Average Error: 17.4 → 0.4
Time: 37.6s
Precision: 64
\[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U\]
↓
\[U + J \cdot \left(\left(\frac{1}{3} \cdot \left(\ell \cdot \left(\ell \cdot \ell\right)\right) + \left(\ell \cdot 2 + \left(\left(\ell \cdot \ell\right) \cdot \left(\ell \cdot \left(\ell \cdot \ell\right)\right)\right) \cdot \frac{1}{60}\right)\right) \cdot \cos \left(\frac{K}{2}\right)\right)\]
\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U
↓
U + J \cdot \left(\left(\frac{1}{3} \cdot \left(\ell \cdot \left(\ell \cdot \ell\right)\right) + \left(\ell \cdot 2 + \left(\left(\ell \cdot \ell\right) \cdot \left(\ell \cdot \left(\ell \cdot \ell\right)\right)\right) \cdot \frac{1}{60}\right)\right) \cdot \cos \left(\frac{K}{2}\right)\right)
double f(double J, double l, double K, double U) {
        double r3115188 = J;
        double r3115189 = l;
        double r3115190 = exp(r3115189);
        double r3115191 = -r3115189;
        double r3115192 = exp(r3115191);
        double r3115193 = r3115190 - r3115192;
        double r3115194 = r3115188 * r3115193;
        double r3115195 = K;
        double r3115196 = 2.0;
        double r3115197 = r3115195 / r3115196;
        double r3115198 = cos(r3115197);
        double r3115199 = r3115194 * r3115198;
        double r3115200 = U;
        double r3115201 = r3115199 + r3115200;
        return r3115201;
}
↓
double f(double J, double l, double K, double U) {
        double r3115202 = U;
        double r3115203 = J;
        double r3115204 = 0.3333333333333333;
        double r3115205 = l;
        double r3115206 = r3115205 * r3115205;
        double r3115207 = r3115205 * r3115206;
        double r3115208 = r3115204 * r3115207;
        double r3115209 = 2.0;
        double r3115210 = r3115205 * r3115209;
        double r3115211 = r3115206 * r3115207;
        double r3115212 = 0.016666666666666666;
        double r3115213 = r3115211 * r3115212;
        double r3115214 = r3115210 + r3115213;
        double r3115215 = r3115208 + r3115214;
        double r3115216 = K;
        double r3115217 = r3115216 / r3115209;
        double r3115218 = cos(r3115217);
        double r3115219 = r3115215 * r3115218;
        double r3115220 = r3115203 * r3115219;
        double r3115221 = r3115202 + r3115220;
        return r3115221;
}

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 17.4

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

  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

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

  5. Applied associate-*l*0.4

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

  6. Simplified0.4

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

  7. Final simplification0.4

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

Reproduce

herbie shell --seed 2019139 
(FPCore (J l K U)
  :name "Maksimov and Kolovsky, Equation (4)"
  (+ (* (* J (- (exp l) (exp (- l)))) (cos (/ K 2))) U))