\[\tan \left(x + \varepsilon\right) - \tan x\]
Test:
NMSE problem 3.3.2
Bits:
128 bits
Bits error versus x
Bits error versus eps
Time: 28.9 s
Input Error: 36.6
Output Error: 26.6
Log: ⚲
Profile: 🕒
\(\begin{cases} \sqrt[3]{\frac{{\left(\frac{\cos x}{\cot \left(x + \varepsilon\right)} - \sin x\right)}^3}{{\left(\cos x\right)}^3}} & \text{when } \varepsilon \le -5.831882313182785 \cdot 10^{-50} \\ \left(\varepsilon + \left(x \cdot x\right) \cdot {\varepsilon}^3\right) + {\varepsilon}^{4} \cdot {x}^3 & \text{when } \varepsilon \le 7.404354464384737 \cdot 10^{-68} \\ \frac{{\left(\cos x\right)}^{3} - {\left(\cot \left(\varepsilon + x\right) \cdot \sin x\right)}^{3}}{\left(\cot \left(x + \varepsilon\right) \cdot \cos x\right) \cdot \left({\left(\cos x\right)}^2 + \left({\left(\cot \left(\varepsilon + x\right) \cdot \sin x\right)}^2 + \cos x \cdot \left(\cot \left(\varepsilon + x\right) \cdot \sin x\right)\right)\right)} & \text{otherwise} \end{cases}\)

