Average Error: 26.3 → 26.3
Time: 9.1s
Precision: 64
\[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\]
↓
\[\begin{array}{l} \mathbf{if}\;y.re \le 1.44736579362970321 \cdot 10^{65}:\\ \;\;\;\;\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}\\ \end{array}\]
\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 1.44736579362970321 \cdot 10^{65}:\\
\;\;\;\;\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\

\mathbf{else}:\\
\;\;\;\;\frac{x.re}{\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 r63442 = x_re;
        double r63443 = y_re;
        double r63444 = r63442 * r63443;
        double r63445 = x_im;
        double r63446 = y_im;
        double r63447 = r63445 * r63446;
        double r63448 = r63444 + r63447;
        double r63449 = r63443 * r63443;
        double r63450 = r63446 * r63446;
        double r63451 = r63449 + r63450;
        double r63452 = r63448 / r63451;
        return r63452;
}
↓
double f(double x_re, double x_im, double y_re, double y_im) {
        double r63453 = y_re;
        double r63454 = 1.4473657936297032e+65;
        bool r63455 = r63453 <= r63454;
        double r63456 = x_re;
        double r63457 = r63456 * r63453;
        double r63458 = x_im;
        double r63459 = y_im;
        double r63460 = r63458 * r63459;
        double r63461 = r63457 + r63460;
        double r63462 = r63453 * r63453;
        double r63463 = r63459 * r63459;
        double r63464 = r63462 + r63463;
        double r63465 = r63461 / r63464;
        double r63466 = sqrt(r63464);
        double r63467 = r63456 / r63466;
        double r63468 = r63455 ? r63465 : r63467;
        return r63468;
}

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 < 1.4473657936297032e+65

    1. Initial program 23.4

      \[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\]

    if 1.4473657936297032e+65 < y.re

    1. Initial program 37.2

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

    3. Applied add-sqr-sqrt37.2

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

    4. Applied associate-/r*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 37.2

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

  4. Final simplification26.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \le 1.44736579362970321 \cdot 10^{65}:\\ \;\;\;\;\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020047 
(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))))