Average Error: 17.1 → 0.4
Time: 8.2s
Precision: 64
\[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U\]
↓
\[\left(J \cdot \left(\frac{1}{3} \cdot {\ell}^{3}\right) + J \cdot \mathsf{fma}\left(\frac{1}{60}, {\ell}^{5}, 2 \cdot \ell\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U\]
\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U
↓
\left(J \cdot \left(\frac{1}{3} \cdot {\ell}^{3}\right) + J \cdot \mathsf{fma}\left(\frac{1}{60}, {\ell}^{5}, 2 \cdot \ell\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U
double f(double J, double l, double K, double U) {
        double r142909 = J;
        double r142910 = l;
        double r142911 = exp(r142910);
        double r142912 = -r142910;
        double r142913 = exp(r142912);
        double r142914 = r142911 - r142913;
        double r142915 = r142909 * r142914;
        double r142916 = K;
        double r142917 = 2.0;
        double r142918 = r142916 / r142917;
        double r142919 = cos(r142918);
        double r142920 = r142915 * r142919;
        double r142921 = U;
        double r142922 = r142920 + r142921;
        return r142922;
}
↓
double f(double J, double l, double K, double U) {
        double r142923 = J;
        double r142924 = 0.3333333333333333;
        double r142925 = l;
        double r142926 = 3.0;
        double r142927 = pow(r142925, r142926);
        double r142928 = r142924 * r142927;
        double r142929 = r142923 * r142928;
        double r142930 = 0.016666666666666666;
        double r142931 = 5.0;
        double r142932 = pow(r142925, r142931);
        double r142933 = 2.0;
        double r142934 = r142933 * r142925;
        double r142935 = fma(r142930, r142932, r142934);
        double r142936 = r142923 * r142935;
        double r142937 = r142929 + r142936;
        double r142938 = K;
        double r142939 = 2.0;
        double r142940 = r142938 / r142939;
        double r142941 = cos(r142940);
        double r142942 = r142937 * r142941;
        double r142943 = U;
        double r142944 = r142942 + r142943;
        return r142944;
}

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. 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 fma-udef0.4

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

  6. Applied distribute-lft-in0.4

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

  7. Final simplification0.4

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

Reproduce

herbie shell --seed 2020060 +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))