Average Error: 17.4 → 0.4
Time: 10.0s
Precision: 64
\[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U\]
↓
\[J \cdot \left(\mathsf{fma}\left(\frac{1}{3}, {\ell}^{3}, \mathsf{fma}\left(\frac{1}{60}, {\ell}^{5}, 2 \cdot \ell\right)\right) \cdot \cos \left(\frac{K}{2}\right)\right) + U\]
\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U
↓
J \cdot \left(\mathsf{fma}\left(\frac{1}{3}, {\ell}^{3}, \mathsf{fma}\left(\frac{1}{60}, {\ell}^{5}, 2 \cdot \ell\right)\right) \cdot \cos \left(\frac{K}{2}\right)\right) + U
double f(double J, double l, double K, double U) {
        double r113844 = J;
        double r113845 = l;
        double r113846 = exp(r113845);
        double r113847 = -r113845;
        double r113848 = exp(r113847);
        double r113849 = r113846 - r113848;
        double r113850 = r113844 * r113849;
        double r113851 = K;
        double r113852 = 2.0;
        double r113853 = r113851 / r113852;
        double r113854 = cos(r113853);
        double r113855 = r113850 * r113854;
        double r113856 = U;
        double r113857 = r113855 + r113856;
        return r113857;
}
↓
double f(double J, double l, double K, double U) {
        double r113858 = J;
        double r113859 = 0.3333333333333333;
        double r113860 = l;
        double r113861 = 3.0;
        double r113862 = pow(r113860, r113861);
        double r113863 = 0.016666666666666666;
        double r113864 = 5.0;
        double r113865 = pow(r113860, r113864);
        double r113866 = 2.0;
        double r113867 = r113866 * r113860;
        double r113868 = fma(r113863, r113865, r113867);
        double r113869 = fma(r113859, r113862, r113868);
        double r113870 = K;
        double r113871 = 2.0;
        double r113872 = r113870 / r113871;
        double r113873 = cos(r113872);
        double r113874 = r113869 * r113873;
        double r113875 = r113858 * r113874;
        double r113876 = U;
        double r113877 = r113875 + r113876;
        return r113877;
}

Error

Bits error versus J

Bits error versus l

Bits error versus K

Bits error versus U

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

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

  3. Simplified0.4

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

  5. Applied associate-*l*0.4

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

  6. Final simplification0.4

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

Reproduce

herbie shell --seed 2020036 +o rules:numerics
(FPCore (J l K U)
  :name "Maksimov and Kolovsky, Equation (4)"
  :precision binary64
  (+ (* (* J (- (exp l) (exp (- l)))) (cos (/ K 2))) U))