Average Error: 17.3 → 0.4
Time: 8.3s
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 r237951 = J;
        double r237952 = l;
        double r237953 = exp(r237952);
        double r237954 = -r237952;
        double r237955 = exp(r237954);
        double r237956 = r237953 - r237955;
        double r237957 = r237951 * r237956;
        double r237958 = K;
        double r237959 = 2.0;
        double r237960 = r237958 / r237959;
        double r237961 = cos(r237960);
        double r237962 = r237957 * r237961;
        double r237963 = U;
        double r237964 = r237962 + r237963;
        return r237964;
}
↓
double f(double J, double l, double K, double U) {
        double r237965 = J;
        double r237966 = 0.3333333333333333;
        double r237967 = l;
        double r237968 = 3.0;
        double r237969 = pow(r237967, r237968);
        double r237970 = 0.016666666666666666;
        double r237971 = 5.0;
        double r237972 = pow(r237967, r237971);
        double r237973 = 2.0;
        double r237974 = r237973 * r237967;
        double r237975 = fma(r237970, r237972, r237974);
        double r237976 = fma(r237966, r237969, r237975);
        double r237977 = r237965 * r237976;
        double r237978 = K;
        double r237979 = 2.0;
        double r237980 = r237978 / r237979;
        double r237981 = cos(r237980);
        double r237982 = U;
        double r237983 = fma(r237977, r237981, r237982);
        return r237983;
}

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. Simplified17.3

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

  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 2020020 +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))