Average Error: 1.1 → 1.1
Time: 10.4s
Precision: 64
\[\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\]
↓
\[\frac{1.0}{\frac{y.re \cdot y.re + y.im \cdot y.im}{x.im \cdot y.re - x.re \cdot y.im}}\]
double f(double x_re, double x_im, double y_re, double y_im) {
        double r1530402 = x_im;
        double r1530403 = y_re;
        double r1530404 = r1530402 * r1530403;
        double r1530405 = x_re;
        double r1530406 = y_im;
        double r1530407 = r1530405 * r1530406;
        double r1530408 = r1530404 - r1530407;
        double r1530409 = r1530403 * r1530403;
        double r1530410 = r1530406 * r1530406;
        double r1530411 = r1530409 + r1530410;
        double r1530412 = r1530408 / r1530411;
        return r1530412;
}
↓
double f(double x_re, double x_im, double y_re, double y_im) {
        double r1530413 = 1.0;
        double r1530414 = y_re;
        double r1530415 = r1530414 * r1530414;
        double r1530416 = y_im;
        double r1530417 = r1530416 * r1530416;
        double r1530418 = r1530415 + r1530417;
        double r1530419 = x_im;
        double r1530420 = r1530419 * r1530414;
        double r1530421 = x_re;
        double r1530422 = r1530421 * r1530416;
        double r1530423 = r1530420 - r1530422;
        double r1530424 = r1530418 / r1530423;
        double r1530425 = r1530413 / r1530424;
        return r1530425;
}


\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}
↓
\frac{1.0}{\frac{y.re \cdot y.re + y.im \cdot y.im}{x.im \cdot y.re - x.re \cdot y.im}}

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 1.1

    \[\frac{\left(\left(x.im \cdot y.re\right) - \left(x.re \cdot y.im\right)\right)}{\left(\frac{\left(y.re \cdot y.re\right)}{\left(y.im \cdot y.im\right)}\right)}\]
  2. Using strategy rm

  3. Applied p16-flip--2.1

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

  5. Applied difference-of-squares2.0

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

  6. Applied associate-/l*1.1

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

  8. Applied associate-/r/1.1

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

  9. Applied associate-/l*1.1

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

  10. Simplified1.1

    \[\leadsto \frac{\color{blue}{\left(1.0\right)}}{\left(\frac{\left(\frac{\left(y.re \cdot y.re\right)}{\left(y.im \cdot y.im\right)}\right)}{\left(\left(x.im \cdot y.re\right) - \left(x.re \cdot y.im\right)\right)}\right)}\]

  11. Final simplification1.1

    \[\leadsto \frac{1.0}{\frac{y.re \cdot y.re + y.im \cdot y.im}{x.im \cdot y.re - x.re \cdot y.im}}\]

Reproduce

herbie shell --seed 2019102 
(FPCore (x.re x.im y.re y.im)
  :name "_divideComplex, imaginary part"
  (/.p16 (-.p16 (*.p16 x.im y.re) (*.p16 x.re y.im)) (+.p16 (*.p16 y.re y.re) (*.p16 y.im y.im))))