Why does the binaryw. format behave differently above width 58? - sas

Question

Consider the following program and output:
data _null_;
input a;
length b $64;
do i = 1 to 64;
fmtname = cats('binary',i);
b = cats(putn(a,fmtname));
put i= b=;
end;
cards;
1
;
run;
Output (SAS 9.1.3, Windows 7 x64):
i=1 b=1
i=2 b=01
i=3 b=001
i=4 b=0001
i=5 b=00001
/*Skipped a few very similar lines*/
i=58 b=0000000000000000000000000000000000000000000000000000000001
i=59 b=11111110000000000000000000000000000000000000000000000000000
i=60 b=111111110000000000000000000000000000000000000000000000000000
i=61 b=1111111110000000000000000000000000000000000000000000000000000
i=62 b=11111111110000000000000000000000000000000000000000000000000000
i=63 b=011111111110000000000000000000000000000000000000000000000000000
i=64 b=0011111111110000000000000000000000000000000000000000000000000000
Last few lines of output from SAS 9.4 on Linux x64:
i=60 b=000000000000000000000000000000000000000000000000000000000001
i=61 b=1111111110000000000000000000000000000000000000000000000000000
i=62 b=11111111110000000000000000000000000000000000000000000000000000
i=63 b=011111111110000000000000000000000000000000000000000000000000000
i=64 b=0011111111110000000000000000000000000000000000000000000000000000
This behaviour is rather unexpected, to me at least, and doesn't seem to be documented on the help page. It agrees with the document I found here for width 64 - standard double precision - but I don't understand why it flips over at width 59.

Answer 1

I don't quite get the same result - mine switches at 61 - but I believe the answer is the same.
Up to some point - 58, 60, somewhere around there - SAS is showing you the fixed-point integer representation of the number. Test this with a decimal, like so:
data _null_;
a=3.14159265358979323846264338327950288419716939937510582;
length b $64;
put a= hex4.;
put a= hex8.;
put a= hex16.;
do i = 1 to 64;
fmtname = cats('binary',i);
b = cats(putn(a,fmtname));
put i= b=;
end;
run;
And you will get a sort-of-surprising result - you see 000...0011 for most of your rows, up through 60. The documentation doesn't explicitly mention this, but it does show it in the example (123.45 and 123 are identical in binary8.).
Then starting at 61, or 59 for you I'm guessing, you see the actual representation of the number as SAS internally stores it (or, arguably, how Intel internally stores it).
The binary documentation doesn't explain this well, but the HEX. documentation does explain it pretty clearly in a tip:
If w< 16, the HEXw. format converts real binary numbers to fixed-point integers before writing them as hexadecimal characters. It also writes negative numbers in two's complement notation, and right aligns digits. If w is 16, HEXw. displays floating-point values in their hexadecimal form.
Binary is doing the same, and on my machine it happens right at the point HEX would also make the change - at 15x4=60. And HEX. shows the same - notice below; hex4. and hex8. show a different result than hex16..
To be clear, the value shown at binary64. is correct, and not any sort of truncation (though 61-63, and in your example 59-60, are left-truncated).
I did find a SAS usage note regarding this, though it's clearly out of date based on our tests:
Beginning with SAS® Version 7, the BINARYw. format was changed to be more consistent with the HEXw. format. When the HEXw. format uses a width of 16, (corresponding to 8 bytes of data), it produces a hexadecimal representation of the floating point value. The BINARYw. format changed so that widths of 57-64 produce a binary representation of the floating point value, since widths of 57-64 correspond to 8 bytes of data.
It also contains a suggestion for how to get consistent results for integers, which may be of use.
BIN_64=PUT(PUT(VALUE,S370FIB8.),$BINARY64.);
S370FIB8. is a format that converts numbers to their fixed integer binary representation, in IBM Mainframe format. (I.e., it writes the integer in Big-Endian format, which is not what you'd get on an Intel machine.)