64(Float64(Float64(-x_46_re) - Float64(y_46_im / Float64(y_46_re / x_46_im))) / hypot(y_46_re, y_46_im));
	elseif (y_46_re <= -4.4e-102)
		tmp = t_0;
	elseif (y_46_re <= 5e-71)
		tmp = Float64(Float64(1.0 / y_46_im) * Float64(x_46_im + Float64(x_46_re / Float64(y_46_im / y_46_re))));
	elseif (y_46_re <= 6.6e+41)
		tmp = t_0;
	elseif (y_46_re <= 3.3e+89)
		tmp = Float64(Float64(x_46_im / y_46_im) + Float64(Float64(y_46_re / y_46_im) * Float64(x_46_re / y_46_im)));
	elseif (y_46_re <= 2.4e+157)
		tmp = t_0;
	else
		tmp = Float64(Float64(x_46_re / y_46_re) + Float64(Float64(y_46_im / y_46_re) * Float64(x_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)
	t_0 = (y_46_im / ((y_46_im / (x_46_im / y_46_im)) + (y_46_re * (y_46_re / x_46_im)))) + ((x_46_re * y_46_re) / ((y_46_re * y_46_re) + (y_46_im * y_46_im)));
	tmp = 0.0;
	if (y_46_re <= -3.1e+109)
		tmp = (-x_46_re - (y_46_im / (y_46_re / x_46_im))) / hypot(y_46_re, y_46_im);
	elseif (y_46_re <= -4.4e-102)
		tmp = t_0;
	elseif (y_46_re <= 5e-71)
		tmp = (1.0 / y_46_im) * (x_46_im + (x_46_re / (y_46_im / y_46_re)));
	elseif (y_46_re <= 6.6e+41)
		tmp = t_0;
	elseif (y_46_re <= 3.3e+89)
		tmp = (x_46_im / y_46_im) + ((y_46_re / y_46_im) * (x_46_re / y_46_im));
	elseif (y_46_re <= 2.4e+157)
		tmp = t_0;
	else
		tmp = (x_46_re / y_46_re) + ((y_46_im / y_46_re) * (x_46_im / y_46_re));
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[(y$46$im / N[(N[(y$46$im / N[(x$46$im / y$46$im), $MachinePrecision]), $MachinePrecision] + N[(y$46$re * N[(y$46$re / x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(x$46$re * y$46$re), $MachinePrecision] / N[(N[(y$46$re * y$46$re), $MachinePrecision] + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$re, -3.1e+109], N[(N[((-x$46$re) - N[(y$46$im / N[(y$46$re / x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, -4.4e-102], t$95$0, If[LessEqual[y$46$re, 5e-71], N[(N[(1.0 / y$46$im), $MachinePrecision] * N[(x$46$im + N[(x$46$re / N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 6.6e+41], t$95$0, If[LessEqual[y$46$re, 3.3e+89], N[(N[(x$46$im / y$46$im), $MachinePrecision] + N[(N[(y$46$re / y$46$im), $MachinePrecision] * N[(x$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 2.4e+157], t$95$0, 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]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if y.re < -3.09999999999999992e109

   1. Initial program 29.3%

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

      1. *-un-lft-identity 29.3%

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

      2. add-sqr-sqrt 29.3%

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

      3. times-frac 29.3%

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

      4. hypot-def 29.3%

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

      5. fma-def 29.3%

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

      6. hypot-def 51.4%

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

   3. Applied egg-rr 51.4%

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

      1. associate-*l/ 51.5%

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

      2. *-un-lft-identity 51.5%

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

   5. Applied egg-rr 51.5%

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

   6. Taylor expanded in y.re around -inf 80.1%

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

      1. +-commutative 80.1%

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

      2. mul-1-neg 80.1%

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

      3. unsub-neg 80.1%

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

      4. mul-1-neg 80.1%

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

      5. associate-/l* 86.4%

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

   8. Simplified 86.4%

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

   if -3.09999999999999992e109 < y.re < -4.40000000000000026e-102 or 4.99999999999999998e-71 < y.re < 6.6000000000000001e41 or 3.29999999999999974e89 < y.re < 2.4e157

   1. Initial program 73.3%

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

      1. *-un-lft-identity 73.3%

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

      2. add-sqr-sqrt 73.3%

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

      3. times-frac 73.3%

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

      4. hypot-def 73.3%

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

      5. fma-def 73.3%

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

      6. hypot-def 79.7%

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

   3. Applied egg-rr 79.7%

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

   4. Taylor expanded in x.re around 0 73.4%

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

      1. associate-/l* 78.7%

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

      2. unpow2 78.7%

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

      3. unpow2 78.7%

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

      4. *-commutative 78.7%

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

      5. unpow2 78.7%

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

      6. unpow2 78.7%

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

   6. Simplified 78.7%

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

   7. Taylor expanded in y.re around 0 78.7%

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

      1. unpow2 78.7%

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

      2. associate-/l* 89.8%

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

      3. unpow2 89.8%

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

      4. associate-/l* 91.1%

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

      5. associate-/r/ 91.1%

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

   9. Simplified 91.1%

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

   if -4.40000000000000026e-102 < y.re < 4.99999999999999998e-71

   1. Initial program 66.9%

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

      1. *-un-lft-identity 66.9%

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

      2. add-sqr-sqrt 66.9%

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

      3. times-frac 67.0%

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

      4. hypot-def 67.0%

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

      5. fma-def 67.0%

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

      6. hypot-def 82.3%

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

   3. Applied egg-rr 82.3%

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

   4. Taylor expanded in y.re around 0 55.2%

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

      1. +-commutative 55.2%

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

      2. associate-/l* 56.1%

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

   6. Simplified 56.1%

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

   7. Taylor expanded in y.re around 0 91.6%

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

   if 6.6000000000000001e41 < y.re < 3.29999999999999974e89

   1. Initial program 27.2%

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

   2. Taylor expanded in y.re around 0 43.8%

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

      1. +-commutative 43.8%

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

      2. *-commutative 43.8%

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

      3. unpow2 43.8%

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

      4. times-frac 75.5%

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

   4. Simplified 75.5%

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

   if 2.4e157 < y.re

   1. Initial program 34.2%

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

   2. Taylor expanded in y.re around inf 85.3%

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

      1. unpow2 85.3%

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

      2. times-frac 89.0%

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

   4. Simplified 89.0%

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

4. Final simplification 89.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -3.1 \cdot 10^{+109}:\\ \;\;\;\;\frac{\left(-x.re\right) - \frac{y.im}{\frac{y.re}{x.im}}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.re \leq -4.4 \cdot 10^{-102}:\\ \;\;\;\;\frac{y.im}{\frac{y.im}{\frac{x.im}{y.im}} + y.re \cdot \frac{y.re}{x.im}} + \frac{x.re \cdot y.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.re \leq 5 \cdot 10^{-71}:\\ \;\;\;\;\frac{1}{y.im} \cdot \left(x.im + \frac{x.re}{\frac{y.im}{y.re}}\right)\\ \mathbf{elif}\;y.re \leq 6.6 \cdot 10^{+41}:\\ \;\;\;\;\frac{y.im}{\frac{y.im}{\frac{x.im}{y.im}} + y.re \cdot \frac{y.re}{x.im}} + \frac{x.re \cdot y.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.re \leq 3.3 \cdot 10^{+89}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re}{y.im} \cdot \frac{x.re}{y.im}\\ \mathbf{elif}\;y.re \leq 2.4 \cdot 10^{+157}:\\ \;\;\;\;\frac{y.im}{\frac{y.im}{\frac{x.im}{y.im}} + y.re \cdot \frac{y.re}{x.im}} + \frac{x.re \cdot y.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{y.im}{y.re} \cdot \frac{x.im}{y.re}\\ \end{array} \]

