_divideComplex, real part

Average Error: 26.8 → 13.4

Time: 19.1s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y.re \le -2.045846669237430472528864761609559364154 \cdot 10^{123}:\\ \;\;\;\;\frac{-x.re}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.re \le 8.747123436813267982750810114699097243072 \cdot 10^{191}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(y.re, x.re, x.im \cdot y.im\right)}{\mathsf{hypot}\left(y.re, y.im\right)}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \end{array}\]

C

double f(double x_re, double x_im, double y_re, double y_im) {
        double r111614 = x_re;
        double r111615 = y_re;
        double r111616 = r111614 * r111615;
        double r111617 = x_im;
        double r111618 = y_im;
        double r111619 = r111617 * r111618;
        double r111620 = r111616 + r111619;
        double r111621 = r111615 * r111615;
        double r111622 = r111618 * r111618;
        double r111623 = r111621 + r111622;
        double r111624 = r111620 / r111623;
        return r111624;
}

↓

double f(double x_re, double x_im, double y_re, double y_im) {
        double r111625 = y_re;
        double r111626 = -2.0458466692374305e+123;
        bool r111627 = r111625 <= r111626;
        double r111628 = x_re;
        double r111629 = -r111628;
        double r111630 = y_im;
        double r111631 = hypot(r111625, r111630);
        double r111632 = r111629 / r111631;
        double r111633 = 8.747123436813268e+191;
        bool r111634 = r111625 <= r111633;
        double r111635 = x_im;
        double r111636 = r111635 * r111630;
        double r111637 = fma(r111625, r111628, r111636);
        double r111638 = r111637 / r111631;
        double r111639 = r111638 / r111631;
        double r111640 = r111628 / r111631;
        double r111641 = r111634 ? r111639 : r111640;
        double r111642 = r111627 ? r111632 : r111641;
        return r111642;
}

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 < -2.0458466692374305e+123

   1. Initial program 41.2

      \[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\]
   2. Using strategy rm

   3. Applied add-sqr-sqrt 41.2

      \[\leadsto \frac{x.re \cdot y.re + x.im \cdot y.im}{\color{blue}{\sqrt{y.re \cdot y.re + y.im \cdot y.im} \cdot \sqrt{y.re \cdot y.re + y.im \cdot y.im}}}\]

   4. Applied *-un-lft-identity 41.2

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

   5. Applied times-frac 41.2

      \[\leadsto \color{blue}{\frac{1}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}} \cdot \frac{x.re \cdot y.re + x.im \cdot y.im}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}}\]

   6. Simplified 41.2

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

   7. Simplified 27.8

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

   9. Applied pow1 27.8

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

   10. Applied pow1 27.8

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

   11. Applied pow-prod-down 27.8

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

   12. Simplified 27.7

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

   13. Taylor expanded around -inf 14.5

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

   14. Simplified 14.5

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

   if -2.0458466692374305e+123 < y.re < 8.747123436813268e+191

   1. Initial program 21.4

      \[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\]
   2. Using strategy rm

   3. Applied add-sqr-sqrt 21.4

      \[\leadsto \frac{x.re \cdot y.re + x.im \cdot y.im}{\color{blue}{\sqrt{y.re \cdot y.re + y.im \cdot y.im} \cdot \sqrt{y.re \cdot y.re + y.im \cdot y.im}}}\]

   4. Applied *-un-lft-identity 21.4

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

   5. Applied times-frac 21.5

      \[\leadsto \color{blue}{\frac{1}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}} \cdot \frac{x.re \cdot y.re + x.im \cdot y.im}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}}\]

   6. Simplified 21.4

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

   7. Simplified 13.4

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

   9. Applied pow1 13.4

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

   10. Applied pow1 13.4

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

   11. Applied pow-prod-down 13.4

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

   12. Simplified 13.3

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

   if 8.747123436813268e+191 < y.re

   1. Initial program 43.0

      \[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\]
   2. Using strategy rm

   3. Applied add-sqr-sqrt 43.0

      \[\leadsto \frac{x.re \cdot y.re + x.im \cdot y.im}{\color{blue}{\sqrt{y.re \cdot y.re + y.im \cdot y.im} \cdot \sqrt{y.re \cdot y.re + y.im \cdot y.im}}}\]

   4. Applied *-un-lft-identity 43.0

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

   5. Applied times-frac 43.0

      \[\leadsto \color{blue}{\frac{1}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}} \cdot \frac{x.re \cdot y.re + x.im \cdot y.im}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}}\]

   6. Simplified 43.0

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

   7. Simplified 30.3

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

   9. Applied pow1 30.3

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

   10. Applied pow1 30.3

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

   11. Applied pow-prod-down 30.3

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

   12. Simplified 30.3

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

   13. Taylor expanded around inf 12.5

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

4. Final simplification 13.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \le -2.045846669237430472528864761609559364154 \cdot 10^{123}:\\ \;\;\;\;\frac{-x.re}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.re \le 8.747123436813267982750810114699097243072 \cdot 10^{191}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(y.re, x.re, x.im \cdot y.im\right)}{\mathsf{hypot}\left(y.re, y.im\right)}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019174 +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))))