Average Error: 16.8 → 0.4
Time: 8.2s
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 r157155 = J;
        double r157156 = l;
        double r157157 = exp(r157156);
        double r157158 = -r157156;
        double r157159 = exp(r157158);
        double r157160 = r157157 - r157159;
        double r157161 = r157155 * r157160;
        double r157162 = K;
        double r157163 = 2.0;
        double r157164 = r157162 / r157163;
        double r157165 = cos(r157164);
        double r157166 = r157161 * r157165;
        double r157167 = U;
        double r157168 = r157166 + r157167;
        return r157168;
}
↓
double f(double J, double l, double K, double U) {
        double r157169 = J;
        double r157170 = 0.3333333333333333;
        double r157171 = l;
        double r157172 = 3.0;
        double r157173 = pow(r157171, r157172);
        double r157174 = 0.016666666666666666;
        double r157175 = 5.0;
        double r157176 = pow(r157171, r157175);
        double r157177 = 2.0;
        double r157178 = r157177 * r157171;
        double r157179 = fma(r157174, r157176, r157178);
        double r157180 = fma(r157170, r157173, r157179);
        double r157181 = r157169 * r157180;
        double r157182 = K;
        double r157183 = 2.0;
        double r157184 = r157182 / r157183;
        double r157185 = cos(r157184);
        double r157186 = U;
        double r157187 = fma(r157181, r157185, r157186);
        return r157187;
}

Error

Bits error versus J

Bits error versus l

Bits error versus K

Bits error versus U

Derivation

  1. Initial program 16.8

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

  2. Simplified16.8

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

  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 2019344 +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))