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Derivation

1. Initial program 14.9

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

2. Initial simplification14.9

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

4. Applied cos-sum0.3

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

6. Applied *-un-lft-identity0.3

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

7. Applied times-frac0.3

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

8. Simplified0.3

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

10. Applied add-cbrt-cube0.4

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

11. Applied add-cbrt-cube0.4

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

12. Applied cbrt-unprod0.4

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

13. Simplified0.4

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

14. Final simplification0.4

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

Runtime

Time bar (total: 25.9s)Debug logProfile

herbie shell --seed 2018352 
(FPCore (r a b)
  :name "r*sin(b)/cos(a+b), A"
  (/ (* r (sin b)) (cos (+ a b))))