Average Error: 17.4 → 0.4
Time: 10.4s
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 r207560 = J;
        double r207561 = l;
        double r207562 = exp(r207561);
        double r207563 = -r207561;
        double r207564 = exp(r207563);
        double r207565 = r207562 - r207564;
        double r207566 = r207560 * r207565;
        double r207567 = K;
        double r207568 = 2.0;
        double r207569 = r207567 / r207568;
        double r207570 = cos(r207569);
        double r207571 = r207566 * r207570;
        double r207572 = U;
        double r207573 = r207571 + r207572;
        return r207573;
}
↓
double f(double J, double l, double K, double U) {
        double r207574 = J;
        double r207575 = 0.3333333333333333;
        double r207576 = l;
        double r207577 = 3.0;
        double r207578 = pow(r207576, r207577);
        double r207579 = 0.016666666666666666;
        double r207580 = 5.0;
        double r207581 = pow(r207576, r207580);
        double r207582 = 2.0;
        double r207583 = r207582 * r207576;
        double r207584 = fma(r207579, r207581, r207583);
        double r207585 = fma(r207575, r207578, r207584);
        double r207586 = r207574 * r207585;
        double r207587 = K;
        double r207588 = 2.0;
        double r207589 = r207587 / r207588;
        double r207590 = cos(r207589);
        double r207591 = U;
        double r207592 = fma(r207586, r207590, r207591);
        return r207592;
}

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 0.4

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

  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 2020018 +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))