_divideComplex, real part

Average Error: 26.1 → 13.0

Time: 16.4s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y.re \le -2.348088248667590665897704881735349807841 \cdot 10^{140}:\\ \;\;\;\;\frac{-x.re}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.re \le 1.777131359138117252196667227028923805467 \cdot 10^{97}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(y.im, x.im, y.re \cdot x.re\right)}{\mathsf{hypot}\left(y.re, y.im\right)}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \end{array}\]

C

double f(double x_re, double x_im, double y_re, double y_im) {
        double r46568 = x_re;
        double r46569 = y_re;
        double r46570 = r46568 * r46569;
        double r46571 = x_im;
        double r46572 = y_im;
        double r46573 = r46571 * r46572;
        double r46574 = r46570 + r46573;
        double r46575 = r46569 * r46569;
        double r46576 = r46572 * r46572;
        double r46577 = r46575 + r46576;
        double r46578 = r46574 / r46577;
        return r46578;
}

↓

double f(double x_re, double x_im, double y_re, double y_im) {
        double r46579 = y_re;
        double r46580 = -2.3480882486675907e+140;
        bool r46581 = r46579 <= r46580;
        double r46582 = x_re;
        double r46583 = -r46582;
        double r46584 = y_im;
        double r46585 = hypot(r46579, r46584);
        double r46586 = r46583 / r46585;
        double r46587 = 1.7771313591381173e+97;
        bool r46588 = r46579 <= r46587;
        double r46589 = x_im;
        double r46590 = r46579 * r46582;
        double r46591 = fma(r46584, r46589, r46590);
        double r46592 = r46591 / r46585;
        double r46593 = r46592 / r46585;
        double r46594 = r46582 / r46585;
        double r46595 = r46588 ? r46593 : r46594;
        double r46596 = r46581 ? r46586 : r46595;
        return r46596;
}

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 < -2.3480882486675907e+140

   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

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

   4. Applied add-sqr-sqrt 42.9

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

   5. Applied *-un-lft-identity 42.9

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

   6. Applied times-frac 42.9

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

   7. Simplified 42.9

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

   8. Simplified 27.2

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

   10. Applied associate-*r/ 27.2

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

   11. Simplified 27.1

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

   12. Taylor expanded around -inf 13.4

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

   13. Simplified 13.4

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

   if -2.3480882486675907e+140 < y.re < 1.7771313591381173e+97

   1. Initial program 19.0

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

   2. Simplified 19.0

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

   4. Applied add-sqr-sqrt 19.0

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

   5. Applied *-un-lft-identity 19.0

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

   6. Applied times-frac 19.0

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

   7. Simplified 18.9

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

   8. Simplified 12.2

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

   10. Applied associate-*r/ 12.1

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

   11. Simplified 12.0

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

   13. Applied *-un-lft-identity 12.0

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

   14. Applied associate-/r* 12.0

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

   15. Simplified 12.0

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

   if 1.7771313591381173e+97 < y.re

   1. Initial program 39.7

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

   2. Simplified 39.7

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

   4. Applied add-sqr-sqrt 39.7

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

   5. Applied *-un-lft-identity 39.7

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

   6. Applied times-frac 39.7

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

   7. Simplified 39.7

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

   8. Simplified 26.4

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

   10. Applied associate-*r/ 26.4

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

   11. Simplified 26.3

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

   12. Taylor expanded around inf 16.2

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

4. Final simplification 13.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \le -2.348088248667590665897704881735349807841 \cdot 10^{140}:\\ \;\;\;\;\frac{-x.re}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.re \le 1.777131359138117252196667227028923805467 \cdot 10^{97}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(y.im, x.im, y.re \cdot x.re\right)}{\mathsf{hypot}\left(y.re, y.im\right)}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019304 +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))))