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Target

Derivation

1. Split input into 3 regimes

2. if eps < -5.453678077661277e-14

   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-sum0.7

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

   6. Applied add-log-exp0.9

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

   7. Applied add-log-exp1.5

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

   8. Applied diff-log1.5

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

   9. Simplified1.5

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

   if -5.453678077661277e-14 < eps < 5.367924384521866e-22

   1. Initial program 45.6

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

   2. Initial simplification45.6

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

   4. Applied tan-sum45.6

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

   5. Taylor expanded around 0 28.7

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

   6. Simplified28.7

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

   if 5.367924384521866e-22 < eps

   1. Initial program 30.5

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

   2. Initial simplification30.5

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

   4. Applied tan-sum1.3

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

   5. Taylor expanded around -inf 1.3

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

4. Final simplification14.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -5.453678077661277 \cdot 10^{-14}:\\ \;\;\;\;\log \left(e^{\frac{\tan \varepsilon + \tan x}{1 - \tan \varepsilon \cdot \tan x} - \tan x}\right)\\ \mathbf{elif}\;\varepsilon \le 5.367924384521866 \cdot 10^{-22}:\\ \;\;\;\;\left(\varepsilon \cdot \varepsilon\right) \cdot \left(x + \varepsilon \cdot \frac{1}{3}\right) + \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan \varepsilon + \tan x}{1 - \frac{\sin \varepsilon \cdot \sin x}{\cos x \cdot \cos \varepsilon}} - \tan x\\ \end{array}\]

Runtime

Time bar (total: 1.1m)Debug logProfile

herbie shell --seed 2018234 
(FPCore (x eps)
  :name "2tan (problem 3.3.2)"

  :herbie-target
  (/ (sin eps) (* (cos x) (cos (+ x eps))))

  (- (tan (+ x eps)) (tan x)))