# [R] Question about R mantissa and number of bits

Nordlund, Dan (DSHS/RDA) NordlDJ at dshs.wa.gov
Tue Nov 8 21:29:17 CET 2011

I am not going through all of your code to understand what you are trying to demonstrate.  R uses the IEEE Standard 754 for Floating Point Numbers.  There is a sign bit, 11 bits for the exponent, and 52 bits for the mantissa.  Because the standard normalizes the mantissa you get an extra bit of precision; i.e. you 53 bits of precision stored in 52 bits.  You might want to read the following

http://www.cse.msu.edu/~cse320/Documents/FloatingPoint.pdf

Dan

Daniel J. Nordlund
Washington State Department of Social and Health Services
Planning, Performance, and Accountability
Research and Data Analysis Division
Olympia, WA 98504-5204

> -----Original Message-----
> From: r-help-bounces at r-project.org [mailto:r-help-bounces at r-
> project.org] On Behalf Of Lafaye de Micheaux
> Sent: Tuesday, November 08, 2011 10:49 AM
> To: r-help at r-project.org
> Subject: [R] Question about R mantissa and number of bits
>
> 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
>  TRUE
>  > a <- 2^150; b <- a + 2^98; b == a
>  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)
>  "0.0001100110011001100110011001100110011001100110011001"
>  >
> formatC(sum(as.numeric(strsplit(dec2bin(0.1,52),"")[][-(1:2)])*2^(-
> (1:52))),50)
>  "0.099999999999999866773237044981215149164199829101562"
>  > formatC(0.1,50)
>  "0.1000000000000000055511151231257827021181583404541"
>  >
> formatC(sum(as.numeric(strsplit(dec2bin(0.1,55),"")[][-(1:2)])*2^(-
> (1:55))),50)
>  "0.1000000000000000055511151231257827021181583404541"
>  > formatC(0.1,50)
>  "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)[]
>
> return(paste(dec2bin.ent(as.numeric(res)),dec2bin.frac(as.numeric(pa
> ste("0.",res,sep="")),prec),sep="."))
> }
>
>
> --
> Pierre Lafaye de Micheaux
>
> Département de Mathématiques et Statistique
> Université de Montréal
> CP 6128, succ. Centre-ville
> Montréal, Québec H3C 3J7
>
> Département de Mathématiques et Statistique
> 2920, chemin de la Tour
> Montréal, Québec H3T 1J4
>
> Tél.: (00-1) 514-343-6607 / Fax: (00-1) 514-343-5700
> lafaye at dms.umontreal.ca
> http://www.biostatisticien.eu
>
> ______________________________________________
> R-help at r-project.org mailing list
> https://stat.ethz.ch/mailman/listinfo/r-help