Average Error: 25.8 → 25.8
Time: 12.8s
Precision: 64
\[\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\]
↓
\[\frac{-x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} + \frac{x.im \cdot y.re}{y.re \cdot y.re + y.im \cdot y.im}\]
\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}
↓
\frac{-x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} + \frac{x.im \cdot y.re}{y.re \cdot y.re + y.im \cdot y.im}
double f(double x_re, double x_im, double y_re, double y_im) {
        double r38371 = x_im;
        double r38372 = y_re;
        double r38373 = r38371 * r38372;
        double r38374 = x_re;
        double r38375 = y_im;
        double r38376 = r38374 * r38375;
        double r38377 = r38373 - r38376;
        double r38378 = r38372 * r38372;
        double r38379 = r38375 * r38375;
        double r38380 = r38378 + r38379;
        double r38381 = r38377 / r38380;
        return r38381;
}
↓
double f(double x_re, double x_im, double y_re, double y_im) {
        double r38382 = x_re;
        double r38383 = y_im;
        double r38384 = r38382 * r38383;
        double r38385 = -r38384;
        double r38386 = y_re;
        double r38387 = r38386 * r38386;
        double r38388 = r38383 * r38383;
        double r38389 = r38387 + r38388;
        double r38390 = r38385 / r38389;
        double r38391 = x_im;
        double r38392 = r38391 * r38386;
        double r38393 = r38392 / r38389;
        double r38394 = r38390 + r38393;
        return r38394;
}

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 25.8

    \[\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\]
  2. Using strategy rm

  3. Applied add-sqr-sqrt25.8

    \[\leadsto \frac{x.im \cdot y.re - x.re \cdot y.im}{\color{blue}{\sqrt{y.re \cdot y.re + y.im \cdot y.im} \cdot \sqrt{y.re \cdot y.re + y.im \cdot y.im}}}\]

  4. Applied associate-/r*25.7

    \[\leadsto \color{blue}{\frac{\frac{x.im \cdot y.re - x.re \cdot y.im}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}}\]
  5. Using strategy rm

  6. Applied div-sub25.7

    \[\leadsto \frac{\color{blue}{\frac{x.im \cdot y.re}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}} - \frac{x.re \cdot y.im}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}}}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}\]
  7. Using strategy rm

  8. Applied add-sqr-sqrt25.7

    \[\leadsto \frac{\frac{x.im \cdot y.re}{\sqrt{\color{blue}{\sqrt{y.re \cdot y.re + y.im \cdot y.im} \cdot \sqrt{y.re \cdot y.re + y.im \cdot y.im}}}} - \frac{x.re \cdot y.im}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}\]

  9. Applied sqrt-prod25.8

    \[\leadsto \frac{\frac{x.im \cdot y.re}{\color{blue}{\sqrt{\sqrt{y.re \cdot y.re + y.im \cdot y.im}} \cdot \sqrt{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}}} - \frac{x.re \cdot y.im}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}\]

  10. Applied times-frac24.1

    \[\leadsto \frac{\color{blue}{\frac{x.im}{\sqrt{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}} \cdot \frac{y.re}{\sqrt{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}}} - \frac{x.re \cdot y.im}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}\]
  11. Using strategy rm

  12. Applied add-sqr-sqrt24.1

    \[\leadsto \frac{\frac{x.im}{\sqrt{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}} \cdot \frac{y.re}{\sqrt{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}} - \frac{x.re \cdot y.im}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}}{\sqrt{\color{blue}{\sqrt{y.re \cdot y.re + y.im \cdot y.im} \cdot \sqrt{y.re \cdot y.re + y.im \cdot y.im}}}}\]

  13. Applied sqrt-prod24.3

    \[\leadsto \frac{\frac{x.im}{\sqrt{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}} \cdot \frac{y.re}{\sqrt{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}} - \frac{x.re \cdot y.im}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}}{\color{blue}{\sqrt{\sqrt{y.re \cdot y.re + y.im \cdot y.im}} \cdot \sqrt{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}}}\]

  14. Final simplification25.8

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

Reproduce

herbie shell --seed 2019303 
(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))))