Average Error: 25.8 → 25.8
Time: 11.9s
Precision: 64
\[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \le 3.7234357932591811 \cdot 10^{232}:\\ \;\;\;\;\frac{\frac{1}{\frac{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}{y.im \cdot x.im + y.re \cdot x.re}}}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-x.re}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}\\ \end{array}\]
\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}\;\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \le 3.7234357932591811 \cdot 10^{232}:\\
\;\;\;\;\frac{\frac{1}{\frac{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}{y.im \cdot x.im + y.re \cdot x.re}}}{\sqrt{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 r59423 = x_re;
        double r59424 = y_re;
        double r59425 = r59423 * r59424;
        double r59426 = x_im;
        double r59427 = y_im;
        double r59428 = r59426 * r59427;
        double r59429 = r59425 + r59428;
        double r59430 = r59424 * r59424;
        double r59431 = r59427 * r59427;
        double r59432 = r59430 + r59431;
        double r59433 = r59429 / r59432;
        return r59433;
}
↓
double f(double x_re, double x_im, double y_re, double y_im) {
        double r59434 = x_re;
        double r59435 = y_re;
        double r59436 = r59434 * r59435;
        double r59437 = x_im;
        double r59438 = y_im;
        double r59439 = r59437 * r59438;
        double r59440 = r59436 + r59439;
        double r59441 = r59435 * r59435;
        double r59442 = r59438 * r59438;
        double r59443 = r59441 + r59442;
        double r59444 = r59440 / r59443;
        double r59445 = 3.723435793259181e+232;
        bool r59446 = r59444 <= r59445;
        double r59447 = 1.0;
        double r59448 = sqrt(r59443);
        double r59449 = r59438 * r59437;
        double r59450 = r59435 * r59434;
        double r59451 = r59449 + r59450;
        double r59452 = r59448 / r59451;
        double r59453 = r59447 / r59452;
        double r59454 = r59453 / r59448;
        double r59455 = -r59434;
        double r59456 = r59455 / r59448;
        double r59457 = r59446 ? r59454 : r59456;
        return r59457;
}

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))) < 3.723435793259181e+232

    1. Initial program 13.9

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

    3. Applied add-sqr-sqrt13.9

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

    4. Applied associate-/r*13.8

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

    6. Applied clear-num13.8

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

    7. Simplified13.8

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

    if 3.723435793259181e+232 < (/ (+ (* x.re y.re) (* x.im y.im)) (+ (* y.re y.re) (* y.im y.im)))

    1. Initial program 60.0

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

    3. Applied add-sqr-sqrt60.0

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

    4. Applied associate-/r*60.0

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

    5. Taylor expanded around -inf 59.9

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

    6. Simplified59.9

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

  4. Final simplification25.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \le 3.7234357932591811 \cdot 10^{232}:\\ \;\;\;\;\frac{\frac{1}{\frac{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}{y.im \cdot x.im + y.re \cdot x.re}}}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-x.re}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020045 
(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))))