Alternative 4: 83.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x.re \cdot y.re}{y.re \cdot y.re + y.im \cdot y.im} + \frac{y.im}{y.re \cdot \frac{y.re}{x.im} + y.im \cdot \frac{y.im}{x.im}}\\ \mathbf{if}\;y.re \leq -2.9 \cdot 10^{+109}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{y.im \cdot \frac{x.im}{y.re}}{y.re}\\ \mathbf{elif}\;y.re \leq -6.6 \cdot 10^{-91}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq 3.15 \cdot 10^{-75}:\\ \;\;\;\;\frac{1}{y.im} \cdot \left(x.im + \frac{x.re}{\frac{y.im}{y.re}}\right)\\ \mathbf{elif}\;y.re \leq 6.6 \cdot 10^{+41}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq 3.3 \cdot 10^{+89}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re}{y.im} \cdot \frac{x.re}{y.im}\\ \mathbf{elif}\;y.re \leq 2.4 \cdot 10^{+157}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{y.im}{y.re} \cdot \frac{x.im}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0
         (+
          (/ (* x.re y.re) (+ (* y.re y.re) (* y.im y.im)))
          (/ y.im (+ (* y.re (/ y.re x.im)) (* y.im (/ y.im x.im)))))))
   (if (<= y.re -2.9e+109)
     (+ (/ x.re y.re) (/ (* y.im (/ x.im y.re)) y.re))
     (if (<= y.re -6.6e-91)
       t_0
       (if (<= y.re 3.15e-75)
         (* (/ 1.0 y.im) (+ x.im (/ x.re (/ y.im y.re))))
         (if (<= y.re 6.6e+41)
           t_0
           (if (<= y.re 3.3e+89)
             (+ (/ x.im y.im) (* (/ y.re y.im) (/ x.re y.im)))
             (if (<= y.re 2.4e+157)
               t_0
               (+ (/ x.re y.re) (* (/ y.im y.re) (/ x.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 = ((x_46_re * y_46_re) / ((y_46_re * y_46_re) + (y_46_im * y_46_im))) + (y_46_im / ((y_46_re * (y_46_re / x_46_im)) + (y_46_im * (y_46_im / x_46_im))));
	double tmp;
	if (y_46_re <= -2.9e+109) {
		tmp = (x_46_re / y_46_re) + ((y_46_im * (x_46_im / y_46_re)) / y_46_re);
	} else if (y_46_re <= -6.6e-91) {
		tmp = t_0;
	} else if (y_46_re <= 3.15e-75) {
		tmp = (1.0 / y_46_im) * (x_46_im + (x_46_re / (y_46_im / y_46_re)));
	} else if (y_46_re <= 6.6e+41) {
		tmp = t_0;
	} else if (y_46_re <= 3.3e+89) {
		tmp = (x_46_im / y_46_im) + ((y_46_re / y_46_im) * (x_46_re / y_46_im));
	} else if (y_46_re <= 2.4e+157) {
		tmp = t_0;
	} else {
		tmp = (x_46_re / y_46_re) + ((y_46_im / y_46_re) * (x_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) :: t_0
    real(8) :: tmp
    t_0 = ((x_46re * y_46re) / ((y_46re * y_46re) + (y_46im * y_46im))) + (y_46im / ((y_46re * (y_46re / x_46im)) + (y_46im * (y_46im / x_46im))))
    if (y_46re <= (-2.9d+109)) then
        tmp = (x_46re / y_46re) + ((y_46im * (x_46im / y_46re)) / y_46re)
    else if (y_46re <= (-6.6d-91)) then
        tmp = t_0
    else if (y_46re <= 3.15d-75) then
        tmp = (1.0d0 / y_46im) * (x_46im + (x_46re / (y_46im / y_46re)))
    else if (y_46re <= 6.6d+41) then
        tmp = t_0
    else if (y_46re <= 3.3d+89) then
        tmp = (x_46im / y_46im) + ((y_46re / y_46im) * (x_46re / y_46im))
    else if (y_46re <= 2.4d+157) then
        tmp = t_0
    else
        tmp = (x_46re / y_46re) + ((y_46im / y_46re) * (x_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 t_0 = ((x_46_re * y_46_re) / ((y_46_re * y_46_re) + (y_46_im * y_46_im))) + (y_46_im / ((y_46_re * (y_46_re / x_46_im)) + (y_46_im * (y_46_im / x_46_im))));
	double tmp;
	if (y_46_re <= -2.9e+109) {
		tmp = (x_46_re / y_46_re) + ((y_46_im * (x_46_im / y_46_re)) / y_46_re);
	} else if (y_46_re <= -6.6e-91) {
		tmp = t_0;
	} else if (y_46_re <= 3.15e-75) {
		tmp = (1.0 / y_46_im) * (x_46_im + (x_46_re / (y_46_im / y_46_re)));
	} else if (y_46_re <= 6.6e+41) {
		tmp = t_0;
	} else if (y_46_re <= 3.3e+89) {
		tmp = (x_46_im / y_46_im) + ((y_46_re / y_46_im) * (x_46_re / y_46_im));
	} else if (y_46_re <= 2.4e+157) {
		tmp = t_0;
	} else {
		tmp = (x_46_re / y_46_re) + ((y_46_im / y_46_re) * (x_46_im / y_46_re));
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = ((x_46_re * y_46_re) / ((y_46_re * y_46_re) + (y_46_im * y_46_im))) + (y_46_im / ((y_46_re * (y_46_re / x_46_im)) + (y_46_im * (y_46_im / x_46_im))))
	tmp = 0
	if y_46_re <= -2.9e+109:
		tmp = (x_46_re / y_46_re) + ((y_46_im * (x_46_im / y_46_re)) / y_46_re)
	elif y_46_re <= -6.6e-91:
		tmp = t_0
	elif y_46_re <= 3.15e-75:
		tmp = (1.0 / y_46_im) * (x_46_im + (x_46_re / (y_46_im / y_46_re)))
	elif y_46_re <= 6.6e+41:
		tmp = t_0
	elif y_46_re <= 3.3e+89:
		tmp = (x_46_im / y_46_im) + ((y_46_re / y_46_im) * (x_46_re / y_46_im))
	elif y_46_re <= 2.4e+157:
		tmp = t_0
	else:
		tmp = (x_46_re / y_46_re) + ((y_46_im / y_46_re) * (x_46_im / y_46_re))
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(Float64(Float64(x_46_re * y_46_re) / Float64(Float64(y_46_re * y_46_re) + Float64(y_46_im * y_46_im))) + Float64(y_46_im / Float64(Float64(y_46_re * Float64(y_46_re / x_46_im)) + Float64(y_46_im * Float64(y_46_im / x_46_im)))))
	tmp = 0.0
	if (y_46_re <= -2.9e+109)
		tmp = Float64(Float64(x_46_re / y_46_re) + Float64(Float64(y_46_im * Float64(x_46_im / y_46_re)) / y_46_re));
	elseif (y_46_re <= -6.6e-91)
		tmp = t_0;
	elseif (y_46_re <= 3.15e-75)
		tmp = Float64(Float64(1.0 / y_46_im) * Float64(x_46_im + Float64(x_46_re / Float64(y_46_im / y_46_re))));
	elseif (y_46_re <= 6.6e+41)
		tmp = t_0;
	elseif (y_46_re <= 3.3e+89)
		tmp = Float64(Float64(x_46_im / y_46_im) + Float64(Float64(y_46_re / y_46_im) * Float64(x_46_re / y_46_im)));
	elseif (y_46_re <= 2.4e+157)
		tmp = t_0;
	else
		tmp = Float64(Float64(x_46_re / y_46_re) + Float64(Float64(y_46_im / y_46_re) * Float64(x_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)
	t_0 = ((x_46_re * y_46_re) / ((y_46_re * y_46_re) + (y_46_im * y_46_im))) + (y_46_im / ((y_46_re * (y_46_re / x_46_im)) + (y_46_im * (y_46_im / x_46_im))));
	tmp = 0.0;
	if (y_46_re <= -2.9e+109)
		tmp = (x_46_re / y_46_re) + ((y_46_im * (x_46_im / y_46_re)) / y_46_re);
	elseif (y_46_re <= -6.6e-91)
		tmp = t_0;
	elseif (y_46_re <= 3.15e-75)
		tmp = (1.0 / y_46_im) * (x_46_im + (x_46_re / (y_46_im / y_46_re)));
	elseif (y_46_re <= 6.6e+41)
		tmp = t_0;
	elseif (y_46_re <= 3.3e+89)
		tmp = (x_46_im / y_46_im) + ((y_46_re / y_46_im) * (x_46_re / y_46_im));
	elseif (y_46_re <= 2.4e+157)
		tmp = t_0;
	else
		tmp = (x_46_re / y_46_re) + ((y_46_im / y_46_re) * (x_46_im / y_46_re));
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[(N[(x$46$re * y$46$re), $MachinePrecision] / N[(N[(y$46$re * y$46$re), $MachinePrecision] + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(y$46$im / N[(N[(y$46$re * N[(y$46$re / x$46$im), $MachinePrecision]), $MachinePrecision] + N[(y$46$im * N[(y$46$im / x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$re, -2.9e+109], N[(N[(x$46$re / y$46$re), $MachinePrecision] + N[(N[(y$46$im * N[(x$46$im / y$46$re), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, -6.6e-91], t$95$0, If[LessEqual[y$46$re, 3.15e-75], N[(N[(1.0 / y$46$im), $MachinePrecision] * N[(x$46$im + N[(x$46$re / N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 6.6e+41], t$95$0, If[LessEqual[y$46$re, 3.3e+89], N[(N[(x$46$im / y$46$im), $MachinePrecision] + N[(N[(y$46$re / y$46$im), $MachinePrecision] * N[(x$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 2.4e+157], t$95$0, 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]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if y.re < -2.9e109

