POWER-OF-2 BOUNDARIES

Transcription

1 Warren.3.fm Page 5 Monday, Jne 17, 5:6 PM CHAPTER 3 POWER-OF- BOUNDARIES 3 1 Ronding Up/Down to a Mltiple of a Known Power of Ronding an nsigned integer down to, for eample, the net smaller mltiple of 8, is trivial: & 8 does it. An alternative is ( >> 3) << 3. These work for signed integers as well, provided rond down means to rond in the negative direction (e.g., ( 37) & ( 8) = ). Ronding p is almost as easy. For eample, an nsigned integer can be ronded p to the net greater mltiple of 8 with either of ( + 7) & 8, or + ( & 7). These epressions are correct for signed integers as well, provided rond p means to rond in the positive direction. The second term of the second epression is sefl if yo want to know how mch yo mst add to to make it a mltiple of 8 [Gold]. To rond a signed integer to the nearest mltiple of 8 toward, yo can combine the two epressions above in an obvios way: s t ( >> 31) & 7; ( + t) & 8 An alternative for the first line is t ( >> ) >> 9, which is sefl if the machine lacks and immediate, or if the constant is too large for its immediate field. Sometimes the ronding factor is given as the log of the alignment amont (e.g., a vale of 3 means to rond to a mltiple of 8). In this case, code sch as the following may be sed, where k = log (alignment amont): rond down: & (( 1) << k) ( >> k) << k rond p: t ( 1 << k) 1; ( + t) & t t ( 1) << k; ( t 1) & t s 5

2 Warren.book Page 6 Monday, Jne 17, :37 PM 6 POWER-OF- BOUNDARIES 3 3 Ronding Up/Down to the Net Power of We define two fnctions that are similar to floor and ceiling, bt which are directed rondings to the closest integral power of, rather than to the closest integer. Mathematically, they are defined by flp() ndefined, <, =, =, clp() log, otherwise; = ndefined, <,, =, log, otherwise. The initial letters of the fnction names are intended to sggest floor and ceiling. Ths, flp() is the greatest power of that is, and clp() is the least power of that is. These definitions make sense even when is not an integer (e.g., flp(.1) =.65). The fnctions satisfy several relations analogos to those involving floor and ceiling, sch as those shown below, where n is an integer. = iff is an integer + n = + n = flp() = clp () iff is a power of or is flp( n ) = n flp() clp() = 1 flp1 ( ), Comptationally, we deal only with the case in which is an integer, and we take it to be nsigned, so the fnctions are well defined for all. We reqire the vale compted to be the arithmetically correct vale modlo 3 (that is, we take clp( ) to be for > ). The fnctions are tablated below for a few vales of flp( ) 1 3 clp( ) 1 8

3 Warren.book Page 7 Monday, Jne 17, :37 PM 3 ROUNDING UP/DOWN TO THE NEXT POWER OF 7 Fnctions flp and clp are connected by the relations shown below. These can be sed to compte one from the other, sbject to the indicated restrictions. clp( ) = flp( 1), 1, = flp( 1), 1, flp( ) = clp( + 1),, = clp( + 1), <. The rond-p and rond-down fnctions can be compted qite easily with the nmber of leading zeros instrction, as shown below. However, for these relations to hold for = and >, the compter mst have its shift instrctions defined to prodce for shift amonts of 1, 3, and 63. Many machines (e.g., PowerPC) have mod 6 shifts, which do this. In the case of 1, it is adeqate if the machine shifts in the opposite direction (that is, a shift left of 1 becomes a shift right of 1). flp() = 1 << ( 31 nlz() ) = 1 << ( nlz() 31) = 8 >> nlz() clp() = 1 << ( 3 nlz( 1) ) Ronding Down Figre 3 1 illstrates a branch-free algorithm that might be sefl if nmber of leading zeros is not available. This algorithm is based on right-propagating the leftmost 1-bit, and eectes in 1 instrctions. Figre 3 shows two simple loops that compte the same fnction. All variables are nsigned integers. The loop on the right keeps trning off the rightmost 1-bit of ntil =, and then retrns the previos vale of. = 8 >> ( nlz( 1) 1) nsigned flp(nsigned ) { = ( >> 1); = ( >> ); = ( >> ); = ( >> 8); = ( >>16); retrn - ( >> 1); } FIGURE 3 1. Greatest power of less than or eqal to, branch-free.

