Maksimov and Kolovsky, Equation (4)

Average Error: 17.1 → 0.6

Time: 5.6s

Precision: 64

Math

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

↓

\[\mathsf{fma}\left(2, \left(J \cdot \cos \left(0.5 \cdot K\right)\right) \cdot \ell, U\right)\]

C

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

↓

double code(double J, double l, double K, double U) {
	return fma(2.0, ((J * cos((0.5 * K))) * l), 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. Simplified 17.1

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

3. Taylor expanded around 0 0.6

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

4. Taylor expanded around inf 0.6

   \[\leadsto \color{blue}{2 \cdot \left(J \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \ell\right)\right) + U}\]

5. Simplified 0.6

   \[\leadsto \color{blue}{\mathsf{fma}\left(2, J \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \ell\right), U\right)}\]
6. Using strategy rm

7. Applied associate-*r* 0.6

   \[\leadsto \mathsf{fma}\left(2, \color{blue}{\left(J \cdot \cos \left(0.5 \cdot K\right)\right) \cdot \ell}, U\right)\]

8. Final simplification 0.6

   \[\leadsto \mathsf{fma}\left(2, \left(J \cdot \cos \left(0.5 \cdot K\right)\right) \cdot \ell, U\right)\]

Reproduce

herbie shell --seed 2020053 +o rules:numerics
(FPCore (J l K U)
  :name "Maksimov and Kolovsky, Equation (4)"
  :precision binary64
  (+ (* (* J (- (exp l) (exp (- l)))) (cos (/ K 2))) U))