Average Error: 14.7 → 1.2
Time: 24.0s
Precision: 64
\[\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)}\]
↓
\[\frac{1}{e^{{\left(\frac{n + m}{2} - M\right)}^{2} + \left(\ell - \left|m - n\right|\right)}}\]
\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{n + m}{2} - M\right)}^{2} + \left(\ell - \left|m - n\right|\right)}}
double f(double K, double m, double n, double M, double l) {
        double r106676 = K;
        double r106677 = m;
        double r106678 = n;
        double r106679 = r106677 + r106678;
        double r106680 = r106676 * r106679;
        double r106681 = 2.0;
        double r106682 = r106680 / r106681;
        double r106683 = M;
        double r106684 = r106682 - r106683;
        double r106685 = cos(r106684);
        double r106686 = r106679 / r106681;
        double r106687 = r106686 - r106683;
        double r106688 = pow(r106687, r106681);
        double r106689 = -r106688;
        double r106690 = l;
        double r106691 = r106677 - r106678;
        double r106692 = fabs(r106691);
        double r106693 = r106690 - r106692;
        double r106694 = r106689 - r106693;
        double r106695 = exp(r106694);
        double r106696 = r106685 * r106695;
        return r106696;
}
↓
double f(double __attribute__((unused)) K, double m, double n, double M, double l) {
        double r106697 = 1.0;
        double r106698 = n;
        double r106699 = m;
        double r106700 = r106698 + r106699;
        double r106701 = 2.0;
        double r106702 = r106700 / r106701;
        double r106703 = M;
        double r106704 = r106702 - r106703;
        double r106705 = pow(r106704, r106701);
        double r106706 = l;
        double r106707 = r106699 - r106698;
        double r106708 = fabs(r106707);
        double r106709 = r106706 - r106708;
        double r106710 = r106705 + r106709;
        double r106711 = exp(r106710);
        double r106712 = r106697 / r106711;
        return r106712;
}

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 14.7

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

  2. Simplified14.7

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

  3. Taylor expanded around 0 1.2

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

  4. Final simplification1.2

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

Reproduce

herbie shell --seed 2019194 +o rules:numerics
(FPCore (K m n M l)
  :name "Maksimov and Kolovsky, Equation (32)"
  (* (cos (- (/ (* K (+ m n)) 2.0) M)) (exp (- (- (pow (- (/ (+ m n) 2.0) M) 2.0)) (- l (fabs (- m n)))))))