   1. Initial program 29.3%

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

   2. Taylor expanded in y.re around inf 76.1%

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

      1. unpow2 76.1%

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

      2. times-frac 82.6%

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

   4. Simplified 82.6%

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

      1. associate-*l/ 83.9%

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

   6. Applied egg-rr 83.9%

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

   if -2.9e109 < y.re < -6.60000000000000023e-91 or 3.14999999999999992e-75 < y.re < 6.6000000000000001e41 or 3.29999999999999974e89 < y.re < 2.4e157

   1. Initial program 72.6%

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

      1. *-un-lft-identity 72.6%

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

      2. add-sqr-sqrt 72.6%

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

      3. times-frac 72.6%

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

      4. hypot-def 72.6%

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

      5. fma-def 72.6%

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

      6. hypot-def 79.2%

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

   3. Applied egg-rr 79.2%

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

   4. Taylor expanded in x.re around 0 72.7%

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

      1. associate-/l* 79.4%

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

      2. unpow2 79.4%

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

      3. unpow2 79.4%

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

      4. *-commutative 79.4%

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

      5. unpow2 79.4%

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

      6. unpow2 79.4%

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

   6. Simplified 79.4%

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

   7. Taylor expanded in y.re around 0 79.4%

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

      1. unpow2 79.4%

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

      2. associate-/l* 90.8%

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

      3. unpow2 90.8%

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

      4. associate-/l* 92.1%

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

      5. associate-/r/ 92.1%

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

   9. Simplified 92.1%

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

       1. associate-/r/ 92.1%

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

   11. Applied egg-rr 92.1%

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

   if -6.60000000000000023e-91 < y.re < 3.14999999999999992e-75

   1. Initial program 67.6%

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

      1. *-un-lft-identity 67.6%

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

      2. add-sqr-sqrt 67.6%

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

      3. times-frac 67.6%

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

      4. hypot-def 67.6%

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

      5. fma-def 67.6%

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

      6. hypot-def 82.7%

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

   3. Applied egg-rr 82.7%

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

   4. Taylor expanded in y.re around 0 55.2%

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

      1. +-commutative 55.2%

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

      2. associate-/l* 56.0%

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

   6. Simplified 56.0%

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

   7. Taylor expanded in y.re around 0 90.8%

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

   if 6.6000000000000001e41 < y.re < 3.29999999999999974e89

   1. Initial program 27.2%

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

   2. Taylor expanded in y.re around 0 43.8%

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

      1. +-commutative 43.8%

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

      2. *-commutative 43.8%

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

      3. unpow2 43.8%

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

      4. times-frac 75.5%

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

   4. Simplified 75.5%

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

   if 2.4e157 < y.re

   1. Initial program 34.2%

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

   2. Taylor expanded in y.re around inf 85.3%

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

      1. unpow2 85.3%

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

      2. times-frac 89.0%

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

   4. Simplified 89.0%

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

4. Final simplification 89.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -2.9 \cdot 10^{+109}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{y.im \cdot \frac{x.im}{y.re}}{y.re}\\ \mathbf{elif}\;y.re \leq -6.6 \cdot 10^{-91}:\\ \;\;\;\;\frac{x.re \cdot y.re}{y.re \cdot y.re + y.im \cdot y.im} + \frac{y.im}{y.re \cdot \frac{y.re}{x.im} + y.im \cdot \frac{y.im}{x.im}}\\ \mathbf{elif}\;y.re \leq 3.15 \cdot 10^{-75}:\\ \;\;\;\;\frac{1}{y.im} \cdot \left(x.im + \frac{x.re}{\frac{y.im}{y.re}}\right)\\ \mathbf{elif}\;y.re \leq 6.6 \cdot 10^{+41}:\\ \;\;\;\;\frac{x.re \cdot y.re}{y.re \cdot y.re + y.im \cdot y.im} + \frac{y.im}{y.re \cdot \frac{y.re}{x.im} + y.im \cdot \frac{y.im}{x.im}}\\ \mathbf{elif}\;y.re \leq 3.3 \cdot 10^{+89}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re}{y.im} \cdot \frac{x.re}{y.im}\\ \mathbf{elif}\;y.re \leq 2.4 \cdot 10^{+157}:\\ \;\;\;\;\frac{x.re \cdot y.re}{y.re \cdot y.re + y.im \cdot y.im} + \frac{y.im}{y.re \cdot \frac{y.re}{x.im} + y.im \cdot \frac{y.im}{x.im}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{y.im}{y.re} \cdot \frac{x.im}{y.re}\\ \end{array} \]

