Average Error: 26.1 → 26.1
Time: 17.7s
Precision: 64
\[\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\]
↓
\[\frac{\frac{1}{\frac{\sqrt{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)}}{x.im \cdot y.re - x.re \cdot y.im}}}{\sqrt{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)}}\]
\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}
↓
\frac{\frac{1}{\frac{\sqrt{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)}}{x.im \cdot y.re - x.re \cdot y.im}}}{\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 r2485950 = x_im;
        double r2485951 = y_re;
        double r2485952 = r2485950 * r2485951;
        double r2485953 = x_re;
        double r2485954 = y_im;
        double r2485955 = r2485953 * r2485954;
        double r2485956 = r2485952 - r2485955;
        double r2485957 = r2485951 * r2485951;
        double r2485958 = r2485954 * r2485954;
        double r2485959 = r2485957 + r2485958;
        double r2485960 = r2485956 / r2485959;
        return r2485960;
}
↓
double f(double x_re, double x_im, double y_re, double y_im) {
        double r2485961 = 1.0;
        double r2485962 = y_im;
        double r2485963 = y_re;
        double r2485964 = r2485963 * r2485963;
        double r2485965 = fma(r2485962, r2485962, r2485964);
        double r2485966 = sqrt(r2485965);
        double r2485967 = x_im;
        double r2485968 = r2485967 * r2485963;
        double r2485969 = x_re;
        double r2485970 = r2485969 * r2485962;
        double r2485971 = r2485968 - r2485970;
        double r2485972 = r2485966 / r2485971;
        double r2485973 = r2485961 / r2485972;
        double r2485974 = r2485973 / r2485966;
        return r2485974;
}

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 26.1

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

  2. Simplified26.1

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

  4. Applied add-sqr-sqrt26.1

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

  5. Applied associate-/r*26.0

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

  7. Applied clear-num26.1

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

  8. Final simplification26.1

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

Reproduce

herbie shell --seed 2019163 +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))))