Average Error: 17.1 → 0.3
Time: 22.7s
Precision: 64
\[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U\]
↓
\[\left(J \cdot \mathsf{fma}\left(\frac{1}{3}, {\ell}^{3}, \mathsf{fma}\left(\frac{1}{60}, {\ell}^{5}, 2 \cdot \ell\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U\]
\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U
↓
\left(J \cdot \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
double f(double J, double l, double K, double U) {
        double r74199 = J;
        double r74200 = l;
        double r74201 = exp(r74200);
        double r74202 = -r74200;
        double r74203 = exp(r74202);
        double r74204 = r74201 - r74203;
        double r74205 = r74199 * r74204;
        double r74206 = K;
        double r74207 = 2.0;
        double r74208 = r74206 / r74207;
        double r74209 = cos(r74208);
        double r74210 = r74205 * r74209;
        double r74211 = U;
        double r74212 = r74210 + r74211;
        return r74212;
}
↓
double f(double J, double l, double K, double U) {
        double r74213 = J;
        double r74214 = 0.3333333333333333;
        double r74215 = l;
        double r74216 = 3.0;
        double r74217 = pow(r74215, r74216);
        double r74218 = 0.016666666666666666;
        double r74219 = 5.0;
        double r74220 = pow(r74215, r74219);
        double r74221 = 2.0;
        double r74222 = r74221 * r74215;
        double r74223 = fma(r74218, r74220, r74222);
        double r74224 = fma(r74214, r74217, r74223);
        double r74225 = r74213 * r74224;
        double r74226 = K;
        double r74227 = 2.0;
        double r74228 = r74226 / r74227;
        double r74229 = cos(r74228);
        double r74230 = r74225 * r74229;
        double r74231 = U;
        double r74232 = r74230 + r74231;
        return r74232;
}

Error

Bits error versus J

Bits error versus l

Bits error versus K

Bits error versus U

Derivation

  1. Initial program 17.1

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

  2. Simplified17.1

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

  3. Taylor expanded around 0 0.3

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

  4. Simplified0.3

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

  6. Applied fma-udef0.3

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

  7. Final simplification0.3

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

Reproduce

herbie shell --seed 2019198 +o rules:numerics
(FPCore (J l K U)
  :name "Maksimov and Kolovsky, Equation (4)"
  (+ (* (* J (- (exp l) (exp (- l)))) (cos (/ K 2.0))) U))