Average Error: 26.1 → 26.1
Time: 16.3s
Precision: 64
\[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\]
↓
\[\frac{(x.re \cdot y.re + \left(x.im \cdot y.im\right))_*}{(y.im \cdot y.im + \left(y.re \cdot y.re\right))_*}\]
double f(double x_re, double x_im, double y_re, double y_im) {
        double r2170147 = x_re;
        double r2170148 = y_re;
        double r2170149 = r2170147 * r2170148;
        double r2170150 = x_im;
        double r2170151 = y_im;
        double r2170152 = r2170150 * r2170151;
        double r2170153 = r2170149 + r2170152;
        double r2170154 = r2170148 * r2170148;
        double r2170155 = r2170151 * r2170151;
        double r2170156 = r2170154 + r2170155;
        double r2170157 = r2170153 / r2170156;
        return r2170157;
}
↓
double f(double x_re, double x_im, double y_re, double y_im) {
        double r2170158 = x_re;
        double r2170159 = y_re;
        double r2170160 = x_im;
        double r2170161 = y_im;
        double r2170162 = r2170160 * r2170161;
        double r2170163 = fma(r2170158, r2170159, r2170162);
        double r2170164 = r2170159 * r2170159;
        double r2170165 = fma(r2170161, r2170161, r2170164);
        double r2170166 = r2170163 / r2170165;
        return r2170166;
}


\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}
↓
\frac{(x.re \cdot y.re + \left(x.im \cdot y.im\right))_*}{(y.im \cdot y.im + \left(y.re \cdot y.re\right))_*}

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 26.1

    \[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\]

  2. Simplified26.1

    \[\leadsto \color{blue}{\frac{(x.re \cdot y.re + \left(x.im \cdot y.im\right))_*}{(y.im \cdot y.im + \left(y.re \cdot y.re\right))_*}}\]

  3. Final simplification26.1

    \[\leadsto \frac{(x.re \cdot y.re + \left(x.im \cdot y.im\right))_*}{(y.im \cdot y.im + \left(y.re \cdot y.re\right))_*}\]

Reproduce

herbie shell --seed 2019102 +o rules:numerics
(FPCore (x.re x.im y.re y.im)
  :name "_divideComplex, real part"
  (/ (+ (* x.re y.re) (* x.im y.im)) (+ (* y.re y.re) (* y.im y.im))))