Error

Target

Derivation

1. Split input into 3 regimes.

2. if eps < -5.668771810976152e-19

   1. Initial program 29.7

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

   3. Applied tan-sum 1.2

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

   5. Applied flip3-- 1.3

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

   6. Applied associate-/r/ 1.3

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

   7. Applied simplify 1.3

      \[\leadsto \color{blue}{\frac{\tan \varepsilon + \tan x}{1 - {\left(\tan \varepsilon \cdot \tan x\right)}^3}} \cdot \left({1}^2 + \left({\left(\tan x \cdot \tan \varepsilon\right)}^2 + 1 \cdot \left(\tan x \cdot \tan \varepsilon\right)\right)\right) - \tan x\]
   8. Using strategy rm

   9. Applied tan-quot 1.3

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

   10. Applied flip-+ 1.4

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

   11. Applied associate-*r/ 1.4

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

   12. Applied frac-sub 1.4

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

   13. Applied simplify 1.5

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

   if -5.668771810976152e-19 < eps < 4.867988538927033e-42

   1. Initial program 44.6

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

   2. Applied taylor 18.4

      \[\leadsto {\varepsilon}^{3} \cdot {x}^2 + \left(\varepsilon + {\varepsilon}^{4} \cdot {x}^{3}\right)\]

   3. Taylor expanded around 0 18.4

      \[\leadsto \color{blue}{{\varepsilon}^{3} \cdot {x}^2 + \left(\varepsilon + {\varepsilon}^{4} \cdot {x}^{3}\right)}\]

   4. Applied simplify 18.4

      \[\leadsto \color{blue}{\left({x}^3 \cdot {\varepsilon}^{4} + {\varepsilon}^3 \cdot \left(x \cdot x\right)\right) + \varepsilon}\]

   if 4.867988538927033e-42 < eps

   1. Initial program 29.6

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

   3. Applied tan-quot 29.5

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

   4. Applied tan-sum 2.9

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

   5. Applied frac-sub 2.9

      \[\leadsto \color{blue}{\frac{\left(\tan x + \tan \varepsilon\right) \cdot \cos x - \left(1 - \tan x \cdot \tan \varepsilon\right) \cdot \sin x}{\left(1 - \tan x \cdot \tan \varepsilon\right) \cdot \cos x}}\]
3. Recombined 3 regimes into one program.
4. Removed slow pow expressions

Runtime

Time bar (total: 1.3m) Debug log

Please include this information when filing a bug report:

herbie shell --seed '#(2277612311 2645429965 1090895633 2857793080 2144184008 3989768357)'
(FPCore (x eps)
  :name "2tan (problem 3.3.2)"
  :herbie-expected 28

  :target
  (/ (sin eps) (* (cos x) (cos (+ x eps))))

  (- (tan (+ x eps)) (tan x)))