Average Error: 26.6 → 27.2
Time: 15.7s
Precision: 64
\[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\]
↓
\[\begin{array}{l} \mathbf{if}\;x.re \le 6.495480430581795464113836761587821513134 \cdot 10^{224}:\\ \;\;\;\;\frac{\frac{x.im \cdot y.im + y.re \cdot x.re}{\sqrt{y.im \cdot y.im + y.re \cdot y.re}}}{\sqrt{y.im \cdot y.im + y.re \cdot y.re}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-x.re}{\sqrt{y.im \cdot y.im + y.re \cdot y.re}}\\ \end{array}\]
\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.re \le 6.495480430581795464113836761587821513134 \cdot 10^{224}:\\
\;\;\;\;\frac{\frac{x.im \cdot y.im + y.re \cdot x.re}{\sqrt{y.im \cdot y.im + y.re \cdot y.re}}}{\sqrt{y.im \cdot y.im + y.re \cdot y.re}}\\

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

\end{array}
double f(double x_re, double x_im, double y_re, double y_im) {
        double r2769093 = x_re;
        double r2769094 = y_re;
        double r2769095 = r2769093 * r2769094;
        double r2769096 = x_im;
        double r2769097 = y_im;
        double r2769098 = r2769096 * r2769097;
        double r2769099 = r2769095 + r2769098;
        double r2769100 = r2769094 * r2769094;
        double r2769101 = r2769097 * r2769097;
        double r2769102 = r2769100 + r2769101;
        double r2769103 = r2769099 / r2769102;
        return r2769103;
}
↓
double f(double x_re, double x_im, double y_re, double y_im) {
        double r2769104 = x_re;
        double r2769105 = 6.4954804305817955e+224;
        bool r2769106 = r2769104 <= r2769105;
        double r2769107 = x_im;
        double r2769108 = y_im;
        double r2769109 = r2769107 * r2769108;
        double r2769110 = y_re;
        double r2769111 = r2769110 * r2769104;
        double r2769112 = r2769109 + r2769111;
        double r2769113 = r2769108 * r2769108;
        double r2769114 = r2769110 * r2769110;
        double r2769115 = r2769113 + r2769114;
        double r2769116 = sqrt(r2769115);
        double r2769117 = r2769112 / r2769116;
        double r2769118 = r2769117 / r2769116;
        double r2769119 = -r2769104;
        double r2769120 = r2769119 / r2769116;
        double r2769121 = r2769106 ? r2769118 : r2769120;
        return r2769121;
}

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.re < 6.4954804305817955e+224

    1. Initial program 25.6

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

    3. Applied add-sqr-sqrt25.6

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

    4. Applied associate-/r*25.5

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

    if 6.4954804305817955e+224 < x.re

    1. Initial program 42.0

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

    3. Applied add-sqr-sqrt42.0

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

    4. Applied associate-/r*41.9

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

    5. Taylor expanded around -inf 52.0

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

    6. Simplified52.0

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

  4. Final simplification27.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \le 6.495480430581795464113836761587821513134 \cdot 10^{224}:\\ \;\;\;\;\frac{\frac{x.im \cdot y.im + y.re \cdot x.re}{\sqrt{y.im \cdot y.im + y.re \cdot y.re}}}{\sqrt{y.im \cdot y.im + y.re \cdot y.re}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-x.re}{\sqrt{y.im \cdot y.im + y.re \cdot y.re}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019171 
(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))))