Error

Derivation

1. Initial program 13.1

   \[x + \left(\tan \left(y + z\right) - \tan a\right)\]
2. Using strategy rm

3. Applied tan-sum0.2

   \[\leadsto x + \left(\color{blue}{\frac{\tan y + \tan z}{1 - \tan y \cdot \tan z}} - \tan a\right)\]
4. Using strategy rm

5. Applied add-cbrt-cube0.3

   \[\leadsto x + \color{blue}{\sqrt[3]{\left(\left(\frac{\tan y + \tan z}{1 - \tan y \cdot \tan z} - \tan a\right) \cdot \left(\frac{\tan y + \tan z}{1 - \tan y \cdot \tan z} - \tan a\right)\right) \cdot \left(\frac{\tan y + \tan z}{1 - \tan y \cdot \tan z} - \tan a\right)}}\]
6. Using strategy rm

7. Applied tan-quot0.3

   \[\leadsto x + \sqrt[3]{\left(\left(\frac{\tan y + \tan z}{1 - \tan y \cdot \tan z} - \tan a\right) \cdot \left(\frac{\tan y + \tan z}{1 - \tan y \cdot \tan z} - \tan a\right)\right) \cdot \left(\frac{\tan y + \tan z}{1 - \tan y \cdot \tan z} - \color{blue}{\frac{\sin a}{\cos a}}\right)}\]

8. Applied frac-sub0.3

   \[\leadsto x + \sqrt[3]{\left(\left(\frac{\tan y + \tan z}{1 - \tan y \cdot \tan z} - \tan a\right) \cdot \left(\frac{\tan y + \tan z}{1 - \tan y \cdot \tan z} - \tan a\right)\right) \cdot \color{blue}{\frac{\left(\tan y + \tan z\right) \cdot \cos a - \left(1 - \tan y \cdot \tan z\right) \cdot \sin a}{\left(1 - \tan y \cdot \tan z\right) \cdot \cos a}}}\]

9. Applied tan-quot0.3

   \[\leadsto x + \sqrt[3]{\left(\left(\frac{\tan y + \tan z}{1 - \tan y \cdot \tan z} - \tan a\right) \cdot \left(\frac{\tan y + \tan z}{1 - \tan y \cdot \tan z} - \color{blue}{\frac{\sin a}{\cos a}}\right)\right) \cdot \frac{\left(\tan y + \tan z\right) \cdot \cos a - \left(1 - \tan y \cdot \tan z\right) \cdot \sin a}{\left(1 - \tan y \cdot \tan z\right) \cdot \cos a}}\]

10. Applied frac-sub0.3

    \[\leadsto x + \sqrt[3]{\left(\left(\frac{\tan y + \tan z}{1 - \tan y \cdot \tan z} - \tan a\right) \cdot \color{blue}{\frac{\left(\tan y + \tan z\right) \cdot \cos a - \left(1 - \tan y \cdot \tan z\right) \cdot \sin a}{\left(1 - \tan y \cdot \tan z\right) \cdot \cos a}}\right) \cdot \frac{\left(\tan y + \tan z\right) \cdot \cos a - \left(1 - \tan y \cdot \tan z\right) \cdot \sin a}{\left(1 - \tan y \cdot \tan z\right) \cdot \cos a}}\]

11. Applied tan-quot0.3

    \[\leadsto x + \sqrt[3]{\left(\left(\frac{\tan y + \tan z}{1 - \tan y \cdot \tan z} - \color{blue}{\frac{\sin a}{\cos a}}\right) \cdot \frac{\left(\tan y + \tan z\right) \cdot \cos a - \left(1 - \tan y \cdot \tan z\right) \cdot \sin a}{\left(1 - \tan y \cdot \tan z\right) \cdot \cos a}\right) \cdot \frac{\left(\tan y + \tan z\right) \cdot \cos a - \left(1 - \tan y \cdot \tan z\right) \cdot \sin a}{\left(1 - \tan y \cdot \tan z\right) \cdot \cos a}}\]

12. Applied frac-sub0.3

    \[\leadsto x + \sqrt[3]{\left(\color{blue}{\frac{\left(\tan y + \tan z\right) \cdot \cos a - \left(1 - \tan y \cdot \tan z\right) \cdot \sin a}{\left(1 - \tan y \cdot \tan z\right) \cdot \cos a}} \cdot \frac{\left(\tan y + \tan z\right) \cdot \cos a - \left(1 - \tan y \cdot \tan z\right) \cdot \sin a}{\left(1 - \tan y \cdot \tan z\right) \cdot \cos a}\right) \cdot \frac{\left(\tan y + \tan z\right) \cdot \cos a - \left(1 - \tan y \cdot \tan z\right) \cdot \sin a}{\left(1 - \tan y \cdot \tan z\right) \cdot \cos a}}\]

