Average Error: 25.6 → 25.7
Time: 24.9s
Precision: 64
\[\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\]
↓
\[\frac{x.im \cdot y.re - y.im \cdot x.re}{\sqrt{(y.im \cdot y.im + \left(y.re \cdot y.re\right))_*}} \cdot \frac{1}{\sqrt{(y.im \cdot y.im + \left(y.re \cdot y.re\right))_*}}\]
\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 y.re - y.im \cdot x.re}{\sqrt{(y.im \cdot y.im + \left(y.re \cdot y.re\right))_*}} \cdot \frac{1}{\sqrt{(y.im \cdot y.im + \left(y.re \cdot y.re\right))_*}}
double f(double x_re, double x_im, double y_re, double y_im) {
        double r1843670 = x_im;
        double r1843671 = y_re;
        double r1843672 = r1843670 * r1843671;
        double r1843673 = x_re;
        double r1843674 = y_im;
        double r1843675 = r1843673 * r1843674;
        double r1843676 = r1843672 - r1843675;
        double r1843677 = r1843671 * r1843671;
        double r1843678 = r1843674 * r1843674;
        double r1843679 = r1843677 + r1843678;
        double r1843680 = r1843676 / r1843679;
        return r1843680;
}
↓
double f(double x_re, double x_im, double y_re, double y_im) {
        double r1843681 = x_im;
        double r1843682 = y_re;
        double r1843683 = r1843681 * r1843682;
        double r1843684 = y_im;
        double r1843685 = x_re;
        double r1843686 = r1843684 * r1843685;
        double r1843687 = r1843683 - r1843686;
        double r1843688 = r1843682 * r1843682;
        double r1843689 = fma(r1843684, r1843684, r1843688);
        double r1843690 = sqrt(r1843689);
        double r1843691 = r1843687 / r1843690;
        double r1843692 = 1.0;
        double r1843693 = r1843692 / r1843690;
        double r1843694 = r1843691 * r1843693;
        return r1843694;
}

Error

Bits error versus x.re

Bits error versus x.im

Bits error versus y.re

Bits error versus y.im

Derivation

  1. Initial program 25.6

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

  2. Simplified25.6

    \[\leadsto \color{blue}{\frac{x.im \cdot y.re - x.re \cdot y.im}{(y.im \cdot y.im + \left(y.re \cdot y.re\right))_*}}\]
  3. Using strategy rm

  4. Applied add-sqr-sqrt25.7

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

  5. Applied *-un-lft-identity25.7

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

  6. Applied times-frac25.7

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

  7. Final simplification25.7

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

Reproduce

herbie shell --seed 2019112 +o rules:numerics
(FPCore (x.re x.im y.re y.im)
  :name "_divideComplex, imaginary part"
  (/ (- (* x.im y.re) (* x.re y.im)) (+ (* y.re y.re) (* y.im y.im))))