Average Error: 16.7 → 0.4
Time: 51.1s
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 r8303950 = J;
        double r8303951 = l;
        double r8303952 = exp(r8303951);
        double r8303953 = -r8303951;
        double r8303954 = exp(r8303953);
        double r8303955 = r8303952 - r8303954;
        double r8303956 = r8303950 * r8303955;
        double r8303957 = K;
        double r8303958 = 2.0;
        double r8303959 = r8303957 / r8303958;
        double r8303960 = cos(r8303959);
        double r8303961 = r8303956 * r8303960;
        double r8303962 = U;
        double r8303963 = r8303961 + r8303962;
        return r8303963;
}
↓
double f(double J, double l, double K, double U) {
        double r8303964 = J;
        double r8303965 = 0.016666666666666666;
        double r8303966 = l;
        double r8303967 = 5.0;
        double r8303968 = pow(r8303966, r8303967);
        double r8303969 = 0.3333333333333333;
        double r8303970 = r8303966 * r8303966;
        double r8303971 = 2.0;
        double r8303972 = fma(r8303969, r8303970, r8303971);
        double r8303973 = r8303966 * r8303972;
        double r8303974 = fma(r8303965, r8303968, r8303973);
        double r8303975 = K;
        double r8303976 = r8303975 / r8303971;
        double r8303977 = cos(r8303976);
        double r8303978 = r8303974 * r8303977;
        double r8303979 = r8303964 * r8303978;
        double r8303980 = U;
        double r8303981 = r8303979 + r8303980;
        return r8303981;
}

Error

Bits error versus J

Bits error versus l

Bits error versus K

Bits error versus U

Derivation

  1. Initial program 16.7

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

  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 2019119 +o rules:numerics
(FPCore (J l K U)
  :name "Maksimov and Kolovsky, Equation (4)"
  (+ (* (* J (- (exp l) (exp (- l)))) (cos (/ K 2))) U))