Error

Target

Derivation

1. Split input into 2 regimes

2. if eps < -2.8765151167705837e-22 or 3.07355218442861e-42 < eps

   1. Initial program 30.0

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

   3. Applied tan-sum2.6

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

   5. Applied tan-quot2.6

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

   6. Applied frac-sub2.6

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

   7. Simplified2.6

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

   if -2.8765151167705837e-22 < eps < 3.07355218442861e-42

   1. Initial program 45.8

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

   3. Applied tan-sum45.8

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

   5. Applied tan-quot45.8

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

   6. Applied tan-quot46.0

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

   7. Applied frac-add46.0

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

   8. Simplified46.0

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

   9. Taylor expanded around 0 27.2

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

   10. 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 simplification13.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -2.8765151167705837 \cdot 10^{-22} \lor \neg \left(\varepsilon \le 3.07355218442861 \cdot 10^{-42}\right):\\ \;\;\;\;\frac{(\left(\cos x\right) \cdot \left(\tan x + \tan \varepsilon\right) + \left(\sin x \cdot (\left(\tan x\right) \cdot \left(\tan \varepsilon\right) + -1)_*\right))_*}{\cos x \cdot \left(1 - \tan x \cdot \tan \varepsilon\right)}\\ \mathbf{else}:\\ \;\;\;\;(\left((\varepsilon \cdot \frac{1}{3} + x)_*\right) \cdot \left(\varepsilon \cdot \varepsilon\right) + \varepsilon)_*\\ \end{array}\]

Runtime

Time bar (total: 1.8m)Debug logProfile

herbie shell --seed 2018258 +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)))