Average Error: 25.1 → 25.1
Time: 18.7s
Precision: 64
\[\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\]
↓
\[\frac{\frac{1}{\sqrt{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)}} \cdot \left(x.im \cdot y.re - x.re \cdot y.im\right)}{\sqrt{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)}}\]
\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}
↓
\frac{\frac{1}{\sqrt{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)}} \cdot \left(x.im \cdot y.re - x.re \cdot y.im\right)}{\sqrt{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)}}
double f(double x_re, double x_im, double y_re, double y_im) {
        double r2411882 = x_im;
        double r2411883 = y_re;
        double r2411884 = r2411882 * r2411883;
        double r2411885 = x_re;
        double r2411886 = y_im;
        double r2411887 = r2411885 * r2411886;
        double r2411888 = r2411884 - r2411887;
        double r2411889 = r2411883 * r2411883;
        double r2411890 = r2411886 * r2411886;
        double r2411891 = r2411889 + r2411890;
        double r2411892 = r2411888 / r2411891;
        return r2411892;
}
↓
double f(double x_re, double x_im, double y_re, double y_im) {
        double r2411893 = 1.0;
        double r2411894 = y_im;
        double r2411895 = y_re;
        double r2411896 = r2411895 * r2411895;
        double r2411897 = fma(r2411894, r2411894, r2411896);
        double r2411898 = sqrt(r2411897);
        double r2411899 = r2411893 / r2411898;
        double r2411900 = x_im;
        double r2411901 = r2411900 * r2411895;
        double r2411902 = x_re;
        double r2411903 = r2411902 * r2411894;
        double r2411904 = r2411901 - r2411903;
        double r2411905 = r2411899 * r2411904;
        double r2411906 = r2411905 / r2411898;
        return r2411906;
}

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 25.1

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

  2. Simplified25.1

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

  4. Applied add-sqr-sqrt25.1

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

  5. Applied associate-/r*25.0

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

  7. Applied div-inv25.1

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

  8. Final simplification25.1

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

Reproduce

herbie shell --seed 2019164 +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))))