Average Error: 26.1 → 24.5
Time: 3.4s
Precision: 64
\[\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\]
↓
\[\frac{\frac{x.im \cdot y.re}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}} - \frac{x.re}{\frac{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}{y.im}}}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}\]
\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}
↓
\frac{\frac{x.im \cdot y.re}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}} - \frac{x.re}{\frac{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}{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 r77340 = x_im;
        double r77341 = y_re;
        double r77342 = r77340 * r77341;
        double r77343 = x_re;
        double r77344 = y_im;
        double r77345 = r77343 * r77344;
        double r77346 = r77342 - r77345;
        double r77347 = r77341 * r77341;
        double r77348 = r77344 * r77344;
        double r77349 = r77347 + r77348;
        double r77350 = r77346 / r77349;
        return r77350;
}
↓
double f(double x_re, double x_im, double y_re, double y_im) {
        double r77351 = x_im;
        double r77352 = y_re;
        double r77353 = r77351 * r77352;
        double r77354 = r77352 * r77352;
        double r77355 = y_im;
        double r77356 = r77355 * r77355;
        double r77357 = r77354 + r77356;
        double r77358 = sqrt(r77357);
        double r77359 = r77353 / r77358;
        double r77360 = x_re;
        double r77361 = r77358 / r77355;
        double r77362 = r77360 / r77361;
        double r77363 = r77359 - r77362;
        double r77364 = r77363 / r77358;
        return r77364;
}

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 26.1

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

  3. Applied add-sqr-sqrt26.1

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

  4. Applied associate-/r*26.1

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

  6. Applied div-sub26.1

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

  8. Applied associate-/l*24.5

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

  9. Final simplification24.5

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

Reproduce

herbie shell --seed 2020062 
(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))))