[R] Question about R mantissa and number of bits

[Lafaye de Micheaux]

Dear all,

I think that every number x in R can be represented in floating point 
arithmetic as:
x = (-1)^s (1+f) 2^(e-1023)
where s is coded on 1 bit, e (positive integer) is coded on 11 bits, and 
f (real in [0,1)) is coded on 52 bits.
Am I right?

We have f=\sum_{i=1}^{52} k_i 2^{-i} for some values k_i in {0,1}.

If this is the case (for the 52 bits), we should have:

The number next to 2^150 should be (-1)^02^150(1+2^(-52))=2^150+2^98
I can check this:
 > a <- 2^150; b <- a + 2^97; b == a
[1] TRUE
 > a <- 2^150; b <- a + 2^98; b == a
[1] FALSE

So it seems that the mantissa is really coded on 52 bits.

But now, if I issue the following commands (using some functions 
provided below to translate from decimal to binary):
 > dec2bin(0.1,52)
[1] "0.0001100110011001100110011001100110011001100110011001"
 > 
formatC(sum(as.numeric(strsplit(dec2bin(0.1,52),"")[[1]][-(1:2)])*2^(-(1:52))),50)
[1] "0.099999999999999866773237044981215149164199829101562"
 > formatC(0.1,50)
[1] "0.1000000000000000055511151231257827021181583404541"
 > 
formatC(sum(as.numeric(strsplit(dec2bin(0.1,55),"")[[1]][-(1:2)])*2^(-(1:55))),50)
[1] "0.1000000000000000055511151231257827021181583404541"
 > formatC(0.1,50)
[1] "0.1000000000000000055511151231257827021181583404541"

So now, using formatC() it seems that f is coded on 55 bits!

Do you have an explanation for this fact?

Many thanks!

Pierre


dec2bin.ent <- function(x) {
   as.integer(paste(rev(as.integer(intToBits(x))), collapse=""))
}

dec2bin.frac <- function(x,prec=52) {
  res <- rep(NA,prec)
  for (i in 1:prec) {
   res[i] <- as.integer(x*2)
   x <- (x*2) %% 1
  }
  return(paste(res,collapse=""))
}

dec2bin <- function(x,prec=52) {
  x <- as.character(x)
  res <- strsplit(x,".",fixed=TRUE)[[1]]
  return(paste(dec2bin.ent(as.numeric(res[1])),dec2bin.frac(as.numeric(paste("0.",res[2],sep="")),prec),sep="."))
}


-- 
Pierre Lafaye de Micheaux

Adresse courrier:
Département de Mathématiques et Statistique
Université de Montréal
CP 6128, succ. Centre-ville
Montréal, Québec H3C 3J7
CANADA

Adresse physique:
Département de Mathématiques et Statistique
Bureau 4249, Pavillon André-Aisenstadt
2920, chemin de la Tour
Montréal, Québec H3T 1J4
CANADA

Tél.: (00-1) 514-343-6607 / Fax: (00-1) 514-343-5700
lafaye at dms.umontreal.ca
http://www.biostatisticien.eu