Average Error: 25.3 → 24.7
Time: 4.4s
Precision: 64
\[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \le 1.134035007520884449694704605899913336371 \cdot 10^{278}:\\ \;\;\;\;\frac{\frac{x.re \cdot y.re + x.im \cdot y.im}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}} \cdot x.re\\ \end{array}\]
\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}\;\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \le 1.134035007520884449694704605899913336371 \cdot 10^{278}:\\
\;\;\;\;\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}}\\

\mathbf{else}:\\
\;\;\;\;\frac{1}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}} \cdot x.re\\

\end{array}
double f(double x_re, double x_im, double y_re, double y_im) {
        double r67396 = x_re;
        double r67397 = y_re;
        double r67398 = r67396 * r67397;
        double r67399 = x_im;
        double r67400 = y_im;
        double r67401 = r67399 * r67400;
        double r67402 = r67398 + r67401;
        double r67403 = r67397 * r67397;
        double r67404 = r67400 * r67400;
        double r67405 = r67403 + r67404;
        double r67406 = r67402 / r67405;
        return r67406;
}
↓
double f(double x_re, double x_im, double y_re, double y_im) {
        double r67407 = x_re;
        double r67408 = y_re;
        double r67409 = r67407 * r67408;
        double r67410 = x_im;
        double r67411 = y_im;
        double r67412 = r67410 * r67411;
        double r67413 = r67409 + r67412;
        double r67414 = r67408 * r67408;
        double r67415 = r67411 * r67411;
        double r67416 = r67414 + r67415;
        double r67417 = r67413 / r67416;
        double r67418 = 1.1340350075208844e+278;
        bool r67419 = r67417 <= r67418;
        double r67420 = sqrt(r67416);
        double r67421 = r67413 / r67420;
        double r67422 = r67421 / r67420;
        double r67423 = 1.0;
        double r67424 = r67423 / r67420;
        double r67425 = r67424 * r67407;
        double r67426 = r67419 ? r67422 : r67425;
        return r67426;
}

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))) < 1.1340350075208844e+278

    1. Initial program 13.5

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

    3. Applied add-sqr-sqrt13.5

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

    4. Applied associate-/r*13.4

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

    if 1.1340350075208844e+278 < (/ (+ (* x.re y.re) (* x.im y.im)) (+ (* y.re y.re) (* y.im y.im)))

    1. Initial program 62.4

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

    3. Applied add-sqr-sqrt62.4

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

    4. Applied *-un-lft-identity62.4

      \[\leadsto \frac{\color{blue}{1 \cdot \left(x.re \cdot y.re + x.im \cdot y.im\right)}}{\sqrt{y.re \cdot y.re + y.im \cdot y.im} \cdot \sqrt{y.re \cdot y.re + y.im \cdot y.im}}\]

    5. Applied times-frac62.4

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

    6. Taylor expanded around inf 60.0

      \[\leadsto \frac{1}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}} \cdot \color{blue}{x.re}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification24.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \le 1.134035007520884449694704605899913336371 \cdot 10^{278}:\\ \;\;\;\;\frac{\frac{x.re \cdot y.re + x.im \cdot y.im}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}} \cdot x.re\\ \end{array}\]

Reproduce

herbie shell --seed 2019352 
(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))))