Average Error: 17.5 → 0.4
Time: 9.2s
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 r169487 = J;
        double r169488 = l;
        double r169489 = exp(r169488);
        double r169490 = -r169488;
        double r169491 = exp(r169490);
        double r169492 = r169489 - r169491;
        double r169493 = r169487 * r169492;
        double r169494 = K;
        double r169495 = 2.0;
        double r169496 = r169494 / r169495;
        double r169497 = cos(r169496);
        double r169498 = r169493 * r169497;
        double r169499 = U;
        double r169500 = r169498 + r169499;
        return r169500;
}
↓
double f(double J, double l, double K, double U) {
        double r169501 = J;
        double r169502 = 0.3333333333333333;
        double r169503 = l;
        double r169504 = 3.0;
        double r169505 = pow(r169503, r169504);
        double r169506 = 0.016666666666666666;
        double r169507 = 5.0;
        double r169508 = pow(r169503, r169507);
        double r169509 = 2.0;
        double r169510 = r169509 * r169503;
        double r169511 = fma(r169506, r169508, r169510);
        double r169512 = fma(r169502, r169505, r169511);
        double r169513 = r169501 * r169512;
        double r169514 = K;
        double r169515 = 2.0;
        double r169516 = r169514 / r169515;
        double r169517 = cos(r169516);
        double r169518 = U;
        double r169519 = fma(r169513, r169517, r169518);
        return r169519;
}

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. Final simplification0.4

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

Reproduce

herbie shell --seed 2019353 +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))