Average Error: 26.7 → 12.8
Time: 4.1s
Precision: 64
\[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\]
↓
\[\begin{array}{l} \mathbf{if}\;y.re \le -1.0042415763270424 \cdot 10^{167}:\\ \;\;\;\;\frac{-1 \cdot x.re}{\mathsf{hypot}\left(y.re, y.im\right) \cdot 1}\\ \mathbf{elif}\;y.re \le 1.5555378499724725 \cdot 10^{174}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}{\mathsf{hypot}\left(y.re, y.im\right)}}{\mathsf{hypot}\left(y.re, y.im\right) \cdot 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{\mathsf{hypot}\left(y.re, y.im\right) \cdot 1}\\ \end{array}\]
\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 -1.0042415763270424 \cdot 10^{167}:\\
\;\;\;\;\frac{-1 \cdot x.re}{\mathsf{hypot}\left(y.re, y.im\right) \cdot 1}\\

\mathbf{elif}\;y.re \le 1.5555378499724725 \cdot 10^{174}:\\
\;\;\;\;\frac{\frac{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}{\mathsf{hypot}\left(y.re, y.im\right)}}{\mathsf{hypot}\left(y.re, y.im\right) \cdot 1}\\

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

\end{array}
double f(double x_re, double x_im, double y_re, double y_im) {
        double r106585 = x_re;
        double r106586 = y_re;
        double r106587 = r106585 * r106586;
        double r106588 = x_im;
        double r106589 = y_im;
        double r106590 = r106588 * r106589;
        double r106591 = r106587 + r106590;
        double r106592 = r106586 * r106586;
        double r106593 = r106589 * r106589;
        double r106594 = r106592 + r106593;
        double r106595 = r106591 / r106594;
        return r106595;
}
↓
double f(double x_re, double x_im, double y_re, double y_im) {
        double r106596 = y_re;
        double r106597 = -1.0042415763270424e+167;
        bool r106598 = r106596 <= r106597;
        double r106599 = -1.0;
        double r106600 = x_re;
        double r106601 = r106599 * r106600;
        double r106602 = y_im;
        double r106603 = hypot(r106596, r106602);
        double r106604 = 1.0;
        double r106605 = r106603 * r106604;
        double r106606 = r106601 / r106605;
        double r106607 = 1.5555378499724725e+174;
        bool r106608 = r106596 <= r106607;
        double r106609 = x_im;
        double r106610 = r106609 * r106602;
        double r106611 = fma(r106600, r106596, r106610);
        double r106612 = r106611 / r106603;
        double r106613 = r106612 / r106605;
        double r106614 = r106600 / r106605;
        double r106615 = r106608 ? r106613 : r106614;
        double r106616 = r106598 ? r106606 : r106615;
        return r106616;
}

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 < -1.0042415763270424e+167

    1. Initial program 45.1

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

    3. Applied add-sqr-sqrt45.1

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

    4. Applied *-un-lft-identity45.1

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

    5. Applied times-frac45.1

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

    6. Simplified45.1

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

    7. Simplified30.1

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

    9. Applied associate-*r/30.1

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

    10. Simplified30.1

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

    11. Taylor expanded around -inf 11.8

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

    if -1.0042415763270424e+167 < y.re < 1.5555378499724725e+174

    1. Initial program 20.9

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

    3. Applied add-sqr-sqrt20.9

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

    4. Applied *-un-lft-identity20.9

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

    5. Applied times-frac21.0

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

    6. Simplified21.0

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

    7. Simplified13.2

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

    9. Applied associate-*r/13.1

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

    10. Simplified13.1

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

    if 1.5555378499724725e+174 < y.re

    1. Initial program 44.5

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

    3. Applied add-sqr-sqrt44.5

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

    4. Applied *-un-lft-identity44.5

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

    5. Applied times-frac44.5

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

    6. Simplified44.5

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

    7. Simplified30.0

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

    9. Applied associate-*r/30.0

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

    10. Simplified29.9

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

    11. Taylor expanded around inf 12.0

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

  4. Final simplification12.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \le -1.0042415763270424 \cdot 10^{167}:\\ \;\;\;\;\frac{-1 \cdot x.re}{\mathsf{hypot}\left(y.re, y.im\right) \cdot 1}\\ \mathbf{elif}\;y.re \le 1.5555378499724725 \cdot 10^{174}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}{\mathsf{hypot}\left(y.re, y.im\right)}}{\mathsf{hypot}\left(y.re, y.im\right) \cdot 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{\mathsf{hypot}\left(y.re, y.im\right) \cdot 1}\\ \end{array}\]

Reproduce

herbie shell --seed 2020036 +o rules:numerics
(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))))