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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 1.5

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

4. Applied *-un-lft-identity1.5

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

5. Applied *-un-lft-identity1.5

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

6. Applied distribute-rgt-neg-in1.5

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

7. Applied distribute-lft-out--1.5

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

8. Applied exp-prod1.5

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

9. Simplified1.5

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

10. Final simplification1.5

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

Runtime

Time bar (total: 52.9s)Debug logProfile

herbie shell --seed 2018304 
(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)))))))