Maksimov and Kolovsky, Equation (32)

1. Initial program 74.1%

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

   1. *-commutative74.1%

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

   2. associate-*r/74.1%

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

   3. associate--r-74.1%

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

   4. +-commutative74.1%

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

   5. associate-+r-74.1%

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

   6. unsub-neg74.1%

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

   7. associate--r+74.1%

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

   8. +-commutative74.1%

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

   9. associate--r+74.1%

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

3. Simplified74.1%

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

4. Taylor expanded in K around 0 97.8%

   \[\leadsto \color{blue}{\cos \left(-M\right)} \cdot e^{\left(\left|m - n\right| - \ell\right) - {\left(\frac{m + n}{2} - M\right)}^{2}} \]
5. Step-by-step derivation

   1. cos-neg97.8%

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

6. Simplified97.8%

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

7. Final simplification97.8%

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

1. Split input into 2 regimes

2. if n < 4.20000000000000009e36

   1. Initial program 75.3%

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

      1. *-commutative75.3%

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

      2. associate-*r/75.3%

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

      3. associate--r-75.3%

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

      4. +-commutative75.3%

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

      5. associate-+r-75.3%

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

      6. unsub-neg75.3%

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

      7. associate--r+75.3%

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

      8. +-commutative75.3%

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

      9. associate--r+75.3%

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

   3. Simplified75.3%

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

   4. Taylor expanded in K around 0 97.1%

      \[\leadsto \color{blue}{\cos \left(-M\right)} \cdot e^{\left(\left|m - n\right| - \ell\right) - {\left(\frac{m + n}{2} - M\right)}^{2}} \]
   5. Step-by-step derivation

      1. cos-neg97.1%

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

   6. Simplified97.1%

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

   7. Taylor expanded in n around 0 78.0%

      \[\leadsto \cos M \cdot e^{\left(\left|m - n\right| - \ell\right) - \color{blue}{\left({\left(0.5 \cdot m - M\right)}^{2} + n \cdot \left(0.5 \cdot m - M\right)\right)}} \]
   8. Step-by-step derivation

      1. unpow278.0%

         \[\leadsto \cos M \cdot e^{\left(\left|m - n\right| - \ell\right) - \left(\color{blue}{\left(0.5 \cdot m - M\right) \cdot \left(0.5 \cdot m - M\right)} + n \cdot \left(0.5 \cdot m - M\right)\right)} \]

      2. distribute-rgt-out82.1%

         \[\leadsto \cos M \cdot e^{\left(\left|m - n\right| - \ell\right) - \color{blue}{\left(0.5 \cdot m - M\right) \cdot \left(\left(0.5 \cdot m - M\right) + n\right)}} \]

   9. Simplified82.1%

      \[\leadsto \cos M \cdot e^{\left(\left|m - n\right| - \ell\right) - \color{blue}{\left(0.5 \cdot m - M\right) \cdot \left(\left(0.5 \cdot m - M\right) + n\right)}} \]

   if 4.20000000000000009e36 < n

   1. Initial program 70.5%

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

      1. *-commutative70.5%

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

      2. associate-*r/70.5%

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

      3. associate--r-70.5%

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

      4. +-commutative70.5%

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

      5. associate-+r-70.5%

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

      6. unsub-neg70.5%

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

      7. associate--r+70.5%

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

      8. +-commutative70.5%

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

      9. associate--r+70.5%

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

   3. Simplified70.5%

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

   4. Taylor expanded in K around 0 100.0%

      \[\leadsto \color{blue}{\cos \left(-M\right)} \cdot e^{\left(\left|m - n\right| - \ell\right) - {\left(\frac{m + n}{2} - M\right)}^{2}} \]
   5. Step-by-step derivation

      1. cos-neg100.0%

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

   6. Simplified100.0%

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

   7. Taylor expanded in n around inf 90.3%

      \[\leadsto \cos M \cdot e^{\left(\left|m - n\right| - \ell\right) - \color{blue}{0.25 \cdot {n}^{2}}} \]
   8. Step-by-step derivation

      1. *-commutative90.3%

         \[\leadsto \cos M \cdot e^{\left(\left|m - n\right| - \ell\right) - \color{blue}{{n}^{2} \cdot 0.25}} \]

      2. unpow290.3%

         \[\leadsto \cos M \cdot e^{\left(\left|m - n\right| - \ell\right) - \color{blue}{\left(n \cdot n\right)} \cdot 0.25} \]

   9. Simplified90.3%

      \[\leadsto \cos M \cdot e^{\left(\left|m - n\right| - \ell\right) - \color{blue}{\left(n \cdot n\right) \cdot 0.25}} \]
3. Recombined 2 regimes into one program.

4. Final simplification84.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq 4.2 \cdot 10^{+36}:\\ \;\;\;\;\cos M \cdot e^{\left(\left|m - n\right| - \ell\right) + \left(n + \left(m \cdot 0.5 - M\right)\right) \cdot \left(M - m \cdot 0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{\left(\left|m - n\right| - \ell\right) - \left(n \cdot n\right) \cdot 0.25}\\ \end{array} \]

1. Split input into 3 regimes

2. if n < 6.49999999999999977e-128

   1. Initial program 72.6%

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

      1. *-commutative72.6%

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

      2. associate-*r/72.6%

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

      3. associate--r-72.6%

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

      4. +-commutative72.6%

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

      5. associate-+r-72.6%

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

      6. unsub-neg72.6%

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

      7. associate--r+72.6%

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

      8. +-commutative72.6%

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

      9. associate--r+72.6%

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

   3. Simplified72.6%

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

   4. Taylor expanded in K around 0 97.1%

      \[\leadsto \color{blue}{\cos \left(-M\right)} \cdot e^{\left(\left|m - n\right| - \ell\right) - {\left(\frac{m + n}{2} - M\right)}^{2}} \]
   5. Step-by-step derivation

      1. cos-neg97.1%

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

   6. Simplified97.1%

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

   7. Taylor expanded in n around 0 74.1%

      \[\leadsto \cos M \cdot e^{\left(\left|m - n\right| - \ell\right) - \color{blue}{\left({\left(0.5 \cdot m - M\right)}^{2} + n \cdot \left(0.5 \cdot m - M\right)\right)}} \]
   8. Step-by-step derivation

      1. unpow274.1%

         \[\leadsto \cos M \cdot e^{\left(\left|m - n\right| - \ell\right) - \left(\color{blue}{\left(0.5 \cdot m - M\right) \cdot \left(0.5 \cdot m - M\right)} + n \cdot \left(0.5 \cdot m - M\right)\right)} \]

      2. distribute-rgt-out79.1%

         \[\leadsto \cos M \cdot e^{\left(\left|m - n\right| - \ell\right) - \color{blue}{\left(0.5 \cdot m - M\right) \cdot \left(\left(0.5 \cdot m - M\right) + n\right)}} \]

   9. Simplified79.1%

      \[\leadsto \cos M \cdot e^{\left(\left|m - n\right| - \ell\right) - \color{blue}{\left(0.5 \cdot m - M\right) \cdot \left(\left(0.5 \cdot m - M\right) + n\right)}} \]

   10. Taylor expanded in l around 0 72.3%

       \[\leadsto \cos M \cdot \color{blue}{e^{\left|m - n\right| - \left(0.5 \cdot m - M\right) \cdot \left(\left(n + 0.5 \cdot m\right) - M\right)}} \]
   11. Step-by-step derivation

