_divideComplex, imaginary part

Average Error: 25.7 → 12.8

Time: 13.7s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y.re \le -1.565167565781187104932955349587597625361 \cdot 10^{162}:\\ \;\;\;\;\frac{-x.im}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.re \le 6.152282888480768325547605387077337858415 \cdot 10^{76}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(y.re, x.im, -y.im \cdot x.re\right)}{\mathsf{hypot}\left(y.re, y.im\right)}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \end{array}\]

C

double f(double x_re, double x_im, double y_re, double y_im) {
        double r59677 = x_im;
        double r59678 = y_re;
        double r59679 = r59677 * r59678;
        double r59680 = x_re;
        double r59681 = y_im;
        double r59682 = r59680 * r59681;
        double r59683 = r59679 - r59682;
        double r59684 = r59678 * r59678;
        double r59685 = r59681 * r59681;
        double r59686 = r59684 + r59685;
        double r59687 = r59683 / r59686;
        return r59687;
}

↓

double f(double x_re, double x_im, double y_re, double y_im) {
        double r59688 = y_re;
        double r59689 = -1.5651675657811871e+162;
        bool r59690 = r59688 <= r59689;
        double r59691 = x_im;
        double r59692 = -r59691;
        double r59693 = y_im;
        double r59694 = hypot(r59688, r59693);
        double r59695 = r59692 / r59694;
        double r59696 = 6.152282888480768e+76;
        bool r59697 = r59688 <= r59696;
        double r59698 = x_re;
        double r59699 = r59693 * r59698;
        double r59700 = -r59699;
        double r59701 = fma(r59688, r59691, r59700);
        double r59702 = r59701 / r59694;
        double r59703 = r59702 / r59694;
        double r59704 = r59691 / r59694;
        double r59705 = r59697 ? r59703 : r59704;
        double r59706 = r59690 ? r59695 : r59705;
        return r59706;
}

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 < -1.5651675657811871e+162

   1. Initial program 45.9

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

   2. Simplified 45.9

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

   4. Applied add-sqr-sqrt 45.9

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

   5. Applied *-un-lft-identity 45.9

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

   6. Applied times-frac 45.9

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

   7. Simplified 45.9

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

   8. Simplified 30.7

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

   10. Applied associate-*r/ 30.7

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

   11. Simplified 30.7

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

   12. Taylor expanded around -inf 13.3

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

   13. Simplified 13.3

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

   if -1.5651675657811871e+162 < y.re < 6.152282888480768e+76

   1. Initial program 18.7

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

   2. Simplified 18.7

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

   4. Applied add-sqr-sqrt 18.7

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

   5. Applied *-un-lft-identity 18.7

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

   6. Applied times-frac 18.7

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

   7. Simplified 18.7

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

   8. Simplified 11.4

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

   10. Applied associate-*r/ 11.4

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

   11. Simplified 11.3

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

   if 6.152282888480768e+76 < y.re

   1. Initial program 37.3

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

   2. Simplified 37.3

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

   4. Applied add-sqr-sqrt 37.3

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

   5. Applied *-un-lft-identity 37.3

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

   6. Applied times-frac 37.3

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

   7. Simplified 37.3

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

   8. Simplified 26.1

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

   10. Applied associate-*r/ 26.1

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

   11. Simplified 26.1

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

   12. Taylor expanded around inf 17.8

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

4. Final simplification 12.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \le -1.565167565781187104932955349587597625361 \cdot 10^{162}:\\ \;\;\;\;\frac{-x.im}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.re \le 6.152282888480768325547605387077337858415 \cdot 10^{76}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(y.re, x.im, -y.im \cdot x.re\right)}{\mathsf{hypot}\left(y.re, y.im\right)}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019209 +o rules:numerics
(FPCore (x.re x.im y.re y.im)
  :name "_divideComplex, imaginary part"
  :precision binary64
  (/ (- (* x.im y.re) (* x.re y.im)) (+ (* y.re y.re) (* y.im y.im))))