在朱莉娅: Float64和BigFloat的平等

Question

在Julia 1.0.0 REPL中，我得到了以下结果：

# Line 1: This make sense.  I did not expect a Float64 to equal a BigFloat.
julia> 26.1 == big"26.1"
false

# Line 2: This surprised me when Line 1 is considered.  Again, like Line 1, I 
# did not expect a Float64 to equal an equivalent BigFloat.
julia> 26.0 == big"26.0"
true

# Line 3: This I expected based on Line 1 behavior.
julia> 26.1 - 0.1 == big"26.1" - 0.1
false

# Line 4: This surprised me based on Line 1 behavior, but it might be 
# explained based on Line 2 behavior.  It seems to imply that if a Float64 
# can be converted to an Integer it will compare equal to an equivalent BigFloat.
julia> 26.1 - 0.1 == big"26.1" - big"0.1"
true

看起来Julia在这里暗地里做了一些与Float64和BigFloat的相等比较，这使得第2行和第4行为真，而第1行和第3行为假。有什么建议吗？

关于"==“的Julia文档似乎没有涵盖这类事情：https://docs.julialang.org/en/v1/base/math/#Base.:==

编辑:基于@EPo下面的一条有用的评论，很容易让上面的所有比较成为现实。例如，第1行和第3行在下面为真，但在上面为假：

# Line 1 is now true.
julia> 26.1 ≈ big"26.1"
true

# Line 3 is now true.
julia> 26.1 - 0.1 ≈ big"26.1" - 0.1
true

Answer 1

一些浮点数可以精确地表示(26.0)，但不是全部，例如：

julia> using Printf
julia> @printf("%.80f",26.0)
26.00000000000000000000000000000000000000000000000000000000000000000000000000000000
julia> @printf("%.80f",0.1)
0.10000000000000000555111512312578270211815834045410156250000000000000000000000000

例如，小数0.5、0.25、0.125也可以用基于二进制的浮点表示精确地表示。举个例子，你有：

julia> 26.125 - 0.125 == big"26.125" - 0.125
true

但是0.1在二进制中是一个周期性的数字，所以它是四舍五入的。

julia> bitstring(0.1)
"0011111110111001100110011001100110011001100110011001100110011010"

最后52位以二进制形式表示分数。(https://en.wikipedia.org/wiki/Double-precision_floating-point_format)

Answer 2

它们不同的原因是因为它们不同

julia> using Printf
julia> string(BigFloat("26.1")-BigFloat("26"))
"1.000000000000000000000000000000000000000000000000000000000000000000000000000553e-01"
julia> @printf("%.17e",Float64(26.1)-Float64(26))
1.00000000000001421e-01
julia> Float64(26.1)-Float64(26) > BigFloat("26.1")-BigFloat("26")
true