Average Error: 17.6 → 0.4
Time: 34.3s
Precision: 64
\[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U\]
↓
\[J \cdot \left(\mathsf{fma}\left(\left(\mathsf{fma}\left(\left(\ell \cdot \ell\right), \frac{1}{3}, 2\right)\right), \ell, \left(\left(\left(\left(\ell \cdot \ell\right) \cdot \ell\right) \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{1}{60}\right)\right) \cdot \cos \left(\frac{K}{2}\right)\right) + U\]
\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U
↓
J \cdot \left(\mathsf{fma}\left(\left(\mathsf{fma}\left(\left(\ell \cdot \ell\right), \frac{1}{3}, 2\right)\right), \ell, \left(\left(\left(\left(\ell \cdot \ell\right) \cdot \ell\right) \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{1}{60}\right)\right) \cdot \cos \left(\frac{K}{2}\right)\right) + U
double f(double J, double l, double K, double U) {
        double r2828935 = J;
        double r2828936 = l;
        double r2828937 = exp(r2828936);
        double r2828938 = -r2828936;
        double r2828939 = exp(r2828938);
        double r2828940 = r2828937 - r2828939;
        double r2828941 = r2828935 * r2828940;
        double r2828942 = K;
        double r2828943 = 2.0;
        double r2828944 = r2828942 / r2828943;
        double r2828945 = cos(r2828944);
        double r2828946 = r2828941 * r2828945;
        double r2828947 = U;
        double r2828948 = r2828946 + r2828947;
        return r2828948;
}
↓
double f(double J, double l, double K, double U) {
        double r2828949 = J;
        double r2828950 = l;
        double r2828951 = r2828950 * r2828950;
        double r2828952 = 0.3333333333333333;
        double r2828953 = 2.0;
        double r2828954 = fma(r2828951, r2828952, r2828953);
        double r2828955 = r2828951 * r2828950;
        double r2828956 = r2828955 * r2828951;
        double r2828957 = 0.016666666666666666;
        double r2828958 = r2828956 * r2828957;
        double r2828959 = fma(r2828954, r2828950, r2828958);
        double r2828960 = K;
        double r2828961 = r2828960 / r2828953;
        double r2828962 = cos(r2828961);
        double r2828963 = r2828959 * r2828962;
        double r2828964 = r2828949 * r2828963;
        double r2828965 = U;
        double r2828966 = r2828964 + r2828965;
        return r2828966;
}

Error

Bits error versus J

Bits error versus l

Bits error versus K

Bits error versus U

Derivation

  1. Initial program 17.6

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

  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. Using strategy rm

  5. Applied associate-*l*0.4

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

  6. Simplified0.4

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

  7. Final simplification0.4

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

Reproduce

herbie shell --seed 2019130 +o rules:numerics
(FPCore (J l K U)
  :name "Maksimov and Kolovsky, Equation (4)"
  (+ (* (* J (- (exp l) (exp (- l)))) (cos (/ K 2))) U))