\[\tan \left(x + \varepsilon\right) - \tan x\]
Test:
NMSE problem 3.3.2
Bits:
128 bits
Bits error versus x
Bits error versus eps
Time: 27.3 s
Input Error: 17.5
Output Error: 3.9
Log: ⚲
Profile: 🕒
\(\begin{cases} \frac{\left(\left(\frac{{\varepsilon}^3 \cdot \frac{4}{3}}{{\left(\frac{\sin x}{\cos x}\right)}^2} + \frac{\frac{\varepsilon}{\frac{\sin x}{\cos x}}}{\frac{\sin x}{\cos x}}\right) + \left(\left(\varepsilon + {\varepsilon}^3 \cdot \frac{1}{3}\right) + \frac{{\left(\cos x\right)}^{4} \cdot {\varepsilon}^3}{{\left(\sin x\right)}^{4}}\right)\right) - \left(\frac{\cos x \cdot {\varepsilon}^2}{\sin x} + \frac{{\varepsilon}^2}{\sin x} \cdot \frac{{\left(\cos x\right)}^3}{\sin x \cdot \sin x}\right)}{\cot \left(\varepsilon + x\right) \cdot \cot x} & \text{when } x \le -1.4391974f-06 \\ \frac{\cot x - \frac{\cos \left(\frac{1}{\varepsilon} + \frac{1}{x}\right)}{\left(\frac{1}{\varepsilon} + \frac{1}{x}\right) - \frac{\frac{1}{6}}{{x}^3}}}{\cot x \cdot \cot \left(\varepsilon + x\right)} & \text{when } x \le 1.8040304f-05 \\ \frac{\left(\left(\frac{{\varepsilon}^3 \cdot \frac{4}{3}}{{\left(\frac{\sin x}{\cos x}\right)}^2} + \frac{\frac{\varepsilon}{\frac{\sin x}{\cos x}}}{\frac{\sin x}{\cos x}}\right) + \left(\left(\varepsilon + {\varepsilon}^3 \cdot \frac{1}{3}\right) + \frac{{\left(\cos x\right)}^{4} \cdot {\varepsilon}^3}{{\left(\sin x\right)}^{4}}\right)\right) - \left(\frac{\cos x \cdot {\varepsilon}^2}{\sin x} + \frac{{\varepsilon}^2}{\sin x} \cdot \frac{{\left(\cos x\right)}^3}{\sin x \cdot \sin x}\right)}{\cot \left(\varepsilon + x\right) \cdot \cot x} & \text{otherwise} \end{cases}\)

    if x < -1.4391974f-06 or 1.8040304f-05 < x

    1. Started with
      \[\tan \left(x + \varepsilon\right) - \tan x\]
      28.5
    2. Using strategy rm
      28.5
    3. Applied tan-cotan to get
      \[\tan \left(x + \varepsilon\right) - \color{red}{\tan x} \leadsto \tan \left(x + \varepsilon\right) - \color{blue}{\frac{1}{\cot x}}\]
      28.8
    4. Applied tan-cotan to get
      \[\color{red}{\tan \left(x + \varepsilon\right)} - \frac{1}{\cot x} \leadsto \color{blue}{\frac{1}{\cot \left(x + \varepsilon\right)}} - \frac{1}{\cot x}\]
      28.5
    5. Applied frac-sub to get
      \[\color{red}{\frac{1}{\cot \left(x + \varepsilon\right)} - \frac{1}{\cot x}} \leadsto \color{blue}{\frac{1 \cdot \cot x - \cot \left(x + \varepsilon\right) \cdot 1}{\cot \left(x + \varepsilon\right) \cdot \cot x}}\]
      28.5
    6. Applied simplify to get
      \[\frac{\color{red}{1 \cdot \cot x - \cot \left(x + \varepsilon\right) \cdot 1}}{\cot \left(x + \varepsilon\right) \cdot \cot x} \leadsto \frac{\color{blue}{\cot x - \cot \left(\varepsilon + x\right)}}{\cot \left(x + \varepsilon\right) \cdot \cot x}\]
