Error

Derivation

1. Initial program 12.9

   \[x + \left(\tan \left(y + z\right) - \tan a\right)\]
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 add-cube-cbrt0.3

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

9. Applied flip--0.3

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

10. Applied associate-/r/0.3

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

11. Applied prod-diff0.3

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

12. Simplified0.2

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

13. Simplified0.2

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

14. Final simplification0.2

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

Runtime

Time bar (total: 1.6m)Debug logProfile

herbie shell --seed 2018252 +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))))