Error

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Derivation

1. Initial program 13.2

   \[x + \left(\tan \left(y + z\right) - \tan a\right)\]
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

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

6. Applied add-log-exp0.2

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

8. Applied flip--0.3

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

9. Final simplification0.3

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

Reproduce

herbie shell --seed 2018365 +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

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