Average Error: 0.3 → 0.4
Time: 32.3s
Precision: 64
Internal Precision: 128
\[\frac{1 - \tan x \cdot \tan x}{1 + \tan x \cdot \tan x}\]
↓
\[\frac{1 - \left(\tan x \cdot \tan x\right) \cdot \left(\left(\tan x \cdot \tan x\right) \cdot \left(\tan x \cdot \tan x\right)\right)}{(\left(\tan x \cdot \tan x\right) \cdot \left(1 + (\left(\tan x \cdot \tan x\right) \cdot \left(\tan x \cdot \tan x\right) + \left(\tan x \cdot \tan x\right))_*\right) + \left(1 + (\left(\tan x \cdot \tan x\right) \cdot \left(\tan x \cdot \tan x\right) + \left(\tan x \cdot \tan x\right))_*\right))_*}\]

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 div-inv0.4

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

  5. Applied flip3--0.4

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

  6. Applied frac-times0.4

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

  7. Simplified0.4

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

  8. Simplified0.4

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

  9. Final simplification0.4

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

Reproduce

herbie shell --seed 2019089 +o rules:numerics
(FPCore (x)
  :name "Trigonometry B"
  (/ (- 1 (* (tan x) (tan x))) (+ 1 (* (tan x) (tan x)))))