Average Error: 17.3 → 0.5
Time: 8.8s
Precision: binary64
\[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
↓
\[\mathsf{fma}\left(J \cdot \mathsf{fma}\left(0.3333333333333333, {\ell}^{3}, \ell \cdot 2\right), \cos \left(\frac{K}{2}\right), U\right) \]
\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(0.3333333333333333, {\ell}^{3}, \ell \cdot 2\right), \cos \left(\frac{K}{2}\right), U\right)

(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 0.3333333333333333 (pow l 3.0) (* l 2.0))) (cos (/ K 2.0)) U))
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(0.3333333333333333, pow(l, 3.0), (l * 2.0))), 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 17.3

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

  2. Taylor expanded in l around 0 0.5

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

  3. Applied egg-rr0.5

    \[\leadsto \color{blue}{{\left(\mathsf{fma}\left(J \cdot \mathsf{fma}\left(0.3333333333333333, {\ell}^{3}, \ell \cdot 2\right), \cos \left(\frac{K}{2}\right), U\right)\right)}^{1}} \]

  4. Final simplification0.5

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

Reproduce

herbie shell --seed 2022129 
(FPCore (J l K U)
  :name "Maksimov and Kolovsky, Equation (4)"
  :precision binary64
  (+ (* (* J (- (exp l) (exp (- l)))) (cos (/ K 2.0))) U))