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Target

Derivation

1. Initial program 36.5

   \[\tan \left(x + \varepsilon\right) - \tan x\]

2. Initial simplification36.5

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

4. Applied tan-sum21.4

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

6. Applied tan-quot21.5

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

7. Applied frac-sub21.5

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

8. Taylor expanded around inf 0.4

   \[\leadsto \frac{\color{blue}{\frac{{\left(\sin x\right)}^{2} \cdot \sin \varepsilon}{\cos x \cdot \cos \varepsilon} + \frac{\cos x \cdot \sin \varepsilon}{\cos \varepsilon}}}{\left(1 - \tan \varepsilon \cdot \tan x\right) \cdot \cos x}\]

9. Simplified0.4

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

10. Final simplification0.4

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

Runtime

Time bar (total: 38.3s)Debug logProfile

herbie shell --seed 2018274 
(FPCore (x eps)
  :name "2tan (problem 3.3.2)"

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

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