_divideComplex, imaginary part

Average Error: 26.2 → 12.9

Time: 19.2s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y.re \le -8.623094169584484727986124738487844232194 \cdot 10^{150}:\\ \;\;\;\;\frac{-x.im}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.re \le 9.991764747534992157625162969102338998239 \cdot 10^{131}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(x.im, y.re, -y.im \cdot x.re\right)}{\mathsf{hypot}\left(y.re, y.im\right)}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \end{array}\]

C

double f(double x_re, double x_im, double y_re, double y_im) {
        double r70592 = x_im;
        double r70593 = y_re;
        double r70594 = r70592 * r70593;
        double r70595 = x_re;
        double r70596 = y_im;
        double r70597 = r70595 * r70596;
        double r70598 = r70594 - r70597;
        double r70599 = r70593 * r70593;
        double r70600 = r70596 * r70596;
        double r70601 = r70599 + r70600;
        double r70602 = r70598 / r70601;
        return r70602;
}

↓

double f(double x_re, double x_im, double y_re, double y_im) {
        double r70603 = y_re;
        double r70604 = -8.623094169584485e+150;
        bool r70605 = r70603 <= r70604;
        double r70606 = x_im;
        double r70607 = -r70606;
        double r70608 = y_im;
        double r70609 = hypot(r70603, r70608);
        double r70610 = r70607 / r70609;
        double r70611 = 9.991764747534992e+131;
        bool r70612 = r70603 <= r70611;
        double r70613 = x_re;
        double r70614 = r70608 * r70613;
        double r70615 = -r70614;
        double r70616 = fma(r70606, r70603, r70615);
        double r70617 = r70616 / r70609;
        double r70618 = r70617 / r70609;
        double r70619 = r70606 / r70609;
        double r70620 = r70612 ? r70618 : r70619;
        double r70621 = r70605 ? r70610 : r70620;
        return r70621;
}

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 < -8.623094169584485e+150

   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 29.0

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

   9. Applied *-un-lft-identity 29.0

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

   10. Applied add-sqr-sqrt 29.0

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

   11. Applied times-frac 29.0

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

   12. Applied associate-*l* 29.0

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

   13. Simplified 28.9

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

   14. Taylor expanded around -inf 12.9

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

   15. Simplified 12.9

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

   if -8.623094169584485e+150 < y.re < 9.991764747534992e+131

   1. Initial program 19.5

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

   3. Applied add-sqr-sqrt 19.5

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

   4. Applied *-un-lft-identity 19.5

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

   5. Applied times-frac 19.5

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

   6. Simplified 19.5

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

   7. Simplified 12.6

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

   9. Applied *-un-lft-identity 12.6

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

   10. Applied add-sqr-sqrt 12.6

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

   11. Applied times-frac 12.6

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

   12. Applied associate-*l* 12.6

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

   13. Simplified 12.4

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

   if 9.991764747534992e+131 < y.re

   1. Initial program 42.1

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

   3. Applied add-sqr-sqrt 42.1

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

   4. Applied *-un-lft-identity 42.1

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

   5. Applied times-frac 42.1

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

   6. Simplified 42.1

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

   7. Simplified 27.4

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

   9. Applied *-un-lft-identity 27.4

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

   10. Applied add-sqr-sqrt 27.4

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

   11. Applied times-frac 27.4

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

   12. Applied associate-*l* 27.4

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

   13. Simplified 27.4

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

   14. Taylor expanded around inf 15.3

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

4. Final simplification 12.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \le -8.623094169584484727986124738487844232194 \cdot 10^{150}:\\ \;\;\;\;\frac{-x.im}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.re \le 9.991764747534992157625162969102338998239 \cdot 10^{131}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(x.im, y.re, -y.im \cdot x.re\right)}{\mathsf{hypot}\left(y.re, y.im\right)}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019325 +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))))