CSE 575 Computer Arithmetic Spring 2003 Mary Jane Irwin (www. cse. psu CSE 575 Computer Arithmetic Spring 2003 Mary Jane Irwin (www.cse.psu.edu/~mji) This set of slides needs more work as well.

A Binary Square Root Algorithm Q square root Q0 = 0 0 1 1 1 0 1 1 0 Z radicand = 11810 0 1 q3 = 1 Q1 = 01  101 ? No so choose 0 0 0 1 1 q2 = 0 0 0 0 Q2 = 010  1001 ? Yes so choose 1 0 1 1 0 1 choose radicand digit so that (4Qi + new radicand digit) does not exceed the partial remainder 10^2 + 18 = 118 (base 10) 1 0 0 1 q1 = 1 Q3 = 0101  10101 ? No so choose 0 0 1 0 0 1 0 0 0 0 0 0 q0 = 0 Q4 = 01010 = 1010 1 0 0 1 0 S remainder = 1810

Shift/Subtract Square Root Square root as repeated shifts & subtracts . Q square root . Z fixed point radicand 1  Z < 4 partial remainder array S scaled remainder (Z - Q2)

Restoring Square Root Recursion Notation Z radicand z1z0 . z-1z-2 … z-l (1  Z < 4) Q square root 1 . q-1q-2 … q-l (1  Q < 2) S scaled rem s1s0 . s-1s-2 … s-l (0  S < 4) Basic recurrence S0 = Z – 1, Q0 = 1, Sl = S and Ql = Q Sj = 2Sj-1 – q-j(2Qj-1 + 2-jq-j) where q-j  (0,1) (Qj-1 =  2-kq-k for k=0 to j-1) Qj-1 is the partially developed root up to its –j-1th digit S1 = 2S0 – q-1(2Q0 + 2-1q-1) = 2(Z-1) – q-1 (2*1 + 2-1q-1) The amount subtracted from 2Si-1 is either 2Qj-1 + 2-j if q-j is 1 or 0 if q-j is 0 Write highlighted iteration on the board for reference in later slides

Square Root Digit Selection First perform the trial subtraction STj = 2Sj-1 – (2Qj-1 + 2-j) 0 if STj < 0  Sj = 2Sj-1 q-j = 1 if STj  0  Sj = STj stop when Sj = 0 or after l iterations Really doing if STj < 0 then q-j= 0 & Sj = STj + (2Qj-1 + 2-j) if STj  0 then q-j= 1 & Sj = STj restore op average number of operations in binary n subtracts (one per digit) n/2 add (restores) on average n one bit left shifts 3n/2 add/subtract cycles NO WAY TO PARALLELIZE since quotient digits are formed serially

Binary Square Root Example Z radicand 0 1. 1 1 0 1 1 0 0 0. 1 1 0 1 1 0 S0 = Z-1 q0 = 1 Q0 = 1. 0 1. 1 0 1 1 0 0 1 0. 1 -(2*(1.)+2-1) 1 1. 0 0 1 1 0 0 q-1 = 0 Q1 = 1.0 restore 0 1. 1 0 1 1 0 0 S1 = S0 1 1. 0 1 1 0 0 0 For lecture 1 0. 0 1 -(2*(1.0)+2-2) 0 1. 0 0 1 0 0 0 q-2 = 1 Q2 = 1.01 S2 1 0. 0 1 0 0 0 0 1 0. 1 0 1 -(2*(1.01)+2-3) 1 1. 1 0 1 0 0 0 q-3 = 0 Q3 = 1.010 restore 1 0. 0 1 0 0 0 0 S3 = S2 . . .

Left Shift Serial Square Root Shift left radicand square root Z or S Z or Q scaled remainder q-j 2Qj-1 n bit adder n bits Control sequencer and root select logic  2-jq-j Add/subt sign

Nonrestoring Square Root Can develop in a similar way a nonrestoring square root algorithm where q-j  (-1,1) and the resulting root can be converted on-the-fly to a binary format Expand this next time!

Higher Radix Square Root To speed up square root can use a similar higher radix approach as used in (SRT) division Sj = rSj-1 – q-j(2Qj-1 + r-jq-j) The problem is, as expected, square root digit selection (a table lookup process). With proper care the same lookup table can be used for QDS and RDS (example for base 4) Division: Sj = 4 Sj-1 – q-jD SqRoot: Sj = 4 Sj-1 – q-j(2Qj-1 + 4-jq-j) with q-j=[-2,2] Expand this next time!

Square Root by Convergence Newton-Raphson method f(X) = X2 –Z that has a root at X = Z Basic iteration of Xi+1 = 0.5(Xi + Z/Xi) Requires a division, an addition, a bit right shift Initial guess of X0 = 2 Xi will remain in range 1  Xi <2 Better initial guesses give faster convergence Quadratic convergence from above Many other schemes Find reciprocal of Z and then multiply by Z to obtain Z using f(X) = 1/X2 – Z that has a root at X = 1/Z

Decimal Example Find Z where Z = 2.410 (Z = 1.549 193 . . .) X0 = (read from table) = 1.5 0  10-1 X1 = 0.5(X0 + 2.4/X0) = 1.550 000 000 1  10-2 X2 = 0.5(X1 + 2.4/X1) = 1.549 193 548 2  10-4 For lecture X3 = 0.5(X2 + 2.4/X2) = 1.549 193 338 2  10-8

Nonrestoring Array Square-Rooter z-1 z-2 1 1 xor z-3 z-4 q-1 1 FA z-5 z-6 q-2 1 derived directly from the dot notation for nonrestoring square root the controlled add/subtract cell depend on the sign of the preceding partial remainder z-7 z-8 q-3 1 q-4 s-1 s-2 s-3 s-4 s-5 s-6 s-7 s-8

Key References Agrawal, High-speed arithmetic arrays, IEEE Trans. on Computers, 28(3):215-224, 1979. Ercegovac and Lang, Division and Square Root: Digit Recurrence Algorithms and Implementations, Kluwer, 1994. Montuschi and Mezzalama, A survey of square rooting algorithms, Proc. IEEE, Vol 137, pp. 31-40, 1990. Parhami, Computer Arithmetic, Oxford Univ. Press, 1999. Schwartz and Flynn, Hardware starting approximation methods and its application to the square root operation, IEEE Trans. on Computers, 45(12):1356-1369, 1996. Zurawski and Gosling, Design of a high-speed square root, multiply and divide unit, IEEE Trans. on Computers, 36(1):13-23, 1987.