\[\tan \left(x + \varepsilon\right) - \tan x\]
Test:
NMSE problem 3.3.2
Bits:
128 bits
Bits error versus x
Bits error versus eps
Time: 9.2 s
Input Error: 16.9
Output Error: 17.4
Log: ⚲
Profile: 🕒
\(\frac{{\left(\log_* (1 + (e^{\tan \left(x + \varepsilon\right)} - 1)^*)\right)}^2 - {\left(\frac{1}{\cot x}\right)}^2}{\frac{1}{\cot x} + \tan \left(x + \varepsilon\right)}\)
  1. Started with
    \[\tan \left(x + \varepsilon\right) - \tan x\]
    16.9
  2. Using strategy rm
    16.9
  3. Applied tan-cotan to get
    \[\tan \left(x + \varepsilon\right) - \color{red}{\tan x} \leadsto \tan \left(x + \varepsilon\right) - \color{blue}{\frac{1}{\cot x}}\]
    17.0
  4. Using strategy rm
    17.0
  5. Applied log1p-expm1-u to get
    \[\color{red}{\tan \left(x + \varepsilon\right)} - \frac{1}{\cot x} \leadsto \color{blue}{\log_* (1 + (e^{\tan \left(x + \varepsilon\right)} - 1)^*)} - \frac{1}{\cot x}\]
    17.3
  6. Using strategy rm
    17.3
  7. Applied flip-- to get
    \[\color{red}{\log_* (1 + (e^{\tan \left(x + \varepsilon\right)} - 1)^*) - \frac{1}{\cot x}} \leadsto \color{blue}{\frac{{\left(\log_* (1 + (e^{\tan \left(x + \varepsilon\right)} - 1)^*)\right)}^2 - {\left(\frac{1}{\cot x}\right)}^2}{\log_* (1 + (e^{\tan \left(x + \varepsilon\right)} - 1)^*) + \frac{1}{\cot x}}}\]
    17.3
  8. Applied simplify to get
    \[\frac{{\left(\log_* (1 + (e^{\tan \left(x + \varepsilon\right)} - 1)^*)\right)}^2 - {\left(\frac{1}{\cot x}\right)}^2}{\color{red}{\log_* (1 + (e^{\tan \left(x + \varepsilon\right)} - 1)^*) + \frac{1}{\cot x}}} \leadsto \frac{{\left(\log_* (1 + (e^{\tan \left(x + \varepsilon\right)} - 1)^*)\right)}^2 - {\left(\frac{1}{\cot x}\right)}^2}{\color{blue}{\frac{1}{\cot x} + \tan \left(x + \varepsilon\right)}}\]
    17.4

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)))))