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. Taylor expanded around -inf 0.2

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

7. Final simplification0.2

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

Reproduce

herbie shell --seed 2019093 
(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))))