Maksimov and Kolovsky, Equation (4)

Average Error: 17.0 → 0.6

Time: 8.3s

Precision: binary64

Math

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

↓

\[\mathsf{fma}\left(J \cdot \mathsf{fma}\left(2, \ell, 0.3333333333333333 \cdot {\ell}^{3}\right), \cos \left(\frac{K}{2}\right), U\right) \]

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
 (fma (* J (fma 2.0 l (* 0.3333333333333333 (pow l 3.0)))) (cos (/ K 2.0)) 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 fma((J * fma(2.0, l, (0.3333333333333333 * pow(l, 3.0)))), 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 17.0

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

2. Simplified 17.0

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

3. Taylor expanded in l around 0 0.6

   \[\leadsto \mathsf{fma}\left(J \cdot \color{blue}{\left(2 \cdot \ell + 0.3333333333333333 \cdot {\ell}^{3}\right)}, \cos \left(\frac{K}{2}\right), U\right) \]

4. Simplified 0.6

   \[\leadsto \mathsf{fma}\left(J \cdot \color{blue}{\mathsf{fma}\left(2, \ell, 0.3333333333333333 \cdot {\ell}^{3}\right)}, \cos \left(\frac{K}{2}\right), U\right) \]

5. Final simplification 0.6

   \[\leadsto \mathsf{fma}\left(J \cdot \mathsf{fma}\left(2, \ell, 0.3333333333333333 \cdot {\ell}^{3}\right), \cos \left(\frac{K}{2}\right), U\right) \]

Reproduce

herbie shell --seed 2021215 
(FPCore (J l K U)
  :name "Maksimov and Kolovsky, Equation (4)"
  :precision binary64
  (+ (* (* J (- (exp l) (exp (- l)))) (cos (/ K 2.0))) U))