Average Error: 17.7 → 0.4
Time: 9.7s
Precision: binary64
\[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
↓
\[\begin{array}{l} t_0 := \cos \left(0.5 \cdot K\right) \cdot J\\ 0.3333333333333333 \cdot \left({\ell}^{3} \cdot t_0\right) + \left(2 \cdot \left(\ell \cdot t_0\right) + U\right) \end{array} \]
\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U
↓
\begin{array}{l}
t_0 := \cos \left(0.5 \cdot K\right) \cdot J\\
0.3333333333333333 \cdot \left({\ell}^{3} \cdot t_0\right) + \left(2 \cdot \left(\ell \cdot t_0\right) + U\right)
\end{array}

(FPCore (J l K U)
 :precision binary64
 (+ (* (* J (- (exp l) (exp (- l)))) (cos (/ K 2.0))) U))
↓
(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (* (cos (* 0.5 K)) J)))
   (+ (* 0.3333333333333333 (* (pow l 3.0) t_0)) (+ (* 2.0 (* l t_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) {
	double t_0 = cos((0.5 * K)) * J;
	return (0.3333333333333333 * (pow(l, 3.0) * t_0)) + ((2.0 * (l * t_0)) + U);
}

Error

Bits error versus J

Bits error versus l

Bits error versus K

Bits error versus U

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 17.7

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

  2. Simplified17.7

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

  3. Taylor expanded in l around 0 0.4

    \[\leadsto \color{blue}{0.3333333333333333 \cdot \left({\ell}^{3} \cdot \left(\cos \left(0.5 \cdot K\right) \cdot J\right)\right) + \left(2 \cdot \left(\ell \cdot \left(\cos \left(0.5 \cdot K\right) \cdot J\right)\right) + U\right)} \]

  4. Final simplification0.4

    \[\leadsto 0.3333333333333333 \cdot \left({\ell}^{3} \cdot \left(\cos \left(0.5 \cdot K\right) \cdot J\right)\right) + \left(2 \cdot \left(\ell \cdot \left(\cos \left(0.5 \cdot K\right) \cdot J\right)\right) + U\right) \]

Reproduce

herbie shell --seed 2022125 
(FPCore (J l K U)
  :name "Maksimov and Kolovsky, Equation (4)"
  :precision binary64
  (+ (* (* J (- (exp l) (exp (- l)))) (cos (/ K 2.0))) U))