    if eps < -5.831882313182785e-50

    1. Started with
      \[\tan \left(x + \varepsilon\right) - \tan x\]
      30.8
    2. Using strategy rm
      30.8
    3. Applied tan-quot to get
      \[\tan \left(x + \varepsilon\right) - \color{red}{\tan x} \leadsto \tan \left(x + \varepsilon\right) - \color{blue}{\frac{\sin x}{\cos x}}\]
      30.6
    4. Applied tan-cotan to get
      \[\color{red}{\tan \left(x + \varepsilon\right)} - \frac{\sin x}{\cos x} \leadsto \color{blue}{\frac{1}{\cot \left(x + \varepsilon\right)}} - \frac{\sin x}{\cos x}\]
      30.6
    5. Applied frac-sub to get
      \[\color{red}{\frac{1}{\cot \left(x + \varepsilon\right)} - \frac{\sin x}{\cos x}} \leadsto \color{blue}{\frac{1 \cdot \cos x - \cot \left(x + \varepsilon\right) \cdot \sin x}{\cot \left(x + \varepsilon\right) \cdot \cos x}}\]
      30.6
    6. Applied simplify to get
      \[\frac{\color{red}{1 \cdot \cos x - \cot \left(x + \varepsilon\right) \cdot \sin x}}{\cot \left(x + \varepsilon\right) \cdot \cos x} \leadsto \frac{\color{blue}{\cos x - \cot \left(\varepsilon + x\right) \cdot \sin x}}{\cot \left(x + \varepsilon\right) \cdot \cos x}\]
      30.6
    7. Using strategy rm
      30.6
    8. Applied add-cbrt-cube to get
      \[\frac{\cos x - \cot \left(\varepsilon + x\right) \cdot \sin x}{\cot \left(x + \varepsilon\right) \cdot \color{red}{\cos x}} \leadsto \frac{\cos x - \cot \left(\varepsilon + x\right) \cdot \sin x}{\cot \left(x + \varepsilon\right) \cdot \color{blue}{\sqrt[3]{{\left(\cos x\right)}^3}}}\]
      30.6
    9. Applied add-cbrt-cube to get
      \[\frac{\cos x - \cot \left(\varepsilon + x\right) \cdot \sin x}{\color{red}{\cot \left(x + \varepsilon\right)} \cdot \sqrt[3]{{\left(\cos x\right)}^3}} \leadsto \frac{\cos x - \cot \left(\varepsilon + x\right) \cdot \sin x}{\color{blue}{\sqrt[3]{{\left(\cot \left(x + \varepsilon\right)\right)}^3}} \cdot \sqrt[3]{{\left(\cos x\right)}^3}}\]
      30.8
    10. Applied cbrt-unprod to get
      \[\frac{\cos x - \cot \left(\varepsilon + x\right) \cdot \sin x}{\color{red}{\sqrt[3]{{\left(\cot \left(x + \varepsilon\right)\right)}^3} \cdot \sqrt[3]{{\left(\cos x\right)}^3}}} \leadsto \frac{\cos x - \cot \left(\varepsilon + x\right) \cdot \sin x}{\color{blue}{\sqrt[3]{{\left(\cot \left(x + \varepsilon\right)\right)}^3 \cdot {\left(\cos x\right)}^3}}}\]
      30.8
    11. Applied add-cbrt-cube to get
      \[\frac{\color{red}{\cos x - \cot \left(\varepsilon + x\right) \cdot \sin x}}{\sqrt[3]{{\left(\cot \left(x + \varepsilon\right)\right)}^3 \cdot {\left(\cos x\right)}^3}} \leadsto \frac{\color{blue}{\sqrt[3]{{\left(\cos x - \cot \left(\varepsilon + x\right) \cdot \sin x\right)}^3}}}{\sqrt[3]{{\left(\cot \left(x + \varepsilon\right)\right)}^3 \cdot {\left(\cos x\right)}^3}}\]
      30.8
    12. Applied cbrt-undiv to get
      \[\color{red}{\frac{\sqrt[3]{{\left(\cos x - \cot \left(\varepsilon + x\right) \cdot \sin x\right)}^3}}{\sqrt[3]{{\left(\cot \left(x + \varepsilon\right)\right)}^3 \cdot {\left(\cos x\right)}^3}}} \leadsto \color{blue}{\sqrt[3]{\frac{{\left(\cos x - \cot \left(\varepsilon + x\right) \cdot \sin x\right)}^3}{{\left(\cot \left(x + \varepsilon\right)\right)}^3 \cdot {\left(\cos x\right)}^3}}}\]
      30.8
    13. Applied simplify to get
      \[\sqrt[3]{\color{red}{\frac{{\left(\cos x - \cot \left(\varepsilon + x\right) \cdot \sin x\right)}^3}{{\left(\cot \left(x + \varepsilon\right)\right)}^3 \cdot {\left(\cos x\right)}^3}}} \leadsto \sqrt[3]{\color{blue}{\frac{{\left(\frac{\cos x}{\cot \left(x + \varepsilon\right)} - 1 \cdot \sin x\right)}^3}{{\left(\cos x\right)}^3}}}\]
      30.8
    14. Applied simplify to get
      \[\sqrt[3]{\frac{\color{red}{{\left(\frac{\cos x}{\cot \left(x + \varepsilon\right)} - 1 \cdot \sin x\right)}^3}}{{\left(\cos x\right)}^3}} \leadsto \sqrt[3]{\frac{\color{blue}{{\left(\frac{\cos x}{\cot \left(x + \varepsilon\right)} - \sin x\right)}^3}}{{\left(\cos x\right)}^3}}\]
      30.8

    if -5.831882313182785e-50 < eps < 7.404354464384737e-68

    1. Started with
      \[\tan \left(x + \varepsilon\right) - \tan x\]
      45.9
    2. Applied taylor to get
      \[\tan \left(x + \varepsilon\right) - \tan x \leadsto \varepsilon + \left({\varepsilon}^{3} \cdot {x}^2 + {\varepsilon}^{4} \cdot {x}^{3}\right)\]
      20.2
    3. Taylor expanded around 0 to get
      \[\color{red}{\varepsilon + \left({\varepsilon}^{3} \cdot {x}^2 + {\varepsilon}^{4} \cdot {x}^{3}\right)} \leadsto \color{blue}{\varepsilon + \left({\varepsilon}^{3} \cdot {x}^2 + {\varepsilon}^{4} \cdot {x}^{3}\right)}\]
      20.2
    4. Applied simplify to get
      \[\color{red}{\varepsilon + \left({\varepsilon}^{3} \cdot {x}^2 + {\varepsilon}^{4} \cdot {x}^{3}\right)} \leadsto \color{blue}{\left(\varepsilon + \left(x \cdot x\right) \cdot {\varepsilon}^3\right) + {\varepsilon}^{4} \cdot {x}^3}\]
      20.2

