Average Error: 16.9 → 0.3
Time: 26.3s
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 r4257038 = J;
        double r4257039 = l;
        double r4257040 = exp(r4257039);
        double r4257041 = -r4257039;
        double r4257042 = exp(r4257041);
        double r4257043 = r4257040 - r4257042;
        double r4257044 = r4257038 * r4257043;
        double r4257045 = K;
        double r4257046 = 2.0;
        double r4257047 = r4257045 / r4257046;
        double r4257048 = cos(r4257047);
        double r4257049 = r4257044 * r4257048;
        double r4257050 = U;
        double r4257051 = r4257049 + r4257050;
        return r4257051;
}
↓
double f(double J, double l, double K, double U) {
        double r4257052 = U;
        double r4257053 = K;
        double r4257054 = 2.0;
        double r4257055 = r4257053 / r4257054;
        double r4257056 = cos(r4257055);
        double r4257057 = l;
        double r4257058 = 5.0;
        double r4257059 = pow(r4257057, r4257058);
        double r4257060 = 0.016666666666666666;
        double r4257061 = 0.3333333333333333;
        double r4257062 = r4257061 * r4257057;
        double r4257063 = 2.0;
        double r4257064 = fma(r4257057, r4257062, r4257063);
        double r4257065 = r4257064 * r4257057;
        double r4257066 = fma(r4257059, r4257060, r4257065);
        double r4257067 = r4257056 * r4257066;
        double r4257068 = J;
        double r4257069 = r4257067 * r4257068;
        double r4257070 = r4257052 + r4257069;
        return r4257070;
}

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

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

  5. Applied associate-*l*0.3

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

  6. Simplified0.3

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

  7. 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))