Average Error: 25.7 → 22.9
Time: 3.2s
Precision: 64
\[\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\]
↓
\[\frac{x.im}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}} \cdot \frac{y.re}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}} - \frac{x.re}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}} \cdot \frac{y.im}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}\]
\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}
↓
\frac{x.im}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}} \cdot \frac{y.re}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}} - \frac{x.re}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}} \cdot \frac{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 r63431 = x_im;
        double r63432 = y_re;
        double r63433 = r63431 * r63432;
        double r63434 = x_re;
        double r63435 = y_im;
        double r63436 = r63434 * r63435;
        double r63437 = r63433 - r63436;
        double r63438 = r63432 * r63432;
        double r63439 = r63435 * r63435;
        double r63440 = r63438 + r63439;
        double r63441 = r63437 / r63440;
        return r63441;
}
↓
double f(double x_re, double x_im, double y_re, double y_im) {
        double r63442 = x_im;
        double r63443 = y_re;
        double r63444 = r63443 * r63443;
        double r63445 = y_im;
        double r63446 = r63445 * r63445;
        double r63447 = r63444 + r63446;
        double r63448 = sqrt(r63447);
        double r63449 = r63442 / r63448;
        double r63450 = r63443 / r63448;
        double r63451 = r63449 * r63450;
        double r63452 = x_re;
        double r63453 = r63452 / r63448;
        double r63454 = r63445 / r63448;
        double r63455 = r63453 * r63454;
        double r63456 = r63451 - r63455;
        return r63456;
}

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 25.7

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

  3. Applied div-sub25.7

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

  5. Applied add-sqr-sqrt25.7

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

  6. Applied times-frac24.5

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

  8. Applied add-sqr-sqrt24.5

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

  9. Applied times-frac22.9

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

  10. Final simplification22.9

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

Reproduce

herbie shell --seed 2020059 
(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))))