\[\tan \left(x + \varepsilon\right) - \tan x\]
Test:
NMSE problem 3.3.2
Bits:
128 bits
Bits error versus x
Bits error versus eps
Time: 7.2 s
Input Error: 16.9
Output Error: 13.0
Log: ⚲
Profile: 🕒
\(\begin{cases} \tan \left(x + \varepsilon\right) - {\left(\sqrt[3]{\tan x}\right)}^3 & \text{when } \varepsilon \le -2.002236f-10 \\ \left(\varepsilon + \left(x \cdot x\right) \cdot {\varepsilon}^3\right) + {\varepsilon}^{4} \cdot {x}^3 & \text{when } \varepsilon \le 5.0320875f-13 \\ \tan \left(x + \varepsilon\right) - {\left(\sqrt[3]{\tan x}\right)}^3 & \text{otherwise} \end{cases}\)

    if eps < -2.002236f-10

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

    if -2.002236f-10 < eps < 5.0320875f-13

    1. Started with
      \[\tan \left(x + \varepsilon\right) - \tan x\]
      21.2
    2. Applied taylor to get
      \[\tan \left(x + \varepsilon\right) - \tan x \leadsto \varepsilon + \left({\varepsilon}^{3} \cdot {x}^2 + {\varepsilon}^{4} \cdot {x}^{3}\right)\]
      10.0
    3. Taylor expanded around 0 to get
      \[\color{red}{\varepsilon + \left({\varepsilon}^{3} \cdot {x}^2 + {\varepsilon}^{4} \cdot {x}^{3}\right)} \leadsto \color{blue}{\varepsilon + \left({\varepsilon}^{3} \cdot {x}^2 + {\varepsilon}^{4} \cdot {x}^{3}\right)}\]
      10.0
    4. Applied simplify to get
      \[\color{red}{\varepsilon + \left({\varepsilon}^{3} \cdot {x}^2 + {\varepsilon}^{4} \cdot {x}^{3}\right)} \leadsto \color{blue}{\left(\varepsilon + \left(x \cdot x\right) \cdot {\varepsilon}^3\right) + {\varepsilon}^{4} \cdot {x}^3}\]
      10.0

    if 5.0320875f-13 < eps

    1. Started with
      \[\tan \left(x + \varepsilon\right) - \tan x\]
      15.1
    2. Using strategy rm
      15.1
    3. Applied add-cube-cbrt to get
      \[\tan \left(x + \varepsilon\right) - \color{red}{\tan x} \leadsto \tan \left(x + \varepsilon\right) - \color{blue}{{\left(\sqrt[3]{\tan x}\right)}^3}\]
      14.8
  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)))))