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Derivation

1. Initial program 15.1

   \[r \cdot \frac{\sin b}{\cos \left(a + b\right)}\]
2. Using strategy rm

3. Applied cos-sum0.3

   \[\leadsto r \cdot \frac{\sin b}{\color{blue}{\cos a \cdot \cos b - \sin a \cdot \sin b}}\]
4. Using strategy rm

5. Applied flip3--0.4

   \[\leadsto r \cdot \frac{\sin b}{\color{blue}{\frac{{\left(\cos a \cdot \cos b\right)}^{3} - {\left(\sin a \cdot \sin b\right)}^{3}}{\left(\cos a \cdot \cos b\right) \cdot \left(\cos a \cdot \cos b\right) + \left(\left(\sin a \cdot \sin b\right) \cdot \left(\sin a \cdot \sin b\right) + \left(\cos a \cdot \cos b\right) \cdot \left(\sin a \cdot \sin b\right)\right)}}}\]

6. Taylor expanded around inf 0.4

   \[\leadsto r \cdot \frac{\sin b}{\frac{{\left(\cos a \cdot \cos b\right)}^{3} - \color{blue}{{\left(\sin b\right)}^{3} \cdot {\left(\sin a\right)}^{3}}}{\left(\cos a \cdot \cos b\right) \cdot \left(\cos a \cdot \cos b\right) + \left(\left(\sin a \cdot \sin b\right) \cdot \left(\sin a \cdot \sin b\right) + \left(\cos a \cdot \cos b\right) \cdot \left(\sin a \cdot \sin b\right)\right)}}\]

7. Final simplification0.4

   \[\leadsto \frac{\sin b}{\frac{{\left(\cos a \cdot \cos b\right)}^{3} - {\left(\sin a\right)}^{3} \cdot {\left(\sin b\right)}^{3}}{\left(\left(\sin b \cdot \sin a\right) \cdot \left(\cos a \cdot \cos b\right) + \left(\sin b \cdot \sin a\right) \cdot \left(\sin b \cdot \sin a\right)\right) + \left(\cos a \cdot \cos b\right) \cdot \left(\cos a \cdot \cos b\right)}} \cdot r\]

Runtime

Time bar (total: 34.8s)Debug logProfile

herbie shell --seed 2018355 +o rules:numerics
(FPCore (r a b)
  :name "r*sin(b)/cos(a+b), B"
  (* r (/ (sin b) (cos (+ a b)))))