Average Error: 1.1 → 1.1
Time: 4.3s
Precision: 64
\[\frac{\left(\frac{\left(x.re \cdot y.re\right)}{\left(x.im \cdot y.im\right)}\right)}{\left(\frac{\left(y.re \cdot y.re\right)}{\left(y.im \cdot y.im\right)}\right)}\]
↓
\[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\]
\frac{\left(\frac{\left(x.re \cdot y.re\right)}{\left(x.im \cdot y.im\right)}\right)}{\left(\frac{\left(y.re \cdot y.re\right)}{\left(y.im \cdot y.im\right)}\right)}
↓
\frac{x.re \cdot y.re + x.im \cdot y.im}{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 r1771350 = x_re;
        double r1771351 = y_re;
        double r1771352 = r1771350 * r1771351;
        double r1771353 = x_im;
        double r1771354 = y_im;
        double r1771355 = r1771353 * r1771354;
        double r1771356 = r1771352 + r1771355;
        double r1771357 = r1771351 * r1771351;
        double r1771358 = r1771354 * r1771354;
        double r1771359 = r1771357 + r1771358;
        double r1771360 = r1771356 / r1771359;
        return r1771360;
}
↓
double f(double x_re, double x_im, double y_re, double y_im) {
        double r1771361 = x_re;
        double r1771362 = y_re;
        double r1771363 = r1771361 * r1771362;
        double r1771364 = x_im;
        double r1771365 = y_im;
        double r1771366 = r1771364 * r1771365;
        double r1771367 = r1771363 + r1771366;
        double r1771368 = r1771362 * r1771362;
        double r1771369 = r1771365 * r1771365;
        double r1771370 = r1771368 + r1771369;
        double r1771371 = r1771367 / r1771370;
        return r1771371;
}

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 1.1

    \[\frac{\left(\frac{\left(x.re \cdot y.re\right)}{\left(x.im \cdot y.im\right)}\right)}{\left(\frac{\left(y.re \cdot y.re\right)}{\left(y.im \cdot y.im\right)}\right)}\]

  2. Final simplification1.1

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

Reproduce

herbie shell --seed 2019128 +o rules:numerics
(FPCore (x.re x.im y.re y.im)
  :name "_divideComplex, real part"
  (/.p16 (+.p16 (*.p16 x.re y.re) (*.p16 x.im y.im)) (+.p16 (*.p16 y.re y.re) (*.p16 y.im y.im))))