Average Error: 17.6 → 0.4
Time: 9.2s
Precision: 64
\[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U\]
↓
\[\left(J \cdot \mathsf{fma}\left(\frac{1}{3}, {\ell}^{3}, \mathsf{fma}\left(\frac{1}{60}, {\ell}^{5}, 2 \cdot \ell\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U\]
\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U
↓
\left(J \cdot \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
double f(double J, double l, double K, double U) {
        double r167639 = J;
        double r167640 = l;
        double r167641 = exp(r167640);
        double r167642 = -r167640;
        double r167643 = exp(r167642);
        double r167644 = r167641 - r167643;
        double r167645 = r167639 * r167644;
        double r167646 = K;
        double r167647 = 2.0;
        double r167648 = r167646 / r167647;
        double r167649 = cos(r167648);
        double r167650 = r167645 * r167649;
        double r167651 = U;
        double r167652 = r167650 + r167651;
        return r167652;
}
↓
double f(double J, double l, double K, double U) {
        double r167653 = J;
        double r167654 = 0.3333333333333333;
        double r167655 = l;
        double r167656 = 3.0;
        double r167657 = pow(r167655, r167656);
        double r167658 = 0.016666666666666666;
        double r167659 = 5.0;
        double r167660 = pow(r167655, r167659);
        double r167661 = 2.0;
        double r167662 = r167661 * r167655;
        double r167663 = fma(r167658, r167660, r167662);
        double r167664 = fma(r167654, r167657, r167663);
        double r167665 = r167653 * r167664;
        double r167666 = K;
        double r167667 = 2.0;
        double r167668 = r167666 / r167667;
        double r167669 = cos(r167668);
        double r167670 = r167665 * r167669;
        double r167671 = U;
        double r167672 = r167670 + r167671;
        return r167672;
}

Error

Bits error versus J

Bits error versus l

Bits error versus K

Bits error versus U

Derivation

  1. Initial program 17.6

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

  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. Final simplification0.4

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

Reproduce

herbie shell --seed 2020083 +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))