_divideComplex, real part

Average Error: 25.8 → 14.6

Time: 16.6s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y.im \le 9.936266958075772 \cdot 10^{+94}:\\ \;\;\;\;\frac{\frac{1}{\frac{\mathsf{hypot}\left(y.im, y.re\right)}{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}}}{\mathsf{hypot}\left(y.im, y.re\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{\mathsf{hypot}\left(y.im, y.re\right)}\\ \end{array}\]

C

double f(double x_re, double x_im, double y_re, double y_im) {
        double r2530744 = x_re;
        double r2530745 = y_re;
        double r2530746 = r2530744 * r2530745;
        double r2530747 = x_im;
        double r2530748 = y_im;
        double r2530749 = r2530747 * r2530748;
        double r2530750 = r2530746 + r2530749;
        double r2530751 = r2530745 * r2530745;
        double r2530752 = r2530748 * r2530748;
        double r2530753 = r2530751 + r2530752;
        double r2530754 = r2530750 / r2530753;
        return r2530754;
}

↓

double f(double x_re, double x_im, double y_re, double y_im) {
        double r2530755 = y_im;
        double r2530756 = 9.936266958075772e+94;
        bool r2530757 = r2530755 <= r2530756;
        double r2530758 = 1.0;
        double r2530759 = y_re;
        double r2530760 = hypot(r2530755, r2530759);
        double r2530761 = x_re;
        double r2530762 = x_im;
        double r2530763 = r2530762 * r2530755;
        double r2530764 = fma(r2530761, r2530759, r2530763);
        double r2530765 = r2530760 / r2530764;
        double r2530766 = r2530758 / r2530765;
        double r2530767 = r2530766 / r2530760;
        double r2530768 = r2530762 / r2530760;
        double r2530769 = r2530757 ? r2530767 : r2530768;
        return r2530769;
}

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 2 regimes

2. if y.im < 9.936266958075772e+94

   1. Initial program 22.9

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

   2. Simplified 22.9

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

   4. Applied add-sqr-sqrt 22.9

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

   5. Applied associate-/r* 22.8

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

   7. Applied fma-udef 22.8

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

   8. Applied hypot-def 22.8

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

   10. Applied fma-udef 22.8

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

   11. Applied hypot-def 14.0

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

   13. Applied *-un-lft-identity 14.0

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

   14. Applied associate-/l* 14.1

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

   if 9.936266958075772e+94 < y.im

   1. Initial program 38.8

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

   2. Simplified 38.8

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

   4. Applied add-sqr-sqrt 38.8

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

   5. Applied associate-/r* 38.7

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

   7. Applied fma-udef 38.7

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

   8. Applied hypot-def 38.7

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

   10. Applied fma-udef 38.7

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

   11. Applied hypot-def 25.9

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

   12. Taylor expanded around 0 16.7

       \[\leadsto \frac{\color{blue}{x.im}}{\mathsf{hypot}\left(y.im, y.re\right)}\]
3. Recombined 2 regimes into one program.

4. Final simplification 14.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \le 9.936266958075772 \cdot 10^{+94}:\\ \;\;\;\;\frac{\frac{1}{\frac{\mathsf{hypot}\left(y.im, y.re\right)}{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}}}{\mathsf{hypot}\left(y.im, y.re\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{\mathsf{hypot}\left(y.im, y.re\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019138 +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))))