\[\tan \left(x + \varepsilon\right) - \tan x\]
Test:
NMSE problem 3.3.2
Bits:
128 bits
Bits error versus x
Bits error versus eps
Time: 1.1 m
Input Error: 36.7
Output Error: 14.6
Log: ⚲
Profile: 🕒
\(\begin{cases} \frac{\cot x \cdot \left(\sin x \cdot \cos \varepsilon\right) + \left(\cot x \cdot \left(\cos x \cdot \sin \varepsilon\right) - \cos \left(x + \varepsilon\right)\right)}{\cos \left(x + \varepsilon\right) \cdot \cot x} & \text{when } \varepsilon \le -9.280756734832815 \cdot 10^{-07} \\ \frac{(\left(\sin \varepsilon\right) * \left(\sin x\right) + \left(\frac{\varepsilon}{\sin x} \cdot \left(\cos x \cdot \cos x\right)\right))_*}{\cos \left(\varepsilon + x\right) \cdot \cot x} & \text{when } \varepsilon \le 1.0802375057316329 \cdot 10^{-14} \\ \frac{\cot x \cdot \left(\sin x \cdot \cos \varepsilon\right) + \left(\cot x \cdot \left(\cos x \cdot \sin \varepsilon\right) - \cos \left(x + \varepsilon\right)\right)}{\cos \left(x + \varepsilon\right) \cdot \cot x} & \text{otherwise} \end{cases}\)

    if eps < -9.280756734832815e-07 or 1.0802375057316329e-14 < eps

    1. Started with
      \[\tan \left(x + \varepsilon\right) - \tan x\]
      29.2
    2. Using strategy rm
      29.2
    3. Applied tan-cotan to get
      \[\tan \left(x + \varepsilon\right) - \color{red}{\tan x} \leadsto \tan \left(x + \varepsilon\right) - \color{blue}{\frac{1}{\cot x}}\]
      29.1
    4. Applied tan-quot to get
      \[\color{red}{\tan \left(x + \varepsilon\right)} - \frac{1}{\cot x} \leadsto \color{blue}{\frac{\sin \left(x + \varepsilon\right)}{\cos \left(x + \varepsilon\right)}} - \frac{1}{\cot x}\]
      29.2
    5. Applied frac-sub to get
      \[\color{red}{\frac{\sin \left(x + \varepsilon\right)}{\cos \left(x + \varepsilon\right)} - \frac{1}{\cot x}} \leadsto \color{blue}{\frac{\sin \left(x + \varepsilon\right) \cdot \cot x - \cos \left(x + \varepsilon\right) \cdot 1}{\cos \left(x + \varepsilon\right) \cdot \cot x}}\]
      29.2
    6. Applied simplify to get
      \[\frac{\color{red}{\sin \left(x + \varepsilon\right) \cdot \cot x - \cos \left(x + \varepsilon\right) \cdot 1}}{\cos \left(x + \varepsilon\right) \cdot \cot x} \leadsto \frac{\color{blue}{\cot x \cdot \sin \left(x + \varepsilon\right) - \cos \left(x + \varepsilon\right)}}{\cos \left(x + \varepsilon\right) \cdot \cot x}\]
      29.2
    7. Using strategy rm
      29.2
    8. Applied sin-sum to get
      \[\frac{\cot x \cdot \color{red}{\sin \left(x + \varepsilon\right)} - \cos \left(x + \varepsilon\right)}{\cos \left(x + \varepsilon\right) \cdot \cot x} \leadsto \frac{\cot x \cdot \color{blue}{\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right)} - \cos \left(x + \varepsilon\right)}{\cos \left(x + \varepsilon\right) \cdot \cot x}\]
      27.5
    9. Applied distribute-lft-in to get
      \[\frac{\color{red}{\cot x \cdot \left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right)} - \cos \left(x + \varepsilon\right)}{\cos \left(x + \varepsilon\right) \cdot \cot x} \leadsto \frac{\color{blue}{\left(\cot x \cdot \left(\sin x \cdot \cos \varepsilon\right) + \cot x \cdot \left(\cos x \cdot \sin \varepsilon\right)\right)} - \cos \left(x + \varepsilon\right)}{\cos \left(x + \varepsilon\right) \cdot \cot x}\]
      27.5
    10. Applied associate--l+ to get
      \[\frac{\color{red}{\left(\cot x \cdot \left(\sin x \cdot \cos \varepsilon\right) + \cot x \cdot \left(\cos x \cdot \sin \varepsilon\right)\right) - \cos \left(x + \varepsilon\right)}}{\cos \left(x + \varepsilon\right) \cdot \cot x} \leadsto \frac{\color{blue}{\cot x \cdot \left(\sin x \cdot \cos \varepsilon\right) + \left(\cot x \cdot \left(\cos x \cdot \sin \varepsilon\right) - \cos \left(x + \varepsilon\right)\right)}}{\cos \left(x + \varepsilon\right) \cdot \cot x}\]
      27.5

