Average Error: 15.6 → 1.4
Time: 43.2s
Precision: 64
\[\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)}\]
↓
\[\left(\sqrt[3]{{e}^{\left(\left(-{\left(\frac{n + m}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)\right)}} \cdot \sqrt[3]{{e}^{\left(\left(-{\left(\frac{n + m}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)\right)}}\right) \cdot \sqrt[3]{e^{\left(-{\left(\frac{n + m}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)}}\]
\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)}
↓
\left(\sqrt[3]{{e}^{\left(\left(-{\left(\frac{n + m}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)\right)}} \cdot \sqrt[3]{{e}^{\left(\left(-{\left(\frac{n + m}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)\right)}}\right) \cdot \sqrt[3]{e^{\left(-{\left(\frac{n + m}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)}}
double f(double K, double m, double n, double M, double l) {
        double r4196495 = K;
        double r4196496 = m;
        double r4196497 = n;
        double r4196498 = r4196496 + r4196497;
        double r4196499 = r4196495 * r4196498;
        double r4196500 = 2.0;
        double r4196501 = r4196499 / r4196500;
        double r4196502 = M;
        double r4196503 = r4196501 - r4196502;
        double r4196504 = cos(r4196503);
        double r4196505 = r4196498 / r4196500;
        double r4196506 = r4196505 - r4196502;
        double r4196507 = pow(r4196506, r4196500);
        double r4196508 = -r4196507;
        double r4196509 = l;
        double r4196510 = r4196496 - r4196497;
        double r4196511 = fabs(r4196510);
        double r4196512 = r4196509 - r4196511;
        double r4196513 = r4196508 - r4196512;
        double r4196514 = exp(r4196513);
        double r4196515 = r4196504 * r4196514;
        return r4196515;
}
↓
double f(double __attribute__((unused)) K, double m, double n, double M, double l) {
        double r4196516 = exp(1.0);
        double r4196517 = n;
        double r4196518 = m;
        double r4196519 = r4196517 + r4196518;
        double r4196520 = 2.0;
        double r4196521 = r4196519 / r4196520;
        double r4196522 = M;
        double r4196523 = r4196521 - r4196522;
        double r4196524 = pow(r4196523, r4196520);
        double r4196525 = -r4196524;
        double r4196526 = l;
        double r4196527 = r4196518 - r4196517;
        double r4196528 = fabs(r4196527);
        double r4196529 = r4196526 - r4196528;
        double r4196530 = r4196525 - r4196529;
        double r4196531 = pow(r4196516, r4196530);
        double r4196532 = cbrt(r4196531);
        double r4196533 = r4196532 * r4196532;
        double r4196534 = exp(r4196530);
        double r4196535 = cbrt(r4196534);
        double r4196536 = r4196533 * r4196535;
        return r4196536;
}

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 15.6

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

  2. Taylor expanded around 0 1.4

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

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

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

  5. Applied exp-prod1.4

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

  6. Simplified1.4

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

  8. Applied add-cube-cbrt1.4

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

  10. Applied e-exp-11.4

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

  11. Applied pow-exp1.4

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

  12. Final simplification1.4

    \[\leadsto \left(\sqrt[3]{{e}^{\left(\left(-{\left(\frac{n + m}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)\right)}} \cdot \sqrt[3]{{e}^{\left(\left(-{\left(\frac{n + m}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)\right)}}\right) \cdot \sqrt[3]{e^{\left(-{\left(\frac{n + m}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)}}\]

Reproduce

herbie shell --seed 2019172 +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)))))))