Average Error: 0.3 → 0.5
Time: 40.6s
Precision: 64
Internal Precision: 576
\[\frac{1 - \tan x \cdot \tan x}{1 + \tan x \cdot \tan x}\]
↓
\[(\left(\sqrt{\frac{1}{(\left(\tan x\right) \cdot \left(\tan x\right) + 1)_*}}\right) \cdot \left(\sqrt{\frac{1}{(\left(\tan x\right) \cdot \left(\tan x\right) + 1)_*}}\right) + \left(\frac{-\tan x \cdot \tan x}{1 + \tan x \cdot \tan x}\right))_*\]

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 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 add-sqr-sqrt0.5

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

  7. Applied fma-neg0.5

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

  8. Final simplification0.5

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

Runtime

Time bar (total: 40.6s)Debug logProfile

herbie shell --seed 2018257 +o rules:numerics
(FPCore (x)
  :name "Trigonometry B"
  (/ (- 1 (* (tan x) (tan x))) (+ 1 (* (tan x) (tan x)))))