Re: Problem with base 2 logs and Floor

• To: mathgroup at smc.vnet.net
• Subject: [mg73020] Re: [mg73001] Problem with base 2 logs and Floor
• From: Bob Hanlon <hanlonr at cox.net>
• Date: Mon, 29 Jan 2007 04:35:38 -0500 (EST)
• Reply-to: hanlonr at cox.net

```f1[r_]:=r - Floor[Log[2, r]] - 1

f2[r_]:=r - Ceiling[Log[2, r]] - 1

Plot[{f1[r],f2[r]},{r,1,17},PlotStyle->{Blue,Red}];

Cases[Table[{r,f1[r],f2[r]},{r,1000}],
x_?(#[[2]]==#[[3]]&):>x[[1]]]

{1,2,4,8,16,32,64,128,256,512}

Simplify[f1[2^n],Element[n, Integers]]

-1 + 2^n - n

Simplify[f1[2^n]==f2[2^n],Element[n, Integers]]

True

Bob Hanlon

---- neillclift at msn.com wrote:
> Hi,
>
> When I use an expression like this:
>
> r - Floor[Log[2, r]] - 1 /. r -> 4
>
> I get precision errors in Mathematica 5.2. If I use an expression like
> this:
>
> N[r - Floor[Log[2, r]] - 1] /. r -> 4
>
> I get a correct result of 1 and no errors. If I use an expression like
> this I get the same result:
>
> N[r - Ceiling[Log[2, r]] - 1] /. r -> 4
>
> This is to be expected as Floor[Log[2, r]]  = Ceiling[Log[2, r]]  when
> r is a power of two.
> Unfortunatly the expessions diverge for r=8:
>
> N[r - Floor[Log[2, r]] - 1] /. r -> 8 gives 5
> N[r - Ceiling[Log[2, r]] - 1] /. r -> 8 gives 4
>
> How can I get exacts results for expessions like this?
> Thanks.
> Neill.
>

```

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