\[\tan \left(x + \varepsilon\right) - \tan x\]
Test:
NMSE problem 3.3.2
Bits:
128 bits
Bits error versus x
Bits error versus eps
Time: 42.8 s
Input Error: 60.3
Output Error: 11.0
Log: ⚲
Profile: 🕒
\(\frac{(\left(\sin x\right) * \left((\frac{1}{3} * \left({\varepsilon}^3\right) + \varepsilon)_*\right) + \left((\frac{4}{3} * \left(\frac{{\varepsilon}^3}{\sin x} \cdot \left(\cos x \cdot \cos x\right)\right) + \left((\left({\left(\frac{\varepsilon}{\sin x}\right)}^3\right) * \left({\left(\cos x\right)}^{4}\right) + \left(\frac{\cos x \cdot \cos x}{\frac{\sin x}{\varepsilon}}\right))_*\right))_*\right))_* - (\left(\cos x\right) * \left(\varepsilon \cdot \varepsilon\right) + \left({\left(\frac{\varepsilon}{\sin x}\right)}^2 \cdot {\left(\cos x\right)}^3\right))_*}{\cos x \cdot \cot \left(x + \varepsilon\right)}\)
  1. Started with
    \[\tan \left(x + \varepsilon\right) - \tan x\]
    60.3
  2. Using strategy rm
    60.3
  3. Applied tan-quot to get
    \[\tan \left(x + \varepsilon\right) - \color{red}{\tan x} \leadsto \tan \left(x + \varepsilon\right) - \color{blue}{\frac{\sin x}{\cos x}}\]
    60.4
  4. Applied tan-cotan to get
    \[\color{red}{\tan \left(x + \varepsilon\right)} - \frac{\sin x}{\cos x} \leadsto \color{blue}{\frac{1}{\cot \left(x + \varepsilon\right)}} - \frac{\sin x}{\cos x}\]
    60.4
  5. Applied frac-sub to get
    \[\color{red}{\frac{1}{\cot \left(x + \varepsilon\right)} - \frac{\sin x}{\cos x}} \leadsto \color{blue}{\frac{1 \cdot \cos x - \cot \left(x + \varepsilon\right) \cdot \sin x}{\cot \left(x + \varepsilon\right) \cdot \cos x}}\]
    60.5
  6. Applied simplify to get
    \[\frac{\color{red}{1 \cdot \cos x - \cot \left(x + \varepsilon\right) \cdot \sin x}}{\cot \left(x + \varepsilon\right) \cdot \cos x} \leadsto \frac{\color{blue}{\cos x - \cot \left(\varepsilon + x\right) \cdot \sin x}}{\cot \left(x + \varepsilon\right) \cdot \cos x}\]
    60.5
  7. Using strategy rm
    60.5
  8. Applied add-cube-cbrt to get
    \[\frac{\color{red}{\cos x - \cot \left(\varepsilon + x\right) \cdot \sin x}}{\cot \left(x + \varepsilon\right) \cdot \cos x} \leadsto \frac{\color{blue}{{\left(\sqrt[3]{\cos x - \cot \left(\varepsilon + x\right) \cdot \sin x}\right)}^3}}{\cot \left(x + \varepsilon\right) \cdot \cos x}\]
    60.5
  9. Applied taylor to get
    \[\frac{{\left(\sqrt[3]{\cos x - \cot \left(\varepsilon + x\right) \cdot \sin x}\right)}^3}{\cot \left(x + \varepsilon\right) \cdot \cos x} \leadsto \frac{\left(\frac{1}{3} \cdot \left({\varepsilon}^{3} \cdot \sin x\right) + \left(\varepsilon \cdot \sin x + \left(\frac{\varepsilon \cdot {\left(\cos x\right)}^2}{\sin x} + \left(\frac{{\varepsilon}^{3} \cdot {\left(\cos x\right)}^{4}}{{\left(\sin x\right)}^{3}} + \frac{4}{3} \cdot \frac{{\varepsilon}^{3} \cdot {\left(\cos x\right)}^2}{\sin x}\right)\right)\right)\right) - \left(\frac{{\varepsilon}^2 \cdot {\left(\cos x\right)}^{3}}{{\left(\sin x\right)}^2} + {\varepsilon}^2 \cdot \cos x\right)}{\cot \left(x + \varepsilon\right) \cdot \cos x}\]
    23.4
  10. Taylor expanded around 0 to get
