Average Error: 17.6 → 0.4
Time: 50.9s
Precision: 64
\[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U\]
↓
\[\mathsf{fma}\left(J \cdot \mathsf{fma}\left(\frac{1}{3}, {\ell}^{3}, \mathsf{fma}\left(\frac{1}{60}, {\ell}^{5}, 2 \cdot \ell\right)\right), \cos \left(\frac{K}{2}\right), U\right)\]
\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U
↓
\mathsf{fma}\left(J \cdot \mathsf{fma}\left(\frac{1}{3}, {\ell}^{3}, \mathsf{fma}\left(\frac{1}{60}, {\ell}^{5}, 2 \cdot \ell\right)\right), \cos \left(\frac{K}{2}\right), U\right)
double f(double J, double l, double K, double U) {
        double r78139 = J;
        double r78140 = l;
        double r78141 = exp(r78140);
        double r78142 = -r78140;
        double r78143 = exp(r78142);
        double r78144 = r78141 - r78143;
        double r78145 = r78139 * r78144;
        double r78146 = K;
        double r78147 = 2.0;
        double r78148 = r78146 / r78147;
        double r78149 = cos(r78148);
        double r78150 = r78145 * r78149;
        double r78151 = U;
        double r78152 = r78150 + r78151;
        return r78152;
}
↓
double f(double J, double l, double K, double U) {
        double r78153 = J;
        double r78154 = 0.3333333333333333;
        double r78155 = l;
        double r78156 = 3.0;
        double r78157 = pow(r78155, r78156);
        double r78158 = 0.016666666666666666;
        double r78159 = 5.0;
        double r78160 = pow(r78155, r78159);
        double r78161 = 2.0;
        double r78162 = r78161 * r78155;
        double r78163 = fma(r78158, r78160, r78162);
        double r78164 = fma(r78154, r78157, r78163);
        double r78165 = r78153 * r78164;
        double r78166 = K;
        double r78167 = 2.0;
        double r78168 = r78166 / r78167;
        double r78169 = cos(r78168);
        double r78170 = U;
        double r78171 = fma(r78165, r78169, r78170);
        return r78171;
}

Error

Bits error versus J

Bits error versus l

Bits error versus K

Bits error versus U

Derivation

  1. Initial program 17.6

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

  2. Simplified17.6

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

  3. Taylor expanded around 0 0.4

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

  4. Simplified0.4

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

  5. Final simplification0.4

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

Reproduce

herbie shell --seed 2019303 +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))