\[\tan \left(x + \varepsilon\right) - \tan x\]
Test:
NMSE problem 3.3.2
Bits:
128 bits
Bits error versus x
Bits error versus eps
Time: 21.5 s
Input Error: 17.2
Output Error: 8.7
Log: ⚲
Profile: 🕒
\(\begin{cases} \frac{\cot x}{1} \cdot \left(\tan \left(x + \varepsilon\right) \cdot \tan x\right) - \frac{1}{\cot x} & \text{when } \varepsilon \le -4.7317332f-09 \\ \frac{(\frac{1}{6} * \left(x \cdot \varepsilon\right) + \left(\frac{\varepsilon}{x}\right))_* - \frac{{\varepsilon}^3}{\frac{x}{\frac{1}{6}}}}{\cos \left(\varepsilon + x\right) \cdot \cot x} & \text{when } \varepsilon \le 1.8738475f-09 \\ \tan \left(x + \varepsilon\right) - \frac{\sin x}{\cos x} & \text{otherwise} \end{cases}\)

    if eps < -4.7317332f-09

    1. Started with
      \[\tan \left(x + \varepsilon\right) - \tan x\]
      13.9
    2. Using strategy rm
      13.9
    3. Applied tan-cotan to get
      \[\tan \left(x + \varepsilon\right) - \color{red}{\tan x} \leadsto \tan \left(x + \varepsilon\right) - \color{blue}{\frac{1}{\cot x}}\]
      13.7
    4. Applied tan-cotan to get
      \[\color{red}{\tan \left(x + \varepsilon\right)} - \frac{1}{\cot x} \leadsto \color{blue}{\frac{1}{\cot \left(x + \varepsilon\right)}} - \frac{1}{\cot x}\]
      13.9
    5. Applied frac-sub to get
      \[\color{red}{\frac{1}{\cot \left(x + \varepsilon\right)} - \frac{1}{\cot x}} \leadsto \color{blue}{\frac{1 \cdot \cot x - \cot \left(x + \varepsilon\right) \cdot 1}{\cot \left(x + \varepsilon\right) \cdot \cot x}}\]
      13.9
    6. Applied simplify to get
      \[\frac{\color{red}{1 \cdot \cot x - \cot \left(x + \varepsilon\right) \cdot 1}}{\cot \left(x + \varepsilon\right) \cdot \cot x} \leadsto \frac{\color{blue}{\cot x - \cot \left(\varepsilon + x\right)}}{\cot \left(x + \varepsilon\right) \cdot \cot x}\]
      13.9
    7. Using strategy rm
      13.9
    8. Applied div-sub to get
      \[\color{red}{\frac{\cot x - \cot \left(\varepsilon + x\right)}{\cot \left(x + \varepsilon\right) \cdot \cot x}} \leadsto \color{blue}{\frac{\cot x}{\cot \left(x + \varepsilon\right) \cdot \cot x} - \frac{\cot \left(\varepsilon + x\right)}{\cot \left(x + \varepsilon\right) \cdot \cot x}}\]
      13.9
    9. Applied simplify to get
      \[\frac{\cot x}{\cot \left(x + \varepsilon\right) \cdot \cot x} - \color{red}{\frac{\cot \left(\varepsilon + x\right)}{\cot \left(x + \varepsilon\right) \cdot \cot x}} \leadsto \frac{\cot x}{\cot \left(x + \varepsilon\right) \cdot \cot x} - \color{blue}{\frac{1}{\cot x}}\]
      13.9
    10. Using strategy rm
      13.9
    11. Applied cotan-tan to get
      \[\frac{\cot x}{\cot \left(x + \varepsilon\right) \cdot \color{red}{\cot x}} - \frac{1}{\cot x} \leadsto \frac{\cot x}{\cot \left(x + \varepsilon\right) \cdot \color{blue}{\frac{1}{\tan x}}} - \frac{1}{\cot x}\]
      13.7
    12. Applied cotan-tan to get
      \[\frac{\cot x}{\color{red}{\cot \left(x + \varepsilon\right)} \cdot \frac{1}{\tan x}} - \frac{1}{\cot x} \leadsto \frac{\cot x}{\color{blue}{\frac{1}{\tan \left(x + \varepsilon\right)}} \cdot \frac{1}{\tan x}} - \frac{1}{\cot x}\]
      13.7
    13. Applied frac-times to get
      \[\frac{\cot x}{\color{red}{\frac{1}{\tan \left(x + \varepsilon\right)} \cdot \frac{1}{\tan x}}} - \frac{1}{\cot x} \leadsto \frac{\cot x}{\color{blue}{\frac{1 \cdot 1}{\tan \left(x + \varepsilon\right) \cdot \tan x}}} - \frac{1}{\cot x}\]
      13.7
    14. Applied associate-/r/ to get
      \[\color{red}{\frac{\cot x}{\frac{1 \cdot 1}{\tan \left(x + \varepsilon\right) \cdot \tan x}}} - \frac{1}{\cot x} \leadsto \color{blue}{\frac{\cot x}{1 \cdot 1} \cdot \left(\tan \left(x + \varepsilon\right) \cdot \tan x\right)} - \frac{1}{\cot x}\]
      13.7
    15. Applied simplify to get
      \[\color{red}{\frac{\cot x}{1 \cdot 1}} \cdot \left(\tan \left(x + \varepsilon\right) \cdot \tan x\right) - \frac{1}{\cot x} \leadsto \color{blue}{\frac{\cot x}{1}} \cdot \left(\tan \left(x + \varepsilon\right) \cdot \tan x\right) - \frac{1}{\cot x}\]
      13.7

