\[\tan \left(x + \varepsilon\right) - \tan x\]
Test:
NMSE problem 3.3.2
Bits:
128 bits
Bits error versus x
Bits error versus eps
Time: 23.9 s
Input Error: 17.0
Output Error: 7.2
Log: ⚲
Profile: 🕒
\(\begin{cases} \frac{\cos x - \frac{\cos \left(\frac{1}{\varepsilon} + \frac{1}{x}\right)}{\frac{\sin \left(\varepsilon + x\right)}{\sin x}}}{\cos x \cdot \cot \left(\varepsilon + x\right)} & \text{when } \varepsilon \le -0.081347406f0 \\ \left(\left(\frac{\varepsilon}{{\left(\frac{\cos x}{\sin x}\right)}^2} + \frac{{\varepsilon}^2}{\cos x} \cdot \sin x\right) + \frac{{\left(\sin x\right)}^3}{\frac{{\left(\cos x\right)}^3}{{\varepsilon}^2}}\right) + \left(\left(\frac{1}{3} \cdot {\varepsilon}^3 + \varepsilon\right) + \left(\frac{\left(\frac{4}{3} \cdot \varepsilon\right) \cdot {\varepsilon}^2}{{\left(\frac{\cos x}{\sin x}\right)}^2} + \frac{{\varepsilon}^3}{\frac{{\left(\cos x\right)}^{4}}{{\left(\sin x\right)}^{4}}}\right)\right) & \text{when } \varepsilon \le 0.07298134f0 \\ \tan \left(x + \varepsilon\right) - \frac{\sin x}{\cos x} & \text{otherwise} \end{cases}\)

    if eps < -0.081347406f0

    1. Started with
      \[\tan \left(x + \varepsilon\right) - \tan x\]
      15.4
    2. Using strategy rm
      15.4
    3. Applied tan-quot to get
      \[\tan \left(x + \varepsilon\right) - \color{red}{\tan x} \leadsto \tan \left(x + \varepsilon\right) - \color{blue}{\frac{\sin x}{\cos x}}\]
      15.3
    4. Applied tan-cotan to get
      \[\color{red}{\tan \left(x + \varepsilon\right)} - \frac{\sin x}{\cos x} \leadsto \color{blue}{\frac{1}{\cot \left(x + \varepsilon\right)}} - \frac{\sin x}{\cos x}\]
      15.3
    5. Applied frac-sub to get
      \[\color{red}{\frac{1}{\cot \left(x + \varepsilon\right)} - \frac{\sin x}{\cos x}} \leadsto \color{blue}{\frac{1 \cdot \cos x - \cot \left(x + \varepsilon\right) \cdot \sin x}{\cot \left(x + \varepsilon\right) \cdot \cos x}}\]
      15.3
    6. Applied simplify to get
      \[\frac{\color{red}{1 \cdot \cos x - \cot \left(x + \varepsilon\right) \cdot \sin x}}{\cot \left(x + \varepsilon\right) \cdot \cos x} \leadsto \frac{\color{blue}{\cos x - \cot \left(\varepsilon + x\right) \cdot \sin x}}{\cot \left(x + \varepsilon\right) \cdot \cos x}\]
      15.3
    7. Using strategy rm
      15.3
    8. Applied cotan-quot to get
      \[\frac{\cos x - \color{red}{\cot \left(\varepsilon + x\right)} \cdot \sin x}{\cot \left(x + \varepsilon\right) \cdot \cos x} \leadsto \frac{\cos x - \color{blue}{\frac{\cos \left(\varepsilon + x\right)}{\sin \left(\varepsilon + x\right)}} \cdot \sin x}{\cot \left(x + \varepsilon\right) \cdot \cos x}\]
      15.4
    9. Applied taylor to get
      \[\frac{\cos x - \frac{\cos \left(\varepsilon + x\right)}{\sin \left(\varepsilon + x\right)} \cdot \sin x}{\cot \left(x + \varepsilon\right) \cdot \cos x} \leadsto \frac{\cos x - \frac{\cos \left(\frac{1}{x} + \frac{1}{\varepsilon}\right)}{\sin \left(\varepsilon + x\right)} \cdot \sin x}{\cot \left(x + \varepsilon\right) \cdot \cos x}\]
      14.3
    10. Taylor expanded around inf to get
      \[\frac{\cos x - \frac{\color{red}{\cos \left(\frac{1}{x} + \frac{1}{\varepsilon}\right)}}{\sin \left(\varepsilon + x\right)} \cdot \sin x}{\cot \left(x + \varepsilon\right) \cdot \cos x} \leadsto \frac{\cos x - \frac{\color{blue}{\cos \left(\frac{1}{x} + \frac{1}{\varepsilon}\right)}}{\sin \left(\varepsilon + x\right)} \cdot \sin x}{\cot \left(x + \varepsilon\right) \cdot \cos x}\]
      14.3
    11. Applied simplify to get
      \[\frac{\cos x - \frac{\cos \left(\frac{1}{x} + \frac{1}{\varepsilon}\right)}{\sin \left(\varepsilon + x\right)} \cdot \sin x}{\cot \left(x + \varepsilon\right) \cdot \cos x} \leadsto \frac{\cos x - \frac{\cos \left(\frac{1}{\varepsilon} + \frac{1}{x}\right)}{\frac{\sin \left(\varepsilon + x\right)}{\sin x}}}{\cos x \cdot \cot \left(\varepsilon + x\right)}\]
      14.3
    12. Applied final simplification

