Maksimov and Kolovsky, Equation (4)

1. Split input into 2 regimes

2. if (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l))) < -inf.0 or 0.0 < (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l)))

   1. Initial program 100.0%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   if -inf.0 < (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l))) < 0.0

   1. Initial program 71.0%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   2. Taylor expanded in l around 0 99.9%

      \[\leadsto \left(J \cdot \color{blue}{\left(0.3333333333333333 \cdot {\ell}^{3} + 2 \cdot \ell\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   3. Taylor expanded in J around 0 99.9%

      \[\leadsto \color{blue}{\cos \left(0.5 \cdot K\right) \cdot \left(\left(0.3333333333333333 \cdot {\ell}^{3} + 2 \cdot \ell\right) \cdot J\right)} + U \]
   4. Step-by-step derivation

      1. associate-*r*99.9%

         \[\leadsto \color{blue}{\left(\cos \left(0.5 \cdot K\right) \cdot \left(0.3333333333333333 \cdot {\ell}^{3} + 2 \cdot \ell\right)\right) \cdot J} + U \]

      2. *-commutative99.9%

         \[\leadsto \left(\cos \color{blue}{\left(K \cdot 0.5\right)} \cdot \left(0.3333333333333333 \cdot {\ell}^{3} + 2 \cdot \ell\right)\right) \cdot J + U \]

      3. fma-def99.9%

         \[\leadsto \left(\cos \left(K \cdot 0.5\right) \cdot \color{blue}{\mathsf{fma}\left(0.3333333333333333, {\ell}^{3}, 2 \cdot \ell\right)}\right) \cdot J + U \]

      4. *-commutative99.9%

         \[\leadsto \left(\cos \left(K \cdot 0.5\right) \cdot \mathsf{fma}\left(0.3333333333333333, {\ell}^{3}, \color{blue}{\ell \cdot 2}\right)\right) \cdot J + U \]

      5. *-commutative99.9%

         \[\leadsto \color{blue}{\left(\mathsf{fma}\left(0.3333333333333333, {\ell}^{3}, \ell \cdot 2\right) \cdot \cos \left(K \cdot 0.5\right)\right)} \cdot J + U \]

      6. *-commutative99.9%

         \[\leadsto \color{blue}{J \cdot \left(\mathsf{fma}\left(0.3333333333333333, {\ell}^{3}, \ell \cdot 2\right) \cdot \cos \left(K \cdot 0.5\right)\right)} + U \]

      7. *-commutative99.9%

         \[\leadsto J \cdot \left(\mathsf{fma}\left(0.3333333333333333, {\ell}^{3}, \ell \cdot 2\right) \cdot \cos \color{blue}{\left(0.5 \cdot K\right)}\right) + U \]

   5. Simplified99.9%

      \[\leadsto \color{blue}{J \cdot \left(\mathsf{fma}\left(0.3333333333333333, {\ell}^{3}, \ell \cdot 2\right) \cdot \cos \left(0.5 \cdot K\right)\right)} + U \]
   6. Step-by-step derivation

      1. fma-udef99.9%

         \[\leadsto J \cdot \left(\color{blue}{\left(0.3333333333333333 \cdot {\ell}^{3} + \ell \cdot 2\right)} \cdot \cos \left(0.5 \cdot K\right)\right) + U \]

   7. Applied egg-rr99.9%

      \[\leadsto J \cdot \left(\color{blue}{\left(0.3333333333333333 \cdot {\ell}^{3} + \ell \cdot 2\right)} \cdot \cos \left(0.5 \cdot K\right)\right) + U \]
3. Recombined 2 regimes into one program.

4. Final simplification100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{\ell} - e^{-\ell} \leq -\infty \lor \neg \left(e^{\ell} - e^{-\ell} \leq 0\right):\\ \;\;\;\;\cos \left(\frac{K}{2}\right) \cdot \left(\left(e^{\ell} - e^{-\ell}\right) \cdot J\right) + U\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(\left(0.3333333333333333 \cdot {\ell}^{3} + \ell \cdot 2\right) \cdot \cos \left(K \cdot 0.5\right)\right)\\ \end{array} \]

1. Split input into 2 regimes

2. if l < -7.79999999999999967e121 or -1e16 < l < 0.048000000000000001 or 5.49999999999999981e102 < l

   1. Initial program 83.4%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   2. Taylor expanded in l around 0 99.0%

      \[\leadsto \left(J \cdot \color{blue}{\left(0.3333333333333333 \cdot {\ell}^{3} + 2 \cdot \ell\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   3. Taylor expanded in J around 0 99.0%

      \[\leadsto \color{blue}{\cos \left(0.5 \cdot K\right) \cdot \left(\left(0.3333333333333333 \cdot {\ell}^{3} + 2 \cdot \ell\right) \cdot J\right)} + U \]
   4. Step-by-step derivation

      1. associate-*r*99.0%

         \[\leadsto \color{blue}{\left(\cos \left(0.5 \cdot K\right) \cdot \left(0.3333333333333333 \cdot {\ell}^{3} + 2 \cdot \ell\right)\right) \cdot J} + U \]

