Average Error: 26.5 → 14.6
Time: 7.3s
Precision: binary64
\[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\]
↓
\[\begin{array}{l} \mathbf{if}\;y.im \leq -6.329466807877134 \cdot 10^{+128}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{elif}\;y.im \leq -5.723038204809748 \cdot 10^{-114}:\\ \;\;\;\;\frac{\left(x.re \cdot y.re + y.im \cdot x.im\right) \cdot \frac{1}{\sqrt{{y.re}^{2} + {y.im}^{2}}}}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}\\ \mathbf{elif}\;y.im \leq 1.5571445232288445 \cdot 10^{-162}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{y.im \cdot x.im}{{y.re}^{2}}\\ \mathbf{elif}\;y.im \leq 8.575139954827566 \cdot 10^{+101}:\\ \;\;\;\;\frac{\left(x.re \cdot y.re + y.im \cdot x.im\right) \cdot \frac{1}{\sqrt{{y.re}^{2} + {y.im}^{2}}}}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \end{array}\]
\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}
↓
\begin{array}{l}
\mathbf{if}\;y.im \leq -6.329466807877134 \cdot 10^{+128}:\\
\;\;\;\;\frac{x.im}{y.im}\\

\mathbf{elif}\;y.im \leq -5.723038204809748 \cdot 10^{-114}:\\
\;\;\;\;\frac{\left(x.re \cdot y.re + y.im \cdot x.im\right) \cdot \frac{1}{\sqrt{{y.re}^{2} + {y.im}^{2}}}}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}\\

\mathbf{elif}\;y.im \leq 1.5571445232288445 \cdot 10^{-162}:\\
\;\;\;\;\frac{x.re}{y.re} + \frac{y.im \cdot x.im}{{y.re}^{2}}\\

\mathbf{elif}\;y.im \leq 8.575139954827566 \cdot 10^{+101}:\\
\;\;\;\;\frac{\left(x.re \cdot y.re + y.im \cdot x.im\right) \cdot \frac{1}{\sqrt{{y.re}^{2} + {y.im}^{2}}}}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}\\

\mathbf{else}:\\
\;\;\;\;\frac{x.im}{y.im}\\

\end{array}
(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))))
↓
(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.im -6.329466807877134e+128)
   (/ x.im y.im)
   (if (<= y.im -5.723038204809748e-114)
     (/
      (*
       (+ (* x.re y.re) (* y.im x.im))
       (/ 1.0 (sqrt (+ (pow y.re 2.0) (pow y.im 2.0)))))
      (sqrt (+ (* y.re y.re) (* y.im y.im))))
     (if (<= y.im 1.5571445232288445e-162)
       (+ (/ x.re y.re) (/ (* y.im x.im) (pow y.re 2.0)))
       (if (<= y.im 8.575139954827566e+101)
         (/
          (*
           (+ (* x.re y.re) (* y.im x.im))
           (/ 1.0 (sqrt (+ (pow y.re 2.0) (pow y.im 2.0)))))
          (sqrt (+ (* y.re y.re) (* y.im y.im))))
         (/ x.im y.im))))))
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));
}
↓
double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_im <= -6.329466807877134e+128) {
		tmp = x_46_im / y_46_im;
	} else if (y_46_im <= -5.723038204809748e-114) {
		tmp = (((x_46_re * y_46_re) + (y_46_im * x_46_im)) * (1.0 / sqrt(pow(y_46_re, 2.0) + pow(y_46_im, 2.0)))) / sqrt((y_46_re * y_46_re) + (y_46_im * y_46_im));
	} else if (y_46_im <= 1.5571445232288445e-162) {
		tmp = (x_46_re / y_46_re) + ((y_46_im * x_46_im) / pow(y_46_re, 2.0));
	} else if (y_46_im <= 8.575139954827566e+101) {
		tmp = (((x_46_re * y_46_re) + (y_46_im * x_46_im)) * (1.0 / sqrt(pow(y_46_re, 2.0) + pow(y_46_im, 2.0)))) / sqrt((y_46_re * y_46_re) + (y_46_im * y_46_im));
	} else {
		tmp = x_46_im / y_46_im;
	}
	return tmp;
}

Error

Bits error versus x.re

Bits error versus x.im

Bits error versus y.re

Bits error versus y.im

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 3 regimes

  2. if y.im < -6.3294668078771342e128 or 8.575139954827566e101 < y.im

    1. Initial program 40.8

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

    2. Taylor expanded around 0 15.7

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

    if -6.3294668078771342e128 < y.im < -5.72303820480974823e-114 or 1.5571445232288445e-162 < y.im < 8.575139954827566e101

    1. Initial program 16.3

      \[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\]
    2. Using strategy rm

    3. Applied add-sqr-sqrt_binary64_112316.3

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

    4. Applied associate-/r*_binary64_104516.2

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

    5. Simplified16.2

      \[\leadsto \frac{\color{blue}{\frac{x.re \cdot y.re + x.im \cdot y.im}{\sqrt{{y.re}^{2} + {y.im}^{2}}}}}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}\]
    6. Using strategy rm

    7. Applied div-inv_binary64_109816.3

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

    if -5.72303820480974823e-114 < y.im < 1.5571445232288445e-162

    1. Initial program 25.1

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

    2. Taylor expanded around inf 10.2

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

    3. Simplified10.2

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

  4. Final simplification14.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -6.329466807877134 \cdot 10^{+128}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{elif}\;y.im \leq -5.723038204809748 \cdot 10^{-114}:\\ \;\;\;\;\frac{\left(x.re \cdot y.re + y.im \cdot x.im\right) \cdot \frac{1}{\sqrt{{y.re}^{2} + {y.im}^{2}}}}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}\\ \mathbf{elif}\;y.im \leq 1.5571445232288445 \cdot 10^{-162}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{y.im \cdot x.im}{{y.re}^{2}}\\ \mathbf{elif}\;y.im \leq 8.575139954827566 \cdot 10^{+101}:\\ \;\;\;\;\frac{\left(x.re \cdot y.re + y.im \cdot x.im\right) \cdot \frac{1}{\sqrt{{y.re}^{2} + {y.im}^{2}}}}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \end{array}\]

Reproduce

herbie shell --seed 2020357 
(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))))