Error

Target

Derivation

1. Initial program 36.7

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

3. Applied tan-sum21.5

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

5. Applied tan-quot21.5

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

6. Applied tan-quot21.5

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

7. Applied frac-times21.5

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

9. Applied flip3--21.5

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

10. Applied associate-/r/21.5

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

11. Applied fma-neg21.5

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

12. Taylor expanded around inf 21.7

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

13. Applied simplify0.4

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

Runtime

Time bar (total: 1.3m)Debug logProfile

herbie shell --seed '#(1064397287 3527694221 3797617954 1138343853 2854031332 1153838279)' +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)))