Average Error: 26.8 → 26.4
Time: 9.3s
Precision: 64
\[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \le 1.3029550565949079 \cdot 10^{271}:\\ \;\;\;\;\frac{\frac{x.re \cdot y.re + x.im \cdot y.im}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{\sqrt{\sqrt{y.re \cdot y.re + y.im \cdot y.im}} \cdot \sqrt{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}}\\ \end{array}\]
\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 1.3029550565949079 \cdot 10^{271}:\\
\;\;\;\;\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}}\\

\mathbf{else}:\\
\;\;\;\;\frac{x.im}{\sqrt{\sqrt{y.re \cdot y.re + y.im \cdot y.im}} \cdot \sqrt{\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 r116517 = x_re;
        double r116518 = y_re;
        double r116519 = r116517 * r116518;
        double r116520 = x_im;
        double r116521 = y_im;
        double r116522 = r116520 * r116521;
        double r116523 = r116519 + r116522;
        double r116524 = r116518 * r116518;
        double r116525 = r116521 * r116521;
        double r116526 = r116524 + r116525;
        double r116527 = r116523 / r116526;
        return r116527;
}
↓
double f(double x_re, double x_im, double y_re, double y_im) {
        double r116528 = x_re;
        double r116529 = y_re;
        double r116530 = r116528 * r116529;
        double r116531 = x_im;
        double r116532 = y_im;
        double r116533 = r116531 * r116532;
        double r116534 = r116530 + r116533;
        double r116535 = r116529 * r116529;
        double r116536 = r116532 * r116532;
        double r116537 = r116535 + r116536;
        double r116538 = r116534 / r116537;
        double r116539 = 1.302955056594908e+271;
        bool r116540 = r116538 <= r116539;
        double r116541 = sqrt(r116537);
        double r116542 = r116534 / r116541;
        double r116543 = r116542 / r116541;
        double r116544 = sqrt(r116541);
        double r116545 = r116544 * r116544;
        double r116546 = r116531 / r116545;
        double r116547 = r116540 ? r116543 : r116546;
        return r116547;
}

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))) < 1.302955056594908e+271

    1. Initial program 14.9

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

    3. Applied add-sqr-sqrt14.9

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

    4. Applied associate-/r*14.8

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

    if 1.302955056594908e+271 < (/ (+ (* x.re y.re) (* x.im y.im)) (+ (* y.re y.re) (* y.im y.im)))

    1. Initial program 61.5

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

    3. Applied add-sqr-sqrt61.5

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

    4. Applied associate-/r*61.5

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

    6. Applied add-sqr-sqrt61.5

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

    7. Applied sqrt-prod61.5

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

    8. Taylor expanded around 0 60.2

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

  4. Final simplification26.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \le 1.3029550565949079 \cdot 10^{271}:\\ \;\;\;\;\frac{\frac{x.re \cdot y.re + x.im \cdot y.im}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{\sqrt{\sqrt{y.re \cdot y.re + y.im \cdot y.im}} \cdot \sqrt{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020042 
(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))))