Maksimov and Kolovsky, Equation (4)

Average Error: 17.1 → 0.1

Time: 7.2s

Precision: binary64

Math

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

↓

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

Error

Bits error versus J

Bits error versus l

Bits error versus K

Bits error versus U

Derivation

1. Initial program 17.1

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

3. Applied sinh-undef 0.1

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

5. Applied associate-*l* 0.1

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

6. Simplified 0.1

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

7. Final simplification 0.1

   \[\leadsto J \cdot \left(2 \cdot \left(\sinh \ell \cdot \cos \left(\frac{K}{2}\right)\right)\right) + U\]

Reproduce

herbie shell --seed 2020184 
(FPCore (J l K U)
  :name "Maksimov and Kolovsky, Equation (4)"
  :precision binary64
  (+ (* (* J (- (exp l) (exp (neg l)))) (cos (/ K 2.0))) U))