Average Error: 17.6 → 0.4
Time: 25.4s
Precision: 64
\[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U\]
↓
\[\mathsf{fma}\left(\cos \left(\frac{K}{2}\right) \cdot J, \mathsf{fma}\left({\ell}^{5}, \frac{1}{60}, \mathsf{fma}\left(\frac{1}{3}, {\ell}^{3}, \ell + \ell\right)\right), U\right)\]
\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U
↓
\mathsf{fma}\left(\cos \left(\frac{K}{2}\right) \cdot J, \mathsf{fma}\left({\ell}^{5}, \frac{1}{60}, \mathsf{fma}\left(\frac{1}{3}, {\ell}^{3}, \ell + \ell\right)\right), U\right)
double f(double J, double l, double K, double U) {
        double r86868 = J;
        double r86869 = l;
        double r86870 = exp(r86869);
        double r86871 = -r86869;
        double r86872 = exp(r86871);
        double r86873 = r86870 - r86872;
        double r86874 = r86868 * r86873;
        double r86875 = K;
        double r86876 = 2.0;
        double r86877 = r86875 / r86876;
        double r86878 = cos(r86877);
        double r86879 = r86874 * r86878;
        double r86880 = U;
        double r86881 = r86879 + r86880;
        return r86881;
}
↓
double f(double J, double l, double K, double U) {
        double r86882 = K;
        double r86883 = 2.0;
        double r86884 = r86882 / r86883;
        double r86885 = cos(r86884);
        double r86886 = J;
        double r86887 = r86885 * r86886;
        double r86888 = l;
        double r86889 = 5.0;
        double r86890 = pow(r86888, r86889);
        double r86891 = 0.016666666666666666;
        double r86892 = 0.3333333333333333;
        double r86893 = 3.0;
        double r86894 = pow(r86888, r86893);
        double r86895 = r86888 + r86888;
        double r86896 = fma(r86892, r86894, r86895);
        double r86897 = fma(r86890, r86891, r86896);
        double r86898 = U;
        double r86899 = fma(r86887, r86897, r86898);
        return r86899;
}

Error

Bits error versus J

Bits error versus l

Bits error versus K

Bits error versus U

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

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

  3. Taylor expanded around 0 0.4

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

  4. Simplified0.4

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

  5. Final simplification0.4

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

Reproduce

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