\[\tan \left(x + \varepsilon\right) - \tan x\]
Test:
NMSE problem 3.3.2
Bits:
128 bits
Bits error versus x
Bits error versus eps
Time: 9.0 s
Input Error: 16.6
Output Error: 16.6
Log: ⚲
Profile: 🕒
\(\frac{{1}^2 \cdot {\left(\cos x\right)}^2 - {\left(\cot \left(x + \varepsilon\right)\right)}^2 \cdot {\left(\sin x\right)}^2}{\left(\frac{1}{\cot \left(x + \varepsilon\right)} + \tan x\right) \cdot \left({\left(\cot \left(x + \varepsilon\right)\right)}^2 \cdot {\left(\cos x\right)}^2\right)}\)
  1. Started with
    \[\tan \left(x + \varepsilon\right) - \tan x\]
    16.6
  2. Using strategy rm
    16.6
  3. Applied tan-cotan to get
    \[\color{red}{\tan \left(x + \varepsilon\right)} - \tan x \leadsto \color{blue}{\frac{1}{\cot \left(x + \varepsilon\right)}} - \tan x\]
    16.6
  4. Using strategy rm
    16.6
  5. Applied flip-- to get
    \[\color{red}{\frac{1}{\cot \left(x + \varepsilon\right)} - \tan x} \leadsto \color{blue}{\frac{{\left(\frac{1}{\cot \left(x + \varepsilon\right)}\right)}^2 - {\left(\tan x\right)}^2}{\frac{1}{\cot \left(x + \varepsilon\right)} + \tan x}}\]
    16.6
  6. Using strategy rm
    16.6
  7. Applied tan-quot to get
    \[\frac{{\left(\frac{1}{\cot \left(x + \varepsilon\right)}\right)}^2 - {\color{red}{\left(\tan x\right)}}^2}{\frac{1}{\cot \left(x + \varepsilon\right)} + \tan x} \leadsto \frac{{\left(\frac{1}{\cot \left(x + \varepsilon\right)}\right)}^2 - {\color{blue}{\left(\frac{\sin x}{\cos x}\right)}}^2}{\frac{1}{\cot \left(x + \varepsilon\right)} + \tan x}\]
    16.6
  8. Applied square-div to get
    \[\frac{{\left(\frac{1}{\cot \left(x + \varepsilon\right)}\right)}^2 - \color{red}{{\left(\frac{\sin x}{\cos x}\right)}^2}}{\frac{1}{\cot \left(x + \varepsilon\right)} + \tan x} \leadsto \frac{{\left(\frac{1}{\cot \left(x + \varepsilon\right)}\right)}^2 - \color{blue}{\frac{{\left(\sin x\right)}^2}{{\left(\cos x\right)}^2}}}{\frac{1}{\cot \left(x + \varepsilon\right)} + \tan x}\]
    16.6
  9. Applied square-div to get
    \[\frac{\color{red}{{\left(\frac{1}{\cot \left(x + \varepsilon\right)}\right)}^2} - \frac{{\left(\sin x\right)}^2}{{\left(\cos x\right)}^2}}{\frac{1}{\cot \left(x + \varepsilon\right)} + \tan x} \leadsto \frac{\color{blue}{\frac{{1}^2}{{\left(\cot \left(x + \varepsilon\right)\right)}^2}} - \frac{{\left(\sin x\right)}^2}{{\left(\cos x\right)}^2}}{\frac{1}{\cot \left(x + \varepsilon\right)} + \tan x}\]
    16.6
  10. Applied frac-sub to get
    \[\frac{\color{red}{\frac{{1}^2}{{\left(\cot \left(x + \varepsilon\right)\right)}^2} - \frac{{\left(\sin x\right)}^2}{{\left(\cos x\right)}^2}}}{\frac{1}{\cot \left(x + \varepsilon\right)} + \tan x} \leadsto \frac{\color{blue}{\frac{{1}^2 \cdot {\left(\cos x\right)}^2 - {\left(\cot \left(x + \varepsilon\right)\right)}^2 \cdot {\left(\sin x\right)}^2}{{\left(\cot \left(x + \varepsilon\right)\right)}^2 \cdot {\left(\cos x\right)}^2}}}{\frac{1}{\cot \left(x + \varepsilon\right)} + \tan x}\]
    16.6
  11. Applied associate-/l/ to get
    \[\color{red}{\frac{\frac{{1}^2 \cdot {\left(\cos x\right)}^2 - {\left(\cot \left(x + \varepsilon\right)\right)}^2 \cdot {\left(\sin x\right)}^2}{{\left(\cot \left(x + \varepsilon\right)\right)}^2 \cdot {\left(\cos x\right)}^2}}{\frac{1}{\cot \left(x + \varepsilon\right)} + \tan x}} \leadsto \color{blue}{\frac{{1}^2 \cdot {\left(\cos x\right)}^2 - {\left(\cot \left(x + \varepsilon\right)\right)}^2 \cdot {\left(\sin x\right)}^2}{\left(\frac{1}{\cot \left(x + \varepsilon\right)} + \tan x\right) \cdot \left({\left(\cot \left(x + \varepsilon\right)\right)}^2 \cdot {\left(\cos x\right)}^2\right)}}\]
    16.6

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)))))