    if -9.280756734832815e-07 < eps < 1.0802375057316329e-14

    1. Started with
      \[\tan \left(x + \varepsilon\right) - \tan x\]
      44.9
    2. Using strategy rm
      44.9
    3. Applied tan-cotan to get
      \[\tan \left(x + \varepsilon\right) - \color{red}{\tan x} \leadsto \tan \left(x + \varepsilon\right) - \color{blue}{\frac{1}{\cot x}}\]
      45.1
    4. Applied tan-quot to get
      \[\color{red}{\tan \left(x + \varepsilon\right)} - \frac{1}{\cot x} \leadsto \color{blue}{\frac{\sin \left(x + \varepsilon\right)}{\cos \left(x + \varepsilon\right)}} - \frac{1}{\cot x}\]
      45.1
    5. Applied frac-sub to get
      \[\color{red}{\frac{\sin \left(x + \varepsilon\right)}{\cos \left(x + \varepsilon\right)} - \frac{1}{\cot x}} \leadsto \color{blue}{\frac{\sin \left(x + \varepsilon\right) \cdot \cot x - \cos \left(x + \varepsilon\right) \cdot 1}{\cos \left(x + \varepsilon\right) \cdot \cot x}}\]
      45.2
    6. Applied simplify to get
      \[\frac{\color{red}{\sin \left(x + \varepsilon\right) \cdot \cot x - \cos \left(x + \varepsilon\right) \cdot 1}}{\cos \left(x + \varepsilon\right) \cdot \cot x} \leadsto \frac{\color{blue}{\cot x \cdot \sin \left(x + \varepsilon\right) - \cos \left(x + \varepsilon\right)}}{\cos \left(x + \varepsilon\right) \cdot \cot x}\]
      45.2
    7. Using strategy rm
      45.2
    8. Applied cos-sum to get
      \[\frac{\cot x \cdot \sin \left(x + \varepsilon\right) - \color{red}{\cos \left(x + \varepsilon\right)}}{\cos \left(x + \varepsilon\right) \cdot \cot x} \leadsto \frac{\cot x \cdot \sin \left(x + \varepsilon\right) - \color{blue}{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right)}}{\cos \left(x + \varepsilon\right) \cdot \cot x}\]
      45.1
    9. Applied associate--r- to get
      \[\frac{\color{red}{\cot x \cdot \sin \left(x + \varepsilon\right) - \left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right)}}{\cos \left(x + \varepsilon\right) \cdot \cot x} \leadsto \frac{\color{blue}{\left(\cot x \cdot \sin \left(x + \varepsilon\right) - \cos x \cdot \cos \varepsilon\right) + \sin x \cdot \sin \varepsilon}}{\cos \left(x + \varepsilon\right) \cdot \cot x}\]
      42.5
    10. Using strategy rm
      42.5
    11. Applied add-cube-cbrt to get
      \[\frac{\color{red}{\left(\cot x \cdot \sin \left(x + \varepsilon\right) - \cos x \cdot \cos \varepsilon\right)} + \sin x \cdot \sin \varepsilon}{\cos \left(x + \varepsilon\right) \cdot \cot x} \leadsto \frac{\color{blue}{{\left(\sqrt[3]{\cot x \cdot \sin \left(x + \varepsilon\right) - \cos x \cdot \cos \varepsilon}\right)}^3} + \sin x \cdot \sin \varepsilon}{\cos \left(x + \varepsilon\right) \cdot \cot x}\]
      42.8
    12. Applied taylor to get
      \[\frac{{\left(\sqrt[3]{\cot x \cdot \sin \left(x + \varepsilon\right) - \cos x \cdot \cos \varepsilon}\right)}^3 + \sin x \cdot \sin \varepsilon}{\cos \left(x + \varepsilon\right) \cdot \cot x} \leadsto \frac{\frac{\varepsilon \cdot {\left(\cos x\right)}^2}{\sin x} + \sin x \cdot \sin \varepsilon}{\cos \left(x + \varepsilon\right) \cdot \cot x}\]
      0.6
    13. Taylor expanded around 0 to get
      \[\frac{\color{red}{\frac{\varepsilon \cdot {\left(\cos x\right)}^2}{\sin x}} + \sin x \cdot \sin \varepsilon}{\cos \left(x + \varepsilon\right) \cdot \cot x} \leadsto \frac{\color{blue}{\frac{\varepsilon \cdot {\left(\cos x\right)}^2}{\sin x}} + \sin x \cdot \sin \varepsilon}{\cos \left(x + \varepsilon\right) \cdot \cot x}\]
      0.6
    14. Applied simplify to get
      \[\frac{\frac{\varepsilon \cdot {\left(\cos x\right)}^2}{\sin x} + \sin x \cdot \sin \varepsilon}{\cos \left(x + \varepsilon\right) \cdot \cot x} \leadsto \frac{(\left(\sin \varepsilon\right) * \left(\sin x\right) + \left(\frac{\varepsilon}{\sin x} \cdot \left(\cos x \cdot \cos x\right)\right))_*}{\cos \left(\varepsilon + x\right) \cdot \cot x}\]
      0.6
    15. 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)))))