Average Error: 17.9 → 0.4
Time: 52.0s
Precision: 64
\[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U\]
↓
\[J \cdot \left((\frac{1}{60} \cdot \left({\ell}^{5}\right) + \left(\ell \cdot (\frac{1}{3} \cdot \left(\ell \cdot \ell\right) + 2)_*\right))_* \cdot \cos \left(\frac{K}{2}\right)\right) + U\]
\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U
↓
J \cdot \left((\frac{1}{60} \cdot \left({\ell}^{5}\right) + \left(\ell \cdot (\frac{1}{3} \cdot \left(\ell \cdot \ell\right) + 2)_*\right))_* \cdot \cos \left(\frac{K}{2}\right)\right) + U
double f(double J, double l, double K, double U) {
        double r32403875 = J;
        double r32403876 = l;
        double r32403877 = exp(r32403876);
        double r32403878 = -r32403876;
        double r32403879 = exp(r32403878);
        double r32403880 = r32403877 - r32403879;
        double r32403881 = r32403875 * r32403880;
        double r32403882 = K;
        double r32403883 = 2.0;
        double r32403884 = r32403882 / r32403883;
        double r32403885 = cos(r32403884);
        double r32403886 = r32403881 * r32403885;
        double r32403887 = U;
        double r32403888 = r32403886 + r32403887;
        return r32403888;
}
↓
double f(double J, double l, double K, double U) {
        double r32403889 = J;
        double r32403890 = 0.016666666666666666;
        double r32403891 = l;
        double r32403892 = 5.0;
        double r32403893 = pow(r32403891, r32403892);
        double r32403894 = 0.3333333333333333;
        double r32403895 = r32403891 * r32403891;
        double r32403896 = 2.0;
        double r32403897 = fma(r32403894, r32403895, r32403896);
        double r32403898 = r32403891 * r32403897;
        double r32403899 = fma(r32403890, r32403893, r32403898);
        double r32403900 = K;
        double r32403901 = r32403900 / r32403896;
        double r32403902 = cos(r32403901);
        double r32403903 = r32403899 * r32403902;
        double r32403904 = r32403889 * r32403903;
        double r32403905 = U;
        double r32403906 = r32403904 + r32403905;
        return r32403906;
}

Error

Bits error versus J

Bits error versus l

Bits error versus K

Bits error versus U

Derivation

  1. Initial program 17.9

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

  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

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

  5. Applied associate-*l*0.4

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

  6. Final simplification0.4

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

Reproduce

herbie shell --seed 2019104 +o rules:numerics
(FPCore (J l K U)
  :name "Maksimov and Kolovsky, Equation (4)"
  (+ (* (* J (- (exp l) (exp (- l)))) (cos (/ K 2))) U))