Average Error: 15.1 → 1.3
Time: 28.9s
Precision: 64
\[\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)}\]
↓
\[\frac{1}{e^{{\left(\frac{m + n}{2} - M\right)}^{2} + \left(\ell - \left|m - n\right|\right)}}\]
\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)}
↓
\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 r87871 = K;
        double r87872 = m;
        double r87873 = n;
        double r87874 = r87872 + r87873;
        double r87875 = r87871 * r87874;
        double r87876 = 2.0;
        double r87877 = r87875 / r87876;
        double r87878 = M;
        double r87879 = r87877 - r87878;
        double r87880 = cos(r87879);
        double r87881 = r87874 / r87876;
        double r87882 = r87881 - r87878;
        double r87883 = pow(r87882, r87876);
        double r87884 = -r87883;
        double r87885 = l;
        double r87886 = r87872 - r87873;
        double r87887 = fabs(r87886);
        double r87888 = r87885 - r87887;
        double r87889 = r87884 - r87888;
        double r87890 = exp(r87889);
        double r87891 = r87880 * r87890;
        return r87891;
}
↓
double f(double __attribute__((unused)) K, double m, double n, double M, double l) {
        double r87892 = 1.0;
        double r87893 = m;
        double r87894 = n;
        double r87895 = r87893 + r87894;
        double r87896 = 2.0;
        double r87897 = r87895 / r87896;
        double r87898 = M;
        double r87899 = r87897 - r87898;
        double r87900 = pow(r87899, r87896);
        double r87901 = l;
        double r87902 = r87893 - r87894;
        double r87903 = fabs(r87902);
        double r87904 = r87901 - r87903;
        double r87905 = r87900 + r87904;
        double r87906 = exp(r87905);
        double r87907 = r87892 / r87906;
        return r87907;
}

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 15.1

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

  2. Simplified15.1

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

  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. Final simplification1.3

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

Reproduce

herbie shell --seed 2019304 +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)))))))