Error

Target

Derivation

1. Split input into 3 regimes

2. if eps < -3.0289062418692434e-15

   1. Initial program 30.4

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

   2. Initial simplification30.4

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

   4. Applied tan-sum0.7

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

   6. Applied div-inv0.7

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

   7. Applied fma-neg0.7

      \[\leadsto \color{blue}{(\left(\tan \varepsilon + \tan x\right) \cdot \left(\frac{1}{1 - \tan \varepsilon \cdot \tan x}\right) + \left(-\tan x\right))_*}\]
   8. Using strategy rm

   9. Applied tan-quot0.8

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

   10. Applied associate-*l/0.8

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

   if -3.0289062418692434e-15 < eps < 8.00925307935431e-68

   1. Initial program 45.8

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

   2. Initial simplification45.8

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

   3. Taylor expanded around 0 27.1

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

   4. Simplified27.1

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

   if 8.00925307935431e-68 < eps

   1. Initial program 29.9

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

   2. Initial simplification29.9

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

   4. Applied tan-sum5.1

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

   6. Applied tan-quot5.1

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

   7. Applied frac-sub5.2

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

   8. Simplified5.1

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

4. Final simplification13.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -3.0289062418692434 \cdot 10^{-15}:\\ \;\;\;\;(\left(\tan \varepsilon + \tan x\right) \cdot \left(\frac{1}{1 - \frac{\sin \varepsilon \cdot \tan x}{\cos \varepsilon}}\right) + \left(-\tan x\right))_*\\ \mathbf{elif}\;\varepsilon \le 8.00925307935431 \cdot 10^{-68}:\\ \;\;\;\;(\left((\varepsilon \cdot \frac{1}{3} + x)_*\right) \cdot \left(\varepsilon \cdot \varepsilon\right) + \varepsilon)_*\\ \mathbf{else}:\\ \;\;\;\;\frac{(\left(\cos x\right) \cdot \left(\tan \varepsilon + \tan x\right) + \left((\left(\tan \varepsilon\right) \cdot \left(\tan x\right) + -1)_* \cdot \sin x\right))_*}{\left(1 - \tan \varepsilon \cdot \tan x\right) \cdot \cos x}\\ \end{array}\]

Runtime

Time bar (total: 49.9s)Debug logProfile

herbie shell --seed 2018339 +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)))