Average Error: 17.4 → 0.4
Time: 8.4s
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 r126694 = J;
        double r126695 = l;
        double r126696 = exp(r126695);
        double r126697 = -r126695;
        double r126698 = exp(r126697);
        double r126699 = r126696 - r126698;
        double r126700 = r126694 * r126699;
        double r126701 = K;
        double r126702 = 2.0;
        double r126703 = r126701 / r126702;
        double r126704 = cos(r126703);
        double r126705 = r126700 * r126704;
        double r126706 = U;
        double r126707 = r126705 + r126706;
        return r126707;
}
↓
double f(double J, double l, double K, double U) {
        double r126708 = J;
        double r126709 = 0.3333333333333333;
        double r126710 = l;
        double r126711 = 3.0;
        double r126712 = pow(r126710, r126711);
        double r126713 = 0.016666666666666666;
        double r126714 = 5.0;
        double r126715 = pow(r126710, r126714);
        double r126716 = 2.0;
        double r126717 = r126716 * r126710;
        double r126718 = fma(r126713, r126715, r126717);
        double r126719 = fma(r126709, r126712, r126718);
        double r126720 = K;
        double r126721 = 2.0;
        double r126722 = r126720 / r126721;
        double r126723 = cos(r126722);
        double r126724 = r126719 * r126723;
        double r126725 = r126708 * r126724;
        double r126726 = U;
        double r126727 = r126725 + r126726;
        return r126727;
}

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))