Average Error: 16.9 → 0.3
Time: 36.6s
Precision: 64
\[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U\]
↓
\[J \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\ell \cdot \ell, \frac{1}{3}, 2\right), \ell, \left(\left(\left(\ell \cdot \ell\right) \cdot \ell\right) \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{1}{60}\right) \cdot \cos \left(\frac{K}{2}\right)\right) + U\]
\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U
↓
J \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\ell \cdot \ell, \frac{1}{3}, 2\right), \ell, \left(\left(\left(\ell \cdot \ell\right) \cdot \ell\right) \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{1}{60}\right) \cdot \cos \left(\frac{K}{2}\right)\right) + U
double f(double J, double l, double K, double U) {
        double r2747243 = J;
        double r2747244 = l;
        double r2747245 = exp(r2747244);
        double r2747246 = -r2747244;
        double r2747247 = exp(r2747246);
        double r2747248 = r2747245 - r2747247;
        double r2747249 = r2747243 * r2747248;
        double r2747250 = K;
        double r2747251 = 2.0;
        double r2747252 = r2747250 / r2747251;
        double r2747253 = cos(r2747252);
        double r2747254 = r2747249 * r2747253;
        double r2747255 = U;
        double r2747256 = r2747254 + r2747255;
        return r2747256;
}
↓
double f(double J, double l, double K, double U) {
        double r2747257 = J;
        double r2747258 = l;
        double r2747259 = r2747258 * r2747258;
        double r2747260 = 0.3333333333333333;
        double r2747261 = 2.0;
        double r2747262 = fma(r2747259, r2747260, r2747261);
        double r2747263 = r2747259 * r2747258;
        double r2747264 = r2747263 * r2747259;
        double r2747265 = 0.016666666666666666;
        double r2747266 = r2747264 * r2747265;
        double r2747267 = fma(r2747262, r2747258, r2747266);
        double r2747268 = K;
        double r2747269 = r2747268 / r2747261;
        double r2747270 = cos(r2747269);
        double r2747271 = r2747267 * r2747270;
        double r2747272 = r2747257 * r2747271;
        double r2747273 = U;
        double r2747274 = r2747272 + r2747273;
        return r2747274;
}

Error

Bits error versus J

Bits error versus l

Bits error versus K

Bits error versus U

Derivation

  1. Initial program 16.9

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

  2. Taylor expanded around 0 0.3

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

  3. Simplified0.3

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

  5. Applied associate-*l*0.3

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

  6. Simplified0.3

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

  7. Final simplification0.3

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

Reproduce

herbie shell --seed 2019138 +o rules:numerics
(FPCore (J l K U)
  :name "Maksimov and Kolovsky, Equation (4)"
  (+ (* (* J (- (exp l) (exp (- l)))) (cos (/ K 2))) U))