Average Error: 25.5 → 25.5
Time: 16.8s
Precision: 64
\[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\]
↓
\[\frac{(x.re \cdot y.re + \left(x.im \cdot y.im\right))_*}{(y.im \cdot y.im + \left(y.re \cdot y.re\right))_*}\]
\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 r2358015 = x_re;
        double r2358016 = y_re;
        double r2358017 = r2358015 * r2358016;
        double r2358018 = x_im;
        double r2358019 = y_im;
        double r2358020 = r2358018 * r2358019;
        double r2358021 = r2358017 + r2358020;
        double r2358022 = r2358016 * r2358016;
        double r2358023 = r2358019 * r2358019;
        double r2358024 = r2358022 + r2358023;
        double r2358025 = r2358021 / r2358024;
        return r2358025;
}
↓
double f(double x_re, double x_im, double y_re, double y_im) {
        double r2358026 = x_re;
        double r2358027 = y_re;
        double r2358028 = x_im;
        double r2358029 = y_im;
        double r2358030 = r2358028 * r2358029;
        double r2358031 = fma(r2358026, r2358027, r2358030);
        double r2358032 = r2358027 * r2358027;
        double r2358033 = fma(r2358029, r2358029, r2358032);
        double r2358034 = r2358031 / r2358033;
        return r2358034;
}

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 25.5

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

  2. Simplified25.5

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

  3. Final simplification25.5

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

Reproduce

herbie shell --seed 2019112 +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))))