Average Error: 0.3 → 0.4
Time: 27.5s
Precision: 64
Internal Precision: 128
\[\frac{1 - \tan x \cdot \tan x}{1 + \tan x \cdot \tan x}\]
↓
\[\frac{1}{\tan x \cdot \tan x + 1} - \frac{\tan x \cdot \tan x}{\tan x \cdot \tan x + 1}\]

Error

Bits error versus x

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 div-sub0.4

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

  4. Final simplification0.4

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

Reproduce

herbie shell --seed 2019051 
(FPCore (x)
  :name "Trigonometry B"
  (/ (- 1 (* (tan x) (tan x))) (+ 1 (* (tan x) (tan x)))))