Average Error: 0.3 → 0.4
Time: 26.5s
Precision: 64
Internal Precision: 576
\[\frac{1 - \tan x \cdot \tan x}{1 + \tan x \cdot \tan x}\]
↓
\[\frac{1}{\frac{(\left(\tan x\right) \cdot \left(\tan x\right) + 1)_*}{1 - \tan x \cdot \tan x}}\]

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 *-un-lft-identity0.3

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

  5. Applied associate-/l*0.4

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

  6. Final simplification0.4

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

Runtime

Time bar (total: 26.5s)Debug logProfile

herbie shell --seed 2018242 +o rules:numerics
(FPCore (x)
  :name "Trigonometry B"
  (/ (- 1 (* (tan x) (tan x))) (+ 1 (* (tan x) (tan x)))))