Average Error: 17.3 → 0.6
Time: 23.8s
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 r1471430 = J;
        double r1471431 = l;
        double r1471432 = exp(r1471431);
        double r1471433 = -r1471431;
        double r1471434 = exp(r1471433);
        double r1471435 = r1471432 - r1471434;
        double r1471436 = r1471430 * r1471435;
        double r1471437 = K;
        double r1471438 = 2.0;
        double r1471439 = r1471437 / r1471438;
        double r1471440 = cos(r1471439);
        double r1471441 = r1471436 * r1471440;
        double r1471442 = U;
        double r1471443 = r1471441 + r1471442;
        return r1471443;
}
↓
double f(double J, double l, double K, double U) {
        double r1471444 = l;
        double r1471445 = J;
        double r1471446 = r1471444 * r1471445;
        double r1471447 = 2.0;
        double r1471448 = r1471446 * r1471447;
        double r1471449 = K;
        double r1471450 = r1471449 / r1471447;
        double r1471451 = cos(r1471450);
        double r1471452 = U;
        double r1471453 = fma(r1471448, r1471451, r1471452);
        return r1471453;
}

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. Simplified17.3

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

  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 2019153 +o rules:numerics
(FPCore (J l K U)
  :name "Maksimov and Kolovsky, Equation (4)"
  (+ (* (* J (- (exp l) (exp (- l)))) (cos (/ K 2))) U))