Error

Derivation

1. Initial program 12.7

   \[x + \left(\tan \left(y + z\right) - \tan a\right)\]
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-sqr-sqrt31.3

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

6. Applied flip--31.3

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

7. Applied associate-/r/31.3

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

8. Applied prod-diff31.3

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

9. Simplified31.3

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

10. Simplified0.2

    \[\leadsto x + \left(\left(\frac{(\left(\tan y + \tan z\right) \cdot \left(\tan z \cdot \tan y\right) + \left(\tan y + \tan z\right))_*}{1 - \left(\tan z \cdot \tan y\right) \cdot \left(\tan z \cdot \tan y\right)} - \tan a\right) + \color{blue}{0}\right)\]
11. Using strategy rm

12. Applied associate-*r*0.2

    \[\leadsto x + \left(\left(\frac{(\left(\tan y + \tan z\right) \cdot \left(\tan z \cdot \tan y\right) + \left(\tan y + \tan z\right))_*}{1 - \color{blue}{\left(\left(\tan z \cdot \tan y\right) \cdot \tan z\right) \cdot \tan y}} - \tan a\right) + 0\right)\]
13. Using strategy rm

14. Applied tan-quot0.2

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

15. Applied associate-*r/0.2

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

16. Applied associate-*l/0.2

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

17. Applied associate-*l/0.2

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

18. Final simplification0.2

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

Runtime

Time bar (total: 1.1m)Debug logProfile

herbie shell --seed 2018354 +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))))