Average Error: 17.2 → 0.4
Time: 46.2s
Precision: 64
\[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U\]
↓
\[\mathsf{fma}\left(J \cdot \mathsf{fma}\left(\frac{1}{3}, {\ell}^{3}, \mathsf{fma}\left(\frac{1}{60}, {\ell}^{5}, 2 \cdot \ell\right)\right), \cos \left(\frac{K}{2}\right), U\right)\]
\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U
↓
\mathsf{fma}\left(J \cdot \mathsf{fma}\left(\frac{1}{3}, {\ell}^{3}, \mathsf{fma}\left(\frac{1}{60}, {\ell}^{5}, 2 \cdot \ell\right)\right), \cos \left(\frac{K}{2}\right), U\right)
double f(double J, double l, double K, double U) {
        double r95976 = J;
        double r95977 = l;
        double r95978 = exp(r95977);
        double r95979 = -r95977;
        double r95980 = exp(r95979);
        double r95981 = r95978 - r95980;
        double r95982 = r95976 * r95981;
        double r95983 = K;
        double r95984 = 2.0;
        double r95985 = r95983 / r95984;
        double r95986 = cos(r95985);
        double r95987 = r95982 * r95986;
        double r95988 = U;
        double r95989 = r95987 + r95988;
        return r95989;
}
↓
double f(double J, double l, double K, double U) {
        double r95990 = J;
        double r95991 = 0.3333333333333333;
        double r95992 = l;
        double r95993 = 3.0;
        double r95994 = pow(r95992, r95993);
        double r95995 = 0.016666666666666666;
        double r95996 = 5.0;
        double r95997 = pow(r95992, r95996);
        double r95998 = 2.0;
        double r95999 = r95998 * r95992;
        double r96000 = fma(r95995, r95997, r95999);
        double r96001 = fma(r95991, r95994, r96000);
        double r96002 = r95990 * r96001;
        double r96003 = K;
        double r96004 = 2.0;
        double r96005 = r96003 / r96004;
        double r96006 = cos(r96005);
        double r96007 = U;
        double r96008 = fma(r96002, r96006, r96007);
        return r96008;
}

Error

Bits error versus J

Bits error versus l

Bits error versus K

Bits error versus U

Derivation

  1. Initial program 17.2

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

  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 fma-def0.4

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

  6. Final simplification0.4

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

Reproduce

herbie shell --seed 2019325 +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))