Average Error: 17.7 → 0.4
Time: 47.4s
Precision: 64
\[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U\]
↓
\[\mathsf{fma}\left(\frac{1}{3} \cdot \left(J \cdot {\ell}^{3}\right) + J \cdot \mathsf{fma}\left(\frac{1}{60}, {\ell}^{5}, 2 \cdot \ell\right), \cos \left(\frac{K}{2}\right), U\right)\]
\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U
↓
\mathsf{fma}\left(\frac{1}{3} \cdot \left(J \cdot {\ell}^{3}\right) + J \cdot \mathsf{fma}\left(\frac{1}{60}, {\ell}^{5}, 2 \cdot \ell\right), \cos \left(\frac{K}{2}\right), U\right)
double f(double J, double l, double K, double U) {
        double r63228 = J;
        double r63229 = l;
        double r63230 = exp(r63229);
        double r63231 = -r63229;
        double r63232 = exp(r63231);
        double r63233 = r63230 - r63232;
        double r63234 = r63228 * r63233;
        double r63235 = K;
        double r63236 = 2.0;
        double r63237 = r63235 / r63236;
        double r63238 = cos(r63237);
        double r63239 = r63234 * r63238;
        double r63240 = U;
        double r63241 = r63239 + r63240;
        return r63241;
}
↓
double f(double J, double l, double K, double U) {
        double r63242 = 0.3333333333333333;
        double r63243 = J;
        double r63244 = l;
        double r63245 = 3.0;
        double r63246 = pow(r63244, r63245);
        double r63247 = r63243 * r63246;
        double r63248 = r63242 * r63247;
        double r63249 = 0.016666666666666666;
        double r63250 = 5.0;
        double r63251 = pow(r63244, r63250);
        double r63252 = 2.0;
        double r63253 = r63252 * r63244;
        double r63254 = fma(r63249, r63251, r63253);
        double r63255 = r63243 * r63254;
        double r63256 = r63248 + r63255;
        double r63257 = K;
        double r63258 = 2.0;
        double r63259 = r63257 / r63258;
        double r63260 = cos(r63259);
        double r63261 = U;
        double r63262 = fma(r63256, r63260, r63261);
        return r63262;
}

Error

Bits error versus J

Bits error versus l

Bits error versus K

Bits error versus U

Derivation

  1. Initial program 17.7

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

  2. Simplified17.7

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

  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. Using strategy rm

  6. Applied fma-udef0.4

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

  7. Applied distribute-lft-in0.4

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

  8. Simplified0.4

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

  9. Final simplification0.4

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

Reproduce

herbie shell --seed 2019323 +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))