Average Error: 17.2 → 0.3
Time: 7.2s
Precision: 64
\[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U\]
↓
\[\mathsf{fma}\left(J, \mathsf{fma}\left(\frac{1}{3}, {\ell}^{3}, \mathsf{fma}\left(\frac{1}{60}, {\ell}^{5}, 2 \cdot \ell\right)\right) \cdot \cos \left(\frac{K}{2}\right), U\right)\]
\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U
↓
\mathsf{fma}\left(J, \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), U\right)
double f(double J, double l, double K, double U) {
        double r173337 = J;
        double r173338 = l;
        double r173339 = exp(r173338);
        double r173340 = -r173338;
        double r173341 = exp(r173340);
        double r173342 = r173339 - r173341;
        double r173343 = r173337 * r173342;
        double r173344 = K;
        double r173345 = 2.0;
        double r173346 = r173344 / r173345;
        double r173347 = cos(r173346);
        double r173348 = r173343 * r173347;
        double r173349 = U;
        double r173350 = r173348 + r173349;
        return r173350;
}
↓
double f(double J, double l, double K, double U) {
        double r173351 = J;
        double r173352 = 0.3333333333333333;
        double r173353 = l;
        double r173354 = 3.0;
        double r173355 = pow(r173353, r173354);
        double r173356 = 0.016666666666666666;
        double r173357 = 5.0;
        double r173358 = pow(r173353, r173357);
        double r173359 = 2.0;
        double r173360 = r173359 * r173353;
        double r173361 = fma(r173356, r173358, r173360);
        double r173362 = fma(r173352, r173355, r173361);
        double r173363 = K;
        double r173364 = 2.0;
        double r173365 = r173363 / r173364;
        double r173366 = cos(r173365);
        double r173367 = r173362 * r173366;
        double r173368 = U;
        double r173369 = fma(r173351, r173367, r173368);
        return r173369;
}

Error

Bits error versus J

Bits error versus l

Bits error versus K

Bits error versus U

Derivation

  1. Initial program 17.2

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

  2. Taylor expanded around 0 0.3

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

  3. Simplified0.3

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

  5. Applied associate-*l*0.3

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

  7. Applied fma-def0.3

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

  8. Final simplification0.3

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

Reproduce

herbie shell --seed 2020024 +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))