Average Error: 26.2 → 22.2
Time: 3.4s
Precision: 64
\[\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\]
↓
\[\frac{x.im \cdot \frac{y.re}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}} - x.re \cdot \frac{y.im}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}\]
\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}
↓
\frac{x.im \cdot \frac{y.re}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}} - x.re \cdot \frac{y.im}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}}{\sqrt{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 r107502 = x_im;
        double r107503 = y_re;
        double r107504 = r107502 * r107503;
        double r107505 = x_re;
        double r107506 = y_im;
        double r107507 = r107505 * r107506;
        double r107508 = r107504 - r107507;
        double r107509 = r107503 * r107503;
        double r107510 = r107506 * r107506;
        double r107511 = r107509 + r107510;
        double r107512 = r107508 / r107511;
        return r107512;
}
↓
double f(double x_re, double x_im, double y_re, double y_im) {
        double r107513 = x_im;
        double r107514 = y_re;
        double r107515 = r107514 * r107514;
        double r107516 = y_im;
        double r107517 = r107516 * r107516;
        double r107518 = r107515 + r107517;
        double r107519 = sqrt(r107518);
        double r107520 = r107514 / r107519;
        double r107521 = r107513 * r107520;
        double r107522 = x_re;
        double r107523 = r107516 / r107519;
        double r107524 = r107522 * r107523;
        double r107525 = r107521 - r107524;
        double r107526 = r107525 / r107519;
        return r107526;
}

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 26.2

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

  3. Applied add-sqr-sqrt26.3

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

  4. Applied associate-/r*26.2

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

  6. Applied div-sub26.2

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

  8. Applied *-un-lft-identity26.2

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

  9. Applied sqrt-prod26.2

    \[\leadsto \frac{\frac{x.im \cdot y.re}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}} - \frac{x.re \cdot y.im}{\color{blue}{\sqrt{1} \cdot \sqrt{y.re \cdot y.re + y.im \cdot y.im}}}}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}\]

  10. Applied times-frac24.3

    \[\leadsto \frac{\frac{x.im \cdot y.re}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}} - \color{blue}{\frac{x.re}{\sqrt{1}} \cdot \frac{y.im}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}}}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}\]

  11. Simplified24.3

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

  13. Applied *-un-lft-identity24.3

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

  14. Applied sqrt-prod24.3

    \[\leadsto \frac{\frac{x.im \cdot y.re}{\color{blue}{\sqrt{1} \cdot \sqrt{y.re \cdot y.re + y.im \cdot y.im}}} - x.re \cdot \frac{y.im}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}\]

  15. Applied times-frac22.2

    \[\leadsto \frac{\color{blue}{\frac{x.im}{\sqrt{1}} \cdot \frac{y.re}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}} - x.re \cdot \frac{y.im}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}\]

  16. Simplified22.2

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

  17. Final simplification22.2

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

Reproduce

herbie shell --seed 2020033 
(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))))