Error

Derivation

1. Initial program 0.3

   \[\frac{1 - \tan x \cdot \tan x}{1 + \tan x \cdot \tan x}\]
2. Using strategy rm

3. Applied flip3-+0.4

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

4. Applied associate-/r/0.4

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

5. Simplified0.4

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

7. Applied tan-quot0.4

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

8. Applied associate-*r/0.4

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

9. Final simplification0.4

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

Runtime

Time bar (total: 15.2s)Debug logProfile

herbie shell --seed 2018290 +o rules:numerics
(FPCore (x)
  :name "Trigonometry B"
  (/ (- 1 (* (tan x) (tan x))) (+ 1 (* (tan x) (tan x)))))