      2. *-commutative99.0%

         \[\leadsto \left(\cos \color{blue}{\left(K \cdot 0.5\right)} \cdot \left(0.3333333333333333 \cdot {\ell}^{3} + 2 \cdot \ell\right)\right) \cdot J + U \]

      3. fma-def99.0%

         \[\leadsto \left(\cos \left(K \cdot 0.5\right) \cdot \color{blue}{\mathsf{fma}\left(0.3333333333333333, {\ell}^{3}, 2 \cdot \ell\right)}\right) \cdot J + U \]

      4. *-commutative99.0%

         \[\leadsto \left(\cos \left(K \cdot 0.5\right) \cdot \mathsf{fma}\left(0.3333333333333333, {\ell}^{3}, \color{blue}{\ell \cdot 2}\right)\right) \cdot J + U \]

      5. *-commutative99.0%

         \[\leadsto \color{blue}{\left(\mathsf{fma}\left(0.3333333333333333, {\ell}^{3}, \ell \cdot 2\right) \cdot \cos \left(K \cdot 0.5\right)\right)} \cdot J + U \]

      6. *-commutative99.0%

         \[\leadsto \color{blue}{J \cdot \left(\mathsf{fma}\left(0.3333333333333333, {\ell}^{3}, \ell \cdot 2\right) \cdot \cos \left(K \cdot 0.5\right)\right)} + U \]

      7. *-commutative99.0%

         \[\leadsto J \cdot \left(\mathsf{fma}\left(0.3333333333333333, {\ell}^{3}, \ell \cdot 2\right) \cdot \cos \color{blue}{\left(0.5 \cdot K\right)}\right) + U \]

   5. Simplified99.0%

      \[\leadsto \color{blue}{J \cdot \left(\mathsf{fma}\left(0.3333333333333333, {\ell}^{3}, \ell \cdot 2\right) \cdot \cos \left(0.5 \cdot K\right)\right)} + U \]
   6. Step-by-step derivation

      1. fma-udef99.0%

         \[\leadsto J \cdot \left(\color{blue}{\left(0.3333333333333333 \cdot {\ell}^{3} + \ell \cdot 2\right)} \cdot \cos \left(0.5 \cdot K\right)\right) + U \]

   7. Applied egg-rr99.0%

      \[\leadsto J \cdot \left(\color{blue}{\left(0.3333333333333333 \cdot {\ell}^{3} + \ell \cdot 2\right)} \cdot \cos \left(0.5 \cdot K\right)\right) + U \]

   if -7.79999999999999967e121 < l < -1e16 or 0.048000000000000001 < l < 5.49999999999999981e102

   1. Initial program 100.0%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   2. Taylor expanded in K around 0 85.4%

      \[\leadsto \color{blue}{\left(e^{\ell} - e^{-\ell}\right) \cdot J} + U \]
3. Recombined 2 regimes into one program.

4. Final simplification96.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -7.8 \cdot 10^{+121} \lor \neg \left(\ell \leq -1 \cdot 10^{+16} \lor \neg \left(\ell \leq 0.048\right) \land \ell \leq 5.5 \cdot 10^{+102}\right):\\ \;\;\;\;U + J \cdot \left(\left(0.3333333333333333 \cdot {\ell}^{3} + \ell \cdot 2\right) \cdot \cos \left(K \cdot 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + \left(e^{\ell} - e^{-\ell}\right) \cdot J\\ \end{array} \]

1. Split input into 3 regimes

2. if l < -2.94999999999999995e94 or -160 < l < 0.17499999999999999 or 5.60000000000000037e102 < l

   1. Initial program 83.5%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   2. Taylor expanded in l around 0 99.9%

      \[\leadsto \left(J \cdot \color{blue}{\left(0.3333333333333333 \cdot {\ell}^{3} + 2 \cdot \ell\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   3. Taylor expanded in J around 0 99.9%

      \[\leadsto \color{blue}{\cos \left(0.5 \cdot K\right) \cdot \left(\left(0.3333333333333333 \cdot {\ell}^{3} + 2 \cdot \ell\right) \cdot J\right)} + U \]
   4. Step-by-step derivation

      1. associate-*r*99.9%

         \[\leadsto \color{blue}{\left(\cos \left(0.5 \cdot K\right) \cdot \left(0.3333333333333333 \cdot {\ell}^{3} + 2 \cdot \ell\right)\right) \cdot J} + U \]

      2. *-commutative99.9%

         \[\leadsto \left(\cos \color{blue}{\left(K \cdot 0.5\right)} \cdot \left(0.3333333333333333 \cdot {\ell}^{3} + 2 \cdot \ell\right)\right) \cdot J + U \]

      3. fma-def99.9%

         \[\leadsto \left(\cos \left(K \cdot 0.5\right) \cdot \color{blue}{\mathsf{fma}\left(0.3333333333333333, {\ell}^{3}, 2 \cdot \ell\right)}\right) \cdot J + U \]

