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

↓

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

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 log1p-expm1-u0.4
\[\leadsto \color{blue}{\log_* (1 + (e^{\frac{1 - \tan x \cdot \tan x}{(\left(\tan x\right) \cdot \left(\tan x\right) + 1)_*}} - 1)^*)}\]
Final simplification0.4
\[\leadsto \log_* (1 + (e^{\frac{1 - \tan x \cdot \tan x}{(\left(\tan x\right) \cdot \left(\tan x\right) + 1)_*}} - 1)^*)\]

Runtime

herbie shell --seed 2018215 +o rules:numerics
(FPCore (x)
  :name "Trigonometry B"
  (/ (- 1 (* (tan x) (tan x))) (+ 1 (* (tan x) (tan x)))))