Average Error: 15.6 → 1.2
Time: 8.0s
Precision: 64
\[\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)}\]
↓
\[e^{\left(-\left(\left(\sqrt[3]{\sqrt[3]{{\left(\frac{m + n}{2} - M\right)}^{2}} \cdot \sqrt[3]{{\left(\frac{m + n}{2} - M\right)}^{2}}} \cdot \sqrt[3]{\sqrt[3]{{\left(\frac{m + n}{2} - M\right)}^{2}} \cdot \sqrt[3]{{\left(\frac{m + n}{2} - M\right)}^{2}}}\right) \cdot \sqrt[3]{\sqrt[3]{{\left(\frac{m + n}{2} - M\right)}^{2}} \cdot \sqrt[3]{{\left(\frac{m + n}{2} - M\right)}^{2}}}\right) \cdot \left(\sqrt[3]{{\left(\sqrt[3]{\frac{m + n}{2} - M} \cdot \sqrt[3]{\frac{m + n}{2} - M}\right)}^{2}} \cdot \sqrt[3]{{\left(\sqrt[3]{\frac{m + n}{2} - M}\right)}^{2}}\right)\right) - \left(\ell - \left|m - n\right|\right)}\]
\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)}
↓
e^{\left(-\left(\left(\sqrt[3]{\sqrt[3]{{\left(\frac{m + n}{2} - M\right)}^{2}} \cdot \sqrt[3]{{\left(\frac{m + n}{2} - M\right)}^{2}}} \cdot \sqrt[3]{\sqrt[3]{{\left(\frac{m + n}{2} - M\right)}^{2}} \cdot \sqrt[3]{{\left(\frac{m + n}{2} - M\right)}^{2}}}\right) \cdot \sqrt[3]{\sqrt[3]{{\left(\frac{m + n}{2} - M\right)}^{2}} \cdot \sqrt[3]{{\left(\frac{m + n}{2} - M\right)}^{2}}}\right) \cdot \left(\sqrt[3]{{\left(\sqrt[3]{\frac{m + n}{2} - M} \cdot \sqrt[3]{\frac{m + n}{2} - M}\right)}^{2}} \cdot \sqrt[3]{{\left(\sqrt[3]{\frac{m + n}{2} - M}\right)}^{2}}\right)\right) - \left(\ell - \left|m - n\right|\right)}
double f(double K, double m, double n, double M, double l) {
        double r162166 = K;
        double r162167 = m;
        double r162168 = n;
        double r162169 = r162167 + r162168;
        double r162170 = r162166 * r162169;
        double r162171 = 2.0;
        double r162172 = r162170 / r162171;
        double r162173 = M;
        double r162174 = r162172 - r162173;
        double r162175 = cos(r162174);
        double r162176 = r162169 / r162171;
        double r162177 = r162176 - r162173;
        double r162178 = pow(r162177, r162171);
        double r162179 = -r162178;
        double r162180 = l;
        double r162181 = r162167 - r162168;
        double r162182 = fabs(r162181);
        double r162183 = r162180 - r162182;
        double r162184 = r162179 - r162183;
        double r162185 = exp(r162184);
        double r162186 = r162175 * r162185;
        return r162186;
}
↓
double f(double __attribute__((unused)) K, double m, double n, double M, double l) {
        double r162187 = m;
        double r162188 = n;
        double r162189 = r162187 + r162188;
        double r162190 = 2.0;
        double r162191 = r162189 / r162190;
        double r162192 = M;
        double r162193 = r162191 - r162192;
        double r162194 = pow(r162193, r162190);
        double r162195 = cbrt(r162194);
        double r162196 = r162195 * r162195;
        double r162197 = cbrt(r162196);
        double r162198 = r162197 * r162197;
        double r162199 = r162198 * r162197;
        double r162200 = cbrt(r162193);
        double r162201 = r162200 * r162200;
        double r162202 = pow(r162201, r162190);
        double r162203 = cbrt(r162202);
        double r162204 = pow(r162200, r162190);
        double r162205 = cbrt(r162204);
        double r162206 = r162203 * r162205;
        double r162207 = r162199 * r162206;
        double r162208 = -r162207;
        double r162209 = l;
        double r162210 = r162187 - r162188;
        double r162211 = fabs(r162210);
        double r162212 = r162209 - r162211;
        double r162213 = r162208 - r162212;
        double r162214 = exp(r162213);
        return r162214;
}

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 1.2

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

  4. Applied add-cube-cbrt1.2

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

  6. Applied add-cube-cbrt1.2

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

  8. Applied add-cube-cbrt1.2

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

  9. Applied unpow-prod-down1.2

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

  10. Applied cbrt-prod1.2

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

  11. Final simplification1.2

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

Reproduce

herbie shell --seed 2020001 +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)))))))