    \[\frac{\color{red}{\left(\frac{1}{3} \cdot \left({\varepsilon}^{3} \cdot \sin x\right) + \left(\varepsilon \cdot \sin x + \left(\frac{\varepsilon \cdot {\left(\cos x\right)}^2}{\sin x} + \left(\frac{{\varepsilon}^{3} \cdot {\left(\cos x\right)}^{4}}{{\left(\sin x\right)}^{3}} + \frac{4}{3} \cdot \frac{{\varepsilon}^{3} \cdot {\left(\cos x\right)}^2}{\sin x}\right)\right)\right)\right) - \left(\frac{{\varepsilon}^2 \cdot {\left(\cos x\right)}^{3}}{{\left(\sin x\right)}^2} + {\varepsilon}^2 \cdot \cos x\right)}}{\cot \left(x + \varepsilon\right) \cdot \cos x} \leadsto \frac{\color{blue}{\left(\frac{1}{3} \cdot \left({\varepsilon}^{3} \cdot \sin x\right) + \left(\varepsilon \cdot \sin x + \left(\frac{\varepsilon \cdot {\left(\cos x\right)}^2}{\sin x} + \left(\frac{{\varepsilon}^{3} \cdot {\left(\cos x\right)}^{4}}{{\left(\sin x\right)}^{3}} + \frac{4}{3} \cdot \frac{{\varepsilon}^{3} \cdot {\left(\cos x\right)}^2}{\sin x}\right)\right)\right)\right) - \left(\frac{{\varepsilon}^2 \cdot {\left(\cos x\right)}^{3}}{{\left(\sin x\right)}^2} + {\varepsilon}^2 \cdot \cos x\right)}}{\cot \left(x + \varepsilon\right) \cdot \cos x}\]
    23.4
  11. Applied simplify to get
    \[\color{red}{\frac{\left(\frac{1}{3} \cdot \left({\varepsilon}^{3} \cdot \sin x\right) + \left(\varepsilon \cdot \sin x + \left(\frac{\varepsilon \cdot {\left(\cos x\right)}^2}{\sin x} + \left(\frac{{\varepsilon}^{3} \cdot {\left(\cos x\right)}^{4}}{{\left(\sin x\right)}^{3}} + \frac{4}{3} \cdot \frac{{\varepsilon}^{3} \cdot {\left(\cos x\right)}^2}{\sin x}\right)\right)\right)\right) - \left(\frac{{\varepsilon}^2 \cdot {\left(\cos x\right)}^{3}}{{\left(\sin x\right)}^2} + {\varepsilon}^2 \cdot \cos x\right)}{\cot \left(x + \varepsilon\right) \cdot \cos x}} \leadsto \color{blue}{\frac{(\left(\sin x\right) * \left((\frac{1}{3} * \left({\varepsilon}^3\right) + \varepsilon)_*\right) + \left((\frac{4}{3} * \left(\frac{{\varepsilon}^3}{\sin x} \cdot \left(\cos x \cdot \cos x\right)\right) + \left((\left(\frac{{\varepsilon}^3}{{\left(\sin x\right)}^3}\right) * \left({\left(\cos x\right)}^{4}\right) + \left(\left(\cos x \cdot \cos x\right) \cdot \frac{\varepsilon}{\sin x}\right))_*\right))_*\right))_* - (\left(\cos x\right) * \left(\varepsilon \cdot \varepsilon\right) + \left(\left(\frac{\varepsilon}{\sin x} \cdot \frac{\varepsilon}{\sin x}\right) \cdot {\left(\cos x\right)}^3\right))_*}{\cos x \cdot \cot \left(x + \varepsilon\right)}}\]
    23.4
  12. Applied simplify to get
    \[\frac{\color{red}{(\left(\sin x\right) * \left((\frac{1}{3} * \left({\varepsilon}^3\right) + \varepsilon)_*\right) + \left((\frac{4}{3} * \left(\frac{{\varepsilon}^3}{\sin x} \cdot \left(\cos x \cdot \cos x\right)\right) + \left((\left(\frac{{\varepsilon}^3}{{\left(\sin x\right)}^3}\right) * \left({\left(\cos x\right)}^{4}\right) + \left(\left(\cos x \cdot \cos x\right) \cdot \frac{\varepsilon}{\sin x}\right))_*\right))_*\right))_* - (\left(\cos x\right) * \left(\varepsilon \cdot \varepsilon\right) + \left(\left(\frac{\varepsilon}{\sin x} \cdot \frac{\varepsilon}{\sin x}\right) \cdot {\left(\cos x\right)}^3\right))_*}}{\cos x \cdot \cot \left(x + \varepsilon\right)} \leadsto \frac{\color{blue}{(\left(\sin x\right) * \left((\frac{1}{3} * \left({\varepsilon}^3\right) + \varepsilon)_*\right) + \left((\frac{4}{3} * \left(\frac{{\varepsilon}^3}{\sin x} \cdot \left(\cos x \cdot \cos x\right)\right) + \left((\left({\left(\frac{\varepsilon}{\sin x}\right)}^3\right) * \left({\left(\cos x\right)}^{4}\right) + \left(\frac{\cos x \cdot \cos x}{\frac{\sin x}{\varepsilon}}\right))_*\right))_*\right))_* - (\left(\cos x\right) * \left(\varepsilon \cdot \varepsilon\right) + \left({\left(\frac{\varepsilon}{\sin x}\right)}^2 \cdot {\left(\cos x\right)}^3\right))_*}}{\cos x \cdot \cot \left(x + \varepsilon\right)}\]
    11.0

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)))))