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 tan-quot0.2

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

6. Applied associate-*l/0.2

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

8. Applied div-inv0.2

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

10. Applied add-sqr-sqrt30.8

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

11. Applied flip3--30.8

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

12. Applied associate-/r/30.8

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

13. Applied prod-diff30.8

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

14. Simplified30.7

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

15. Simplified0.2

    \[\leadsto x + \left((\left(\frac{\tan y + \tan z}{(\left({\left(\tan z \cdot \sin y\right)}^{3}\right) \cdot \left(\frac{-1}{{\left(\cos y\right)}^{3}}\right) + 1)_*}\right) \cdot \left((\left((\left(\frac{\tan z}{\cos y}\right) \cdot \left(\sin y\right) + 1)_*\right) \cdot \left(\frac{\tan z}{\cos y} \cdot \sin y\right) + 1)_*\right) + \left(-\tan a\right))_* + \color{blue}{0}\right)\]

16. Final simplification0.2

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

Runtime

Time bar (total: 1.4m)Debug logProfile

herbie shell --seed 2018297 +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))))