Error

Derivation

1. Initial program 15.1

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

3. Applied cos-sum0.3

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

5. Applied flip3--0.4

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

6. Applied associate-/r/0.5

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

7. Final simplification0.5

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

Reproduce

herbie shell --seed 2019090 
(FPCore (r a b)
  :name "r*sin(b)/cos(a+b), A"
  (/ (* r (sin b)) (cos (+ a b))))