Average Error: 17.3 → 0.6
Time: 25.2s
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 r1521218 = J;
        double r1521219 = l;
        double r1521220 = exp(r1521219);
        double r1521221 = -r1521219;
        double r1521222 = exp(r1521221);
        double r1521223 = r1521220 - r1521222;
        double r1521224 = r1521218 * r1521223;
        double r1521225 = K;
        double r1521226 = 2.0;
        double r1521227 = r1521225 / r1521226;
        double r1521228 = cos(r1521227);
        double r1521229 = r1521224 * r1521228;
        double r1521230 = U;
        double r1521231 = r1521229 + r1521230;
        return r1521231;
}
↓
double f(double J, double l, double K, double U) {
        double r1521232 = l;
        double r1521233 = J;
        double r1521234 = r1521232 * r1521233;
        double r1521235 = 2.0;
        double r1521236 = r1521234 * r1521235;
        double r1521237 = K;
        double r1521238 = r1521237 / r1521235;
        double r1521239 = cos(r1521238);
        double r1521240 = U;
        double r1521241 = fma(r1521236, r1521239, r1521240);
        return r1521241;
}

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))