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 r1410993 = x_re;
        double r1410994 = y_re;
        double r1410995 = r1410993 * r1410994;
        double r1410996 = x_im;
        double r1410997 = y_im;
        double r1410998 = r1410996 * r1410997;
        double r1410999 = r1410995 + r1410998;
        double r1411000 = r1410994 * r1410994;
        double r1411001 = r1410997 * r1410997;
        double r1411002 = r1411000 + r1411001;
        double r1411003 = r1410999 / r1411002;
        return r1411003;
}
↓
double f(double x_re, double x_im, double y_re, double y_im) {
        double r1411004 = x_re;
        double r1411005 = y_re;
        double r1411006 = r1411004 * r1411005;
        double r1411007 = x_im;
        double r1411008 = y_im;
        double r1411009 = r1411007 * r1411008;
        double r1411010 = r1411006 + r1411009;
        double r1411011 = r1411005 * r1411005;
        double r1411012 = r1411008 * r1411008;
        double r1411013 = r1411011 + r1411012;
        double r1411014 = r1411010 / r1411013;
        return r1411014;
}

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 +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))))