An interesting, perhaps non-obvious issue here, is: Why do floating point numbers have 2^53 as their limiting value when there's only 52 bits of mantissa?
It's because an implied leading 1 digit is always assumed for non-zero floating point numbers, so it's not represented in the mantissa bits. (This is spelled out in the page I linked, in the paragraph immediately above the diagram which shows the layout of the bits. But it's easy to miss if you are quickly skimming over the content.) There's other interesting quirks of this format (like: integer less than and greater than mechanisms can be used when comparing floating point values). Some people put real thought into it. FYI, -- Raul On Sat, Sep 25, 2021 at 4:52 PM Raul Miller <[email protected]> wrote: > > Actually, I think the limit value for raw integers is > > 2^63x > 9223372036854775808 > > Because we use signed integer values, so one bit is reserved for > distinguishing positive and negative values. > > (Technically, it's going to be one less than that for positive > integers and exactly that for negative integers, because we also have > to represent the value 0.) > > And, once that's exceeded, we use > https://en.wikipedia.org/wiki/Double-precision_floating-point_format#IEEE_754_double-precision_binary_floating-point_format:_binary64 > which means that the mantissa's precision is limited to > > x:2^53 > 9007199254740992 > > So ... intermediate results arrived at using different numbers are > likely going to be different. > > FYI, > > -- > Raul > > On Sat, Sep 25, 2021 at 4:44 PM Henry Rich <[email protected]> wrote: > > > > 2^64x > > 18446744073709551616 > > > > 135301852344706760704 > > > > Henry Rich > > > > > > On 9/25/2021 4:32 PM, Skip Cave wrote: > > > From: https://code.jsoftware.com/wiki/Essays/Fibonacci_Sequence > > > > > > f0b=: (-&2 +&$: -&1) ^: (1&<) M. > > > > > > f2a=: 3 : '{. +/\@|.^:y 0 1x' > > > > > > x:f0b 98 > > > > > > 135301852344706760704 > > > > > > x:f2a 98 > > > > > > 135301852344706746049 > > > > > > > > > (x:f0b 98) -: x:f2a 98 > > > > > > 0 > > > > > > > > > Why the discrepancy? > > > > > > > > > Skip > > > > > > > > > Skip Cave > > > Cave Consulting LLC > > > ---------------------------------------------------------------------- > > > For information about J forums see http://www.jsoftware.com/forums.htm > > > > > > -- > > This email has been checked for viruses by AVG. > > https://www.avg.com > > > > ---------------------------------------------------------------------- > > For information about J forums see http://www.jsoftware.com/forums.htm ---------------------------------------------------------------------- For information about J forums see http://www.jsoftware.com/forums.htm