Alternative 5: 83.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y.im}{\frac{y.im}{\frac{x.im}{y.im}} + y.re \cdot \frac{y.re}{x.im}} + \frac{x.re \cdot y.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{if}\;y.re \leq -6 \cdot 10^{+105}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{y.im \cdot \frac{x.im}{y.re}}{y.re}\\ \mathbf{elif}\;y.re \leq -1.28 \cdot 10^{-100}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq 3 \cdot 10^{-80}:\\ \;\;\;\;\frac{1}{y.im} \cdot \left(x.im + \frac{x.re}{\frac{y.im}{y.re}}\right)\\ \mathbf{elif}\;y.re \leq 6.6 \cdot 10^{+41}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq 3.3 \cdot 10^{+89}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re}{y.im} \cdot \frac{x.re}{y.im}\\ \mathbf{elif}\;y.re \leq 2.4 \cdot 10^{+157}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{y.im}{y.re} \cdot \frac{x.im}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0
         (+
          (/ y.im (+ (/ y.im (/ x.im y.im)) (* y.re (/ y.re x.im))))
          (/ (* x.re y.re) (+ (* y.re y.re) (* y.im y.im))))))
   (if (<= y.re -6e+105)
     (+ (/ x.re y.re) (/ (* y.im (/ x.im y.re)) y.re))
     (if (<= y.re -1.28e-100)
       t_0
       (if (<= y.re 3e-80)
         (* (/ 1.0 y.im) (+ x.im (/ x.re (/ y.im y.re))))
         (if (<= y.re 6.6e+41)
           t_0
           (if (<= y.re 3.3e+89)
             (+ (/ x.im y.im) (* (/ y.re y.im) (/ x.re y.im)))
             (if (<= y.re 2.4e+157)
               t_0
               (+ (/ x.re y.re) (* (/ y.im y.re) (/ x.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 = (y_46_im / ((y_46_im / (x_46_im / y_46_im)) + (y_46_re * (y_46_re / x_46_im)))) + ((x_46_re * y_46_re) / ((y_46_re * y_46_re) + (y_46_im * y_46_im)));
	double tmp;
	if (y_46_re <= -6e+105) {
		tmp = (x_46_re / y_46_re) + ((y_46_im * (x_46_im / y_46_re)) / y_46_re);
	} else if (y_46_re <= -1.28e-100) {
		tmp = t_0;
	} else if (y_46_re <= 3e-80) {
		tmp = (1.0 / y_46_im) * (x_46_im + (x_46_re / (y_46_im / y_46_re)));
	} else if (y_46_re <= 6.6e+41) {
		tmp = t_0;
	} else if (y_46_re <= 3.3e+89) {
		tmp = (x_46_im / y_46_im) + ((y_46_re / y_46_im) * (x_46_re / y_46_im));
	} else if (y_46_re <= 2.4e+157) {
		tmp = t_0;
	} else {
		tmp = (x_46_re / y_46_re) + ((y_46_im / y_46_re) * (x_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) :: t_0
    real(8) :: tmp
    t_0 = (y_46im / ((y_46im / (x_46im / y_46im)) + (y_46re * (y_46re / x_46im)))) + ((x_46re * y_46re) / ((y_46re * y_46re) + (y_46im * y_46im)))
    if (y_46re <= (-6d+105)) then
        tmp = (x_46re / y_46re) + ((y_46im * (x_46im / y_46re)) / y_46re)
    else if (y_46re <= (-1.28d-100)) then
        tmp = t_0
    else if (y_46re <= 3d-80) then
        tmp = (1.0d0 / y_46im) * (x_46im + (x_46re / (y_46im / y_46re)))
    else if (y_46re <= 6.6d+41) then
        tmp = t_0
    else if (y_46re <= 3.3d+89) then
        tmp = (x_46im / y_46im) + ((y_46re / y_46im) * (x_46re / y_46im))
    else if (y_46re <= 2.4d+157) then
        tmp = t_0
    else
        tmp = (x_46re / y_46re) + ((y_46im / y_46re) * (x_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 t_0 = (y_46_im / ((y_46_im / (x_46_im / y_46_im)) + (y_46_re * (y_46_re / x_46_im)))) + ((x_46_re * y_46_re) / ((y_46_re * y_46_re) + (y_46_im * y_46_im)));
	double tmp;
	if (y_46_re <= -6e+105) {
		tmp = (x_46_re / y_46_re) + ((y_46_im * (x_46_im / y_46_re)) / y_46_re);
	} else if (y_46_re <= -1.28e-100) {
		tmp = t_0;
	} else if (y_46_re <= 3e-80) {
		tmp = (1.0 / y_46_im) * (x_46_im + (x_46_re / (y_46_im / y_46_re)));
	} else if (y_46_re <= 6.6e+41) {
		tmp = t_0;
	} else if (y_46_re <= 3.3e+89) {
		tmp = (x_46_im / y_46_im) + ((y_46_re / y_46_im) * (x_46_re / y_46_im));
	} else if (y_46_re <= 2.4e+157) {
		tmp = t_0;
	} else {
		tmp = (x_46_re / y_46_re) + ((y_46_im / y_46_re) * (x_46_im / y_46_re));
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = (y_46_im / ((y_46_im / (x_46_im / y_46_im)) + (y_46_re * (y_46_re / x_46_im)))) + ((x_46_re * y_46_re) / ((y_46_re * y_46_re) + (y_46_im * y_46_im)))
	tmp = 0
	if y_46_re <= -6e+105:
		tmp = (x_46_re / y_46_re) + ((y_46_im * (x_46_im / y_46_re)) / y_46_re)
	elif y_46_re <= -1.28e-100:
		tmp = t_0
	elif y_46_re <= 3e-80:
		tmp = (1.0 / y_46_im) * (x_46_im + (x_46_re / (y_46_im / y_46_re)))
	elif y_46_re <= 6.6e+41:
		tmp = t_0
	elif y_46_re <= 3.3e+89:
		tmp = (x_46_im / y_46_im) + ((y_46_re / y_46_im) * (x_46_re / y_46_im))
	elif y_46_re <= 2.4e+157:
		tmp = t_0
	else:
		tmp = (x_46_re / y_46_re) + ((y_46_im / y_46_re) * (x_46_im / y_46_re))
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(Float64(y_46_im / Float64(Float64(y_46_im / Float64(x_46_im / y_46_im)) + Float64(y_46_re * Float64(y_46_re / x_46_im)))) + Float64(Float64(x_46_re * y_46_re) / Float64(Float64(y_46_re * y_46_re) + Float64(y_46_im * y_46_im))))
	tmp = 0.0
	if (y_46_re <= -6e+105)
		tmp = Float64(Float64(x_46_re / y_46_re) + Float64(Float64(y_46_im * Float64(x_46_im / y_46_re)) / y_46_re));
	elseif (y_46_re <= -1.28e-100)
		tmp = t_0;
	elseif (y_46_re <= 3e-80)
		tmp = Float64(Float64(1.0 / y_46_im) * Float64(x_46_im + Float64(x_46_re / Float64(y_46_im / y_46_re))));
	elseif (y_46_re <= 6.6e+41)
		tmp = t_0;
	elseif (y_46_re <= 3.3e+89)
		tmp = Float64(Float64(x_46_im / y_46_im) + Float64(Float64(y_46_re / y_46_im) * Float64(x_46_re / y_46_im)));
	elseif (y_46_re <= 2.4e+157)
		tmp = t_0;
	else
		tmp = Float64(Float64(x_46_re / y_46_re) + Float64(Float64(y_46_im / y_46_re) * Float64(x_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)
	t_0 = (y_46_im / ((y_46_im / (x_46_im / y_46_im)) + (y_46_re * (y_46_re / x_46_im)))) + ((x_46_re * y_46_re) / ((y_46_re * y_46_re) + (y_46_im * y_46_im)));
	tmp = 0.0;
	if (y_46_re <= -6e+105)
		tmp = (x_46_re / y_46_re) + ((y_46_im * (x_46_im / y_46_re)) / y_46_re);
	elseif (y_46_re <= -1.28e-100)
		tmp = t_0;
	elseif (y_46_re <= 3e-80)
		tmp = (1.0 / y_46_im) * (x_46_im + (x_46_re / (y_46_im / y_46_re)));
	elseif (y_46_re <= 6.6e+41)
		tmp = t_0;
	elseif (y_46_re <= 3.3e+89)
		tmp = (x_46_im / y_46_im) + ((y_46_re / y_46_im) * (x_46_re / y_46_im));
	elseif (y_46_re <= 2.4e+157)
		tmp = t_0;
	else
		tmp = (x_46_re / y_46_re) + ((y_46_im / y_46_re) * (x_46_im / y_46_re));
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[(y$46$im / N[(N[(y$46$im / N[(x$46$im / y$46$im), $MachinePrecision]), $MachinePrecision] + N[(y$46$re * N[(y$46$re / x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(x$46$re * y$46$re), $MachinePrecision] / N[(N[(y$46$re * y$46$re), $MachinePrecision] + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$re, -6e+105], N[(N[(x$46$re / y$46$re), $MachinePrecision] + N[(N[(y$46$im * N[(x$46$im / y$46$re), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, -1.28e-100], t$95$0, If[LessEqual[y$46$re, 3e-80], N[(N[(1.0 / y$46$im), $MachinePrecision] * N[(x$46$im + N[(x$46$re / N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 6.6e+41], t$95$0, If[LessEqual[y$46$re, 3.3e+89], N[(N[(x$46$im / y$46$im), $MachinePrecision] + N[(N[(y$46$re / y$46$im), $MachinePrecision] * N[(x$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 2.4e+157], t$95$0, 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]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if y.re < -6.0000000000000001e105

   1. Initial program 29.3%

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

   2. Taylor expanded in y.re around inf 76.1%

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

      1. unpow2 76.1%

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

      2. times-frac 82.6%

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

   4. Simplified 82.6%

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

      1. associate-*l/ 83.9%

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

   6. Applied egg-rr 83.9%

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

   if -6.0000000000000001e105 < y.re < -1.27999999999999991e-100 or 3.00000000000000007e-80 < y.re < 6.6000000000000001e41 or 3.29999999999999974e89 < y.re < 2.4e157

