Error

Target

Derivation

1. Initial program 36.6

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

3. Applied tan-quot36.6

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

4. Applied tan-sum21.6

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

5. Applied frac-sub21.7

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

6. Taylor expanded around -inf 0.4

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

7. Applied simplify0.4

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

9. Applied quot-tan0.4

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

Runtime

Time bar (total: 1.2m)Debug logProfile

herbie shell --seed '#(1070131407 1246090267 3027482374 2150728003 2026520792 2347815650)' +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)))