_divideComplex, imaginary part

Average Error: 26.4 → 6.4

Time: 12.2s

Precision: 64

• Falling back on racket egraph because egg-herbie package not installed

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y.im \le -1.954211085394229 \cdot 10^{231}:\\ \;\;\;\;\frac{x.re}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.im \le 2.03024456745374313 \cdot 10^{152}:\\ \;\;\;\;\frac{\frac{y.im \cdot x.re}{-\mathsf{hypot}\left(y.re, y.im\right)} - \frac{y.re}{-\mathsf{hypot}\left(y.re, y.im\right)} \cdot x.im}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-x.re}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \end{array}\]

C

double f(double x_re, double x_im, double y_re, double y_im) {
        double r70605 = x_im;
        double r70606 = y_re;
        double r70607 = r70605 * r70606;
        double r70608 = x_re;
        double r70609 = y_im;
        double r70610 = r70608 * r70609;
        double r70611 = r70607 - r70610;
        double r70612 = r70606 * r70606;
        double r70613 = r70609 * r70609;
        double r70614 = r70612 + r70613;
        double r70615 = r70611 / r70614;
        return r70615;
}

↓

double f(double x_re, double x_im, double y_re, double y_im) {
        double r70616 = y_im;
        double r70617 = -1.9542110853942293e+231;
        bool r70618 = r70616 <= r70617;
        double r70619 = x_re;
        double r70620 = y_re;
        double r70621 = hypot(r70620, r70616);
        double r70622 = r70619 / r70621;
        double r70623 = 2.0302445674537431e+152;
        bool r70624 = r70616 <= r70623;
        double r70625 = r70616 * r70619;
        double r70626 = -r70621;
        double r70627 = r70625 / r70626;
        double r70628 = r70620 / r70626;
        double r70629 = x_im;
        double r70630 = r70628 * r70629;
        double r70631 = r70627 - r70630;
        double r70632 = r70631 / r70621;
        double r70633 = -r70619;
        double r70634 = r70633 / r70621;
        double r70635 = r70624 ? r70632 : r70634;
        double r70636 = r70618 ? r70622 : r70635;
        return r70636;
}

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.im < -1.9542110853942293e+231

   1. Initial program 41.3

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

   3. Applied add-sqr-sqrt 41.3

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

   4. Applied *-un-lft-identity 41.3

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

   5. Applied times-frac 41.3

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

   6. Simplified 41.3

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

   7. Simplified 33.9

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

   9. Applied associate-*r/ 33.9

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

   10. Simplified 33.9

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

   12. Applied frac-2neg 33.9

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

   13. Simplified 33.9

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

   14. Taylor expanded around -inf 8.0

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

   if -1.9542110853942293e+231 < y.im < 2.0302445674537431e+152

   1. Initial program 22.2

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

   3. Applied add-sqr-sqrt 22.2

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

   4. Applied *-un-lft-identity 22.2

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

   5. Applied times-frac 22.2

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

   6. Simplified 22.2

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

   7. Simplified 14.1

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

   9. Applied associate-*r/ 14.1

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

   10. Simplified 14.0

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

   12. Applied frac-2neg 14.0

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

   13. Simplified 14.0

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

   15. Applied div-sub 14.0

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

   16. Simplified 5.1

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

   if 2.0302445674537431e+152 < y.im

   1. Initial program 44.3

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

   3. Applied add-sqr-sqrt 44.3

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

   4. Applied *-un-lft-identity 44.3

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

   5. Applied times-frac 44.3

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

   6. Simplified 44.3

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

   7. Simplified 28.6

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

   9. Applied associate-*r/ 28.6

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

   10. Simplified 28.6

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

   11. Taylor expanded around 0 13.5

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

   12. Simplified 13.5

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

4. Final simplification 6.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \le -1.954211085394229 \cdot 10^{231}:\\ \;\;\;\;\frac{x.re}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.im \le 2.03024456745374313 \cdot 10^{152}:\\ \;\;\;\;\frac{\frac{y.im \cdot x.re}{-\mathsf{hypot}\left(y.re, y.im\right)} - \frac{y.re}{-\mathsf{hypot}\left(y.re, y.im\right)} \cdot x.im}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-x.re}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020043 +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))))