Error

Derivation

1. Initial program 12.9

   \[x + \left(\tan \left(y + z\right) - \tan a\right)\]
2. Using strategy rm

3. Applied tan-quot12.9

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

4. Applied tan-sum0.2

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

5. Applied frac-sub0.2

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

6. Simplified0.2

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

8. Applied add-cbrt-cube0.2

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

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

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

11. Applied times-frac0.2

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

12. Simplified0.2

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

13. Final simplification0.2

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

Reproduce

herbie shell --seed 2019022 +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))))

Details

Time bar (total: 54.0s)Debug log