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 tan-quot0.4

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

4. Applied associate-*r/0.4

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

6. Applied add-cbrt-cube0.4

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

7. Applied add-cbrt-cube0.5

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

8. Applied cbrt-unprod0.4

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

9. Applied simplify0.4

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

10. Taylor expanded around inf 0.5

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

11. Applied simplify0.4

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

Runtime

Time bar (total: 48.0s)Debug logProfile

herbie shell --seed 2018167 +o rules:numerics
(FPCore (x)
  :name "Trigonometry B"
  (/ (- 1 (* (tan x) (tan x))) (+ 1 (* (tan x) (tan x)))))