From savicky at cs.cas.cz  Mon Sep  3 16:39:57 2007
From: savicky at cs.cas.cz (Petr Savicky)
Date: Mon, 3 Sep 2007 16:39:57 +0200
Subject: [Rd] When 1+2 != 3 (PR#9895)
In-Reply-To: <20070903065922.DA71966911@slim.kubism.ku.dk>
References: <20070903065922.DA71966911@slim.kubism.ku.dk>
Message-ID: <20070903143957.GA19039@cs.cas.cz>

On Mon, Sep 03, 2007 at 08:59:22AM +0200, marco.vicentini at gmail.com wrote:
> Full_Name: Marco Vicentini, University of Verona
> Version: 2.4.1 & 2.5.1
> OS: OsX & WinXP
> Submission from: (NULL) (157.27.253.46)
> 
> 
> When I proceed to test the following equation 1 + 2 == 3, I obviously obtain the
> value TRUE. But when I tryed to do the same using real number (i.e. 0.1 + 0.2 ==
> 0.3)  I obtained an unusual FALSE.
> In the online help there are some tricks for this problem. It suggests to use
> identical(...) which again answer FALSE. Only using isTRUE(all.equal(0.3, 0.1 +
> 0.2)) I can obtain the true value TRUE.

A rational number has a finite binary expansion iff its denominator is a power
of 2. Numbers 0.1 and 0.2 are 1/10 and 1/5, so they have 5 in their denominator.
Their binary expansion is
 0.1  = .0001100110011001100110011001100110...
 0.2  = .0011001100110011001100110011001100...
A double variable stores the numbers rounded to 53 significant binary digits.
Hence, they are not exactly 0.1 and 0.2, as may be seen in
  formatC(0.1,digits=30) # [1] "0.100000000000000005551115123126"
  formatC(0.2,digits=30) # [1] "0.200000000000000011102230246252"

In order to compare numbers with some tolerance, the function all.equal
may be used, which you also mention below. See its help page, which
specifies the tolerance to be  .Machine$double.eps ^ 0.5.

> But the problem does not concern only the operator ==. Many other functions,
> among over:  sort, order, unique, duplicate, identical are not able to deal with
> this problem. This is very dangerous because no advice are provide by the online
> help, and anybody can use these functions no think to unusual results.
> 
> I think that the problem is due to how double number are store by the C
> compiler.

Not C compiler, but the hardware.

Petr Savicky.

> If it may be usefull, I have written to small function (Unique and isEqual)
> which can deal with this problem of the double numbers.
> 
> I also add some other conditions for the same problem.
> 
> 	0.3 == 0.15 + 0.15
>     0.3 == 0.1 + 0.2
> 1 - 0.7 == 0.3
>     0.1 == 1 - 0.9
> 
>     0.2 == 1 - 0.2 - 0.2 - 0.2 - 0.2
>    -0.2 == 1 - 0.2 - 0.2 - 0.2 - 0.2 - 0.2 - 0.2
> 
> identical (0.3, 0.1 + 0.2)
> all.equal (0.3, 0.1 + 0.2)
> 
> identical (-0.2 , 1 - 0.2 - 0.2 - 0.2 - 0.2 - 0.2 - 0.2)
> all.equal (-0.2 , 1 - 0.2 - 0.2 - 0.2 - 0.2 - 0.2 - 0.2)
> 
> isTRUE( all.equal (-0.2 , 1 - 0.2 - 0.2 - 0.2 - 0.2 - 0.2 - 0.2) )
> 
> 
>    -0.2 == 1 - 0.2 - 0.2 - 0.2 - 0.2 - 0.2 - 0.2
> 
> a= -0.2 
> b=         1 - 0.2 - 0.2 - 0.2 - 0.2 - 0.2 - 0.2
> 
> x<-c(a,b)
> sprintf("%.15f",x)
> sprintf("%.50f",x)
> 
> 
> 
> Unique <- function(x, digits = 8, fast = TRUE) {
> 
> 	if (fast) {
> 		unique (round(x * 10^digits)) / 10^digits	
> 	} else {
> 		x = sort(x)
> 		for (i in 1:(length(x)-1))
> 			if (isTRUE(all.equal(x[i],x[i+1]))) x[i] = NaN
> 		x [ which (!is.nan(x)) ]
> 	}}
> 
> isEqual <- function (object, x, tol = 1e-9) {
> 	if (!is.vector(object)) stop("Object must be a vector")
> 	if (is.character(object)) stop("Object can not be a character")
> 	if (!is.real(x)) stop("x must be a real number")
> 	if (any(is.na(c(object,x)))) stop("NA is not supported")
> 	if (length(x) != 1) stop("length x must equal to 1")
> 
> 	ifelse (abs(object - x) < tol, TRUE,FALSE) 
> #	.Call("isEqual",as.real(object),as.real(x),as.real(tol), PACKAGE="mvUtils")
> }
> 
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>



[Rd] When 1+2 != 3 (PR#9895)
Petr Savicky