Average Error: 17.4 → 0.5
Time: 7.4s
Precision: 64
\[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U\]
↓
\[\mathsf{fma}\left(J \cdot \mathsf{fma}\left(\frac{1}{3}, {\ell}^{3}, \mathsf{fma}\left(\frac{1}{60}, {\ell}^{5}, 2 \cdot \ell\right)\right), \cos \left(\frac{K}{2}\right), U\right)\]
\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U
↓
\mathsf{fma}\left(J \cdot \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)
double f(double J, double l, double K, double U) {
        double r163410 = J;
        double r163411 = l;
        double r163412 = exp(r163411);
        double r163413 = -r163411;
        double r163414 = exp(r163413);
        double r163415 = r163412 - r163414;
        double r163416 = r163410 * r163415;
        double r163417 = K;
        double r163418 = 2.0;
        double r163419 = r163417 / r163418;
        double r163420 = cos(r163419);
        double r163421 = r163416 * r163420;
        double r163422 = U;
        double r163423 = r163421 + r163422;
        return r163423;
}
↓
double f(double J, double l, double K, double U) {
        double r163424 = J;
        double r163425 = 0.3333333333333333;
        double r163426 = l;
        double r163427 = 3.0;
        double r163428 = pow(r163426, r163427);
        double r163429 = 0.016666666666666666;
        double r163430 = 5.0;
        double r163431 = pow(r163426, r163430);
        double r163432 = 2.0;
        double r163433 = r163432 * r163426;
        double r163434 = fma(r163429, r163431, r163433);
        double r163435 = fma(r163425, r163428, r163434);
        double r163436 = r163424 * r163435;
        double r163437 = K;
        double r163438 = 2.0;
        double r163439 = r163437 / r163438;
        double r163440 = cos(r163439);
        double r163441 = U;
        double r163442 = fma(r163436, r163440, r163441);
        return r163442;
}

Error

Bits error versus J

Bits error versus l

Bits error versus K

Bits error versus U

Derivation

  1. Initial program 17.4

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

  2. Simplified17.4

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

  3. Taylor expanded around 0 0.5

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

  4. Simplified0.5

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

  5. Final simplification0.5

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

Reproduce

herbie shell --seed 2020021 +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))