Average Error: 15.3 → 1.4
Time: 30.5s
Precision: 64
\[\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)}\]
↓
\[\frac{1}{{e}^{\left({\left(\frac{m + n}{2} - M\right)}^{2} + \left(\ell - \left|m - n\right|\right)\right)}}\]
\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({\left(\frac{m + n}{2} - M\right)}^{2} + \left(\ell - \left|m - n\right|\right)\right)}}
double f(double K, double m, double n, double M, double l) {
        double r122351 = K;
        double r122352 = m;
        double r122353 = n;
        double r122354 = r122352 + r122353;
        double r122355 = r122351 * r122354;
        double r122356 = 2.0;
        double r122357 = r122355 / r122356;
        double r122358 = M;
        double r122359 = r122357 - r122358;
        double r122360 = cos(r122359);
        double r122361 = r122354 / r122356;
        double r122362 = r122361 - r122358;
        double r122363 = pow(r122362, r122356);
        double r122364 = -r122363;
        double r122365 = l;
        double r122366 = r122352 - r122353;
        double r122367 = fabs(r122366);
        double r122368 = r122365 - r122367;
        double r122369 = r122364 - r122368;
        double r122370 = exp(r122369);
        double r122371 = r122360 * r122370;
        return r122371;
}
↓
double f(double __attribute__((unused)) K, double m, double n, double M, double l) {
        double r122372 = 1.0;
        double r122373 = exp(1.0);
        double r122374 = m;
        double r122375 = n;
        double r122376 = r122374 + r122375;
        double r122377 = 2.0;
        double r122378 = r122376 / r122377;
        double r122379 = M;
        double r122380 = r122378 - r122379;
        double r122381 = pow(r122380, r122377);
        double r122382 = l;
        double r122383 = r122374 - r122375;
        double r122384 = fabs(r122383);
        double r122385 = r122382 - r122384;
        double r122386 = r122381 + r122385;
        double r122387 = pow(r122373, r122386);
        double r122388 = r122372 / r122387;
        return r122388;
}

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 1.4

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

  5. Applied *-un-lft-identity1.4

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

  6. Applied exp-prod1.4

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

  7. Simplified1.4

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

  8. Final simplification1.4

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

Reproduce

herbie shell --seed 2019325 +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)))))))