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 r1743474 = x_re;
        double r1743475 = y_re;
        double r1743476 = r1743474 * r1743475;
        double r1743477 = x_im;
        double r1743478 = y_im;
        double r1743479 = r1743477 * r1743478;
        double r1743480 = r1743476 + r1743479;
        double r1743481 = r1743475 * r1743475;
        double r1743482 = r1743478 * r1743478;
        double r1743483 = r1743481 + r1743482;
        double r1743484 = r1743480 / r1743483;
        return r1743484;
}
↓
double f(double x_re, double x_im, double y_re, double y_im) {
        double r1743485 = x_re;
        double r1743486 = y_re;
        double r1743487 = r1743485 * r1743486;
        double r1743488 = x_im;
        double r1743489 = y_im;
        double r1743490 = r1743488 * r1743489;
        double r1743491 = r1743487 + r1743490;
        double r1743492 = r1743486 * r1743486;
        double r1743493 = r1743489 * r1743489;
        double r1743494 = r1743492 + r1743493;
        double r1743495 = r1743491 / r1743494;
        return r1743495;
}

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 2019135 
(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))))