Error

Target

Derivation

1. Initial program 36.8

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

3. Applied tan-sum21.8

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

5. Applied flip--21.9

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

6. Applied associate-/r/21.9

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

7. Applied fma-neg21.9

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

8. Taylor expanded around inf 22.0

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

9. Applied simplify0.6

   \[\leadsto \color{blue}{(\left(\frac{\frac{\sin x}{\cos x} \cdot \frac{\sin x}{\cos x}}{1 - \frac{\left(\sin x \cdot \sin \varepsilon\right) \cdot \left(\sin x \cdot \sin \varepsilon\right)}{\left(\cos x \cdot \cos \varepsilon\right) \cdot \left(\cos x \cdot \cos \varepsilon\right)}}\right) \cdot \left(\frac{\sin \varepsilon}{\cos \varepsilon}\right) + \left(\frac{\frac{\sin \varepsilon}{\cos \varepsilon}}{1 - \frac{\left(\sin x \cdot \sin \varepsilon\right) \cdot \left(\sin x \cdot \sin \varepsilon\right)}{\left(\cos x \cdot \cos \varepsilon\right) \cdot \left(\cos x \cdot \cos \varepsilon\right)}}\right))_* + \left(\frac{(\left(\frac{\sin \varepsilon}{\cos \varepsilon}\right) \cdot \left(\frac{\sin \varepsilon}{\cos \varepsilon}\right) + 1)_* \cdot \frac{\sin x}{\cos x}}{1 - \frac{\left(\sin x \cdot \sin \varepsilon\right) \cdot \left(\sin x \cdot \sin \varepsilon\right)}{\left(\cos x \cdot \cos \varepsilon\right) \cdot \left(\cos x \cdot \cos \varepsilon\right)}} - \frac{\sin x}{\cos x}\right)}\]
10. Using strategy rm

11. Applied frac-times0.6

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

Runtime

Time bar (total: 1.1m)Debug logProfile

herbie shell --seed '#(1064300848 3212030778 2049303162 3567222883 2277747821 1384278011)' +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)))