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

⬇

\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -2.1454937753948827 \cdot 10^{-23}:\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - \sqrt[3]{{\left(\tan x\right)}^{3} \cdot {\left(\tan \varepsilon\right)}^{3}}} - \tan x\\ \mathbf{if}\;\varepsilon \le 6.683176783612011 \cdot 10^{-65}:\\ \;\;\;\;\varepsilon + \left({\varepsilon}^{3} \cdot {x}^{2} + {\varepsilon}^{2} \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - \sqrt[3]{{\left(\tan x\right)}^{3} \cdot {\left(\tan \varepsilon\right)}^{3}}} - \tan x\\ \end{array}\]

Original	36.7
Comparison	27.0
Herbie	8.7

Derivation

Split input into 2 regimes.
if eps < -2.1454937753948827e-23 or 6.683176783612011e-65 < eps
1. Initial program 30.6
  \[\tan \left(x + \varepsilon\right) - \tan x\]
2. Using strategy rm
3. Applied tan-sum 3.7
  \[\leadsto \color{blue}{\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon}} - \tan x\]
4. Using strategy rm
5. Applied add-cbrt-cube 3.8
  \[\leadsto \frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \color{blue}{\sqrt[3]{\left(\tan \varepsilon \cdot \tan \varepsilon\right) \cdot \tan \varepsilon}}} - \tan x\]
6. Applied add-cbrt-cube 3.8
  \[\leadsto \frac{\tan x + \tan \varepsilon}{1 - \color{blue}{\sqrt[3]{\left(\tan x \cdot \tan x\right) \cdot \tan x}} \cdot \sqrt[3]{\left(\tan \varepsilon \cdot \tan \varepsilon\right) \cdot \tan \varepsilon}} - \tan x\]
7. Applied cbrt-unprod 3.8
  \[\leadsto \frac{\tan x + \tan \varepsilon}{1 - \color{blue}{\sqrt[3]{\left(\left(\tan x \cdot \tan x\right) \cdot \tan x\right) \cdot \left(\left(\tan \varepsilon \cdot \tan \varepsilon\right) \cdot \tan \varepsilon\right)}}} - \tan x\]
8. Applied simplify 3.8
  \[\leadsto \frac{\tan x + \tan \varepsilon}{1 - \sqrt[3]{\color{blue}{{\left(\tan x\right)}^{3} \cdot {\left(\tan \varepsilon\right)}^{3}}}} - \tan x\]
if -2.1454937753948827e-23 < eps < 6.683176783612011e-65
1. Initial program 45.1
  \[\tan \left(x + \varepsilon\right) - \tan x\]
2. Applied taylor 15.3
  \[\leadsto \varepsilon + \left({\varepsilon}^{3} \cdot {x}^{2} + {\varepsilon}^{2} \cdot x\right)\]
3. Taylor expanded around 0 15.3
  
  \[\leadsto \color{blue}{\varepsilon + \left({\varepsilon}^{3} \cdot {x}^{2} + {\varepsilon}^{2} \cdot x\right)}\]
Recombined 2 regimes into one program.
Removed slow pow expressions

Runtime

Please include this information when filing a bug report:

herbie shell --seed '#(1068028399 4028058041 2917032441 2563479541 765645300 1132738916)'
(FPCore (x eps)
  :name "2tan (problem 3.3.2)"
  :herbie-expected 28

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

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

Error

Target

Derivation

`if eps < -2.1454937753948827e-23 or 6.683176783612011e-65 < eps`

`if -2.1454937753948827e-23 < eps < 6.683176783612011e-65`

Runtime