Average Error: 17.1 → 0.4
Time: 8.0s
Precision: 64
\[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U\]
↓
\[\left(J \cdot \mathsf{fma}\left(\frac{1}{3}, {\ell}^{3}, \mathsf{fma}\left(\frac{1}{60}, {\ell}^{5}, 2 \cdot \ell\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U\]
\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U
↓
\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) + U
double f(double J, double l, double K, double U) {
        double r166260 = J;
        double r166261 = l;
        double r166262 = exp(r166261);
        double r166263 = -r166261;
        double r166264 = exp(r166263);
        double r166265 = r166262 - r166264;
        double r166266 = r166260 * r166265;
        double r166267 = K;
        double r166268 = 2.0;
        double r166269 = r166267 / r166268;
        double r166270 = cos(r166269);
        double r166271 = r166266 * r166270;
        double r166272 = U;
        double r166273 = r166271 + r166272;
        return r166273;
}
↓
double f(double J, double l, double K, double U) {
        double r166274 = J;
        double r166275 = 0.3333333333333333;
        double r166276 = l;
        double r166277 = 3.0;
        double r166278 = pow(r166276, r166277);
        double r166279 = 0.016666666666666666;
        double r166280 = 5.0;
        double r166281 = pow(r166276, r166280);
        double r166282 = 2.0;
        double r166283 = r166282 * r166276;
        double r166284 = fma(r166279, r166281, r166283);
        double r166285 = fma(r166275, r166278, r166284);
        double r166286 = r166274 * r166285;
        double r166287 = K;
        double r166288 = 2.0;
        double r166289 = r166287 / r166288;
        double r166290 = cos(r166289);
        double r166291 = r166286 * r166290;
        double r166292 = U;
        double r166293 = r166291 + r166292;
        return r166293;
}

Error

Bits error versus J

Bits error versus l

Bits error versus K

Bits error versus U

Derivation

  1. Initial program 17.1

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

  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. Final simplification0.4

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

Reproduce

herbie shell --seed 2020057 +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))