       1. *-commutative72.3%

          \[\leadsto \cos M \cdot e^{\left|m - n\right| - \color{blue}{\left(\left(n + 0.5 \cdot m\right) - M\right) \cdot \left(0.5 \cdot m - M\right)}} \]

       2. *-commutative72.3%

          \[\leadsto \cos M \cdot e^{\left|m - n\right| - \left(\left(n + \color{blue}{m \cdot 0.5}\right) - M\right) \cdot \left(0.5 \cdot m - M\right)} \]

       3. associate-+r-72.3%

          \[\leadsto \cos M \cdot e^{\left|m - n\right| - \color{blue}{\left(n + \left(m \cdot 0.5 - M\right)\right)} \cdot \left(0.5 \cdot m - M\right)} \]

       4. sub-neg72.3%

          \[\leadsto \cos M \cdot e^{\left|m - n\right| - \left(n + \color{blue}{\left(m \cdot 0.5 + \left(-M\right)\right)}\right) \cdot \left(0.5 \cdot m - M\right)} \]

       5. *-commutative72.3%

          \[\leadsto \cos M \cdot e^{\left|m - n\right| - \left(n + \left(\color{blue}{0.5 \cdot m} + \left(-M\right)\right)\right) \cdot \left(0.5 \cdot m - M\right)} \]

       6. +-commutative72.3%

          \[\leadsto \cos M \cdot e^{\left|m - n\right| - \left(n + \color{blue}{\left(\left(-M\right) + 0.5 \cdot m\right)}\right) \cdot \left(0.5 \cdot m - M\right)} \]

       7. mul-1-neg72.3%

          \[\leadsto \cos M \cdot e^{\left|m - n\right| - \left(n + \left(\color{blue}{-1 \cdot M} + 0.5 \cdot m\right)\right) \cdot \left(0.5 \cdot m - M\right)} \]

       8. sub-neg72.3%

          \[\leadsto \cos M \cdot e^{\left|m - n\right| - \left(n + \left(-1 \cdot M + 0.5 \cdot m\right)\right) \cdot \color{blue}{\left(0.5 \cdot m + \left(-M\right)\right)}} \]

       9. +-commutative72.3%

          \[\leadsto \cos M \cdot e^{\left|m - n\right| - \left(n + \left(-1 \cdot M + 0.5 \cdot m\right)\right) \cdot \color{blue}{\left(\left(-M\right) + 0.5 \cdot m\right)}} \]

       10. distribute-rgt-in66.1%

           \[\leadsto \cos M \cdot e^{\left|m - n\right| - \color{blue}{\left(\left(-M\right) \cdot \left(n + \left(-1 \cdot M + 0.5 \cdot m\right)\right) + \left(0.5 \cdot m\right) \cdot \left(n + \left(-1 \cdot M + 0.5 \cdot m\right)\right)\right)}} \]

       11. associate--r+66.1%

           \[\leadsto \cos M \cdot e^{\color{blue}{\left(\left|m - n\right| - \left(-M\right) \cdot \left(n + \left(-1 \cdot M + 0.5 \cdot m\right)\right)\right) - \left(0.5 \cdot m\right) \cdot \left(n + \left(-1 \cdot M + 0.5 \cdot m\right)\right)}} \]

       12. mul-1-neg66.1%

           \[\leadsto \cos M \cdot e^{\left(\left|m - n\right| - \left(-M\right) \cdot \left(n + \left(-1 \cdot M + 0.5 \cdot m\right)\right)\right) - \left(0.5 \cdot m\right) \cdot \left(n + \left(\color{blue}{\left(-M\right)} + 0.5 \cdot m\right)\right)} \]

   12. Simplified72.3%

       \[\leadsto \cos M \cdot \color{blue}{e^{\left|n - m\right| - \left(m \cdot 0.5 - M\right) \cdot \left(\left(n - M\right) + m \cdot 0.5\right)}} \]

   if 6.49999999999999977e-128 < n < 2.49999999999999988e36

   1. Initial program 88.5%

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

      1. *-commutative88.5%

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

      2. associate-*r/88.5%

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

      3. associate--r-88.5%

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

      4. +-commutative88.5%

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

      5. associate-+r-88.5%

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

      6. unsub-neg88.5%

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

      7. associate--r+88.5%

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

      8. +-commutative88.5%

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

      9. associate--r+88.5%

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

   3. Simplified88.5%

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

   4. Taylor expanded in K around 0 97.0%

      \[\leadsto \color{blue}{\cos \left(-M\right)} \cdot e^{\left(\left|m - n\right| - \ell\right) - {\left(\frac{m + n}{2} - M\right)}^{2}} \]
   5. Step-by-step derivation

      1. cos-neg97.0%

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

   6. Simplified97.0%

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

   7. Taylor expanded in M around inf 76.1%

      \[\leadsto \cos M \cdot e^{\left(\left|m - n\right| - \ell\right) - \color{blue}{{M}^{2}}} \]
   8. Step-by-step derivation

      1. unpow276.1%

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

   9. Simplified76.1%

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

   if 2.49999999999999988e36 < n

   1. Initial program 70.5%

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

      1. *-commutative70.5%

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

      2. associate-*r/70.5%

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

      3. associate--r-70.5%

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

      4. +-commutative70.5%

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

      5. associate-+r-70.5%

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

      6. unsub-neg70.5%

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

      7. associate--r+70.5%

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

      8. +-commutative70.5%

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

      9. associate--r+70.5%

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

   3. Simplified70.5%

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

   4. Taylor expanded in K around 0 100.0%

      \[\leadsto \color{blue}{\cos \left(-M\right)} \cdot e^{\left(\left|m - n\right| - \ell\right) - {\left(\frac{m + n}{2} - M\right)}^{2}} \]
   5. Step-by-step derivation

      1. cos-neg100.0%

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

   6. Simplified100.0%

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

   7. Taylor expanded in n around inf 90.3%

      \[\leadsto \cos M \cdot e^{\left(\left|m - n\right| - \ell\right) - \color{blue}{0.25 \cdot {n}^{2}}} \]
   8. Step-by-step derivation

      1. *-commutative90.3%

         \[\leadsto \cos M \cdot e^{\left(\left|m - n\right| - \ell\right) - \color{blue}{{n}^{2} \cdot 0.25}} \]

      2. unpow290.3%

         \[\leadsto \cos M \cdot e^{\left(\left|m - n\right| - \ell\right) - \color{blue}{\left(n \cdot n\right)} \cdot 0.25} \]

   9. Simplified90.3%

      \[\leadsto \cos M \cdot e^{\left(\left|m - n\right| - \ell\right) - \color{blue}{\left(n \cdot n\right) \cdot 0.25}} \]
3. Recombined 3 regimes into one program.

