Average Error: 26.6 → 13.6
Time: 9.5s
Precision: 64
\[\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \le -1033969720.49365747:\\ \;\;\;\;\frac{x.im}{\frac{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)}{y.re}} - \frac{x.re}{\frac{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)}{y.im}}\\ \mathbf{elif}\;\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \le 7.8735811135706561 \cdot 10^{304}:\\ \;\;\;\;\sqrt{\frac{1}{\mathsf{hypot}\left(y.re, y.im\right) \cdot 1}} \cdot \left(\sqrt{\frac{1}{\mathsf{hypot}\left(y.re, y.im\right) \cdot 1}} \cdot \frac{x.im \cdot y.re - x.re \cdot y.im}{\mathsf{hypot}\left(y.re, y.im\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \frac{-1 \cdot x.re}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \end{array}\]
\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}\;\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \le -1033969720.49365747:\\
\;\;\;\;\frac{x.im}{\frac{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)}{y.re}} - \frac{x.re}{\frac{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)}{y.im}}\\

\mathbf{elif}\;\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \le 7.8735811135706561 \cdot 10^{304}:\\
\;\;\;\;\sqrt{\frac{1}{\mathsf{hypot}\left(y.re, y.im\right) \cdot 1}} \cdot \left(\sqrt{\frac{1}{\mathsf{hypot}\left(y.re, y.im\right) \cdot 1}} \cdot \frac{x.im \cdot y.re - x.re \cdot y.im}{\mathsf{hypot}\left(y.re, y.im\right)}\right)\\

\mathbf{else}:\\
\;\;\;\;1 \cdot \frac{-1 \cdot x.re}{\mathsf{hypot}\left(y.re, y.im\right)}\\

\end{array}
double f(double x_re, double x_im, double y_re, double y_im) {
        double r358 = x_im;
        double r359 = y_re;
        double r360 = r358 * r359;
        double r361 = x_re;
        double r362 = y_im;
        double r363 = r361 * r362;
        double r364 = r360 - r363;
        double r365 = r359 * r359;
        double r366 = r362 * r362;
        double r367 = r365 + r366;
        double r368 = r364 / r367;
        return r368;
}
↓
double f(double x_re, double x_im, double y_re, double y_im) {
        double r369 = x_im;
        double r370 = y_re;
        double r371 = r369 * r370;
        double r372 = x_re;
        double r373 = y_im;
        double r374 = r372 * r373;
        double r375 = r371 - r374;
        double r376 = r370 * r370;
        double r377 = r373 * r373;
        double r378 = r376 + r377;
        double r379 = r375 / r378;
        double r380 = -1033969720.4936575;
        bool r381 = r379 <= r380;
        double r382 = fma(r370, r370, r377);
        double r383 = r382 / r370;
        double r384 = r369 / r383;
        double r385 = r382 / r373;
        double r386 = r372 / r385;
        double r387 = r384 - r386;
        double r388 = 7.873581113570656e+304;
        bool r389 = r379 <= r388;
        double r390 = 1.0;
        double r391 = hypot(r370, r373);
        double r392 = r391 * r390;
        double r393 = r390 / r392;
        double r394 = sqrt(r393);
        double r395 = r375 / r391;
        double r396 = r394 * r395;
        double r397 = r394 * r396;
        double r398 = -1.0;
        double r399 = r398 * r372;
        double r400 = r399 / r391;
        double r401 = r390 * r400;
        double r402 = r389 ? r397 : r401;
        double r403 = r381 ? r387 : r402;
        return r403;
}

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 3 regimes

  2. if (/ (- (* x.im y.re) (* x.re y.im)) (+ (* y.re y.re) (* y.im y.im))) < -1033969720.4936575

    1. Initial program 17.1

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

    3. Applied div-sub17.1

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

    4. Simplified11.1

      \[\leadsto \color{blue}{\frac{x.im}{\frac{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)}{y.re}}} - \frac{x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\]

