pdftex bug

[Nelson H. F. Beebe]

Nobert Preining observes:

>>  maybe the specification of type1 fonts requires a certain maximum precision.

I just checked p. 26 of the black-and-white covered Adobe Type 1 Font Format
book, and found that it demands that coordinates lie in [-2000, +2000].

Thus, at most 11 bits (because 2**11 = 2048) are required to represent
their precision, and David W. Matula's formula from
	
	The Base Conversion Theorem
	Proceedings of the American Mathematical Society 19(3) 716--723 June 1968
	https://doi.org/10.1090/s0002-9939-1968-0234908-9

which can be written in hoc like this

	 func matula(p) { return ceil( p * log10(2) + 1 ) }

shows that at most 5 decimal digits are needed to recover the original
binary value of a coordinate. That of course, does not count leading
zeros, so the example of

	/FontMatrix [0.000833333333333 0 0 0.000833333333333 0 0]

could have been shortened to

	/FontMatrix [0.00083333 0 0 0.00083333 0 0]

Still, software should be able to correctly handle excess digits on
input values.

PostScript was defined with the IEEE 754 32-bit floating-point format
and, for integers, 32-bit two's complement values.  The floating-point
value has 24 bits in the significand, for which Matula's formula
requires 9 decimal digits for exact round-trip conversion binary ->
decimal -> binary.

The PDF Reference Specification version 1.3 in Appendix C.1 shows
32-bit two's complement integers, but curiously, nonzero real
magnitudes are limited to the range [1/65536, 32767].  The book does
not mention "floating-point" in its index, and the given range
suggests that Adobe implemented PDF with 32-bit fixed-point numbers,
with a 1-bit sign, 15-bit integer value, and 16 bit fraction.  That is
a 31-bit magnitude for which Matula's formula requires 11 decimal digits.

I don't have books with the TrueType and FontType specifications, so I
cannot easily check their coordinate limits.  Their Wikipedia articles
do not mention numeric limits.  Perhaps someone else on this list has
the necessary documents, and can report back.

----------------------------------------

For reader convenience, here is multicolumn table of bits and decimal
digits for Matula's formula, filtered with "pr -c7-f":

	hoc> for (p = 1; p < 32; ++p) println p, matula(p)
	 1  2	   6  3	     11	 5     16  6	 20  8	   24  9     28	 10
	 2  2	   7  4	     12	 5     17  7	 21  8	   25  9     29	 10
	 3  2	   8  4	     13	 5     18  7	 22  8	   26  9     30	 11
	 4  3	   9  4	     14	 6     19  7	 23  8	   27  10    31	 11
	 5  3	  10  5      15	 6

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- Nelson H. F. Beebe                    Tel: +1 801 581 5254                  -
- University of Utah                                                          -
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