Maksimov and Kolovsky, Equation (3)

Average Error: 18.0 → 8.1

Time: 8.9s

Precision: 64

Math

\[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}}\]

↓

\[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \mathsf{hypot}\left(\sqrt{1}, {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{\left(\frac{2}{2}\right)}\right)\]

C

double f(double J, double K, double U) {
        double r192700 = -2.0;
        double r192701 = J;
        double r192702 = r192700 * r192701;
        double r192703 = K;
        double r192704 = 2.0;
        double r192705 = r192703 / r192704;
        double r192706 = cos(r192705);
        double r192707 = r192702 * r192706;
        double r192708 = 1.0;
        double r192709 = U;
        double r192710 = r192704 * r192701;
        double r192711 = r192710 * r192706;
        double r192712 = r192709 / r192711;
        double r192713 = pow(r192712, r192704);
        double r192714 = r192708 + r192713;
        double r192715 = sqrt(r192714);
        double r192716 = r192707 * r192715;
        return r192716;
}

↓

double f(double J, double K, double U) {
        double r192717 = -2.0;
        double r192718 = J;
        double r192719 = r192717 * r192718;
        double r192720 = K;
        double r192721 = 2.0;
        double r192722 = r192720 / r192721;
        double r192723 = cos(r192722);
        double r192724 = r192719 * r192723;
        double r192725 = 1.0;
        double r192726 = sqrt(r192725);
        double r192727 = U;
        double r192728 = r192721 * r192718;
        double r192729 = r192728 * r192723;
        double r192730 = r192727 / r192729;
        double r192731 = 2.0;
        double r192732 = r192721 / r192731;
        double r192733 = pow(r192730, r192732);
        double r192734 = hypot(r192726, r192733);
        double r192735 = r192724 * r192734;
        return r192735;
}

Error

Bits error versus J

Bits error versus K

Bits error versus U

Derivation

1. Initial program 18.0

   \[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}}\]
2. Using strategy rm

3. Applied sqr-pow 18.0

   \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + \color{blue}{{\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{\left(\frac{2}{2}\right)} \cdot {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{\left(\frac{2}{2}\right)}}}\]

4. Applied add-sqr-sqrt 18.0

   \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{\color{blue}{\sqrt{1} \cdot \sqrt{1}} + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{\left(\frac{2}{2}\right)} \cdot {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{\left(\frac{2}{2}\right)}}\]

5. Applied hypot-def 8.1

   \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \color{blue}{\mathsf{hypot}\left(\sqrt{1}, {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{\left(\frac{2}{2}\right)}\right)}\]
6. Using strategy rm

7. Applied add-sqr-sqrt 8.3

   \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \color{blue}{\left(\sqrt{\mathsf{hypot}\left(\sqrt{1}, {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{\left(\frac{2}{2}\right)}\right)} \cdot \sqrt{\mathsf{hypot}\left(\sqrt{1}, {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{\left(\frac{2}{2}\right)}\right)}\right)}\]

8. Applied associate-*r* 8.3

   \[\leadsto \color{blue}{\left(\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{\mathsf{hypot}\left(\sqrt{1}, {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{\left(\frac{2}{2}\right)}\right)}\right) \cdot \sqrt{\mathsf{hypot}\left(\sqrt{1}, {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{\left(\frac{2}{2}\right)}\right)}}\]
9. Using strategy rm

10. Applied pow1 8.3

    \[\leadsto \left(\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{\mathsf{hypot}\left(\sqrt{1}, {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{\left(\frac{2}{2}\right)}\right)}\right) \cdot \color{blue}{{\left(\sqrt{\mathsf{hypot}\left(\sqrt{1}, {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{\left(\frac{2}{2}\right)}\right)}\right)}^{1}}\]

11. Applied pow1 8.3

    \[\leadsto \left(\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \color{blue}{{\left(\sqrt{\mathsf{hypot}\left(\sqrt{1}, {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{\left(\frac{2}{2}\right)}\right)}\right)}^{1}}\right) \cdot {\left(\sqrt{\mathsf{hypot}\left(\sqrt{1}, {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{\left(\frac{2}{2}\right)}\right)}\right)}^{1}\]

