Average Error: 25.9 → 25.7
Time: 14.4s
Precision: 64
\[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{x.im \cdot y.im + x.re \cdot y.re}{y.re \cdot y.re + y.im \cdot y.im} \le 5.836032336515321 \cdot 10^{+282}:\\ \;\;\;\;\frac{\frac{x.im \cdot y.im + x.re \cdot y.re}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}\\ \end{array}\]
\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}\;\frac{x.im \cdot y.im + x.re \cdot y.re}{y.re \cdot y.re + y.im \cdot y.im} \le 5.836032336515321 \cdot 10^{+282}:\\
\;\;\;\;\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}}\\

\mathbf{else}:\\
\;\;\;\;\frac{x.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 r1754852 = x_re;
        double r1754853 = y_re;
        double r1754854 = r1754852 * r1754853;
        double r1754855 = x_im;
        double r1754856 = y_im;
        double r1754857 = r1754855 * r1754856;
        double r1754858 = r1754854 + r1754857;
        double r1754859 = r1754853 * r1754853;
        double r1754860 = r1754856 * r1754856;
        double r1754861 = r1754859 + r1754860;
        double r1754862 = r1754858 / r1754861;
        return r1754862;
}
↓
double f(double x_re, double x_im, double y_re, double y_im) {
        double r1754863 = x_im;
        double r1754864 = y_im;
        double r1754865 = r1754863 * r1754864;
        double r1754866 = x_re;
        double r1754867 = y_re;
        double r1754868 = r1754866 * r1754867;
        double r1754869 = r1754865 + r1754868;
        double r1754870 = r1754867 * r1754867;
        double r1754871 = r1754864 * r1754864;
        double r1754872 = r1754870 + r1754871;
        double r1754873 = r1754869 / r1754872;
        double r1754874 = 5.836032336515321e+282;
        bool r1754875 = r1754873 <= r1754874;
        double r1754876 = sqrt(r1754872);
        double r1754877 = r1754869 / r1754876;
        double r1754878 = r1754877 / r1754876;
        double r1754879 = r1754863 / r1754876;
        double r1754880 = r1754875 ? r1754878 : r1754879;
        return r1754880;
}

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 y.re) (* x.im y.im)) (+ (* y.re y.re) (* y.im y.im))) < 5.836032336515321e+282

    1. Initial program 14.1

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

    3. Applied add-sqr-sqrt14.1

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

    4. Applied associate-/r*14.0

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

    if 5.836032336515321e+282 < (/ (+ (* x.re y.re) (* x.im y.im)) (+ (* y.re y.re) (* y.im y.im)))

    1. Initial program 61.0

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

    3. Applied add-sqr-sqrt61.0

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

    4. Applied associate-/r*61.0

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

    5. Taylor expanded around 0 60.2

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

  4. Final simplification25.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x.im \cdot y.im + x.re \cdot y.re}{y.re \cdot y.re + y.im \cdot y.im} \le 5.836032336515321 \cdot 10^{+282}:\\ \;\;\;\;\frac{\frac{x.im \cdot y.im + x.re \cdot y.re}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019149 
(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))))