Average Error: 25.1 → 25.1
Time: 25.6s
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 r2798899 = x_im;
        double r2798900 = y_re;
        double r2798901 = r2798899 * r2798900;
        double r2798902 = x_re;
        double r2798903 = y_im;
        double r2798904 = r2798902 * r2798903;
        double r2798905 = r2798901 - r2798904;
        double r2798906 = r2798900 * r2798900;
        double r2798907 = r2798903 * r2798903;
        double r2798908 = r2798906 + r2798907;
        double r2798909 = r2798905 / r2798908;
        return r2798909;
}
↓
double f(double x_re, double x_im, double y_re, double y_im) {
        double r2798910 = 1.0;
        double r2798911 = y_im;
        double r2798912 = y_re;
        double r2798913 = r2798912 * r2798912;
        double r2798914 = fma(r2798911, r2798911, r2798913);
        double r2798915 = sqrt(r2798914);
        double r2798916 = r2798910 / r2798915;
        double r2798917 = x_im;
        double r2798918 = r2798917 * r2798912;
        double r2798919 = x_re;
        double r2798920 = r2798919 * r2798911;
        double r2798921 = r2798918 - r2798920;
        double r2798922 = r2798916 * r2798921;
        double r2798923 = r2798922 / r2798915;
        return r2798923;
}

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 2019158 +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))))