Average Error: 17.6 → 0.4
Time: 23.8s
Precision: 64
\[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U\]
↓
\[U + J \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \left(\left(\left(\ell + \ell\right) + {\ell}^{3} \cdot \frac{1}{3}\right) + {\ell}^{5} \cdot \frac{1}{60}\right)\right)\]
\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U
↓
U + J \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \left(\left(\left(\ell + \ell\right) + {\ell}^{3} \cdot \frac{1}{3}\right) + {\ell}^{5} \cdot \frac{1}{60}\right)\right)
double f(double J, double l, double K, double U) {
        double r84976 = J;
        double r84977 = l;
        double r84978 = exp(r84977);
        double r84979 = -r84977;
        double r84980 = exp(r84979);
        double r84981 = r84978 - r84980;
        double r84982 = r84976 * r84981;
        double r84983 = K;
        double r84984 = 2.0;
        double r84985 = r84983 / r84984;
        double r84986 = cos(r84985);
        double r84987 = r84982 * r84986;
        double r84988 = U;
        double r84989 = r84987 + r84988;
        return r84989;
}
↓
double f(double J, double l, double K, double U) {
        double r84990 = U;
        double r84991 = J;
        double r84992 = K;
        double r84993 = 2.0;
        double r84994 = r84992 / r84993;
        double r84995 = cos(r84994);
        double r84996 = l;
        double r84997 = r84996 + r84996;
        double r84998 = 3.0;
        double r84999 = pow(r84996, r84998);
        double r85000 = 0.3333333333333333;
        double r85001 = r84999 * r85000;
        double r85002 = r84997 + r85001;
        double r85003 = 5.0;
        double r85004 = pow(r84996, r85003);
        double r85005 = 0.016666666666666666;
        double r85006 = r85004 * r85005;
        double r85007 = r85002 + r85006;
        double r85008 = r84995 * r85007;
        double r85009 = r84991 * r85008;
        double r85010 = r84990 + r85009;
        return r85010;
}

Error

Bits error versus J

Bits error versus l

Bits error versus K

Bits error versus U

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

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. Simplified17.6

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

  3. Taylor expanded around 0 0.4

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

  4. Simplified0.4

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

  5. Final simplification0.4

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

Reproduce

herbie shell --seed 2019174 
(FPCore (J l K U)
  :name "Maksimov and Kolovsky, Equation (4)"
  (+ (* (* J (- (exp l) (exp (- l)))) (cos (/ K 2.0))) U))