Average Error: 26.4 → 25.7
Time: 7.7s
Precision: 64
\[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \le 3.29064580259523516 \cdot 10^{293}:\\ \;\;\;\;\frac{x.re \cdot y.re + x.im \cdot y.im}{\sqrt{y.re \cdot y.re + y.im \cdot y.im} \cdot \sqrt{y.re \cdot y.re + y.im \cdot y.im}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-x.re}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}\\ \end{array}\]
\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}\;\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \le 3.29064580259523516 \cdot 10^{293}:\\
\;\;\;\;\frac{x.re \cdot y.re + x.im \cdot y.im}{\sqrt{y.re \cdot y.re + y.im \cdot y.im} \cdot \sqrt{y.re \cdot y.re + y.im \cdot y.im}}\\

\mathbf{else}:\\
\;\;\;\;\frac{-x.re}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}\\

\end{array}
double f(double x_re, double x_im, double y_re, double y_im) {
        double r66238 = x_re;
        double r66239 = y_re;
        double r66240 = r66238 * r66239;
        double r66241 = x_im;
        double r66242 = y_im;
        double r66243 = r66241 * r66242;
        double r66244 = r66240 + r66243;
        double r66245 = r66239 * r66239;
        double r66246 = r66242 * r66242;
        double r66247 = r66245 + r66246;
        double r66248 = r66244 / r66247;
        return r66248;
}
↓
double f(double x_re, double x_im, double y_re, double y_im) {
        double r66249 = x_re;
        double r66250 = y_re;
        double r66251 = r66249 * r66250;
        double r66252 = x_im;
        double r66253 = y_im;
        double r66254 = r66252 * r66253;
        double r66255 = r66251 + r66254;
        double r66256 = r66250 * r66250;
        double r66257 = r66253 * r66253;
        double r66258 = r66256 + r66257;
        double r66259 = r66255 / r66258;
        double r66260 = 3.290645802595235e+293;
        bool r66261 = r66259 <= r66260;
        double r66262 = sqrt(r66258);
        double r66263 = r66262 * r66262;
        double r66264 = r66255 / r66263;
        double r66265 = -r66249;
        double r66266 = r66265 / r66262;
        double r66267 = r66261 ? r66264 : r66266;
        return r66267;
}

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 2 regimes

  2. if (/ (+ (* x.re y.re) (* x.im y.im)) (+ (* y.re y.re) (* y.im y.im))) < 3.290645802595235e+293

    1. Initial program 14.4

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

    3. Applied add-sqr-sqrt14.4

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

    if 3.290645802595235e+293 < (/ (+ (* x.re y.re) (* x.im y.im)) (+ (* y.re y.re) (* y.im y.im)))

    1. Initial program 63.2

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

    3. Applied add-sqr-sqrt63.2

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

    4. Applied associate-/r*63.2

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

    5. Taylor expanded around -inf 60.2

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

    6. Simplified60.2

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

  4. Final simplification25.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \le 3.29064580259523516 \cdot 10^{293}:\\ \;\;\;\;\frac{x.re \cdot y.re + x.im \cdot y.im}{\sqrt{y.re \cdot y.re + y.im \cdot y.im} \cdot \sqrt{y.re \cdot y.re + y.im \cdot y.im}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-x.re}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020046 
(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))))