Average Error: 0.3 → 0.3
Time: 46.6s
Precision: 64
Internal Precision: 576
\[\frac{1 - \tan x \cdot \tan x}{1 + \tan x \cdot \tan x}\]
↓
\[\frac{(\left(\tan x\right) \cdot \left(-\tan x\right) + 1)_*}{(\left(\tan x\right) \cdot \left(\tan x\right) + 1)_*}\]

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-sub0.4

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

  4. Simplified0.4

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

  6. Applied div-inv0.4

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

  7. Applied add-sqr-sqrt0.5

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

  8. Applied prod-diff0.4

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

  9. Simplified0.3

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

  10. Simplified0.3

    \[\leadsto \frac{(\left(\tan x\right) \cdot \left(-\tan x\right) + 1)_*}{(\left(\tan x\right) \cdot \left(\tan x\right) + 1)_*} + \color{blue}{0}\]

  11. Final simplification0.3

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

Runtime

Time bar (total: 46.6s)Debug logProfile

herbie shell --seed 2018248 +o rules:numerics
(FPCore (x)
  :name "Trigonometry B"
  (/ (- 1 (* (tan x) (tan x))) (+ 1 (* (tan x) (tan x)))))