Average Error: 15.3 → 1.1
Time: 7.5s
Precision: 64
\[\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)}\]
↓
\[\frac{1}{e^{{\left(\frac{m + n}{2} - M\right)}^{2} + \left(\ell - \left|m - n\right|\right)}}\]
\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 r174501 = K;
        double r174502 = m;
        double r174503 = n;
        double r174504 = r174502 + r174503;
        double r174505 = r174501 * r174504;
        double r174506 = 2.0;
        double r174507 = r174505 / r174506;
        double r174508 = M;
        double r174509 = r174507 - r174508;
        double r174510 = cos(r174509);
        double r174511 = r174504 / r174506;
        double r174512 = r174511 - r174508;
        double r174513 = pow(r174512, r174506);
        double r174514 = -r174513;
        double r174515 = l;
        double r174516 = r174502 - r174503;
        double r174517 = fabs(r174516);
        double r174518 = r174515 - r174517;
        double r174519 = r174514 - r174518;
        double r174520 = exp(r174519);
        double r174521 = r174510 * r174520;
        return r174521;
}
↓
double f(double __attribute__((unused)) K, double m, double n, double M, double l) {
        double r174522 = 1.0;
        double r174523 = m;
        double r174524 = n;
        double r174525 = r174523 + r174524;
        double r174526 = 2.0;
        double r174527 = r174525 / r174526;
        double r174528 = M;
        double r174529 = r174527 - r174528;
        double r174530 = pow(r174529, r174526);
        double r174531 = l;
        double r174532 = r174523 - r174524;
        double r174533 = fabs(r174532);
        double r174534 = r174531 - r174533;
        double r174535 = r174530 + r174534;
        double r174536 = exp(r174535);
        double r174537 = r174522 / r174536;
        return r174537;
}

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 1.1

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

  4. Final simplification1.1

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

Reproduce

herbie shell --seed 2020056 +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)))))))