    if -0.081347406f0 < eps < 0.07298134f0

    1. Started with
      \[\tan \left(x + \varepsilon\right) - \tan x\]
      19.7
    2. Using strategy rm
      19.7
    3. Applied tan-cotan to get
      \[\color{red}{\tan \left(x + \varepsilon\right)} - \tan x \leadsto \color{blue}{\frac{1}{\cot \left(x + \varepsilon\right)}} - \tan x\]
      20.1
    4. Using strategy rm
      20.1
    5. Applied add-cube-cbrt to get
      \[\color{red}{\frac{1}{\cot \left(x + \varepsilon\right)} - \tan x} \leadsto \color{blue}{{\left(\sqrt[3]{\frac{1}{\cot \left(x + \varepsilon\right)} - \tan x}\right)}^3}\]
      20.2
    6. Applied taylor to get
      \[{\left(\sqrt[3]{\frac{1}{\cot \left(x + \varepsilon\right)} - \tan x}\right)}^3 \leadsto \frac{\varepsilon \cdot {\left(\sin x\right)}^2}{{\left(\cos x\right)}^2} + \left(\frac{{\varepsilon}^2 \cdot \sin x}{\cos x} + \left(\frac{{\varepsilon}^2 \cdot {\left(\sin x\right)}^{3}}{{\left(\cos x\right)}^{3}} + \left(\frac{1}{3} \cdot {\varepsilon}^{3} + \left(\varepsilon + \left(\frac{4}{3} \cdot \frac{{\varepsilon}^{3} \cdot {\left(\sin x\right)}^2}{{\left(\cos x\right)}^2} + \frac{{\varepsilon}^{3} \cdot {\left(\sin x\right)}^{4}}{{\left(\cos x\right)}^{4}}\right)\right)\right)\right)\right)\]
      0.2
    7. Taylor expanded around 0 to get
      \[\color{red}{\frac{\varepsilon \cdot {\left(\sin x\right)}^2}{{\left(\cos x\right)}^2} + \left(\frac{{\varepsilon}^2 \cdot \sin x}{\cos x} + \left(\frac{{\varepsilon}^2 \cdot {\left(\sin x\right)}^{3}}{{\left(\cos x\right)}^{3}} + \left(\frac{1}{3} \cdot {\varepsilon}^{3} + \left(\varepsilon + \left(\frac{4}{3} \cdot \frac{{\varepsilon}^{3} \cdot {\left(\sin x\right)}^2}{{\left(\cos x\right)}^2} + \frac{{\varepsilon}^{3} \cdot {\left(\sin x\right)}^{4}}{{\left(\cos x\right)}^{4}}\right)\right)\right)\right)\right)} \leadsto \color{blue}{\frac{\varepsilon \cdot {\left(\sin x\right)}^2}{{\left(\cos x\right)}^2} + \left(\frac{{\varepsilon}^2 \cdot \sin x}{\cos x} + \left(\frac{{\varepsilon}^2 \cdot {\left(\sin x\right)}^{3}}{{\left(\cos x\right)}^{3}} + \left(\frac{1}{3} \cdot {\varepsilon}^{3} + \left(\varepsilon + \left(\frac{4}{3} \cdot \frac{{\varepsilon}^{3} \cdot {\left(\sin x\right)}^2}{{\left(\cos x\right)}^2} + \frac{{\varepsilon}^{3} \cdot {\left(\sin x\right)}^{4}}{{\left(\cos x\right)}^{4}}\right)\right)\right)\right)\right)}\]
      0.2
    8. Applied simplify to get
      \[\frac{\varepsilon \cdot {\left(\sin x\right)}^2}{{\left(\cos x\right)}^2} + \left(\frac{{\varepsilon}^2 \cdot \sin x}{\cos x} + \left(\frac{{\varepsilon}^2 \cdot {\left(\sin x\right)}^{3}}{{\left(\cos x\right)}^{3}} + \left(\frac{1}{3} \cdot {\varepsilon}^{3} + \left(\varepsilon + \left(\frac{4}{3} \cdot \frac{{\varepsilon}^{3} \cdot {\left(\sin x\right)}^2}{{\left(\cos x\right)}^2} + \frac{{\varepsilon}^{3} \cdot {\left(\sin x\right)}^{4}}{{\left(\cos x\right)}^{4}}\right)\right)\right)\right)\right) \leadsto \left(\left(\frac{\varepsilon}{{\left(\frac{\cos x}{\sin x}\right)}^2} + \frac{{\varepsilon}^2}{\cos x} \cdot \sin x\right) + \frac{{\left(\sin x\right)}^3}{\frac{{\left(\cos x\right)}^3}{{\varepsilon}^2}}\right) + \left(\left(\frac{1}{3} \cdot {\varepsilon}^3 + \varepsilon\right) + \left(\frac{\left(\frac{4}{3} \cdot \varepsilon\right) \cdot {\varepsilon}^2}{{\left(\frac{\cos x}{\sin x}\right)}^2} + \frac{{\varepsilon}^3}{\frac{{\left(\cos x\right)}^{4}}{{\left(\sin x\right)}^{4}}}\right)\right)\]
      0.2
    9. Applied final simplification

    if 0.07298134f0 < eps

    1. Started with
      \[\tan \left(x + \varepsilon\right) - \tan x\]
      13.4
    2. Using strategy rm
      13.4
    3. Applied tan-quot to get
      \[\tan \left(x + \varepsilon\right) - \color{red}{\tan x} \leadsto \tan \left(x + \varepsilon\right) - \color{blue}{\frac{\sin x}{\cos x}}\]
      13.3
  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)))))