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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 associate-*r/0.3

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

7. Applied flip--0.4

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

9. Applied *-un-lft-identity0.4

   \[\leadsto \frac{r \cdot \sin b}{\frac{\left(\cos a \cdot \cos b\right) \cdot \left(\cos a \cdot \cos b\right) - \left(\sin a \cdot \sin b\right) \cdot \left(\sin a \cdot \sin b\right)}{\color{blue}{1 \cdot \left(\cos a \cdot \cos b + \sin a \cdot \sin b\right)}}}\]

10. Applied difference-of-squares0.4

    \[\leadsto \frac{r \cdot \sin b}{\frac{\color{blue}{\left(\cos a \cdot \cos b + \sin a \cdot \sin b\right) \cdot \left(\cos a \cdot \cos b - \sin a \cdot \sin b\right)}}{1 \cdot \left(\cos a \cdot \cos b + \sin a \cdot \sin b\right)}}\]

11. Applied times-frac0.3

    \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\frac{\cos a \cdot \cos b + \sin a \cdot \sin b}{1} \cdot \frac{\cos a \cdot \cos b - \sin a \cdot \sin b}{\cos a \cdot \cos b + \sin a \cdot \sin b}}}\]

12. Simplified0.3

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

13. Final simplification0.3

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

Reproduce

herbie shell --seed 2019002 
(FPCore (r a b)
  :name "r*sin(b)/cos(a+b), B"
  (* r (/ (sin b) (cos (+ a b)))))

Details

Time bar (total: 33.5s)Debug log