Average Error: 25.7 → 12.3
Time: 14.2s
Precision: 64
\[\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\]
↓
\[\begin{array}{l} \mathbf{if}\;y.re \le -4.78970221325173 \cdot 10^{+151}:\\ \;\;\;\;\frac{-x.im}{\mathsf{hypot}\left(y.im, y.re\right)}\\ \mathbf{elif}\;y.re \le 7.209078410675665 \cdot 10^{+132}:\\ \;\;\;\;\frac{\frac{x.im \cdot y.re}{\mathsf{hypot}\left(y.im, y.re\right)}}{\mathsf{hypot}\left(y.im, y.re\right)} - \frac{\frac{y.im \cdot x.re}{\mathsf{hypot}\left(y.im, y.re\right)}}{\mathsf{hypot}\left(y.im, y.re\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{\mathsf{hypot}\left(y.im, y.re\right)}\\ \end{array}\]
\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}
↓
\begin{array}{l}
\mathbf{if}\;y.re \le -4.78970221325173 \cdot 10^{+151}:\\
\;\;\;\;\frac{-x.im}{\mathsf{hypot}\left(y.im, y.re\right)}\\

\mathbf{elif}\;y.re \le 7.209078410675665 \cdot 10^{+132}:\\
\;\;\;\;\frac{\frac{x.im \cdot y.re}{\mathsf{hypot}\left(y.im, y.re\right)}}{\mathsf{hypot}\left(y.im, y.re\right)} - \frac{\frac{y.im \cdot x.re}{\mathsf{hypot}\left(y.im, y.re\right)}}{\mathsf{hypot}\left(y.im, y.re\right)}\\

\mathbf{else}:\\
\;\;\;\;\frac{x.im}{\mathsf{hypot}\left(y.im, y.re\right)}\\

\end{array}
double f(double x_re, double x_im, double y_re, double y_im) {
        double r2521612 = x_im;
        double r2521613 = y_re;
        double r2521614 = r2521612 * r2521613;
        double r2521615 = x_re;
        double r2521616 = y_im;
        double r2521617 = r2521615 * r2521616;
        double r2521618 = r2521614 - r2521617;
        double r2521619 = r2521613 * r2521613;
        double r2521620 = r2521616 * r2521616;
        double r2521621 = r2521619 + r2521620;
        double r2521622 = r2521618 / r2521621;
        return r2521622;
}
↓
double f(double x_re, double x_im, double y_re, double y_im) {
        double r2521623 = y_re;
        double r2521624 = -4.78970221325173e+151;
        bool r2521625 = r2521623 <= r2521624;
        double r2521626 = x_im;
        double r2521627 = -r2521626;
        double r2521628 = y_im;
        double r2521629 = hypot(r2521628, r2521623);
        double r2521630 = r2521627 / r2521629;
        double r2521631 = 7.209078410675665e+132;
        bool r2521632 = r2521623 <= r2521631;
        double r2521633 = r2521626 * r2521623;
        double r2521634 = r2521633 / r2521629;
        double r2521635 = r2521634 / r2521629;
        double r2521636 = x_re;
        double r2521637 = r2521628 * r2521636;
        double r2521638 = r2521637 / r2521629;
        double r2521639 = r2521638 / r2521629;
        double r2521640 = r2521635 - r2521639;
        double r2521641 = r2521626 / r2521629;
        double r2521642 = r2521632 ? r2521640 : r2521641;
        double r2521643 = r2521625 ? r2521630 : r2521642;
        return r2521643;
}

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.re < -4.78970221325173e+151

    1. Initial program 44.1

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

    2. Simplified44.1

      \[\leadsto \color{blue}{\frac{x.im \cdot y.re - x.re \cdot y.im}{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)}}\]
    3. Using strategy rm

    4. Applied add-sqr-sqrt44.1

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

    5. Applied associate-/r*44.1

      \[\leadsto \color{blue}{\frac{\frac{x.im \cdot y.re - x.re \cdot y.im}{\sqrt{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)}}}{\sqrt{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)}}}\]
    6. Using strategy rm

    7. Applied fma-udef44.1

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

    8. Applied hypot-def44.1

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

    9. Taylor expanded around -inf 13.0

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

    10. Simplified13.0

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

    if -4.78970221325173e+151 < y.re < 7.209078410675665e+132

    1. Initial program 18.7

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

    2. Simplified18.7

      \[\leadsto \color{blue}{\frac{x.im \cdot y.re - x.re \cdot y.im}{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)}}\]
    3. Using strategy rm

    4. Applied add-sqr-sqrt18.8

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

    5. Applied associate-/r*18.7

      \[\leadsto \color{blue}{\frac{\frac{x.im \cdot y.re - x.re \cdot y.im}{\sqrt{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)}}}{\sqrt{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)}}}\]
    6. Using strategy rm

    7. Applied fma-udef18.7

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

    8. Applied hypot-def18.6

      \[\leadsto \frac{\frac{x.im \cdot y.re - x.re \cdot y.im}{\sqrt{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)}}}{\color{blue}{\mathsf{hypot}\left(y.im, y.re\right)}}\]
    9. Using strategy rm

    10. Applied *-un-lft-identity18.6

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

    11. Applied associate-/r*18.6

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

    12. Simplified11.7

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

    14. Applied div-sub11.7

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

    15. Applied div-sub11.7

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

    if 7.209078410675665e+132 < y.re

    1. Initial program 41.8

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

    2. Simplified41.8

      \[\leadsto \color{blue}{\frac{x.im \cdot y.re - x.re \cdot y.im}{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)}}\]
    3. Using strategy rm

    4. Applied add-sqr-sqrt41.8

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

    5. Applied associate-/r*41.8

      \[\leadsto \color{blue}{\frac{\frac{x.im \cdot y.re - x.re \cdot y.im}{\sqrt{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)}}}{\sqrt{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)}}}\]
    6. Using strategy rm

    7. Applied fma-udef41.8

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

    8. Applied hypot-def41.8

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

    9. Taylor expanded around inf 14.4

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

  4. Final simplification12.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \le -4.78970221325173 \cdot 10^{+151}:\\ \;\;\;\;\frac{-x.im}{\mathsf{hypot}\left(y.im, y.re\right)}\\ \mathbf{elif}\;y.re \le 7.209078410675665 \cdot 10^{+132}:\\ \;\;\;\;\frac{\frac{x.im \cdot y.re}{\mathsf{hypot}\left(y.im, y.re\right)}}{\mathsf{hypot}\left(y.im, y.re\right)} - \frac{\frac{y.im \cdot x.re}{\mathsf{hypot}\left(y.im, y.re\right)}}{\mathsf{hypot}\left(y.im, y.re\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{\mathsf{hypot}\left(y.im, y.re\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019168 +o rules:numerics
(FPCore (x.re x.im y.re y.im)
  :name "_divideComplex, imaginary part"
  (/ (- (* x.im y.re) (* x.re y.im)) (+ (* y.re y.re) (* y.im y.im))))