Average Error: 0.3 → 0.4
Time: 1.2m
Precision: 64
Internal Precision: 576
\[\frac{1 - \tan x \cdot \tan x}{1 + \tan x \cdot \tan x}\]
↓
\[\frac{1}{(\left(\tan x\right) \cdot \left(\tan x\right) + 1)_*} - \frac{\tan x \cdot \tan x}{(\left(\tan x\right) \cdot \left(\tan x\right) + 1)_*}\]

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. Applied simplify0.4

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

  5. Applied simplify0.4

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

Runtime

Time bar (total: 1.2m)Debug logProfile

herbie shell --seed '#(1071246582 2318319007 2683472949 3810440501 3233274817 2724848749)' +o rules:numerics
(FPCore (x)
  :name "Trigonometry B"
  (/ (- 1 (* (tan x) (tan x))) (+ 1 (* (tan x) (tan x)))))