Average Error: 25.7 → 25.7
Time: 13.9s
Precision: 64
\[\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\]
↓
\[\frac{\frac{1}{\frac{\sqrt{\mathsf{fma}\left(y.im, y.im, \left(y.re \cdot y.re\right)\right)}}{x.im \cdot y.re - x.re \cdot y.im}}}{\sqrt{\mathsf{fma}\left(y.im, y.im, \left(y.re \cdot y.re\right)\right)}}\]
\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, \left(y.re \cdot y.re\right)\right)}}{x.im \cdot y.re - x.re \cdot y.im}}}{\sqrt{\mathsf{fma}\left(y.im, y.im, \left(y.re \cdot y.re\right)\right)}}
double f(double x_re, double x_im, double y_re, double y_im) {
        double r1391297 = x_im;
        double r1391298 = y_re;
        double r1391299 = r1391297 * r1391298;
        double r1391300 = x_re;
        double r1391301 = y_im;
        double r1391302 = r1391300 * r1391301;
        double r1391303 = r1391299 - r1391302;
        double r1391304 = r1391298 * r1391298;
        double r1391305 = r1391301 * r1391301;
        double r1391306 = r1391304 + r1391305;
        double r1391307 = r1391303 / r1391306;
        return r1391307;
}
↓
double f(double x_re, double x_im, double y_re, double y_im) {
        double r1391308 = 1.0;
        double r1391309 = y_im;
        double r1391310 = y_re;
        double r1391311 = r1391310 * r1391310;
        double r1391312 = fma(r1391309, r1391309, r1391311);
        double r1391313 = sqrt(r1391312);
        double r1391314 = x_im;
        double r1391315 = r1391314 * r1391310;
        double r1391316 = x_re;
        double r1391317 = r1391316 * r1391309;
        double r1391318 = r1391315 - r1391317;
        double r1391319 = r1391313 / r1391318;
        double r1391320 = r1391308 / r1391319;
        double r1391321 = r1391320 / r1391313;
        return r1391321;
}

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 25.7

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

  2. Simplified25.7

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

  4. Applied add-sqr-sqrt25.7

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

  5. Applied associate-/r*25.6

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

  7. Applied clear-num25.7

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

  8. Final simplification25.7

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

Reproduce

herbie shell --seed 2019130 +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))))