Average Error: 17.2 → 0.4
Time: 9.2s
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 r161698 = J;
        double r161699 = l;
        double r161700 = exp(r161699);
        double r161701 = -r161699;
        double r161702 = exp(r161701);
        double r161703 = r161700 - r161702;
        double r161704 = r161698 * r161703;
        double r161705 = K;
        double r161706 = 2.0;
        double r161707 = r161705 / r161706;
        double r161708 = cos(r161707);
        double r161709 = r161704 * r161708;
        double r161710 = U;
        double r161711 = r161709 + r161710;
        return r161711;
}
↓
double f(double J, double l, double K, double U) {
        double r161712 = J;
        double r161713 = 0.3333333333333333;
        double r161714 = l;
        double r161715 = 3.0;
        double r161716 = pow(r161714, r161715);
        double r161717 = 0.016666666666666666;
        double r161718 = 5.0;
        double r161719 = pow(r161714, r161718);
        double r161720 = 2.0;
        double r161721 = r161720 * r161714;
        double r161722 = fma(r161717, r161719, r161721);
        double r161723 = fma(r161713, r161716, r161722);
        double r161724 = r161712 * r161723;
        double r161725 = K;
        double r161726 = 2.0;
        double r161727 = r161725 / r161726;
        double r161728 = cos(r161727);
        double r161729 = U;
        double r161730 = fma(r161724, r161728, r161729);
        return r161730;
}

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. Simplified17.2

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

  3. Taylor expanded around 0 0.4

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

  4. Simplified0.4

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

  5. 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 2019356 +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))