   1. Initial program 73.3%

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

      1. *-un-lft-identity 73.3%

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

      2. add-sqr-sqrt 73.3%

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

      3. times-frac 73.3%

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

      4. hypot-def 73.3%

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

      5. fma-def 73.3%

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

      6. hypot-def 79.7%

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

   3. Applied egg-rr 79.7%

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

   4. Taylor expanded in x.re around 0 73.4%

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

      1. associate-/l* 78.7%

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

      2. unpow2 78.7%

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

      3. unpow2 78.7%

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

      4. *-commutative 78.7%

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

      5. unpow2 78.7%

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

      6. unpow2 78.7%

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

   6. Simplified 78.7%

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

   7. Taylor expanded in y.re around 0 78.7%

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

      1. unpow2 78.7%

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

      2. associate-/l* 89.8%

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

      3. unpow2 89.8%

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

      4. associate-/l* 91.1%

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

      5. associate-/r/ 91.1%

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

   9. Simplified 91.1%

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

   if -1.27999999999999991e-100 < y.re < 3.00000000000000007e-80

   1. Initial program 66.9%

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

      1. *-un-lft-identity 66.9%

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

      2. add-sqr-sqrt 66.9%

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

      3. times-frac 67.0%

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

      4. hypot-def 67.0%

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

      5. fma-def 67.0%

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

      6. hypot-def 82.3%

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

   3. Applied egg-rr 82.3%

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

   4. Taylor expanded in y.re around 0 55.2%

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

      1. +-commutative 55.2%

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

      2. associate-/l* 56.1%

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

   6. Simplified 56.1%

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

   7. Taylor expanded in y.re around 0 91.6%

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

   if 6.6000000000000001e41 < y.re < 3.29999999999999974e89

   1. Initial program 27.2%

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

   2. Taylor expanded in y.re around 0 43.8%

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

      1. +-commutative 43.8%

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

      2. *-commutative 43.8%

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

      3. unpow2 43.8%

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

      4. times-frac 75.5%

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

   4. Simplified 75.5%

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

   if 2.4e157 < y.re

   1. Initial program 34.2%

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

   2. Taylor expanded in y.re around inf 85.3%

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

      1. unpow2 85.3%

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

      2. times-frac 89.0%

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

   4. Simplified 89.0%

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

4. Final simplification 89.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -6 \cdot 10^{+105}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{y.im \cdot \frac{x.im}{y.re}}{y.re}\\ \mathbf{elif}\;y.re \leq -1.28 \cdot 10^{-100}:\\ \;\;\;\;\frac{y.im}{\frac{y.im}{\frac{x.im}{y.im}} + y.re \cdot \frac{y.re}{x.im}} + \frac{x.re \cdot y.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.re \leq 3 \cdot 10^{-80}:\\ \;\;\;\;\frac{1}{y.im} \cdot \left(x.im + \frac{x.re}{\frac{y.im}{y.re}}\right)\\ \mathbf{elif}\;y.re \leq 6.6 \cdot 10^{+41}:\\ \;\;\;\;\frac{y.im}{\frac{y.im}{\frac{x.im}{y.im}} + y.re \cdot \frac{y.re}{x.im}} + \frac{x.re \cdot y.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.re \leq 3.3 \cdot 10^{+89}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re}{y.im} \cdot \frac{x.re}{y.im}\\ \mathbf{elif}\;y.re \leq 2.4 \cdot 10^{+157}:\\ \;\;\;\;\frac{y.im}{\frac{y.im}{\frac{x.im}{y.im}} + y.re \cdot \frac{y.re}{x.im}} + \frac{x.re \cdot y.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{y.im}{y.re} \cdot \frac{x.im}{y.re}\\ \end{array} \]

Alternative 6: 82.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ t_1 := \frac{x.im}{y.im} + \frac{y.re}{y.im} \cdot \frac{x.re}{y.im}\\ \mathbf{if}\;y.im \leq -1.7 \cdot 10^{+74}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.im \leq -8.5 \cdot 10^{-115}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq 3.9 \cdot 10^{-176}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{x.im \cdot \frac{y.im}{y.re}}{y.re}\\ \mathbf{elif}\;y.im \leq 2.4 \cdot 10^{-163}:\\ \;\;\;\;\frac{1}{y.im} \cdot \left(x.im + \frac{x.re \cdot y.re}{y.im}\right)\\ \mathbf{elif}\;y.im \leq 1.16 \cdot 10^{+56}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0
         (/ (+ (* x.re y.re) (* x.im y.im)) (+ (* y.re y.re) (* y.im y.im))))
        (t_1 (+ (/ x.im y.im) (* (/ y.re y.im) (/ x.re y.im)))))
   (if (<= y.im -1.7e+74)
     t_1
     (if (<= y.im -8.5e-115)
       t_0
       (if (<= y.im 3.9e-176)
         (+ (/ x.re y.re) (/ (* x.im (/ y.im y.re)) y.re))
         (if (<= y.im 2.4e-163)
           (* (/ 1.0 y.im) (+ x.im (/ (* x.re y.re) y.im)))
           (if (<= y.im 1.16e+56) t_0 t_1)))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = ((x_46_re * y_46_re) + (x_46_im * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	double t_1 = (x_46_im / y_46_im) + ((y_46_re / y_46_im) * (x_46_re / y_46_im));
	double tmp;
	if (y_46_im <= -1.7e+74) {
		tmp = t_1;
	} else if (y_46_im <= -8.5e-115) {
		tmp = t_0;
	} else if (y_46_im <= 3.9e-176) {
		tmp = (x_46_re / y_46_re) + ((x_46_im * (y_46_im / y_46_re)) / y_46_re);
	} else if (y_46_im <= 2.4e-163) {
		tmp = (1.0 / y_46_im) * (x_46_im + ((x_46_re * y_46_re) / y_46_im));
	} else if (y_46_im <= 1.16e+56) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = ((x_46re * y_46re) + (x_46im * y_46im)) / ((y_46re * y_46re) + (y_46im * y_46im))
    t_1 = (x_46im / y_46im) + ((y_46re / y_46im) * (x_46re / y_46im))
    if (y_46im <= (-1.7d+74)) then
        tmp = t_1
    else if (y_46im <= (-8.5d-115)) then
        tmp = t_0
    else if (y_46im <= 3.9d-176) then
        tmp = (x_46re / y_46re) + ((x_46im * (y_46im / y_46re)) / y_46re)
    else if (y_46im <= 2.4d-163) then
        tmp = (1.0d0 / y_46im) * (x_46im + ((x_46re * y_46re) / y_46im))
    else if (y_46im <= 1.16d+56) then
        tmp = t_0
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = ((x_46_re * y_46_re) + (x_46_im * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	double t_1 = (x_46_im / y_46_im) + ((y_46_re / y_46_im) * (x_46_re / y_46_im));
	double tmp;
	if (y_46_im <= -1.7e+74) {
		tmp = t_1;
	} else if (y_46_im <= -8.5e-115) {
		tmp = t_0;
	} else if (y_46_im <= 3.9e-176) {
		tmp = (x_46_re / y_46_re) + ((x_46_im * (y_46_im / y_46_re)) / y_46_re);
	} else if (y_46_im <= 2.4e-163) {
		tmp = (1.0 / y_46_im) * (x_46_im + ((x_46_re * y_46_re) / y_46_im));
	} else if (y_46_im <= 1.16e+56) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = ((x_46_re * y_46_re) + (x_46_im * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im))
	t_1 = (x_46_im / y_46_im) + ((y_46_re / y_46_im) * (x_46_re / y_46_im))
	tmp = 0
	if y_46_im <= -1.7e+74:
		tmp = t_1
	elif y_46_im <= -8.5e-115:
		tmp = t_0
	elif y_46_im <= 3.9e-176:
		tmp = (x_46_re / y_46_re) + ((x_46_im * (y_46_im / y_46_re)) / y_46_re)
	elif y_46_im <= 2.4e-163:
		tmp = (1.0 / y_46_im) * (x_46_im + ((x_46_re * y_46_re) / y_46_im))
	elif y_46_im <= 1.16e+56:
		tmp = t_0
	else:
		tmp = t_1
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(Float64(Float64(x_46_re * y_46_re) + Float64(x_46_im * y_46_im)) / Float64(Float64(y_46_re * y_46_re) + Float64(y_46_im * y_46_im)))
	t_1 = Float64(Float64(x_46_im / y_46_im) + Float64(Float64(y_46_re / y_46_im) * Float64(x_46_re / y_46_im)))
	tmp = 0.0
	if (y_46_im <= -1.7e+74)
		tmp = t_1;
	elseif (y_46_im <= -8.5e-115)
		tmp = t_0;
	elseif (y_46_im <= 3.9e-176)
		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 <= 2.4e-163)
		tmp = Float64(Float64(1.0 / y_46_im) * Float64(x_46_im + Float64(Float64(x_46_re * y_46_re) / y_46_im)));
	elseif (y_46_im <= 1.16e+56)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = ((x_46_re * y_46_re) + (x_46_im * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	t_1 = (x_46_im / y_46_im) + ((y_46_re / y_46_im) * (x_46_re / y_46_im));
	tmp = 0.0;
	if (y_46_im <= -1.7e+74)
		tmp = t_1;
	elseif (y_46_im <= -8.5e-115)
		tmp = t_0;
	elseif (y_46_im <= 3.9e-176)
		tmp = (x_46_re / y_46_re) + ((x_46_im * (y_46_im / y_46_re)) / y_46_re);
	elseif (y_46_im <= 2.4e-163)
		tmp = (1.0 / y_46_im) * (x_46_im + ((x_46_re * y_46_re) / y_46_im));
	elseif (y_46_im <= 1.16e+56)
		tmp = t_0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[(N[(x$46$re * y$46$re), $MachinePrecision] + N[(x$46$im * y$46$im), $MachinePrecision]), $MachinePrecision] / N[(N[(y$46$re * y$46$re), $MachinePrecision] + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x$46$im / y$46$im), $MachinePrecision] + N[(N[(y$46$re / y$46$im), $MachinePrecision] * N[(x$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$im, -1.7e+74], t$95$1, If[LessEqual[y$46$im, -8.5e-115], t$95$0, If[LessEqual[y$46$im, 3.9e-176], 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, 2.4e-163], N[(N[(1.0 / y$46$im), $MachinePrecision] * N[(x$46$im + N[(N[(x$46$re * y$46$re), $MachinePrecision] / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, 1.16e+56], t$95$0, t$95$1]]]]]]]