4. Final simplification77.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq 6.5 \cdot 10^{-128}:\\ \;\;\;\;\cos M \cdot e^{\left|m - n\right| + \left(m \cdot 0.5 - M\right) \cdot \left(\left(M - n\right) - m \cdot 0.5\right)}\\ \mathbf{elif}\;n \leq 2.5 \cdot 10^{+36}:\\ \;\;\;\;\cos M \cdot e^{\left(\left|m - n\right| - \ell\right) - M \cdot M}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{\left(\left|m - n\right| - \ell\right) - \left(n \cdot n\right) \cdot 0.25}\\ \end{array} \]

1. Split input into 3 regimes

2. if n < 2.1499999999999999e-112

   1. Initial program 72.7%

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

      1. *-commutative72.7%

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

      2. associate-*r/72.7%

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

      3. associate--r-72.7%

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

      4. +-commutative72.7%

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

      5. associate-+r-72.7%

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

      6. unsub-neg72.7%

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

      7. associate--r+72.7%

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

      8. +-commutative72.7%

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

      9. associate--r+72.7%

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

   3. Simplified72.7%

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

   4. Taylor expanded in K around 0 97.1%

      \[\leadsto \color{blue}{\cos \left(-M\right)} \cdot e^{\left(\left|m - n\right| - \ell\right) - {\left(\frac{m + n}{2} - M\right)}^{2}} \]
   5. Step-by-step derivation

      1. cos-neg97.1%

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

   6. Simplified97.1%

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

   7. Taylor expanded in m around inf 61.8%

      \[\leadsto \cos M \cdot e^{\left(\left|m - n\right| - \ell\right) - \color{blue}{0.25 \cdot {m}^{2}}} \]
   8. Step-by-step derivation

      1. *-commutative61.8%

         \[\leadsto \cos M \cdot e^{\left(\left|m - n\right| - \ell\right) - \color{blue}{{m}^{2} \cdot 0.25}} \]

      2. unpow261.8%

         \[\leadsto \cos M \cdot e^{\left(\left|m - n\right| - \ell\right) - \color{blue}{\left(m \cdot m\right)} \cdot 0.25} \]

   9. Simplified61.8%

      \[\leadsto \cos M \cdot e^{\left(\left|m - n\right| - \ell\right) - \color{blue}{\left(m \cdot m\right) \cdot 0.25}} \]

   if 2.1499999999999999e-112 < n < 1.0499999999999999e35

   1. Initial program 88.1%

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

      1. *-commutative88.1%

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

      2. associate-*r/88.1%

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

      3. associate--r-88.1%

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

      4. +-commutative88.1%

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

      5. associate-+r-88.1%

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

      6. unsub-neg88.1%

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

      7. associate--r+88.1%

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

      8. +-commutative88.1%

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

      9. associate--r+88.1%

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

   3. Simplified88.1%

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

   4. Taylor expanded in M around inf 72.7%

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

      1. unpow275.4%

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

   6. Simplified72.7%

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

   7. Taylor expanded in n around inf 75.4%

      \[\leadsto \cos \color{blue}{\left(0.5 \cdot \left(n \cdot K\right)\right)} \cdot e^{\left(\left|m - n\right| - \ell\right) - M \cdot M} \]

   if 1.0499999999999999e35 < n

   1. Initial program 70.5%

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

      1. *-commutative70.5%

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

      2. associate-*r/70.5%

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

      3. associate--r-70.5%

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

      4. +-commutative70.5%

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

      5. associate-+r-70.5%

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

      6. unsub-neg70.5%

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

      7. associate--r+70.5%

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

      8. +-commutative70.5%

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

      9. associate--r+70.5%

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

   3. Simplified70.5%

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

   4. Taylor expanded in K around 0 100.0%

      \[\leadsto \color{blue}{\cos \left(-M\right)} \cdot e^{\left(\left|m - n\right| - \ell\right) - {\left(\frac{m + n}{2} - M\right)}^{2}} \]
   5. Step-by-step derivation

      1. cos-neg100.0%

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

   6. Simplified100.0%

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

   7. Taylor expanded in n around inf 90.3%

      \[\leadsto \cos M \cdot e^{\left(\left|m - n\right| - \ell\right) - \color{blue}{0.25 \cdot {n}^{2}}} \]
   8. Step-by-step derivation

      1. *-commutative90.3%

         \[\leadsto \cos M \cdot e^{\left(\left|m - n\right| - \ell\right) - \color{blue}{{n}^{2} \cdot 0.25}} \]

      2. unpow290.3%

         \[\leadsto \cos M \cdot e^{\left(\left|m - n\right| - \ell\right) - \color{blue}{\left(n \cdot n\right)} \cdot 0.25} \]

   9. Simplified90.3%

      \[\leadsto \cos M \cdot e^{\left(\left|m - n\right| - \ell\right) - \color{blue}{\left(n \cdot n\right) \cdot 0.25}} \]
3. Recombined 3 regimes into one program.

4. Final simplification70.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq 2.15 \cdot 10^{-112}:\\ \;\;\;\;\cos M \cdot e^{\left(\left|m - n\right| - \ell\right) - 0.25 \cdot \left(m \cdot m\right)}\\ \mathbf{elif}\;n \leq 1.05 \cdot 10^{+35}:\\ \;\;\;\;e^{\left(\left|m - n\right| - \ell\right) - M \cdot M} \cdot \cos \left(0.5 \cdot \left(n \cdot K\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{\left(\left|m - n\right| - \ell\right) - \left(n \cdot n\right) \cdot 0.25}\\ \end{array} \]

1. Split input into 3 regimes

2. if n < 5.79999999999999985e-112

   1. Initial program 72.7%

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

      1. *-commutative72.7%

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

      2. associate-*r/72.7%

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

      3. associate--r-72.7%

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

      4. +-commutative72.7%

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

      5. associate-+r-72.7%

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

      6. unsub-neg72.7%

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

      7. associate--r+72.7%

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

      8. +-commutative72.7%

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

      9. associate--r+72.7%

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

   3. Simplified72.7%

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

   4. Taylor expanded in K around 0 97.1%

      \[\leadsto \color{blue}{\cos \left(-M\right)} \cdot e^{\left(\left|m - n\right| - \ell\right) - {\left(\frac{m + n}{2} - M\right)}^{2}} \]
   5. Step-by-step derivation

      1. cos-neg97.1%

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

   6. Simplified97.1%

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

   7. Taylor expanded in n around 0 74.3%

      \[\leadsto \cos M \cdot e^{\left(\left|m - n\right| - \ell\right) - \color{blue}{\left({\left(0.5 \cdot m - M\right)}^{2} + n \cdot \left(0.5 \cdot m - M\right)\right)}} \]
   8. Step-by-step derivation

      1. unpow274.3%

         \[\leadsto \cos M \cdot e^{\left(\left|m - n\right| - \ell\right) - \left(\color{blue}{\left(0.5 \cdot m - M\right) \cdot \left(0.5 \cdot m - M\right)} + n \cdot \left(0.5 \cdot m - M\right)\right)} \]

      2. distribute-rgt-out79.2%

         \[\leadsto \cos M \cdot e^{\left(\left|m - n\right| - \ell\right) - \color{blue}{\left(0.5 \cdot m - M\right) \cdot \left(\left(0.5 \cdot m - M\right) + n\right)}} \]

   9. Simplified79.2%

      \[\leadsto \cos M \cdot e^{\left(\left|m - n\right| - \ell\right) - \color{blue}{\left(0.5 \cdot m - M\right) \cdot \left(\left(0.5 \cdot m - M\right) + n\right)}} \]

