Maksimov and Kolovsky, Equation (4)

Average Error: 17.6 → 0.5

Time: 6.2s

Precision: binary64

Math

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

↓

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

FPCore

(FPCore (J l K U)
 :precision binary64
 (+ (* (* J (- (exp l) (exp (- l)))) (cos (/ K 2.0))) U))

↓

(FPCore (J l K U) :precision binary64 (+ (* J (* (+ l l) (cos (/ K 2.0)))) U))

C

double code(double J, double l, double K, double U) {
	return ((double) (((double) (((double) (J * ((double) (((double) exp(l)) - ((double) exp(((double) -(l)))))))) * ((double) cos((K / 2.0))))) + U));
}

↓

double code(double J, double l, double K, double U) {
	return ((double) (((double) (J * ((double) (((double) (l + l)) * ((double) cos((K / 2.0))))))) + U));
}

Error

Bits error versus J

Bits error versus l

Bits error versus K

Bits error versus U

Derivation

1. Initial program 17.6

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

2. Taylor expanded around 0 18.0

   \[\leadsto \left(J \cdot \color{blue}{\left(\left(e^{-0} \cdot \ell + \left(\ell + \left(0.5 \cdot {\ell}^{2} + 1\right)\right)\right) - \left(0.5 \cdot \left(e^{-0} \cdot {\ell}^{2}\right) + e^{-0}\right)\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U\]

3. Simplified 0.5

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

5. Applied associate-*l*_binary64 0.5

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

6. Final simplification 0.5

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

Reproduce

herbie shell --seed 2020204 
(FPCore (J l K U)
  :name "Maksimov and Kolovsky, Equation (4)"
  :precision binary64
  (+ (* (* J (- (exp l) (exp (- l)))) (cos (/ K 2.0))) U))