_divideComplex, real part

Average Error: 25.8 → 13.4

Time: 19.6s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y.im \le -2.686314195490964030784246986594388998529 \cdot 10^{247}:\\ \;\;\;\;x.im \cdot \frac{-1}{\mathsf{hypot}\left(y.im, y.re\right)}\\ \mathbf{elif}\;y.im \le 2.619423696010160209014178018647314634761 \cdot 10^{118}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}{\mathsf{hypot}\left(y.im, y.re\right)}}{\mathsf{hypot}\left(y.im, y.re\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{\mathsf{hypot}\left(y.im, y.re\right)}\\ \end{array}\]

C

double f(double x_re, double x_im, double y_re, double y_im) {
        double r60242 = x_re;
        double r60243 = y_re;
        double r60244 = r60242 * r60243;
        double r60245 = x_im;
        double r60246 = y_im;
        double r60247 = r60245 * r60246;
        double r60248 = r60244 + r60247;
        double r60249 = r60243 * r60243;
        double r60250 = r60246 * r60246;
        double r60251 = r60249 + r60250;
        double r60252 = r60248 / r60251;
        return r60252;
}

↓

double f(double x_re, double x_im, double y_re, double y_im) {
        double r60253 = y_im;
        double r60254 = -2.686314195490964e+247;
        bool r60255 = r60253 <= r60254;
        double r60256 = x_im;
        double r60257 = -1.0;
        double r60258 = y_re;
        double r60259 = hypot(r60253, r60258);
        double r60260 = r60257 / r60259;
        double r60261 = r60256 * r60260;
        double r60262 = 2.61942369601016e+118;
        bool r60263 = r60253 <= r60262;
        double r60264 = x_re;
        double r60265 = r60256 * r60253;
        double r60266 = fma(r60264, r60258, r60265);
        double r60267 = r60266 / r60259;
        double r60268 = r60267 / r60259;
        double r60269 = r60256 / r60259;
        double r60270 = r60263 ? r60268 : r60269;
        double r60271 = r60255 ? r60261 : r60270;
        return r60271;
}

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 < -2.686314195490964e+247

   1. Initial program 39.4

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

   2. Simplified 39.4

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

   4. Applied add-sqr-sqrt 39.4

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

   5. Applied *-un-lft-identity 39.4

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

   6. Applied times-frac 39.4

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

   7. Simplified 39.4

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

   8. Simplified 32.4

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

   10. Applied clear-num 32.4

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

   11. Taylor expanded around -inf 7.5

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

   12. Simplified 7.5

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

   if -2.686314195490964e+247 < y.im < 2.61942369601016e+118

   1. Initial program 21.8

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

   2. Simplified 21.8

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

   4. Applied add-sqr-sqrt 21.8

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

   5. Applied *-un-lft-identity 21.8

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

   6. Applied times-frac 21.9

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

   7. Simplified 21.8

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

   8. Simplified 13.5

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

   10. Applied *-un-lft-identity 13.5

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

   11. Applied *-un-lft-identity 13.5

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

   12. Applied times-frac 13.5

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

   13. Applied associate-*l* 13.5

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

   14. Simplified 13.4

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

   if 2.61942369601016e+118 < y.im

   1. Initial program 40.6

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

   2. Simplified 40.6

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

   4. Applied add-sqr-sqrt 40.6

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

   5. Applied *-un-lft-identity 40.6

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

   6. Applied times-frac 40.6

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

   7. Simplified 40.6

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

   8. Simplified 27.3

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

   10. Applied *-un-lft-identity 27.3

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

   11. Applied *-un-lft-identity 27.3

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

   12. Applied times-frac 27.3

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

   13. Applied associate-*l* 27.3

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

   14. Simplified 27.2

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

   15. Taylor expanded around 0 15.1

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

4. Final simplification 13.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \le -2.686314195490964030784246986594388998529 \cdot 10^{247}:\\ \;\;\;\;x.im \cdot \frac{-1}{\mathsf{hypot}\left(y.im, y.re\right)}\\ \mathbf{elif}\;y.im \le 2.619423696010160209014178018647314634761 \cdot 10^{118}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}{\mathsf{hypot}\left(y.im, y.re\right)}}{\mathsf{hypot}\left(y.im, y.re\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{\mathsf{hypot}\left(y.im, y.re\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019323 +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))))