   10. Taylor expanded in M around 0 59.4%

       \[\leadsto \cos M \cdot \color{blue}{e^{\left|m - n\right| - \left(0.5 \cdot \left(\left(n + 0.5 \cdot m\right) \cdot m\right) + \ell\right)}} \]

   if 5.79999999999999985e-112 < n < 6.7999999999999999e34

   1. Initial program 88.1%

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

      1. *-commutative88.1%

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

      2. associate-*r/88.1%

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

      3. associate--r-88.1%

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

      4. +-commutative88.1%

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

      5. associate-+r-88.1%

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

      6. unsub-neg88.1%

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

      7. associate--r+88.1%

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

      8. +-commutative88.1%

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

      9. associate--r+88.1%

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

   3. Simplified88.1%

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

   4. Taylor expanded in M around inf 72.7%

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

      1. unpow275.4%

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

   6. Simplified72.7%

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

   7. Taylor expanded in n around inf 75.4%

      \[\leadsto \cos \color{blue}{\left(0.5 \cdot \left(n \cdot K\right)\right)} \cdot e^{\left(\left|m - n\right| - \ell\right) - M \cdot M} \]

   if 6.7999999999999999e34 < n

   1. Initial program 70.5%

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

      1. *-commutative70.5%

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

      2. associate-*r/70.5%

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

      3. associate--r-70.5%

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

      4. +-commutative70.5%

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

      5. associate-+r-70.5%

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

      6. unsub-neg70.5%

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

      7. associate--r+70.5%

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

      8. +-commutative70.5%

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

      9. associate--r+70.5%

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

   3. Simplified70.5%

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

   4. Taylor expanded in K around 0 100.0%

      \[\leadsto \color{blue}{\cos \left(-M\right)} \cdot e^{\left(\left|m - n\right| - \ell\right) - {\left(\frac{m + n}{2} - M\right)}^{2}} \]
   5. Step-by-step derivation

      1. cos-neg100.0%

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

   6. Simplified100.0%

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

   7. Taylor expanded in n around inf 90.3%

      \[\leadsto \cos M \cdot e^{\left(\left|m - n\right| - \ell\right) - \color{blue}{0.25 \cdot {n}^{2}}} \]
   8. Step-by-step derivation

      1. *-commutative90.3%

         \[\leadsto \cos M \cdot e^{\left(\left|m - n\right| - \ell\right) - \color{blue}{{n}^{2} \cdot 0.25}} \]

      2. unpow290.3%

         \[\leadsto \cos M \cdot e^{\left(\left|m - n\right| - \ell\right) - \color{blue}{\left(n \cdot n\right)} \cdot 0.25} \]

   9. Simplified90.3%

      \[\leadsto \cos M \cdot e^{\left(\left|m - n\right| - \ell\right) - \color{blue}{\left(n \cdot n\right) \cdot 0.25}} \]
3. Recombined 3 regimes into one program.

4. Final simplification68.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq 5.8 \cdot 10^{-112}:\\ \;\;\;\;\cos M \cdot e^{\left|m - n\right| - \left(0.5 \cdot \left(m \cdot \left(n + m \cdot 0.5\right)\right) + \ell\right)}\\ \mathbf{elif}\;n \leq 6.8 \cdot 10^{+34}:\\ \;\;\;\;e^{\left(\left|m - n\right| - \ell\right) - M \cdot M} \cdot \cos \left(0.5 \cdot \left(n \cdot K\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{\left(\left|m - n\right| - \ell\right) - \left(n \cdot n\right) \cdot 0.25}\\ \end{array} \]

1. Split input into 3 regimes

2. if n < 3.8000000000000002e-128

   1. Initial program 72.6%

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

      1. *-commutative72.6%

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

      2. associate-*r/72.6%

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

      3. associate--r-72.6%

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

      4. +-commutative72.6%

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

      5. associate-+r-72.6%

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

      6. unsub-neg72.6%

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

      7. associate--r+72.6%

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

      8. +-commutative72.6%

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

      9. associate--r+72.6%

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

   3. Simplified72.6%

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

   4. Taylor expanded in K around 0 97.1%

      \[\leadsto \color{blue}{\cos \left(-M\right)} \cdot e^{\left(\left|m - n\right| - \ell\right) - {\left(\frac{m + n}{2} - M\right)}^{2}} \]
   5. Step-by-step derivation

      1. cos-neg97.1%

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

   6. Simplified97.1%

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

   7. Taylor expanded in m around inf 61.5%

      \[\leadsto \cos M \cdot e^{\left(\left|m - n\right| - \ell\right) - \color{blue}{0.25 \cdot {m}^{2}}} \]
   8. Step-by-step derivation

      1. *-commutative61.5%

         \[\leadsto \cos M \cdot e^{\left(\left|m - n\right| - \ell\right) - \color{blue}{{m}^{2} \cdot 0.25}} \]

      2. unpow261.5%

         \[\leadsto \cos M \cdot e^{\left(\left|m - n\right| - \ell\right) - \color{blue}{\left(m \cdot m\right)} \cdot 0.25} \]

   9. Simplified61.5%

      \[\leadsto \cos M \cdot e^{\left(\left|m - n\right| - \ell\right) - \color{blue}{\left(m \cdot m\right) \cdot 0.25}} \]

   if 3.8000000000000002e-128 < n < 1.00000000000000004e36

   1. Initial program 88.5%

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

      1. *-commutative88.5%

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

      2. associate-*r/88.5%

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

      3. associate--r-88.5%

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

      4. +-commutative88.5%

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

      5. associate-+r-88.5%

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

      6. unsub-neg88.5%

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

      7. associate--r+88.5%

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

      8. +-commutative88.5%

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

      9. associate--r+88.5%

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

   3. Simplified88.5%

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

   4. Taylor expanded in K around 0 97.0%

      \[\leadsto \color{blue}{\cos \left(-M\right)} \cdot e^{\left(\left|m - n\right| - \ell\right) - {\left(\frac{m + n}{2} - M\right)}^{2}} \]
   5. Step-by-step derivation

      1. cos-neg97.0%

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

   6. Simplified97.0%

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

   7. Taylor expanded in M around inf 76.1%

      \[\leadsto \cos M \cdot e^{\left(\left|m - n\right| - \ell\right) - \color{blue}{{M}^{2}}} \]
   8. Step-by-step derivation

      1. unpow276.1%

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

   9. Simplified76.1%

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

   if 1.00000000000000004e36 < n

   1. Initial program 70.5%

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

      1. *-commutative70.5%

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

      2. associate-*r/70.5%

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

      3. associate--r-70.5%

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

      4. +-commutative70.5%

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

      5. associate-+r-70.5%

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

      6. unsub-neg70.5%

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

      7. associate--r+70.5%

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

      8. +-commutative70.5%

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

      9. associate--r+70.5%

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

   3. Simplified70.5%

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

   4. Taylor expanded in K around 0 100.0%

      \[\leadsto \color{blue}{\cos \left(-M\right)} \cdot e^{\left(\left|m - n\right| - \ell\right) - {\left(\frac{m + n}{2} - M\right)}^{2}} \]
   5. Step-by-step derivation

      1. cos-neg100.0%

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

   6. Simplified100.0%

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

   7. Taylor expanded in n around inf 90.3%

      \[\leadsto \cos M \cdot e^{\left(\left|m - n\right| - \ell\right) - \color{blue}{0.25 \cdot {n}^{2}}} \]
   8. Step-by-step derivation

      1. *-commutative90.3%

         \[\leadsto \cos M \cdot e^{\left(\left|m - n\right| - \ell\right) - \color{blue}{{n}^{2} \cdot 0.25}} \]

      2. unpow290.3%

         \[\leadsto \cos M \cdot e^{\left(\left|m - n\right| - \ell\right) - \color{blue}{\left(n \cdot n\right)} \cdot 0.25} \]

   9. Simplified90.3%

      \[\leadsto \cos M \cdot e^{\left(\left|m - n\right| - \ell\right) - \color{blue}{\left(n \cdot n\right) \cdot 0.25}} \]
3. Recombined 3 regimes into one program.

