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

⬇

\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -2.3749642250710625 \cdot 10^{-44}:\\ \;\;\;\;\frac{\tan \varepsilon + \tan x}{1 - {\left(\tan \varepsilon\right)}^3 \cdot {\left(\tan x\right)}^3} \cdot \left({1}^2 + \left({\left(\tan x \cdot \tan \varepsilon\right)}^2 + 1 \cdot \left(\tan x \cdot \tan \varepsilon\right)\right)\right) - \tan x\\ \mathbf{if}\;\varepsilon \le 1.8344586189103676 \cdot 10^{-35}:\\ \;\;\;\;\left({x}^2 \cdot {\varepsilon}^3 + {\varepsilon}^{4} \cdot {x}^3\right) + \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan \varepsilon + \tan x}{1 - {\left(\tan \varepsilon\right)}^3 \cdot {\left(\tan x\right)}^3} \cdot \left({1}^2 + \left({\left(\tan x \cdot \tan \varepsilon\right)}^2 + 1 \cdot \left(\tan x \cdot \tan \varepsilon\right)\right)\right) - \tan x\\ \end{array}\]

Original	37.0
Comparison	26.5
Herbie	9.7

Derivation

Split input into 2 regimes.
if eps < -2.3749642250710625e-44 or 1.8344586189103676e-35 < eps
1. Initial program 30.4
  \[\tan \left(x + \varepsilon\right) - \tan x\]
2. Using strategy rm
3. Applied tan-sum 3.3
  \[\leadsto \color{blue}{\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon}} - \tan x\]
4. Using strategy rm
5. Applied flip3-- 3.4
  \[\leadsto \frac{\tan x + \tan \varepsilon}{\color{blue}{\frac{{1}^{3} - {\left(\tan x \cdot \tan \varepsilon\right)}^{3}}{{1}^2 + \left({\left(\tan x \cdot \tan \varepsilon\right)}^2 + 1 \cdot \left(\tan x \cdot \tan \varepsilon\right)\right)}}} - \tan x\]
6. Applied associate-/r/ 3.4
  \[\leadsto \color{blue}{\frac{\tan x + \tan \varepsilon}{{1}^{3} - {\left(\tan x \cdot \tan \varepsilon\right)}^{3}} \cdot \left({1}^2 + \left({\left(\tan x \cdot \tan \varepsilon\right)}^2 + 1 \cdot \left(\tan x \cdot \tan \varepsilon\right)\right)\right)} - \tan x\]
7. Applied simplify 3.4
  \[\leadsto \color{blue}{\frac{\tan \varepsilon + \tan x}{1 - {\left(\tan \varepsilon \cdot \tan x\right)}^3}} \cdot \left({1}^2 + \left({\left(\tan x \cdot \tan \varepsilon\right)}^2 + 1 \cdot \left(\tan x \cdot \tan \varepsilon\right)\right)\right) - \tan x\]
8. Using strategy rm
9. Applied cube-prod 3.4
  \[\leadsto \frac{\tan \varepsilon + \tan x}{1 - \color{blue}{{\left(\tan \varepsilon\right)}^3 \cdot {\left(\tan x\right)}^3}} \cdot \left({1}^2 + \left({\left(\tan x \cdot \tan \varepsilon\right)}^2 + 1 \cdot \left(\tan x \cdot \tan \varepsilon\right)\right)\right) - \tan x\]
if -2.3749642250710625e-44 < eps < 1.8344586189103676e-35
1. Initial program 45.8
  \[\tan \left(x + \varepsilon\right) - \tan x\]
2. Applied taylor 18.1
  \[\leadsto \varepsilon + \left({\varepsilon}^{4} \cdot {x}^{3} + {\varepsilon}^{3} \cdot {x}^2\right)\]
3. Taylor expanded around 0 18.1
  
  \[\leadsto \color{blue}{\varepsilon + \left({\varepsilon}^{4} \cdot {x}^{3} + {\varepsilon}^{3} \cdot {x}^2\right)}\]
4. Applied simplify 18.1
  \[\leadsto \color{blue}{\left({x}^2 \cdot {\varepsilon}^3 + {\varepsilon}^{4} \cdot {x}^3\right) + \varepsilon}\]
Recombined 2 regimes into one program.
Removed slow pow expressions

Runtime

Please include this information when filing a bug report:

herbie shell --seed '#(1065033997 2389885643 4100569014 2620012693 26800780 3144211646)'
(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 < -2.3749642250710625e-44 or 1.8344586189103676e-35 < eps`

`if -2.3749642250710625e-44 < eps < 1.8344586189103676e-35`

Runtime