_divideComplex, real part

Average Error: 26.4 → 12.8

Time: 4.1s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y.re \le -2.9459980091776752 \cdot 10^{158}:\\ \;\;\;\;\frac{-1 \cdot x.re}{\mathsf{hypot}\left(y.re, y.im\right) \cdot 1}\\ \mathbf{elif}\;y.re \le -86223460658051.5:\\ \;\;\;\;\left(\frac{\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)}}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot y.re\right) \cdot x.re + \left(\frac{\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)}}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot y.im\right) \cdot x.im\\ \mathbf{elif}\;y.re \le 2.2413672752256256 \cdot 10^{125}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}{\mathsf{hypot}\left(y.re, y.im\right)}}{\mathsf{hypot}\left(y.re, y.im\right) \cdot 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{\mathsf{hypot}\left(y.re, y.im\right) \cdot 1}\\ \end{array}\]

C

double f(double x_re, double x_im, double y_re, double y_im) {
        double r69690 = x_re;
        double r69691 = y_re;
        double r69692 = r69690 * r69691;
        double r69693 = x_im;
        double r69694 = y_im;
        double r69695 = r69693 * r69694;
        double r69696 = r69692 + r69695;
        double r69697 = r69691 * r69691;
        double r69698 = r69694 * r69694;
        double r69699 = r69697 + r69698;
        double r69700 = r69696 / r69699;
        return r69700;
}

↓

double f(double x_re, double x_im, double y_re, double y_im) {
        double r69701 = y_re;
        double r69702 = -2.945998009177675e+158;
        bool r69703 = r69701 <= r69702;
        double r69704 = -1.0;
        double r69705 = x_re;
        double r69706 = r69704 * r69705;
        double r69707 = y_im;
        double r69708 = hypot(r69701, r69707);
        double r69709 = 1.0;
        double r69710 = r69708 * r69709;
        double r69711 = r69706 / r69710;
        double r69712 = -86223460658051.5;
        bool r69713 = r69701 <= r69712;
        double r69714 = r69709 / r69708;
        double r69715 = r69714 / r69708;
        double r69716 = r69715 * r69701;
        double r69717 = r69716 * r69705;
        double r69718 = r69715 * r69707;
        double r69719 = x_im;
        double r69720 = r69718 * r69719;
        double r69721 = r69717 + r69720;
        double r69722 = 2.2413672752256256e+125;
        bool r69723 = r69701 <= r69722;
        double r69724 = r69719 * r69707;
        double r69725 = fma(r69705, r69701, r69724);
        double r69726 = r69725 / r69708;
        double r69727 = r69726 / r69710;
        double r69728 = r69705 / r69710;
        double r69729 = r69723 ? r69727 : r69728;
        double r69730 = r69713 ? r69721 : r69729;
        double r69731 = r69703 ? r69711 : r69730;
        return r69731;
}

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 4 regimes

2. if y.re < -2.945998009177675e+158

   1. Initial program 45.8

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

   3. Applied add-sqr-sqrt 45.8

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

   4. Applied *-un-lft-identity 45.8

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

   5. Applied times-frac 45.8

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

   6. Simplified 45.8

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

   7. Simplified 29.5

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

   9. Applied associate-*r/ 29.5

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

   10. Simplified 29.5

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

   11. Taylor expanded around -inf 12.9

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

   if -2.945998009177675e+158 < y.re < -86223460658051.5

   1. Initial program 22.3

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

   3. Applied add-sqr-sqrt 22.3

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

   4. Applied *-un-lft-identity 22.3

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

   5. Applied times-frac 22.3

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

   6. Simplified 22.3

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

   7. Simplified 17.2

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

   9. Applied *-un-lft-identity 17.2

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

   10. Applied times-frac 17.3

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

   11. Simplified 17.3

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

   13. Applied fma-udef 17.3

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

   14. Applied distribute-lft-in 17.3

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

   15. Applied distribute-lft-in 17.3

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

   16. Simplified 12.0

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

   17. Simplified 15.5

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

   if -86223460658051.5 < y.re < 2.2413672752256256e+125

   1. Initial program 18.7

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

   3. Applied add-sqr-sqrt 18.7

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

   4. Applied *-un-lft-identity 18.7

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

   5. Applied times-frac 18.7

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

   6. Simplified 18.7

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

   7. Simplified 11.9

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

   9. Applied associate-*r/ 11.8

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

   10. Simplified 11.7

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

   if 2.2413672752256256e+125 < y.re

   1. Initial program 42.7

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

   3. Applied add-sqr-sqrt 42.7

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

   4. Applied *-un-lft-identity 42.7

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

   5. Applied times-frac 42.7

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

   6. Simplified 42.7

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

   7. Simplified 28.1

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

   9. Applied associate-*r/ 28.1

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

   10. Simplified 28.1

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

   11. Taylor expanded around inf 14.9

       \[\leadsto \frac{\color{blue}{x.re}}{\mathsf{hypot}\left(y.re, y.im\right) \cdot 1}\]
3. Recombined 4 regimes into one program.

4. Final simplification 12.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \le -2.9459980091776752 \cdot 10^{158}:\\ \;\;\;\;\frac{-1 \cdot x.re}{\mathsf{hypot}\left(y.re, y.im\right) \cdot 1}\\ \mathbf{elif}\;y.re \le -86223460658051.5:\\ \;\;\;\;\left(\frac{\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)}}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot y.re\right) \cdot x.re + \left(\frac{\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)}}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot y.im\right) \cdot x.im\\ \mathbf{elif}\;y.re \le 2.2413672752256256 \cdot 10^{125}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}{\mathsf{hypot}\left(y.re, y.im\right)}}{\mathsf{hypot}\left(y.re, y.im\right) \cdot 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{\mathsf{hypot}\left(y.re, y.im\right) \cdot 1}\\ \end{array}\]

Reproduce

herbie shell --seed 2020065 +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))))