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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. Taylor expanded around 0 1.5

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

4. Applied add-cube-cbrt1.5

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

5. Applied unpow-prod-down1.5

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

6. Simplified1.5

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

8. Applied add-cube-cbrt1.5

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

9. Applied associate-*r*1.5

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

10. Final simplification1.5

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

Reproduce

herbie shell --seed 2019002 +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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