Average Error: 25.6 → 25.5
Time: 33.2s
Precision: 64
\[\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\]
↓
\[\begin{array}{l} \mathbf{if}\;y.re \le 6.443252805869156 \cdot 10^{+104}:\\ \;\;\;\;\frac{\frac{x.im \cdot y.re - y.im \cdot x.re}{\sqrt{(y.im \cdot y.im + \left(y.re \cdot y.re\right))_*}}}{\sqrt{(y.im \cdot y.im + \left(y.re \cdot y.re\right))_*}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{\sqrt{(y.im \cdot y.im + \left(y.re \cdot y.re\right))_*}}\\ \end{array}\]
\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}
↓
\begin{array}{l}
\mathbf{if}\;y.re \le 6.443252805869156 \cdot 10^{+104}:\\
\;\;\;\;\frac{\frac{x.im \cdot y.re - y.im \cdot x.re}{\sqrt{(y.im \cdot y.im + \left(y.re \cdot y.re\right))_*}}}{\sqrt{(y.im \cdot y.im + \left(y.re \cdot y.re\right))_*}}\\

\mathbf{else}:\\
\;\;\;\;\frac{x.im}{\sqrt{(y.im \cdot y.im + \left(y.re \cdot y.re\right))_*}}\\

\end{array}
double f(double x_re, double x_im, double y_re, double y_im) {
        double r6424392 = x_im;
        double r6424393 = y_re;
        double r6424394 = r6424392 * r6424393;
        double r6424395 = x_re;
        double r6424396 = y_im;
        double r6424397 = r6424395 * r6424396;
        double r6424398 = r6424394 - r6424397;
        double r6424399 = r6424393 * r6424393;
        double r6424400 = r6424396 * r6424396;
        double r6424401 = r6424399 + r6424400;
        double r6424402 = r6424398 / r6424401;
        return r6424402;
}
↓
double f(double x_re, double x_im, double y_re, double y_im) {
        double r6424403 = y_re;
        double r6424404 = 6.443252805869156e+104;
        bool r6424405 = r6424403 <= r6424404;
        double r6424406 = x_im;
        double r6424407 = r6424406 * r6424403;
        double r6424408 = y_im;
        double r6424409 = x_re;
        double r6424410 = r6424408 * r6424409;
        double r6424411 = r6424407 - r6424410;
        double r6424412 = r6424403 * r6424403;
        double r6424413 = fma(r6424408, r6424408, r6424412);
        double r6424414 = sqrt(r6424413);
        double r6424415 = r6424411 / r6424414;
        double r6424416 = r6424415 / r6424414;
        double r6424417 = r6424406 / r6424414;
        double r6424418 = r6424405 ? r6424416 : r6424417;
        return r6424418;
}

Error

Bits error versus x.re

Bits error versus x.im

Bits error versus y.re

Bits error versus y.im

Derivation

  1. Split input into 2 regimes

  2. if y.re < 6.443252805869156e+104

    1. Initial program 22.4

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

    2. Simplified22.4

      \[\leadsto \color{blue}{\frac{x.im \cdot y.re - x.re \cdot y.im}{(y.im \cdot y.im + \left(y.re \cdot y.re\right))_*}}\]
    3. Using strategy rm

    4. Applied add-sqr-sqrt22.4

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

    5. Applied *-un-lft-identity22.4

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

    6. Applied times-frac22.4

      \[\leadsto \color{blue}{\frac{1}{\sqrt{(y.im \cdot y.im + \left(y.re \cdot y.re\right))_*}} \cdot \frac{x.im \cdot y.re - x.re \cdot y.im}{\sqrt{(y.im \cdot y.im + \left(y.re \cdot y.re\right))_*}}}\]
    7. Using strategy rm

    8. Applied associate-*l/22.3

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

    if 6.443252805869156e+104 < y.re

    1. Initial program 41.1

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

    2. Simplified41.1

      \[\leadsto \color{blue}{\frac{x.im \cdot y.re - x.re \cdot y.im}{(y.im \cdot y.im + \left(y.re \cdot y.re\right))_*}}\]
    3. Using strategy rm

    4. Applied add-sqr-sqrt41.1

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

    5. Applied associate-/r*41.0

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

    6. Taylor expanded around inf 40.7

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

  4. Final simplification25.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \le 6.443252805869156 \cdot 10^{+104}:\\ \;\;\;\;\frac{\frac{x.im \cdot y.re - y.im \cdot x.re}{\sqrt{(y.im \cdot y.im + \left(y.re \cdot y.re\right))_*}}}{\sqrt{(y.im \cdot y.im + \left(y.re \cdot y.re\right))_*}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{\sqrt{(y.im \cdot y.im + \left(y.re \cdot y.re\right))_*}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019104 +o rules:numerics
(FPCore (x.re x.im y.re y.im)
  :name "_divideComplex, imaginary part"
  (/ (- (* x.im y.re) (* x.re y.im)) (+ (* y.re y.re) (* y.im y.im))))