Floating Point Representation

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Stuart 2011-8-7

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Suppose we are in a floating-point regime where the mantissa had 3 digits and the exponent has one digit.

Then, 0 = .000e-9 is relatively isolated from next two positive numbers .100e-9 and .101e-9. Which are 10e-12 apart.

This is from Hamming's book on numerical methods, but how is .100e-9 the next positive number, why not .001e-9?

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James Tursa 2011-8-7

@Walter: OP has stated a decimal digit system, so hidden bit & normalization is not applicable.

Walter Roberson 2011-8-7

Okay on the hidden bit, but I think normalization is still a factor.

You seldom want a system that has a distinct zero for every different exponent value (e.g., 000E-9 000E-10).

You seldom want a system that has three different representations for some values, such as 100E-12 010E-11 001E-10

IEEE 754, referenced in the tags, *requires* normalization of binary values, except for the extremely small denormalized numbers, which it has special rules for.

IEEE 754 did adopt an optional decimal representation; my recollection at the moment is that it uses normalization as well, but I would need to cross-check that.

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Answer 1

James Tursa 2011-8-7

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1) What does this have to do with MATLAB?

2) How can you have numbers in the e-9 and e-12 range with only one exponent digit? Is this some type of decimal digit floating point system you are talking about? If so, what is the exponent bias? How can you get negative exponents in e-9 range with only one digit?

In short, please explain the floating point format you are talking about in detail.

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Stuart 2011-8-7

Decimal digits, rest isn't relevant to answer.

James Tursa 2011-8-7

Obviously you don't think so, but I do. Why isn't 0.001e-10 or 0.001e-11 or 0.001e-12 the next positive number? The exponent bias *does* make a difference to me, and I can't answer your question without knowing this.

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