Average Error: 17.4 → 0.4
Time: 51.8s
Precision: 64
Internal Precision: 1344
\[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U\]
↓
\[U + J \cdot \left(\left(\ell \cdot 2 + \left(\frac{1}{60} \cdot {\ell}^{5} + {\ell}^{3} \cdot \frac{1}{3}\right)\right) \cdot \cos \left(\frac{K}{2}\right)\right)\]

Error

Bits error versus J

Bits error versus l

Bits error versus K

Bits error versus U

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 17.4

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

  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. Using strategy rm

  4. Applied associate-*l*0.4

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

  5. Final simplification0.4

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

Runtime

Time bar (total: 51.8s)Debug logProfile

herbie shell --seed 2018217 
(FPCore (J l K U)
  :name "Maksimov and Kolovsky, Equation (4)"
  (+ (* (* J (- (exp l) (exp (- l)))) (cos (/ K 2))) U))