      4. *-commutative99.9%

         \[\leadsto \left(\cos \left(K \cdot 0.5\right) \cdot \mathsf{fma}\left(0.3333333333333333, {\ell}^{3}, \color{blue}{\ell \cdot 2}\right)\right) \cdot J + U \]

      5. *-commutative99.9%

         \[\leadsto \color{blue}{\left(\mathsf{fma}\left(0.3333333333333333, {\ell}^{3}, \ell \cdot 2\right) \cdot \cos \left(K \cdot 0.5\right)\right)} \cdot J + U \]

      6. *-commutative99.9%

         \[\leadsto \color{blue}{J \cdot \left(\mathsf{fma}\left(0.3333333333333333, {\ell}^{3}, \ell \cdot 2\right) \cdot \cos \left(K \cdot 0.5\right)\right)} + U \]

      7. *-commutative99.9%

         \[\leadsto J \cdot \left(\mathsf{fma}\left(0.3333333333333333, {\ell}^{3}, \ell \cdot 2\right) \cdot \cos \color{blue}{\left(0.5 \cdot K\right)}\right) + U \]

   5. Simplified99.9%

      \[\leadsto \color{blue}{J \cdot \left(\mathsf{fma}\left(0.3333333333333333, {\ell}^{3}, \ell \cdot 2\right) \cdot \cos \left(0.5 \cdot K\right)\right)} + U \]
   6. Step-by-step derivation

      1. fma-udef99.9%

         \[\leadsto J \cdot \left(\color{blue}{\left(0.3333333333333333 \cdot {\ell}^{3} + \ell \cdot 2\right)} \cdot \cos \left(0.5 \cdot K\right)\right) + U \]

   7. Applied egg-rr99.9%

      \[\leadsto J \cdot \left(\color{blue}{\left(0.3333333333333333 \cdot {\ell}^{3} + \ell \cdot 2\right)} \cdot \cos \left(0.5 \cdot K\right)\right) + U \]

   if -2.94999999999999995e94 < l < -160

   1. Initial program 100.0%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   2. Taylor expanded in K around 0 0.0%

      \[\leadsto \color{blue}{\left(\left(e^{\ell} - e^{-\ell}\right) \cdot J + -0.125 \cdot \left({K}^{2} \cdot \left(\left(e^{\ell} - e^{-\ell}\right) \cdot J\right)\right)\right)} + U \]
   3. Step-by-step derivation

      1. associate-*r*0.0%

         \[\leadsto \left(\left(e^{\ell} - e^{-\ell}\right) \cdot J + \color{blue}{\left(-0.125 \cdot {K}^{2}\right) \cdot \left(\left(e^{\ell} - e^{-\ell}\right) \cdot J\right)}\right) + U \]

      2. distribute-rgt1-in70.8%

         \[\leadsto \color{blue}{\left(-0.125 \cdot {K}^{2} + 1\right) \cdot \left(\left(e^{\ell} - e^{-\ell}\right) \cdot J\right)} + U \]

      3. unpow270.8%

         \[\leadsto \left(-0.125 \cdot \color{blue}{\left(K \cdot K\right)} + 1\right) \cdot \left(\left(e^{\ell} - e^{-\ell}\right) \cdot J\right) + U \]

   4. Simplified70.8%

      \[\leadsto \color{blue}{\left(-0.125 \cdot \left(K \cdot K\right) + 1\right) \cdot \left(\left(e^{\ell} - e^{-\ell}\right) \cdot J\right)} + U \]

   if 0.17499999999999999 < l < 5.60000000000000037e102

   1. Initial program 100.0%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   2. Taylor expanded in K around 0 95.7%

      \[\leadsto \color{blue}{\left(e^{\ell} - e^{-\ell}\right) \cdot J} + U \]
3. Recombined 3 regimes into one program.

4. Final simplification96.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -2.95 \cdot 10^{+94}:\\ \;\;\;\;U + J \cdot \left(\left(0.3333333333333333 \cdot {\ell}^{3} + \ell \cdot 2\right) \cdot \cos \left(K \cdot 0.5\right)\right)\\ \mathbf{elif}\;\ell \leq -160:\\ \;\;\;\;U + \left(-0.125 \cdot \left(K \cdot K\right) + 1\right) \cdot \left(\left(e^{\ell} - e^{-\ell}\right) \cdot J\right)\\ \mathbf{elif}\;\ell \leq 0.175 \lor \neg \left(\ell \leq 5.6 \cdot 10^{+102}\right):\\ \;\;\;\;U + J \cdot \left(\left(0.3333333333333333 \cdot {\ell}^{3} + \ell \cdot 2\right) \cdot \cos \left(K \cdot 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + \left(e^{\ell} - e^{-\ell}\right) \cdot J\\ \end{array} \]

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K 2)) < 0.110000000000000001

   1. Initial program 79.8%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   2. Taylor expanded in l around 0 89.5%

      \[\leadsto \left(J \cdot \color{blue}{\left(0.3333333333333333 \cdot {\ell}^{3} + 2 \cdot \ell\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   3. Taylor expanded in J around 0 89.5%

      \[\leadsto \color{blue}{\cos \left(0.5 \cdot K\right) \cdot \left(\left(0.3333333333333333 \cdot {\ell}^{3} + 2 \cdot \ell\right) \cdot J\right)} + U \]
   4. Step-by-step derivation

