Error

Target

Derivation

1. Initial program 37.2

   \[\tan \left(x + \varepsilon\right) - \tan x\]
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 add-cube-cbrt22.1

   \[\leadsto \frac{\tan x + \tan \varepsilon}{1 - \color{blue}{\left(\sqrt[3]{\tan x \cdot \tan \varepsilon} \cdot \sqrt[3]{\tan x \cdot \tan \varepsilon}\right) \cdot \sqrt[3]{\tan x \cdot \tan \varepsilon}}} - \tan x\]
6. Using strategy rm

7. Applied tan-quot22.2

   \[\leadsto \frac{\tan x + \tan \varepsilon}{1 - \left(\sqrt[3]{\tan x \cdot \tan \varepsilon} \cdot \sqrt[3]{\tan x \cdot \tan \varepsilon}\right) \cdot \sqrt[3]{\tan x \cdot \tan \varepsilon}} - \color{blue}{\frac{\sin x}{\cos x}}\]

8. Applied frac-sub22.2

   \[\leadsto \color{blue}{\frac{\left(\tan x + \tan \varepsilon\right) \cdot \cos x - \left(1 - \left(\sqrt[3]{\tan x \cdot \tan \varepsilon} \cdot \sqrt[3]{\tan x \cdot \tan \varepsilon}\right) \cdot \sqrt[3]{\tan x \cdot \tan \varepsilon}\right) \cdot \sin x}{\left(1 - \left(\sqrt[3]{\tan x \cdot \tan \varepsilon} \cdot \sqrt[3]{\tan x \cdot \tan \varepsilon}\right) \cdot \sqrt[3]{\tan x \cdot \tan \varepsilon}\right) \cdot \cos x}}\]

9. Applied simplify22.3

   \[\leadsto \frac{\color{blue}{\cos x \cdot \left(\tan \varepsilon + \tan x\right) - \left(\sin x - \left(\tan \varepsilon \cdot \tan x\right) \cdot \sin x\right)}}{\left(1 - \left(\sqrt[3]{\tan x \cdot \tan \varepsilon} \cdot \sqrt[3]{\tan x \cdot \tan \varepsilon}\right) \cdot \sqrt[3]{\tan x \cdot \tan \varepsilon}\right) \cdot \cos x}\]

10. Applied simplify22.1

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

11. Taylor expanded around inf 0.4

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

12. Applied simplify0.4

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

Runtime

Time bar (total: 1.8m)Debug logProfile

herbie shell --seed '#(1072840222 1305617769 1692503039 1353360431 4178980589 1488672652)' 
(FPCore (x eps)
  :name "2tan (problem 3.3.2)"

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

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