Average Error: 25.9 → 1.0
Time: 4.2s
Precision: 64
\[\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\]
↓
\[\frac{\mathsf{fma}\left(\frac{x.im}{\sqrt{\mathsf{hypot}\left(y.re, y.im\right)}}, \frac{y.re}{\sqrt{\mathsf{hypot}\left(y.re, y.im\right)}}, -\frac{y.im}{\sqrt{\mathsf{hypot}\left(y.re, y.im\right)}} \cdot \frac{x.re}{\sqrt{\mathsf{hypot}\left(y.re, y.im\right)}}\right) + \frac{y.im}{\sqrt{\mathsf{hypot}\left(y.re, y.im\right)}} \cdot \left(\left(-\frac{x.re}{\sqrt{\mathsf{hypot}\left(y.re, y.im\right)}}\right) + \frac{x.re}{\sqrt{\mathsf{hypot}\left(y.re, y.im\right)}}\right)}{\mathsf{hypot}\left(y.re, y.im\right)}\]
\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}
↓
\frac{\mathsf{fma}\left(\frac{x.im}{\sqrt{\mathsf{hypot}\left(y.re, y.im\right)}}, \frac{y.re}{\sqrt{\mathsf{hypot}\left(y.re, y.im\right)}}, -\frac{y.im}{\sqrt{\mathsf{hypot}\left(y.re, y.im\right)}} \cdot \frac{x.re}{\sqrt{\mathsf{hypot}\left(y.re, y.im\right)}}\right) + \frac{y.im}{\sqrt{\mathsf{hypot}\left(y.re, y.im\right)}} \cdot \left(\left(-\frac{x.re}{\sqrt{\mathsf{hypot}\left(y.re, y.im\right)}}\right) + \frac{x.re}{\sqrt{\mathsf{hypot}\left(y.re, y.im\right)}}\right)}{\mathsf{hypot}\left(y.re, y.im\right)}
double f(double x_re, double x_im, double y_re, double y_im) {
        double r72217 = x_im;
        double r72218 = y_re;
        double r72219 = r72217 * r72218;
        double r72220 = x_re;
        double r72221 = y_im;
        double r72222 = r72220 * r72221;
        double r72223 = r72219 - r72222;
        double r72224 = r72218 * r72218;
        double r72225 = r72221 * r72221;
        double r72226 = r72224 + r72225;
        double r72227 = r72223 / r72226;
        return r72227;
}
↓
double f(double x_re, double x_im, double y_re, double y_im) {
        double r72228 = x_im;
        double r72229 = y_re;
        double r72230 = y_im;
        double r72231 = hypot(r72229, r72230);
        double r72232 = sqrt(r72231);
        double r72233 = r72228 / r72232;
        double r72234 = r72229 / r72232;
        double r72235 = r72230 / r72232;
        double r72236 = x_re;
        double r72237 = r72236 / r72232;
        double r72238 = r72235 * r72237;
        double r72239 = -r72238;
        double r72240 = fma(r72233, r72234, r72239);
        double r72241 = -r72237;
        double r72242 = r72241 + r72237;
        double r72243 = r72235 * r72242;
        double r72244 = r72240 + r72243;
        double r72245 = r72244 / r72231;
        return r72245;
}

Error

Bits error versus x.re

Bits error versus x.im

Bits error versus y.re

Bits error versus y.im

Derivation

  1. Initial program 25.9

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

  3. Applied add-sqr-sqrt25.9

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

  4. Applied *-un-lft-identity25.9

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

  5. Applied times-frac25.9

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

  6. Simplified25.9

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

  7. Simplified17.0

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

  9. Applied *-un-lft-identity17.0

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

  10. Applied associate-*l*17.0

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

  11. Simplified16.9

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

  13. Applied div-sub16.9

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

  15. Applied add-sqr-sqrt17.0

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

  16. Applied times-frac9.4

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

  17. Applied add-sqr-sqrt9.5

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

  18. Applied times-frac1.0

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

  19. Applied prod-diff1.0

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

  20. Simplified1.0

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

  21. Final simplification1.0

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

Reproduce

herbie shell --seed 2020001 +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))))