Error

Target

Derivation

1. Initial program 36.9

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

2. Initial simplification36.9

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

4. Applied tan-sum22.1

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

6. Applied flip--22.1

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

7. Applied associate-/r/22.1

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

8. Applied fma-neg22.1

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

9. Simplified22.1

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

10. Taylor expanded around inf 22.3

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

11. Simplified0.6

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

13. Applied add-log-exp0.8

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

14. Final simplification0.8

    \[\leadsto (\left(\frac{\sin x}{\cos x}\right) \cdot \left(\frac{\frac{\sin \varepsilon}{\cos \varepsilon} \cdot \frac{\sin \varepsilon}{\cos \varepsilon}}{1 - \left(\frac{\sin x}{\cos \varepsilon} \cdot \frac{\sin x}{\cos \varepsilon}\right) \cdot \left(\frac{\sin \varepsilon}{\cos x} \cdot \frac{\sin \varepsilon}{\cos x}\right)}\right) + \left(\frac{(\left(\frac{\sin x}{\cos x}\right) \cdot \left(\frac{\sin x}{\cos x}\right) + 1)_* \cdot \frac{\sin \varepsilon}{\cos \varepsilon}}{1 - \left(\frac{\sin \varepsilon}{\cos x} \cdot \frac{\sin \varepsilon}{\cos x}\right) \cdot \log \left(e^{\frac{\sin x}{\cos \varepsilon} \cdot \frac{\sin x}{\cos \varepsilon}}\right)}\right))_* + \left(\frac{\frac{\sin x}{\cos x}}{1 - \left(\frac{\sin x}{\cos \varepsilon} \cdot \frac{\sin x}{\cos \varepsilon}\right) \cdot \left(\frac{\sin \varepsilon}{\cos x} \cdot \frac{\sin \varepsilon}{\cos x}\right)} - \frac{\sin x}{\cos x}\right)\]

Runtime

Time bar (total: 5.8m)Debug logProfile

herbie shell --seed 2018252 +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)))