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Target

Derivation

1. Initial program 36.8

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

3. Applied tan-sum22.0

   \[\leadsto \color{blue}{\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon}} - \tan x\]
4. Using strategy rm

5. Applied div-inv22.0

   \[\leadsto \color{blue}{\left(\tan x + \tan \varepsilon\right) \cdot \frac{1}{1 - \tan x \cdot \tan \varepsilon}} - \tan x\]
6. Using strategy rm

7. Applied tan-quot22.1

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

8. Applied associate-*r/22.1

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

9. Applied frac-sub22.2

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

10. Simplified20.8

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

11. Taylor expanded around -inf 0.4

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

12. Final simplification0.4

    \[\leadsto \frac{\left(\tan \varepsilon \cdot \tan x\right) \cdot \sin x + \frac{\cos x \cdot \sin \varepsilon}{\cos \varepsilon}}{\left(1 - \tan \varepsilon \cdot \tan x\right) \cdot \cos x}\]

Runtime

Time bar (total: 49.3s)Debug logProfile

herbie shell --seed 2018348 
(FPCore (x eps)
  :name "2tan (problem 3.3.2)"

  :herbie-target
  (/ (sin eps) (* (cos x) (cos (+ x eps))))

  (- (tan (+ x eps)) (tan x)))