\[\tan \left(x + \varepsilon\right) - \tan x\]
Test:
NMSE problem 3.3.2
Bits:
128 bits
Bits error versus x
Bits error versus eps
Time: 6.2 s
Input Error: 17.0
Output Error: 15.2
Log: ⚲
Profile: 🕒
\(\frac{\cot x \cdot \left(\sin x \cdot \cos \varepsilon\right) + \left(\cot x \cdot \left(\cos x \cdot \sin \varepsilon\right) - \cos \left(x + \varepsilon\right)\right)}{\cos \left(x + \varepsilon\right) \cdot \cot x}\)
  1. Started with
    \[\tan \left(x + \varepsilon\right) - \tan x\]
    17.0
  2. Using strategy rm
    17.0
  3. Applied tan-cotan to get
    \[\tan \left(x + \varepsilon\right) - \color{red}{\tan x} \leadsto \tan \left(x + \varepsilon\right) - \color{blue}{\frac{1}{\cot x}}\]
    17.1
  4. Applied tan-quot to get
    \[\color{red}{\tan \left(x + \varepsilon\right)} - \frac{1}{\cot x} \leadsto \color{blue}{\frac{\sin \left(x + \varepsilon\right)}{\cos \left(x + \varepsilon\right)}} - \frac{1}{\cot x}\]
    17.2
  5. Applied frac-sub to get
    \[\color{red}{\frac{\sin \left(x + \varepsilon\right)}{\cos \left(x + \varepsilon\right)} - \frac{1}{\cot x}} \leadsto \color{blue}{\frac{\sin \left(x + \varepsilon\right) \cdot \cot x - \cos \left(x + \varepsilon\right) \cdot 1}{\cos \left(x + \varepsilon\right) \cdot \cot x}}\]
    17.2
  6. Applied simplify to get
    \[\frac{\color{red}{\sin \left(x + \varepsilon\right) \cdot \cot x - \cos \left(x + \varepsilon\right) \cdot 1}}{\cos \left(x + \varepsilon\right) \cdot \cot x} \leadsto \frac{\color{blue}{\cot x \cdot \sin \left(x + \varepsilon\right) - \cos \left(x + \varepsilon\right)}}{\cos \left(x + \varepsilon\right) \cdot \cot x}\]
    17.2
  7. Using strategy rm
    17.2
  8. Applied sin-sum to get
    \[\frac{\cot x \cdot \color{red}{\sin \left(x + \varepsilon\right)} - \cos \left(x + \varepsilon\right)}{\cos \left(x + \varepsilon\right) \cdot \cot x} \leadsto \frac{\cot x \cdot \color{blue}{\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right)} - \cos \left(x + \varepsilon\right)}{\cos \left(x + \varepsilon\right) \cdot \cot x}\]
    15.3
  9. Applied distribute-lft-in to get
    \[\frac{\color{red}{\cot x \cdot \left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right)} - \cos \left(x + \varepsilon\right)}{\cos \left(x + \varepsilon\right) \cdot \cot x} \leadsto \frac{\color{blue}{\left(\cot x \cdot \left(\sin x \cdot \cos \varepsilon\right) + \cot x \cdot \left(\cos x \cdot \sin \varepsilon\right)\right)} - \cos \left(x + \varepsilon\right)}{\cos \left(x + \varepsilon\right) \cdot \cot x}\]
    15.2
  10. Applied associate--l+ to get
    \[\frac{\color{red}{\left(\cot x \cdot \left(\sin x \cdot \cos \varepsilon\right) + \cot x \cdot \left(\cos x \cdot \sin \varepsilon\right)\right) - \cos \left(x + \varepsilon\right)}}{\cos \left(x + \varepsilon\right) \cdot \cot x} \leadsto \frac{\color{blue}{\cot x \cdot \left(\sin x \cdot \cos \varepsilon\right) + \left(\cot x \cdot \left(\cos x \cdot \sin \varepsilon\right) - \cos \left(x + \varepsilon\right)\right)}}{\cos \left(x + \varepsilon\right) \cdot \cot x}\]
    15.2

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)))))