Average Error: 17.5 → 0.4
Time: 8.6s
Precision: binary64
\[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U\]
↓
\[\left(\ell + \left(\ell + \left(\frac{1}{3} \cdot {\ell}^{3} + \frac{1}{60} \cdot {\ell}^{5}\right)\right)\right) \cdot \left(J \cdot \cos \left(\frac{K}{2}\right)\right) + U\]

Error

Bits error versus J

Bits error versus l

Bits error versus K

Bits error versus U

Derivation

  1. Initial program 17.5

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

  2. Simplified17.5

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

  3. Taylor expanded around 0 0.4

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

  4. Simplified0.4

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

  6. Applied pow10.4

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

  7. Applied pow10.4

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

  8. Applied pow-prod-down0.4

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

  9. Applied pow10.4

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

  10. Applied pow-prod-down0.4

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

  11. Simplified0.4

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

  12. Final simplification0.4

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

Reproduce

herbie shell --seed 2020179 
(FPCore (J l K U)
  :name "Maksimov and Kolovsky, Equation (4)"
  :precision binary64
  (+ (* (* J (- (exp l) (exp (neg l)))) (cos (/ K 2.0))) U))