_divideComplex, real part

Average Error: 25.7 → 13.0

Time: 18.9s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y.im \le -7.67400831190257 \cdot 10^{+96}:\\ \;\;\;\;\frac{-x.im}{\mathsf{hypot}\left(y.im, y.re\right)}\\ \mathbf{elif}\;y.im \le 4.711008584896772 \cdot 10^{+178}:\\ \;\;\;\;\frac{\frac{1}{\frac{\mathsf{hypot}\left(y.im, y.re\right)}{\mathsf{fma}\left(x.im, y.im, x.re \cdot y.re\right)}}}{\mathsf{hypot}\left(y.im, y.re\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{\mathsf{hypot}\left(y.im, y.re\right)}\\ \end{array}\]

C

double f(double x_re, double x_im, double y_re, double y_im) {
        double r1038472 = x_re;
        double r1038473 = y_re;
        double r1038474 = r1038472 * r1038473;
        double r1038475 = x_im;
        double r1038476 = y_im;
        double r1038477 = r1038475 * r1038476;
        double r1038478 = r1038474 + r1038477;
        double r1038479 = r1038473 * r1038473;
        double r1038480 = r1038476 * r1038476;
        double r1038481 = r1038479 + r1038480;
        double r1038482 = r1038478 / r1038481;
        return r1038482;
}

↓

double f(double x_re, double x_im, double y_re, double y_im) {
        double r1038483 = y_im;
        double r1038484 = -7.67400831190257e+96;
        bool r1038485 = r1038483 <= r1038484;
        double r1038486 = x_im;
        double r1038487 = -r1038486;
        double r1038488 = y_re;
        double r1038489 = hypot(r1038483, r1038488);
        double r1038490 = r1038487 / r1038489;
        double r1038491 = 4.711008584896772e+178;
        bool r1038492 = r1038483 <= r1038491;
        double r1038493 = 1.0;
        double r1038494 = x_re;
        double r1038495 = r1038494 * r1038488;
        double r1038496 = fma(r1038486, r1038483, r1038495);
        double r1038497 = r1038489 / r1038496;
        double r1038498 = r1038493 / r1038497;
        double r1038499 = r1038498 / r1038489;
        double r1038500 = r1038486 / r1038489;
        double r1038501 = r1038492 ? r1038499 : r1038500;
        double r1038502 = r1038485 ? r1038490 : r1038501;
        return r1038502;
}

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 < -7.67400831190257e+96

   1. Initial program 38.9

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

   2. Simplified 38.9

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

   4. Applied add-sqr-sqrt 38.9

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

   5. Applied associate-/r* 38.8

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

   7. Applied *-un-lft-identity 38.8

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

   8. Applied sqrt-prod 38.8

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

   9. Applied *-un-lft-identity 38.8

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

   10. Applied *-un-lft-identity 38.8

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

   11. Applied times-frac 38.8

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

   12. Applied times-frac 38.8

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

   13. Simplified 38.8

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

   14. Simplified 25.9

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

   15. Taylor expanded around -inf 17.2

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

   16. Simplified 17.2

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

   if -7.67400831190257e+96 < y.im < 4.711008584896772e+178

   1. Initial program 19.6

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

   2. Simplified 19.6

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

   4. Applied add-sqr-sqrt 19.6

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

   5. Applied associate-/r* 19.5

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

   7. Applied *-un-lft-identity 19.5

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

   8. Applied sqrt-prod 19.5

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

   9. Applied *-un-lft-identity 19.5

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

   10. Applied *-un-lft-identity 19.5

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

   11. Applied times-frac 19.5

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

   12. Applied times-frac 19.5

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

   13. Simplified 19.5

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

   14. Simplified 12.1

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

   16. Applied *-un-lft-identity 12.1

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

   17. Applied associate-/l* 12.1

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

   if 4.711008584896772e+178 < y.im

   1. Initial program 44.1

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

   2. Simplified 44.1

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

   4. Applied add-sqr-sqrt 44.1

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

   5. Applied associate-/r* 44.1

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

   7. Applied *-un-lft-identity 44.1

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

   8. Applied sqrt-prod 44.1

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

   9. Applied *-un-lft-identity 44.1

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

   10. Applied *-un-lft-identity 44.1

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

   11. Applied times-frac 44.1

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

   12. Applied times-frac 44.1

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

   13. Simplified 44.1

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

   14. Simplified 30.5

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

   16. Applied *-un-lft-identity 30.5

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

   17. Applied associate-/l* 30.6

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

   18. Taylor expanded around inf 11.9

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

4. Final simplification 13.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \le -7.67400831190257 \cdot 10^{+96}:\\ \;\;\;\;\frac{-x.im}{\mathsf{hypot}\left(y.im, y.re\right)}\\ \mathbf{elif}\;y.im \le 4.711008584896772 \cdot 10^{+178}:\\ \;\;\;\;\frac{\frac{1}{\frac{\mathsf{hypot}\left(y.im, y.re\right)}{\mathsf{fma}\left(x.im, y.im, x.re \cdot y.re\right)}}}{\mathsf{hypot}\left(y.im, y.re\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{\mathsf{hypot}\left(y.im, y.re\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019152 +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))))