Average Error: 17.2 → 0.6
Time: 26.1s
Precision: 64
Internal Precision: 128
\[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U\]
↓
\[U + \left({\left({\left(\cos \left(\frac{1}{2} \cdot K\right)\right)}^{2}\right)}^{\frac{1}{3}} \cdot \left(\left(\ell \cdot J\right) \cdot 2\right)\right) \cdot \log \left(e^{\sqrt[3]{\cos \left(\frac{K}{2}\right)}}\right)\]

Error

Bits error versus J

Bits error versus l

Bits error versus K

Bits error versus U

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

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 0.5

    \[\leadsto \color{blue}{\left(2 \cdot \left(J \cdot \ell\right)\right)} \cdot \cos \left(\frac{K}{2}\right) + U\]
  3. Using strategy rm

  4. Applied add-cube-cbrt0.7

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

  5. Applied associate-*r*0.7

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

  6. Taylor expanded around inf 0.6

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

  8. Applied add-log-exp0.6

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

  9. Final simplification0.6

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

Reproduce

herbie shell --seed 2019021 
(FPCore (J l K U)
  :name "Maksimov and Kolovsky, Equation (4)"
  (+ (* (* J (- (exp l) (exp (- l)))) (cos (/ K 2))) U))