Average Error: 17.1 → 0.4
Time: 59.0s
Precision: 64
Internal Precision: 128
\[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U\]
↓
\[(J \cdot \left((\frac{1}{60} \cdot \left({\ell}^{5}\right) + \left(\ell \cdot (\frac{1}{3} \cdot \left(\ell \cdot \ell\right) + 2)_*\right))_* \cdot \cos \left(\frac{K}{2}\right)\right) + U)_*\]

Error

Bits error versus J

Bits error versus l

Bits error versus K

Bits error versus U

Derivation

  1. Initial program 17.1

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

  2. Taylor expanded around 0 0.4

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

  3. Simplified0.4

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

  5. Applied *-un-lft-identity0.4

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

  6. Applied associate-*r*0.4

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

  7. Applied associate-*l*0.4

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

  8. Applied fma-def0.4

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

  9. Simplified0.4

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

  10. Final simplification0.4

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

Reproduce

herbie shell --seed 2019072 +o rules:numerics
(FPCore (J l K U)
  :name "Maksimov and Kolovsky, Equation (4)"
  (+ (* (* J (- (exp l) (exp (- l)))) (cos (/ K 2))) U))