Average Error: 17.5 → 0.4
Time: 29.9s
Precision: 64
\[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U\]
↓
\[\left(\frac{1}{3} \cdot \left(J \cdot {\ell}^{3}\right) + \mathsf{fma}\left(\frac{1}{60}, {\ell}^{5}, 2 \cdot \ell\right) \cdot J\right) \cdot \cos \left(\frac{K}{2}\right) + U\]
\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U
↓
\left(\frac{1}{3} \cdot \left(J \cdot {\ell}^{3}\right) + \mathsf{fma}\left(\frac{1}{60}, {\ell}^{5}, 2 \cdot \ell\right) \cdot J\right) \cdot \cos \left(\frac{K}{2}\right) + U
double f(double J, double l, double K, double U) {
        double r123220 = J;
        double r123221 = l;
        double r123222 = exp(r123221);
        double r123223 = -r123221;
        double r123224 = exp(r123223);
        double r123225 = r123222 - r123224;
        double r123226 = r123220 * r123225;
        double r123227 = K;
        double r123228 = 2.0;
        double r123229 = r123227 / r123228;
        double r123230 = cos(r123229);
        double r123231 = r123226 * r123230;
        double r123232 = U;
        double r123233 = r123231 + r123232;
        return r123233;
}
↓
double f(double J, double l, double K, double U) {
        double r123234 = 0.3333333333333333;
        double r123235 = J;
        double r123236 = l;
        double r123237 = 3.0;
        double r123238 = pow(r123236, r123237);
        double r123239 = r123235 * r123238;
        double r123240 = r123234 * r123239;
        double r123241 = 0.016666666666666666;
        double r123242 = 5.0;
        double r123243 = pow(r123236, r123242);
        double r123244 = 2.0;
        double r123245 = r123244 * r123236;
        double r123246 = fma(r123241, r123243, r123245);
        double r123247 = r123246 * r123235;
        double r123248 = r123240 + r123247;
        double r123249 = K;
        double r123250 = 2.0;
        double r123251 = r123249 / r123250;
        double r123252 = cos(r123251);
        double r123253 = r123248 * r123252;
        double r123254 = U;
        double r123255 = r123253 + r123254;
        return r123255;
}

Error

Bits error versus J

Bits error versus l

Bits error versus K

Bits error versus U

Derivation

  1. Initial program 17.5

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

  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 fma-udef0.4

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

  6. Applied distribute-lft-in0.4

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

  7. Simplified0.4

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

  8. Simplified0.4

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

  9. Final simplification0.4

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

Reproduce

herbie shell --seed 2019350 +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))