BigFloat converted to string returns different number?

Specific Domains Numerics
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1

Hi,

I am new to Julia and in the documentation I came across this:

julia> big"1.23456789012345678901"
1.234567890123456789010000000000000000000000000000000000000000000000000000000004

At first I thought that this is some sort of displaying error but then I tried this:

julia> string(big"1.23456789012345678901")
"1.234567890123456789010000000000000000000000000000000000000000000000000000000004"

How does “1.23456789012345678901” turn into “1.234567890123456789010000000000000000000000000000000000000000000000000000000004”?

I know that normal floats aren’t precise because they can’t be represented exactly but isn’t BigFloat supposed to fix this?

Can someone please explain what is going on here?

Thanks

2

BigFloat is still a binary format, just with more bits (256 by default). Most decimal expansions have infinite length in binary. If you want to represent decimal numbers exactly, you need to use a different number type.

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3

To slightly elaborate on Oscar’s answer and more directly address the question “what is going on here?”, 1.234567890123456789010000000000000000000000000000000000000000000000000000000004 is the closest arbitrary-precision (well, whatever precision you’re currently using, presumably 256-bit, the default) that can represent the real number 1.23456789012345678901. You can see that both the previous and the following floating-point numbers are much further away from 1.23456789012345678901 than 1.234567890123456789010000000000000000000000000000000000000000000000000000000004 is:

julia> x = big"1.23456789012345678901"
1.234567890123456789010000000000000000000000000000000000000000000000000000000004

julia> prevfloat(x)
1.234567890123456789009999999999999999999999999999999999999999999999999999999987

julia> nextfloat(x)
1.234567890123456789010000000000000000000000000000000000000000000000000000000021
1 Like
4

To add a simple example to the previous answers: The underlying problem is approximating a decimal fraction in binary terms can be infinite, like this one:
\frac{1}{10} =\frac{1}{16} + \frac{1}{32} + \frac{1}{256} + \frac{1}{512} + \frac{1}{4096} + ...

If you like math, you can even proof that, e.g. if \frac{1}{10} = \frac{n}{2^k} then 2^k = 10n and then 2 | 5.)

5

Thanks. Then I misunderstood.
I thought it would be an exact decimal representation.
Do you know a different number type in Julia that implements it exactly or should I use rationals instead for calculations and convert the result back into a BigFloat?

1 Like
6

If you want decimal arithmetic, you should check out GitHub - JuliaMath/Decimals.jl: Pure Julia decimal arithmetic library.. Rational arithmetic also works, but can overflow in some situations.

7

Thanks.
I didn’t know that prevfloat and nextfloat existed.

8

Thanks for the example that makes it obvious. But what do you mean with the 2|5?

9

Ops, “2 | 5” was my notation for “2 divides 5”.