Average Error: 26.5 → 4.3
Time: 14.8s
Precision: 64
\[\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\]
↓
\[\begin{array}{l} \mathbf{if}\;x.im \le -61.10607895990863625002020853571593761444 \lor \neg \left(x.im \le 1.310939369483501924587354661839021785547 \cdot 10^{-16}\right):\\ \;\;\;\;\frac{x.im}{y.re + \frac{y.im}{\frac{y.re}{y.im}}} - \frac{x.re}{y.im + \frac{y.re \cdot y.re}{y.im}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{\frac{y.im \cdot y.im}{y.re} + y.re} - \frac{x.re}{y.im + y.re \cdot \frac{y.re}{y.im}}\\ \end{array}\]
\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}\;x.im \le -61.10607895990863625002020853571593761444 \lor \neg \left(x.im \le 1.310939369483501924587354661839021785547 \cdot 10^{-16}\right):\\
\;\;\;\;\frac{x.im}{y.re + \frac{y.im}{\frac{y.re}{y.im}}} - \frac{x.re}{y.im + \frac{y.re \cdot y.re}{y.im}}\\

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

\end{array}
double f(double x_re, double x_im, double y_re, double y_im) {
        double r59718 = x_im;
        double r59719 = y_re;
        double r59720 = r59718 * r59719;
        double r59721 = x_re;
        double r59722 = y_im;
        double r59723 = r59721 * r59722;
        double r59724 = r59720 - r59723;
        double r59725 = r59719 * r59719;
        double r59726 = r59722 * r59722;
        double r59727 = r59725 + r59726;
        double r59728 = r59724 / r59727;
        return r59728;
}
↓
double f(double x_re, double x_im, double y_re, double y_im) {
        double r59729 = x_im;
        double r59730 = -61.106078959908636;
        bool r59731 = r59729 <= r59730;
        double r59732 = 1.310939369483502e-16;
        bool r59733 = r59729 <= r59732;
        double r59734 = !r59733;
        bool r59735 = r59731 || r59734;
        double r59736 = y_re;
        double r59737 = y_im;
        double r59738 = r59736 / r59737;
        double r59739 = r59737 / r59738;
        double r59740 = r59736 + r59739;
        double r59741 = r59729 / r59740;
        double r59742 = x_re;
        double r59743 = r59736 * r59736;
        double r59744 = r59743 / r59737;
        double r59745 = r59737 + r59744;
        double r59746 = r59742 / r59745;
        double r59747 = r59741 - r59746;
        double r59748 = r59737 * r59737;
        double r59749 = r59748 / r59736;
        double r59750 = r59749 + r59736;
        double r59751 = r59729 / r59750;
        double r59752 = r59736 * r59738;
        double r59753 = r59737 + r59752;
        double r59754 = r59742 / r59753;
        double r59755 = r59751 - r59754;
        double r59756 = r59735 ? r59747 : r59755;
        return r59756;
}

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.im < -61.106078959908636 or 1.310939369483502e-16 < x.im

    1. Initial program 32.8

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

    3. Applied div-sub32.8

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

    4. Simplified30.2

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

    5. Simplified28.4

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

    6. Taylor expanded around 0 16.0

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

    7. Simplified16.0

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

    8. Taylor expanded around 0 8.8

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

    9. Simplified8.8

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

    11. Applied associate-/l*4.9

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

    if -61.106078959908636 < x.im < 1.310939369483502e-16

    1. Initial program 20.2

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

    3. Applied div-sub20.2

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

    4. Simplified19.7

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

    5. Simplified18.3

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

    6. Taylor expanded around 0 15.8

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

    7. Simplified15.8

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

    8. Taylor expanded around 0 7.3

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

    9. Simplified7.3

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

    11. Applied *-un-lft-identity7.3

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

    12. Applied times-frac3.8

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

    13. Simplified3.8

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

  4. Final simplification4.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \le -61.10607895990863625002020853571593761444 \lor \neg \left(x.im \le 1.310939369483501924587354661839021785547 \cdot 10^{-16}\right):\\ \;\;\;\;\frac{x.im}{y.re + \frac{y.im}{\frac{y.re}{y.im}}} - \frac{x.re}{y.im + \frac{y.re \cdot y.re}{y.im}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{\frac{y.im \cdot y.im}{y.re} + y.re} - \frac{x.re}{y.im + y.re \cdot \frac{y.re}{y.im}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019196 
(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))))