    if 7.404354464384737e-68 < eps

    1. Started with
      \[\tan \left(x + \varepsilon\right) - \tan x\]
      30.6
    2. Using strategy rm
      30.6
    3. Applied tan-quot to get
      \[\tan \left(x + \varepsilon\right) - \color{red}{\tan x} \leadsto \tan \left(x + \varepsilon\right) - \color{blue}{\frac{\sin x}{\cos x}}\]
      30.4
    4. Applied tan-cotan to get
      \[\color{red}{\tan \left(x + \varepsilon\right)} - \frac{\sin x}{\cos x} \leadsto \color{blue}{\frac{1}{\cot \left(x + \varepsilon\right)}} - \frac{\sin x}{\cos x}\]
      30.4
    5. Applied frac-sub to get
      \[\color{red}{\frac{1}{\cot \left(x + \varepsilon\right)} - \frac{\sin x}{\cos x}} \leadsto \color{blue}{\frac{1 \cdot \cos x - \cot \left(x + \varepsilon\right) \cdot \sin x}{\cot \left(x + \varepsilon\right) \cdot \cos x}}\]
      30.4
    6. Applied simplify to get
      \[\frac{\color{red}{1 \cdot \cos x - \cot \left(x + \varepsilon\right) \cdot \sin x}}{\cot \left(x + \varepsilon\right) \cdot \cos x} \leadsto \frac{\color{blue}{\cos x - \cot \left(\varepsilon + x\right) \cdot \sin x}}{\cot \left(x + \varepsilon\right) \cdot \cos x}\]
      30.4
    7. Using strategy rm
      30.4
    8. Applied flip3-- to get
      \[\frac{\color{red}{\cos x - \cot \left(\varepsilon + x\right) \cdot \sin x}}{\cot \left(x + \varepsilon\right) \cdot \cos x} \leadsto \frac{\color{blue}{\frac{{\left(\cos x\right)}^{3} - {\left(\cot \left(\varepsilon + x\right) \cdot \sin x\right)}^{3}}{{\left(\cos x\right)}^2 + \left({\left(\cot \left(\varepsilon + x\right) \cdot \sin x\right)}^2 + \cos x \cdot \left(\cot \left(\varepsilon + x\right) \cdot \sin x\right)\right)}}}{\cot \left(x + \varepsilon\right) \cdot \cos x}\]
      30.4
    9. Applied associate-/l/ to get
      \[\color{red}{\frac{\frac{{\left(\cos x\right)}^{3} - {\left(\cot \left(\varepsilon + x\right) \cdot \sin x\right)}^{3}}{{\left(\cos x\right)}^2 + \left({\left(\cot \left(\varepsilon + x\right) \cdot \sin x\right)}^2 + \cos x \cdot \left(\cot \left(\varepsilon + x\right) \cdot \sin x\right)\right)}}{\cot \left(x + \varepsilon\right) \cdot \cos x}} \leadsto \color{blue}{\frac{{\left(\cos x\right)}^{3} - {\left(\cot \left(\varepsilon + x\right) \cdot \sin x\right)}^{3}}{\left(\cot \left(x + \varepsilon\right) \cdot \cos x\right) \cdot \left({\left(\cos x\right)}^2 + \left({\left(\cot \left(\varepsilon + x\right) \cdot \sin x\right)}^2 + \cos x \cdot \left(\cot \left(\varepsilon + x\right) \cdot \sin x\right)\right)\right)}}\]
      30.4
  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)))))