Error

Target

Derivation

1. Split input into 2 regimes

2. if eps < -6.985217187054404e-99 or 3.5691640258255704e-51 < eps

   1. Initial program 29.9

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

   3. Applied tan-sum5.6

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

   5. Applied tan-quot5.6

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

   6. Applied frac-sub5.6

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

   7. Simplified5.6

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

   if -6.985217187054404e-99 < eps < 3.5691640258255704e-51

   1. Initial program 47.5

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

   3. Applied tan-sum47.5

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

   5. Applied flip3--47.5

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

   6. Applied associate-/r/47.5

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

   7. Applied fma-neg47.5

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

   8. Taylor expanded around 0 30.5

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

   9. Simplified30.5

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

4. Final simplification15.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -6.985217187054404 \cdot 10^{-99} \lor \neg \left(\varepsilon \le 3.5691640258255704 \cdot 10^{-51}\right):\\ \;\;\;\;\frac{(\left(\cos x\right) \cdot \left(\tan x + \tan \varepsilon\right) + \left(\sin x \cdot (\left(\tan x\right) \cdot \left(\tan \varepsilon\right) + -1)_*\right))_*}{\cos x \cdot \left(1 - \tan x \cdot \tan \varepsilon\right)}\\ \mathbf{else}:\\ \;\;\;\;(\left(x \cdot \varepsilon\right) \cdot \left(x + \varepsilon\right) + \varepsilon)_*\\ \end{array}\]

Runtime

Time bar (total: 54.7s)Debug logProfile

herbie shell --seed 2018274 +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)))