Average Error: 25.7 → 25.6
Time: 15.2s
Precision: 64
\[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\]
↓
\[\begin{array}{l} \mathbf{if}\;y.im \le 6.646276389058437 \cdot 10^{+117}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(x.re, y.re, \left(x.im \cdot y.im\right)\right)}{\sqrt{\mathsf{fma}\left(y.im, y.im, \left(y.re \cdot y.re\right)\right)}}}{\sqrt{\mathsf{fma}\left(y.im, y.im, \left(y.re \cdot y.re\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{\sqrt{\mathsf{fma}\left(y.im, y.im, \left(y.re \cdot y.re\right)\right)}}\\ \end{array}\]
\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}
↓
\begin{array}{l}
\mathbf{if}\;y.im \le 6.646276389058437 \cdot 10^{+117}:\\
\;\;\;\;\frac{\frac{\mathsf{fma}\left(x.re, y.re, \left(x.im \cdot y.im\right)\right)}{\sqrt{\mathsf{fma}\left(y.im, y.im, \left(y.re \cdot y.re\right)\right)}}}{\sqrt{\mathsf{fma}\left(y.im, y.im, \left(y.re \cdot y.re\right)\right)}}\\

\mathbf{else}:\\
\;\;\;\;\frac{x.im}{\sqrt{\mathsf{fma}\left(y.im, y.im, \left(y.re \cdot y.re\right)\right)}}\\

\end{array}
double f(double x_re, double x_im, double y_re, double y_im) {
        double r1436149 = x_re;
        double r1436150 = y_re;
        double r1436151 = r1436149 * r1436150;
        double r1436152 = x_im;
        double r1436153 = y_im;
        double r1436154 = r1436152 * r1436153;
        double r1436155 = r1436151 + r1436154;
        double r1436156 = r1436150 * r1436150;
        double r1436157 = r1436153 * r1436153;
        double r1436158 = r1436156 + r1436157;
        double r1436159 = r1436155 / r1436158;
        return r1436159;
}
↓
double f(double x_re, double x_im, double y_re, double y_im) {
        double r1436160 = y_im;
        double r1436161 = 6.646276389058437e+117;
        bool r1436162 = r1436160 <= r1436161;
        double r1436163 = x_re;
        double r1436164 = y_re;
        double r1436165 = x_im;
        double r1436166 = r1436165 * r1436160;
        double r1436167 = fma(r1436163, r1436164, r1436166);
        double r1436168 = r1436164 * r1436164;
        double r1436169 = fma(r1436160, r1436160, r1436168);
        double r1436170 = sqrt(r1436169);
        double r1436171 = r1436167 / r1436170;
        double r1436172 = r1436171 / r1436170;
        double r1436173 = r1436165 / r1436170;
        double r1436174 = r1436162 ? r1436172 : r1436173;
        return r1436174;
}

Error

Bits error versus x.re

Bits error versus x.im

Bits error versus y.re

Bits error versus y.im

Derivation

  1. Split input into 2 regimes

  2. if y.im < 6.646276389058437e+117

    1. Initial program 22.5

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

    2. Simplified22.5

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

    4. Applied add-sqr-sqrt22.5

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

    5. Applied associate-/r*22.5

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

    if 6.646276389058437e+117 < y.im

    1. Initial program 42.3

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

    2. Simplified42.3

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

    4. Applied add-sqr-sqrt42.3

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

    5. Applied associate-/r*42.2

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

    6. Taylor expanded around 0 41.5

      \[\leadsto \frac{\color{blue}{x.im}}{\sqrt{\mathsf{fma}\left(y.im, y.im, \left(y.re \cdot y.re\right)\right)}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification25.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \le 6.646276389058437 \cdot 10^{+117}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(x.re, y.re, \left(x.im \cdot y.im\right)\right)}{\sqrt{\mathsf{fma}\left(y.im, y.im, \left(y.re \cdot y.re\right)\right)}}}{\sqrt{\mathsf{fma}\left(y.im, y.im, \left(y.re \cdot y.re\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{\sqrt{\mathsf{fma}\left(y.im, y.im, \left(y.re \cdot y.re\right)\right)}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019107 +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))))