Derivation

1. Split input into 4 regimes

2. if y.im < -1.7e74 or 1.1599999999999999e56 < y.im

   1. Initial program 35.8%

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

   2. Taylor expanded in y.re around 0 75.4%

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

      1. +-commutative 75.4%

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

      2. *-commutative 75.4%

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

      3. unpow2 75.4%

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

      4. times-frac 83.5%

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

   4. Simplified 83.5%

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

   if -1.7e74 < y.im < -8.49999999999999953e-115 or 2.4000000000000001e-163 < y.im < 1.1599999999999999e56

   1. Initial program 85.5%

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

   if -8.49999999999999953e-115 < y.im < 3.8999999999999997e-176

   1. Initial program 65.5%

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

   2. Taylor expanded in y.re around inf 85.4%

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

      1. unpow2 85.4%

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

      2. times-frac 87.7%

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

   4. Simplified 87.7%

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

      1. associate-*r/ 90.9%

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

   6. Applied egg-rr 90.9%

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

   if 3.8999999999999997e-176 < y.im < 2.4000000000000001e-163

   1. Initial program 4.6%

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

      1. *-un-lft-identity 4.6%

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

      2. add-sqr-sqrt 4.6%

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

      3. times-frac 4.6%

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

      4. hypot-def 4.6%

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

      5. fma-def 4.6%

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

      6. hypot-def 99.4%

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

   3. Applied egg-rr 99.4%

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

   4. Taylor expanded in y.re around 0 89.7%

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

      1. +-commutative 89.7%

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

      2. associate-/l* 89.7%

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

   6. Simplified 89.7%

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

   7. Taylor expanded in y.re around 0 89.4%

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

   8. Taylor expanded in x.im around 0 89.4%

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

4. Final simplification 85.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -1.7 \cdot 10^{+74}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re}{y.im} \cdot \frac{x.re}{y.im}\\ \mathbf{elif}\;y.im \leq -8.5 \cdot 10^{-115}:\\ \;\;\;\;\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.im \leq 3.9 \cdot 10^{-176}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{x.im \cdot \frac{y.im}{y.re}}{y.re}\\ \mathbf{elif}\;y.im \leq 2.4 \cdot 10^{-163}:\\ \;\;\;\;\frac{1}{y.im} \cdot \left(x.im + \frac{x.re \cdot y.re}{y.im}\right)\\ \mathbf{elif}\;y.im \leq 1.16 \cdot 10^{+56}:\\ \;\;\;\;\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re}{y.im} \cdot \frac{x.re}{y.im}\\ \end{array} \]

Alternative 7: 76.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x.re}{y.re} + \frac{y.im}{y.re} \cdot \frac{x.im}{y.re}\\ \mathbf{if}\;y.re \leq -4.8 \cdot 10^{+108}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq 3.7 \cdot 10^{-57}:\\ \;\;\;\;\frac{1}{y.im} \cdot \left(x.im + \frac{x.re}{\frac{y.im}{y.re}}\right)\\ \mathbf{elif}\;y.re \leq 6.5 \cdot 10^{-36} \lor \neg \left(y.re \leq 3.3 \cdot 10^{+89}\right):\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re}{y.im} \cdot \frac{x.re}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (+ (/ x.re y.re) (* (/ y.im y.re) (/ x.im y.re)))))
   (if (<= y.re -4.8e+108)
     t_0
     (if (<= y.re 3.7e-57)
       (* (/ 1.0 y.im) (+ x.im (/ x.re (/ y.im y.re))))
       (if (or (<= y.re 6.5e-36) (not (<= y.re 3.3e+89)))
         t_0
         (+ (/ x.im y.im) (* (/ y.re y.im) (/ x.re y.im))))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = (x_46_re / y_46_re) + ((y_46_im / y_46_re) * (x_46_im / y_46_re));
	double tmp;
	if (y_46_re <= -4.8e+108) {
		tmp = t_0;
	} else if (y_46_re <= 3.7e-57) {
		tmp = (1.0 / y_46_im) * (x_46_im + (x_46_re / (y_46_im / y_46_re)));
	} else if ((y_46_re <= 6.5e-36) || !(y_46_re <= 3.3e+89)) {
		tmp = t_0;
	} else {
		tmp = (x_46_im / y_46_im) + ((y_46_re / y_46_im) * (x_46_re / y_46_im));
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (x_46re / y_46re) + ((y_46im / y_46re) * (x_46im / y_46re))
    if (y_46re <= (-4.8d+108)) then
        tmp = t_0
    else if (y_46re <= 3.7d-57) then
        tmp = (1.0d0 / y_46im) * (x_46im + (x_46re / (y_46im / y_46re)))
    else if ((y_46re <= 6.5d-36) .or. (.not. (y_46re <= 3.3d+89))) then
        tmp = t_0
    else
        tmp = (x_46im / y_46im) + ((y_46re / y_46im) * (x_46re / y_46im))
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = (x_46_re / y_46_re) + ((y_46_im / y_46_re) * (x_46_im / y_46_re));
	double tmp;
	if (y_46_re <= -4.8e+108) {
		tmp = t_0;
	} else if (y_46_re <= 3.7e-57) {
		tmp = (1.0 / y_46_im) * (x_46_im + (x_46_re / (y_46_im / y_46_re)));
	} else if ((y_46_re <= 6.5e-36) || !(y_46_re <= 3.3e+89)) {
		tmp = t_0;
	} else {
		tmp = (x_46_im / y_46_im) + ((y_46_re / y_46_im) * (x_46_re / y_46_im));
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = (x_46_re / y_46_re) + ((y_46_im / y_46_re) * (x_46_im / y_46_re))
	tmp = 0
	if y_46_re <= -4.8e+108:
		tmp = t_0
	elif y_46_re <= 3.7e-57:
		tmp = (1.0 / y_46_im) * (x_46_im + (x_46_re / (y_46_im / y_46_re)))
	elif (y_46_re <= 6.5e-36) or not (y_46_re <= 3.3e+89):
		tmp = t_0
	else:
		tmp = (x_46_im / y_46_im) + ((y_46_re / y_46_im) * (x_46_re / y_46_im))
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(Float64(x_46_re / y_46_re) + Float64(Float64(y_46_im / y_46_re) * Float64(x_46_im / y_46_re)))
	tmp = 0.0
	if (y_46_re <= -4.8e+108)
		tmp = t_0;
	elseif (y_46_re <= 3.7e-57)
		tmp = Float64(Float64(1.0 / y_46_im) * Float64(x_46_im + Float64(x_46_re / Float64(y_46_im / y_46_re))));
	elseif ((y_46_re <= 6.5e-36) || !(y_46_re <= 3.3e+89))
		tmp = t_0;
	else
		tmp = Float64(Float64(x_46_im / y_46_im) + Float64(Float64(y_46_re / y_46_im) * Float64(x_46_re / y_46_im)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = (x_46_re / y_46_re) + ((y_46_im / y_46_re) * (x_46_im / y_46_re));
	tmp = 0.0;
	if (y_46_re <= -4.8e+108)
		tmp = t_0;
	elseif (y_46_re <= 3.7e-57)
		tmp = (1.0 / y_46_im) * (x_46_im + (x_46_re / (y_46_im / y_46_re)));
	elseif ((y_46_re <= 6.5e-36) || ~((y_46_re <= 3.3e+89)))
		tmp = t_0;
	else
		tmp = (x_46_im / y_46_im) + ((y_46_re / y_46_im) * (x_46_re / y_46_im));
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = 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]}, If[LessEqual[y$46$re, -4.8e+108], t$95$0, If[LessEqual[y$46$re, 3.7e-57], N[(N[(1.0 / y$46$im), $MachinePrecision] * N[(x$46$im + N[(x$46$re / N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[y$46$re, 6.5e-36], N[Not[LessEqual[y$46$re, 3.3e+89]], $MachinePrecision]], t$95$0, N[(N[(x$46$im / y$46$im), $MachinePrecision] + N[(N[(y$46$re / y$46$im), $MachinePrecision] * N[(x$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -4.80000000000000037e108 or 3.7e-57 < y.re < 6.50000000000000012e-36 or 3.29999999999999974e89 < y.re

