Average Error: 25.4 → 25.4
Time: 22.3s
Precision: 64
\[\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\]
↓
\[\frac{\frac{1}{\sqrt{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)}} \cdot \mathsf{fma}\left(x.im, y.re, \left(-x.re\right) \cdot y.im\right)}{\sqrt{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)}}\]
\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 \mathsf{fma}\left(x.im, y.re, \left(-x.re\right) \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 r1576633 = x_im;
        double r1576634 = y_re;
        double r1576635 = r1576633 * r1576634;
        double r1576636 = x_re;
        double r1576637 = y_im;
        double r1576638 = r1576636 * r1576637;
        double r1576639 = r1576635 - r1576638;
        double r1576640 = r1576634 * r1576634;
        double r1576641 = r1576637 * r1576637;
        double r1576642 = r1576640 + r1576641;
        double r1576643 = r1576639 / r1576642;
        return r1576643;
}
↓
double f(double x_re, double x_im, double y_re, double y_im) {
        double r1576644 = 1.0;
        double r1576645 = y_im;
        double r1576646 = y_re;
        double r1576647 = r1576646 * r1576646;
        double r1576648 = fma(r1576645, r1576645, r1576647);
        double r1576649 = sqrt(r1576648);
        double r1576650 = r1576644 / r1576649;
        double r1576651 = x_im;
        double r1576652 = x_re;
        double r1576653 = -r1576652;
        double r1576654 = r1576653 * r1576645;
        double r1576655 = fma(r1576651, r1576646, r1576654);
        double r1576656 = r1576650 * r1576655;
        double r1576657 = r1576656 / r1576649;
        return r1576657;
}

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 25.4

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

  2. Simplified25.4

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

  4. Applied add-sqr-sqrt25.4

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

  5. Applied associate-/r*25.3

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

  7. Applied fma-neg25.3

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

  9. Applied div-inv25.4

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

  10. Final simplification25.4

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

Reproduce

herbie shell --seed 2019152 +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))))