Average Error: 26.2 → 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 r69560 = x_im;
        double r69561 = y_re;
        double r69562 = r69560 * r69561;
        double r69563 = x_re;
        double r69564 = y_im;
        double r69565 = r69563 * r69564;
        double r69566 = r69562 - r69565;
        double r69567 = r69561 * r69561;
        double r69568 = r69564 * r69564;
        double r69569 = r69567 + r69568;
        double r69570 = r69566 / r69569;
        return r69570;
}
↓
double f(double x_re, double x_im, double y_re, double y_im) {
        double r69571 = x_im;
        double r69572 = y_re;
        double r69573 = r69572 * r69572;
        double r69574 = y_im;
        double r69575 = r69574 * r69574;
        double r69576 = r69573 + r69575;
        double r69577 = sqrt(r69576);
        double r69578 = r69571 / r69577;
        double r69579 = r69572 / r69577;
        double r69580 = r69578 * r69579;
        double r69581 = x_re;
        double r69582 = r69581 / r69577;
        double r69583 = r69574 / r69577;
        double r69584 = r69582 * r69583;
        double r69585 = r69580 - r69584;
        return r69585;
}

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 div-sub26.2

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

  5. Applied add-sqr-sqrt26.2

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

  6. Applied times-frac24.7

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

  8. Applied add-sqr-sqrt24.7

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

  9. Applied times-frac22.9

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

  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 2020049 
(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))))