Average Error: 17.5 → 0.4
Time: 9.6s
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 r160228 = J;
        double r160229 = l;
        double r160230 = exp(r160229);
        double r160231 = -r160229;
        double r160232 = exp(r160231);
        double r160233 = r160230 - r160232;
        double r160234 = r160228 * r160233;
        double r160235 = K;
        double r160236 = 2.0;
        double r160237 = r160235 / r160236;
        double r160238 = cos(r160237);
        double r160239 = r160234 * r160238;
        double r160240 = U;
        double r160241 = r160239 + r160240;
        return r160241;
}
↓
double f(double J, double l, double K, double U) {
        double r160242 = J;
        double r160243 = 0.3333333333333333;
        double r160244 = l;
        double r160245 = 3.0;
        double r160246 = pow(r160244, r160245);
        double r160247 = 0.016666666666666666;
        double r160248 = 5.0;
        double r160249 = pow(r160244, r160248);
        double r160250 = 2.0;
        double r160251 = r160250 * r160244;
        double r160252 = fma(r160247, r160249, r160251);
        double r160253 = fma(r160243, r160246, r160252);
        double r160254 = K;
        double r160255 = 2.0;
        double r160256 = r160254 / r160255;
        double r160257 = cos(r160256);
        double r160258 = r160253 * r160257;
        double r160259 = r160242 * r160258;
        double r160260 = U;
        double r160261 = r160259 + r160260;
        return r160261;
}

Error

Bits error versus J

Bits error versus l

Bits error versus K

Bits error versus U

Derivation

  1. Initial program 17.5

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

  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 2020035 +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))