Average Error: 0.3 → 0.5
Time: 1.1m
Precision: 64
Internal Precision: 576
\[\frac{1 - \tan x \cdot \tan x}{1 + \tan x \cdot \tan x}\]
↓
\[\frac{{1}^{3} - {\left(\tan x\right)}^{3} \cdot {\left(\tan x\right)}^{3}}{(\left(\tan x \cdot \tan x\right) \cdot \left((\left(\tan x\right) \cdot \left(\tan x\right) + 1)_* \cdot (\left(\tan x\right) \cdot \left(\tan x\right) + 1)_*\right) + \left((\left(\tan x\right) \cdot \left(\tan x\right) + 1)_*\right))_*}\]

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. Applied simplify0.3

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

  4. Applied flip3--0.4

    \[\leadsto \frac{\color{blue}{\frac{{1}^{3} - {\left(\tan x \cdot \tan x\right)}^{3}}{1 \cdot 1 + \left(\left(\tan x \cdot \tan x\right) \cdot \left(\tan x \cdot \tan x\right) + 1 \cdot \left(\tan x \cdot \tan x\right)\right)}}}{(\left(\tan x\right) \cdot \left(\tan x\right) + 1)_*}\]

  5. Applied associate-/l/0.4

    \[\leadsto \color{blue}{\frac{{1}^{3} - {\left(\tan x \cdot \tan x\right)}^{3}}{(\left(\tan x\right) \cdot \left(\tan x\right) + 1)_* \cdot \left(1 \cdot 1 + \left(\left(\tan x \cdot \tan x\right) \cdot \left(\tan x \cdot \tan x\right) + 1 \cdot \left(\tan x \cdot \tan x\right)\right)\right)}}\]

  6. Applied simplify0.4

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

  8. Applied unpow-prod-down0.5

    \[\leadsto \frac{{1}^{3} - \color{blue}{{\left(\tan x\right)}^{3} \cdot {\left(\tan x\right)}^{3}}}{(\left(\tan x \cdot \tan x\right) \cdot \left((\left(\tan x\right) \cdot \left(\tan x\right) + 1)_* \cdot (\left(\tan x\right) \cdot \left(\tan x\right) + 1)_*\right) + \left((\left(\tan x\right) \cdot \left(\tan x\right) + 1)_*\right))_*}\]

Runtime

Time bar (total: 1.1m)Debug logProfile

herbie shell --seed '#(1071373924 2949776965 1885069702 3247780810 90874544 2263903749)' +o rules:numerics
(FPCore (x)
  :name "Trigonometry B"
  (/ (- 1 (* (tan x) (tan x))) (+ 1 (* (tan x) (tan x)))))