Average Error: 17.9 → 0.5
Time: 11.4s
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 r163743 = J;
        double r163744 = l;
        double r163745 = exp(r163744);
        double r163746 = -r163744;
        double r163747 = exp(r163746);
        double r163748 = r163745 - r163747;
        double r163749 = r163743 * r163748;
        double r163750 = K;
        double r163751 = 2.0;
        double r163752 = r163750 / r163751;
        double r163753 = cos(r163752);
        double r163754 = r163749 * r163753;
        double r163755 = U;
        double r163756 = r163754 + r163755;
        return r163756;
}
↓
double f(double J, double l, double K, double U) {
        double r163757 = J;
        double r163758 = 0.3333333333333333;
        double r163759 = l;
        double r163760 = 3.0;
        double r163761 = pow(r163759, r163760);
        double r163762 = 0.016666666666666666;
        double r163763 = 5.0;
        double r163764 = pow(r163759, r163763);
        double r163765 = 2.0;
        double r163766 = r163765 * r163759;
        double r163767 = fma(r163762, r163764, r163766);
        double r163768 = fma(r163758, r163761, r163767);
        double r163769 = r163757 * r163768;
        double r163770 = K;
        double r163771 = 2.0;
        double r163772 = r163770 / r163771;
        double r163773 = cos(r163772);
        double r163774 = U;
        double r163775 = fma(r163769, r163773, r163774);
        return r163775;
}

Error

Bits error versus J

Bits error versus l

Bits error versus K

Bits error versus U

Derivation

  1. Initial program 17.9

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

  2. Simplified17.9

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

  3. Taylor expanded around 0 0.5

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

  4. Simplified0.5

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

  5. Final simplification0.5

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

Reproduce

herbie shell --seed 2020001 +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))