      28.5
    7. Using strategy rm
      28.5
    8. Applied add-log-exp to get
      \[\frac{\cot x - \color{red}{\cot \left(\varepsilon + x\right)}}{\cot \left(x + \varepsilon\right) \cdot \cot x} \leadsto \frac{\cot x - \color{blue}{\log \left(e^{\cot \left(\varepsilon + x\right)}\right)}}{\cot \left(x + \varepsilon\right) \cdot \cot x}\]
      29.2
    9. Applied add-log-exp to get
      \[\frac{\color{red}{\cot x} - \log \left(e^{\cot \left(\varepsilon + x\right)}\right)}{\cot \left(x + \varepsilon\right) \cdot \cot x} \leadsto \frac{\color{blue}{\log \left(e^{\cot x}\right)} - \log \left(e^{\cot \left(\varepsilon + x\right)}\right)}{\cot \left(x + \varepsilon\right) \cdot \cot x}\]
      28.9
    10. Applied diff-log to get
      \[\frac{\color{red}{\log \left(e^{\cot x}\right) - \log \left(e^{\cot \left(\varepsilon + x\right)}\right)}}{\cot \left(x + \varepsilon\right) \cdot \cot x} \leadsto \frac{\color{blue}{\log \left(\frac{e^{\cot x}}{e^{\cot \left(\varepsilon + x\right)}}\right)}}{\cot \left(x + \varepsilon\right) \cdot \cot x}\]
      28.9
    11. Applied simplify to get
      \[\frac{\log \color{red}{\left(\frac{e^{\cot x}}{e^{\cot \left(\varepsilon + x\right)}}\right)}}{\cot \left(x + \varepsilon\right) \cdot \cot x} \leadsto \frac{\log \color{blue}{\left(e^{\cot x - \cot \left(\varepsilon + x\right)}\right)}}{\cot \left(x + \varepsilon\right) \cdot \cot x}\]
      28.6
    12. Applied taylor to get
      \[\frac{\log \left(e^{\cot x - \cot \left(\varepsilon + x\right)}\right)}{\cot \left(x + \varepsilon\right) \cdot \cot x} \leadsto \frac{\left(\frac{\varepsilon \cdot {\left(\cos x\right)}^2}{{\left(\sin x\right)}^2} + \left(\frac{4}{3} \cdot \frac{{\varepsilon}^{3} \cdot {\left(\cos x\right)}^2}{{\left(\sin x\right)}^2} + \left(\frac{1}{3} \cdot {\varepsilon}^{3} + \left(\varepsilon + \frac{{\varepsilon}^{3} \cdot {\left(\cos x\right)}^{4}}{{\left(\sin x\right)}^{4}}\right)\right)\right)\right) - \left(\frac{{\varepsilon}^2 \cdot \cos x}{\sin x} + \frac{{\varepsilon}^2 \cdot {\left(\cos x\right)}^{3}}{{\left(\sin x\right)}^{3}}\right)}{\cot \left(x + \varepsilon\right) \cdot \cot x}\]
      7.9
    13. Taylor expanded around 0 to get
      \[\frac{\color{red}{\left(\frac{\varepsilon \cdot {\left(\cos x\right)}^2}{{\left(\sin x\right)}^2} + \left(\frac{4}{3} \cdot \frac{{\varepsilon}^{3} \cdot {\left(\cos x\right)}^2}{{\left(\sin x\right)}^2} + \left(\frac{1}{3} \cdot {\varepsilon}^{3} + \left(\varepsilon + \frac{{\varepsilon}^{3} \cdot {\left(\cos x\right)}^{4}}{{\left(\sin x\right)}^{4}}\right)\right)\right)\right) - \left(\frac{{\varepsilon}^2 \cdot \cos x}{\sin x} + \frac{{\varepsilon}^2 \cdot {\left(\cos x\right)}^{3}}{{\left(\sin x\right)}^{3}}\right)}}{\cot \left(x + \varepsilon\right) \cdot \cot x} \leadsto \frac{\color{blue}{\left(\frac{\varepsilon \cdot {\left(\cos x\right)}^2}{{\left(\sin x\right)}^2} + \left(\frac{4}{3} \cdot \frac{{\varepsilon}^{3} \cdot {\left(\cos x\right)}^2}{{\left(\sin x\right)}^2} + \left(\frac{1}{3} \cdot {\varepsilon}^{3} + \left(\varepsilon + \frac{{\varepsilon}^{3} \cdot {\left(\cos x\right)}^{4}}{{\left(\sin x\right)}^{4}}\right)\right)\right)\right) - \left(\frac{{\varepsilon}^2 \cdot \cos x}{\sin x} + \frac{{\varepsilon}^2 \cdot {\left(\cos x\right)}^{3}}{{\left(\sin x\right)}^{3}}\right)}}{\cot \left(x + \varepsilon\right) \cdot \cot x}\]
      7.9
    14. Applied simplify to get
      \[\frac{\left(\frac{\varepsilon \cdot {\left(\cos x\right)}^2}{{\left(\sin x\right)}^2} + \left(\frac{4}{3} \cdot \frac{{\varepsilon}^{3} \cdot {\left(\cos x\right)}^2}{{\left(\sin x\right)}^2} + \left(\frac{1}{3} \cdot {\varepsilon}^{3} + \left(\varepsilon + \frac{{\varepsilon}^{3} \cdot {\left(\cos x\right)}^{4}}{{\left(\sin x\right)}^{4}}\right)\right)\right)\right) - \left(\frac{{\varepsilon}^2 \cdot \cos x}{\sin x} + \frac{{\varepsilon}^2 \cdot {\left(\cos x\right)}^{3}}{{\left(\sin x\right)}^{3}}\right)}{\cot \left(x + \varepsilon\right) \cdot \cot x} \leadsto \frac{\left(\left(\frac{{\varepsilon}^3 \cdot \frac{4}{3}}{{\left(\frac{\sin x}{\cos x}\right)}^2} + \frac{\frac{\varepsilon}{\frac{\sin x}{\cos x}}}{\frac{\sin x}{\cos x}}\right) + \left(\left(\varepsilon + {\varepsilon}^3 \cdot \frac{1}{3}\right) + \frac{{\left(\cos x\right)}^{4} \cdot {\varepsilon}^3}{{\left(\sin x\right)}^{4}}\right)\right) - \left(\frac{\cos x \cdot {\varepsilon}^2}{\sin x} + \frac{{\varepsilon}^2}{\sin x} \cdot \frac{{\left(\cos x\right)}^3}{\sin x \cdot \sin x}\right)}{\cot \left(\varepsilon + x\right) \cdot \cot x}\]
      7.9
    15. Applied final simplification

    if -1.4391974f-06 < x < 1.8040304f-05

    1. Started with
      \[\tan \left(x + \varepsilon\right) - \tan x\]
      7.7
    2. Using strategy rm
      7.7
    3. Applied tan-cotan to get
      \[\tan \left(x + \varepsilon\right) - \color{red}{\tan x} \leadsto \tan \left(x + \varepsilon\right) - \color{blue}{\frac{1}{\cot x}}\]
      7.8
    4. Applied tan-cotan to get
      \[\color{red}{\tan \left(x + \varepsilon\right)} - \frac{1}{\cot x} \leadsto \color{blue}{\frac{1}{\cot \left(x + \varepsilon\right)}} - \frac{1}{\cot x}\]
      7.7
    5. Applied frac-sub to get
      \[\color{red}{\frac{1}{\cot \left(x + \varepsilon\right)} - \frac{1}{\cot x}} \leadsto \color{blue}{\frac{1 \cdot \cot x - \cot \left(x + \varepsilon\right) \cdot 1}{\cot \left(x + \varepsilon\right) \cdot \cot x}}\]
