I will never understand float point arithmetic… julia> a = 10000 * 7 /100 700.0 julia> b = 10000 * 0.07 700.0000000000001 julia> c = 1000 * 0.69999999999999999 700.0 (but at list I am aware of the danger…)

[image] sylvaticus: (but at list I am aware of the danger…) As we say in France: “un homme averti en vaut 1.999999999999999” (Literally, the saying says something like “a forewarned person is worth two”)

Also, I’m not sure you were really looking for an explanation, but maybe it will help to note that: julia> 7/100 0.07 So this is merely an example of non-associativity of the floating-point operations: julia> 10000 * (7 / 100) 700.0000000000001 julia> (10000 * 7) / 100 700.0

Look at this excellent presentation by Avik Sengupta Everything You Wanted to Know About Floating Point Numbers, And Didn't Know... Everything You Wanted to Know About Floating Point Numbers, And Didn't Know Who to Ask - JuliaHub

[image] ffevotte: julia> 7/100 0.07 Part of the confusion is that, although this is printed as 0.07, it’s actually a slightly different number. (Binary) floating-point values are integers times powers of two, so 7/100 is not exactly representable and has to be rounded to: julia> big(7/100) 0.070000000000000006661338147750939242541790008544921875 When you realize that a rounding operation has already occurred here, then the fact that multiplying this by 10000 results in something different from 700 is not so surprising. In contrast, when you compute 10000 * 7 / 100 it is parsed as (10000 * 7) / 100, and each of these operations can be performed exactly (the result of each operation is an integer value) so there is no rounding (even if you write 1e4 * 7 / 100 so that the first multiplication is done in floating point). The reason that 7/100 prints as 0.07 is tricky — printing floating-point values in decimal has a long history, and basically all of computer science has coalesced around the principle that you should print the shortest decimal number that rounds to the same value when converted back to floating point. (0.07 will print the same way in Python, C, …)

It’s not so hard to understand floating-point arithmetic, actually! Trefethen and Bau’s book “Numerical Linear Algebra” covers the essentials in a couple lectures (each 4 or 5 pages of text). It’ll change your reaction to the above from “Wha…?” to “Yep, that’s how it works, no mystery about it.” Well worth reading.

[image] stevengj: when you compute 10000 * 7 / 100 it is parsed as (10000 * 7) / 100, and each of these operations can be performed exactly (the result of each operation is an integer value) so there is no rounding What is the rule here, is for a / b, when both a and b (or at least b) is an integer (or non-approx), and the result should be, then you get exact, i.e. integer? About “it is parsed as `(10000 * 7) / 100”, for constants, is it allowed to parse as 7 * (10000 / 100)"; e.g. if 7 had been not a constant?

If you want that calculation, I recommend to use rational number representation: julia> 10000*7//100 700//1

To complement a bit, IEEE754 Float64 are stored as M × 2E, where M is the 53-bit integer mantissa and E the 11-bit exponent (some further details omitted). 700.0 becomes M=700 and E=0, as integer values between and -2^52 and 2^52-1 are stored with E=0. 0.07 becomes M=6157265115545601 and E=-43, in terms of bits very different from an integer value. BTW, is there a package for Julia that would reveal mantissa and exponent of IEEE754 floats?

[image] Palli: What is the rule here, is for a / b, when both a and b (or at least b) is an integer (or non-approx), and the result should be, then you get exact, i.e. integer? Yes. The result of each elementary fp operation (+,-,*,/) is as if you had done that operation exactly and then rounded to the closest representable value. This means that if the exact result is representable, it will be returned exactly.

[image] stephancb: BTW, is there a package for Julia that would reveal mantissa and exponent of IEEE754 floats? There are built-in functions: How to get the significand and the exponent of a floating point number?

I will never understand floating-point arithmetic

Offtopic

stevengj March 8, 2024, 1:30pm 12

There are built-in functions: How to get the significand and the exponent of a floating point number?

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I will never understand floating-point arithmetic

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