Average Error: 16.8 → 0.7
Time: 33.3s
Precision: 64
Internal Precision: 1344
\[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U\]
↓
\[\left(\left(\left(\ell \cdot J\right) \cdot 2\right) \cdot {\left(\cos \left(\frac{K}{2}\right) \cdot \cos \left(\frac{K}{2}\right)\right)}^{\frac{1}{3}}\right) \cdot \sqrt[3]{\cos \left(\frac{K}{2}\right)} + U\]

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 16.8

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

  2. Taylor expanded around 0 0.7

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

  4. Applied add-cube-cbrt0.8

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

  5. Applied associate-*r*0.8

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

  7. Applied pow1/316.4

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

  8. Applied pow1/316.3

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

  9. Applied pow-prod-down0.7

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

  10. Final simplification0.7

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

Runtime

Time bar (total: 33.3s)Debug logProfile

herbie shell --seed 2018230 
(FPCore (J l K U)
  :name "Maksimov and Kolovsky, Equation (4)"
  (+ (* (* J (- (exp l) (exp (- l)))) (cos (/ K 2))) U))