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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 clear-num0.4

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

5. Applied add-cbrt-cube0.4

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

6. Applied add-cbrt-cube0.5

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

7. Applied cbrt-unprod0.4

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

8. Simplified0.4

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

10. Applied tan-quot0.5

    \[\leadsto \frac{1}{\frac{1 + \sqrt[3]{{\left(\tan x\right)}^{6}}}{1 - \tan x \cdot \color{blue}{\frac{\sin x}{\cos x}}}}\]

11. Applied associate-*r/0.5

    \[\leadsto \frac{1}{\frac{1 + \sqrt[3]{{\left(\tan x\right)}^{6}}}{1 - \color{blue}{\frac{\tan x \cdot \sin x}{\cos x}}}}\]

12. Final simplification0.5

    \[\leadsto \frac{1}{\frac{\sqrt[3]{{\left(\tan x\right)}^{6}} + 1}{1 - \frac{\sin x \cdot \tan x}{\cos x}}}\]

Runtime

Time bar (total: 31.0s)Debug logProfile

herbie shell --seed 2018304 
(FPCore (x)
  :name "Trigonometry B"
  (/ (- 1 (* (tan x) (tan x))) (+ 1 (* (tan x) (tan x)))))