Average Error: 17.4 → 0.3
Time: 25.2s
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 r78992 = J;
        double r78993 = l;
        double r78994 = exp(r78993);
        double r78995 = -r78993;
        double r78996 = exp(r78995);
        double r78997 = r78994 - r78996;
        double r78998 = r78992 * r78997;
        double r78999 = K;
        double r79000 = 2.0;
        double r79001 = r78999 / r79000;
        double r79002 = cos(r79001);
        double r79003 = r78998 * r79002;
        double r79004 = U;
        double r79005 = r79003 + r79004;
        return r79005;
}
↓
double f(double J, double l, double K, double U) {
        double r79006 = K;
        double r79007 = 2.0;
        double r79008 = r79006 / r79007;
        double r79009 = cos(r79008);
        double r79010 = J;
        double r79011 = r79009 * r79010;
        double r79012 = l;
        double r79013 = 5.0;
        double r79014 = pow(r79012, r79013);
        double r79015 = 0.016666666666666666;
        double r79016 = 0.3333333333333333;
        double r79017 = 3.0;
        double r79018 = pow(r79012, r79017);
        double r79019 = r79012 + r79012;
        double r79020 = fma(r79016, r79018, r79019);
        double r79021 = fma(r79014, r79015, r79020);
        double r79022 = U;
        double r79023 = fma(r79011, r79021, r79022);
        return r79023;
}

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. Simplified17.4

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

  3. Taylor expanded around 0 0.3

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

  4. Simplified0.3

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

  5. Final simplification0.3

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

Reproduce

herbie shell --seed 2019179 +o rules:numerics
(FPCore (J l K U)
  :name "Maksimov and Kolovsky, Equation (4)"
  (+ (* (* J (- (exp l) (exp (- l)))) (cos (/ K 2.0))) U))