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

⬇

\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -1.1324884610140404 \cdot 10^{-37}:\\ \;\;\;\;\frac{\cos x - {\left(\sqrt[3]{\cot \left(x + \varepsilon\right)} \cdot \sqrt[3]{\sin x}\right)}^3}{\cot \left(x + \varepsilon\right) \cdot \cos x}\\ \mathbf{if}\;\varepsilon \le -8.343087553984244 \cdot 10^{-186}:\\ \;\;\;\;\frac{\left(\sin x \cdot \varepsilon + \left(\frac{{\varepsilon}^{3} \cdot {\left(\cos x\right)}^{4}}{{\left(\sin x\right)}^{3}} + \left(\frac{\varepsilon \cdot {\left(\cos x\right)}^2}{\sin x} + \left(\frac{1}{3} \cdot \left(\sin x \cdot {\varepsilon}^{3}\right) + \frac{4}{3} \cdot \frac{{\varepsilon}^{3} \cdot {\left(\cos x\right)}^2}{\sin x}\right)\right)\right)\right) - \left(\frac{{\varepsilon}^2 \cdot {\left(\cos x\right)}^{3}}{{\left(\sin x\right)}^2} + {\varepsilon}^2 \cdot \cos x\right)}{\cot \left(x + \varepsilon\right) \cdot \cos x}\\ \mathbf{if}\;\varepsilon \le -1.4036887096279737 \cdot 10^{-262}:\\ \;\;\;\;\left(\varepsilon + {\varepsilon}^{4} \cdot {x}^3\right) + \left(x \cdot x\right) \cdot {\varepsilon}^3\\ \mathbf{if}\;\varepsilon \le 4.0170806835946265 \cdot 10^{-42}:\\ \;\;\;\;\frac{\left(\sin x \cdot \varepsilon + \left(\frac{{\varepsilon}^{3} \cdot {\left(\cos x\right)}^{4}}{{\left(\sin x\right)}^{3}} + \left(\frac{\varepsilon \cdot {\left(\cos x\right)}^2}{\sin x} + \left(\frac{1}{3} \cdot \left(\sin x \cdot {\varepsilon}^{3}\right) + \frac{4}{3} \cdot \frac{{\varepsilon}^{3} \cdot {\left(\cos x\right)}^2}{\sin x}\right)\right)\right)\right) - \left(\frac{{\varepsilon}^2 \cdot {\left(\cos x\right)}^{3}}{{\left(\sin x\right)}^2} + {\varepsilon}^2 \cdot \cos x\right)}{\cot \left(x + \varepsilon\right) \cdot \cos x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos x - {\left(\sqrt[3]{\cot \left(x + \varepsilon\right)} \cdot \sqrt[3]{\sin x}\right)}^3}{\cot \left(x + \varepsilon\right) \cdot \cos x}\\ \end{array}\]

