Average Error: 25.9 → 0.9
Time: 18.5s
Precision: 64
\[\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\]
↓
\[\frac{\frac{y.re}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot x.im - \frac{x.re}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot y.im}{\mathsf{hypot}\left(y.re, y.im\right)}\]
\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}
↓
\frac{\frac{y.re}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot x.im - \frac{x.re}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot y.im}{\mathsf{hypot}\left(y.re, y.im\right)}
double f(double x_re, double x_im, double y_re, double y_im) {
        double r54461 = x_im;
        double r54462 = y_re;
        double r54463 = r54461 * r54462;
        double r54464 = x_re;
        double r54465 = y_im;
        double r54466 = r54464 * r54465;
        double r54467 = r54463 - r54466;
        double r54468 = r54462 * r54462;
        double r54469 = r54465 * r54465;
        double r54470 = r54468 + r54469;
        double r54471 = r54467 / r54470;
        return r54471;
}
↓
double f(double x_re, double x_im, double y_re, double y_im) {
        double r54472 = y_re;
        double r54473 = y_im;
        double r54474 = hypot(r54472, r54473);
        double r54475 = r54472 / r54474;
        double r54476 = x_im;
        double r54477 = r54475 * r54476;
        double r54478 = x_re;
        double r54479 = r54478 / r54474;
        double r54480 = r54479 * r54473;
        double r54481 = r54477 - r54480;
        double r54482 = r54481 / r54474;
        return r54482;
}

Error

Bits error versus x.re

Bits error versus x.im

Bits error versus y.re

Bits error versus y.im

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 25.9

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

  2. Simplified25.9

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

  4. Applied add-sqr-sqrt25.9

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

  5. Applied *-un-lft-identity25.9

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

  6. Applied times-frac25.9

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

  7. Simplified25.9

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

  8. Simplified16.8

    \[\leadsto \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \color{blue}{\frac{x.im \cdot y.re - x.re \cdot y.im}{\mathsf{hypot}\left(y.re, y.im\right)}}\]
  9. Using strategy rm

  10. Applied *-un-lft-identity16.8

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

  11. Applied *-un-lft-identity16.8

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

  12. Applied times-frac16.8

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

  13. Applied associate-*l*16.8

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

  14. Simplified16.7

    \[\leadsto \frac{1}{1} \cdot \color{blue}{\frac{\frac{x.im \cdot y.re - x.re \cdot y.im}{\mathsf{hypot}\left(y.re, y.im\right)}}{\mathsf{hypot}\left(y.re, y.im\right)}}\]
  15. Using strategy rm

  16. Applied div-sub16.7

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

  17. Simplified9.0

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

  18. Simplified0.9

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

  19. Final simplification0.9

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

Reproduce

herbie shell --seed 2019323 +o rules:numerics
(FPCore (x.re x.im y.re y.im)
  :name "_divideComplex, imaginary part"
  :precision binary64
  (/ (- (* x.im y.re) (* x.re y.im)) (+ (* y.re y.re) (* y.im y.im))))