Error

Target

Derivation

1. Split input into 2 regimes

2. if eps < -1.1247731385407306e-34 or 4.605063170740122e-29 < eps

   1. Initial program 30.2

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

   2. Initial simplification30.2

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

   4. Applied tan-sum2.3

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

   6. Applied tan-quot2.3

      \[\leadsto \frac{\tan \varepsilon + \tan x}{1 - \tan \varepsilon \cdot \tan x} - \color{blue}{\frac{\sin x}{\cos x}}\]

   7. Applied frac-sub2.4

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

   8. Simplified2.3

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

   if -1.1247731385407306e-34 < eps < 4.605063170740122e-29

   1. Initial program 46.1

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

   2. Initial simplification46.1

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

   4. Applied tan-sum46.1

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

   5. Taylor expanded around 0 27.2

      \[\leadsto \color{blue}{x \cdot {\varepsilon}^{2} + \left(\frac{1}{3} \cdot {\varepsilon}^{3} + \varepsilon\right)}\]

   6. Simplified27.2

      \[\leadsto \color{blue}{(\left((\varepsilon \cdot \frac{1}{3} + x)_*\right) \cdot \left(\varepsilon \cdot \varepsilon\right) + \varepsilon)_*}\]
3. Recombined 2 regimes into one program.

4. Final simplification13.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -1.1247731385407306 \cdot 10^{-34} \lor \neg \left(\varepsilon \le 4.605063170740122 \cdot 10^{-29}\right):\\ \;\;\;\;\frac{(\left(\cos x\right) \cdot \left(\tan \varepsilon + \tan x\right) + \left(\sin x \cdot (\left(\tan \varepsilon\right) \cdot \left(\tan x\right) + -1)_*\right))_*}{\cos x \cdot \left(1 - \tan \varepsilon \cdot \tan x\right)}\\ \mathbf{else}:\\ \;\;\;\;(\left((\varepsilon \cdot \frac{1}{3} + x)_*\right) \cdot \left(\varepsilon \cdot \varepsilon\right) + \varepsilon)_*\\ \end{array}\]

Runtime

Time bar (total: 1.2m)Debug logProfile

herbie shell --seed 2018355 +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)))