Maksimov and Kolovsky, Equation (4)

Average Error: 17.2 → 0.7

Time: 30.8s

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 r3914803 = J;
        double r3914804 = l;
        double r3914805 = exp(r3914804);
        double r3914806 = -r3914804;
        double r3914807 = exp(r3914806);
        double r3914808 = r3914805 - r3914807;
        double r3914809 = r3914803 * r3914808;
        double r3914810 = K;
        double r3914811 = 2.0;
        double r3914812 = r3914810 / r3914811;
        double r3914813 = cos(r3914812);
        double r3914814 = r3914809 * r3914813;
        double r3914815 = U;
        double r3914816 = r3914814 + r3914815;
        return r3914816;
}

↓

double f(double J, double l, double K, double U) {
        double r3914817 = 2.0;
        double r3914818 = J;
        double r3914819 = r3914817 * r3914818;
        double r3914820 = 0.5;
        double r3914821 = K;
        double r3914822 = r3914820 * r3914821;
        double r3914823 = cos(r3914822);
        double r3914824 = l;
        double r3914825 = r3914823 * r3914824;
        double r3914826 = r3914819 * r3914825;
        double r3914827 = U;
        double r3914828 = r3914826 + r3914827;
        return r3914828;
}

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))