Average Error: 17.5 → 0.4
Time: 8.1s
Precision: 64
\[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U\]
↓
\[J \cdot \left(\mathsf{fma}\left(\frac{1}{3}, {\ell}^{3}, \mathsf{fma}\left(\frac{1}{60}, {\ell}^{5}, 2 \cdot \ell\right)\right) \cdot \cos \left(\frac{K}{2}\right)\right) + U\]
\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U
↓
J \cdot \left(\mathsf{fma}\left(\frac{1}{3}, {\ell}^{3}, \mathsf{fma}\left(\frac{1}{60}, {\ell}^{5}, 2 \cdot \ell\right)\right) \cdot \cos \left(\frac{K}{2}\right)\right) + U
double f(double J, double l, double K, double U) {
        double r113275 = J;
        double r113276 = l;
        double r113277 = exp(r113276);
        double r113278 = -r113276;
        double r113279 = exp(r113278);
        double r113280 = r113277 - r113279;
        double r113281 = r113275 * r113280;
        double r113282 = K;
        double r113283 = 2.0;
        double r113284 = r113282 / r113283;
        double r113285 = cos(r113284);
        double r113286 = r113281 * r113285;
        double r113287 = U;
        double r113288 = r113286 + r113287;
        return r113288;
}
↓
double f(double J, double l, double K, double U) {
        double r113289 = J;
        double r113290 = 0.3333333333333333;
        double r113291 = l;
        double r113292 = 3.0;
        double r113293 = pow(r113291, r113292);
        double r113294 = 0.016666666666666666;
        double r113295 = 5.0;
        double r113296 = pow(r113291, r113295);
        double r113297 = 2.0;
        double r113298 = r113297 * r113291;
        double r113299 = fma(r113294, r113296, r113298);
        double r113300 = fma(r113290, r113293, r113299);
        double r113301 = K;
        double r113302 = 2.0;
        double r113303 = r113301 / r113302;
        double r113304 = cos(r113303);
        double r113305 = r113300 * r113304;
        double r113306 = r113289 * r113305;
        double r113307 = U;
        double r113308 = r113306 + r113307;
        return r113308;
}

Error

Bits error versus J

Bits error versus l

Bits error versus K

Bits error versus U

Derivation

  1. Initial program 17.5

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

  2. Simplified17.5

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

  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. Using strategy rm

  6. Applied fma-udef0.4

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

  8. Applied associate-*l*0.4

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

  9. Final simplification0.4

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

Reproduce

herbie shell --seed 2019347 +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))