Average Error: 26.3 → 27.2
Time: 4.1s
Precision: 64
\[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\]
↓
\[\begin{array}{l} \mathbf{if}\;x.im \le -1.83333069166780128 \cdot 10^{209}:\\ \;\;\;\;\frac{x.im}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re \cdot y.re + x.im \cdot y.im}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}} \cdot \frac{1}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}\\ \end{array}\]
\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}\;x.im \le -1.83333069166780128 \cdot 10^{209}:\\
\;\;\;\;\frac{x.im}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}\\

\mathbf{else}:\\
\;\;\;\;\frac{x.re \cdot y.re + x.im \cdot y.im}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}} \cdot \frac{1}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}\\

\end{array}
double f(double x_re, double x_im, double y_re, double y_im) {
        double r56860 = x_re;
        double r56861 = y_re;
        double r56862 = r56860 * r56861;
        double r56863 = x_im;
        double r56864 = y_im;
        double r56865 = r56863 * r56864;
        double r56866 = r56862 + r56865;
        double r56867 = r56861 * r56861;
        double r56868 = r56864 * r56864;
        double r56869 = r56867 + r56868;
        double r56870 = r56866 / r56869;
        return r56870;
}
↓
double f(double x_re, double x_im, double y_re, double y_im) {
        double r56871 = x_im;
        double r56872 = -1.8333306916678013e+209;
        bool r56873 = r56871 <= r56872;
        double r56874 = y_re;
        double r56875 = r56874 * r56874;
        double r56876 = y_im;
        double r56877 = r56876 * r56876;
        double r56878 = r56875 + r56877;
        double r56879 = sqrt(r56878);
        double r56880 = r56871 / r56879;
        double r56881 = x_re;
        double r56882 = r56881 * r56874;
        double r56883 = r56871 * r56876;
        double r56884 = r56882 + r56883;
        double r56885 = r56884 / r56879;
        double r56886 = 1.0;
        double r56887 = r56886 / r56879;
        double r56888 = r56885 * r56887;
        double r56889 = r56873 ? r56880 : r56888;
        return r56889;
}

Error

Bits error versus x.re

Bits error versus x.im

Bits error versus y.re

Bits error versus y.im

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 2 regimes

  2. if x.im < -1.8333306916678013e+209

    1. Initial program 40.6

      \[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\]
    2. Using strategy rm

    3. Applied add-sqr-sqrt40.6

      \[\leadsto \frac{x.re \cdot y.re + x.im \cdot y.im}{\color{blue}{\sqrt{y.re \cdot y.re + y.im \cdot y.im} \cdot \sqrt{y.re \cdot y.re + y.im \cdot y.im}}}\]

    4. Applied associate-/r*40.6

      \[\leadsto \color{blue}{\frac{\frac{x.re \cdot y.re + x.im \cdot y.im}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}}\]

    5. Taylor expanded around 0 52.0

      \[\leadsto \frac{\color{blue}{x.im}}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}\]

    if -1.8333306916678013e+209 < x.im

    1. Initial program 25.2

      \[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\]
    2. Using strategy rm

    3. Applied add-sqr-sqrt25.2

      \[\leadsto \frac{x.re \cdot y.re + x.im \cdot y.im}{\color{blue}{\sqrt{y.re \cdot y.re + y.im \cdot y.im} \cdot \sqrt{y.re \cdot y.re + y.im \cdot y.im}}}\]

    4. Applied associate-/r*25.1

      \[\leadsto \color{blue}{\frac{\frac{x.re \cdot y.re + x.im \cdot y.im}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}}\]
    5. Using strategy rm

    6. Applied div-inv25.2

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

  4. Final simplification27.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \le -1.83333069166780128 \cdot 10^{209}:\\ \;\;\;\;\frac{x.im}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re \cdot y.re + x.im \cdot y.im}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}} \cdot \frac{1}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020056 
(FPCore (x.re x.im y.re y.im)
  :name "_divideComplex, real part"
  :precision binary64
  (/ (+ (* x.re y.re) (* x.im y.im)) (+ (* y.re y.re) (* y.im y.im))))