Average Error: 17.3 → 0.4
Time: 50.4s
Precision: 64
\[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U\]
↓
\[J \cdot \left(\mathsf{fma}\left(\frac{1}{3}, {\ell}^{3}, \mathsf{fma}\left(\frac{1}{60}, {\ell}^{5}, 2 \cdot \ell\right)\right) \cdot \cos \left(\frac{K}{2}\right)\right) + U\]
\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U
↓
J \cdot \left(\mathsf{fma}\left(\frac{1}{3}, {\ell}^{3}, \mathsf{fma}\left(\frac{1}{60}, {\ell}^{5}, 2 \cdot \ell\right)\right) \cdot \cos \left(\frac{K}{2}\right)\right) + U
double f(double J, double l, double K, double U) {
        double r73493 = J;
        double r73494 = l;
        double r73495 = exp(r73494);
        double r73496 = -r73494;
        double r73497 = exp(r73496);
        double r73498 = r73495 - r73497;
        double r73499 = r73493 * r73498;
        double r73500 = K;
        double r73501 = 2.0;
        double r73502 = r73500 / r73501;
        double r73503 = cos(r73502);
        double r73504 = r73499 * r73503;
        double r73505 = U;
        double r73506 = r73504 + r73505;
        return r73506;
}
↓
double f(double J, double l, double K, double U) {
        double r73507 = J;
        double r73508 = 0.3333333333333333;
        double r73509 = l;
        double r73510 = 3.0;
        double r73511 = pow(r73509, r73510);
        double r73512 = 0.016666666666666666;
        double r73513 = 5.0;
        double r73514 = pow(r73509, r73513);
        double r73515 = 2.0;
        double r73516 = r73515 * r73509;
        double r73517 = fma(r73512, r73514, r73516);
        double r73518 = fma(r73508, r73511, r73517);
        double r73519 = K;
        double r73520 = 2.0;
        double r73521 = r73519 / r73520;
        double r73522 = cos(r73521);
        double r73523 = r73518 * r73522;
        double r73524 = r73507 * r73523;
        double r73525 = U;
        double r73526 = r73524 + r73525;
        return r73526;
}

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. Taylor expanded around 0 0.4

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

  3. Simplified0.4

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

  5. Applied associate-*l*0.4

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

  6. Final simplification0.4

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

Reproduce

herbie shell --seed 2019306 +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))