13. Applied frac-times0.3

    \[\leadsto x + \sqrt[3]{\color{blue}{\frac{\left(\left(\tan y + \tan z\right) \cdot \cos a - \left(1 - \tan y \cdot \tan z\right) \cdot \sin a\right) \cdot \left(\left(\tan y + \tan z\right) \cdot \cos a - \left(1 - \tan y \cdot \tan z\right) \cdot \sin a\right)}{\left(\left(1 - \tan y \cdot \tan z\right) \cdot \cos a\right) \cdot \left(\left(1 - \tan y \cdot \tan z\right) \cdot \cos a\right)}} \cdot \frac{\left(\tan y + \tan z\right) \cdot \cos a - \left(1 - \tan y \cdot \tan z\right) \cdot \sin a}{\left(1 - \tan y \cdot \tan z\right) \cdot \cos a}}\]

14. Applied frac-times0.3

    \[\leadsto x + \sqrt[3]{\color{blue}{\frac{\left(\left(\left(\tan y + \tan z\right) \cdot \cos a - \left(1 - \tan y \cdot \tan z\right) \cdot \sin a\right) \cdot \left(\left(\tan y + \tan z\right) \cdot \cos a - \left(1 - \tan y \cdot \tan z\right) \cdot \sin a\right)\right) \cdot \left(\left(\tan y + \tan z\right) \cdot \cos a - \left(1 - \tan y \cdot \tan z\right) \cdot \sin a\right)}{\left(\left(\left(1 - \tan y \cdot \tan z\right) \cdot \cos a\right) \cdot \left(\left(1 - \tan y \cdot \tan z\right) \cdot \cos a\right)\right) \cdot \left(\left(1 - \tan y \cdot \tan z\right) \cdot \cos a\right)}}}\]

15. Applied cbrt-div0.3

    \[\leadsto x + \color{blue}{\frac{\sqrt[3]{\left(\left(\left(\tan y + \tan z\right) \cdot \cos a - \left(1 - \tan y \cdot \tan z\right) \cdot \sin a\right) \cdot \left(\left(\tan y + \tan z\right) \cdot \cos a - \left(1 - \tan y \cdot \tan z\right) \cdot \sin a\right)\right) \cdot \left(\left(\tan y + \tan z\right) \cdot \cos a - \left(1 - \tan y \cdot \tan z\right) \cdot \sin a\right)}}{\sqrt[3]{\left(\left(\left(1 - \tan y \cdot \tan z\right) \cdot \cos a\right) \cdot \left(\left(1 - \tan y \cdot \tan z\right) \cdot \cos a\right)\right) \cdot \left(\left(1 - \tan y \cdot \tan z\right) \cdot \cos a\right)}}}\]

16. Simplified0.3

    \[\leadsto x + \frac{\color{blue}{(\left(\sin a\right) \cdot \left(\tan z \cdot \tan y + -1\right) + \left(\cos a \cdot \left(\tan y + \tan z\right)\right))_*}}{\sqrt[3]{\left(\left(\left(1 - \tan y \cdot \tan z\right) \cdot \cos a\right) \cdot \left(\left(1 - \tan y \cdot \tan z\right) \cdot \cos a\right)\right) \cdot \left(\left(1 - \tan y \cdot \tan z\right) \cdot \cos a\right)}}\]

17. Final simplification0.3

    \[\leadsto x + \frac{(\left(\sin a\right) \cdot \left(\tan y \cdot \tan z + -1\right) + \left(\cos a \cdot \left(\tan y + \tan z\right)\right))_*}{\sqrt[3]{\left(\left(\left(1 - \tan y \cdot \tan z\right) \cdot \cos a\right) \cdot \left(\left(1 - \tan y \cdot \tan z\right) \cdot \cos a\right)\right) \cdot \left(\left(1 - \tan y \cdot \tan z\right) \cdot \cos a\right)}}\]

Runtime

Time bar (total: 4.5m)Debug logProfile

herbie shell --seed 2018274 +o rules:numerics
(FPCore (x y z a)
  :name "(+ x (- (tan (+ y z)) (tan a)))"
  :pre (and (or (== x 0) (<= 0.5884142 x 505.5909)) (or (<= -1.796658e+308 y -9.425585e-310) (<= 1.284938e-309 y 1.751224e+308)) (or (<= -1.776707e+308 z -8.599796e-310) (<= 3.293145e-311 z 1.725154e+308)) (or (<= -1.796658e+308 a -9.425585e-310) (<= 1.284938e-309 a 1.751224e+308)))
  (+ x (- (tan (+ y z)) (tan a))))