Average Error: 15.5 → 1.3
Time: 8.3s
Precision: 64
\[\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)}\]
↓
\[\frac{1}{e^{\left(\sqrt[3]{\sqrt[3]{{\left({\left(\frac{m + n}{2} - M\right)}^{2}\right)}^{3}} + \left(\ell - \left|m - n\right|\right)} \cdot \sqrt[3]{\sqrt[3]{{\left({\left(\frac{m + n}{2} - M\right)}^{2}\right)}^{3}} + \left(\ell - \left|m - n\right|\right)}\right) \cdot \sqrt[3]{\sqrt[3]{{\left({\left(\frac{m + n}{2} - M\right)}^{2}\right)}^{3}} + \left(\ell - \left|m - n\right|\right)}}}\]
\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(\sqrt[3]{\sqrt[3]{{\left({\left(\frac{m + n}{2} - M\right)}^{2}\right)}^{3}} + \left(\ell - \left|m - n\right|\right)} \cdot \sqrt[3]{\sqrt[3]{{\left({\left(\frac{m + n}{2} - M\right)}^{2}\right)}^{3}} + \left(\ell - \left|m - n\right|\right)}\right) \cdot \sqrt[3]{\sqrt[3]{{\left({\left(\frac{m + n}{2} - M\right)}^{2}\right)}^{3}} + \left(\ell - \left|m - n\right|\right)}}}
double code(double K, double m, double n, double M, double l) {
	return (cos((((K * (m + n)) / 2.0) - M)) * exp((-pow((((m + n) / 2.0) - M), 2.0) - (l - fabs((m - n))))));
}
↓
double code(double K, double m, double n, double M, double l) {
	return (1.0 / exp(((cbrt((cbrt(pow(pow((((m + n) / 2.0) - M), 2.0), 3.0)) + (l - fabs((m - n))))) * cbrt((cbrt(pow(pow((((m + n) / 2.0) - M), 2.0), 3.0)) + (l - fabs((m - n)))))) * cbrt((cbrt(pow(pow((((m + n) / 2.0) - M), 2.0), 3.0)) + (l - fabs((m - n))))))));
}

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 15.5

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

  2. Simplified15.5

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

  3. Taylor expanded around 0 1.3

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

  5. Applied add-cbrt-cube1.3

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

  6. Simplified1.3

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

  8. Applied add-cube-cbrt1.3

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

  9. Final simplification1.3

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

Reproduce

herbie shell --seed 2020075 
(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)))))))