Average Error: 0.3 → 0.4
Time: 28.3s
Precision: 64
Internal Precision: 384
\[\frac{1 - \tan x \cdot \tan x}{1 + \tan x \cdot \tan x}\]
↓
\[\frac{1}{1 + {\left({\left(\tan x\right)}^{3} \cdot {\left(\tan x\right)}^{3}\right)}^{\frac{1}{3}}} - \frac{\tan x \cdot \tan x}{1 + \tan x \cdot \tan x}\]

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. 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. Using strategy rm

  5. Applied add-cbrt-cube0.4

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

  6. Applied simplify0.4

    \[\leadsto \frac{1}{1 + \sqrt[3]{\color{blue}{{\left(\tan x\right)}^{3} \cdot {\left(\tan x\right)}^{3}}}} - \frac{\tan x \cdot \tan x}{1 + \tan x \cdot \tan x}\]
  7. Using strategy rm

  8. Applied pow1/30.4

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

Runtime

Time bar (total: 28.3s)Debug logProfile

herbie shell --seed '#(1064269945 2896236262 301053905 1701069080 1701464310 1614783279)' 
(FPCore (x)
  :name "Trigonometry B"
  (/ (- 1 (* (tan x) (tan x))) (+ 1 (* (tan x) (tan x)))))