Average Error: 17.3 → 0.4
Time: 22.6s
Precision: 64
\[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U\]
↓
\[U + J \cdot \left(\mathsf{fma}\left({\ell}^{3}, \frac{1}{3}, \mathsf{fma}\left(\frac{1}{60}, {\ell}^{5}, 2 \cdot \ell\right)\right) \cdot \cos \left(\frac{K}{2}\right)\right)\]
\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U
↓
U + J \cdot \left(\mathsf{fma}\left({\ell}^{3}, \frac{1}{3}, \mathsf{fma}\left(\frac{1}{60}, {\ell}^{5}, 2 \cdot \ell\right)\right) \cdot \cos \left(\frac{K}{2}\right)\right)
double f(double J, double l, double K, double U) {
        double r79212 = J;
        double r79213 = l;
        double r79214 = exp(r79213);
        double r79215 = -r79213;
        double r79216 = exp(r79215);
        double r79217 = r79214 - r79216;
        double r79218 = r79212 * r79217;
        double r79219 = K;
        double r79220 = 2.0;
        double r79221 = r79219 / r79220;
        double r79222 = cos(r79221);
        double r79223 = r79218 * r79222;
        double r79224 = U;
        double r79225 = r79223 + r79224;
        return r79225;
}
↓
double f(double J, double l, double K, double U) {
        double r79226 = U;
        double r79227 = J;
        double r79228 = l;
        double r79229 = 3.0;
        double r79230 = pow(r79228, r79229);
        double r79231 = 0.3333333333333333;
        double r79232 = 0.016666666666666666;
        double r79233 = 5.0;
        double r79234 = pow(r79228, r79233);
        double r79235 = 2.0;
        double r79236 = r79235 * r79228;
        double r79237 = fma(r79232, r79234, r79236);
        double r79238 = fma(r79230, r79231, r79237);
        double r79239 = K;
        double r79240 = 2.0;
        double r79241 = r79239 / r79240;
        double r79242 = cos(r79241);
        double r79243 = r79238 * r79242;
        double r79244 = r79227 * r79243;
        double r79245 = r79226 + r79244;
        return r79245;
}

Error

Bits error versus J

Bits error versus l

Bits error versus K

Bits error versus U

Derivation

  1. Initial program 17.3

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

  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 pow10.4

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

  6. Applied pow10.4

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

  7. Applied pow10.4

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

  8. Applied pow-prod-down0.4

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

  9. Applied pow-prod-down0.4

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

  10. Simplified0.4

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

  11. Final simplification0.4

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

Reproduce

herbie shell --seed 2019208 +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))