\[\tan \left(x + \varepsilon\right) - \tan x\]
Test:
NMSE problem 3.3.2
Bits:
128 bits
Bits error versus x
Bits error versus eps
Time: 5.2 s
Input Error: 16.8
Output Error: 16.8
Log: ⚲
Profile: 🕒
\(\frac{{\left(\frac{1}{\frac{1}{\tan \left(x + \varepsilon\right)}}\right)}^2 - {\left(\tan x\right)}^2}{\tan x + \tan \left(\varepsilon + x\right)}\)
  1. Started with
    \[\tan \left(x + \varepsilon\right) - \tan x\]
    16.8
  2. Using strategy rm
    16.8
  3. Applied tan-cotan to get
    \[\color{red}{\tan \left(x + \varepsilon\right)} - \tan x \leadsto \color{blue}{\frac{1}{\cot \left(x + \varepsilon\right)}} - \tan x\]
    16.9
  4. Using strategy rm
    16.9
  5. Applied cotan-tan to get
    \[\frac{1}{\color{red}{\cot \left(x + \varepsilon\right)}} - \tan x \leadsto \frac{1}{\color{blue}{\frac{1}{\tan \left(x + \varepsilon\right)}}} - \tan x\]
    16.8
  6. Using strategy rm
    16.8
  7. Applied flip-- to get
    \[\color{red}{\frac{1}{\frac{1}{\tan \left(x + \varepsilon\right)}} - \tan x} \leadsto \color{blue}{\frac{{\left(\frac{1}{\frac{1}{\tan \left(x + \varepsilon\right)}}\right)}^2 - {\left(\tan x\right)}^2}{\frac{1}{\frac{1}{\tan \left(x + \varepsilon\right)}} + \tan x}}\]
    16.8
  8. Applied simplify to get
    \[\frac{{\left(\frac{1}{\frac{1}{\tan \left(x + \varepsilon\right)}}\right)}^2 - {\left(\tan x\right)}^2}{\color{red}{\frac{1}{\frac{1}{\tan \left(x + \varepsilon\right)}} + \tan x}} \leadsto \frac{{\left(\frac{1}{\frac{1}{\tan \left(x + \varepsilon\right)}}\right)}^2 - {\left(\tan x\right)}^2}{\color{blue}{\tan x + \tan \left(\varepsilon + x\right)}}\]
    16.8

Original test:


(lambda ((x default) (eps default))
  #:name "NMSE problem 3.3.2"
  (- (tan (+ x eps)) (tan x))
  #:target
  (/ (sin eps) (* (cos x) (cos (+ x eps)))))