Average Error: 26.1 → 12.6
Time: 34.2s
Precision: 64
\[\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\]
↓
\[\begin{array}{l} \mathbf{if}\;y.re \le -4.78970221325173028480304050242606063264 \cdot 10^{151}:\\ \;\;\;\;\frac{-x.im}{\mathsf{hypot}\left(y.im, y.re\right)}\\ \mathbf{elif}\;y.re \le 7.209078410675664983392874709740943245737 \cdot 10^{132}:\\ \;\;\;\;\frac{\frac{x.im \cdot y.re}{\mathsf{hypot}\left(y.im, y.re\right)}}{\mathsf{hypot}\left(y.im, y.re\right)} - \frac{\frac{y.im \cdot x.re}{\mathsf{hypot}\left(y.im, y.re\right)}}{\mathsf{hypot}\left(y.im, y.re\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{\mathsf{hypot}\left(y.im, y.re\right)}\\ \end{array}\]
\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 -4.78970221325173028480304050242606063264 \cdot 10^{151}:\\
\;\;\;\;\frac{-x.im}{\mathsf{hypot}\left(y.im, y.re\right)}\\

\mathbf{elif}\;y.re \le 7.209078410675664983392874709740943245737 \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 r2545182 = x_im;
        double r2545183 = y_re;
        double r2545184 = r2545182 * r2545183;
        double r2545185 = x_re;
        double r2545186 = y_im;
        double r2545187 = r2545185 * r2545186;
        double r2545188 = r2545184 - r2545187;
        double r2545189 = r2545183 * r2545183;
        double r2545190 = r2545186 * r2545186;
        double r2545191 = r2545189 + r2545190;
        double r2545192 = r2545188 / r2545191;
        return r2545192;
}
↓
double f(double x_re, double x_im, double y_re, double y_im) {
        double r2545193 = y_re;
        double r2545194 = -4.78970221325173e+151;
        bool r2545195 = r2545193 <= r2545194;
        double r2545196 = x_im;
        double r2545197 = -r2545196;
        double r2545198 = y_im;
        double r2545199 = hypot(r2545198, r2545193);
        double r2545200 = r2545197 / r2545199;
        double r2545201 = 7.209078410675665e+132;
        bool r2545202 = r2545193 <= r2545201;
        double r2545203 = r2545196 * r2545193;
        double r2545204 = r2545203 / r2545199;
        double r2545205 = r2545204 / r2545199;
        double r2545206 = x_re;
        double r2545207 = r2545198 * r2545206;
        double r2545208 = r2545207 / r2545199;
        double r2545209 = r2545208 / r2545199;
        double r2545210 = r2545205 - r2545209;
        double r2545211 = r2545196 / r2545199;
        double r2545212 = r2545202 ? r2545210 : r2545211;
        double r2545213 = r2545195 ? r2545200 : r2545212;
        return r2545213;
}

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 44.6

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

    2. Simplified44.6

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

    4. Applied add-sqr-sqrt44.6

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

    5. Applied associate-/r*44.6

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

    7. Applied fma-udef44.6

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

    8. Applied hypot-def44.6

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

    9. 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 19.2

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

    2. Simplified19.2

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

    4. Applied add-sqr-sqrt19.2

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

    5. Applied associate-/r*19.1

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

    7. Applied fma-udef19.1

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

    8. Applied hypot-def19.1

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

    10. Applied *-un-lft-identity19.1

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

    11. Applied associate-/r*19.1

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

    12. Simplified12.1

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

    14. Applied div-sub12.1

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

    15. Applied div-sub12.1

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

    if 7.209078410675665e+132 < y.re

    1. Initial program 42.3

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

    2. Simplified42.3

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

    4. Applied add-sqr-sqrt42.3

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

    5. Applied associate-/r*42.3

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

    7. Applied fma-udef42.3

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

    8. Applied hypot-def42.3

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

    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 simplification12.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \le -4.78970221325173028480304050242606063264 \cdot 10^{151}:\\ \;\;\;\;\frac{-x.im}{\mathsf{hypot}\left(y.im, y.re\right)}\\ \mathbf{elif}\;y.re \le 7.209078410675664983392874709740943245737 \cdot 10^{132}:\\ \;\;\;\;\frac{\frac{x.im \cdot y.re}{\mathsf{hypot}\left(y.im, y.re\right)}}{\mathsf{hypot}\left(y.im, y.re\right)} - \frac{\frac{y.im \cdot x.re}{\mathsf{hypot}\left(y.im, y.re\right)}}{\mathsf{hypot}\left(y.im, y.re\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{\mathsf{hypot}\left(y.im, y.re\right)}\\ \end{array}\]

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))))