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Re: Loss of precision

• To: mathgroup at smc.vnet.net
• Subject: [mg126684] Re: Loss of precision
• From: danl at wolfram.com
• Date: Thu, 31 May 2012 02:46:32 -0400 (EDT)
• Delivered-to: l-mathgroup@mail-archive0.wolfram.com
• References: <jpnhi8$qa4$1@smc.vnet.net> <jpq6t7$6vs$1@smc.vnet.net> <jq265m$6jt$1@smc.vnet.net>

On Tuesday, May 29, 2012 4:50:46 AM UTC-5, Richard Fateman wrote:
> On 5/26/2012 2:14 AM, danl at ... wrote:
> > On Friday, May 25, 2012 3:57:44 AM UTC-5, sam.... at yahoo.com wrote:
> >> Hi,
> >>
> >> I understand intuitively why Sin[Large Number] cannot be computed too accurately. For example
> >>
> >> Precision[Sin[SetPrecision[10^10, 100]]] = 89.75
> >>
> >> We can I find an explanation of precisely why and how 100 becomes 89.75.
>
> Sin[Large Number]  can be computed accurately to any desired number of
> digits.  Just as one can compute pi to any specified number of digits,
> one can compute 10^10  or 10^100 modulo pi/2  to any specified number of
> digits.
>
> Daniel does describe what Mathematica does, which is hardly
> mathematically or computationally inevitable.
> RJF

The operand in question was not 10^10, but rather a finite precision approximation thereto. Once you pin the precision of the operand to a finite value, you are subject to precision loss. If your convention is that, for example, .001 "really" means .001000000... then of course it is a different matter, because you have not pinned the precision to something finite. As you observe, tacitly, that's not what Mathematica does.

Daniel Lichtblau
Wolfram Research