Average Error: 26.1 → 26.1
Time: 17.5s
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 r2036395 = x_im;
        double r2036396 = y_re;
        double r2036397 = r2036395 * r2036396;
        double r2036398 = x_re;
        double r2036399 = y_im;
        double r2036400 = r2036398 * r2036399;
        double r2036401 = r2036397 - r2036400;
        double r2036402 = r2036396 * r2036396;
        double r2036403 = r2036399 * r2036399;
        double r2036404 = r2036402 + r2036403;
        double r2036405 = r2036401 / r2036404;
        return r2036405;
}
↓
double f(double x_re, double x_im, double y_re, double y_im) {
        double r2036406 = 1.0;
        double r2036407 = y_im;
        double r2036408 = y_re;
        double r2036409 = r2036408 * r2036408;
        double r2036410 = fma(r2036407, r2036407, r2036409);
        double r2036411 = sqrt(r2036410);
        double r2036412 = x_im;
        double r2036413 = r2036412 * r2036408;
        double r2036414 = x_re;
        double r2036415 = r2036414 * r2036407;
        double r2036416 = r2036413 - r2036415;
        double r2036417 = r2036411 / r2036416;
        double r2036418 = r2036406 / r2036417;
        double r2036419 = r2036418 / r2036411;
        return r2036419;
}

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))))