Average Error: 25.8 → 25.1
Time: 12.7s
Precision: 64
\[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \le 4.737122355537421671534518653596175011386 \cdot 10^{285}:\\ \;\;\;\;\frac{\frac{x.re \cdot y.re + x.im \cdot y.im}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-x.re}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}\\ \end{array}\]
\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 4.737122355537421671534518653596175011386 \cdot 10^{285}:\\
\;\;\;\;\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.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 r55544 = x_re;
        double r55545 = y_re;
        double r55546 = r55544 * r55545;
        double r55547 = x_im;
        double r55548 = y_im;
        double r55549 = r55547 * r55548;
        double r55550 = r55546 + r55549;
        double r55551 = r55545 * r55545;
        double r55552 = r55548 * r55548;
        double r55553 = r55551 + r55552;
        double r55554 = r55550 / r55553;
        return r55554;
}
↓
double f(double x_re, double x_im, double y_re, double y_im) {
        double r55555 = x_re;
        double r55556 = y_re;
        double r55557 = r55555 * r55556;
        double r55558 = x_im;
        double r55559 = y_im;
        double r55560 = r55558 * r55559;
        double r55561 = r55557 + r55560;
        double r55562 = r55556 * r55556;
        double r55563 = r55559 * r55559;
        double r55564 = r55562 + r55563;
        double r55565 = r55561 / r55564;
        double r55566 = 4.7371223555374217e+285;
        bool r55567 = r55565 <= r55566;
        double r55568 = sqrt(r55564);
        double r55569 = r55561 / r55568;
        double r55570 = r55569 / r55568;
        double r55571 = -r55555;
        double r55572 = r55571 / r55568;
        double r55573 = r55567 ? r55570 : r55572;
        return r55573;
}

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))) < 4.7371223555374217e+285

    1. Initial program 13.7

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

    3. Applied add-sqr-sqrt13.7

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

    4. Applied associate-/r*13.6

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

    if 4.7371223555374217e+285 < (/ (+ (* x.re y.re) (* x.im y.im)) (+ (* y.re y.re) (* y.im y.im)))

    1. Initial program 62.6

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

    3. Applied add-sqr-sqrt62.7

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

    4. Applied associate-/r*62.6

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

    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 simplification25.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \le 4.737122355537421671534518653596175011386 \cdot 10^{285}:\\ \;\;\;\;\frac{\frac{x.re \cdot y.re + x.im \cdot y.im}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-x.re}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019323 
(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))))