    5. Simplified5.8

      \[\leadsto \frac{x.im}{\frac{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)}{y.re}} - \color{blue}{\frac{x.re}{\frac{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)}{y.im}}}\]

    if -1033969720.4936575 < (/ (- (* x.im y.re) (* x.re y.im)) (+ (* y.re y.re) (* y.im y.im))) < 7.873581113570656e+304

    1. Initial program 13.4

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

    3. Applied add-sqr-sqrt13.4

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

    4. Applied *-un-lft-identity13.4

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

    5. Applied times-frac13.4

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

    6. Simplified13.4

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

    7. Simplified0.9

      \[\leadsto \frac{1}{\mathsf{hypot}\left(y.re, y.im\right) \cdot 1} \cdot \color{blue}{\frac{x.im \cdot y.re - x.re \cdot y.im}{\mathsf{hypot}\left(y.re, y.im\right)}}\]
    8. Using strategy rm

    9. Applied add-sqr-sqrt1.1

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

    10. Applied associate-*l*1.1

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

    if 7.873581113570656e+304 < (/ (- (* x.im y.re) (* x.re y.im)) (+ (* y.re y.re) (* y.im y.im)))

    1. Initial program 63.7

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

    3. Applied add-sqr-sqrt63.7

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

    4. Applied *-un-lft-identity63.7

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

    5. Applied times-frac63.7

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

    6. Simplified63.7

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

    7. Simplified61.2

      \[\leadsto \frac{1}{\mathsf{hypot}\left(y.re, y.im\right) \cdot 1} \cdot \color{blue}{\frac{x.im \cdot y.re - x.re \cdot y.im}{\mathsf{hypot}\left(y.re, y.im\right)}}\]
    8. Using strategy rm

    9. Applied *-un-lft-identity61.2

      \[\leadsto \color{blue}{\left(1 \cdot \frac{1}{\mathsf{hypot}\left(y.re, y.im\right) \cdot 1}\right)} \cdot \frac{x.im \cdot y.re - x.re \cdot y.im}{\mathsf{hypot}\left(y.re, y.im\right)}\]

    10. Applied associate-*l*61.2

      \[\leadsto \color{blue}{1 \cdot \left(\frac{1}{\mathsf{hypot}\left(y.re, y.im\right) \cdot 1} \cdot \frac{x.im \cdot y.re - x.re \cdot y.im}{\mathsf{hypot}\left(y.re, y.im\right)}\right)}\]

    11. Simplified61.2

      \[\leadsto 1 \cdot \color{blue}{\frac{\frac{x.im \cdot y.re - x.re \cdot y.im}{\mathsf{hypot}\left(y.re, y.im\right)}}{\mathsf{hypot}\left(y.re, y.im\right)}}\]

    12. Taylor expanded around 0 48.5

      \[\leadsto 1 \cdot \frac{\color{blue}{-1 \cdot x.re}}{\mathsf{hypot}\left(y.re, y.im\right)}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification13.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \le -1033969720.49365747:\\ \;\;\;\;\frac{x.im}{\frac{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)}{y.re}} - \frac{x.re}{\frac{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)}{y.im}}\\ \mathbf{elif}\;\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \le 7.8735811135706561 \cdot 10^{304}:\\ \;\;\;\;\sqrt{\frac{1}{\mathsf{hypot}\left(y.re, y.im\right) \cdot 1}} \cdot \left(\sqrt{\frac{1}{\mathsf{hypot}\left(y.re, y.im\right) \cdot 1}} \cdot \frac{x.im \cdot y.re - x.re \cdot y.im}{\mathsf{hypot}\left(y.re, y.im\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \frac{-1 \cdot x.re}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020025 +o rules:numerics
(FPCore (x.re x.im y.re y.im)
  :name "_divideComplex, imaginary part"
  :precision binary64
  (/ (- (* x.im y.re) (* x.re y.im)) (+ (* y.re y.re) (* y.im y.im))))