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>  # macheps.r                         J F Monahan, June 2001
>  #                                   revised August 2007, 2009
>  # compute machine epsilon
>  #   *** these results on R version 2.9.0 on Linux       ***
>  #   *** Vista version 2.9.1 gives old 2**(-53)+2**(-63) ***
>  #
>  # first try some values to get in ballpark
>   eps <- 2**(-56)
>   "2**(-56)"
[1] "2**(-56)"
>   eps
[1] 1.387779e-17
>   "Is 1+eps bigger than 1 for eps = j*2**(-56), j = 1 to 16"
[1] "Is 1+eps bigger than 1 for eps = j*2**(-56), j = 1 to 16"
>   (1!=(1+eps*(1:16)))      # F if different, T if not 
 [1] FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE  TRUE  TRUE  TRUE  TRUE
[13]  TRUE  TRUE  TRUE  TRUE
>  #
>  # look at representations with sprintf and "%a"
>   sprintf("%a",(1+eps*(1:16)))
 [1] "0x1p+0"               "0x1p+0"               "0x1p+0"              
 [4] "0x1p+0"               "0x1p+0"               "0x1p+0"              
 [7] "0x1p+0"               "0x1p+0"               "0x1.0000000000001p+0"
[10] "0x1.0000000000001p+0" "0x1.0000000000001p+0" "0x1.0000000000001p+0"
[13] "0x1.0000000000001p+0" "0x1.0000000000001p+0" "0x1.0000000000001p+0"
[16] "0x1.0000000000001p+0"
>  #
>  # at this point it's around 8*2**(-56) or 2**(-53)
>  #  or ... 64*2**(-59), so try 60 to 70
>   "Is 1+eps bigger than 1 for eps = j*2**(-59), j = 60 to 70"
[1] "Is 1+eps bigger than 1 for eps = j*2**(-59), j = 60 to 70"
>   eps <- 2**(-59)
>   (1!=(1+eps*(60:70)))
 [1] FALSE FALSE FALSE FALSE FALSE  TRUE  TRUE  TRUE  TRUE  TRUE  TRUE
>  #
>  # look at representations 
>   sprintf("%a",(1+eps*(60:70)))
 [1] "0x1p+0"               "0x1p+0"               "0x1p+0"              
 [4] "0x1p+0"               "0x1p+0"               "0x1.0000000000001p+0"
 [7] "0x1.0000000000001p+0" "0x1.0000000000001p+0" "0x1.0000000000001p+0"
[10] "0x1.0000000000001p+0" "0x1.0000000000001p+0"
>  #
>  # another try
>   eps <- 2**(-53)          # first guess + smaller
>   (1!=(1+eps*(1 + 2**(-51:-55))))
[1]  TRUE  TRUE FALSE FALSE FALSE
>  #
>  # look at representations
>   sprintf("%a",(1+eps*(1 + 2**(-51:-55))))
[1] "0x1.0000000000001p+0" "0x1.0000000000001p+0" "0x1p+0"              
[4] "0x1p+0"               "0x1p+0"              
>  #
>  # this is different -- the real machine epsilon
>   "Is 1+eps bigger than 1 for eps = 2**(-53) + 2**(-105)"
[1] "Is 1+eps bigger than 1 for eps = 2**(-53) + 2**(-105)"
>   eps <- 2**(-53) + 2**(-105)
>   eps
[1] 1.110223e-16
>  # this should do it
>   sprintf("%a",eps)
[1] "0x1.0000000000001p-53"
>   (1!=(1+eps))
[1] TRUE
>  #
>  # this is the same -- just a hair too small
>   "Is 1+eps bigger than 1 for eps = 2**(-53) + 2**(-106)"
[1] "Is 1+eps bigger than 1 for eps = 2**(-53) + 2**(-106)"
>   eps <- 2**(-53) + 2**(-106)
>   eps
[1] 1.110223e-16
>  # this should be too much 
>   sprintf("%a",eps)
[1] "0x1p-53"
>  # test it
>   (1!=(1+eps))
[1] FALSE
>  #
>   rm(list=ls())
>   q()