From be9aa9a652ea92ba5459c9337b6738de61b54eae Mon Sep 17 00:00:00 2001 From: Ben Pfaff Date: Tue, 20 Dec 2005 22:15:44 +0000 Subject: [PATCH] Suggest 17.14 instead of 21.10 because the extra precision may be necessary for some implementations. Thanks to Matt Page and "Jeff Sun" for the suggestion. --- doc/44bsd.texi | 42 ++++++++++++++++++++++-------------------- 1 file changed, 22 insertions(+), 20 deletions(-) diff --git a/doc/44bsd.texi b/doc/44bsd.texi index 18c5a61..e53e829 100644 --- a/doc/44bsd.texi +++ b/doc/44bsd.texi @@ -272,31 +272,33 @@ difficult, but many students do not know how to do it. This section explains the basics. The fundamental idea is to treat the rightmost bits of an integer as -representing a fraction. For example, we can designate the lowest 10 +representing a fraction. For example, we can designate the lowest 14 bits of a signed 32-bit integer as fractional bits, so that an integer -@var{x} represents the real number +@m{x} represents the real number @iftex -@m{x/2^{10}}. +@m{x/2^{14}}. @end iftex @ifnottex -@m{x/(2**10)}, where ** represents exponentiation. +@m{x/(2**14)}, where ** represents exponentiation. @end ifnottex -This is called a 21.10 fixed-point number representation, because there -are 21 bits before the decimal point, 10 bits after it, and one sign +This is called a 17.14 fixed-point number representation, because there +are 17 bits before the decimal point, 14 bits after it, and one sign bit.@footnote{Because we are working in binary, the ``decimal'' point might more correctly be called the ``binary'' point, but the meaning -should be clear.} A number in 21.10 format represents, at maximum, a -value of @am{(2^{31} - 1) / 2^{10} \approx, (2**31 - 1)/(2**10) = -approx.} 2,097,151.999. +should be clear.} A number in 17.14 format represents, at maximum, a +value of @am{(2^{31} - 1) / 2^{14} \approx, (2**31 - 1)/(2**14) = +approx.} 131,071.999. Suppose that we are using a @m{p.q} fixed-point format, and let @am{f = 2^q, f = 2**q}. By the definition above, we can convert an integer or real number into @m{p.q} format by multiplying with @m{f}. For example, -in 21.10 format the fraction 59/60 used in the calculation of -@var{load_avg}, above, is @am{(59/60)2^{10}, 59/60*(2**10)} = 1,007 +in 17.14 format the fraction 59/60 used in the calculation of +@var{load_avg}, above, is @am{(59/60)2^{14}, 59/60*(2**14)} = 16,111 (rounded to nearest). To convert a fixed-point value back to an integer, divide by @m{f}. (The normal @samp{/} operator in C rounds -down. To round to nearest, add @m{f / 2} before dividing.) +toward zero, that is, it rounds positive numbers down and negative +numbers up. To round to nearest, add @m{f / 2} to a positive number, or +subtract it from a negative number, before dividing.) Many operations on fixed-point numbers are straightforward. Let @code{x} and @code{y} be fixed-point numbers, and let @code{n} be an @@ -308,19 +310,19 @@ quotient, @code{x / n}. Multiplying two fixed-point values has two complications. First, the decimal point of the result is @m{q} bits too far to the left. Consider that @am{(59/60)(59/60), (59/60)*(59/60)} should be slightly less than -1, but @tm{1,007\times 1,007}@nm{1,007*1,007} = 1,014,049 is much -greater than @am{2^{10},2**10} = 1,024. Shifting @m{q} bits right, we -get @tm{1,014,049/2^{10}}@nm{1,014,049/(2**10)} = 990, or about 0.97, +1, but @tm{16,111\times 16,111}@nm{16,111*16,111} = 259,564,321 is much +greater than @am{2^{14},2**14} = 16,384. Shifting @m{q} bits right, we +get @tm{259,564,321/2^{14}}@nm{259,564,321/(2**14)} = 15,842, or about 0.97, the correct answer. Second, the multiplication can overflow even though -the answer is representable. For example, 128 in 21.10 format is -@am{128 \times 2^{10}, 128*(2**10)} = 131,072 and its square @am{128^2, -128**2} = 16,384 is well within the 21.10 range, but @tm{131,072^2 = -2^{34}}@nm{131,072**2 = 2**34}, greater than the maximum signed 32-bit +the answer is representable. For example, 64 in 17.14 format is +@am{64 \times 2^{14}, 64*(2**14)} = 1,048,576 and its square @am{64^2, +64**2} = 4,096 is well within the 17.14 range, but @tm{1,048,576^2 = +2^{40}}@nm{1,048,576**2 = 2**40}, greater than the maximum signed 32-bit integer value @am{2^{31} - 1, 2**31 - 1}. An easy solution is to do the multiplication as a 64-bit operation. The product of @code{x} and @code{y} is then @code{((int64_t) x) * y / f}. -Dividing two fixed-point values has the opposite complications. The +Dividing two fixed-point values has opposite issues. The decimal point will be too far to the right, which we fix by shifting the dividend @m{q} bits to the left before the division. The left shift discards the top @m{q} bits of the dividend, which we can again fix by -- 2.30.2