\[\tan \left(x + \varepsilon\right) - \tan x\]
Test:
NMSE problem 3.3.2
Bits:
128 bits
Bits error versus x
Bits error versus eps
Time: 21.9 s
Input Error: 36.9
Output Error: 36.9
Log: ⚲
Profile: 🕒
\(\frac{{\left(\cos x\right)}^{3} - {\left(\cot \left(\varepsilon + x\right) \cdot \sin x\right)}^{3}}{\left(\cot \left(x + \varepsilon\right) \cdot \cos x\right) \cdot \left({\left(\cos x\right)}^2 + \left({\left(\cot \left(\varepsilon + x\right) \cdot \sin x\right)}^2 + \cos x \cdot \left(\cot \left(\varepsilon + x\right) \cdot \sin x\right)\right)\right)}\)
  1. Started with
    \[\tan \left(x + \varepsilon\right) - \tan x\]
    36.9
  2. Using strategy rm
    36.9
  3. Applied tan-quot to get
    \[\tan \left(x + \varepsilon\right) - \color{red}{\tan x} \leadsto \tan \left(x + \varepsilon\right) - \color{blue}{\frac{\sin x}{\cos x}}\]
    36.9
  4. Applied tan-cotan to get
    \[\color{red}{\tan \left(x + \varepsilon\right)} - \frac{\sin x}{\cos x} \leadsto \color{blue}{\frac{1}{\cot \left(x + \varepsilon\right)}} - \frac{\sin x}{\cos x}\]
    36.9
  5. Applied frac-sub to get
    \[\color{red}{\frac{1}{\cot \left(x + \varepsilon\right)} - \frac{\sin x}{\cos x}} \leadsto \color{blue}{\frac{1 \cdot \cos x - \cot \left(x + \varepsilon\right) \cdot \sin x}{\cot \left(x + \varepsilon\right) \cdot \cos x}}\]
    36.9
  6. Applied simplify to get
    \[\frac{\color{red}{1 \cdot \cos x - \cot \left(x + \varepsilon\right) \cdot \sin x}}{\cot \left(x + \varepsilon\right) \cdot \cos x} \leadsto \frac{\color{blue}{\cos x - \cot \left(\varepsilon + x\right) \cdot \sin x}}{\cot \left(x + \varepsilon\right) \cdot \cos x}\]
    36.9
  7. Using strategy rm
    36.9
  8. Applied flip3-- to get
    \[\frac{\color{red}{\cos x - \cot \left(\varepsilon + x\right) \cdot \sin x}}{\cot \left(x + \varepsilon\right) \cdot \cos x} \leadsto \frac{\color{blue}{\frac{{\left(\cos x\right)}^{3} - {\left(\cot \left(\varepsilon + x\right) \cdot \sin x\right)}^{3}}{{\left(\cos x\right)}^2 + \left({\left(\cot \left(\varepsilon + x\right) \cdot \sin x\right)}^2 + \cos x \cdot \left(\cot \left(\varepsilon + x\right) \cdot \sin x\right)\right)}}}{\cot \left(x + \varepsilon\right) \cdot \cos x}\]
    36.9
  9. Applied associate-/l/ to get
    \[\color{red}{\frac{\frac{{\left(\cos x\right)}^{3} - {\left(\cot \left(\varepsilon + x\right) \cdot \sin x\right)}^{3}}{{\left(\cos x\right)}^2 + \left({\left(\cot \left(\varepsilon + x\right) \cdot \sin x\right)}^2 + \cos x \cdot \left(\cot \left(\varepsilon + x\right) \cdot \sin x\right)\right)}}{\cot \left(x + \varepsilon\right) \cdot \cos x}} \leadsto \color{blue}{\frac{{\left(\cos x\right)}^{3} - {\left(\cot \left(\varepsilon + x\right) \cdot \sin x\right)}^{3}}{\left(\cot \left(x + \varepsilon\right) \cdot \cos x\right) \cdot \left({\left(\cos x\right)}^2 + \left({\left(\cot \left(\varepsilon + x\right) \cdot \sin x\right)}^2 + \cos x \cdot \left(\cot \left(\varepsilon + x\right) \cdot \sin x\right)\right)\right)}}\]
    36.9

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)))))