在 Julia 中：Float64 和 BigFloat 的相等性

Question

在 Julia 1.0.0 REPL 中，我得到以下结果：

# 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

似乎 Julia 在幕后做了一些事情来与 Float64 和 BigFloat 进行相等性比较，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 文档似乎没有涵盖此类内容： https://docs.julialang.org/en/v1/base/math/#Base.:==

编辑：根据下面@EPo 的有用评论，很容易使上面的所有比较都成为现实。例如，下面的第 1 行和第 3 行是正确的，尽管它们在上面是错误的：

# 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

有些浮点数可以精确表示（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法精确表示。例如你有：

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

但是0.1在二进制中是周期数，所以四舍五入。

julia> bitstring(0.1)
"0011111110111001100110011001100110011001100110011001100110011010"

后52位代表二进制的小数。 (https://en.wikipedia.org/wiki/Double-precision_floating-point_format)

Answer 2

之所以不一样是因为它们不一样

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

在 Julia 中：Float64 和 BigFloat 的相等性

In Julia: Equality of Float64 and BigFloat

precision

arbitrary-precision

julia