Average Error: 15.2 → 1.3
Time: 20.1s
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 r96719 = K;
        double r96720 = m;
        double r96721 = n;
        double r96722 = r96720 + r96721;
        double r96723 = r96719 * r96722;
        double r96724 = 2.0;
        double r96725 = r96723 / r96724;
        double r96726 = M;
        double r96727 = r96725 - r96726;
        double r96728 = cos(r96727);
        double r96729 = r96722 / r96724;
        double r96730 = r96729 - r96726;
        double r96731 = pow(r96730, r96724);
        double r96732 = -r96731;
        double r96733 = l;
        double r96734 = r96720 - r96721;
        double r96735 = fabs(r96734);
        double r96736 = r96733 - r96735;
        double r96737 = r96732 - r96736;
        double r96738 = exp(r96737);
        double r96739 = r96728 * r96738;
        return r96739;
}
↓
double f(double __attribute__((unused)) K, double m, double n, double M, double l) {
        double r96740 = 1.0;
        double r96741 = m;
        double r96742 = n;
        double r96743 = r96741 + r96742;
        double r96744 = 2.0;
        double r96745 = r96743 / r96744;
        double r96746 = M;
        double r96747 = r96745 - r96746;
        double r96748 = pow(r96747, r96744);
        double r96749 = l;
        double r96750 = r96741 - r96742;
        double r96751 = fabs(r96750);
        double r96752 = r96749 - r96751;
        double r96753 = r96748 + r96752;
        double r96754 = exp(r96753);
        double r96755 = r96740 / r96754;
        return r96755;
}

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 15.2

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

  2. Simplified15.2

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

  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 2019305 +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)))))))