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Target

Derivation

1. Initial program 37.0

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

3. Applied tan-sum21.6

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

5. Applied div-inv21.7

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

6. Applied fma-neg21.6

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

8. Applied frac-2neg21.6

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

9. Applied simplify21.6

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

10. Taylor expanded around -inf 13.1

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

Runtime

Time bar (total: 1.3m)Debug logProfile

herbie shell --seed '#(1072936661 1621281212 3440817831 3219514234 460296804 1258167384)' +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)))