Maksimov and Kolovsky, Equation (4)

Average Error: 17.2 → 0.7

Time: 27.8s

Precision: 64

Math

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

↓

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

C

double f(double J, double l, double K, double U) {
        double r3913435 = J;
        double r3913436 = l;
        double r3913437 = exp(r3913436);
        double r3913438 = -r3913436;
        double r3913439 = exp(r3913438);
        double r3913440 = r3913437 - r3913439;
        double r3913441 = r3913435 * r3913440;
        double r3913442 = K;
        double r3913443 = 2.0;
        double r3913444 = r3913442 / r3913443;
        double r3913445 = cos(r3913444);
        double r3913446 = r3913441 * r3913445;
        double r3913447 = U;
        double r3913448 = r3913446 + r3913447;
        return r3913448;
}

↓

double f(double J, double l, double K, double U) {
        double r3913449 = 2.0;
        double r3913450 = J;
        double r3913451 = r3913449 * r3913450;
        double r3913452 = 0.5;
        double r3913453 = K;
        double r3913454 = r3913452 * r3913453;
        double r3913455 = cos(r3913454);
        double r3913456 = l;
        double r3913457 = r3913455 * r3913456;
        double r3913458 = r3913451 * r3913457;
        double r3913459 = U;
        double r3913460 = r3913458 + r3913459;
        return r3913460;
}

Error

Bits error versus J

Bits error versus l

Bits error versus K

Bits error versus U

Derivation

1. Initial program 17.2

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

2. Taylor expanded around 0 0.7

   \[\leadsto \color{blue}{\left(2 \cdot \left(J \cdot \ell\right)\right)} \cdot \cos \left(\frac{K}{2}\right) + U\]
3. Using strategy rm

4. Applied add-sqr-sqrt 1.0

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

5. Applied associate-*l* 0.9

   \[\leadsto \color{blue}{\left(\sqrt{2} \cdot \left(\sqrt{2} \cdot \left(J \cdot \ell\right)\right)\right)} \cdot \cos \left(\frac{K}{2}\right) + U\]
6. Using strategy rm

7. Applied pow1 0.9

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

8. Applied pow1 0.9

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

9. Applied pow1 0.9

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

10. Applied pow-prod-down 0.9

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

11. Applied pow1 0.9

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

12. Applied pow-prod-down 0.9

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

13. Applied pow1 0.9

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

14. Applied pow-prod-down 0.9

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

15. Applied pow-prod-down 0.9

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

16. Simplified 0.7

    \[\leadsto {\color{blue}{\left(\left(\cos \left(\frac{K}{2}\right) \cdot \left(2 \cdot J\right)\right) \cdot \ell\right)}}^{1} + U\]
17. Using strategy rm

18. Applied *-un-lft-identity 0.7

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

19. Applied *-un-lft-identity 0.7

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

20. Applied distribute-lft-out 0.7

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

21. Simplified 0.7

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

22. Final simplification 0.7

    \[\leadsto \left(2 \cdot J\right) \cdot \left(\cos \left(\frac{1}{2} \cdot K\right) \cdot \ell\right) + U\]

Reproduce

herbie shell --seed 2019162 
(FPCore (J l K U)
  :name "Maksimov and Kolovsky, Equation (4)"
  (+ (* (* J (- (exp l) (exp (- l)))) (cos (/ K 2))) U))