Average Error: 0.3 → 0.5
Time: 38.1s
Precision: 64
Internal Precision: 384
\[\frac{1 - \tan x \cdot \tan x}{1 + \tan x \cdot \tan x}\]
↓
\[\frac{{1}^{3} - {\left(\tan x\right)}^{3} \cdot {\left(\tan x\right)}^{3}}{(\left((\left(\tan x\right) \cdot \left(\tan x\right) + 1)_* \cdot (\left(\tan x\right) \cdot \left(\tan x\right) + 1)_*\right) \cdot \left(\tan x \cdot \tan x\right) + \left((\left(\tan x\right) \cdot \left(\tan x\right) + 1)_*\right))_*}\]

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. Using strategy rm

  3. Applied flip3--0.4

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

  4. Applied associate-/l/0.4

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

  5. Applied simplify0.4

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

  7. Applied unpow-prod-down0.5

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

Runtime

Time bar (total: 38.1s)Debug logProfile

herbie shell --seed '#(1070355188 2193211668 3977393919 3454156579 3755371326 1656365382)' +o rules:numerics
(FPCore (x)
  :name "Trigonometry B"
  (/ (- 1 (* (tan x) (tan x))) (+ 1 (* (tan x) (tan x)))))