4. Final simplification70.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq 3.8 \cdot 10^{-128}:\\ \;\;\;\;\cos M \cdot e^{\left(\left|m - n\right| - \ell\right) - 0.25 \cdot \left(m \cdot m\right)}\\ \mathbf{elif}\;n \leq 10^{+36}:\\ \;\;\;\;\cos M \cdot e^{\left(\left|m - n\right| - \ell\right) - M \cdot M}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{\left(\left|m - n\right| - \ell\right) - \left(n \cdot n\right) \cdot 0.25}\\ \end{array} \]

1. Split input into 2 regimes

2. if M < -1.6e20 or 2.05e43 < M

   1. Initial program 78.8%

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

      1. *-commutative78.8%

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

      2. associate-*r/78.8%

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

      3. associate--r-78.8%

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

      4. +-commutative78.8%

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

      5. associate-+r-78.8%

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

      6. unsub-neg78.8%

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

      7. associate--r+78.8%

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

      8. +-commutative78.8%

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

      9. associate--r+78.8%

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

   3. Simplified78.8%

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

   4. Taylor expanded in K around 0 100.0%

      \[\leadsto \color{blue}{\cos \left(-M\right)} \cdot e^{\left(\left|m - n\right| - \ell\right) - {\left(\frac{m + n}{2} - M\right)}^{2}} \]
   5. Step-by-step derivation

      1. cos-neg100.0%

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

   6. Simplified100.0%

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

   7. Taylor expanded in M around inf 83.3%

      \[\leadsto \cos M \cdot e^{\left(\left|m - n\right| - \ell\right) - \color{blue}{{M}^{2}}} \]
   8. Step-by-step derivation

      1. unpow283.3%

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

   9. Simplified83.3%

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

   10. Taylor expanded in l around 0 81.6%

       \[\leadsto \cos M \cdot \color{blue}{e^{\left|m - n\right| - {M}^{2}}} \]
   11. Step-by-step derivation

       1. fabs-sub81.6%

          \[\leadsto \cos M \cdot e^{\color{blue}{\left|n - m\right|} - {M}^{2}} \]

       2. unpow281.6%

          \[\leadsto \cos M \cdot e^{\left|n - m\right| - \color{blue}{M \cdot M}} \]

   12. Simplified81.6%

       \[\leadsto \cos M \cdot \color{blue}{e^{\left|n - m\right| - M \cdot M}} \]

   if -1.6e20 < M < 2.05e43

   1. Initial program 70.1%

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

      1. *-commutative70.1%

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

      2. associate-*r/70.1%

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

      3. associate--r-70.1%

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

      4. +-commutative70.1%

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

      5. associate-+r-70.1%

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

      6. unsub-neg70.1%

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

      7. associate--r+70.1%

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

      8. +-commutative70.1%

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

      9. associate--r+70.1%

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

   3. Simplified70.1%

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

   4. Taylor expanded in K around 0 95.9%

      \[\leadsto \color{blue}{\cos \left(-M\right)} \cdot e^{\left(\left|m - n\right| - \ell\right) - {\left(\frac{m + n}{2} - M\right)}^{2}} \]
   5. Step-by-step derivation

      1. cos-neg95.9%

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

   6. Simplified95.9%

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

   7. Taylor expanded in M around inf 33.2%

      \[\leadsto \cos M \cdot e^{\left(\left|m - n\right| - \ell\right) - \color{blue}{{M}^{2}}} \]
   8. Step-by-step derivation

      1. unpow233.2%

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

   9. Simplified33.2%

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

   10. Taylor expanded in M around 0 31.8%

       \[\leadsto \cos M \cdot \color{blue}{e^{\left|m - n\right| - \ell}} \]
   11. Step-by-step derivation

       1. fabs-sub31.8%

          \[\leadsto \cos M \cdot e^{\color{blue}{\left|n - m\right|} - \ell} \]

   12. Simplified31.8%

       \[\leadsto \cos M \cdot \color{blue}{e^{\left|n - m\right| - \ell}} \]

   13. Taylor expanded in M around 0 22.1%

       \[\leadsto \color{blue}{-0.5 \cdot \left({M}^{2} \cdot e^{\left|n - m\right| - \ell}\right) + e^{\left|n - m\right| - \ell}} \]
   14. Step-by-step derivation

       1. +-commutative22.1%

          \[\leadsto \color{blue}{e^{\left|n - m\right| - \ell} + -0.5 \cdot \left({M}^{2} \cdot e^{\left|n - m\right| - \ell}\right)} \]

       2. *-lft-identity22.1%

          \[\leadsto \color{blue}{1 \cdot e^{\left|n - m\right| - \ell}} + -0.5 \cdot \left({M}^{2} \cdot e^{\left|n - m\right| - \ell}\right) \]

       3. associate-*r*22.1%

          \[\leadsto 1 \cdot e^{\left|n - m\right| - \ell} + \color{blue}{\left(-0.5 \cdot {M}^{2}\right) \cdot e^{\left|n - m\right| - \ell}} \]

       4. distribute-rgt-out32.6%

          \[\leadsto \color{blue}{e^{\left|n - m\right| - \ell} \cdot \left(1 + -0.5 \cdot {M}^{2}\right)} \]

       5. unpow232.6%

          \[\leadsto e^{\left|n - m\right| - \ell} \cdot \left(1 + -0.5 \cdot \color{blue}{\left(M \cdot M\right)}\right) \]

   15. Simplified32.6%

       \[\leadsto \color{blue}{e^{\left|n - m\right| - \ell} \cdot \left(1 + -0.5 \cdot \left(M \cdot M\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification55.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq -1.6 \cdot 10^{+20} \lor \neg \left(M \leq 2.05 \cdot 10^{+43}\right):\\ \;\;\;\;\cos M \cdot e^{\left|m - n\right| - M \cdot M}\\ \mathbf{else}:\\ \;\;\;\;e^{\left|m - n\right| - \ell} \cdot \left(1 + \left(M \cdot M\right) \cdot -0.5\right)\\ \end{array} \]

1. Split input into 2 regimes

2. if m < -2.9e-4

   1. Initial program 61.7%

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

      1. *-commutative61.7%

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

      2. associate-*r/61.7%

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

      3. associate--r-61.7%

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

      4. +-commutative61.7%

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

      5. associate-+r-61.7%

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

      6. unsub-neg61.7%

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

      7. associate--r+61.7%

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

      8. +-commutative61.7%

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

      9. associate--r+61.7%

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

   3. Simplified61.7%

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

   4. Taylor expanded in K around 0 97.0%

      \[\leadsto \color{blue}{\cos \left(-M\right)} \cdot e^{\left(\left|m - n\right| - \ell\right) - {\left(\frac{m + n}{2} - M\right)}^{2}} \]
   5. Step-by-step derivation

