Average Error: 17.6 → 0.7
Time: 22.7s
Precision: 64
\[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U\]
↓
\[\mathsf{fma}\left(\left(\ell \cdot J\right) \cdot 2, \cos \left(\frac{K}{2}\right), U\right)\]
\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U
↓
\mathsf{fma}\left(\left(\ell \cdot J\right) \cdot 2, \cos \left(\frac{K}{2}\right), U\right)
double f(double J, double l, double K, double U) {
        double r3592275 = J;
        double r3592276 = l;
        double r3592277 = exp(r3592276);
        double r3592278 = -r3592276;
        double r3592279 = exp(r3592278);
        double r3592280 = r3592277 - r3592279;
        double r3592281 = r3592275 * r3592280;
        double r3592282 = K;
        double r3592283 = 2.0;
        double r3592284 = r3592282 / r3592283;
        double r3592285 = cos(r3592284);
        double r3592286 = r3592281 * r3592285;
        double r3592287 = U;
        double r3592288 = r3592286 + r3592287;
        return r3592288;
}
↓
double f(double J, double l, double K, double U) {
        double r3592289 = l;
        double r3592290 = J;
        double r3592291 = r3592289 * r3592290;
        double r3592292 = 2.0;
        double r3592293 = r3592291 * r3592292;
        double r3592294 = K;
        double r3592295 = 2.0;
        double r3592296 = r3592294 / r3592295;
        double r3592297 = cos(r3592296);
        double r3592298 = U;
        double r3592299 = fma(r3592293, r3592297, r3592298);
        return r3592299;
}

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

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

  3. Taylor expanded around 0 0.7

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

  4. Final simplification0.7

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

Reproduce

herbie shell --seed 2019171 +o rules:numerics
(FPCore (J l K U)
  :name "Maksimov and Kolovsky, Equation (4)"
  (+ (* (* J (- (exp l) (exp (- l)))) (cos (/ K 2.0))) U))