Average Error: 15.3 → 1.5
Time: 7.6m
Precision: 64
\[\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)}\]
↓
\[{e}^{\left(\left(-{\left(\frac{n + m}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)\right)}\]
\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)}
↓
{e}^{\left(\left(-{\left(\frac{n + m}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)\right)}
double f(double K, double m, double n, double M, double l) {
        double r69720511 = K;
        double r69720512 = m;
        double r69720513 = n;
        double r69720514 = r69720512 + r69720513;
        double r69720515 = r69720511 * r69720514;
        double r69720516 = 2.0;
        double r69720517 = r69720515 / r69720516;
        double r69720518 = M;
        double r69720519 = r69720517 - r69720518;
        double r69720520 = cos(r69720519);
        double r69720521 = r69720514 / r69720516;
        double r69720522 = r69720521 - r69720518;
        double r69720523 = pow(r69720522, r69720516);
        double r69720524 = -r69720523;
        double r69720525 = l;
        double r69720526 = r69720512 - r69720513;
        double r69720527 = fabs(r69720526);
        double r69720528 = r69720525 - r69720527;
        double r69720529 = r69720524 - r69720528;
        double r69720530 = exp(r69720529);
        double r69720531 = r69720520 * r69720530;
        return r69720531;
}
↓
double f(double __attribute__((unused)) K, double m, double n, double M, double l) {
        double r69720532 = exp(1.0);
        double r69720533 = n;
        double r69720534 = m;
        double r69720535 = r69720533 + r69720534;
        double r69720536 = 2.0;
        double r69720537 = r69720535 / r69720536;
        double r69720538 = M;
        double r69720539 = r69720537 - r69720538;
        double r69720540 = pow(r69720539, r69720536);
        double r69720541 = -r69720540;
        double r69720542 = l;
        double r69720543 = r69720534 - r69720533;
        double r69720544 = fabs(r69720543);
        double r69720545 = r69720542 - r69720544;
        double r69720546 = r69720541 - r69720545;
        double r69720547 = pow(r69720532, r69720546);
        return r69720547;
}

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. Taylor expanded around 0 1.5

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

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

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

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

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

  6. Applied distribute-rgt-neg-in1.5

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

  7. Applied distribute-lft-out--1.5

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

  8. Applied exp-prod1.5

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

  9. Simplified1.5

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

  10. Final simplification1.5

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

Reproduce

herbie shell --seed 2019128 +o rules:numerics
(FPCore (K m n M l)
  :name "Maksimov and Kolovsky, Equation (32)"
  (* (cos (- (/ (* K (+ m n)) 2) M)) (exp (- (- (pow (- (/ (+ m n) 2) M) 2)) (- l (fabs (- m n)))))))