    if -4.7317332f-09 < eps < 1.8738475f-09

    1. Started with
      \[\tan \left(x + \varepsilon\right) - \tan x\]
      21.5
    2. Using strategy rm
      21.5
    3. Applied tan-cotan to get
      \[\tan \left(x + \varepsilon\right) - \color{red}{\tan x} \leadsto \tan \left(x + \varepsilon\right) - \color{blue}{\frac{1}{\cot x}}\]
      21.4
    4. Applied tan-quot to get
      \[\color{red}{\tan \left(x + \varepsilon\right)} - \frac{1}{\cot x} \leadsto \color{blue}{\frac{\sin \left(x + \varepsilon\right)}{\cos \left(x + \varepsilon\right)}} - \frac{1}{\cot x}\]
      21.4
    5. Applied frac-sub to get
      \[\color{red}{\frac{\sin \left(x + \varepsilon\right)}{\cos \left(x + \varepsilon\right)} - \frac{1}{\cot x}} \leadsto \color{blue}{\frac{\sin \left(x + \varepsilon\right) \cdot \cot x - \cos \left(x + \varepsilon\right) \cdot 1}{\cos \left(x + \varepsilon\right) \cdot \cot x}}\]
      21.4
    6. Applied simplify to get
      \[\frac{\color{red}{\sin \left(x + \varepsilon\right) \cdot \cot x - \cos \left(x + \varepsilon\right) \cdot 1}}{\cos \left(x + \varepsilon\right) \cdot \cot x} \leadsto \frac{\color{blue}{\cot x \cdot \sin \left(x + \varepsilon\right) - \cos \left(x + \varepsilon\right)}}{\cos \left(x + \varepsilon\right) \cdot \cot x}\]
      21.4
    7. Applied taylor to get
      \[\frac{\cot x \cdot \sin \left(x + \varepsilon\right) - \cos \left(x + \varepsilon\right)}{\cos \left(x + \varepsilon\right) \cdot \cot x} \leadsto \frac{\left(\frac{\varepsilon}{x} + \frac{1}{6} \cdot \left(\varepsilon \cdot x\right)\right) - \frac{1}{6} \cdot \frac{{\varepsilon}^{3}}{x}}{\cos \left(x + \varepsilon\right) \cdot \cot x}\]
      0.3
    8. Taylor expanded around 0 to get
      \[\frac{\color{red}{\left(\frac{\varepsilon}{x} + \frac{1}{6} \cdot \left(\varepsilon \cdot x\right)\right) - \frac{1}{6} \cdot \frac{{\varepsilon}^{3}}{x}}}{\cos \left(x + \varepsilon\right) \cdot \cot x} \leadsto \frac{\color{blue}{\left(\frac{\varepsilon}{x} + \frac{1}{6} \cdot \left(\varepsilon \cdot x\right)\right) - \frac{1}{6} \cdot \frac{{\varepsilon}^{3}}{x}}}{\cos \left(x + \varepsilon\right) \cdot \cot x}\]
      0.3
    9. Applied simplify to get
      \[\frac{\left(\frac{\varepsilon}{x} + \frac{1}{6} \cdot \left(\varepsilon \cdot x\right)\right) - \frac{1}{6} \cdot \frac{{\varepsilon}^{3}}{x}}{\cos \left(x + \varepsilon\right) \cdot \cot x} \leadsto \frac{(\frac{1}{6} * \left(x \cdot \varepsilon\right) + \left(\frac{\varepsilon}{x}\right))_* - \frac{{\varepsilon}^3}{\frac{x}{\frac{1}{6}}}}{\cos \left(\varepsilon + x\right) \cdot \cot x}\]
      0.3
    10. Applied final simplification

    if 1.8738475f-09 < eps

    1. Started with
      \[\tan \left(x + \varepsilon\right) - \tan x\]
      14.9
    2. Using strategy rm
      14.9
    3. Applied tan-quot to get
      \[\tan \left(x + \varepsilon\right) - \color{red}{\tan x} \leadsto \tan \left(x + \varepsilon\right) - \color{blue}{\frac{\sin x}{\cos x}}\]
      14.7
  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)))))