How does Matlab store this large of a number with only 64 bits?

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The largest number in double precision (that is, 64 bit) floating point arithmetic is 1.111...110 x 2^(512) (where there are 51 1's after the radix point). This number is less than 2 x 2^(512) == 2^(513) == 8^(171) < 10^(171). Therefore, when I assign x = 10^(171), I expect that x will be stored as Inf. However, this is not the case. Calling x in the interactive console displays 1.0000e+171. The only explanation I could think of is that Matlab uses more than 64 bits to store x. But a quick check of whos x reveals that x is stored in 8 bytes.

In fact, the largest power of 10 which will not be stored as Inf is 10^308.

Can someone please explain what is going on here?

precision floating-point matlab
1 Answers

I'm sorry, I made a simple mistake here. In 64 bit arithmetic, 11 bits are used to encode the exponent. Therefore we have 2^(11) = 2048 possible exponents, and so they range from -1023 to 1024, not from -511 to 512 like I thought. Therefore the largest number in 64 bit arithmetic is $1.111...110 x 2^(1024)$, which is in fact (with the exponent having 3 significant digits) 10^(308.6), corroborating my experimental results.

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