      1. associate-*r*89.5%

         \[\leadsto \color{blue}{\left(\cos \left(0.5 \cdot K\right) \cdot \left(0.3333333333333333 \cdot {\ell}^{3} + 2 \cdot \ell\right)\right) \cdot J} + U \]

      2. *-commutative89.5%

         \[\leadsto \left(\cos \color{blue}{\left(K \cdot 0.5\right)} \cdot \left(0.3333333333333333 \cdot {\ell}^{3} + 2 \cdot \ell\right)\right) \cdot J + U \]

      3. fma-def89.5%

         \[\leadsto \left(\cos \left(K \cdot 0.5\right) \cdot \color{blue}{\mathsf{fma}\left(0.3333333333333333, {\ell}^{3}, 2 \cdot \ell\right)}\right) \cdot J + U \]

      4. *-commutative89.5%

         \[\leadsto \left(\cos \left(K \cdot 0.5\right) \cdot \mathsf{fma}\left(0.3333333333333333, {\ell}^{3}, \color{blue}{\ell \cdot 2}\right)\right) \cdot J + U \]

      5. *-commutative89.5%

         \[\leadsto \color{blue}{\left(\mathsf{fma}\left(0.3333333333333333, {\ell}^{3}, \ell \cdot 2\right) \cdot \cos \left(K \cdot 0.5\right)\right)} \cdot J + U \]

      6. *-commutative89.5%

         \[\leadsto \color{blue}{J \cdot \left(\mathsf{fma}\left(0.3333333333333333, {\ell}^{3}, \ell \cdot 2\right) \cdot \cos \left(K \cdot 0.5\right)\right)} + U \]

      7. *-commutative89.5%

         \[\leadsto J \cdot \left(\mathsf{fma}\left(0.3333333333333333, {\ell}^{3}, \ell \cdot 2\right) \cdot \cos \color{blue}{\left(0.5 \cdot K\right)}\right) + U \]

   5. Simplified89.5%

      \[\leadsto \color{blue}{J \cdot \left(\mathsf{fma}\left(0.3333333333333333, {\ell}^{3}, \ell \cdot 2\right) \cdot \cos \left(0.5 \cdot K\right)\right)} + U \]

   6. Taylor expanded in l around 0 68.7%

      \[\leadsto J \cdot \left(\color{blue}{\left(2 \cdot \ell\right)} \cdot \cos \left(0.5 \cdot K\right)\right) + U \]
   7. Step-by-step derivation

      1. *-commutative68.7%

         \[\leadsto J \cdot \left(\color{blue}{\left(\ell \cdot 2\right)} \cdot \cos \left(0.5 \cdot K\right)\right) + U \]

   8. Simplified68.7%

      \[\leadsto J \cdot \left(\color{blue}{\left(\ell \cdot 2\right)} \cdot \cos \left(0.5 \cdot K\right)\right) + U \]

   if 0.110000000000000001 < (cos.f64 (/.f64 K 2))

   1. Initial program 89.2%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   2. Taylor expanded in K around 0 88.8%

      \[\leadsto \color{blue}{\left(e^{\ell} - e^{-\ell}\right) \cdot J} + U \]

   3. Taylor expanded in l around 0 79.8%

      \[\leadsto \color{blue}{\left(0.3333333333333333 \cdot {\ell}^{3} + 2 \cdot \ell\right)} \cdot J + U \]
3. Recombined 2 regimes into one program.

4. Final simplification76.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq 0.11:\\ \;\;\;\;U + J \cdot \left(\left(\ell \cdot 2\right) \cdot \cos \left(K \cdot 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(0.3333333333333333 \cdot {\ell}^{3} + \ell \cdot 2\right)\\ \end{array} \]

1. Split input into 2 regimes

2. if l < -1e16 or 0.0058 < l

   1. Initial program 100.0%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   2. Taylor expanded in K around 0 79.3%

      \[\leadsto \color{blue}{\left(e^{\ell} - e^{-\ell}\right) \cdot J} + U \]

   if -1e16 < l < 0.0058

   1. Initial program 71.5%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   2. Taylor expanded in l around 0 98.3%

      \[\leadsto \left(J \cdot \color{blue}{\left(0.3333333333333333 \cdot {\ell}^{3} + 2 \cdot \ell\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   3. Taylor expanded in l around 0 98.0%

      \[\leadsto \color{blue}{2 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \left(\ell \cdot J\right)\right)} + U \]
   4. Step-by-step derivation

      1. associate-*r*98.0%

         \[\leadsto \color{blue}{\left(2 \cdot \cos \left(0.5 \cdot K\right)\right) \cdot \left(\ell \cdot J\right)} + U \]

      2. *-commutative98.0%

         \[\leadsto \left(2 \cdot \cos \left(0.5 \cdot K\right)\right) \cdot \color{blue}{\left(J \cdot \ell\right)} + U \]

