Average Error: 0.3 → 0.4
Time: 31.0s
Precision: 64
Internal Precision: 576
\[\frac{1 - \tan x \cdot \tan x}{1 + \tan x \cdot \tan x}\]
↓
\[\frac{\frac{1 - \left(\tan x \cdot \tan x\right) \cdot \left(\tan x \cdot \tan x\right)}{1 + \tan x \cdot \tan x}}{(\left(\tan x\right) \cdot \left(\tan x\right) + 1)_*}\]

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. Initial simplification0.3

    \[\leadsto \frac{1 - \tan x \cdot \tan x}{(\left(\tan x\right) \cdot \left(\tan x\right) + 1)_*}\]
  3. Using strategy rm

  4. Applied flip--0.4

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

  6. Applied *-commutative0.4

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

  7. Final simplification0.4

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

Runtime

Time bar (total: 31.0s)Debug logProfile

herbie shell --seed 2018250 +o rules:numerics
(FPCore (x)
  :name "Trigonometry B"
  (/ (- 1 (* (tan x) (tan x))) (+ 1 (* (tan x) (tan x)))))