Average Error: 17.1 → 0.4
Time: 50.7s
Precision: 64
\[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U\]
↓
\[\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot (\frac{1}{60} \cdot \left({\ell}^{5}\right) + \left(\ell \cdot (\frac{1}{3} \cdot \left(\ell \cdot \ell\right) + 2)_*\right))_*\right) + U\]
double f(double J, double l, double K, double U) {
        double r8939814 = J;
        double r8939815 = l;
        double r8939816 = exp(r8939815);
        double r8939817 = -r8939815;
        double r8939818 = exp(r8939817);
        double r8939819 = r8939816 - r8939818;
        double r8939820 = r8939814 * r8939819;
        double r8939821 = K;
        double r8939822 = 2.0;
        double r8939823 = r8939821 / r8939822;
        double r8939824 = cos(r8939823);
        double r8939825 = r8939820 * r8939824;
        double r8939826 = U;
        double r8939827 = r8939825 + r8939826;
        return r8939827;
}
↓
double f(double J, double l, double K, double U) {
        double r8939828 = K;
        double r8939829 = 2.0;
        double r8939830 = r8939828 / r8939829;
        double r8939831 = cos(r8939830);
        double r8939832 = J;
        double r8939833 = 0.016666666666666666;
        double r8939834 = l;
        double r8939835 = 5.0;
        double r8939836 = pow(r8939834, r8939835);
        double r8939837 = 0.3333333333333333;
        double r8939838 = r8939834 * r8939834;
        double r8939839 = fma(r8939837, r8939838, r8939829);
        double r8939840 = r8939834 * r8939839;
        double r8939841 = fma(r8939833, r8939836, r8939840);
        double r8939842 = r8939832 * r8939841;
        double r8939843 = r8939831 * r8939842;
        double r8939844 = U;
        double r8939845 = r8939843 + r8939844;
        return r8939845;
}


\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U
↓
\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot (\frac{1}{60} \cdot \left({\ell}^{5}\right) + \left(\ell \cdot (\frac{1}{3} \cdot \left(\ell \cdot \ell\right) + 2)_*\right))_*\right) + U

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 0.4

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

  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. Final simplification0.4

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

Reproduce

herbie shell --seed 2019101 +o rules:numerics
(FPCore (J l K U)
  :name "Maksimov and Kolovsky, Equation (4)"
  (+ (* (* J (- (exp l) (exp (- l)))) (cos (/ K 2))) U))