      3. associate-*r*98.0%

         \[\leadsto \color{blue}{\left(\left(2 \cdot \cos \left(0.5 \cdot K\right)\right) \cdot J\right) \cdot \ell} + U \]

      4. *-commutative98.0%

         \[\leadsto \left(\left(2 \cdot \cos \color{blue}{\left(K \cdot 0.5\right)}\right) \cdot J\right) \cdot \ell + U \]

      5. *-commutative98.0%

         \[\leadsto \left(\color{blue}{\left(\cos \left(K \cdot 0.5\right) \cdot 2\right)} \cdot J\right) \cdot \ell + U \]

      6. associate-*r*98.0%

         \[\leadsto \color{blue}{\left(\cos \left(K \cdot 0.5\right) \cdot \left(2 \cdot J\right)\right)} \cdot \ell + U \]

      7. *-commutative98.0%

         \[\leadsto \left(\cos \left(K \cdot 0.5\right) \cdot \color{blue}{\left(J \cdot 2\right)}\right) \cdot \ell + U \]

      8. *-commutative98.0%

         \[\leadsto \color{blue}{\ell \cdot \left(\cos \left(K \cdot 0.5\right) \cdot \left(J \cdot 2\right)\right)} + U \]

      9. *-commutative98.0%

         \[\leadsto \ell \cdot \color{blue}{\left(\left(J \cdot 2\right) \cdot \cos \left(K \cdot 0.5\right)\right)} + U \]

      10. *-commutative98.0%

          \[\leadsto \ell \cdot \left(\color{blue}{\left(2 \cdot J\right)} \cdot \cos \left(K \cdot 0.5\right)\right) + U \]

      11. *-commutative98.0%

          \[\leadsto \ell \cdot \left(\left(2 \cdot J\right) \cdot \cos \color{blue}{\left(0.5 \cdot K\right)}\right) + U \]

   5. Simplified98.0%

      \[\leadsto \color{blue}{\ell \cdot \left(\left(2 \cdot J\right) \cdot \cos \left(0.5 \cdot K\right)\right)} + U \]
3. Recombined 2 regimes into one program.

4. Final simplification88.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1 \cdot 10^{+16} \lor \neg \left(\ell \leq 0.0058\right):\\ \;\;\;\;U + \left(e^{\ell} - e^{-\ell}\right) \cdot J\\ \mathbf{else}:\\ \;\;\;\;U + \ell \cdot \left(\cos \left(K \cdot 0.5\right) \cdot \left(J \cdot 2\right)\right)\\ \end{array} \]

1. Split input into 2 regimes

2. if l < -1.45e16 or 66000 < l

   1. Initial program 100.0%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   2. Taylor expanded in K around 0 79.3%

      \[\leadsto \color{blue}{\left(e^{\ell} - e^{-\ell}\right) \cdot J} + U \]

   3. Taylor expanded in l around 0 56.6%

      \[\leadsto \color{blue}{\left(2 \cdot \left(\ell \cdot J\right) + 0.3333333333333333 \cdot \left({\ell}^{3} \cdot J\right)\right)} + U \]

   4. Taylor expanded in l around inf 56.6%

      \[\leadsto \color{blue}{0.3333333333333333 \cdot \left({\ell}^{3} \cdot J\right)} + U \]

   if -1.45e16 < l < 66000

   1. Initial program 71.5%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   2. Taylor expanded in l around 0 98.3%

      \[\leadsto \left(J \cdot \color{blue}{\left(0.3333333333333333 \cdot {\ell}^{3} + 2 \cdot \ell\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   3. Taylor expanded in J around 0 98.3%

      \[\leadsto \color{blue}{\cos \left(0.5 \cdot K\right) \cdot \left(\left(0.3333333333333333 \cdot {\ell}^{3} + 2 \cdot \ell\right) \cdot J\right)} + U \]
   4. Step-by-step derivation

      1. associate-*r*98.3%

         \[\leadsto \color{blue}{\left(\cos \left(0.5 \cdot K\right) \cdot \left(0.3333333333333333 \cdot {\ell}^{3} + 2 \cdot \ell\right)\right) \cdot J} + U \]

      2. *-commutative98.3%

         \[\leadsto \left(\cos \color{blue}{\left(K \cdot 0.5\right)} \cdot \left(0.3333333333333333 \cdot {\ell}^{3} + 2 \cdot \ell\right)\right) \cdot J + U \]

      3. fma-def98.3%

         \[\leadsto \left(\cos \left(K \cdot 0.5\right) \cdot \color{blue}{\mathsf{fma}\left(0.3333333333333333, {\ell}^{3}, 2 \cdot \ell\right)}\right) \cdot J + U \]

      4. *-commutative98.3%

         \[\leadsto \left(\cos \left(K \cdot 0.5\right) \cdot \mathsf{fma}\left(0.3333333333333333, {\ell}^{3}, \color{blue}{\ell \cdot 2}\right)\right) \cdot J + U \]

      5. *-commutative98.3%

         \[\leadsto \color{blue}{\left(\mathsf{fma}\left(0.3333333333333333, {\ell}^{3}, \ell \cdot 2\right) \cdot \cos \left(K \cdot 0.5\right)\right)} \cdot J + U \]

