Average Error: 17.1 → 0.1
Time: 21.4s
Precision: binary64
\[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
↓
\[\mathsf{fma}\left(J, \left(2 \cdot \sinh \ell\right) \cdot \cos \left(\frac{K}{2}\right), U\right) \]
\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U
↓
\mathsf{fma}\left(J, \left(2 \cdot \sinh \ell\right) \cdot \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 (* (* 2.0 (sinh l)) (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, ((2.0 * sinh(l)) * 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 17.1

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

  2. Applied sinh-undef_binary640.1

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

  3. Applied associate-*l*_binary640.1

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

  4. Applied fma-def_binary640.1

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

  5. Final simplification0.1

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

Reproduce

herbie shell --seed 2021357 
(FPCore (J l K U)
  :name "Maksimov and Kolovsky, Equation (4)"
  :precision binary64
  (+ (* (* J (- (exp l) (exp (- l)))) (cos (/ K 2.0))) U))