Average Error: 15.0 → 1.3
Time: 9.2s
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 r262 = K;
        double r263 = m;
        double r264 = n;
        double r265 = r263 + r264;
        double r266 = r262 * r265;
        double r267 = 2.0;
        double r268 = r266 / r267;
        double r269 = M;
        double r270 = r268 - r269;
        double r271 = cos(r270);
        double r272 = r265 / r267;
        double r273 = r272 - r269;
        double r274 = pow(r273, r267);
        double r275 = -r274;
        double r276 = l;
        double r277 = r263 - r264;
        double r278 = fabs(r277);
        double r279 = r276 - r278;
        double r280 = r275 - r279;
        double r281 = exp(r280);
        double r282 = r271 * r281;
        return r282;
}
↓
double f(double __attribute__((unused)) K, double m, double n, double M, double l) {
        double r283 = 1.0;
        double r284 = m;
        double r285 = n;
        double r286 = r284 + r285;
        double r287 = 2.0;
        double r288 = r286 / r287;
        double r289 = M;
        double r290 = r288 - r289;
        double r291 = pow(r290, r287);
        double r292 = l;
        double r293 = r284 - r285;
        double r294 = fabs(r293);
        double r295 = r292 - r294;
        double r296 = r291 + r295;
        double r297 = exp(r296);
        double r298 = r283 / r297;
        return r298;
}

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 15.0

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

  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 2020025 +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)))))))