Average Error: 16.9 → 0.4
Time: 7.7s
Precision: 64
\[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U\]
↓
\[\mathsf{fma}\left(J, \mathsf{fma}\left(\frac{1}{3}, {\ell}^{3}, \mathsf{fma}\left(\frac{1}{60}, {\ell}^{5}, 2 \cdot \ell\right)\right) \cdot \cos \left(\frac{K}{2}\right), U\right)\]
\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U
↓
\mathsf{fma}\left(J, \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), U\right)
double f(double J, double l, double K, double U) {
        double r102306 = J;
        double r102307 = l;
        double r102308 = exp(r102307);
        double r102309 = -r102307;
        double r102310 = exp(r102309);
        double r102311 = r102308 - r102310;
        double r102312 = r102306 * r102311;
        double r102313 = K;
        double r102314 = 2.0;
        double r102315 = r102313 / r102314;
        double r102316 = cos(r102315);
        double r102317 = r102312 * r102316;
        double r102318 = U;
        double r102319 = r102317 + r102318;
        return r102319;
}
↓
double f(double J, double l, double K, double U) {
        double r102320 = J;
        double r102321 = 0.3333333333333333;
        double r102322 = l;
        double r102323 = 3.0;
        double r102324 = pow(r102322, r102323);
        double r102325 = 0.016666666666666666;
        double r102326 = 5.0;
        double r102327 = pow(r102322, r102326);
        double r102328 = 2.0;
        double r102329 = r102328 * r102322;
        double r102330 = fma(r102325, r102327, r102329);
        double r102331 = fma(r102321, r102324, r102330);
        double r102332 = K;
        double r102333 = 2.0;
        double r102334 = r102332 / r102333;
        double r102335 = cos(r102334);
        double r102336 = r102331 * r102335;
        double r102337 = U;
        double r102338 = fma(r102320, r102336, r102337);
        return r102338;
}

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. Simplified16.9

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

  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. Using strategy rm

  10. Applied fma-def0.4

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

  11. Final simplification0.4

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

Reproduce

herbie shell --seed 2020056 +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))