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Re: Error in Subtraction with V4.0??

• To: mathgroup at smc.vnet.net
• Subject: [mg22537] Re: [mg22463] Error in Subtraction with V4.0??
• From: Daniel Lichtblau <danl at wolfram.com>
• Date: Thu, 9 Mar 2000 03:24:24 -0500 (EST)
• References: <200003080722.CAA13113@smc.vnet.net>
• Sender: owner-wri-mathgroup at wolfram.com

BEIntInc wrote:
> 
> Can someone please explain why
> 
> 1. - .999999999999
> 
> results in
> 9.999778782798785*10^-13
> 
> The old way (by hand) gives me
> 
> 1x10^-12
> 
> Is this an error or am I missing something?
> 
> TIA,
> 
> John Brooks

The result is correct for machine arithmetic and the short explanation
is that you are seeing massive cancellation at machine precision. To see
in more detail what is happenning, we might do as follows.

First we'll check the bit representation of (machine precision) 1 and
999999999999/10^12. To avoid introducing sources of error from
binary-to-decimal conversion we'll numericize the exact entity for the
latter.

In[33]:= ??$NumberBits
$NumberBits[x] gives a list containing the sign, a list of the bits
thought to
   be correct, a list of the bits not thought to be correct, and the
binary
   exponent. The number x must have a head of Real.

Attributes[$NumberBits] = {Listable, Protected}

In[34]:= e1 = 1.; e2 = N[999999999999/10^12];

In[36]:= InputForm[$NumberBits[e1]]

Out[36]//InputForm= 
{1, {}, {1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0, 
  0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0, 
  0, 0, 0, 0, 0}, 1}

In[37]:= InputForm[$NumberBits[e2]]

Out[37]//InputForm= 
{1, {}, {1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,
1, 1, 
  1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 0, 0, 1,
1, 0, 
  1, 0, 0, 0, 1}, 0}

When we subtract in binary arithmetic from a power of two we can get the
same effect by complementing the bit-string for the second argument and
multiplying by appropriate powers of two, except that the lowest '1' bit
is uncomplemented. (If this is not clear, consider subtracting (binary)
1101 from 10000. You get 11.)

d2 = $NumberBits[e2][[3]];
bitComplement[bit_] := Mod[1+bit,2]
d3 = Map[bitComplement, d2];
d3 = MapAt[bitComplement, d3, -1]

In[115]:= InputForm[num = d3 . 2^Range[-1,-53,-1]]
Out[115]//InputForm= 9007/9007199254740992

In[116]:= InputForm[num2 = N[num]]
Out[116]//InputForm= 9.999778782798785*^-13


Daniel Lichtblau
Wolfram Research

• References:
  • Error in Subtraction with V4.0??
    • From: beintinc@aol.com (BEIntInc)