Average Error: 16.9 → 0.3
Time: 28.1s
Precision: 64
\[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U\]
↓
\[U + \left(\cos \left(\frac{K}{2}\right) \cdot \mathsf{fma}\left({\ell}^{5}, \frac{1}{60}, \mathsf{fma}\left(\ell, \frac{1}{3} \cdot \ell, 2\right) \cdot \ell\right)\right) \cdot J\]
\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U
↓
U + \left(\cos \left(\frac{K}{2}\right) \cdot \mathsf{fma}\left({\ell}^{5}, \frac{1}{60}, \mathsf{fma}\left(\ell, \frac{1}{3} \cdot \ell, 2\right) \cdot \ell\right)\right) \cdot J
double f(double J, double l, double K, double U) {
        double r4230231 = J;
        double r4230232 = l;
        double r4230233 = exp(r4230232);
        double r4230234 = -r4230232;
        double r4230235 = exp(r4230234);
        double r4230236 = r4230233 - r4230235;
        double r4230237 = r4230231 * r4230236;
        double r4230238 = K;
        double r4230239 = 2.0;
        double r4230240 = r4230238 / r4230239;
        double r4230241 = cos(r4230240);
        double r4230242 = r4230237 * r4230241;
        double r4230243 = U;
        double r4230244 = r4230242 + r4230243;
        return r4230244;
}
↓
double f(double J, double l, double K, double U) {
        double r4230245 = U;
        double r4230246 = K;
        double r4230247 = 2.0;
        double r4230248 = r4230246 / r4230247;
        double r4230249 = cos(r4230248);
        double r4230250 = l;
        double r4230251 = 5.0;
        double r4230252 = pow(r4230250, r4230251);
        double r4230253 = 0.016666666666666666;
        double r4230254 = 0.3333333333333333;
        double r4230255 = r4230254 * r4230250;
        double r4230256 = 2.0;
        double r4230257 = fma(r4230250, r4230255, r4230256);
        double r4230258 = r4230257 * r4230250;
        double r4230259 = fma(r4230252, r4230253, r4230258);
        double r4230260 = r4230249 * r4230259;
        double r4230261 = J;
        double r4230262 = r4230260 * r4230261;
        double r4230263 = r4230245 + r4230262;
        return r4230263;
}

Error

Bits error versus J

Bits error versus l

Bits error versus K

Bits error versus U

Derivation

  1. Initial program 16.9

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

  2. Taylor expanded around 0 0.3

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

  3. Simplified0.3

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

  5. Applied associate-*l*0.3

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

  6. Final simplification0.3

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

Reproduce

herbie shell --seed 2019172 +o rules:numerics
(FPCore (J l K U)
  :name "Maksimov and Kolovsky, Equation (4)"
  (+ (* (* J (- (exp l) (exp (- l)))) (cos (/ K 2.0))) U))