Average Error: 17.3 → 0.4
Time: 28.1s
Precision: 64
\[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U\]
↓
\[J \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \mathsf{fma}\left(\frac{1}{60}, {\ell}^{5}, \mathsf{fma}\left(\ell \cdot \ell, \frac{1}{3}, 2\right) \cdot \ell\right)\right) + U\]
\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U
↓
J \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \mathsf{fma}\left(\frac{1}{60}, {\ell}^{5}, \mathsf{fma}\left(\ell \cdot \ell, \frac{1}{3}, 2\right) \cdot \ell\right)\right) + U
double f(double J, double l, double K, double U) {
        double r3858439 = J;
        double r3858440 = l;
        double r3858441 = exp(r3858440);
        double r3858442 = -r3858440;
        double r3858443 = exp(r3858442);
        double r3858444 = r3858441 - r3858443;
        double r3858445 = r3858439 * r3858444;
        double r3858446 = K;
        double r3858447 = 2.0;
        double r3858448 = r3858446 / r3858447;
        double r3858449 = cos(r3858448);
        double r3858450 = r3858445 * r3858449;
        double r3858451 = U;
        double r3858452 = r3858450 + r3858451;
        return r3858452;
}
↓
double f(double J, double l, double K, double U) {
        double r3858453 = J;
        double r3858454 = K;
        double r3858455 = 2.0;
        double r3858456 = r3858454 / r3858455;
        double r3858457 = cos(r3858456);
        double r3858458 = 0.016666666666666666;
        double r3858459 = l;
        double r3858460 = 5.0;
        double r3858461 = pow(r3858459, r3858460);
        double r3858462 = r3858459 * r3858459;
        double r3858463 = 0.3333333333333333;
        double r3858464 = fma(r3858462, r3858463, r3858455);
        double r3858465 = r3858464 * r3858459;
        double r3858466 = fma(r3858458, r3858461, r3858465);
        double r3858467 = r3858457 * r3858466;
        double r3858468 = r3858453 * r3858467;
        double r3858469 = U;
        double r3858470 = r3858468 + r3858469;
        return r3858470;
}

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

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

  3. Simplified0.4

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

  5. Applied associate-*l*0.4

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

  6. Simplified0.4

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

  7. Final simplification0.4

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

Reproduce

herbie shell --seed 2019164 +o rules:numerics
(FPCore (J l K U)
  :name "Maksimov and Kolovsky, Equation (4)"
  (+ (* (* J (- (exp l) (exp (- l)))) (cos (/ K 2))) U))