Average Error: 17.1 → 9.7
Time: 36.5s
Precision: 64
Internal Precision: 128
\[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U\]
↓
\[\begin{array}{l} \mathbf{if}\;K \le -16916119.220738366:\\ \;\;\;\;(\left(\frac{{\left(\cos \left(\frac{K}{2}\right) \cdot e^{\ell}\right)}^{3} - {\left(\frac{\cos \left(\frac{K}{2}\right)}{e^{\ell}}\right)}^{3}}{\left(\cos \left(\frac{K}{2}\right) \cdot e^{\ell}\right) \cdot \left(\cos \left(\frac{K}{2}\right) \cdot e^{\ell}\right) + \left(\frac{\cos \left(\frac{K}{2}\right)}{e^{\ell}} \cdot \frac{\cos \left(\frac{K}{2}\right)}{e^{\ell}} + \frac{\cos \left(\frac{K}{2}\right)}{e^{\ell}} \cdot \left(\cos \left(\frac{K}{2}\right) \cdot e^{\ell}\right)\right)}\right) \cdot J + U)_*\\ \mathbf{elif}\;K \le 256217.53622618708:\\ \;\;\;\;(\left((\left(\ell \cdot \ell\right) \cdot \frac{1}{3} + \left((\frac{-1}{4} \cdot \left(K \cdot K\right) + 2)_*\right))_* \cdot \ell\right) \cdot J + U)_*\\ \mathbf{else}:\\ \;\;\;\;(\left(\cos \left(\frac{K}{2}\right) \cdot e^{\ell} - \log_* (1 + (e^{\frac{\cos \left(\frac{K}{2}\right)}{e^{\ell}}} - 1)^*)\right) \cdot J + U)_*\\ \end{array}\]

Error

Bits error versus J

Bits error versus l

Bits error versus K

Bits error versus U

Derivation

  1. Split input into 3 regimes

  2. if K < -16916119.220738366

    1. Initial program 17.2

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

    2. Simplified17.2

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

    4. Applied flip3--17.3

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

    if -16916119.220738366 < K < 256217.53622618708

    1. Initial program 16.9

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

    2. Simplified16.9

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

    3. Taylor expanded around 0 1.0

      \[\leadsto (\color{blue}{\left(\left(\frac{1}{3} \cdot {\ell}^{3} + 2 \cdot \ell\right) - \frac{1}{4} \cdot \left({K}^{2} \cdot \ell\right)\right)} \cdot J + U)_*\]

    4. Simplified1.0

      \[\leadsto (\color{blue}{\left((\left(\ell \cdot \ell\right) \cdot \frac{1}{3} + \left((\frac{-1}{4} \cdot \left(K \cdot K\right) + 2)_*\right))_* \cdot \ell\right)} \cdot J + U)_*\]

    if 256217.53622618708 < K

    1. Initial program 17.6

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

    2. Simplified17.6

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

    4. Applied log1p-expm1-u20.2

      \[\leadsto (\left(e^{\ell} \cdot \cos \left(\frac{K}{2}\right) - \color{blue}{\log_* (1 + (e^{\frac{\cos \left(\frac{K}{2}\right)}{e^{\ell}}} - 1)^*)}\right) \cdot J + U)_*\]
  3. Recombined 3 regimes into one program.

  4. Final simplification9.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;K \le -16916119.220738366:\\ \;\;\;\;(\left(\frac{{\left(\cos \left(\frac{K}{2}\right) \cdot e^{\ell}\right)}^{3} - {\left(\frac{\cos \left(\frac{K}{2}\right)}{e^{\ell}}\right)}^{3}}{\left(\cos \left(\frac{K}{2}\right) \cdot e^{\ell}\right) \cdot \left(\cos \left(\frac{K}{2}\right) \cdot e^{\ell}\right) + \left(\frac{\cos \left(\frac{K}{2}\right)}{e^{\ell}} \cdot \frac{\cos \left(\frac{K}{2}\right)}{e^{\ell}} + \frac{\cos \left(\frac{K}{2}\right)}{e^{\ell}} \cdot \left(\cos \left(\frac{K}{2}\right) \cdot e^{\ell}\right)\right)}\right) \cdot J + U)_*\\ \mathbf{elif}\;K \le 256217.53622618708:\\ \;\;\;\;(\left((\left(\ell \cdot \ell\right) \cdot \frac{1}{3} + \left((\frac{-1}{4} \cdot \left(K \cdot K\right) + 2)_*\right))_* \cdot \ell\right) \cdot J + U)_*\\ \mathbf{else}:\\ \;\;\;\;(\left(\cos \left(\frac{K}{2}\right) \cdot e^{\ell} - \log_* (1 + (e^{\frac{\cos \left(\frac{K}{2}\right)}{e^{\ell}}} - 1)^*)\right) \cdot J + U)_*\\ \end{array}\]

Reproduce

herbie shell --seed 2019068 +o rules:numerics
(FPCore (J l K U)
  :name "Maksimov and Kolovsky, Equation (4)"
  (+ (* (* J (- (exp l) (exp (- l)))) (cos (/ K 2))) U))