      7.7
    6. Applied simplify to get
      \[\frac{\color{red}{1 \cdot \cot x - \cot \left(x + \varepsilon\right) \cdot 1}}{\cot \left(x + \varepsilon\right) \cdot \cot x} \leadsto \frac{\color{blue}{\cot x - \cot \left(\varepsilon + x\right)}}{\cot \left(x + \varepsilon\right) \cdot \cot x}\]
      7.7
    7. Applied taylor to get
      \[\frac{\cot x - \cot \left(\varepsilon + x\right)}{\cot \left(x + \varepsilon\right) \cdot \cot x} \leadsto \frac{\cot x - \frac{\cos \left(\frac{1}{x} + \frac{1}{\varepsilon}\right)}{\sin \left(\frac{1}{x} + \frac{1}{\varepsilon}\right)}}{\cot \left(x + \varepsilon\right) \cdot \cot x}\]
      0.4
    8. Taylor expanded around inf to get
      \[\frac{\cot x - \color{red}{\frac{\cos \left(\frac{1}{x} + \frac{1}{\varepsilon}\right)}{\sin \left(\frac{1}{x} + \frac{1}{\varepsilon}\right)}}}{\cot \left(x + \varepsilon\right) \cdot \cot x} \leadsto \frac{\cot x - \color{blue}{\frac{\cos \left(\frac{1}{x} + \frac{1}{\varepsilon}\right)}{\sin \left(\frac{1}{x} + \frac{1}{\varepsilon}\right)}}}{\cot \left(x + \varepsilon\right) \cdot \cot x}\]
      0.4
    9. Applied taylor to get
      \[\frac{\cot x - \frac{\cos \left(\frac{1}{x} + \frac{1}{\varepsilon}\right)}{\sin \left(\frac{1}{x} + \frac{1}{\varepsilon}\right)}}{\cot \left(x + \varepsilon\right) \cdot \cot x} \leadsto \frac{\cot x - \frac{\cos \left(\frac{1}{x} + \frac{1}{\varepsilon}\right)}{\left(\frac{1}{x} + \frac{1}{\varepsilon}\right) - \frac{1}{6} \cdot \frac{1}{{x}^{3}}}}{\cot \left(x + \varepsilon\right) \cdot \cot x}\]
      0.2
    10. Taylor expanded around inf to get
      \[\frac{\cot x - \frac{\cos \left(\frac{1}{x} + \frac{1}{\varepsilon}\right)}{\color{red}{\left(\frac{1}{x} + \frac{1}{\varepsilon}\right) - \frac{1}{6} \cdot \frac{1}{{x}^{3}}}}}{\cot \left(x + \varepsilon\right) \cdot \cot x} \leadsto \frac{\cot x - \frac{\cos \left(\frac{1}{x} + \frac{1}{\varepsilon}\right)}{\color{blue}{\left(\frac{1}{x} + \frac{1}{\varepsilon}\right) - \frac{1}{6} \cdot \frac{1}{{x}^{3}}}}}{\cot \left(x + \varepsilon\right) \cdot \cot x}\]
      0.2
    11. Applied simplify to get
      \[\frac{\cot x - \frac{\cos \left(\frac{1}{x} + \frac{1}{\varepsilon}\right)}{\left(\frac{1}{x} + \frac{1}{\varepsilon}\right) - \frac{1}{6} \cdot \frac{1}{{x}^{3}}}}{\cot \left(x + \varepsilon\right) \cdot \cot x} \leadsto \frac{\cot x - \frac{\cos \left(\frac{1}{\varepsilon} + \frac{1}{x}\right)}{\left(\frac{1}{\varepsilon} + \frac{1}{x}\right) - \frac{\frac{1}{6}}{{x}^3}}}{\cot x \cdot \cot \left(\varepsilon + x\right)}\]
      0.2
    12. Applied final simplification
  1. Removed slow pow expressions

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)))))