Average Error: 26.2 → 26.1
Time: 11.8s
Precision: 64
\[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\]
↓
\[\begin{array}{l} \mathbf{if}\;y.re \le -1.0148544823210362 \cdot 10^{+76}:\\ \;\;\;\;\frac{-x.re}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x.im \cdot y.im + x.re \cdot y.re}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}\\ \end{array}\]
\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}\;y.re \le -1.0148544823210362 \cdot 10^{+76}:\\
\;\;\;\;\frac{-x.re}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}\\

\mathbf{else}:\\
\;\;\;\;\frac{\frac{x.im \cdot y.im + x.re \cdot y.re}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}}{\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 r1510024 = x_re;
        double r1510025 = y_re;
        double r1510026 = r1510024 * r1510025;
        double r1510027 = x_im;
        double r1510028 = y_im;
        double r1510029 = r1510027 * r1510028;
        double r1510030 = r1510026 + r1510029;
        double r1510031 = r1510025 * r1510025;
        double r1510032 = r1510028 * r1510028;
        double r1510033 = r1510031 + r1510032;
        double r1510034 = r1510030 / r1510033;
        return r1510034;
}
↓
double f(double x_re, double x_im, double y_re, double y_im) {
        double r1510035 = y_re;
        double r1510036 = -1.0148544823210362e+76;
        bool r1510037 = r1510035 <= r1510036;
        double r1510038 = x_re;
        double r1510039 = -r1510038;
        double r1510040 = r1510035 * r1510035;
        double r1510041 = y_im;
        double r1510042 = r1510041 * r1510041;
        double r1510043 = r1510040 + r1510042;
        double r1510044 = sqrt(r1510043);
        double r1510045 = r1510039 / r1510044;
        double r1510046 = x_im;
        double r1510047 = r1510046 * r1510041;
        double r1510048 = r1510038 * r1510035;
        double r1510049 = r1510047 + r1510048;
        double r1510050 = r1510049 / r1510044;
        double r1510051 = r1510050 / r1510044;
        double r1510052 = r1510037 ? r1510045 : r1510051;
        return r1510052;
}

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 y.re < -1.0148544823210362e+76

    1. Initial program 37.3

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

    3. Applied add-sqr-sqrt37.3

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

    4. Applied associate-/r*37.2

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

    5. Taylor expanded around -inf 37.3

      \[\leadsto \frac{\color{blue}{-1 \cdot x.re}}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}\]

    6. Simplified37.3

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

    if -1.0148544823210362e+76 < y.re

    1. Initial program 23.3

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

    3. Applied add-sqr-sqrt23.3

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

    4. Applied associate-/r*23.2

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

  4. Final simplification26.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \le -1.0148544823210362 \cdot 10^{+76}:\\ \;\;\;\;\frac{-x.re}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x.im \cdot y.im + x.re \cdot y.re}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019153 
(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))))