Average Error: 17.9 → 0.4
Time: 28.4s
Precision: 64
\[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U\]
↓
\[\left(J \cdot \mathsf{fma}\left(\frac{1}{3}, {\ell}^{3}, \mathsf{fma}\left(\frac{1}{60}, {\ell}^{5}, 2 \cdot \ell\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U\]
\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U
↓
\left(J \cdot \mathsf{fma}\left(\frac{1}{3}, {\ell}^{3}, \mathsf{fma}\left(\frac{1}{60}, {\ell}^{5}, 2 \cdot \ell\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U
double f(double J, double l, double K, double U) {
        double r235951 = J;
        double r235952 = l;
        double r235953 = exp(r235952);
        double r235954 = -r235952;
        double r235955 = exp(r235954);
        double r235956 = r235953 - r235955;
        double r235957 = r235951 * r235956;
        double r235958 = K;
        double r235959 = 2.0;
        double r235960 = r235958 / r235959;
        double r235961 = cos(r235960);
        double r235962 = r235957 * r235961;
        double r235963 = U;
        double r235964 = r235962 + r235963;
        return r235964;
}
↓
double f(double J, double l, double K, double U) {
        double r235965 = J;
        double r235966 = 0.3333333333333333;
        double r235967 = l;
        double r235968 = 3.0;
        double r235969 = pow(r235967, r235968);
        double r235970 = 0.016666666666666666;
        double r235971 = 5.0;
        double r235972 = pow(r235967, r235971);
        double r235973 = 2.0;
        double r235974 = r235973 * r235967;
        double r235975 = fma(r235970, r235972, r235974);
        double r235976 = fma(r235966, r235969, r235975);
        double r235977 = r235965 * r235976;
        double r235978 = K;
        double r235979 = 2.0;
        double r235980 = r235978 / r235979;
        double r235981 = cos(r235980);
        double r235982 = r235977 * r235981;
        double r235983 = U;
        double r235984 = r235982 + r235983;
        return r235984;
}

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 0.4

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

  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. Using strategy rm

  6. Applied fma-udef0.4

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

  7. Final simplification0.4

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

Reproduce

herbie shell --seed 2020042 +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))