Average Error: 17.3 → 0.4
Time: 33.5s
Precision: 64
\[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U\]
↓
\[\left(\mathsf{fma}\left(\frac{1}{3} \cdot \ell, \ell, 2\right) \cdot \left(J \cdot \ell\right) + \left({\ell}^{5} \cdot \frac{1}{60}\right) \cdot J\right) \cdot \cos \left(\frac{K}{2}\right) + U\]
\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U
↓
\left(\mathsf{fma}\left(\frac{1}{3} \cdot \ell, \ell, 2\right) \cdot \left(J \cdot \ell\right) + \left({\ell}^{5} \cdot \frac{1}{60}\right) \cdot J\right) \cdot \cos \left(\frac{K}{2}\right) + U
double f(double J, double l, double K, double U) {
        double r4535640 = J;
        double r4535641 = l;
        double r4535642 = exp(r4535641);
        double r4535643 = -r4535641;
        double r4535644 = exp(r4535643);
        double r4535645 = r4535642 - r4535644;
        double r4535646 = r4535640 * r4535645;
        double r4535647 = K;
        double r4535648 = 2.0;
        double r4535649 = r4535647 / r4535648;
        double r4535650 = cos(r4535649);
        double r4535651 = r4535646 * r4535650;
        double r4535652 = U;
        double r4535653 = r4535651 + r4535652;
        return r4535653;
}
↓
double f(double J, double l, double K, double U) {
        double r4535654 = 0.3333333333333333;
        double r4535655 = l;
        double r4535656 = r4535654 * r4535655;
        double r4535657 = 2.0;
        double r4535658 = fma(r4535656, r4535655, r4535657);
        double r4535659 = J;
        double r4535660 = r4535659 * r4535655;
        double r4535661 = r4535658 * r4535660;
        double r4535662 = 5.0;
        double r4535663 = pow(r4535655, r4535662);
        double r4535664 = 0.016666666666666666;
        double r4535665 = r4535663 * r4535664;
        double r4535666 = r4535665 * r4535659;
        double r4535667 = r4535661 + r4535666;
        double r4535668 = K;
        double r4535669 = r4535668 / r4535657;
        double r4535670 = cos(r4535669);
        double r4535671 = r4535667 * r4535670;
        double r4535672 = U;
        double r4535673 = r4535671 + r4535672;
        return r4535673;
}

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

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

  5. Applied fma-udef0.4

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

  6. Applied distribute-lft-in0.4

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

  7. Simplified0.4

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

  8. Final simplification0.4

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

Reproduce

herbie shell --seed 2019134 +o rules:numerics
(FPCore (J l K U)
  :name "Maksimov and Kolovsky, Equation (4)"
  (+ (* (* J (- (exp l) (exp (- l)))) (cos (/ K 2))) U))