Error

Target

Derivation

1. Initial program 36.7

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

3. Applied tan-quot36.7

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

4. Applied tan-sum21.8

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

5. Applied frac-sub21.8

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

6. Taylor expanded around inf 0.4

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

7. Applied simplify0.4

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

Runtime

Time bar (total: 2.0m)Debug logProfile

herbie shell --seed '#(1071119240 1686926585 3481876196 78132896 2080707795 3185793749)' +o rules:numerics
(FPCore (x eps)
  :name "2tan (problem 3.3.2)"

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

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