Average Error: 17.9 → 0.5
Time: 7.0s
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 r148458 = J;
        double r148459 = l;
        double r148460 = exp(r148459);
        double r148461 = -r148459;
        double r148462 = exp(r148461);
        double r148463 = r148460 - r148462;
        double r148464 = r148458 * r148463;
        double r148465 = K;
        double r148466 = 2.0;
        double r148467 = r148465 / r148466;
        double r148468 = cos(r148467);
        double r148469 = r148464 * r148468;
        double r148470 = U;
        double r148471 = r148469 + r148470;
        return r148471;
}
↓
double f(double J, double l, double K, double U) {
        double r148472 = J;
        double r148473 = 0.3333333333333333;
        double r148474 = l;
        double r148475 = 3.0;
        double r148476 = pow(r148474, r148475);
        double r148477 = 0.016666666666666666;
        double r148478 = 5.0;
        double r148479 = pow(r148474, r148478);
        double r148480 = 2.0;
        double r148481 = r148480 * r148474;
        double r148482 = fma(r148477, r148479, r148481);
        double r148483 = fma(r148473, r148476, r148482);
        double r148484 = r148472 * r148483;
        double r148485 = K;
        double r148486 = 2.0;
        double r148487 = r148485 / r148486;
        double r148488 = cos(r148487);
        double r148489 = U;
        double r148490 = fma(r148484, r148488, r148489);
        return r148490;
}

Error

Bits error versus J

Bits error versus l

Bits error versus K

Bits error versus U

Derivation

  1. Initial program 17.9

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

  2. Taylor expanded around 0 0.5

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

  3. Simplified0.5

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

  5. Applied fma-def0.5

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

  6. 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 2020001 +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))