_divideComplex, real part

Average Error: 25.6 → 12.6

Time: 16.7s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y.re \le -1.264168133681399 \cdot 10^{+152}:\\ \;\;\;\;\frac{-x.re}{\mathsf{hypot}\left(y.im, y.re\right)}\\ \mathbf{elif}\;y.re \le 1.4672736653528514 \cdot 10^{+187}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(x.im, y.im, x.re \cdot y.re\right)}{\mathsf{hypot}\left(y.im, y.re\right)}}{\mathsf{hypot}\left(y.im, y.re\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{\mathsf{hypot}\left(y.im, y.re\right)}\\ \end{array}\]

C

double f(double x_re, double x_im, double y_re, double y_im) {
        double r1810176 = x_re;
        double r1810177 = y_re;
        double r1810178 = r1810176 * r1810177;
        double r1810179 = x_im;
        double r1810180 = y_im;
        double r1810181 = r1810179 * r1810180;
        double r1810182 = r1810178 + r1810181;
        double r1810183 = r1810177 * r1810177;
        double r1810184 = r1810180 * r1810180;
        double r1810185 = r1810183 + r1810184;
        double r1810186 = r1810182 / r1810185;
        return r1810186;
}

↓

double f(double x_re, double x_im, double y_re, double y_im) {
        double r1810187 = y_re;
        double r1810188 = -1.264168133681399e+152;
        bool r1810189 = r1810187 <= r1810188;
        double r1810190 = x_re;
        double r1810191 = -r1810190;
        double r1810192 = y_im;
        double r1810193 = hypot(r1810192, r1810187);
        double r1810194 = r1810191 / r1810193;
        double r1810195 = 1.4672736653528514e+187;
        bool r1810196 = r1810187 <= r1810195;
        double r1810197 = x_im;
        double r1810198 = r1810190 * r1810187;
        double r1810199 = fma(r1810197, r1810192, r1810198);
        double r1810200 = r1810199 / r1810193;
        double r1810201 = r1810200 / r1810193;
        double r1810202 = r1810190 / r1810193;
        double r1810203 = r1810196 ? r1810201 : r1810202;
        double r1810204 = r1810189 ? r1810194 : r1810203;
        return r1810204;
}

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 < -1.264168133681399e+152

   1. Initial program 43.7

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

   2. Simplified 43.7

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

   4. Applied add-sqr-sqrt 43.7

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

   5. Applied associate-/r* 43.7

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

   7. Applied fma-udef 43.7

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

   8. Applied hypot-def 43.7

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

   10. Applied fma-udef 43.7

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

   11. Applied hypot-def 28.2

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

   12. Taylor expanded around -inf 13.6

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

   13. Simplified 13.6

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

   if -1.264168133681399e+152 < y.re < 1.4672736653528514e+187

   1. Initial program 19.9

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

   2. Simplified 19.9

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

   4. Applied add-sqr-sqrt 19.9

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

   5. Applied associate-/r* 19.8

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

   7. Applied fma-udef 19.8

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

   8. Applied hypot-def 19.8

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

   10. Applied fma-udef 19.8

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

   11. Applied hypot-def 12.6

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

   13. Applied *-un-lft-identity 12.6

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

   14. Applied associate-/r* 12.6

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

   15. Simplified 12.6

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

   if 1.4672736653528514e+187 < y.re

   1. Initial program 42.9

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

   2. Simplified 42.9

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

   4. Applied add-sqr-sqrt 42.9

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

   5. Applied associate-/r* 42.9

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

   7. Applied fma-udef 42.9

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

   8. Applied hypot-def 42.9

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

   9. Taylor expanded around inf 11.6

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

4. Final simplification 12.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \le -1.264168133681399 \cdot 10^{+152}:\\ \;\;\;\;\frac{-x.re}{\mathsf{hypot}\left(y.im, y.re\right)}\\ \mathbf{elif}\;y.re \le 1.4672736653528514 \cdot 10^{+187}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(x.im, y.im, x.re \cdot y.re\right)}{\mathsf{hypot}\left(y.im, y.re\right)}}{\mathsf{hypot}\left(y.im, y.re\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{\mathsf{hypot}\left(y.im, y.re\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019135 +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))))