Average Error: 25.7 → 14.2
Time: 4.1s
Precision: 64
\[\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\]
↓
\[\begin{array}{l} \mathbf{if}\;y.im \le -1.33199508007579152 \cdot 10^{154}:\\ \;\;\;\;\frac{\left(x.im \cdot y.re - x.re \cdot y.im\right) \cdot \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.im \le -0.29009036449311909:\\ \;\;\;\;\frac{x.im}{\frac{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)}{y.re}} - \frac{x.re}{\frac{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)}{y.im}}\\ \mathbf{elif}\;y.im \le 1.37238418162211356 \cdot 10^{-46}:\\ \;\;\;\;\frac{\frac{1}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{x.im \cdot y.re - x.re \cdot y.im}}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.im \le 1.24061954906872438 \cdot 10^{154}:\\ \;\;\;\;\frac{x.im}{\frac{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)}{y.re}} - \frac{x.re}{\frac{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)}{y.im}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1 \cdot x.re}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \end{array}\]
\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.im \le -1.33199508007579152 \cdot 10^{154}:\\
\;\;\;\;\frac{\left(x.im \cdot y.re - x.re \cdot y.im\right) \cdot \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)}}{\mathsf{hypot}\left(y.re, y.im\right)}\\

\mathbf{elif}\;y.im \le -0.29009036449311909:\\
\;\;\;\;\frac{x.im}{\frac{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)}{y.re}} - \frac{x.re}{\frac{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)}{y.im}}\\

\mathbf{elif}\;y.im \le 1.37238418162211356 \cdot 10^{-46}:\\
\;\;\;\;\frac{\frac{1}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{x.im \cdot y.re - x.re \cdot y.im}}}{\mathsf{hypot}\left(y.re, y.im\right)}\\

\mathbf{elif}\;y.im \le 1.24061954906872438 \cdot 10^{154}:\\
\;\;\;\;\frac{x.im}{\frac{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)}{y.re}} - \frac{x.re}{\frac{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)}{y.im}}\\

\mathbf{else}:\\
\;\;\;\;\frac{-1 \cdot x.re}{\mathsf{hypot}\left(y.re, y.im\right)}\\

\end{array}
double f(double x_re, double x_im, double y_re, double y_im) {
        double r52852 = x_im;
        double r52853 = y_re;
        double r52854 = r52852 * r52853;
        double r52855 = x_re;
        double r52856 = y_im;
        double r52857 = r52855 * r52856;
        double r52858 = r52854 - r52857;
        double r52859 = r52853 * r52853;
        double r52860 = r52856 * r52856;
        double r52861 = r52859 + r52860;
        double r52862 = r52858 / r52861;
        return r52862;
}
↓
double f(double x_re, double x_im, double y_re, double y_im) {
        double r52863 = y_im;
        double r52864 = -1.3319950800757915e+154;
        bool r52865 = r52863 <= r52864;
        double r52866 = x_im;
        double r52867 = y_re;
        double r52868 = r52866 * r52867;
        double r52869 = x_re;
        double r52870 = r52869 * r52863;
        double r52871 = r52868 - r52870;
        double r52872 = 1.0;
        double r52873 = hypot(r52867, r52863);
        double r52874 = r52872 / r52873;
        double r52875 = r52871 * r52874;
        double r52876 = r52875 / r52873;
        double r52877 = -0.2900903644931191;
        bool r52878 = r52863 <= r52877;
        double r52879 = r52863 * r52863;
        double r52880 = fma(r52867, r52867, r52879);
        double r52881 = r52880 / r52867;
        double r52882 = r52866 / r52881;
        double r52883 = r52880 / r52863;
        double r52884 = r52869 / r52883;
        double r52885 = r52882 - r52884;
        double r52886 = 1.3723841816221136e-46;
        bool r52887 = r52863 <= r52886;
        double r52888 = r52873 / r52871;
        double r52889 = r52872 / r52888;
        double r52890 = r52889 / r52873;
        double r52891 = 1.2406195490687244e+154;
        bool r52892 = r52863 <= r52891;
        double r52893 = -1.0;
        double r52894 = r52893 * r52869;
        double r52895 = r52894 / r52873;
        double r52896 = r52892 ? r52885 : r52895;
        double r52897 = r52887 ? r52890 : r52896;
        double r52898 = r52878 ? r52885 : r52897;
        double r52899 = r52865 ? r52876 : r52898;
        return r52899;
}

Error

Bits error versus x.re

Bits error versus x.im

Bits error versus y.re

Bits error versus y.im

Derivation

  1. Split input into 4 regimes

  2. if y.im < -1.3319950800757915e+154

    1. Initial program 44.3

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

    3. Applied add-sqr-sqrt44.3

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

    4. Applied *-un-lft-identity44.3

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

    5. Applied times-frac44.3

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

    6. Simplified44.3

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

    7. Simplified28.6

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

    9. Applied associate-*r/28.6

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

    10. Simplified28.6

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

    12. Applied div-inv28.6

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

    if -1.3319950800757915e+154 < y.im < -0.2900903644931191 or 1.3723841816221136e-46 < y.im < 1.2406195490687244e+154

    1. Initial program 18.8

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

    3. Applied div-sub18.8

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

    4. Simplified18.3

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

    5. Simplified12.9

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

    if -0.2900903644931191 < y.im < 1.3723841816221136e-46

    1. Initial program 18.9

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

    3. Applied add-sqr-sqrt18.9

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

    4. Applied *-un-lft-identity18.9

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

    5. Applied times-frac18.9

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

    6. Simplified18.9

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

    7. Simplified11.0

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

    9. Applied associate-*r/11.0

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

    10. Simplified10.9

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

    12. Applied clear-num11.0

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

    if 1.2406195490687244e+154 < y.im

    1. Initial program 45.2

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

    3. Applied add-sqr-sqrt45.2

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

    4. Applied *-un-lft-identity45.2

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

    5. Applied times-frac45.2

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

    6. Simplified45.2

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

    7. Simplified29.7

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

    9. Applied associate-*r/29.7

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

    10. Simplified29.7

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

    11. Taylor expanded around 0 13.4

      \[\leadsto \frac{\color{blue}{-1 \cdot x.re}}{\mathsf{hypot}\left(y.re, y.im\right)}\]
  3. Recombined 4 regimes into one program.

  4. Final simplification14.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \le -1.33199508007579152 \cdot 10^{154}:\\ \;\;\;\;\frac{\left(x.im \cdot y.re - x.re \cdot y.im\right) \cdot \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.im \le -0.29009036449311909:\\ \;\;\;\;\frac{x.im}{\frac{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)}{y.re}} - \frac{x.re}{\frac{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)}{y.im}}\\ \mathbf{elif}\;y.im \le 1.37238418162211356 \cdot 10^{-46}:\\ \;\;\;\;\frac{\frac{1}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{x.im \cdot y.re - x.re \cdot y.im}}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.im \le 1.24061954906872438 \cdot 10^{154}:\\ \;\;\;\;\frac{x.im}{\frac{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)}{y.re}} - \frac{x.re}{\frac{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)}{y.im}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1 \cdot x.re}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020003 +o rules:numerics
(FPCore (x.re x.im y.re y.im)
  :name "_divideComplex, imaginary part"
  :precision binary64
  (/ (- (* x.im y.re) (* x.re y.im)) (+ (* y.re y.re) (* y.im y.im))))