Average Error: 15.3 → 1.3
Time: 10.9s
Precision: 64
\[\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)}\]
↓
\[\left(\sqrt[3]{\frac{1}{e^{{\left(\frac{m + n}{2} - M\right)}^{2} + \left(\ell - \left|m - n\right|\right)}}} \cdot \sqrt[3]{\frac{1}{e^{{\left(\frac{m + n}{2} - M\right)}^{2} + \left(\ell - \left|m - n\right|\right)}}}\right) \cdot \sqrt[3]{\frac{1}{e^{{\left(\frac{m + n}{2} - M\right)}^{2} + \left(\ell - \left|m - n\right|\right)}}}\]
\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)}
↓
\left(\sqrt[3]{\frac{1}{e^{{\left(\frac{m + n}{2} - M\right)}^{2} + \left(\ell - \left|m - n\right|\right)}}} \cdot \sqrt[3]{\frac{1}{e^{{\left(\frac{m + n}{2} - M\right)}^{2} + \left(\ell - \left|m - n\right|\right)}}}\right) \cdot \sqrt[3]{\frac{1}{e^{{\left(\frac{m + n}{2} - M\right)}^{2} + \left(\ell - \left|m - n\right|\right)}}}
double f(double K, double m, double n, double M, double l) {
        double r208849 = K;
        double r208850 = m;
        double r208851 = n;
        double r208852 = r208850 + r208851;
        double r208853 = r208849 * r208852;
        double r208854 = 2.0;
        double r208855 = r208853 / r208854;
        double r208856 = M;
        double r208857 = r208855 - r208856;
        double r208858 = cos(r208857);
        double r208859 = r208852 / r208854;
        double r208860 = r208859 - r208856;
        double r208861 = pow(r208860, r208854);
        double r208862 = -r208861;
        double r208863 = l;
        double r208864 = r208850 - r208851;
        double r208865 = fabs(r208864);
        double r208866 = r208863 - r208865;
        double r208867 = r208862 - r208866;
        double r208868 = exp(r208867);
        double r208869 = r208858 * r208868;
        return r208869;
}
↓
double f(double __attribute__((unused)) K, double m, double n, double M, double l) {
        double r208870 = 1.0;
        double r208871 = m;
        double r208872 = n;
        double r208873 = r208871 + r208872;
        double r208874 = 2.0;
        double r208875 = r208873 / r208874;
        double r208876 = M;
        double r208877 = r208875 - r208876;
        double r208878 = pow(r208877, r208874);
        double r208879 = l;
        double r208880 = r208871 - r208872;
        double r208881 = fabs(r208880);
        double r208882 = r208879 - r208881;
        double r208883 = r208878 + r208882;
        double r208884 = exp(r208883);
        double r208885 = r208870 / r208884;
        double r208886 = cbrt(r208885);
        double r208887 = r208886 * r208886;
        double r208888 = r208887 * r208886;
        return r208888;
}

Error

Bits error versus K

Bits error versus m

Bits error versus n

Bits error versus M

Bits error versus l

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 15.3

    \[\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)}\]

  2. Simplified15.3

    \[\leadsto \color{blue}{\frac{\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right)}{e^{{\left(\frac{m + n}{2} - M\right)}^{2} + \left(\ell - \left|m - n\right|\right)}}}\]

  3. Taylor expanded around 0 1.3

    \[\leadsto \frac{\color{blue}{1}}{e^{{\left(\frac{m + n}{2} - M\right)}^{2} + \left(\ell - \left|m - n\right|\right)}}\]
  4. Using strategy rm

  5. Applied add-cube-cbrt1.3

    \[\leadsto \color{blue}{\left(\sqrt[3]{\frac{1}{e^{{\left(\frac{m + n}{2} - M\right)}^{2} + \left(\ell - \left|m - n\right|\right)}}} \cdot \sqrt[3]{\frac{1}{e^{{\left(\frac{m + n}{2} - M\right)}^{2} + \left(\ell - \left|m - n\right|\right)}}}\right) \cdot \sqrt[3]{\frac{1}{e^{{\left(\frac{m + n}{2} - M\right)}^{2} + \left(\ell - \left|m - n\right|\right)}}}}\]

  6. Final simplification1.3

    \[\leadsto \left(\sqrt[3]{\frac{1}{e^{{\left(\frac{m + n}{2} - M\right)}^{2} + \left(\ell - \left|m - n\right|\right)}}} \cdot \sqrt[3]{\frac{1}{e^{{\left(\frac{m + n}{2} - M\right)}^{2} + \left(\ell - \left|m - n\right|\right)}}}\right) \cdot \sqrt[3]{\frac{1}{e^{{\left(\frac{m + n}{2} - M\right)}^{2} + \left(\ell - \left|m - n\right|\right)}}}\]

Reproduce

herbie shell --seed 2020036 +o rules:numerics
(FPCore (K m n M l)
  :name "Maksimov and Kolovsky, Equation (32)"
  :precision binary64
  (* (cos (- (/ (* K (+ m n)) 2) M)) (exp (- (- (pow (- (/ (+ m n) 2) M) 2)) (- l (fabs (- m n)))))))