      6. *-commutative98.3%

         \[\leadsto \color{blue}{J \cdot \left(\mathsf{fma}\left(0.3333333333333333, {\ell}^{3}, \ell \cdot 2\right) \cdot \cos \left(K \cdot 0.5\right)\right)} + U \]

      7. *-commutative98.3%

         \[\leadsto J \cdot \left(\mathsf{fma}\left(0.3333333333333333, {\ell}^{3}, \ell \cdot 2\right) \cdot \cos \color{blue}{\left(0.5 \cdot K\right)}\right) + U \]

   5. Simplified98.3%

      \[\leadsto \color{blue}{J \cdot \left(\mathsf{fma}\left(0.3333333333333333, {\ell}^{3}, \ell \cdot 2\right) \cdot \cos \left(0.5 \cdot K\right)\right)} + U \]

   6. Taylor expanded in l around 0 98.0%

      \[\leadsto J \cdot \left(\color{blue}{\left(2 \cdot \ell\right)} \cdot \cos \left(0.5 \cdot K\right)\right) + U \]
   7. Step-by-step derivation

      1. *-commutative98.0%

         \[\leadsto J \cdot \left(\color{blue}{\left(\ell \cdot 2\right)} \cdot \cos \left(0.5 \cdot K\right)\right) + U \]

   8. Simplified98.0%

      \[\leadsto J \cdot \left(\color{blue}{\left(\ell \cdot 2\right)} \cdot \cos \left(0.5 \cdot K\right)\right) + U \]
3. Recombined 2 regimes into one program.

4. Final simplification76.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1.45 \cdot 10^{+16} \lor \neg \left(\ell \leq 66000\right):\\ \;\;\;\;U + 0.3333333333333333 \cdot \left(J \cdot {\ell}^{3}\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(\left(\ell \cdot 2\right) \cdot \cos \left(K \cdot 0.5\right)\right)\\ \end{array} \]

1. Split input into 2 regimes

2. if l < -1.4e16 or 1.2e5 < l

   1. Initial program 100.0%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   2. Taylor expanded in K around 0 79.3%

      \[\leadsto \color{blue}{\left(e^{\ell} - e^{-\ell}\right) \cdot J} + U \]

   3. Taylor expanded in l around 0 56.6%

      \[\leadsto \color{blue}{\left(2 \cdot \left(\ell \cdot J\right) + 0.3333333333333333 \cdot \left({\ell}^{3} \cdot J\right)\right)} + U \]

   4. Taylor expanded in l around inf 56.6%

      \[\leadsto \color{blue}{0.3333333333333333 \cdot \left({\ell}^{3} \cdot J\right)} + U \]

   if -1.4e16 < l < 1.2e5

   1. Initial program 71.5%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   2. Taylor expanded in l around 0 98.3%

      \[\leadsto \left(J \cdot \color{blue}{\left(0.3333333333333333 \cdot {\ell}^{3} + 2 \cdot \ell\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   3. Taylor expanded in l around 0 98.0%

      \[\leadsto \color{blue}{2 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \left(\ell \cdot J\right)\right)} + U \]
   4. Step-by-step derivation

      1. associate-*r*98.0%

         \[\leadsto \color{blue}{\left(2 \cdot \cos \left(0.5 \cdot K\right)\right) \cdot \left(\ell \cdot J\right)} + U \]

      2. *-commutative98.0%

         \[\leadsto \left(2 \cdot \cos \left(0.5 \cdot K\right)\right) \cdot \color{blue}{\left(J \cdot \ell\right)} + U \]

      3. associate-*r*98.0%

         \[\leadsto \color{blue}{\left(\left(2 \cdot \cos \left(0.5 \cdot K\right)\right) \cdot J\right) \cdot \ell} + U \]

      4. *-commutative98.0%

         \[\leadsto \left(\left(2 \cdot \cos \color{blue}{\left(K \cdot 0.5\right)}\right) \cdot J\right) \cdot \ell + U \]

      5. *-commutative98.0%

         \[\leadsto \left(\color{blue}{\left(\cos \left(K \cdot 0.5\right) \cdot 2\right)} \cdot J\right) \cdot \ell + U \]

      6. associate-*r*98.0%

         \[\leadsto \color{blue}{\left(\cos \left(K \cdot 0.5\right) \cdot \left(2 \cdot J\right)\right)} \cdot \ell + U \]

      7. *-commutative98.0%

         \[\leadsto \left(\cos \left(K \cdot 0.5\right) \cdot \color{blue}{\left(J \cdot 2\right)}\right) \cdot \ell + U \]

      8. *-commutative98.0%

         \[\leadsto \color{blue}{\ell \cdot \left(\cos \left(K \cdot 0.5\right) \cdot \left(J \cdot 2\right)\right)} + U \]

      9. *-commutative98.0%

         \[\leadsto \ell \cdot \color{blue}{\left(\left(J \cdot 2\right) \cdot \cos \left(K \cdot 0.5\right)\right)} + U \]

      10. *-commutative98.0%

          \[\leadsto \ell \cdot \left(\color{blue}{\left(2 \cdot J\right)} \cdot \cos \left(K \cdot 0.5\right)\right) + U \]

      11. *-commutative98.0%

          \[\leadsto \ell \cdot \left(\left(2 \cdot J\right) \cdot \cos \color{blue}{\left(0.5 \cdot K\right)}\right) + U \]

   5. Simplified98.0%

      \[\leadsto \color{blue}{\ell \cdot \left(\left(2 \cdot J\right) \cdot \cos \left(0.5 \cdot K\right)\right)} + U \]
3. Recombined 2 regimes into one program.

