Average Error: 17.4 → 0.4
Time: 13.8s
Precision: 64
\[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U\]
↓
\[\mathsf{fma}\left(J, \cos \left(\frac{K}{2}\right) \cdot \mathsf{fma}\left(\frac{1}{3}, {\ell}^{3}, \mathsf{fma}\left(\frac{1}{60}, {\ell}^{5}, 2 \cdot \ell\right)\right), U\right)\]
\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U
↓
\mathsf{fma}\left(J, \cos \left(\frac{K}{2}\right) \cdot \mathsf{fma}\left(\frac{1}{3}, {\ell}^{3}, \mathsf{fma}\left(\frac{1}{60}, {\ell}^{5}, 2 \cdot \ell\right)\right), U\right)
double f(double J, double l, double K, double U) {
        double r253343 = J;
        double r253344 = l;
        double r253345 = exp(r253344);
        double r253346 = -r253344;
        double r253347 = exp(r253346);
        double r253348 = r253345 - r253347;
        double r253349 = r253343 * r253348;
        double r253350 = K;
        double r253351 = 2.0;
        double r253352 = r253350 / r253351;
        double r253353 = cos(r253352);
        double r253354 = r253349 * r253353;
        double r253355 = U;
        double r253356 = r253354 + r253355;
        return r253356;
}
↓
double f(double J, double l, double K, double U) {
        double r253357 = J;
        double r253358 = K;
        double r253359 = 2.0;
        double r253360 = r253358 / r253359;
        double r253361 = cos(r253360);
        double r253362 = 0.3333333333333333;
        double r253363 = l;
        double r253364 = 3.0;
        double r253365 = pow(r253363, r253364);
        double r253366 = 0.016666666666666666;
        double r253367 = 5.0;
        double r253368 = pow(r253363, r253367);
        double r253369 = 2.0;
        double r253370 = r253369 * r253363;
        double r253371 = fma(r253366, r253368, r253370);
        double r253372 = fma(r253362, r253365, r253371);
        double r253373 = r253361 * r253372;
        double r253374 = U;
        double r253375 = fma(r253357, r253373, r253374);
        return r253375;
}

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. Taylor expanded around 0 0.4

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

  3. Simplified0.4

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

  5. Applied associate-*l*0.4

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

  6. Simplified0.4

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

  8. Applied fma-def0.4

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

  9. Final simplification0.4

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

Reproduce

herbie shell --seed 2020046 +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))