Average Error: 1.1 → 1.1
Time: 8.9s
Precision: 64
\[\frac{\left(\left(x.im \cdot y.re\right) - \left(x.re \cdot y.im\right)\right)}{\left(\frac{\left(y.re \cdot y.re\right)}{\left(y.im \cdot y.im\right)}\right)}\]
↓
\[\frac{\frac{x.im \cdot y.re + x.re \cdot y.im}{\frac{x.im \cdot y.re + x.re \cdot y.im}{x.im \cdot y.re - x.re \cdot y.im}}}{y.re \cdot y.re + y.im \cdot y.im}\]
\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)}
↓
\frac{\frac{x.im \cdot y.re + x.re \cdot y.im}{\frac{x.im \cdot y.re + x.re \cdot y.im}{x.im \cdot y.re - x.re \cdot y.im}}}{y.re \cdot y.re + y.im \cdot y.im}
double f(double x_re, double x_im, double y_re, double y_im) {
        double r1422917 = x_im;
        double r1422918 = y_re;
        double r1422919 = r1422917 * r1422918;
        double r1422920 = x_re;
        double r1422921 = y_im;
        double r1422922 = r1422920 * r1422921;
        double r1422923 = r1422919 - r1422922;
        double r1422924 = r1422918 * r1422918;
        double r1422925 = r1422921 * r1422921;
        double r1422926 = r1422924 + r1422925;
        double r1422927 = r1422923 / r1422926;
        return r1422927;
}
↓
double f(double x_re, double x_im, double y_re, double y_im) {
        double r1422928 = x_im;
        double r1422929 = y_re;
        double r1422930 = r1422928 * r1422929;
        double r1422931 = x_re;
        double r1422932 = y_im;
        double r1422933 = r1422931 * r1422932;
        double r1422934 = r1422930 + r1422933;
        double r1422935 = r1422930 - r1422933;
        double r1422936 = r1422934 / r1422935;
        double r1422937 = r1422934 / r1422936;
        double r1422938 = r1422929 * r1422929;
        double r1422939 = r1422932 * r1422932;
        double r1422940 = r1422938 + r1422939;
        double r1422941 = r1422937 / r1422940;
        return r1422941;
}

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. Final simplification1.1

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

Reproduce

herbie shell --seed 2019121 
(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))))