Average Error: 17.7 → 0.3
Time: 36.9s
Precision: 64
Internal Precision: 1344
\[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U\]
↓
\[\left(J \cdot \cos \left(\frac{K}{2}\right)\right) \cdot (\left((\frac{1}{3} \cdot \left(\ell \cdot \ell\right) + 2)_*\right) \cdot \ell + \left(\frac{1}{60} \cdot {\ell}^{5}\right))_* + U\]

Error

Bits error versus J

Bits error versus l

Bits error versus K

Bits error versus U

Derivation

  1. Initial program 17.7

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

  2. Taylor expanded around 0 0.3

    \[\leadsto \left(J \cdot \color{blue}{\left(2 \cdot \ell + \left(\frac{1}{3} \cdot {\ell}^{3} + \frac{1}{60} \cdot {\ell}^{5}\right)\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U\]

  3. Simplified0.3

    \[\leadsto \left(J \cdot \color{blue}{(\left((\frac{1}{3} \cdot \left(\ell \cdot \ell\right) + 2)_*\right) \cdot \ell + \left({\ell}^{5} \cdot \frac{1}{60}\right))_*}\right) \cdot \cos \left(\frac{K}{2}\right) + U\]
  4. Using strategy rm

  5. Applied fma-udef0.3

    \[\leadsto \left(J \cdot \color{blue}{\left((\frac{1}{3} \cdot \left(\ell \cdot \ell\right) + 2)_* \cdot \ell + {\ell}^{5} \cdot \frac{1}{60}\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U\]

  6. Applied distribute-lft-in0.3

    \[\leadsto \color{blue}{\left(J \cdot \left((\frac{1}{3} \cdot \left(\ell \cdot \ell\right) + 2)_* \cdot \ell\right) + J \cdot \left({\ell}^{5} \cdot \frac{1}{60}\right)\right)} \cdot \cos \left(\frac{K}{2}\right) + U\]
  7. Using strategy rm

  8. Applied pow10.3

    \[\leadsto \left(J \cdot \left((\frac{1}{3} \cdot \left(\ell \cdot \ell\right) + 2)_* \cdot \ell\right) + J \cdot \left({\ell}^{5} \cdot \frac{1}{60}\right)\right) \cdot \color{blue}{{\left(\cos \left(\frac{K}{2}\right)\right)}^{1}} + U\]

  9. Applied pow10.3

    \[\leadsto \color{blue}{{\left(J \cdot \left((\frac{1}{3} \cdot \left(\ell \cdot \ell\right) + 2)_* \cdot \ell\right) + J \cdot \left({\ell}^{5} \cdot \frac{1}{60}\right)\right)}^{1}} \cdot {\left(\cos \left(\frac{K}{2}\right)\right)}^{1} + U\]

  10. Applied pow-prod-down0.3

    \[\leadsto \color{blue}{{\left(\left(J \cdot \left((\frac{1}{3} \cdot \left(\ell \cdot \ell\right) + 2)_* \cdot \ell\right) + J \cdot \left({\ell}^{5} \cdot \frac{1}{60}\right)\right) \cdot \cos \left(\frac{K}{2}\right)\right)}^{1}} + U\]

  11. Simplified0.3

    \[\leadsto {\color{blue}{\left((\left((\frac{1}{3} \cdot \left(\ell \cdot \ell\right) + 2)_*\right) \cdot \ell + \left({\ell}^{5} \cdot \frac{1}{60}\right))_* \cdot \left(J \cdot \cos \left(\frac{K}{2}\right)\right)\right)}}^{1} + U\]

  12. Final simplification0.3

    \[\leadsto \left(J \cdot \cos \left(\frac{K}{2}\right)\right) \cdot (\left((\frac{1}{3} \cdot \left(\ell \cdot \ell\right) + 2)_*\right) \cdot \ell + \left(\frac{1}{60} \cdot {\ell}^{5}\right))_* + U\]

Runtime

Time bar (total: 36.9s)Debug logProfile

herbie shell --seed 2018220 +o rules:numerics
(FPCore (J l K U)
  :name "Maksimov and Kolovsky, Equation (4)"
  (+ (* (* J (- (exp l) (exp (- l)))) (cos (/ K 2))) U))