Average Error: 26.6 → 27.2
Time: 16.3s
Precision: 64
\[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\]
↓
\[\begin{array}{l} \mathbf{if}\;x.re \le 6.495480430581795464113836761587821513134 \cdot 10^{224}:\\ \;\;\;\;\frac{\frac{x.im \cdot y.im + y.re \cdot x.re}{\sqrt{y.im \cdot y.im + y.re \cdot y.re}}}{\sqrt{y.im \cdot y.im + y.re \cdot y.re}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-x.re}{\sqrt{y.im \cdot y.im + y.re \cdot y.re}}\\ \end{array}\]
\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 r3026371 = x_re;
        double r3026372 = y_re;
        double r3026373 = r3026371 * r3026372;
        double r3026374 = x_im;
        double r3026375 = y_im;
        double r3026376 = r3026374 * r3026375;
        double r3026377 = r3026373 + r3026376;
        double r3026378 = r3026372 * r3026372;
        double r3026379 = r3026375 * r3026375;
        double r3026380 = r3026378 + r3026379;
        double r3026381 = r3026377 / r3026380;
        return r3026381;
}
↓
double f(double x_re, double x_im, double y_re, double y_im) {
        double r3026382 = x_re;
        double r3026383 = 6.4954804305817955e+224;
        bool r3026384 = r3026382 <= r3026383;
        double r3026385 = x_im;
        double r3026386 = y_im;
        double r3026387 = r3026385 * r3026386;
        double r3026388 = y_re;
        double r3026389 = r3026388 * r3026382;
        double r3026390 = r3026387 + r3026389;
        double r3026391 = r3026386 * r3026386;
        double r3026392 = r3026388 * r3026388;
        double r3026393 = r3026391 + r3026392;
        double r3026394 = sqrt(r3026393);
        double r3026395 = r3026390 / r3026394;
        double r3026396 = r3026395 / r3026394;
        double r3026397 = -r3026382;
        double r3026398 = r3026397 / r3026394;
        double r3026399 = r3026384 ? r3026396 : r3026398;
        return r3026399;
}

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))))