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Derivation

1. Initial program 15.6

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

2. Initial simplification15.6

   \[\leadsto \frac{r \cdot \sin b}{\cos \left(b + a\right)}\]
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 add-cbrt-cube0.4

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

7. Applied add-cbrt-cube0.4

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

8. Applied cbrt-unprod0.4

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

9. Simplified0.4

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

11. Applied cube-prod0.4

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

12. Applied cbrt-prod0.4

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

13. Simplified0.4

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

14. Final simplification0.4

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

Runtime

Time bar (total: 31.4s)Debug logProfile

herbie shell --seed 2018254 +o rules:numerics
(FPCore (r a b)
  :name "r*sin(b)/cos(a+b), B"
  (* r (/ (sin b) (cos (+ a b)))))