Average Error: 26.7 → 13.2
Time: 40.2s
Precision: 64
\[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\]
↓
\[\begin{array}{l} \mathbf{if}\;y.im \le -1.258889379708196059634998409540390148429 \cdot 10^{166}:\\ \;\;\;\;\frac{-x.im}{\mathsf{hypot}\left(y.im, y.re\right)}\\ \mathbf{elif}\;y.im \le 2.476328044238816658061320815262943904384 \cdot 10^{164}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x.im, y.im, y.re \cdot x.re\right)}{\mathsf{hypot}\left(y.im, y.re\right)} \cdot \frac{1}{\mathsf{hypot}\left(y.im, y.re\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{\mathsf{hypot}\left(y.im, y.re\right)}\\ \end{array}\]
\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}
↓
\begin{array}{l}
\mathbf{if}\;y.im \le -1.258889379708196059634998409540390148429 \cdot 10^{166}:\\
\;\;\;\;\frac{-x.im}{\mathsf{hypot}\left(y.im, y.re\right)}\\

\mathbf{elif}\;y.im \le 2.476328044238816658061320815262943904384 \cdot 10^{164}:\\
\;\;\;\;\frac{\mathsf{fma}\left(x.im, y.im, y.re \cdot x.re\right)}{\mathsf{hypot}\left(y.im, y.re\right)} \cdot \frac{1}{\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 r2646302 = x_re;
        double r2646303 = y_re;
        double r2646304 = r2646302 * r2646303;
        double r2646305 = x_im;
        double r2646306 = y_im;
        double r2646307 = r2646305 * r2646306;
        double r2646308 = r2646304 + r2646307;
        double r2646309 = r2646303 * r2646303;
        double r2646310 = r2646306 * r2646306;
        double r2646311 = r2646309 + r2646310;
        double r2646312 = r2646308 / r2646311;
        return r2646312;
}
↓
double f(double x_re, double x_im, double y_re, double y_im) {
        double r2646313 = y_im;
        double r2646314 = -1.258889379708196e+166;
        bool r2646315 = r2646313 <= r2646314;
        double r2646316 = x_im;
        double r2646317 = -r2646316;
        double r2646318 = y_re;
        double r2646319 = hypot(r2646313, r2646318);
        double r2646320 = r2646317 / r2646319;
        double r2646321 = 2.4763280442388167e+164;
        bool r2646322 = r2646313 <= r2646321;
        double r2646323 = x_re;
        double r2646324 = r2646318 * r2646323;
        double r2646325 = fma(r2646316, r2646313, r2646324);
        double r2646326 = r2646325 / r2646319;
        double r2646327 = 1.0;
        double r2646328 = r2646327 / r2646319;
        double r2646329 = r2646326 * r2646328;
        double r2646330 = r2646316 / r2646319;
        double r2646331 = r2646322 ? r2646329 : r2646330;
        double r2646332 = r2646315 ? r2646320 : r2646331;
        return r2646332;
}

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 3 regimes

  2. if y.im < -1.258889379708196e+166

    1. Initial program 45.9

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

    2. Simplified45.9

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

    4. Applied add-sqr-sqrt45.9

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

    5. Applied associate-/r*45.9

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

    7. Applied fma-udef45.9

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

    8. Applied hypot-def45.9

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

    10. Applied *-un-lft-identity45.9

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

    11. Applied associate-/r*45.9

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

    12. Simplified29.4

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

    13. Taylor expanded around -inf 13.0

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

    14. Simplified13.0

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

    if -1.258889379708196e+166 < y.im < 2.4763280442388167e+164

    1. Initial program 20.5

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

    2. Simplified20.5

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

    4. Applied add-sqr-sqrt20.5

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

    5. Applied associate-/r*20.4

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

    7. Applied fma-udef20.4

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

    8. Applied hypot-def20.4

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

    10. Applied *-un-lft-identity20.4

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

    11. Applied associate-/r*20.4

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

    12. Simplified13.2

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

    14. Applied div-inv13.3

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

    if 2.4763280442388167e+164 < y.im

    1. Initial program 45.3

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

    2. Simplified45.3

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

    4. Applied add-sqr-sqrt45.3

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

    5. Applied associate-/r*45.3

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

    7. Applied fma-udef45.3

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

    8. Applied hypot-def45.3

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

    9. Taylor expanded around 0 13.0

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

  4. Final simplification13.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \le -1.258889379708196059634998409540390148429 \cdot 10^{166}:\\ \;\;\;\;\frac{-x.im}{\mathsf{hypot}\left(y.im, y.re\right)}\\ \mathbf{elif}\;y.im \le 2.476328044238816658061320815262943904384 \cdot 10^{164}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x.im, y.im, y.re \cdot x.re\right)}{\mathsf{hypot}\left(y.im, y.re\right)} \cdot \frac{1}{\mathsf{hypot}\left(y.im, y.re\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{\mathsf{hypot}\left(y.im, y.re\right)}\\ \end{array}\]

Reproduce

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