Average Error: 17.6 → 0.4
Time: 51.1s
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 r77374 = J;
        double r77375 = l;
        double r77376 = exp(r77375);
        double r77377 = -r77375;
        double r77378 = exp(r77377);
        double r77379 = r77376 - r77378;
        double r77380 = r77374 * r77379;
        double r77381 = K;
        double r77382 = 2.0;
        double r77383 = r77381 / r77382;
        double r77384 = cos(r77383);
        double r77385 = r77380 * r77384;
        double r77386 = U;
        double r77387 = r77385 + r77386;
        return r77387;
}
↓
double f(double J, double l, double K, double U) {
        double r77388 = J;
        double r77389 = 0.3333333333333333;
        double r77390 = l;
        double r77391 = 3.0;
        double r77392 = pow(r77390, r77391);
        double r77393 = 0.016666666666666666;
        double r77394 = 5.0;
        double r77395 = pow(r77390, r77394);
        double r77396 = 2.0;
        double r77397 = r77396 * r77390;
        double r77398 = fma(r77393, r77395, r77397);
        double r77399 = fma(r77389, r77392, r77398);
        double r77400 = r77388 * r77399;
        double r77401 = K;
        double r77402 = 2.0;
        double r77403 = r77401 / r77402;
        double r77404 = cos(r77403);
        double r77405 = U;
        double r77406 = fma(r77400, r77404, r77405);
        return r77406;
}

Error

Bits error versus J

Bits error versus l

Bits error versus K

Bits error versus U

Derivation

  1. Initial program 17.6

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

  2. Simplified17.6

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

  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 2019303 +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))