\[\tan \left(x + \varepsilon\right) - \tan x\]
Test:
NMSE problem 3.3.2
Bits:
128 bits
Bits error versus x
Bits error versus eps
Time: 9.4 s
Input Error: 17.0
Output Error: 12.6
Log: ⚲
Profile: 🕒
\(\begin{cases} \frac{1}{\cot \left(x + \varepsilon\right)} - \tan x & \text{when } \varepsilon \le -4.9196116f-11 \\ \left(\varepsilon + \left(x \cdot x\right) \cdot {\varepsilon}^3\right) + {\varepsilon}^{4} \cdot {x}^3 & \text{when } \varepsilon \le 5.877158f-09 \\ \tan \left(x + \varepsilon\right) - {\left(\frac{\sqrt[3]{\sin x}}{\sqrt[3]{\cos x}}\right)}^3 & \text{otherwise} \end{cases}\)

    if eps < -4.9196116f-11

    1. Started with
      \[\tan \left(x + \varepsilon\right) - \tan x\]
      14.6
    2. Using strategy rm
      14.6
    3. Applied tan-cotan to get
      \[\color{red}{\tan \left(x + \varepsilon\right)} - \tan x \leadsto \color{blue}{\frac{1}{\cot \left(x + \varepsilon\right)}} - \tan x\]
      14.4

    if -4.9196116f-11 < eps < 5.877158f-09

    1. Started with
      \[\tan \left(x + \varepsilon\right) - \tan x\]
      21.1
    2. Applied taylor to get
      \[\tan \left(x + \varepsilon\right) - \tan x \leadsto \varepsilon + \left({\varepsilon}^{3} \cdot {x}^2 + {\varepsilon}^{4} \cdot {x}^{3}\right)\]
      9.8
    3. Taylor expanded around 0 to get
      \[\color{red}{\varepsilon + \left({\varepsilon}^{3} \cdot {x}^2 + {\varepsilon}^{4} \cdot {x}^{3}\right)} \leadsto \color{blue}{\varepsilon + \left({\varepsilon}^{3} \cdot {x}^2 + {\varepsilon}^{4} \cdot {x}^{3}\right)}\]
      9.8
    4. Applied simplify to get
      \[\color{red}{\varepsilon + \left({\varepsilon}^{3} \cdot {x}^2 + {\varepsilon}^{4} \cdot {x}^{3}\right)} \leadsto \color{blue}{\left(\varepsilon + \left(x \cdot x\right) \cdot {\varepsilon}^3\right) + {\varepsilon}^{4} \cdot {x}^3}\]
      9.8

    if 5.877158f-09 < eps

    1. Started with
      \[\tan \left(x + \varepsilon\right) - \tan x\]
      14.3
    2. Using strategy rm
      14.3
    3. Applied tan-quot to get
      \[\tan \left(x + \varepsilon\right) - \color{red}{\tan x} \leadsto \tan \left(x + \varepsilon\right) - \color{blue}{\frac{\sin x}{\cos x}}\]
      14.1
    4. Using strategy rm
      14.1
    5. Applied add-cube-cbrt to get
      \[\tan \left(x + \varepsilon\right) - \frac{\sin x}{\color{red}{\cos x}} \leadsto \tan \left(x + \varepsilon\right) - \frac{\sin x}{\color{blue}{{\left(\sqrt[3]{\cos x}\right)}^3}}\]
      14.1
    6. Applied add-cube-cbrt to get
      \[\tan \left(x + \varepsilon\right) - \frac{\color{red}{\sin x}}{{\left(\sqrt[3]{\cos x}\right)}^3} \leadsto \tan \left(x + \varepsilon\right) - \frac{\color{blue}{{\left(\sqrt[3]{\sin x}\right)}^3}}{{\left(\sqrt[3]{\cos x}\right)}^3}\]
      14.1
    7. Applied cube-undiv to get
      \[\tan \left(x + \varepsilon\right) - \color{red}{\frac{{\left(\sqrt[3]{\sin x}\right)}^3}{{\left(\sqrt[3]{\cos x}\right)}^3}} \leadsto \tan \left(x + \varepsilon\right) - \color{blue}{{\left(\frac{\sqrt[3]{\sin x}}{\sqrt[3]{\cos x}}\right)}^3}\]
      14.1
  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)))))