      1. cos-neg97.0%

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

   6. Simplified97.0%

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

   7. Taylor expanded in n around 0 80.4%

      \[\leadsto \cos M \cdot e^{\left(\left|m - n\right| - \ell\right) - \color{blue}{\left({\left(0.5 \cdot m - M\right)}^{2} + n \cdot \left(0.5 \cdot m - M\right)\right)}} \]
   8. Step-by-step derivation

      1. unpow280.4%

         \[\leadsto \cos M \cdot e^{\left(\left|m - n\right| - \ell\right) - \left(\color{blue}{\left(0.5 \cdot m - M\right) \cdot \left(0.5 \cdot m - M\right)} + n \cdot \left(0.5 \cdot m - M\right)\right)} \]

      2. distribute-rgt-out87.2%

         \[\leadsto \cos M \cdot e^{\left(\left|m - n\right| - \ell\right) - \color{blue}{\left(0.5 \cdot m - M\right) \cdot \left(\left(0.5 \cdot m - M\right) + n\right)}} \]

   9. Simplified87.2%

      \[\leadsto \cos M \cdot e^{\left(\left|m - n\right| - \ell\right) - \color{blue}{\left(0.5 \cdot m - M\right) \cdot \left(\left(0.5 \cdot m - M\right) + n\right)}} \]

   10. Taylor expanded in l around 0 83.9%

       \[\leadsto \cos M \cdot \color{blue}{e^{\left|m - n\right| - \left(0.5 \cdot m - M\right) \cdot \left(\left(n + 0.5 \cdot m\right) - M\right)}} \]
   11. Step-by-step derivation

       1. *-commutative83.9%

          \[\leadsto \cos M \cdot e^{\left|m - n\right| - \color{blue}{\left(\left(n + 0.5 \cdot m\right) - M\right) \cdot \left(0.5 \cdot m - M\right)}} \]

       2. *-commutative83.9%

          \[\leadsto \cos M \cdot e^{\left|m - n\right| - \left(\left(n + \color{blue}{m \cdot 0.5}\right) - M\right) \cdot \left(0.5 \cdot m - M\right)} \]

       3. associate-+r-83.9%

          \[\leadsto \cos M \cdot e^{\left|m - n\right| - \color{blue}{\left(n + \left(m \cdot 0.5 - M\right)\right)} \cdot \left(0.5 \cdot m - M\right)} \]

       4. sub-neg83.9%

          \[\leadsto \cos M \cdot e^{\left|m - n\right| - \left(n + \color{blue}{\left(m \cdot 0.5 + \left(-M\right)\right)}\right) \cdot \left(0.5 \cdot m - M\right)} \]

       5. *-commutative83.9%

          \[\leadsto \cos M \cdot e^{\left|m - n\right| - \left(n + \left(\color{blue}{0.5 \cdot m} + \left(-M\right)\right)\right) \cdot \left(0.5 \cdot m - M\right)} \]

       6. +-commutative83.9%

          \[\leadsto \cos M \cdot e^{\left|m - n\right| - \left(n + \color{blue}{\left(\left(-M\right) + 0.5 \cdot m\right)}\right) \cdot \left(0.5 \cdot m - M\right)} \]

       7. mul-1-neg83.9%

          \[\leadsto \cos M \cdot e^{\left|m - n\right| - \left(n + \left(\color{blue}{-1 \cdot M} + 0.5 \cdot m\right)\right) \cdot \left(0.5 \cdot m - M\right)} \]

       8. sub-neg83.9%

          \[\leadsto \cos M \cdot e^{\left|m - n\right| - \left(n + \left(-1 \cdot M + 0.5 \cdot m\right)\right) \cdot \color{blue}{\left(0.5 \cdot m + \left(-M\right)\right)}} \]

       9. +-commutative83.9%

          \[\leadsto \cos M \cdot e^{\left|m - n\right| - \left(n + \left(-1 \cdot M + 0.5 \cdot m\right)\right) \cdot \color{blue}{\left(\left(-M\right) + 0.5 \cdot m\right)}} \]

       10. distribute-rgt-in65.6%

           \[\leadsto \cos M \cdot e^{\left|m - n\right| - \color{blue}{\left(\left(-M\right) \cdot \left(n + \left(-1 \cdot M + 0.5 \cdot m\right)\right) + \left(0.5 \cdot m\right) \cdot \left(n + \left(-1 \cdot M + 0.5 \cdot m\right)\right)\right)}} \]

       11. associate--r+65.6%

           \[\leadsto \cos M \cdot e^{\color{blue}{\left(\left|m - n\right| - \left(-M\right) \cdot \left(n + \left(-1 \cdot M + 0.5 \cdot m\right)\right)\right) - \left(0.5 \cdot m\right) \cdot \left(n + \left(-1 \cdot M + 0.5 \cdot m\right)\right)}} \]

       12. mul-1-neg65.6%

           \[\leadsto \cos M \cdot e^{\left(\left|m - n\right| - \left(-M\right) \cdot \left(n + \left(-1 \cdot M + 0.5 \cdot m\right)\right)\right) - \left(0.5 \cdot m\right) \cdot \left(n + \left(\color{blue}{\left(-M\right)} + 0.5 \cdot m\right)\right)} \]

   12. Simplified83.9%

       \[\leadsto \cos M \cdot \color{blue}{e^{\left|n - m\right| - \left(m \cdot 0.5 - M\right) \cdot \left(\left(n - M\right) + m \cdot 0.5\right)}} \]

   13. Taylor expanded in m around inf 88.8%

       \[\leadsto \cos M \cdot e^{\left|n - m\right| - \color{blue}{0.25 \cdot {m}^{2}}} \]
   14. Step-by-step derivation

       1. unpow288.8%

          \[\leadsto \cos M \cdot e^{\left|n - m\right| - 0.25 \cdot \color{blue}{\left(m \cdot m\right)}} \]

   15. Simplified88.8%

       \[\leadsto \cos M \cdot e^{\left|n - m\right| - \color{blue}{0.25 \cdot \left(m \cdot m\right)}} \]

   if -2.9e-4 < m

   1. Initial program 77.9%

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

      1. *-commutative77.9%

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

      2. associate-*r/77.9%

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

      3. associate--r-77.9%

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

      4. +-commutative77.9%

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

      5. associate-+r-77.9%

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

      6. unsub-neg77.9%

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

      7. associate--r+77.9%

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

      8. +-commutative77.9%

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

      9. associate--r+77.9%

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

   3. Simplified77.9%

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

   4. Taylor expanded in K around 0 98.0%

      \[\leadsto \color{blue}{\cos \left(-M\right)} \cdot e^{\left(\left|m - n\right| - \ell\right) - {\left(\frac{m + n}{2} - M\right)}^{2}} \]
   5. Step-by-step derivation

      1. cos-neg98.0%

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

   6. Simplified98.0%

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

   7. Taylor expanded in M around inf 61.0%

      \[\leadsto \cos M \cdot e^{\left(\left|m - n\right| - \ell\right) - \color{blue}{{M}^{2}}} \]
   8. Step-by-step derivation

      1. unpow261.0%

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

   9. Simplified61.0%

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

   10. Taylor expanded in l around 0 50.4%

       \[\leadsto \cos M \cdot \color{blue}{e^{\left|m - n\right| - {M}^{2}}} \]
   11. Step-by-step derivation

       1. fabs-sub50.4%

          \[\leadsto \cos M \cdot e^{\color{blue}{\left|n - m\right|} - {M}^{2}} \]

       2. unpow250.4%

          \[\leadsto \cos M \cdot e^{\left|n - m\right| - \color{blue}{M \cdot M}} \]

   12. Simplified50.4%

       \[\leadsto \cos M \cdot \color{blue}{e^{\left|n - m\right| - M \cdot M}} \]
3. Recombined 2 regimes into one program.

