Maksimov and Kolovsky, Equation (4)

Average Error: 16.7 → 0.3

Time: 5.9s

Precision: 64

Math

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

↓

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

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 ((((0.3333333333333333 * pow(l, 3.0)) + ((0.016666666666666666 * pow(l, 5.0)) + (2.0 * l))) * (J * 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 16.7

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

2. Taylor expanded around 0 0.3

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

4. Applied *-commutative 0.3

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

5. Applied associate-*l* 0.3

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

6. Final simplification 0.3

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

Reproduce

herbie shell --seed 2020071 
(FPCore (J l K U)
  :name "Maksimov and Kolovsky, Equation (4)"
  :precision binary64
  (+ (* (* J (- (exp l) (exp (- l)))) (cos (/ K 2))) U))