Error

Target

Derivation

1. Split input into 3 regimes

2. if eps < -3.4079781252857947e-40

   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-quot30.2

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

   5. Applied tan-sum3.1

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

   6. Applied frac-sub3.1

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

   7. Simplified3.1

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

   if -3.4079781252857947e-40 < eps < 4.8537106354764405e-18

   1. Initial program 45.9

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

   2. Initial simplification45.9

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

   3. Taylor expanded around 0 28.4

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

   4. Simplified28.4

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

   if 4.8537106354764405e-18 < eps

   1. Initial program 30.3

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

   2. Initial simplification30.3

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

   4. Applied tan-sum1.0

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

   6. Applied add-sqr-sqrt31.5

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

   7. Applied flip--31.6

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

   8. Applied associate-/r/31.6

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

   9. Applied prod-diff31.5

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

   10. Simplified31.5

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

   11. Simplified1.1

       \[\leadsto \left(\frac{(\left(\tan x + \tan \varepsilon\right) \cdot \left(\tan \varepsilon \cdot \tan x\right) + \left(\tan x + \tan \varepsilon\right))_*}{1 - \left(\tan \varepsilon \cdot \tan x\right) \cdot \left(\tan \varepsilon \cdot \tan x\right)} - \tan x\right) + \color{blue}{0}\]
3. Recombined 3 regimes into one program.

4. Final simplification14.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -3.4079781252857947 \cdot 10^{-40}:\\ \;\;\;\;\frac{(\left(\cos x\right) \cdot \left(\tan \varepsilon + \tan x\right) + \left(\sin x \cdot (\left(\tan \varepsilon\right) \cdot \left(\tan x\right) + -1)_*\right))_*}{\cos x \cdot \left(1 - \tan \varepsilon \cdot \tan x\right)}\\ \mathbf{elif}\;\varepsilon \le 4.8537106354764405 \cdot 10^{-18}:\\ \;\;\;\;(\left((\varepsilon \cdot \frac{1}{3} + x)_*\right) \cdot \left(\varepsilon \cdot \varepsilon\right) + \varepsilon)_*\\ \mathbf{else}:\\ \;\;\;\;\frac{(\left(\tan \varepsilon + \tan x\right) \cdot \left(\tan \varepsilon \cdot \tan x\right) + \left(\tan \varepsilon + \tan x\right))_*}{1 - \left(\tan \varepsilon \cdot \tan x\right) \cdot \left(\tan \varepsilon \cdot \tan x\right)} - \tan x\\ \end{array}\]

Runtime

Time bar (total: 1.7m)Debug logProfile

herbie shell --seed 2018346 +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)))