4. Final simplification59.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -0.00029:\\ \;\;\;\;\cos M \cdot e^{\left|m - n\right| - 0.25 \cdot \left(m \cdot m\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{\left|m - n\right| - M \cdot M}\\ \end{array} \]

1. Split input into 2 regimes

2. if m < -4.49999999999999998e55

   1. Initial program 58.0%

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

      1. *-commutative58.0%

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

      2. associate-*r/58.0%

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

      3. associate--r-58.0%

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

      4. +-commutative58.0%

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

      5. associate-+r-58.0%

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

      6. unsub-neg58.0%

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

      7. associate--r+58.0%

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

      8. +-commutative58.0%

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

      9. associate--r+58.0%

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

   3. Simplified58.0%

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

   4. Taylor expanded in K around 0 98.0%

      \[\leadsto \color{blue}{\cos \left(-M\right)} \cdot e^{\left(\left|m - n\right| - \ell\right) - {\left(\frac{m + n}{2} - M\right)}^{2}} \]
   5. Step-by-step derivation

      1. cos-neg98.0%

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

   6. Simplified98.0%

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

   7. Taylor expanded in n around 0 78.1%

      \[\leadsto \cos M \cdot e^{\left(\left|m - n\right| - \ell\right) - \color{blue}{\left({\left(0.5 \cdot m - M\right)}^{2} + n \cdot \left(0.5 \cdot m - M\right)\right)}} \]
   8. Step-by-step derivation

      1. unpow278.1%

         \[\leadsto \cos M \cdot e^{\left(\left|m - n\right| - \ell\right) - \left(\color{blue}{\left(0.5 \cdot m - M\right) \cdot \left(0.5 \cdot m - M\right)} + n \cdot \left(0.5 \cdot m - M\right)\right)} \]

      2. distribute-rgt-out86.2%

         \[\leadsto \cos M \cdot e^{\left(\left|m - n\right| - \ell\right) - \color{blue}{\left(0.5 \cdot m - M\right) \cdot \left(\left(0.5 \cdot m - M\right) + n\right)}} \]

   9. Simplified86.2%

      \[\leadsto \cos M \cdot e^{\left(\left|m - n\right| - \ell\right) - \color{blue}{\left(0.5 \cdot m - M\right) \cdot \left(\left(0.5 \cdot m - M\right) + n\right)}} \]

   10. Taylor expanded in l around 0 86.2%

       \[\leadsto \cos M \cdot \color{blue}{e^{\left|m - n\right| - \left(0.5 \cdot m - M\right) \cdot \left(\left(n + 0.5 \cdot m\right) - M\right)}} \]
   11. Step-by-step derivation

       1. *-commutative86.2%

          \[\leadsto \cos M \cdot e^{\left|m - n\right| - \color{blue}{\left(\left(n + 0.5 \cdot m\right) - M\right) \cdot \left(0.5 \cdot m - M\right)}} \]

       2. *-commutative86.2%

          \[\leadsto \cos M \cdot e^{\left|m - n\right| - \left(\left(n + \color{blue}{m \cdot 0.5}\right) - M\right) \cdot \left(0.5 \cdot m - M\right)} \]

       3. associate-+r-86.2%

          \[\leadsto \cos M \cdot e^{\left|m - n\right| - \color{blue}{\left(n + \left(m \cdot 0.5 - M\right)\right)} \cdot \left(0.5 \cdot m - M\right)} \]

       4. sub-neg86.2%

          \[\leadsto \cos M \cdot e^{\left|m - n\right| - \left(n + \color{blue}{\left(m \cdot 0.5 + \left(-M\right)\right)}\right) \cdot \left(0.5 \cdot m - M\right)} \]

       5. *-commutative86.2%

          \[\leadsto \cos M \cdot e^{\left|m - n\right| - \left(n + \left(\color{blue}{0.5 \cdot m} + \left(-M\right)\right)\right) \cdot \left(0.5 \cdot m - M\right)} \]

       6. +-commutative86.2%

          \[\leadsto \cos M \cdot e^{\left|m - n\right| - \left(n + \color{blue}{\left(\left(-M\right) + 0.5 \cdot m\right)}\right) \cdot \left(0.5 \cdot m - M\right)} \]

       7. mul-1-neg86.2%

          \[\leadsto \cos M \cdot e^{\left|m - n\right| - \left(n + \left(\color{blue}{-1 \cdot M} + 0.5 \cdot m\right)\right) \cdot \left(0.5 \cdot m - M\right)} \]

       8. sub-neg86.2%

          \[\leadsto \cos M \cdot e^{\left|m - n\right| - \left(n + \left(-1 \cdot M + 0.5 \cdot m\right)\right) \cdot \color{blue}{\left(0.5 \cdot m + \left(-M\right)\right)}} \]

       9. +-commutative86.2%

          \[\leadsto \cos M \cdot e^{\left|m - n\right| - \left(n + \left(-1 \cdot M + 0.5 \cdot m\right)\right) \cdot \color{blue}{\left(\left(-M\right) + 0.5 \cdot m\right)}} \]

       10. distribute-rgt-in64.2%

           \[\leadsto \cos M \cdot e^{\left|m - n\right| - \color{blue}{\left(\left(-M\right) \cdot \left(n + \left(-1 \cdot M + 0.5 \cdot m\right)\right) + \left(0.5 \cdot m\right) \cdot \left(n + \left(-1 \cdot M + 0.5 \cdot m\right)\right)\right)}} \]

       11. associate--r+64.2%

           \[\leadsto \cos M \cdot e^{\color{blue}{\left(\left|m - n\right| - \left(-M\right) \cdot \left(n + \left(-1 \cdot M + 0.5 \cdot m\right)\right)\right) - \left(0.5 \cdot m\right) \cdot \left(n + \left(-1 \cdot M + 0.5 \cdot m\right)\right)}} \]

       12. mul-1-neg64.2%

           \[\leadsto \cos M \cdot e^{\left(\left|m - n\right| - \left(-M\right) \cdot \left(n + \left(-1 \cdot M + 0.5 \cdot m\right)\right)\right) - \left(0.5 \cdot m\right) \cdot \left(n + \left(\color{blue}{\left(-M\right)} + 0.5 \cdot m\right)\right)} \]

   12. Simplified86.2%

       \[\leadsto \cos M \cdot \color{blue}{e^{\left|n - m\right| - \left(m \cdot 0.5 - M\right) \cdot \left(\left(n - M\right) + m \cdot 0.5\right)}} \]

   13. Taylor expanded in m around inf 94.1%

       \[\leadsto \cos M \cdot e^{\left|n - m\right| - \color{blue}{0.25 \cdot {m}^{2}}} \]
   14. Step-by-step derivation

