\[\tan \left(x + \varepsilon\right) - \tan x\]
Test:
NMSE problem 3.3.2
Bits:
128 bits
Bits error versus x
Bits error versus eps
Time: 7.0 s
Input Error: 16.8
Output Error: 16.9
Log: ⚲
Profile: 🕒
\(\frac{{\left(\tan \left(x + \varepsilon\right)\right)}^2 - {\left(\log_* (1 + (e^{\tan x} - 1)^*)\right)}^2}{\tan x + \tan \left(x + \varepsilon\right)}\)
  1. Started with
    \[\tan \left(x + \varepsilon\right) - \tan x\]
    16.8
  2. Using strategy rm
    16.8
  3. Applied log1p-expm1-u to get
    \[\tan \left(x + \varepsilon\right) - \color{red}{\tan x} \leadsto \tan \left(x + \varepsilon\right) - \color{blue}{\log_* (1 + (e^{\tan x} - 1)^*)}\]
    16.9
  4. Using strategy rm
    16.9
  5. Applied flip-- to get
    \[\color{red}{\tan \left(x + \varepsilon\right) - \log_* (1 + (e^{\tan x} - 1)^*)} \leadsto \color{blue}{\frac{{\left(\tan \left(x + \varepsilon\right)\right)}^2 - {\left(\log_* (1 + (e^{\tan x} - 1)^*)\right)}^2}{\tan \left(x + \varepsilon\right) + \log_* (1 + (e^{\tan x} - 1)^*)}}\]
    16.9
  6. Applied simplify to get
    \[\frac{{\left(\tan \left(x + \varepsilon\right)\right)}^2 - {\left(\log_* (1 + (e^{\tan x} - 1)^*)\right)}^2}{\color{red}{\tan \left(x + \varepsilon\right) + \log_* (1 + (e^{\tan x} - 1)^*)}} \leadsto \frac{{\left(\tan \left(x + \varepsilon\right)\right)}^2 - {\left(\log_* (1 + (e^{\tan x} - 1)^*)\right)}^2}{\color{blue}{\tan x + \tan \left(x + \varepsilon\right)}}\]
    16.9

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)))))