Average Error: 26.2 → 13.2
Time: 15.0s
Precision: 64
\[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\]
↓
\[\begin{array}{l} \mathbf{if}\;y.re \le -6.99838818051005 \cdot 10^{+73}:\\ \;\;\;\;\frac{-x.re}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.re \le 1.938821746780901 \cdot 10^{+120}:\\ \;\;\;\;\frac{\frac{1}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{\mathsf{fma}\left(x.re, y.re, y.im \cdot x.im\right)}}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \end{array}\]
\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 -6.99838818051005 \cdot 10^{+73}:\\
\;\;\;\;\frac{-x.re}{\mathsf{hypot}\left(y.re, y.im\right)}\\

\mathbf{elif}\;y.re \le 1.938821746780901 \cdot 10^{+120}:\\
\;\;\;\;\frac{\frac{1}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{\mathsf{fma}\left(x.re, y.re, y.im \cdot x.im\right)}}}{\mathsf{hypot}\left(y.re, y.im\right)}\\

\mathbf{else}:\\
\;\;\;\;\frac{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 r944375 = x_re;
        double r944376 = y_re;
        double r944377 = r944375 * r944376;
        double r944378 = x_im;
        double r944379 = y_im;
        double r944380 = r944378 * r944379;
        double r944381 = r944377 + r944380;
        double r944382 = r944376 * r944376;
        double r944383 = r944379 * r944379;
        double r944384 = r944382 + r944383;
        double r944385 = r944381 / r944384;
        return r944385;
}
↓
double f(double x_re, double x_im, double y_re, double y_im) {
        double r944386 = y_re;
        double r944387 = -6.99838818051005e+73;
        bool r944388 = r944386 <= r944387;
        double r944389 = x_re;
        double r944390 = -r944389;
        double r944391 = y_im;
        double r944392 = hypot(r944386, r944391);
        double r944393 = r944390 / r944392;
        double r944394 = 1.938821746780901e+120;
        bool r944395 = r944386 <= r944394;
        double r944396 = 1.0;
        double r944397 = x_im;
        double r944398 = r944391 * r944397;
        double r944399 = fma(r944389, r944386, r944398);
        double r944400 = r944392 / r944399;
        double r944401 = r944396 / r944400;
        double r944402 = r944401 / r944392;
        double r944403 = r944389 / r944392;
        double r944404 = r944395 ? r944402 : r944403;
        double r944405 = r944388 ? r944393 : r944404;
        return r944405;
}

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 y.re < -6.99838818051005e+73

    1. Initial program 37.1

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

    3. Applied add-sqr-sqrt37.1

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

    4. Applied associate-/r*37.0

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

    6. Applied hypot-def37.0

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

    7. Taylor expanded around -inf 17.1

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

    8. Simplified17.1

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

    if -6.99838818051005e+73 < y.re < 1.938821746780901e+120

    1. Initial program 18.7

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

    3. Applied add-sqr-sqrt18.7

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

    4. Applied associate-/r*18.6

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

    6. Applied hypot-def18.6

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

    8. Applied clear-num18.7

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

    9. Simplified11.6

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

    if 1.938821746780901e+120 < y.re

    1. Initial program 41.2

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

    3. Applied add-sqr-sqrt41.2

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

    4. Applied associate-/r*41.1

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

    6. Applied hypot-def41.1

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

    7. Taylor expanded around inf 14.6

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

  4. Final simplification13.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \le -6.99838818051005 \cdot 10^{+73}:\\ \;\;\;\;\frac{-x.re}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.re \le 1.938821746780901 \cdot 10^{+120}:\\ \;\;\;\;\frac{\frac{1}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{\mathsf{fma}\left(x.re, y.re, y.im \cdot x.im\right)}}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019153 +o rules:numerics
(FPCore (x.re x.im y.re y.im)
  :name "_divideComplex, real part"
  (/ (+ (* x.re y.re) (* x.im y.im)) (+ (* y.re y.re) (* y.im y.im))))