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

↓

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

Derivation

Initial program 0.3
\[\frac{1 - \tan x \cdot \tan x}{1 + \tan x \cdot \tan x}\]
Using strategy rm
Applied div-sub0.4
\[\leadsto \color{blue}{\frac{1}{1 + \tan x \cdot \tan x} - \frac{\tan x \cdot \tan x}{1 + \tan x \cdot \tan x}}\]
Simplified0.4
\[\leadsto \color{blue}{\frac{1}{(\left(\tan x\right) \cdot \left(\tan x\right) + 1)_*}} - \frac{\tan x \cdot \tan x}{1 + \tan x \cdot \tan x}\]
Final simplification0.4
\[\leadsto \frac{1}{(\left(\tan x\right) \cdot \left(\tan x\right) + 1)_*} - \frac{\tan x \cdot \tan x}{\tan x \cdot \tan x + 1}\]

Runtime

	Baseline	Herbie	Oracle	Span	%
Regimes	0.4	0.4	0.1	0.4	0%

herbie shell --seed 2018297 +o rules:numerics
(FPCore (x)
  :name "Trigonometry B"
  (/ (- 1 (* (tan x) (tan x))) (+ 1 (* (tan x) (tan x)))))