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Derivation

1. Initial program 15.2

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

2. Initial simplification15.2

   \[\leadsto \frac{r \cdot \sin b}{\cos \left(b + a\right)}\]
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 fma-neg0.3

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

8. Applied add-log-exp0.4

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

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

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

11. Applied times-frac0.4

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

12. Simplified0.4

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

13. Simplified0.3

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

14. Final simplification0.3

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

Runtime

Time bar (total: 25.4s)Debug logProfile

herbie shell --seed 2018353 +o rules:numerics
(FPCore (r a b)
  :name "r*sin(b)/cos(a+b), A"
  (/ (* r (sin b)) (cos (+ a b))))