       1. unpow294.1%

          \[\leadsto \cos M \cdot e^{\left|n - m\right| - 0.25 \cdot \color{blue}{\left(m \cdot m\right)}} \]

   15. Simplified94.1%

       \[\leadsto \cos M \cdot e^{\left|n - m\right| - \color{blue}{0.25 \cdot \left(m \cdot m\right)}} \]

   if -4.49999999999999998e55 < m

   1. Initial program 78.0%

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

      1. *-commutative78.0%

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

      2. associate-*r/78.0%

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

      3. associate--r-78.0%

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

      4. +-commutative78.0%

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

      5. associate-+r-78.0%

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

      6. unsub-neg78.0%

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

      7. associate--r+78.0%

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

      8. +-commutative78.0%

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

      9. associate--r+78.0%

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

   3. Simplified78.0%

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

   4. Taylor expanded in K around 0 97.7%

      \[\leadsto \color{blue}{\cos \left(-M\right)} \cdot e^{\left(\left|m - n\right| - \ell\right) - {\left(\frac{m + n}{2} - M\right)}^{2}} \]
   5. Step-by-step derivation

      1. cos-neg97.7%

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

   6. Simplified97.7%

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

   7. Taylor expanded in M around inf 61.5%

      \[\leadsto \cos M \cdot e^{\left(\left|m - n\right| - \ell\right) - \color{blue}{{M}^{2}}} \]
   8. Step-by-step derivation

      1. unpow261.5%

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

   9. Simplified61.5%

      \[\leadsto \cos M \cdot e^{\left(\left|m - n\right| - \ell\right) - \color{blue}{M \cdot M}} \]
3. Recombined 2 regimes into one program.

4. Final simplification67.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -4.5 \cdot 10^{+55}:\\ \;\;\;\;\cos M \cdot e^{\left|m - n\right| - 0.25 \cdot \left(m \cdot m\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{\left(\left|m - n\right| - \ell\right) - M \cdot M}\\ \end{array} \]

1. Initial program 74.1%

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

   1. *-commutative74.1%

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

   2. associate-*r/74.1%

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

   3. associate--r-74.1%

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

   4. +-commutative74.1%

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

   5. associate-+r-74.1%

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

   6. unsub-neg74.1%

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

   7. associate--r+74.1%

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

   8. +-commutative74.1%

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

   9. associate--r+74.1%

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

3. Simplified74.1%

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

4. Taylor expanded in K around 0 97.8%

   \[\leadsto \color{blue}{\cos \left(-M\right)} \cdot e^{\left(\left|m - n\right| - \ell\right) - {\left(\frac{m + n}{2} - M\right)}^{2}} \]
5. Step-by-step derivation

   1. cos-neg97.8%

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

6. Simplified97.8%

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

7. Taylor expanded in M around inf 56.3%

   \[\leadsto \cos M \cdot e^{\left(\left|m - n\right| - \ell\right) - \color{blue}{{M}^{2}}} \]
8. Step-by-step derivation

   1. unpow256.3%

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

9. Simplified56.3%

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

10. Taylor expanded in M around 0 26.0%

    \[\leadsto \cos M \cdot \color{blue}{e^{\left|m - n\right| - \ell}} \]
11. Step-by-step derivation

    1. fabs-sub26.0%

       \[\leadsto \cos M \cdot e^{\color{blue}{\left|n - m\right|} - \ell} \]

12. Simplified26.0%

    \[\leadsto \cos M \cdot \color{blue}{e^{\left|n - m\right| - \ell}} \]

13. Final simplification26.0%

    \[\leadsto \cos M \cdot e^{\left|m - n\right| - \ell} \]

1. Initial program 74.1%

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

   1. *-commutative74.1%

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

   2. associate-*r/74.1%

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

   3. associate--r-74.1%

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

   4. +-commutative74.1%

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

   5. associate-+r-74.1%

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

   6. unsub-neg74.1%

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

   7. associate--r+74.1%

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

   8. +-commutative74.1%

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

   9. associate--r+74.1%

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

3. Simplified74.1%

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

4. Taylor expanded in K around 0 97.8%

   \[\leadsto \color{blue}{\cos \left(-M\right)} \cdot e^{\left(\left|m - n\right| - \ell\right) - {\left(\frac{m + n}{2} - M\right)}^{2}} \]
5. Step-by-step derivation

   1. cos-neg97.8%

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

6. Simplified97.8%

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

7. Taylor expanded in M around inf 56.3%

   \[\leadsto \cos M \cdot e^{\left(\left|m - n\right| - \ell\right) - \color{blue}{{M}^{2}}} \]
8. Step-by-step derivation

   1. unpow256.3%

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

9. Simplified56.3%

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

10. Taylor expanded in M around 0 26.0%

    \[\leadsto \cos M \cdot \color{blue}{e^{\left|m - n\right| - \ell}} \]
11. Step-by-step derivation

    1. fabs-sub26.0%

       \[\leadsto \cos M \cdot e^{\color{blue}{\left|n - m\right|} - \ell} \]

12. Simplified26.0%

    \[\leadsto \cos M \cdot \color{blue}{e^{\left|n - m\right| - \ell}} \]

13. Taylor expanded in M around 0 15.2%

    \[\leadsto \color{blue}{-0.5 \cdot \left({M}^{2} \cdot e^{\left|n - m\right| - \ell}\right) + e^{\left|n - m\right| - \ell}} \]
14. Step-by-step derivation

    1. +-commutative15.2%

       \[\leadsto \color{blue}{e^{\left|n - m\right| - \ell} + -0.5 \cdot \left({M}^{2} \cdot e^{\left|n - m\right| - \ell}\right)} \]

    2. *-lft-identity15.2%

       \[\leadsto \color{blue}{1 \cdot e^{\left|n - m\right| - \ell}} + -0.5 \cdot \left({M}^{2} \cdot e^{\left|n - m\right| - \ell}\right) \]

    3. associate-*r*15.2%

       \[\leadsto 1 \cdot e^{\left|n - m\right| - \ell} + \color{blue}{\left(-0.5 \cdot {M}^{2}\right) \cdot e^{\left|n - m\right| - \ell}} \]

    4. distribute-rgt-out21.3%

       \[\leadsto \color{blue}{e^{\left|n - m\right| - \ell} \cdot \left(1 + -0.5 \cdot {M}^{2}\right)} \]

    5. unpow221.3%

       \[\leadsto e^{\left|n - m\right| - \ell} \cdot \left(1 + -0.5 \cdot \color{blue}{\left(M \cdot M\right)}\right) \]

15. Simplified21.3%

    \[\leadsto \color{blue}{e^{\left|n - m\right| - \ell} \cdot \left(1 + -0.5 \cdot \left(M \cdot M\right)\right)} \]

16. Final simplification21.3%

    \[\leadsto e^{\left|m - n\right| - \ell} \cdot \left(1 + \left(M \cdot M\right) \cdot -0.5\right) \]

herbie shell --seed 2023166 
(FPCore (K m n M l)
  :name "Maksimov and Kolovsky, Equation (32)"
  :precision binary64
  (* (cos (- (/ (* K (+ m n)) 2.0) M)) (exp (- (- (pow (- (/ (+ m n) 2.0) M) 2.0)) (- l (fabs (- m n)))))))