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. Simplified0.3

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

16. Simplified0.3

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

17. Taylor expanded around -inf 0.2

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

18. Simplified0.2

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

19. Final simplification0.2

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

Runtime

Time bar (total: 5.4m)Debug logProfile

herbie shell --seed 2018255 +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))))