Average Error: 25.8 → 25.8
Time: 16.6s
Precision: 64
\[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\]
↓
\[\frac{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)}\]
\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}
↓
\frac{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)}
double f(double x_re, double x_im, double y_re, double y_im) {
        double r2356947 = x_re;
        double r2356948 = y_re;
        double r2356949 = r2356947 * r2356948;
        double r2356950 = x_im;
        double r2356951 = y_im;
        double r2356952 = r2356950 * r2356951;
        double r2356953 = r2356949 + r2356952;
        double r2356954 = r2356948 * r2356948;
        double r2356955 = r2356951 * r2356951;
        double r2356956 = r2356954 + r2356955;
        double r2356957 = r2356953 / r2356956;
        return r2356957;
}
↓
double f(double x_re, double x_im, double y_re, double y_im) {
        double r2356958 = x_re;
        double r2356959 = y_re;
        double r2356960 = x_im;
        double r2356961 = y_im;
        double r2356962 = r2356960 * r2356961;
        double r2356963 = fma(r2356958, r2356959, r2356962);
        double r2356964 = r2356959 * r2356959;
        double r2356965 = fma(r2356961, r2356961, r2356964);
        double r2356966 = r2356963 / r2356965;
        return r2356966;
}

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 25.8

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

  2. Simplified25.8

    \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)}}\]

  3. Final simplification25.8

    \[\leadsto \frac{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)}\]

Reproduce

herbie shell --seed 2019163 +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))))