Average Error: 17.0 → 0.6
Time: 20.6s
Precision: 64
\[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U\]
↓
\[\mathsf{fma}\left(\left(\ell \cdot J\right) \cdot 2, \cos \left(\frac{K}{2}\right), U\right)\]
\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U
↓
\mathsf{fma}\left(\left(\ell \cdot J\right) \cdot 2, \cos \left(\frac{K}{2}\right), U\right)
double f(double J, double l, double K, double U) {
        double r1476600 = J;
        double r1476601 = l;
        double r1476602 = exp(r1476601);
        double r1476603 = -r1476601;
        double r1476604 = exp(r1476603);
        double r1476605 = r1476602 - r1476604;
        double r1476606 = r1476600 * r1476605;
        double r1476607 = K;
        double r1476608 = 2.0;
        double r1476609 = r1476607 / r1476608;
        double r1476610 = cos(r1476609);
        double r1476611 = r1476606 * r1476610;
        double r1476612 = U;
        double r1476613 = r1476611 + r1476612;
        return r1476613;
}
↓
double f(double J, double l, double K, double U) {
        double r1476614 = l;
        double r1476615 = J;
        double r1476616 = r1476614 * r1476615;
        double r1476617 = 2.0;
        double r1476618 = r1476616 * r1476617;
        double r1476619 = K;
        double r1476620 = r1476619 / r1476617;
        double r1476621 = cos(r1476620);
        double r1476622 = U;
        double r1476623 = fma(r1476618, r1476621, r1476622);
        return r1476623;
}

Error

Bits error versus J

Bits error versus l

Bits error versus K

Bits error versus U

Derivation

  1. Initial program 17.0

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

  2. Simplified17.0

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

  3. Taylor expanded around 0 0.6

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

  4. Final simplification0.6

    \[\leadsto \mathsf{fma}\left(\left(\ell \cdot J\right) \cdot 2, \cos \left(\frac{K}{2}\right), U\right)\]

Reproduce

herbie shell --seed 2019156 +o rules:numerics
(FPCore (J l K U)
  :name "Maksimov and Kolovsky, Equation (4)"
  (+ (* (* J (- (exp l) (exp (- l)))) (cos (/ K 2))) U))