\[\tan \left(x + \varepsilon\right) - \tan x\]
Test:
NMSE problem 3.3.2
Bits:
128 bits
Bits error versus x
Bits error versus eps
Time: 9.0 s
Input Error: 16.7
Output Error: 12.3
Log: ⚲
Profile: 🕒
\(\begin{cases} \tan \left(x + \varepsilon\right) - {\left(\sqrt[3]{\tan x}\right)}^3 & \text{when } \varepsilon \le -0.00010876831f0 \\ \left(\varepsilon + \left(x \cdot x\right) \cdot {\varepsilon}^3\right) + {\varepsilon}^{4} \cdot {x}^3 & \text{when } \varepsilon \le 3.6911058f-10 \\ \sqrt[3]{{\left(\frac{1}{\cot \left(x + \varepsilon\right)}\right)}^3} - \tan x & \text{otherwise} \end{cases}\)

    if eps < -0.00010876831f0

    1. Started with
      \[\tan \left(x + \varepsilon\right) - \tan x\]
      14.1
    2. Using strategy rm
      14.1
    3. Applied add-cube-cbrt to get
      \[\tan \left(x + \varepsilon\right) - \color{red}{\tan x} \leadsto \tan \left(x + \varepsilon\right) - \color{blue}{{\left(\sqrt[3]{\tan x}\right)}^3}\]
      13.9

    if -0.00010876831f0 < eps < 3.6911058f-10

    1. Started with
      \[\tan \left(x + \varepsilon\right) - \tan x\]
      20.2
    2. Applied taylor to get
      \[\tan \left(x + \varepsilon\right) - \tan x \leadsto \varepsilon + \left({\varepsilon}^{3} \cdot {x}^2 + {\varepsilon}^{4} \cdot {x}^{3}\right)\]
      9.2
    3. Taylor expanded around 0 to get
      \[\color{red}{\varepsilon + \left({\varepsilon}^{3} \cdot {x}^2 + {\varepsilon}^{4} \cdot {x}^{3}\right)} \leadsto \color{blue}{\varepsilon + \left({\varepsilon}^{3} \cdot {x}^2 + {\varepsilon}^{4} \cdot {x}^{3}\right)}\]
      9.2
    4. Applied simplify to get
      \[\color{red}{\varepsilon + \left({\varepsilon}^{3} \cdot {x}^2 + {\varepsilon}^{4} \cdot {x}^{3}\right)} \leadsto \color{blue}{\left(\varepsilon + \left(x \cdot x\right) \cdot {\varepsilon}^3\right) + {\varepsilon}^{4} \cdot {x}^3}\]
      9.2

    if 3.6911058f-10 < 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
      \[\color{red}{\tan \left(x + \varepsilon\right)} - \tan x \leadsto \color{blue}{\frac{1}{\cot \left(x + \varepsilon\right)}} - \tan x\]
      14.3
    4. Using strategy rm
      14.3
    5. Applied add-cbrt-cube to get
      \[\color{red}{\frac{1}{\cot \left(x + \varepsilon\right)}} - \tan x \leadsto \color{blue}{\sqrt[3]{{\left(\frac{1}{\cot \left(x + \varepsilon\right)}\right)}^3}} - \tan x\]
      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)))))