\[\tan \left(x + \varepsilon\right) - \tan x\]
Test:
NMSE problem 3.3.2
Bits:
128 bits
Bits error versus x
Bits error versus eps
Time: 7.3 s
Input Error: 17.0
Output Error: 17.0
Log: ⚲
Profile: 🕒
\(\sin x \cdot \left(\frac{\cot x}{\cot \left(x + \varepsilon\right) \cdot \cos x} - \frac{1}{\cos x}\right)\)
  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-cotan to get
    \[\color{red}{\tan \left(x + \varepsilon\right)} - \frac{1}{\cot x} \leadsto \color{blue}{\frac{1}{\cot \left(x + \varepsilon\right)}} - \frac{1}{\cot x}\]
    16.9
  5. Applied frac-sub to get
    \[\color{red}{\frac{1}{\cot \left(x + \varepsilon\right)} - \frac{1}{\cot x}} \leadsto \color{blue}{\frac{1 \cdot \cot x - \cot \left(x + \varepsilon\right) \cdot 1}{\cot \left(x + \varepsilon\right) \cdot \cot x}}\]
    16.9
  6. Applied simplify to get
    \[\frac{\color{red}{1 \cdot \cot x - \cot \left(x + \varepsilon\right) \cdot 1}}{\cot \left(x + \varepsilon\right) \cdot \cot x} \leadsto \frac{\color{blue}{\cot x - \cot \left(\varepsilon + x\right)}}{\cot \left(x + \varepsilon\right) \cdot \cot x}\]
    16.9
  7. Using strategy rm
    16.9
  8. Applied div-sub to get
    \[\color{red}{\frac{\cot x - \cot \left(\varepsilon + x\right)}{\cot \left(x + \varepsilon\right) \cdot \cot x}} \leadsto \color{blue}{\frac{\cot x}{\cot \left(x + \varepsilon\right) \cdot \cot x} - \frac{\cot \left(\varepsilon + x\right)}{\cot \left(x + \varepsilon\right) \cdot \cot x}}\]
    16.9
  9. Applied simplify to get
    \[\frac{\cot x}{\cot \left(x + \varepsilon\right) \cdot \cot x} - \color{red}{\frac{\cot \left(\varepsilon + x\right)}{\cot \left(x + \varepsilon\right) \cdot \cot x}} \leadsto \frac{\cot x}{\cot \left(x + \varepsilon\right) \cdot \cot x} - \color{blue}{\frac{1}{\cot x}}\]
    17.0
  10. Using strategy rm
    17.0
  11. Applied cotan-quot to get
    \[\frac{\cot x}{\cot \left(x + \varepsilon\right) \cdot \cot x} - \frac{1}{\color{red}{\cot x}} \leadsto \frac{\cot x}{\cot \left(x + \varepsilon\right) \cdot \cot x} - \frac{1}{\color{blue}{\frac{\cos x}{\sin x}}}\]
    17.0
  12. Applied associate-/r/ to get
    \[\frac{\cot x}{\cot \left(x + \varepsilon\right) \cdot \cot x} - \color{red}{\frac{1}{\frac{\cos x}{\sin x}}} \leadsto \frac{\cot x}{\cot \left(x + \varepsilon\right) \cdot \cot x} - \color{blue}{\frac{1}{\cos x} \cdot \sin x}\]
    17.0
  13. Applied cotan-quot to get
    \[\frac{\cot x}{\cot \left(x + \varepsilon\right) \cdot \color{red}{\cot x}} - \frac{1}{\cos x} \cdot \sin x \leadsto \frac{\cot x}{\cot \left(x + \varepsilon\right) \cdot \color{blue}{\frac{\cos x}{\sin x}}} - \frac{1}{\cos x} \cdot \sin x\]
    17.0
  14. Applied associate-*r/ to get
    \[\frac{\cot x}{\color{red}{\cot \left(x + \varepsilon\right) \cdot \frac{\cos x}{\sin x}}} - \frac{1}{\cos x} \cdot \sin x \leadsto \frac{\cot x}{\color{blue}{\frac{\cot \left(x + \varepsilon\right) \cdot \cos x}{\sin x}}} - \frac{1}{\cos x} \cdot \sin x\]
    17.0
  15. Applied associate-/r/ to get
    \[\color{red}{\frac{\cot x}{\frac{\cot \left(x + \varepsilon\right) \cdot \cos x}{\sin x}}} - \frac{1}{\cos x} \cdot \sin x \leadsto \color{blue}{\frac{\cot x}{\cot \left(x + \varepsilon\right) \cdot \cos x} \cdot \sin x} - \frac{1}{\cos x} \cdot \sin x\]
    17.0
  16. Applied distribute-rgt-out-- to get
    \[\color{red}{\frac{\cot x}{\cot \left(x + \varepsilon\right) \cdot \cos x} \cdot \sin x - \frac{1}{\cos x} \cdot \sin x} \leadsto \color{blue}{\sin x \cdot \left(\frac{\cot x}{\cot \left(x + \varepsilon\right) \cdot \cos x} - \frac{1}{\cos x}\right)}\]
    17.0

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)))))