Average Error: 1.1 → 1.1
Time: 5.1s
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 r230201 = x_re;
        double r230202 = y_re;
        double r230203 = r230201 * r230202;
        double r230204 = x_im;
        double r230205 = y_im;
        double r230206 = r230204 * r230205;
        double r230207 = r230203 + r230206;
        double r230208 = r230202 * r230202;
        double r230209 = r230205 * r230205;
        double r230210 = r230208 + r230209;
        double r230211 = r230207 / r230210;
        return r230211;
}
↓
double f(double x_re, double x_im, double y_re, double y_im) {
        double r230212 = x_re;
        double r230213 = y_re;
        double r230214 = r230212 * r230213;
        double r230215 = x_im;
        double r230216 = y_im;
        double r230217 = r230215 * r230216;
        double r230218 = r230214 + r230217;
        double r230219 = r230213 * r230213;
        double r230220 = r230216 * r230216;
        double r230221 = r230219 + r230220;
        double r230222 = r230218 / r230221;
        return r230222;
}

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 2019134 +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))))