Error

Derivation

1. Initial program 0.3

   \[\frac{1 - \tan x \cdot \tan x}{1 + \tan x \cdot \tan x}\]

2. Initial simplification0.3

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

4. Applied div-sub0.4

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

6. Applied flip--0.4

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

8. Applied add-exp-log0.4

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

9. Applied add-exp-log0.4

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

10. Applied prod-exp0.4

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

11. Final simplification0.4

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

Runtime

Time bar (total: 52.3s)Debug logProfile

herbie shell --seed 2018346 +o rules:numerics
(FPCore (x)
  :name "Trigonometry B"
  (/ (- 1 (* (tan x) (tan x))) (+ 1 (* (tan x) (tan x)))))