\[\tan \left(x + \varepsilon\right) - \tan x\]
Test:
NMSE problem 3.3.2
Bits:
128 bits
Bits error versus x
Bits error versus eps
Time: 10.4 s
Input Error: 16.3
Output Error: 11.8
Log: ⚲
Profile: 🕒
\(\begin{cases} {\left(\sqrt[3]{\tan \left(x + \varepsilon\right)}\right)}^3 - \tan x & \text{when } \varepsilon \le -0.00030713418f0 \\ \varepsilon + \left({\varepsilon}^{3} \cdot {x}^2 + {\varepsilon}^{4} \cdot {x}^{3}\right) & \text{when } \varepsilon \le 3.6169324f-06 \\ \frac{\cos x - \cot \left(\varepsilon + x\right) \cdot \sin x}{\cot \left(x + \varepsilon\right) \cdot \cos x} & \text{otherwise} \end{cases}\)

    if eps < -0.00030713418f0

    1. Started with
      \[\tan \left(x + \varepsilon\right) - \tan x\]
      14.0
    2. Using strategy rm
      14.0
    3. Applied add-cube-cbrt to get
      \[\color{red}{\tan \left(x + \varepsilon\right)} - \tan x \leadsto \color{blue}{{\left(\sqrt[3]{\tan \left(x + \varepsilon\right)}\right)}^3} - \tan x\]
      14.1

    if -0.00030713418f0 < eps < 3.6169324f-06

    1. Started with
      \[\tan \left(x + \varepsilon\right) - \tan x\]
      19.6
    2. Applied taylor to get
      \[\tan \left(x + \varepsilon\right) - \tan x \leadsto \varepsilon + \left({\varepsilon}^{3} \cdot {x}^2 + {\varepsilon}^{4} \cdot {x}^{3}\right)\]
      9.0
    3. Taylor expanded around 0 to get
      \[\color{red}{\varepsilon + \left({\varepsilon}^{3} \cdot {x}^2 + {\varepsilon}^{4} \cdot {x}^{3}\right)} \leadsto \color{blue}{\varepsilon + \left({\varepsilon}^{3} \cdot {x}^2 + {\varepsilon}^{4} \cdot {x}^{3}\right)}\]
      9.0

    if 3.6169324f-06 < eps

    1. Started with
      \[\tan \left(x + \varepsilon\right) - \tan x\]
      13.7
    2. Using strategy rm
      13.7
    3. Applied tan-quot to get
      \[\tan \left(x + \varepsilon\right) - \color{red}{\tan x} \leadsto \tan \left(x + \varepsilon\right) - \color{blue}{\frac{\sin x}{\cos x}}\]
      13.5
    4. Applied tan-cotan to get
      \[\color{red}{\tan \left(x + \varepsilon\right)} - \frac{\sin x}{\cos x} \leadsto \color{blue}{\frac{1}{\cot \left(x + \varepsilon\right)}} - \frac{\sin x}{\cos x}\]
      13.5
    5. Applied frac-sub to get
      \[\color{red}{\frac{1}{\cot \left(x + \varepsilon\right)} - \frac{\sin x}{\cos x}} \leadsto \color{blue}{\frac{1 \cdot \cos x - \cot \left(x + \varepsilon\right) \cdot \sin x}{\cot \left(x + \varepsilon\right) \cdot \cos x}}\]
      13.5
    6. Applied simplify to get
      \[\frac{\color{red}{1 \cdot \cos x - \cot \left(x + \varepsilon\right) \cdot \sin x}}{\cot \left(x + \varepsilon\right) \cdot \cos x} \leadsto \frac{\color{blue}{\cos x - \cot \left(\varepsilon + x\right) \cdot \sin x}}{\cot \left(x + \varepsilon\right) \cdot \cos x}\]
      13.5
  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)))))