4 Warren.book Page 8 Monday, Jne 17, :37 PM 8 POWER-OF- BOUNDARIES 3 y = 8; do { while (y > ) y = ; y = y >> 1; = & ( - 1); retrn } while(!= ); retrn y; The loop on the left eectes in nlz( ) + 3 instrctions. The loop on the right, for, eectes in pop() instrctions, 1 if the comparison to is zerocost. Ronding Up The right-propagation trick yields a good algorithm for ronding p to the net power of. This algorithm, shown in Figre 3 3, is branch-free and rns in 1 instrctions. An attempt to compte this with the obvios loop does not work ot very well: y = 1; FIGURE 3. Greatest power of less than or eqal to, simple loops. while (y < ) y = *y; retrn y; // Unsigned comparison. This code retrns 1 for =, which is probably not what yo want, loops forever for, and eectes in n + 3 instrctions, where n is the power of of the retrned integer. Ths, it is slower than the branch-free code, in terms of instrctions eected, for n 3 ( 8). nsigned clp(nsigned ) { = - 1; = ( >> 1); = ( >> ); = ( >> ); = ( >> 8); = ( >>16); retrn + 1; } FIGURE 3 3. Least power of greater than or eqal to. 1. pop() is the nmber of 1-bits in.

5 Warren.book Page 9 Monday, Jne 17, :37 PM 3 3 DETECTING A POWER-OF- BOUNDARY CROSSING Detecting a Power-of- Bondary Crossing Assme memory is divided into blocks that are a power of in size, starting at address. The blocks may be words, doblewords, pages, and so on. Then, given a starting address a and a length l, we wish to determine whether or not the address range from a to a + l 1, l, crosses a block bondary. The qantities a and l are nsigned and any vales that fit in a register are possible. If l = or 1, a bondary crossing does not occr, regardless of a. If l eceeds the block size, a bondary crossing does occr, regardless of a. For very large vales of l (wraparond is possible), a bondary crossing can occr even if the first and last bytes of the address range are in the same block. There is a srprisingly concise way to detect bondary crossings on the IBM System/37 [CJS]. This method is illstrated below for a block size of 96 bytes (a common page size). O RA,=A(-96) ALR RA,RL BO CROSSES The first instrction forms the logical or of RA (which contains the starting address a) and the nmber FFFFF. The second instrction adds in the length, and sets the machine s -bit condition code. For the add logical instrction, the first bit of the condition code is set to 1 if a carry occrred, and the second bit is set to 1 if the 3-bit register reslt is nonzero. The last instrction branches if both bits are set. At the branch target, RA will contain the length that etends beyond the first page (this is an etra featre that was not asked for). If, for eample, a = and l = 96, a carry occrs bt the register reslt is, so the program properly does not branch to label CROSSES. Let s see how this method can be adapted to RISC machines, which generally do not have branch on carry and register reslt nonzero. Using a block size of 8 for notational simplicity, the method of [CJS] branches to CROSSES if a carry occrred (( a 8) + l 3 ) and the register reslt is nonzero (( a 8) + l 3 ). Ths, it is eqivalent to the predicate ( a 8) + l > 3. This in trn is eqivalent to getting a carry in the final addition in evalating (( a 8) 1) + l. If the machine has branch on carry, this can be sed directly, giving a soltion in abot five instrctions conting a load of the constant 8. If the machine does not have branch on carry, we can se the fact that carry occrs in + y iff < y (see Unsigned Add/Sbtract on page 9) to obtain the epression (( a 8) 1) l. <

6 Warren.book Page 5 Monday, Jne 17, :37 PM 5 POWER-OF- BOUNDARIES 3 3 Using varios identities sch as ( 1) = gives the following eqivalent epressions for the bondary crossed predicate: ( a 8) l These can be evalated in five or si instrctions on most RISC compters. Using another tack, clearly an 8-byte bondary is crossed iff This cannot be directly evalated becase of the possibility of overflow (which occrs if l is very large), bt it is easily rearranged to 8 ( a & 7) < l, which can be directly evalated on the compter (no part of it overflows). This gives the epression which can be evalated in five instrctions on most RISCs (for if it has sbtract from immediate). If a bondary crossing occrs, the length that etends beyond the first block is given by l ( 8 ( a & 7) ), which can be calclated with one additional instrction (sbtract). < ( a 8) + 1 l < ( a & 7) + 1 l < ( a & 7) + l ( a & 7) < l,