   1. Initial program 41.1%

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

   2. Taylor expanded in y.re around inf 80.3%

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

      1. unpow2 80.3%

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

      2. times-frac 84.7%

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

   4. Simplified 84.7%

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

   if -4.80000000000000037e108 < y.re < 3.7e-57

   1. Initial program 68.1%

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

      1. *-un-lft-identity 68.1%

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

      2. add-sqr-sqrt 68.1%

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

      3. times-frac 68.1%

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

      4. hypot-def 68.1%

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

      5. fma-def 68.1%

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

      6. hypot-def 80.7%

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

   3. Applied egg-rr 80.7%

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

   4. Taylor expanded in y.re around 0 50.4%

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

      1. +-commutative 50.4%

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

      2. associate-/l* 51.0%

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

   6. Simplified 51.0%

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

   7. Taylor expanded in y.re around 0 82.4%

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

   if 6.50000000000000012e-36 < y.re < 3.29999999999999974e89

   1. Initial program 53.1%

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

   2. Taylor expanded in y.re around 0 50.0%

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

      1. +-commutative 50.0%

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

      2. *-commutative 50.0%

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

      3. unpow2 50.0%

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

      4. times-frac 67.8%

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

   4. Simplified 67.8%

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

4. Final simplification 81.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -4.8 \cdot 10^{+108}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{y.im}{y.re} \cdot \frac{x.im}{y.re}\\ \mathbf{elif}\;y.re \leq 3.7 \cdot 10^{-57}:\\ \;\;\;\;\frac{1}{y.im} \cdot \left(x.im + \frac{x.re}{\frac{y.im}{y.re}}\right)\\ \mathbf{elif}\;y.re \leq 6.5 \cdot 10^{-36} \lor \neg \left(y.re \leq 3.3 \cdot 10^{+89}\right):\\ \;\;\;\;\frac{x.re}{y.re} + \frac{y.im}{y.re} \cdot \frac{x.im}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re}{y.im} \cdot \frac{x.re}{y.im}\\ \end{array} \]

Alternative 8: 76.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -3.3 \cdot 10^{+105}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{y.im \cdot \frac{x.im}{y.re}}{y.re}\\ \mathbf{elif}\;y.re \leq 4 \cdot 10^{-57}:\\ \;\;\;\;\frac{1}{y.im} \cdot \left(x.im + \frac{x.re}{\frac{y.im}{y.re}}\right)\\ \mathbf{elif}\;y.re \leq 3.4 \cdot 10^{-40} \lor \neg \left(y.re \leq 3.3 \cdot 10^{+89}\right):\\ \;\;\;\;\frac{x.re}{y.re} + \frac{y.im}{y.re} \cdot \frac{x.im}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re}{y.im} \cdot \frac{x.re}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.re -3.3e+105)
   (+ (/ x.re y.re) (/ (* y.im (/ x.im y.re)) y.re))
   (if (<= y.re 4e-57)
     (* (/ 1.0 y.im) (+ x.im (/ x.re (/ y.im y.re))))
     (if (or (<= y.re 3.4e-40) (not (<= y.re 3.3e+89)))
       (+ (/ x.re y.re) (* (/ y.im y.re) (/ x.im y.re)))
       (+ (/ x.im y.im) (* (/ y.re y.im) (/ x.re y.im)))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_re <= -3.3e+105) {
		tmp = (x_46_re / y_46_re) + ((y_46_im * (x_46_im / y_46_re)) / y_46_re);
	} else if (y_46_re <= 4e-57) {
		tmp = (1.0 / y_46_im) * (x_46_im + (x_46_re / (y_46_im / y_46_re)));
	} else if ((y_46_re <= 3.4e-40) || !(y_46_re <= 3.3e+89)) {
		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) + ((y_46_re / y_46_im) * (x_46_re / y_46_im));
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if (y_46re <= (-3.3d+105)) then
        tmp = (x_46re / y_46re) + ((y_46im * (x_46im / y_46re)) / y_46re)
    else if (y_46re <= 4d-57) then
        tmp = (1.0d0 / y_46im) * (x_46im + (x_46re / (y_46im / y_46re)))
    else if ((y_46re <= 3.4d-40) .or. (.not. (y_46re <= 3.3d+89))) then
        tmp = (x_46re / y_46re) + ((y_46im / y_46re) * (x_46im / y_46re))
    else
        tmp = (x_46im / y_46im) + ((y_46re / y_46im) * (x_46re / y_46im))
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_re <= -3.3e+105) {
		tmp = (x_46_re / y_46_re) + ((y_46_im * (x_46_im / y_46_re)) / y_46_re);
	} else if (y_46_re <= 4e-57) {
		tmp = (1.0 / y_46_im) * (x_46_im + (x_46_re / (y_46_im / y_46_re)));
	} else if ((y_46_re <= 3.4e-40) || !(y_46_re <= 3.3e+89)) {
		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) + ((y_46_re / y_46_im) * (x_46_re / y_46_im));
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if y_46_re <= -3.3e+105:
		tmp = (x_46_re / y_46_re) + ((y_46_im * (x_46_im / y_46_re)) / y_46_re)
	elif y_46_re <= 4e-57:
		tmp = (1.0 / y_46_im) * (x_46_im + (x_46_re / (y_46_im / y_46_re)))
	elif (y_46_re <= 3.4e-40) or not (y_46_re <= 3.3e+89):
		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) + ((y_46_re / y_46_im) * (x_46_re / y_46_im))
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (y_46_re <= -3.3e+105)
		tmp = Float64(Float64(x_46_re / y_46_re) + Float64(Float64(y_46_im * Float64(x_46_im / y_46_re)) / y_46_re));
	elseif (y_46_re <= 4e-57)
		tmp = Float64(Float64(1.0 / y_46_im) * Float64(x_46_im + Float64(x_46_re / Float64(y_46_im / y_46_re))));
	elseif ((y_46_re <= 3.4e-40) || !(y_46_re <= 3.3e+89))
		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(y_46_re / y_46_im) * Float64(x_46_re / y_46_im)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if (y_46_re <= -3.3e+105)
		tmp = (x_46_re / y_46_re) + ((y_46_im * (x_46_im / y_46_re)) / y_46_re);
	elseif (y_46_re <= 4e-57)
		tmp = (1.0 / y_46_im) * (x_46_im + (x_46_re / (y_46_im / y_46_re)));
	elseif ((y_46_re <= 3.4e-40) || ~((y_46_re <= 3.3e+89)))
		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) + ((y_46_re / y_46_im) * (x_46_re / y_46_im));
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[y$46$re, -3.3e+105], N[(N[(x$46$re / y$46$re), $MachinePrecision] + N[(N[(y$46$im * N[(x$46$im / y$46$re), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 4e-57], N[(N[(1.0 / y$46$im), $MachinePrecision] * N[(x$46$im + N[(x$46$re / N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[y$46$re, 3.4e-40], N[Not[LessEqual[y$46$re, 3.3e+89]], $MachinePrecision]], 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[(y$46$re / y$46$im), $MachinePrecision] * N[(x$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if y.re < -3.29999999999999997e105

   1. Initial program 29.3%

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

   2. Taylor expanded in y.re around inf 76.1%

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

      1. unpow2 76.1%

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

      2. times-frac 82.6%

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

   4. Simplified 82.6%

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

      1. associate-*l/ 83.9%

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

   6. Applied egg-rr 83.9%

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

   if -3.29999999999999997e105 < y.re < 3.99999999999999982e-57

   1. Initial program 68.1%

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

      1. *-un-lft-identity 68.1%

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

      2. add-sqr-sqrt 68.1%

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

      3. times-frac 68.1%

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

      4. hypot-def 68.1%

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

      5. fma-def 68.1%

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

      6. hypot-def 80.7%

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

   3. Applied egg-rr 80.7%

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

   4. Taylor expanded in y.re around 0 50.4%

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

      1. +-commutative 50.4%

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

      2. associate-/l* 51.0%

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

   6. Simplified 51.0%

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

   7. Taylor expanded in y.re around 0 82.4%

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

   if 3.99999999999999982e-57 < y.re < 3.39999999999999984e-40 or 3.29999999999999974e89 < y.re

