\[\tan \left(x + \varepsilon\right) - \tan x\]
Test:
NMSE problem 3.3.2
Bits:
128 bits
Bits error versus x
Bits error versus eps
Time: 9.7 s
Input Error: 16.8
Output Error: 9.0
Log: ⚲
Profile: 🕒
\(\begin{cases} \frac{\frac{\cos x \cdot \tan \left(x + \varepsilon\right) - \sin x}{\sin x \cdot \tan \left(\varepsilon + x\right)}}{\cot \left(x + \varepsilon\right) \cdot \cot x} & \text{when } \varepsilon \le -4.7317332f-09 \\ \left(\varepsilon + x\right) \cdot \left(x \cdot \varepsilon\right) + \varepsilon & \text{when } \varepsilon \le 1.8738475f-09 \\ \tan \left(x + \varepsilon\right) - \frac{\sin x}{\cos x} & \text{otherwise} \end{cases}\)

    if eps < -4.7317332f-09

    1. Started with
      \[\tan \left(x + \varepsilon\right) - \tan x\]
      14.0
    2. Using strategy rm
      14.0
    3. Applied tan-cotan to get
      \[\tan \left(x + \varepsilon\right) - \color{red}{\tan x} \leadsto \tan \left(x + \varepsilon\right) - \color{blue}{\frac{1}{\cot x}}\]
      13.8
    4. Applied tan-cotan to get
      \[\color{red}{\tan \left(x + \varepsilon\right)} - \frac{1}{\cot x} \leadsto \color{blue}{\frac{1}{\cot \left(x + \varepsilon\right)}} - \frac{1}{\cot x}\]
      14.0
    5. Applied frac-sub to get
      \[\color{red}{\frac{1}{\cot \left(x + \varepsilon\right)} - \frac{1}{\cot x}} \leadsto \color{blue}{\frac{1 \cdot \cot x - \cot \left(x + \varepsilon\right) \cdot 1}{\cot \left(x + \varepsilon\right) \cdot \cot x}}\]
      14.0
    6. Applied simplify to get
      \[\frac{\color{red}{1 \cdot \cot x - \cot \left(x + \varepsilon\right) \cdot 1}}{\cot \left(x + \varepsilon\right) \cdot \cot x} \leadsto \frac{\color{blue}{\cot x - \cot \left(\varepsilon + x\right)}}{\cot \left(x + \varepsilon\right) \cdot \cot x}\]
      14.0
    7. Using strategy rm
      14.0
    8. Applied cotan-tan to get
      \[\frac{\cot x - \color{red}{\cot \left(\varepsilon + x\right)}}{\cot \left(x + \varepsilon\right) \cdot \cot x} \leadsto \frac{\cot x - \color{blue}{\frac{1}{\tan \left(\varepsilon + x\right)}}}{\cot \left(x + \varepsilon\right) \cdot \cot x}\]
      13.8
    9. Applied cotan-quot to get
      \[\frac{\color{red}{\cot x} - \frac{1}{\tan \left(\varepsilon + x\right)}}{\cot \left(x + \varepsilon\right) \cdot \cot x} \leadsto \frac{\color{blue}{\frac{\cos x}{\sin x}} - \frac{1}{\tan \left(\varepsilon + x\right)}}{\cot \left(x + \varepsilon\right) \cdot \cot x}\]
      13.8
    10. Applied frac-sub to get
      \[\frac{\color{red}{\frac{\cos x}{\sin x} - \frac{1}{\tan \left(\varepsilon + x\right)}}}{\cot \left(x + \varepsilon\right) \cdot \cot x} \leadsto \frac{\color{blue}{\frac{\cos x \cdot \tan \left(\varepsilon + x\right) - \sin x \cdot 1}{\sin x \cdot \tan \left(\varepsilon + x\right)}}}{\cot \left(x + \varepsilon\right) \cdot \cot x}\]
      13.8
    11. Applied simplify to get
      \[\frac{\frac{\color{red}{\cos x \cdot \tan \left(\varepsilon + x\right) - \sin x \cdot 1}}{\sin x \cdot \tan \left(\varepsilon + x\right)}}{\cot \left(x + \varepsilon\right) \cdot \cot x} \leadsto \frac{\frac{\color{blue}{\cos x \cdot \tan \left(x + \varepsilon\right) - \sin x}}{\sin x \cdot \tan \left(\varepsilon + x\right)}}{\cot \left(x + \varepsilon\right) \cdot \cot x}\]
      13.8

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

    1. Started with
      \[\tan \left(x + \varepsilon\right) - \tan x\]
      20.8
    2. Using strategy rm
      20.8
    3. Applied tan-cotan to get
      \[\tan \left(x + \varepsilon\right) - \color{red}{\tan x} \leadsto \tan \left(x + \varepsilon\right) - \color{blue}{\frac{1}{\cot x}}\]
      20.8
    4. Applied tan-cotan to get
      \[\color{red}{\tan \left(x + \varepsilon\right)} - \frac{1}{\cot x} \leadsto \color{blue}{\frac{1}{\cot \left(x + \varepsilon\right)}} - \frac{1}{\cot x}\]
      20.8
    5. Applied frac-sub to get
      \[\color{red}{\frac{1}{\cot \left(x + \varepsilon\right)} - \frac{1}{\cot x}} \leadsto \color{blue}{\frac{1 \cdot \cot x - \cot \left(x + \varepsilon\right) \cdot 1}{\cot \left(x + \varepsilon\right) \cdot \cot x}}\]
      20.8
    6. Applied simplify to get
      \[\frac{\color{red}{1 \cdot \cot x - \cot \left(x + \varepsilon\right) \cdot 1}}{\cot \left(x + \varepsilon\right) \cdot \cot x} \leadsto \frac{\color{blue}{\cot x - \cot \left(\varepsilon + x\right)}}{\cot \left(x + \varepsilon\right) \cdot \cot x}\]
      20.8
    7. Applied taylor to get
      \[\frac{\cot x - \cot \left(\varepsilon + x\right)}{\cot \left(x + \varepsilon\right) \cdot \cot x} \leadsto \varepsilon + \left(\varepsilon \cdot {x}^2 + {\varepsilon}^2 \cdot x\right)\]
      4.9
    8. Taylor expanded around 0 to get
      \[\color{red}{\varepsilon + \left(\varepsilon \cdot {x}^2 + {\varepsilon}^2 \cdot x\right)} \leadsto \color{blue}{\varepsilon + \left(\varepsilon \cdot {x}^2 + {\varepsilon}^2 \cdot x\right)}\]
      4.9
    9. Applied simplify to get
      \[\color{red}{\varepsilon + \left(\varepsilon \cdot {x}^2 + {\varepsilon}^2 \cdot x\right)} \leadsto \color{blue}{\left(\varepsilon + x\right) \cdot \left(x \cdot \varepsilon\right) + \varepsilon}\]
      0.1

    if 1.8738475f-09 < eps

    1. Started with
      \[\tan \left(x + \varepsilon\right) - \tan x\]
      14.8
    2. Using strategy rm
      14.8
    3. Applied tan-quot to get
      \[\tan \left(x + \varepsilon\right) - \color{red}{\tan x} \leadsto \tan \left(x + \varepsilon\right) - \color{blue}{\frac{\sin x}{\cos x}}\]
      14.6
  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)))))