Average Error: 17.6 → 0.4
Time: 24.7s
Precision: 64
\[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U\]
↓
\[\mathsf{fma}\left(J \cdot \mathsf{fma}\left(\frac{1}{3}, {\ell}^{3}, \mathsf{fma}\left(\frac{1}{60}, {\ell}^{5}, 2 \cdot \ell\right)\right), \cos \left(\frac{K}{2}\right), U\right)\]
\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(\frac{1}{3}, {\ell}^{3}, \mathsf{fma}\left(\frac{1}{60}, {\ell}^{5}, 2 \cdot \ell\right)\right), \cos \left(\frac{K}{2}\right), U\right)
double f(double J, double l, double K, double U) {
        double r89765 = J;
        double r89766 = l;
        double r89767 = exp(r89766);
        double r89768 = -r89766;
        double r89769 = exp(r89768);
        double r89770 = r89767 - r89769;
        double r89771 = r89765 * r89770;
        double r89772 = K;
        double r89773 = 2.0;
        double r89774 = r89772 / r89773;
        double r89775 = cos(r89774);
        double r89776 = r89771 * r89775;
        double r89777 = U;
        double r89778 = r89776 + r89777;
        return r89778;
}
↓
double f(double J, double l, double K, double U) {
        double r89779 = J;
        double r89780 = 0.3333333333333333;
        double r89781 = l;
        double r89782 = 3.0;
        double r89783 = pow(r89781, r89782);
        double r89784 = 0.016666666666666666;
        double r89785 = 5.0;
        double r89786 = pow(r89781, r89785);
        double r89787 = 2.0;
        double r89788 = r89787 * r89781;
        double r89789 = fma(r89784, r89786, r89788);
        double r89790 = fma(r89780, r89783, r89789);
        double r89791 = r89779 * r89790;
        double r89792 = K;
        double r89793 = 2.0;
        double r89794 = r89792 / r89793;
        double r89795 = cos(r89794);
        double r89796 = U;
        double r89797 = fma(r89791, r89795, r89796);
        return r89797;
}

Error

Bits error versus J

Bits error versus l

Bits error versus K

Bits error versus U

Derivation

  1. Initial program 17.6

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

  2. Simplified17.6

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

  3. Taylor expanded around 0 0.4

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

  4. Simplified0.4

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

  5. Final simplification0.4

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

Reproduce

herbie shell --seed 2019305 +o rules:numerics
(FPCore (J l K U)
  :name "Maksimov and Kolovsky, Equation (4)"
  :precision binary64
  (+ (* (* J (- (exp l) (exp (- l)))) (cos (/ K 2))) U))