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Derivation

1. Initial program 13.6

   \[x + \left(\tan \left(y + z\right) - \tan a\right)\]
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. Taylor expanded around inf 0.2

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

6. Applied flip3--0.2

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

7. Applied associate-/r/0.2

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

8. Simplified0.2

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

10. Applied add-cube-cbrt0.2

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

11. Applied *-un-lft-identity0.2

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

12. Applied times-frac0.2

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

13. Applied div-inv0.2

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

14. Applied times-frac0.2

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

15. Simplified0.2

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

16. Simplified0.2

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

17. Final simplification0.2

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

Runtime

Time bar (total: 2.7m)Debug logProfile

herbie shell --seed 2018251 
(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))))