4. Final simplification76.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1.4 \cdot 10^{+16} \lor \neg \left(\ell \leq 120000\right):\\ \;\;\;\;U + 0.3333333333333333 \cdot \left(J \cdot {\ell}^{3}\right)\\ \mathbf{else}:\\ \;\;\;\;U + \ell \cdot \left(\cos \left(K \cdot 0.5\right) \cdot \left(J \cdot 2\right)\right)\\ \end{array} \]

1. Split input into 2 regimes

2. if l < -1.16e16 or 2.39999999999999991 < l

   1. Initial program 100.0%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   2. Taylor expanded in K around 0 79.3%

      \[\leadsto \color{blue}{\left(e^{\ell} - e^{-\ell}\right) \cdot J} + U \]

   3. Taylor expanded in l around 0 56.6%

      \[\leadsto \color{blue}{\left(2 \cdot \left(\ell \cdot J\right) + 0.3333333333333333 \cdot \left({\ell}^{3} \cdot J\right)\right)} + U \]

   4. Taylor expanded in l around inf 56.6%

      \[\leadsto \color{blue}{0.3333333333333333 \cdot \left({\ell}^{3} \cdot J\right)} + U \]

   if -1.16e16 < l < 2.39999999999999991

   1. Initial program 71.5%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   2. Taylor expanded in K around 0 69.2%

      \[\leadsto \color{blue}{\left(e^{\ell} - e^{-\ell}\right) \cdot J} + U \]

   3. Taylor expanded in l around 0 80.2%

      \[\leadsto \color{blue}{2 \cdot \left(\ell \cdot J\right)} + U \]
   4. Step-by-step derivation

      1. *-commutative80.2%

         \[\leadsto \color{blue}{\left(\ell \cdot J\right) \cdot 2} + U \]

   5. Simplified80.2%

      \[\leadsto \color{blue}{\left(\ell \cdot J\right) \cdot 2} + U \]
3. Recombined 2 regimes into one program.

4. Final simplification67.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1.16 \cdot 10^{+16} \lor \neg \left(\ell \leq 2.4\right):\\ \;\;\;\;U + 0.3333333333333333 \cdot \left(J \cdot {\ell}^{3}\right)\\ \mathbf{else}:\\ \;\;\;\;U + 2 \cdot \left(\ell \cdot J\right)\\ \end{array} \]

1. Split input into 3 regimes

2. if l < -420 or 3.4e124 < l

   1. Initial program 100.0%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   2. Taylor expanded in l around 0 29.9%

      \[\leadsto \left(J \cdot \color{blue}{\left(2 \cdot \ell\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   3. Taylor expanded in K around 0 13.8%

      \[\leadsto \color{blue}{\left(2 \cdot \left(\ell \cdot J\right) + -0.25 \cdot \left({K}^{2} \cdot \left(\ell \cdot J\right)\right)\right)} + U \]
   4. Step-by-step derivation

      1. +-commutative13.8%

         \[\leadsto \color{blue}{\left(-0.25 \cdot \left({K}^{2} \cdot \left(\ell \cdot J\right)\right) + 2 \cdot \left(\ell \cdot J\right)\right)} + U \]

      2. associate-*r*13.8%

         \[\leadsto \left(\color{blue}{\left(-0.25 \cdot {K}^{2}\right) \cdot \left(\ell \cdot J\right)} + 2 \cdot \left(\ell \cdot J\right)\right) + U \]

      3. distribute-rgt-out36.0%

         \[\leadsto \color{blue}{\left(\ell \cdot J\right) \cdot \left(-0.25 \cdot {K}^{2} + 2\right)} + U \]

      4. *-commutative36.0%

         \[\leadsto \left(\ell \cdot J\right) \cdot \left(\color{blue}{{K}^{2} \cdot -0.25} + 2\right) + U \]

      5. unpow236.0%

         \[\leadsto \left(\ell \cdot J\right) \cdot \left(\color{blue}{\left(K \cdot K\right)} \cdot -0.25 + 2\right) + U \]

   5. Simplified36.0%

      \[\leadsto \color{blue}{\left(\ell \cdot J\right) \cdot \left(\left(K \cdot K\right) \cdot -0.25 + 2\right)} + U \]

   if -420 < l < 125000

   1. Initial program 71.2%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   2. Taylor expanded in K around 0 70.6%

      \[\leadsto \color{blue}{\left(e^{\ell} - e^{-\ell}\right) \cdot J} + U \]

