Average Error: 1.1 → 1.1
Time: 5.0s
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 r1944686 = x_re;
        double r1944687 = y_re;
        double r1944688 = r1944686 * r1944687;
        double r1944689 = x_im;
        double r1944690 = y_im;
        double r1944691 = r1944689 * r1944690;
        double r1944692 = r1944688 + r1944691;
        double r1944693 = r1944687 * r1944687;
        double r1944694 = r1944690 * r1944690;
        double r1944695 = r1944693 + r1944694;
        double r1944696 = r1944692 / r1944695;
        return r1944696;
}
↓
double f(double x_re, double x_im, double y_re, double y_im) {
        double r1944697 = x_re;
        double r1944698 = y_re;
        double r1944699 = r1944697 * r1944698;
        double r1944700 = x_im;
        double r1944701 = y_im;
        double r1944702 = r1944700 * r1944701;
        double r1944703 = r1944699 + r1944702;
        double r1944704 = r1944698 * r1944698;
        double r1944705 = r1944701 * r1944701;
        double r1944706 = r1944704 + r1944705;
        double r1944707 = r1944703 / r1944706;
        return r1944707;
}

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 2019130 +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))))