Average Error: 17.0 → 0.4
Time: 9.3s
Precision: 64
\[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U\]
↓
\[J \cdot \left(\mathsf{fma}\left(\frac{1}{3}, {\ell}^{3}, \mathsf{fma}\left(\frac{1}{60}, {\ell}^{5}, 2 \cdot \ell\right)\right) \cdot \cos \left(\frac{K}{2}\right)\right) + U\]
\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U
↓
J \cdot \left(\mathsf{fma}\left(\frac{1}{3}, {\ell}^{3}, \mathsf{fma}\left(\frac{1}{60}, {\ell}^{5}, 2 \cdot \ell\right)\right) \cdot \cos \left(\frac{K}{2}\right)\right) + U
double f(double J, double l, double K, double U) {
        double r138108 = J;
        double r138109 = l;
        double r138110 = exp(r138109);
        double r138111 = -r138109;
        double r138112 = exp(r138111);
        double r138113 = r138110 - r138112;
        double r138114 = r138108 * r138113;
        double r138115 = K;
        double r138116 = 2.0;
        double r138117 = r138115 / r138116;
        double r138118 = cos(r138117);
        double r138119 = r138114 * r138118;
        double r138120 = U;
        double r138121 = r138119 + r138120;
        return r138121;
}
↓
double f(double J, double l, double K, double U) {
        double r138122 = J;
        double r138123 = 0.3333333333333333;
        double r138124 = l;
        double r138125 = 3.0;
        double r138126 = pow(r138124, r138125);
        double r138127 = 0.016666666666666666;
        double r138128 = 5.0;
        double r138129 = pow(r138124, r138128);
        double r138130 = 2.0;
        double r138131 = r138130 * r138124;
        double r138132 = fma(r138127, r138129, r138131);
        double r138133 = fma(r138123, r138126, r138132);
        double r138134 = K;
        double r138135 = 2.0;
        double r138136 = r138134 / r138135;
        double r138137 = cos(r138136);
        double r138138 = r138133 * r138137;
        double r138139 = r138122 * r138138;
        double r138140 = U;
        double r138141 = r138139 + r138140;
        return r138141;
}

Error

Bits error versus J

Bits error versus l

Bits error versus K

Bits error versus U

Derivation

  1. Initial program 17.0

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

  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 associate-*l*0.4

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

  6. Final simplification0.4

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

Reproduce

herbie shell --seed 2020034 +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))