Average Error: 0.3 → 0.5

Time: 45.6s

Precision: 64

Internal Precision: 576

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

↓

\[(e^{\log_* (1 + \frac{1 - \frac{\tan x \cdot \sin x}{\cos x}}{1 + \tan x \cdot \tan x})} - 1)^*\]

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Initial program 0.3
\[\frac{1 - \tan x \cdot \tan x}{1 + \tan x \cdot \tan x}\]
Using strategy rm
Applied tan-quot0.4
\[\leadsto \frac{1 - \tan x \cdot \color{blue}{\frac{\sin x}{\cos x}}}{1 + \tan x \cdot \tan x}\]
Applied associate-*r/0.4
\[\leadsto \frac{1 - \color{blue}{\frac{\tan x \cdot \sin x}{\cos x}}}{1 + \tan x \cdot \tan x}\]
Using strategy rm
Applied expm1-log1p-u0.5
\[\leadsto \color{blue}{(e^{\log_* (1 + \frac{1 - \frac{\tan x \cdot \sin x}{\cos x}}{1 + \tan x \cdot \tan x})} - 1)^*}\]
Final simplification0.5
\[\leadsto (e^{\log_* (1 + \frac{1 - \frac{\tan x \cdot \sin x}{\cos x}}{1 + \tan x \cdot \tan x})} - 1)^*\]

	Baseline	Herbie	Oracle	Span	%
Regimes	0.5	0.5	0.1	0.4	0%

herbie shell --seed 2018295 +o rules:numerics
(FPCore (x)
  :name "Trigonometry B"
  (/ (- 1 (* (tan x) (tan x))) (+ 1 (* (tan x) (tan x)))))