Average Error: 17.4 → 0.4
Time: 8.9s
Precision: 64
\[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U\]
↓
\[\mathsf{fma}\left(J \cdot \mathsf{fma}\left(\frac{1}{3}, {\ell}^{3}, \mathsf{fma}\left(\frac{1}{60}, {\ell}^{5}, 2 \cdot \ell\right)\right), \cos \left(\frac{K}{2}\right), U\right)\]
\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U
↓
\mathsf{fma}\left(J \cdot \mathsf{fma}\left(\frac{1}{3}, {\ell}^{3}, \mathsf{fma}\left(\frac{1}{60}, {\ell}^{5}, 2 \cdot \ell\right)\right), \cos \left(\frac{K}{2}\right), U\right)
double f(double J, double l, double K, double U) {
        double r135313 = J;
        double r135314 = l;
        double r135315 = exp(r135314);
        double r135316 = -r135314;
        double r135317 = exp(r135316);
        double r135318 = r135315 - r135317;
        double r135319 = r135313 * r135318;
        double r135320 = K;
        double r135321 = 2.0;
        double r135322 = r135320 / r135321;
        double r135323 = cos(r135322);
        double r135324 = r135319 * r135323;
        double r135325 = U;
        double r135326 = r135324 + r135325;
        return r135326;
}
↓
double f(double J, double l, double K, double U) {
        double r135327 = J;
        double r135328 = 0.3333333333333333;
        double r135329 = l;
        double r135330 = 3.0;
        double r135331 = pow(r135329, r135330);
        double r135332 = 0.016666666666666666;
        double r135333 = 5.0;
        double r135334 = pow(r135329, r135333);
        double r135335 = 2.0;
        double r135336 = r135335 * r135329;
        double r135337 = fma(r135332, r135334, r135336);
        double r135338 = fma(r135328, r135331, r135337);
        double r135339 = r135327 * r135338;
        double r135340 = K;
        double r135341 = 2.0;
        double r135342 = r135340 / r135341;
        double r135343 = cos(r135342);
        double r135344 = U;
        double r135345 = fma(r135339, r135343, r135344);
        return r135345;
}

Error

Bits error versus J

Bits error versus l

Bits error versus K

Bits error versus U

Derivation

  1. Initial program 17.4

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

  2. Simplified17.4

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

  3. Taylor expanded around 0 0.4

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

  4. Simplified0.4

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

  5. Final simplification0.4

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

Reproduce

herbie shell --seed 2020064 +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))