\[\tan \left(x + \varepsilon\right) - \tan x\]
Test:
NMSE problem 3.3.2
Bits:
128 bits
Bits error versus x
Bits error versus eps
Time: 32.9 s
Input Error: 16.8
Output Error: 7.4
Log: ⚲
Profile: 🕒
\(\begin{cases} \frac{{\left(\frac{1}{\log \left(e^{\cot \left(x + \varepsilon\right)}\right)}\right)}^2 - {\left(\tan x\right)}^2}{\frac{1}{\cot \left(x + \varepsilon\right)} + \tan x} & \text{when } \varepsilon \le -0.09829997f0 \\ \left((\left(\frac{\varepsilon}{\cos x}\right) * \left(\frac{\sin x}{\frac{\cos x}{\sin x}}\right) + \left(\sin x \cdot \frac{{\varepsilon}^2}{\cos x}\right))_* + (\left(\frac{{\varepsilon}^2}{{\left(\cos x\right)}^3}\right) * \left({\left(\sin x\right)}^3\right) + \left(\frac{1}{3} \cdot {\varepsilon}^3\right))_*\right) + (\left(\frac{{\varepsilon}^3}{{\left(\cos x\right)}^{4}}\right) * \left({\left(\sin x\right)}^{4}\right) + \left((\left(\frac{\frac{4}{3}}{\cos x}\right) * \left(\frac{\sin x \cdot \sin x}{\frac{\cos x}{{\varepsilon}^3}}\right) + \varepsilon)_*\right))_* & \text{when } \varepsilon \le 0.07828799f0 \\ \frac{\cot x \cdot \sin \left(x + \varepsilon\right) - \cos \left(\frac{1}{\varepsilon} + \frac{1}{x}\right)}{\cos \left(x + \varepsilon\right) \cdot \cot x} & \text{otherwise} \end{cases}\)

    if eps < -0.09829997f0

    1. Started with
      \[\tan \left(x + \varepsilon\right) - \tan x\]
      14.0
    2. Using strategy rm
      14.0
    3. Applied tan-cotan to get
      \[\color{red}{\tan \left(x + \varepsilon\right)} - \tan x \leadsto \color{blue}{\frac{1}{\cot \left(x + \varepsilon\right)}} - \tan x\]
      13.8
    4. Using strategy rm
      13.8
    5. Applied flip-- to get
      \[\color{red}{\frac{1}{\cot \left(x + \varepsilon\right)} - \tan x} \leadsto \color{blue}{\frac{{\left(\frac{1}{\cot \left(x + \varepsilon\right)}\right)}^2 - {\left(\tan x\right)}^2}{\frac{1}{\cot \left(x + \varepsilon\right)} + \tan x}}\]
      13.8
    6. Using strategy rm
      13.8
    7. Applied add-log-exp to get
      \[\frac{{\left(\frac{1}{\color{red}{\cot \left(x + \varepsilon\right)}}\right)}^2 - {\left(\tan x\right)}^2}{\frac{1}{\cot \left(x + \varepsilon\right)} + \tan x} \leadsto \frac{{\left(\frac{1}{\color{blue}{\log \left(e^{\cot \left(x + \varepsilon\right)}\right)}}\right)}^2 - {\left(\tan x\right)}^2}{\frac{1}{\cot \left(x + \varepsilon\right)} + \tan x}\]
      14.4

