Average Error: 17.4 → 0.4
Time: 28.7s
Precision: 64
\[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U\]
↓
\[J \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \mathsf{fma}\left(\ell, \mathsf{fma}\left(\frac{1}{3}, \ell \cdot \ell, 2\right), {\ell}^{5} \cdot \frac{1}{60}\right)\right) + U\]
\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U
↓
J \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \mathsf{fma}\left(\ell, \mathsf{fma}\left(\frac{1}{3}, \ell \cdot \ell, 2\right), {\ell}^{5} \cdot \frac{1}{60}\right)\right) + U
double f(double J, double l, double K, double U) {
        double r1526942 = J;
        double r1526943 = l;
        double r1526944 = exp(r1526943);
        double r1526945 = -r1526943;
        double r1526946 = exp(r1526945);
        double r1526947 = r1526944 - r1526946;
        double r1526948 = r1526942 * r1526947;
        double r1526949 = K;
        double r1526950 = 2.0;
        double r1526951 = r1526949 / r1526950;
        double r1526952 = cos(r1526951);
        double r1526953 = r1526948 * r1526952;
        double r1526954 = U;
        double r1526955 = r1526953 + r1526954;
        return r1526955;
}
↓
double f(double J, double l, double K, double U) {
        double r1526956 = J;
        double r1526957 = K;
        double r1526958 = 2.0;
        double r1526959 = r1526957 / r1526958;
        double r1526960 = cos(r1526959);
        double r1526961 = l;
        double r1526962 = 0.3333333333333333;
        double r1526963 = r1526961 * r1526961;
        double r1526964 = fma(r1526962, r1526963, r1526958);
        double r1526965 = 5.0;
        double r1526966 = pow(r1526961, r1526965);
        double r1526967 = 0.016666666666666666;
        double r1526968 = r1526966 * r1526967;
        double r1526969 = fma(r1526961, r1526964, r1526968);
        double r1526970 = r1526960 * r1526969;
        double r1526971 = r1526956 * r1526970;
        double r1526972 = U;
        double r1526973 = r1526971 + r1526972;
        return r1526973;
}

Error

Bits error versus J

Bits error versus l

Bits error versus K

Bits error versus U

Derivation

  1. Initial program 17.4

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

  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

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

  5. Applied associate-*l*0.4

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

  6. Simplified0.4

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

  7. Final simplification0.4

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

Reproduce

herbie shell --seed 2019151 +o rules:numerics
(FPCore (J l K U)
  :name "Maksimov and Kolovsky, Equation (4)"
  (+ (* (* J (- (exp l) (exp (- l)))) (cos (/ K 2))) U))