Error

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. Initial simplification15.3

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

3. Taylor expanded around 0 1.5

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

5. Applied add-cube-cbrt1.5

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

6. Applied associate-/r*1.5

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

7. Final simplification1.5

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

Reproduce

herbie shell --seed 2018360 +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)))))))

Details

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