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Derivation

1. Initial program 14.9

   \[r \cdot \frac{\sin b}{\cos \left(a + b\right)}\]
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. Applied associate-/r/0.4

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

7. Applied associate-*r*0.5

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

8. Simplified0.5

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

10. Applied add-cbrt-cube0.7

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

11. Applied add-cbrt-cube0.9

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

12. Applied cbrt-unprod0.9

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

13. Applied rem-cube-cbrt0.5

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

14. Simplified0.5

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

15. Final simplification0.5

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

Runtime

Time bar (total: 34.0s)Debug logProfile

herbie shell --seed 2018251 
(FPCore (r a b)
  :name "r*sin(b)/cos(a+b), B"
  (* r (/ (sin b) (cos (+ a b)))))