Average Error: 0.3 → 0.4
Time: 38.9s
Precision: 64
\[\frac{1 - \tan x \cdot \tan x}{1 + \tan x \cdot \tan x}\]
↓
\[\left(\frac{1}{1 + \tan x \cdot \tan x} - \frac{\tan x}{1 + \tan x \cdot \tan x}\right) \cdot \left(1 + \tan x\right)\]

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 *-un-lft-identity0.3

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

  4. Applied *-un-lft-identity0.3

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

  5. Applied difference-of-squares0.4

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

  6. Applied times-frac0.4

    \[\leadsto \color{blue}{\frac{1 + \tan x}{1} \cdot \frac{1 - \tan x}{1 + \tan x \cdot \tan x}}\]

  7. Simplified0.4

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

  9. Applied div-sub0.4

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

  10. Final simplification0.4

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

Reproduce

herbie shell --seed 2019100 
(FPCore (x)
  :name "Trigonometry B"
  (/ (- 1 (* (tan x) (tan x))) (+ 1 (* (tan x) (tan x)))))