Initial program 37.2
\[\tan \left(x + \varepsilon\right) - \tan x\]
- Using strategy
rm Applied tan-sum21.5
\[\leadsto \color{blue}{\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon}} - \tan x\]
- Using strategy
rm Applied flip3--21.5
\[\leadsto \frac{\tan x + \tan \varepsilon}{\color{blue}{\frac{{1}^{3} - {\left(\tan x \cdot \tan \varepsilon\right)}^{3}}{1 \cdot 1 + \left(\left(\tan x \cdot \tan \varepsilon\right) \cdot \left(\tan x \cdot \tan \varepsilon\right) + 1 \cdot \left(\tan x \cdot \tan \varepsilon\right)\right)}}} - \tan x\]
Applied associate-/r/21.5
\[\leadsto \color{blue}{\frac{\tan x + \tan \varepsilon}{{1}^{3} - {\left(\tan x \cdot \tan \varepsilon\right)}^{3}} \cdot \left(1 \cdot 1 + \left(\left(\tan x \cdot \tan \varepsilon\right) \cdot \left(\tan x \cdot \tan \varepsilon\right) + 1 \cdot \left(\tan x \cdot \tan \varepsilon\right)\right)\right)} - \tan x\]
Applied fma-neg21.5
\[\leadsto \color{blue}{(\left(\frac{\tan x + \tan \varepsilon}{{1}^{3} - {\left(\tan x \cdot \tan \varepsilon\right)}^{3}}\right) \cdot \left(1 \cdot 1 + \left(\left(\tan x \cdot \tan \varepsilon\right) \cdot \left(\tan x \cdot \tan \varepsilon\right) + 1 \cdot \left(\tan x \cdot \tan \varepsilon\right)\right)\right) + \left(-\tan x\right))_*}\]
Taylor expanded around inf 21.6
\[\leadsto \color{blue}{\left(\frac{{\left(\sin \varepsilon\right)}^{2} \cdot \sin x}{{\left(\cos \varepsilon\right)}^{2} \cdot \left(\cos x \cdot \left(1 - \frac{{\left(\sin \varepsilon\right)}^{3} \cdot {\left(\sin x\right)}^{3}}{{\left(\cos \varepsilon\right)}^{3} \cdot {\left(\cos x\right)}^{3}}\right)\right)} + \left(\frac{\sin \varepsilon}{\cos \varepsilon \cdot \left(1 - \frac{{\left(\sin \varepsilon\right)}^{3} \cdot {\left(\sin x\right)}^{3}}{{\left(\cos \varepsilon\right)}^{3} \cdot {\left(\cos x\right)}^{3}}\right)} + \left(\frac{\sin \varepsilon \cdot {\left(\sin x\right)}^{2}}{\cos \varepsilon \cdot \left({\left(\cos x\right)}^{2} \cdot \left(1 - \frac{{\left(\sin \varepsilon\right)}^{3} \cdot {\left(\sin x\right)}^{3}}{{\left(\cos \varepsilon\right)}^{3} \cdot {\left(\cos x\right)}^{3}}\right)\right)} + \left(\frac{{\left(\sin \varepsilon\right)}^{2} \cdot {\left(\sin x\right)}^{3}}{{\left(\cos \varepsilon\right)}^{2} \cdot \left({\left(\cos x\right)}^{3} \cdot \left(1 - \frac{{\left(\sin \varepsilon\right)}^{3} \cdot {\left(\sin x\right)}^{3}}{{\left(\cos \varepsilon\right)}^{3} \cdot {\left(\cos x\right)}^{3}}\right)\right)} + \left(\frac{\sin x}{\cos x \cdot \left(1 - \frac{{\left(\sin \varepsilon\right)}^{3} \cdot {\left(\sin x\right)}^{3}}{{\left(\cos \varepsilon\right)}^{3} \cdot {\left(\cos x\right)}^{3}}\right)} + \frac{{\left(\sin \varepsilon\right)}^{3} \cdot {\left(\sin x\right)}^{2}}{{\left(\cos \varepsilon\right)}^{3} \cdot \left({\left(\cos x\right)}^{2} \cdot \left(1 - \frac{{\left(\sin \varepsilon\right)}^{3} \cdot {\left(\sin x\right)}^{3}}{{\left(\cos \varepsilon\right)}^{3} \cdot {\left(\cos x\right)}^{3}}\right)\right)}\right)\right)\right)\right)\right) - \frac{\sin x}{\cos x}}\]
