Average Error: 25.7 → 25.6
Time: 25.3s
Precision: 64
\[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\]
↓
\[\begin{array}{l} \mathbf{if}\;y.re \le -2.9064590122309793 \cdot 10^{+107}:\\ \;\;\;\;\frac{-x.re}{\sqrt{(y.im \cdot y.im + \left(y.re \cdot y.re\right))_*}}\\ \mathbf{else}:\\ \;\;\;\;\frac{(x.re \cdot y.re + \left(y.im \cdot x.im\right))_* \cdot \frac{1}{\sqrt{(y.im \cdot y.im + \left(y.re \cdot y.re\right))_*}}}{\sqrt{(y.im \cdot y.im + \left(y.re \cdot y.re\right))_*}}\\ \end{array}\]
\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.re \le -2.9064590122309793 \cdot 10^{+107}:\\
\;\;\;\;\frac{-x.re}{\sqrt{(y.im \cdot y.im + \left(y.re \cdot y.re\right))_*}}\\

\mathbf{else}:\\
\;\;\;\;\frac{(x.re \cdot y.re + \left(y.im \cdot x.im\right))_* \cdot \frac{1}{\sqrt{(y.im \cdot y.im + \left(y.re \cdot y.re\right))_*}}}{\sqrt{(y.im \cdot y.im + \left(y.re \cdot y.re\right))_*}}\\

\end{array}
double f(double x_re, double x_im, double y_re, double y_im) {
        double r2075862 = x_re;
        double r2075863 = y_re;
        double r2075864 = r2075862 * r2075863;
        double r2075865 = x_im;
        double r2075866 = y_im;
        double r2075867 = r2075865 * r2075866;
        double r2075868 = r2075864 + r2075867;
        double r2075869 = r2075863 * r2075863;
        double r2075870 = r2075866 * r2075866;
        double r2075871 = r2075869 + r2075870;
        double r2075872 = r2075868 / r2075871;
        return r2075872;
}
↓
double f(double x_re, double x_im, double y_re, double y_im) {
        double r2075873 = y_re;
        double r2075874 = -2.9064590122309793e+107;
        bool r2075875 = r2075873 <= r2075874;
        double r2075876 = x_re;
        double r2075877 = -r2075876;
        double r2075878 = y_im;
        double r2075879 = r2075873 * r2075873;
        double r2075880 = fma(r2075878, r2075878, r2075879);
        double r2075881 = sqrt(r2075880);
        double r2075882 = r2075877 / r2075881;
        double r2075883 = x_im;
        double r2075884 = r2075878 * r2075883;
        double r2075885 = fma(r2075876, r2075873, r2075884);
        double r2075886 = 1.0;
        double r2075887 = r2075886 / r2075881;
        double r2075888 = r2075885 * r2075887;
        double r2075889 = r2075888 / r2075881;
        double r2075890 = r2075875 ? r2075882 : r2075889;
        return r2075890;
}

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.re < -2.9064590122309793e+107

    1. Initial program 41.1

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

    2. Simplified41.1

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

    4. Applied add-sqr-sqrt41.1

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

    5. Applied associate-/r*41.1

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

    6. Taylor expanded around -inf 40.4

      \[\leadsto \frac{\color{blue}{-1 \cdot x.re}}{\sqrt{(y.im \cdot y.im + \left(y.re \cdot y.re\right))_*}}\]

    7. Simplified40.4

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

    if -2.9064590122309793e+107 < y.re

    1. Initial program 22.6

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

    2. Simplified22.6

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

    4. Applied add-sqr-sqrt22.6

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

    5. Applied associate-/r*22.5

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

    7. Applied div-inv22.6

      \[\leadsto \frac{\color{blue}{(x.re \cdot y.re + \left(x.im \cdot y.im\right))_* \cdot \frac{1}{\sqrt{(y.im \cdot y.im + \left(y.re \cdot y.re\right))_*}}}}{\sqrt{(y.im \cdot y.im + \left(y.re \cdot y.re\right))_*}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification25.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \le -2.9064590122309793 \cdot 10^{+107}:\\ \;\;\;\;\frac{-x.re}{\sqrt{(y.im \cdot y.im + \left(y.re \cdot y.re\right))_*}}\\ \mathbf{else}:\\ \;\;\;\;\frac{(x.re \cdot y.re + \left(y.im \cdot x.im\right))_* \cdot \frac{1}{\sqrt{(y.im \cdot y.im + \left(y.re \cdot y.re\right))_*}}}{\sqrt{(y.im \cdot y.im + \left(y.re \cdot y.re\right))_*}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019104 +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))))