Average Error: 17.9 → 0.6
Time: 45.1s
Precision: 64
Internal Precision: 128
\[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U\]
↓
\[\left(\left(\left(\log \left(e^{\frac{1}{60} \cdot {\ell}^{5}}\right) + {\ell}^{3} \cdot \frac{1}{3}\right) + \ell \cdot 2\right) \cdot J\right) \cdot \cos \left(\frac{K}{2}\right) + U\]

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 17.9

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

  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 add-log-exp0.6

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

  5. Final simplification0.6

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

Runtime

Time bar (total: 45.1s)Debug logProfile

herbie shell --seed 2018304 
(FPCore (J l K U)
  :name "Maksimov and Kolovsky, Equation (4)"
  (+ (* (* J (- (exp l) (exp (- l)))) (cos (/ K 2))) U))