Average Error: 17.1 → 0.3
Time: 27.6s
Precision: 64
\[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U\]
↓
\[U + \left(\cos \left(\frac{K}{2}\right) \cdot \mathsf{fma}\left(\frac{1}{60}, {\ell}^{5}, \mathsf{fma}\left(\ell \cdot \left(\ell \cdot \ell\right), \frac{1}{3}, \ell + \ell\right)\right)\right) \cdot J\]
\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U
↓
U + \left(\cos \left(\frac{K}{2}\right) \cdot \mathsf{fma}\left(\frac{1}{60}, {\ell}^{5}, \mathsf{fma}\left(\ell \cdot \left(\ell \cdot \ell\right), \frac{1}{3}, \ell + \ell\right)\right)\right) \cdot J
double f(double J, double l, double K, double U) {
        double r1452895 = J;
        double r1452896 = l;
        double r1452897 = exp(r1452896);
        double r1452898 = -r1452896;
        double r1452899 = exp(r1452898);
        double r1452900 = r1452897 - r1452899;
        double r1452901 = r1452895 * r1452900;
        double r1452902 = K;
        double r1452903 = 2.0;
        double r1452904 = r1452902 / r1452903;
        double r1452905 = cos(r1452904);
        double r1452906 = r1452901 * r1452905;
        double r1452907 = U;
        double r1452908 = r1452906 + r1452907;
        return r1452908;
}
↓
double f(double J, double l, double K, double U) {
        double r1452909 = U;
        double r1452910 = K;
        double r1452911 = 2.0;
        double r1452912 = r1452910 / r1452911;
        double r1452913 = cos(r1452912);
        double r1452914 = 0.016666666666666666;
        double r1452915 = l;
        double r1452916 = 5.0;
        double r1452917 = pow(r1452915, r1452916);
        double r1452918 = r1452915 * r1452915;
        double r1452919 = r1452915 * r1452918;
        double r1452920 = 0.3333333333333333;
        double r1452921 = r1452915 + r1452915;
        double r1452922 = fma(r1452919, r1452920, r1452921);
        double r1452923 = fma(r1452914, r1452917, r1452922);
        double r1452924 = r1452913 * r1452923;
        double r1452925 = J;
        double r1452926 = r1452924 * r1452925;
        double r1452927 = r1452909 + r1452926;
        return r1452927;
}

Error

Bits error versus J

Bits error versus l

Bits error versus K

Bits error versus U

Derivation

  1. Initial program 17.1

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

  2. Taylor expanded around 0 0.3

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

  3. Simplified0.3

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

  5. Applied associate-*l*0.3

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

  6. Final simplification0.3

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

Reproduce

herbie shell --seed 2019152 +o rules:numerics
(FPCore (J l K U)
  :name "Maksimov and Kolovsky, Equation (4)"
  (+ (* (* J (- (exp l) (exp (- l)))) (cos (/ K 2))) U))