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

↓

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

Derivation

Initial program 0.3
\[\frac{1 - \tan x \cdot \tan x}{1 + \tan x \cdot \tan x}\]
Initial simplification0.3
\[\leadsto \frac{1 - \tan x \cdot \tan x}{(\left(\tan x\right) \cdot \left(\tan x\right) + 1)_*}\]
Using strategy rm
Applied expm1-log1p-u0.4
\[\leadsto \color{blue}{(e^{\log_* (1 + \frac{1 - \tan x \cdot \tan x}{(\left(\tan x\right) \cdot \left(\tan x\right) + 1)_*})} - 1)^*}\]
Final simplification0.4
\[\leadsto (e^{\log_* (1 + \frac{1 - \tan x \cdot \tan x}{(\left(\tan x\right) \cdot \left(\tan x\right) + 1)_*})} - 1)^*\]

Runtime

herbie shell --seed 2018214 +o rules:numerics
(FPCore (x)
  :name "Trigonometry B"
  (/ (- 1 (* (tan x) (tan x))) (+ 1 (* (tan x) (tan x)))))