Average Error: 17.2 → 0.4
Time: 35.2s
Precision: 64
\[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U\]
↓
\[U + \cos \left(\frac{K}{2}\right) \cdot \left(\mathsf{fma}\left(\left({\ell}^{5}\right), \frac{1}{60}, \left(\ell \cdot \left(2 + \left(\ell \cdot \ell\right) \cdot \frac{1}{3}\right)\right)\right) \cdot J\right)\]
\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U
↓
U + \cos \left(\frac{K}{2}\right) \cdot \left(\mathsf{fma}\left(\left({\ell}^{5}\right), \frac{1}{60}, \left(\ell \cdot \left(2 + \left(\ell \cdot \ell\right) \cdot \frac{1}{3}\right)\right)\right) \cdot J\right)
double f(double J, double l, double K, double U) {
        double r3559965 = J;
        double r3559966 = l;
        double r3559967 = exp(r3559966);
        double r3559968 = -r3559966;
        double r3559969 = exp(r3559968);
        double r3559970 = r3559967 - r3559969;
        double r3559971 = r3559965 * r3559970;
        double r3559972 = K;
        double r3559973 = 2.0;
        double r3559974 = r3559972 / r3559973;
        double r3559975 = cos(r3559974);
        double r3559976 = r3559971 * r3559975;
        double r3559977 = U;
        double r3559978 = r3559976 + r3559977;
        return r3559978;
}
↓
double f(double J, double l, double K, double U) {
        double r3559979 = U;
        double r3559980 = K;
        double r3559981 = 2.0;
        double r3559982 = r3559980 / r3559981;
        double r3559983 = cos(r3559982);
        double r3559984 = l;
        double r3559985 = 5.0;
        double r3559986 = pow(r3559984, r3559985);
        double r3559987 = 0.016666666666666666;
        double r3559988 = r3559984 * r3559984;
        double r3559989 = 0.3333333333333333;
        double r3559990 = r3559988 * r3559989;
        double r3559991 = r3559981 + r3559990;
        double r3559992 = r3559984 * r3559991;
        double r3559993 = fma(r3559986, r3559987, r3559992);
        double r3559994 = J;
        double r3559995 = r3559993 * r3559994;
        double r3559996 = r3559983 * r3559995;
        double r3559997 = r3559979 + r3559996;
        return r3559997;
}

Error

Bits error versus J

Bits error versus l

Bits error versus K

Bits error versus U

Derivation

  1. Initial program 17.2

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

  2. Taylor expanded around 0 0.4

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

  3. Simplified0.4

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

  4. Final simplification0.4

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

Reproduce

herbie shell --seed 2019132 +o rules:numerics
(FPCore (J l K U)
  :name "Maksimov and Kolovsky, Equation (4)"
  (+ (* (* J (- (exp l) (exp (- l)))) (cos (/ K 2))) U))