Average Error: 1.1 → 1.1
Time: 4.6s
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 r732161 = x_re;
        double r732162 = y_re;
        double r732163 = r732161 * r732162;
        double r732164 = x_im;
        double r732165 = y_im;
        double r732166 = r732164 * r732165;
        double r732167 = r732163 + r732166;
        double r732168 = r732162 * r732162;
        double r732169 = r732165 * r732165;
        double r732170 = r732168 + r732169;
        double r732171 = r732167 / r732170;
        return r732171;
}
↓
double f(double x_re, double x_im, double y_re, double y_im) {
        double r732172 = x_re;
        double r732173 = y_re;
        double r732174 = r732172 * r732173;
        double r732175 = x_im;
        double r732176 = y_im;
        double r732177 = r732175 * r732176;
        double r732178 = r732174 + r732177;
        double r732179 = r732173 * r732173;
        double r732180 = r732176 * r732176;
        double r732181 = r732179 + r732180;
        double r732182 = r732178 / r732181;
        return r732182;
}

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 2019107 
(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))))