Average Error: 17.4 → 0.5
Time: 1.4m
Precision: 64
Internal Precision: 1344
\[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U\]
↓
\[(\left((\left((\left(\ell \cdot \ell\right) \cdot \frac{1}{3} + 2)_*\right) \cdot \ell + \left(\frac{1}{60} \cdot {\ell}^{5}\right))_* \cdot J\right) \cdot \left({\left(\sqrt[3]{\cos \left(\frac{K}{2}\right)}\right)}^{3}\right) + U)_*\]

Error

Bits error versus J

Bits error versus l

Bits error versus K

Bits error versus U

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

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

  3. Applied simplify0.4

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

  5. Applied add-cube-cbrt0.5

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

  6. Applied associate-*r*0.5

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

  8. Applied add-cube-cbrt0.6

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

  9. Taylor expanded around 0 0.6

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

  10. Applied simplify0.5

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

Runtime

Time bar (total: 1.4m)Debug logProfile

herbie shell --seed 2020178 +o rules:numerics
(FPCore (J l K U)
  :name "Maksimov and Kolovsky, Equation (4)"
  (+ (* (* J (- (exp l) (exp (- l)))) (cos (/ K 2))) U))