   1. Initial program 51.2%

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

   2. Taylor expanded in y.re around inf 83.9%

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

      1. unpow2 83.9%

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

      2. times-frac 86.6%

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

   4. Simplified 86.6%

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

   if 3.39999999999999984e-40 < y.re < 3.29999999999999974e89

   1. Initial program 53.1%

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

   2. Taylor expanded in y.re around 0 50.0%

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

      1. +-commutative 50.0%

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

      2. *-commutative 50.0%

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

      3. unpow2 50.0%

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

      4. times-frac 67.8%

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

   4. Simplified 67.8%

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

4. Final simplification 81.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -3.3 \cdot 10^{+105}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{y.im \cdot \frac{x.im}{y.re}}{y.re}\\ \mathbf{elif}\;y.re \leq 4 \cdot 10^{-57}:\\ \;\;\;\;\frac{1}{y.im} \cdot \left(x.im + \frac{x.re}{\frac{y.im}{y.re}}\right)\\ \mathbf{elif}\;y.re \leq 3.4 \cdot 10^{-40} \lor \neg \left(y.re \leq 3.3 \cdot 10^{+89}\right):\\ \;\;\;\;\frac{x.re}{y.re} + \frac{y.im}{y.re} \cdot \frac{x.im}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re}{y.im} \cdot \frac{x.re}{y.im}\\ \end{array} \]

Alternative 9: 72.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -2.9 \cdot 10^{+108}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{elif}\;y.re \leq 1.5 \cdot 10^{+90}:\\ \;\;\;\;\frac{1}{y.im} \cdot \left(x.im + \frac{x.re}{\frac{y.im}{y.re}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.re -2.9e+108)
   (/ x.re y.re)
   (if (<= y.re 1.5e+90)
     (* (/ 1.0 y.im) (+ x.im (/ x.re (/ y.im y.re))))
     (/ x.re y.re))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_re <= -2.9e+108) {
		tmp = x_46_re / y_46_re;
	} else if (y_46_re <= 1.5e+90) {
		tmp = (1.0 / y_46_im) * (x_46_im + (x_46_re / (y_46_im / y_46_re)));
	} else {
		tmp = x_46_re / y_46_re;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if (y_46re <= (-2.9d+108)) then
        tmp = x_46re / y_46re
    else if (y_46re <= 1.5d+90) then
        tmp = (1.0d0 / y_46im) * (x_46im + (x_46re / (y_46im / y_46re)))
    else
        tmp = x_46re / y_46re
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_re <= -2.9e+108) {
		tmp = x_46_re / y_46_re;
	} else if (y_46_re <= 1.5e+90) {
		tmp = (1.0 / y_46_im) * (x_46_im + (x_46_re / (y_46_im / y_46_re)));
	} else {
		tmp = x_46_re / y_46_re;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if y_46_re <= -2.9e+108:
		tmp = x_46_re / y_46_re
	elif y_46_re <= 1.5e+90:
		tmp = (1.0 / y_46_im) * (x_46_im + (x_46_re / (y_46_im / y_46_re)))
	else:
		tmp = x_46_re / y_46_re
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (y_46_re <= -2.9e+108)
		tmp = Float64(x_46_re / y_46_re);
	elseif (y_46_re <= 1.5e+90)
		tmp = Float64(Float64(1.0 / y_46_im) * Float64(x_46_im + Float64(x_46_re / Float64(y_46_im / y_46_re))));
	else
		tmp = Float64(x_46_re / y_46_re);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if (y_46_re <= -2.9e+108)
		tmp = x_46_re / y_46_re;
	elseif (y_46_re <= 1.5e+90)
		tmp = (1.0 / y_46_im) * (x_46_im + (x_46_re / (y_46_im / y_46_re)));
	else
		tmp = x_46_re / y_46_re;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[y$46$re, -2.9e+108], N[(x$46$re / y$46$re), $MachinePrecision], If[LessEqual[y$46$re, 1.5e+90], N[(N[(1.0 / y$46$im), $MachinePrecision] * N[(x$46$im + N[(x$46$re / N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x$46$re / y$46$re), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y.re < -2.90000000000000007e108 or 1.49999999999999989e90 < y.re

   1. Initial program 35.6%

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

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

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

   if -2.90000000000000007e108 < y.re < 1.49999999999999989e90

   1. Initial program 67.1%

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

      1. *-un-lft-identity 67.1%

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

      2. add-sqr-sqrt 67.1%

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

      3. times-frac 67.1%

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

      4. hypot-def 67.1%

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

      5. fma-def 67.1%

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

      6. hypot-def 79.9%

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

   3. Applied egg-rr 79.9%

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

   4. Taylor expanded in y.re around 0 46.7%

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

      1. +-commutative 46.7%

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

      2. associate-/l* 47.8%

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

   6. Simplified 47.8%

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

   7. Taylor expanded in y.re around 0 77.7%

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

4. Final simplification 78.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -2.9 \cdot 10^{+108}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{elif}\;y.re \leq 1.5 \cdot 10^{+90}:\\ \;\;\;\;\frac{1}{y.im} \cdot \left(x.im + \frac{x.re}{\frac{y.im}{y.re}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \end{array} \]

Alternative 10: 61.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -2.8 \cdot 10^{+108}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{elif}\;y.re \leq 1.12 \cdot 10^{-71} \lor \neg \left(y.re \leq 1.15 \cdot 10^{-16}\right) \land y.re \leq 2.7 \cdot 10^{+90}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.re -2.8e+108)
   (/ x.re y.re)
   (if (or (<= y.re 1.12e-71) (and (not (<= y.re 1.15e-16)) (<= y.re 2.7e+90)))
     (/ x.im y.im)
     (/ x.re y.re))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_re <= -2.8e+108) {
		tmp = x_46_re / y_46_re;
	} else if ((y_46_re <= 1.12e-71) || (!(y_46_re <= 1.15e-16) && (y_46_re <= 2.7e+90))) {
		tmp = x_46_im / y_46_im;
	} else {
		tmp = x_46_re / y_46_re;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if (y_46re <= (-2.8d+108)) then
        tmp = x_46re / y_46re
    else if ((y_46re <= 1.12d-71) .or. (.not. (y_46re <= 1.15d-16)) .and. (y_46re <= 2.7d+90)) then
        tmp = x_46im / y_46im
    else
        tmp = x_46re / y_46re
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_re <= -2.8e+108) {
		tmp = x_46_re / y_46_re;
	} else if ((y_46_re <= 1.12e-71) || (!(y_46_re <= 1.15e-16) && (y_46_re <= 2.7e+90))) {
		tmp = x_46_im / y_46_im;
	} else {
		tmp = x_46_re / y_46_re;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if y_46_re <= -2.8e+108:
		tmp = x_46_re / y_46_re
	elif (y_46_re <= 1.12e-71) or (not (y_46_re <= 1.15e-16) and (y_46_re <= 2.7e+90)):
		tmp = x_46_im / y_46_im
	else:
		tmp = x_46_re / y_46_re
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (y_46_re <= -2.8e+108)
		tmp = Float64(x_46_re / y_46_re);
	elseif ((y_46_re <= 1.12e-71) || (!(y_46_re <= 1.15e-16) && (y_46_re <= 2.7e+90)))
		tmp = Float64(x_46_im / y_46_im);
	else
		tmp = Float64(x_46_re / y_46_re);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if (y_46_re <= -2.8e+108)
		tmp = x_46_re / y_46_re;
	elseif ((y_46_re <= 1.12e-71) || (~((y_46_re <= 1.15e-16)) && (y_46_re <= 2.7e+90)))
		tmp = x_46_im / y_46_im;
	else
		tmp = x_46_re / y_46_re;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[y$46$re, -2.8e+108], N[(x$46$re / y$46$re), $MachinePrecision], If[Or[LessEqual[y$46$re, 1.12e-71], And[N[Not[LessEqual[y$46$re, 1.15e-16]], $MachinePrecision], LessEqual[y$46$re, 2.7e+90]]], N[(x$46$im / y$46$im), $MachinePrecision], N[(x$46$re / y$46$re), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y.re < -2.7999999999999998e108 or 1.1199999999999999e-71 < y.re < 1.15e-16 or 2.7e90 < y.re

   1. Initial program 44.4%

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

   2. Taylor expanded in y.re around inf 78.5%

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

   if -2.7999999999999998e108 < y.re < 1.1199999999999999e-71 or 1.15e-16 < y.re < 2.7e90

   1. Initial program 64.8%

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

   2. Taylor expanded in y.re around 0 64.5%

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

4. Final simplification 69.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -2.8 \cdot 10^{+108}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{elif}\;y.re \leq 1.12 \cdot 10^{-71} \lor \neg \left(y.re \leq 1.15 \cdot 10^{-16}\right) \land y.re \leq 2.7 \cdot 10^{+90}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \end{array} \]

Alternative 11: 43.8% 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 57.9%

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

2. Taylor expanded in y.re around 0 48.0%

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

3. Final simplification 48.0%

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

Reproduce

herbie shell --seed 2023228 
(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))))