    if -0.09829997f0 < eps < 0.07828799f0

    1. Started with
      \[\tan \left(x + \varepsilon\right) - \tan x\]
      19.5
    2. Using strategy rm
      19.5
    3. Applied add-cube-cbrt to get
      \[\color{red}{\tan \left(x + \varepsilon\right) - \tan x} \leadsto \color{blue}{{\left(\sqrt[3]{\tan \left(x + \varepsilon\right) - \tan x}\right)}^3}\]
      19.7
    4. Applied taylor to get
      \[{\left(\sqrt[3]{\tan \left(x + \varepsilon\right) - \tan x}\right)}^3 \leadsto \frac{\varepsilon \cdot {\left(\sin x\right)}^2}{{\left(\cos x\right)}^2} + \left(\frac{{\varepsilon}^2 \cdot \sin x}{\cos x} + \left(\frac{{\varepsilon}^2 \cdot {\left(\sin x\right)}^{3}}{{\left(\cos x\right)}^{3}} + \left(\frac{1}{3} \cdot {\varepsilon}^{3} + \left(\varepsilon + \left(\frac{4}{3} \cdot \frac{{\varepsilon}^{3} \cdot {\left(\sin x\right)}^2}{{\left(\cos x\right)}^2} + \frac{{\varepsilon}^{3} \cdot {\left(\sin x\right)}^{4}}{{\left(\cos x\right)}^{4}}\right)\right)\right)\right)\right)\]
      0.2
    5. 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
    6. Applied simplify to get
      \[\frac{\varepsilon \cdot {\left(\sin x\right)}^2}{{\left(\cos x\right)}^2} + \left(\frac{{\varepsilon}^2 \cdot \sin x}{\cos x} + \left(\frac{{\varepsilon}^2 \cdot {\left(\sin x\right)}^{3}}{{\left(\cos x\right)}^{3}} + \left(\frac{1}{3} \cdot {\varepsilon}^{3} + \left(\varepsilon + \left(\frac{4}{3} \cdot \frac{{\varepsilon}^{3} \cdot {\left(\sin x\right)}^2}{{\left(\cos x\right)}^2} + \frac{{\varepsilon}^{3} \cdot {\left(\sin x\right)}^{4}}{{\left(\cos x\right)}^{4}}\right)\right)\right)\right)\right) \leadsto \left((\left(\frac{\varepsilon}{\cos x}\right) * \left(\frac{\sin x}{\frac{\cos x}{\sin x}}\right) + \left(\sin x \cdot \frac{{\varepsilon}^2}{\cos x}\right))_* + (\left(\frac{{\varepsilon}^2}{{\left(\cos x\right)}^3}\right) * \left({\left(\sin x\right)}^3\right) + \left(\frac{1}{3} \cdot {\varepsilon}^3\right))_*\right) + (\left(\frac{{\varepsilon}^3}{{\left(\cos x\right)}^{4}}\right) * \left({\left(\sin x\right)}^{4}\right) + \left((\left(\frac{\frac{4}{3}}{\cos x}\right) * \left(\frac{\sin x \cdot \sin x}{\frac{\cos x}{{\varepsilon}^3}}\right) + \varepsilon)_*\right))_*\]
      0.2
    7. Applied final simplification

    if 0.07828799f0 < eps

    1. Started with
      \[\tan \left(x + \varepsilon\right) - \tan x\]
      14.6
    2. Using strategy rm
      14.6
    3. Applied tan-cotan to get
      \[\tan \left(x + \varepsilon\right) - \color{red}{\tan x} \leadsto \tan \left(x + \varepsilon\right) - \color{blue}{\frac{1}{\cot x}}\]
      14.5
    4. Applied tan-quot to get
      \[\color{red}{\tan \left(x + \varepsilon\right)} - \frac{1}{\cot x} \leadsto \color{blue}{\frac{\sin \left(x + \varepsilon\right)}{\cos \left(x + \varepsilon\right)}} - \frac{1}{\cot x}\]
      14.6
    5. Applied frac-sub to get
      \[\color{red}{\frac{\sin \left(x + \varepsilon\right)}{\cos \left(x + \varepsilon\right)} - \frac{1}{\cot x}} \leadsto \color{blue}{\frac{\sin \left(x + \varepsilon\right) \cdot \cot x - \cos \left(x + \varepsilon\right) \cdot 1}{\cos \left(x + \varepsilon\right) \cdot \cot x}}\]
      14.6
    6. Applied simplify to get
      \[\frac{\color{red}{\sin \left(x + \varepsilon\right) \cdot \cot x - \cos \left(x + \varepsilon\right) \cdot 1}}{\cos \left(x + \varepsilon\right) \cdot \cot x} \leadsto \frac{\color{blue}{\cot x \cdot \sin \left(x + \varepsilon\right) - \cos \left(x + \varepsilon\right)}}{\cos \left(x + \varepsilon\right) \cdot \cot x}\]
      14.6
    7. Applied taylor to get
      \[\frac{\cot x \cdot \sin \left(x + \varepsilon\right) - \cos \left(x + \varepsilon\right)}{\cos \left(x + \varepsilon\right) \cdot \cot x} \leadsto \frac{\cot x \cdot \sin \left(x + \varepsilon\right) - \cos \left(\frac{1}{x} + \frac{1}{\varepsilon}\right)}{\cos \left(x + \varepsilon\right) \cdot \cot x}\]
      13.8
    8. Taylor expanded around inf to get
      \[\frac{\cot x \cdot \sin \left(x + \varepsilon\right) - \color{red}{\cos \left(\frac{1}{x} + \frac{1}{\varepsilon}\right)}}{\cos \left(x + \varepsilon\right) \cdot \cot x} \leadsto \frac{\cot x \cdot \sin \left(x + \varepsilon\right) - \color{blue}{\cos \left(\frac{1}{x} + \frac{1}{\varepsilon}\right)}}{\cos \left(x + \varepsilon\right) \cdot \cot x}\]
      13.8
    9. Applied simplify to get
      \[\frac{\cot x \cdot \sin \left(x + \varepsilon\right) - \cos \left(\frac{1}{x} + \frac{1}{\varepsilon}\right)}{\cos \left(x + \varepsilon\right) \cdot \cot x} \leadsto \frac{\cot x \cdot \sin \left(x + \varepsilon\right) - \cos \left(\frac{1}{\varepsilon} + \frac{1}{x}\right)}{\cos \left(x + \varepsilon\right) \cdot \cot x}\]
      13.8
    10. 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)))))