Average Error: 1.1 → 1.1
Time: 3.8s
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 r377085 = x_re;
        double r377086 = y_re;
        double r377087 = r377085 * r377086;
        double r377088 = x_im;
        double r377089 = y_im;
        double r377090 = r377088 * r377089;
        double r377091 = r377087 + r377090;
        double r377092 = r377086 * r377086;
        double r377093 = r377089 * r377089;
        double r377094 = r377092 + r377093;
        double r377095 = r377091 / r377094;
        return r377095;
}
↓
double f(double x_re, double x_im, double y_re, double y_im) {
        double r377096 = x_re;
        double r377097 = y_re;
        double r377098 = r377096 * r377097;
        double r377099 = x_im;
        double r377100 = y_im;
        double r377101 = r377099 * r377100;
        double r377102 = r377098 + r377101;
        double r377103 = r377097 * r377097;
        double r377104 = r377100 * r377100;
        double r377105 = r377103 + r377104;
        double r377106 = r377102 / r377105;
        return r377106;
}

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 2019121 +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))))