   3. Taylor expanded in l around 0 80.8%

      \[\leadsto \color{blue}{2 \cdot \left(\ell \cdot J\right)} + U \]
   4. Step-by-step derivation

      1. *-commutative80.8%

         \[\leadsto \color{blue}{\left(\ell \cdot J\right) \cdot 2} + U \]

   5. Simplified80.8%

      \[\leadsto \color{blue}{\left(\ell \cdot J\right) \cdot 2} + U \]

   if 125000 < l < 3.4e124

   1. Initial program 100.0%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   3. Applied egg-rr1.6%

      \[\leadsto \left(J \cdot \color{blue}{-8}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   5. Applied egg-rr33.3%

      \[\leadsto \color{blue}{U \cdot U} \]
3. Recombined 3 regimes into one program.

4. Final simplification56.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -420:\\ \;\;\;\;U + \left(\ell \cdot J\right) \cdot \left(2 + \left(K \cdot K\right) \cdot -0.25\right)\\ \mathbf{elif}\;\ell \leq 125000:\\ \;\;\;\;U + 2 \cdot \left(\ell \cdot J\right)\\ \mathbf{elif}\;\ell \leq 3.4 \cdot 10^{+124}:\\ \;\;\;\;U \cdot U\\ \mathbf{else}:\\ \;\;\;\;U + \left(\ell \cdot J\right) \cdot \left(2 + \left(K \cdot K\right) \cdot -0.25\right)\\ \end{array} \]

1. Split input into 2 regimes

2. if l < -3.79999999999999969e104 or 1.9e5 < l

   1. Initial program 100.0%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   3. Applied egg-rr2.1%

      \[\leadsto \left(J \cdot \color{blue}{-8}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   5. Applied egg-rr19.4%

      \[\leadsto \color{blue}{U \cdot U} \]

   if -3.79999999999999969e104 < l < 1.9e5

   1. Initial program 76.2%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   3. Applied egg-rr36.4%

      \[\leadsto \left(J \cdot \color{blue}{-8}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   4. Taylor expanded in J around 0 57.3%

      \[\leadsto \color{blue}{U} \]
3. Recombined 2 regimes into one program.

4. Final simplification40.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -3.8 \cdot 10^{+104}:\\ \;\;\;\;U \cdot U\\ \mathbf{elif}\;\ell \leq 190000:\\ \;\;\;\;U\\ \mathbf{else}:\\ \;\;\;\;U \cdot U\\ \end{array} \]

1. Split input into 3 regimes

2. if l < -4.2e20

   1. Initial program 100.0%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   3. Applied egg-rr3.1%

      \[\leadsto \left(J \cdot \color{blue}{-8}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   5. Applied egg-rr17.3%

      \[\leadsto \color{blue}{\left(U \cdot -134217728\right) \cdot U} \]

   if -4.2e20 < l < 1.9e5

   1. Initial program 72.1%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   3. Applied egg-rr42.0%

      \[\leadsto \left(J \cdot \color{blue}{-8}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   4. Taylor expanded in J around 0 66.6%

      \[\leadsto \color{blue}{U} \]

   if 1.9e5 < l

   1. Initial program 100.0%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   3. Applied egg-rr1.6%

      \[\leadsto \left(J \cdot \color{blue}{-8}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   5. Applied egg-rr20.5%

      \[\leadsto \color{blue}{U \cdot U} \]
3. Recombined 3 regimes into one program.

4. Final simplification42.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -4.2 \cdot 10^{+20}:\\ \;\;\;\;U \cdot \left(U \cdot -134217728\right)\\ \mathbf{elif}\;\ell \leq 190000:\\ \;\;\;\;U\\ \mathbf{else}:\\ \;\;\;\;U \cdot U\\ \end{array} \]

1. Initial program 86.5%

   \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

2. Taylor expanded in K around 0 74.5%

   \[\leadsto \color{blue}{\left(e^{\ell} - e^{-\ell}\right) \cdot J} + U \]

3. Taylor expanded in l around 0 49.7%

   \[\leadsto \color{blue}{2 \cdot \left(\ell \cdot J\right)} + U \]
4. Step-by-step derivation

   1. *-commutative49.7%

      \[\leadsto \color{blue}{\left(\ell \cdot J\right) \cdot 2} + U \]

5. Simplified49.7%

   \[\leadsto \color{blue}{\left(\ell \cdot J\right) \cdot 2} + U \]

6. Final simplification49.7%

   \[\leadsto U + 2 \cdot \left(\ell \cdot J\right) \]

1. Initial program 86.5%

   \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

3. Applied egg-rr21.5%

   \[\leadsto \left(J \cdot \color{blue}{-8}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

4. Taylor expanded in J around 0 33.3%

   \[\leadsto \color{blue}{U} \]

5. Final simplification33.3%

   \[\leadsto U \]

herbie shell --seed 2023166 
(FPCore (J l K U)
  :name "Maksimov and Kolovsky, Equation (4)"
  :precision binary64
  (+ (* (* J (- (exp l) (exp (- l)))) (cos (/ K 2.0))) U))