Average Error: 17.3 → 0.3
Time: 34.0s
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 r75466 = J;
        double r75467 = l;
        double r75468 = exp(r75467);
        double r75469 = -r75467;
        double r75470 = exp(r75469);
        double r75471 = r75468 - r75470;
        double r75472 = r75466 * r75471;
        double r75473 = K;
        double r75474 = 2.0;
        double r75475 = r75473 / r75474;
        double r75476 = cos(r75475);
        double r75477 = r75472 * r75476;
        double r75478 = U;
        double r75479 = r75477 + r75478;
        return r75479;
}
↓
double f(double J, double l, double K, double U) {
        double r75480 = J;
        double r75481 = 0.3333333333333333;
        double r75482 = l;
        double r75483 = 3.0;
        double r75484 = pow(r75482, r75483);
        double r75485 = 0.016666666666666666;
        double r75486 = 5.0;
        double r75487 = pow(r75482, r75486);
        double r75488 = 2.0;
        double r75489 = r75488 * r75482;
        double r75490 = fma(r75485, r75487, r75489);
        double r75491 = fma(r75481, r75484, r75490);
        double r75492 = r75480 * r75491;
        double r75493 = K;
        double r75494 = 2.0;
        double r75495 = r75493 / r75494;
        double r75496 = cos(r75495);
        double r75497 = U;
        double r75498 = fma(r75492, r75496, r75497);
        return r75498;
}

Error

Bits error versus J

Bits error versus l

Bits error versus K

Bits error versus U

Derivation

  1. Initial program 17.3

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

  2. Simplified17.3

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

  3. Taylor expanded around 0 0.3

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

  4. Simplified0.3

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

  5. Final simplification0.3

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

Reproduce

herbie shell --seed 2019199 +o rules:numerics
(FPCore (J l K U)
  :name "Maksimov and Kolovsky, Equation (4)"
  (+ (* (* J (- (exp l) (exp (- l)))) (cos (/ K 2.0))) U))