Original	42.0
Comparison	21.0
Herbie	24.5

Derivation

Split input into 3 regimes.
if eps < -1.1324884610140404e-37 or 4.0170806835946265e-42 < eps
1. Initial program 29.6
  \[\tan \left(x + \varepsilon\right) - \tan x\]
2. Using strategy rm
3. Applied tan-quot 29.5
  \[\leadsto \tan \left(x + \varepsilon\right) - \color{blue}{\frac{\sin x}{\cos x}}\]
4. Applied tan-cotan 29.5
  \[\leadsto \color{blue}{\frac{1}{\cot \left(x + \varepsilon\right)}} - \frac{\sin x}{\cos x}\]
5. Applied frac-sub 29.4
  \[\leadsto \color{blue}{\frac{1 \cdot \cos x - \cot \left(x + \varepsilon\right) \cdot \sin x}{\cot \left(x + \varepsilon\right) \cdot \cos x}}\]
6. Applied simplify 29.4
  \[\leadsto \frac{\color{blue}{\cos x - \cot \left(x + \varepsilon\right) \cdot \sin x}}{\cot \left(x + \varepsilon\right) \cdot \cos x}\]
7. Using strategy rm
8. Applied add-cube-cbrt 29.3
  \[\leadsto \frac{\cos x - \cot \left(x + \varepsilon\right) \cdot \color{blue}{{\left(\sqrt[3]{\sin x}\right)}^3}}{\cot \left(x + \varepsilon\right) \cdot \cos x}\]
9. Applied add-cube-cbrt 29.3
  \[\leadsto \frac{\cos x - \color{blue}{{\left(\sqrt[3]{\cot \left(x + \varepsilon\right)}\right)}^3} \cdot {\left(\sqrt[3]{\sin x}\right)}^3}{\cot \left(x + \varepsilon\right) \cdot \cos x}\]
10. Applied cube-unprod 29.3
  \[\leadsto \frac{\cos x - \color{blue}{{\left(\sqrt[3]{\cot \left(x + \varepsilon\right)} \cdot \sqrt[3]{\sin x}\right)}^3}}{\cot \left(x + \varepsilon\right) \cdot \cos x}\]
if -1.1324884610140404e-37 < eps < -8.343087553984244e-186 or -1.4036887096279737e-262 < eps < 4.0170806835946265e-42
1. Initial program 59.9
  \[\tan \left(x + \varepsilon\right) - \tan x\]
2. Using strategy rm
3. Applied tan-quot 60.2
  \[\leadsto \tan \left(x + \varepsilon\right) - \color{blue}{\frac{\sin x}{\cos x}}\]
4. Applied tan-cotan 60.3
  \[\leadsto \color{blue}{\frac{1}{\cot \left(x + \varepsilon\right)}} - \frac{\sin x}{\cos x}\]
5. Applied frac-sub 60.3
  \[\leadsto \color{blue}{\frac{1 \cdot \cos x - \cot \left(x + \varepsilon\right) \cdot \sin x}{\cot \left(x + \varepsilon\right) \cdot \cos x}}\]
6. Applied simplify 60.3
  \[\leadsto \frac{\color{blue}{\cos x - \cot \left(x + \varepsilon\right) \cdot \sin x}}{\cot \left(x + \varepsilon\right) \cdot \cos x}\]
7. Using strategy rm
8. Applied add-cbrt-cube 60.3
  \[\leadsto \frac{\color{blue}{\sqrt[3]{{\left(\cos x - \cot \left(x + \varepsilon\right) \cdot \sin x\right)}^3}}}{\cot \left(x + \varepsilon\right) \cdot \cos x}\]
9. Applied taylor 17.2
  \[\leadsto \frac{\left(\sin x \cdot \varepsilon + \left(\frac{{\varepsilon}^{3} \cdot {\left(\cos x\right)}^{4}}{{\left(\sin x\right)}^{3}} + \left(\frac{\varepsilon \cdot {\left(\cos x\right)}^2}{\sin x} + \left(\frac{1}{3} \cdot \left(\sin x \cdot {\varepsilon}^{3}\right) + \frac{4}{3} \cdot \frac{{\varepsilon}^{3} \cdot {\left(\cos x\right)}^2}{\sin x}\right)\right)\right)\right) - \left(\frac{{\varepsilon}^2 \cdot {\left(\cos x\right)}^{3}}{{\left(\sin x\right)}^2} + {\varepsilon}^2 \cdot \cos x\right)}{\cot \left(x + \varepsilon\right) \cdot \cos x}\]
10. Taylor expanded around 0 17.2
  
  \[\leadsto \frac{\color{blue}{\left(\sin x \cdot \varepsilon + \left(\frac{{\varepsilon}^{3} \cdot {\left(\cos x\right)}^{4}}{{\left(\sin x\right)}^{3}} + \left(\frac{\varepsilon \cdot {\left(\cos x\right)}^2}{\sin x} + \left(\frac{1}{3} \cdot \left(\sin x \cdot {\varepsilon}^{3}\right) + \frac{4}{3} \cdot \frac{{\varepsilon}^{3} \cdot {\left(\cos x\right)}^2}{\sin x}\right)\right)\right)\right) - \left(\frac{{\varepsilon}^2 \cdot {\left(\cos x\right)}^{3}}{{\left(\sin x\right)}^2} + {\varepsilon}^2 \cdot \cos x\right)}}{\cot \left(x + \varepsilon\right) \cdot \cos x}\]
if -8.343087553984244e-186 < eps < -1.4036887096279737e-262
1. Initial program 51.0
  \[\tan \left(x + \varepsilon\right) - \tan x\]
2. Applied taylor 23.0
  \[\leadsto \varepsilon + \left({\varepsilon}^{4} \cdot {x}^{3} + {\varepsilon}^{3} \cdot {x}^2\right)\]
3. Taylor expanded around 0 23.0
  
  \[\leadsto \color{blue}{\varepsilon + \left({\varepsilon}^{4} \cdot {x}^{3} + {\varepsilon}^{3} \cdot {x}^2\right)}\]
4. Applied simplify 23.0
  \[\leadsto \color{blue}{\left(\varepsilon + {\varepsilon}^{4} \cdot {x}^3\right) + \left(x \cdot x\right) \cdot {\varepsilon}^3}\]
Recombined 3 regimes into one program.
Removed slow pow expressions

Error

Target

Derivation

`if eps < -1.1324884610140404e-37 or 4.0170806835946265e-42 < eps`

`if -1.1324884610140404e-37 < eps < -8.343087553984244e-186 or -1.4036887096279737e-262 < eps < 4.0170806835946265e-42`

`if -8.343087553984244e-186 < eps < -1.4036887096279737e-262`

Runtime