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Derivation

1. Initial program 0.3

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

2. Initial simplification0.3

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

4. Applied tan-quot0.4

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

5. Applied associate-*r/0.4

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

7. Applied flip-+0.4

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

8. Applied associate-/r/0.4

   \[\leadsto \color{blue}{\frac{1 - \frac{\tan x \cdot \sin x}{\cos x}}{\left(\tan x \cdot \tan x\right) \cdot \left(\tan x \cdot \tan x\right) - 1 \cdot 1} \cdot \left(\tan x \cdot \tan x - 1\right)}\]
9. Using strategy rm

10. Applied tan-quot0.4

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

11. Applied associate-*r/0.4

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

12. Final simplification0.4

    \[\leadsto \left(\tan x \cdot \tan x - 1\right) \cdot \frac{1 - \frac{\tan x \cdot \sin x}{\cos x}}{\frac{\tan x \cdot \sin x}{\cos x} \cdot \left(\tan x \cdot \tan x\right) - 1}\]

Runtime

Time bar (total: 27.3s)Debug logProfile

herbie shell --seed 2018263 
(FPCore (x)
  :name "Trigonometry B"
  (/ (- 1 (* (tan x) (tan x))) (+ 1 (* (tan x) (tan x)))))