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Target

Derivation

1. Split input into 2 regimes

2. if eps < -5.128890368545333e-20 or 3.618880331423569e-16 < eps

   1. Initial program 29.7

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

   2. Initial simplification29.7

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

   4. Applied tan-sum0.9

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

   6. Applied flip3--1.0

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

   7. Applied associate-/r/1.0

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

   8. Simplified1.0

      \[\leadsto \frac{\tan \varepsilon + \tan x}{{1}^{3} - {\left(\tan \varepsilon \cdot \tan x\right)}^{3}} \cdot \color{blue}{\left(\left(1 + \tan x \cdot \tan \varepsilon\right) + \left(\tan x \cdot \tan \varepsilon\right) \cdot \left(\tan x \cdot \tan \varepsilon\right)\right)} - \tan x\]
   9. Using strategy rm

   10. Applied flip--1.1

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

   if -5.128890368545333e-20 < eps < 3.618880331423569e-16

   1. Initial program 45.3

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

   2. Initial simplification45.3

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

   4. Applied tan-sum45.2

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

   5. Taylor expanded around 0 30.6

      \[\leadsto \color{blue}{\varepsilon + \left({x}^{3} \cdot {\varepsilon}^{2} + \left({x}^{2} \cdot \varepsilon + \left(\frac{1}{3} \cdot {\varepsilon}^{3} + \left(\frac{4}{3} \cdot \left({x}^{2} \cdot {\varepsilon}^{3}\right) + {x}^{4} \cdot {\varepsilon}^{3}\right)\right)\right)\right)}\]

   6. Simplified30.6

      \[\leadsto \color{blue}{\left(\left(\varepsilon + {x}^{3} \cdot \left(\varepsilon \cdot \varepsilon\right)\right) + \left(\left(x \cdot x\right) \cdot \varepsilon + \left(\frac{1}{3} \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right) + \left(\left(x \cdot \frac{4}{3}\right) \cdot x + {x}^{4}\right) \cdot {\varepsilon}^{3}}\]
3. Recombined 2 regimes into one program.

4. Final simplification14.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -5.128890368545333 \cdot 10^{-20} \lor \neg \left(\varepsilon \le 3.618880331423569 \cdot 10^{-16}\right):\\ \;\;\;\;\frac{\left(\left(\left(\tan \varepsilon \cdot \tan x + 1\right) + \left(\tan \varepsilon \cdot \tan x\right) \cdot \left(\tan \varepsilon \cdot \tan x\right)\right) \cdot \frac{\tan \varepsilon + \tan x}{{1}^{3} - {\left(\tan \varepsilon \cdot \tan x\right)}^{3}}\right) \cdot \left(\left(\left(\tan \varepsilon \cdot \tan x + 1\right) + \left(\tan \varepsilon \cdot \tan x\right) \cdot \left(\tan \varepsilon \cdot \tan x\right)\right) \cdot \frac{\tan \varepsilon + \tan x}{{1}^{3} - {\left(\tan \varepsilon \cdot \tan x\right)}^{3}}\right) - \tan x \cdot \tan x}{\left(\left(\tan \varepsilon \cdot \tan x + 1\right) + \left(\tan \varepsilon \cdot \tan x\right) \cdot \left(\tan \varepsilon \cdot \tan x\right)\right) \cdot \frac{\tan \varepsilon + \tan x}{{1}^{3} - {\left(\tan \varepsilon \cdot \tan x\right)}^{3}} + \tan x}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\varepsilon \cdot \left(x \cdot x\right) + \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\frac{1}{3} \cdot \varepsilon\right)\right) + \left(\varepsilon + \left(\varepsilon \cdot \varepsilon\right) \cdot {x}^{3}\right)\right) + \left(\left(x \cdot \frac{4}{3}\right) \cdot x + {x}^{4}\right) \cdot {\varepsilon}^{3}\\ \end{array}\]

Runtime

Time bar (total: 7.6m)Debug logProfile

herbie shell --seed 2018227 
(FPCore (x eps)
  :name "2tan (problem 3.3.2)"

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

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