Applied simplify0.5
\[\leadsto \color{blue}{(\left(\frac{\frac{\sin \varepsilon}{\cos \varepsilon} \cdot \frac{\sin \varepsilon}{\cos \varepsilon}}{\cos x}\right) \cdot \left(\frac{\sin x}{1 - {\left(\frac{\sin \varepsilon}{\cos \varepsilon}\right)}^{3} \cdot \frac{{\left(\sin x\right)}^{3}}{{\left(\cos x\right)}^{3}}}\right) + \left(\frac{\frac{\sin \varepsilon}{\cos \varepsilon}}{1 - {\left(\frac{\sin \varepsilon}{\cos \varepsilon}\right)}^{3} \cdot \frac{{\left(\sin x\right)}^{3}}{{\left(\cos x\right)}^{3}}}\right))_* + \left((\left(\frac{\sin \varepsilon}{\cos \varepsilon}\right) \cdot \left(\frac{\frac{\sin x}{\cos x}}{1 - {\left(\frac{\sin \varepsilon}{\cos \varepsilon}\right)}^{3} \cdot \frac{{\left(\sin x\right)}^{3}}{{\left(\cos x\right)}^{3}}} \cdot \frac{\sin x}{\cos x}\right) + \left(\frac{\left(\frac{\sin \varepsilon}{\cos \varepsilon} \cdot \frac{\sin \varepsilon}{\cos \varepsilon}\right) \cdot {\left(\sin x\right)}^{3}}{{\left(\cos x\right)}^{3} \cdot \left(1 - {\left(\frac{\sin \varepsilon}{\cos \varepsilon}\right)}^{3} \cdot \frac{{\left(\sin x\right)}^{3}}{{\left(\cos x\right)}^{3}}\right)}\right))_* + \left(\left(\frac{\left(\frac{\sin \varepsilon}{{\left(\cos \varepsilon\right)}^{3}} \cdot \frac{\sin \varepsilon \cdot \sin \varepsilon}{\cos x \cdot \cos x}\right) \cdot \left(\sin x \cdot \sin x\right)}{1 - {\left(\frac{\sin \varepsilon}{\cos \varepsilon}\right)}^{3} \cdot \frac{{\left(\sin x\right)}^{3}}{{\left(\cos x\right)}^{3}}} + \frac{\frac{\sin x}{\cos x}}{1 - {\left(\frac{\sin \varepsilon}{\cos \varepsilon}\right)}^{3} \cdot \frac{{\left(\sin x\right)}^{3}}{{\left(\cos x\right)}^{3}}}\right) - \frac{\sin x}{\cos x}\right)\right)}\]
Taylor expanded around -inf 0.5
\[\leadsto (\left(\frac{\frac{\sin \varepsilon}{\cos \varepsilon} \cdot \frac{\sin \varepsilon}{\cos \varepsilon}}{\cos x}\right) \cdot \left(\frac{\sin x}{1 - {\left(\frac{\sin \varepsilon}{\cos \varepsilon}\right)}^{3} \cdot \frac{{\left(\sin x\right)}^{3}}{{\left(\cos x\right)}^{3}}}\right) + \left(\frac{\frac{\sin \varepsilon}{\cos \varepsilon}}{1 - {\left(\frac{\sin \varepsilon}{\cos \varepsilon}\right)}^{3} \cdot \frac{{\left(\sin x\right)}^{3}}{{\left(\cos x\right)}^{3}}}\right))_* + \left((\left(\frac{\sin \varepsilon}{\cos \varepsilon}\right) \cdot \left(\frac{\frac{\sin x}{\cos x}}{1 - {\left(\frac{\sin \varepsilon}{\cos \varepsilon}\right)}^{3} \cdot \frac{{\left(\sin x\right)}^{3}}{{\left(\cos x\right)}^{3}}} \cdot \frac{\sin x}{\cos x}\right) + \left(\frac{\left(\frac{\sin \varepsilon}{\cos \varepsilon} \cdot \frac{\sin \varepsilon}{\cos \varepsilon}\right) \cdot {\left(\sin x\right)}^{3}}{{\left(\cos x\right)}^{3} \cdot \left(1 - {\left(\frac{\sin \varepsilon}{\cos \varepsilon}\right)}^{3} \cdot \frac{{\left(\sin x\right)}^{3}}{{\left(\cos x\right)}^{3}}\right)}\right))_* + \color{blue}{\left(\left(\frac{\sin x}{\cos x \cdot \left(1 - \frac{{\left(\sin x\right)}^{3} \cdot {\left(\sin \varepsilon\right)}^{3}}{{\left(\cos \varepsilon\right)}^{3} \cdot {\left(\cos x\right)}^{3}}\right)} + \frac{{\left(\sin x\right)}^{2} \cdot {\left(\sin \varepsilon\right)}^{3}}{{\left(\cos \varepsilon\right)}^{3} \cdot \left({\left(\cos x\right)}^{2} \cdot \left(1 - \frac{{\left(\sin x\right)}^{3} \cdot {\left(\sin \varepsilon\right)}^{3}}{{\left(\cos \varepsilon\right)}^{3} \cdot {\left(\cos x\right)}^{3}}\right)\right)}\right) - \frac{\sin x}{\cos x}\right)}\right)\]
Applied simplify0.5
\[\leadsto \color{blue}{\left((\left(\frac{\frac{\sin \varepsilon}{\cos \varepsilon}}{\frac{\cos x}{\frac{\sin \varepsilon}{\cos \varepsilon}}}\right) \cdot \left(\frac{\sin x}{1 - \frac{{\left(\sin x\right)}^{3}}{{\left(\cos x\right)}^{3}} \cdot {\left(\frac{\sin \varepsilon}{\cos \varepsilon}\right)}^{3}}\right) + \left(\frac{\frac{\sin \varepsilon}{\cos \varepsilon}}{1 - \frac{{\left(\sin x\right)}^{3}}{{\left(\cos x\right)}^{3}} \cdot {\left(\frac{\sin \varepsilon}{\cos \varepsilon}\right)}^{3}}\right))_* + (\left(\frac{\sin \varepsilon}{\cos \varepsilon}\right) \cdot \left(\frac{\frac{\sin x}{\cos x} \cdot \frac{\sin x}{\cos x}}{1 - \frac{{\left(\sin x\right)}^{3}}{{\left(\cos x\right)}^{3}} \cdot {\left(\frac{\sin \varepsilon}{\cos \varepsilon}\right)}^{3}}\right) + \left(\frac{\left(\frac{\sin \varepsilon}{\cos \varepsilon} \cdot \frac{\sin \varepsilon}{\cos \varepsilon}\right) \cdot \frac{{\left(\sin x\right)}^{3}}{{\left(\cos x\right)}^{3}}}{1 - \frac{{\left(\sin x\right)}^{3}}{{\left(\cos x\right)}^{3}} \cdot {\left(\frac{\sin \varepsilon}{\cos \varepsilon}\right)}^{3}}\right))_*\right) + \left({\left(\frac{\sin \varepsilon}{\cos \varepsilon}\right)}^{3} \cdot \frac{\frac{\sin x}{\cos x} \cdot \frac{\sin x}{\cos x}}{1 - \frac{{\left(\sin x\right)}^{3}}{{\left(\cos x\right)}^{3}} \cdot {\left(\frac{\sin \varepsilon}{\cos \varepsilon}\right)}^{3}} - \left(\frac{\sin x}{\cos x} - \frac{\frac{\sin x}{\cos x}}{1 - \frac{{\left(\sin x\right)}^{3}}{{\left(\cos x\right)}^{3}} \cdot {\left(\frac{\sin \varepsilon}{\cos \varepsilon}\right)}^{3}}\right)\right)}\]