12. Applied pow1 8.3

    \[\leadsto \left(\left(\left(-2 \cdot J\right) \cdot \color{blue}{{\left(\cos \left(\frac{K}{2}\right)\right)}^{1}}\right) \cdot {\left(\sqrt{\mathsf{hypot}\left(\sqrt{1}, {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{\left(\frac{2}{2}\right)}\right)}\right)}^{1}\right) \cdot {\left(\sqrt{\mathsf{hypot}\left(\sqrt{1}, {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{\left(\frac{2}{2}\right)}\right)}\right)}^{1}\]

13. Applied pow1 8.3

    \[\leadsto \left(\left(\left(-2 \cdot \color{blue}{{J}^{1}}\right) \cdot {\left(\cos \left(\frac{K}{2}\right)\right)}^{1}\right) \cdot {\left(\sqrt{\mathsf{hypot}\left(\sqrt{1}, {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{\left(\frac{2}{2}\right)}\right)}\right)}^{1}\right) \cdot {\left(\sqrt{\mathsf{hypot}\left(\sqrt{1}, {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{\left(\frac{2}{2}\right)}\right)}\right)}^{1}\]

14. Applied pow1 8.3

    \[\leadsto \left(\left(\left(\color{blue}{{-2}^{1}} \cdot {J}^{1}\right) \cdot {\left(\cos \left(\frac{K}{2}\right)\right)}^{1}\right) \cdot {\left(\sqrt{\mathsf{hypot}\left(\sqrt{1}, {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{\left(\frac{2}{2}\right)}\right)}\right)}^{1}\right) \cdot {\left(\sqrt{\mathsf{hypot}\left(\sqrt{1}, {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{\left(\frac{2}{2}\right)}\right)}\right)}^{1}\]

15. Applied pow-prod-down 8.3

    \[\leadsto \left(\left(\color{blue}{{\left(-2 \cdot J\right)}^{1}} \cdot {\left(\cos \left(\frac{K}{2}\right)\right)}^{1}\right) \cdot {\left(\sqrt{\mathsf{hypot}\left(\sqrt{1}, {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{\left(\frac{2}{2}\right)}\right)}\right)}^{1}\right) \cdot {\left(\sqrt{\mathsf{hypot}\left(\sqrt{1}, {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{\left(\frac{2}{2}\right)}\right)}\right)}^{1}\]

16. Applied pow-prod-down 8.3

    \[\leadsto \left(\color{blue}{{\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right)}^{1}} \cdot {\left(\sqrt{\mathsf{hypot}\left(\sqrt{1}, {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{\left(\frac{2}{2}\right)}\right)}\right)}^{1}\right) \cdot {\left(\sqrt{\mathsf{hypot}\left(\sqrt{1}, {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{\left(\frac{2}{2}\right)}\right)}\right)}^{1}\]

17. Applied pow-prod-down 8.3

    \[\leadsto \color{blue}{{\left(\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{\mathsf{hypot}\left(\sqrt{1}, {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{\left(\frac{2}{2}\right)}\right)}\right)}^{1}} \cdot {\left(\sqrt{\mathsf{hypot}\left(\sqrt{1}, {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{\left(\frac{2}{2}\right)}\right)}\right)}^{1}\]

18. Applied pow-prod-down 8.3

    \[\leadsto \color{blue}{{\left(\left(\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{\mathsf{hypot}\left(\sqrt{1}, {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{\left(\frac{2}{2}\right)}\right)}\right) \cdot \sqrt{\mathsf{hypot}\left(\sqrt{1}, {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{\left(\frac{2}{2}\right)}\right)}\right)}^{1}}\]

19. Simplified 8.1

    \[\leadsto {\color{blue}{\left(\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \mathsf{hypot}\left(\sqrt{1}, {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{\left(\frac{2}{2}\right)}\right)\right)}}^{1}\]

20. Final simplification 8.1

    \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \mathsf{hypot}\left(\sqrt{1}, {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{\left(\frac{2}{2}\right)}\right)\]

Reproduce

herbie shell --seed 2020064 +o rules:numerics
(FPCore (J K U)
  :name "Maksimov and Kolovsky, Equation (3)"
  :precision binary64
  (* (* (* -2 J) (cos (/ K 2))) (sqrt (+ 1 (pow (/ U (* (* 2 J) (cos (/ K 2)))) 2)))))