Average Error: 25.4 → 25.5
Time: 42.9s
Precision: 64
\[\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\]
↓
\[\frac{1}{\mathsf{fma}\left(y.im, y.im, \left(y.re \cdot y.re\right)\right)} \cdot \left(x.im \cdot y.re - x.re \cdot y.im\right)\]
\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}
↓
\frac{1}{\mathsf{fma}\left(y.im, y.im, \left(y.re \cdot y.re\right)\right)} \cdot \left(x.im \cdot y.re - x.re \cdot y.im\right)
double f(double x_re, double x_im, double y_re, double y_im) {
        double r3763628 = x_im;
        double r3763629 = y_re;
        double r3763630 = r3763628 * r3763629;
        double r3763631 = x_re;
        double r3763632 = y_im;
        double r3763633 = r3763631 * r3763632;
        double r3763634 = r3763630 - r3763633;
        double r3763635 = r3763629 * r3763629;
        double r3763636 = r3763632 * r3763632;
        double r3763637 = r3763635 + r3763636;
        double r3763638 = r3763634 / r3763637;
        return r3763638;
}
↓
double f(double x_re, double x_im, double y_re, double y_im) {
        double r3763639 = 1.0;
        double r3763640 = y_im;
        double r3763641 = y_re;
        double r3763642 = r3763641 * r3763641;
        double r3763643 = fma(r3763640, r3763640, r3763642);
        double r3763644 = r3763639 / r3763643;
        double r3763645 = x_im;
        double r3763646 = r3763645 * r3763641;
        double r3763647 = x_re;
        double r3763648 = r3763647 * r3763640;
        double r3763649 = r3763646 - r3763648;
        double r3763650 = r3763644 * r3763649;
        return r3763650;
}

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 25.4

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

  2. Simplified25.4

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

  4. Applied div-inv25.5

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

  5. Final simplification25.5

    \[\leadsto \frac{1}{\mathsf{fma}\left(y.im, y.im, \left(y.re \cdot y.re\right)\right)} \cdot \left(x.im \cdot y.re - x.re \cdot y.im\right)\]

Reproduce

herbie shell --seed 2019125 +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))))