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with older people, while younger people accrue very little value from the typical DB plan. So, employers with DB plans are voting with their dollars and saying they want to target older employees and provide larger retirement dollars to them. The plan freezes, like that of HP, are clear reactions to this skewing. HP actually allowed current employees to stay in the existing DB plan, basically freezing eligibility. For new employees, they will participate in a new DC plan. This type of generational freeze is used because of concerns about changing the deal with existing employees and in reaction to the significant negative reaction at other large companies who made changes to retirement plans, for example, IBM s trouble with its cash balance conversion. Ignoring these transitional issues for a moment, the HP change represents a significant change in where HP will be targeting their retirement dollars in the future. By switching to a DC plan, it says that HP will pay comparable dollars regardless of an employee s age. So, for example, a 30-year-old will get a 5 percent of pay contribution and a 60-year-old will also get a 5 percent of pay contribution. Whereas, if these same two employees were in a traditional DB plan, the 30-year-old would accrue a benefit costing 1 percent of pay, whereas the 60 year old employee s benefit would cost nearly 10 percent of pay.

If more space can be afforded and faster divide operations are required between arbitrary numbers, more sophisticated techniques are required as described in the following section.

n-bit Montgomery multiplier (figure 15.6)

The Goldschmidt method can be used when high-speed division operations need to be pipelined to maximize throughput. This method falls into a class of successive approximation algorithms (sometimes called iterative, but not to be confused with the implementation from the previous section) that approach the true quotient with each recursion of the algorithm. The nice feature about algorithms such as Goldschmidt is that they can be pipelined in an ef cient manner. In other words, the hardware can be constructed such that one division operation can be completed on each edge of the clock and can be implemented with less area than would be required for a brute-force unrolling of the iterative method discussed in the previous section.

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01, 10, 11

DEFINED BENEFIT PLANS AS WEALTH ACCUMULATION VEHICLES For many small, closely held firms with owners over the age of 40, this age skewing of DB plan contributions can allow them to shelter significant dollars into a DB plan compared with a traditional DC plan. Figure 6.1 illustrates this difference in the maximum tax deductible contributions of DB and DC plans.

The Goldschmidt method provides a way to pipeline the division process in a manner that is much more ef cient than unrolling the loop of the iterative method.

architecture circuit of exp_data_path is signal mult_in1, mult_in2, mult_out, reg_e_in, e_t, y_t, exp_n, exp_2n: std_logic_vector(n-1 downto 0); component Montgomery_multiplier...end component; begin exp_2n<=conv_std_logic_vector(int_exp_2n, n); with sel select mult_in1<=y when "00", e_t when others; with sel select mult_in2<=exp_2n when "00", e_t when "01", y_t when "10", one when others; multiplier: Montgomery_multiplier port map (mult_in1, mult_in2, mult_out); exp_n<=conv_std_logic_vector(int_exp_n, n); with sel select reg_e_in<=exp_n when "00", mult_out when others; process(clk) begin if clk event and clk= 1 then if enable_e= 1 then e_t<=reg_e_in; end if; end if;

(continues)

The idea behind the Goldschmidt algorithm is to calculate Q N/D by approximating the value 1/D for multiplication with N, and then approach the true quotient through successive approximations. This is useful for large numbers such as those represented by the IEEE 754 oating-point standard, which has 64-bit operators. Clearly, a look-up table of the order 250 bits (potential size of a oating-point mantissa) is infeasible, but drastically reduced look-up tables of the order 210 (a typical value for a Goldschmidt implementation) can be practical. There are a number of papers that provide a ne theoretical foundation for the Goldschmidt algorithm. Of particular interest to the design engineer is the number of iterations required to produce an error within acceptable bounds. The steps to approximate Q N/D using the Goldschmidt algorithm are as follows: 1. Move the xed point for N and D to locations such that N ! 1 and D , 2. In the context of oating-point operations, this is referred to as normalizing the numerator and denominator. 2. Start with an initial approximation to 1/D by use of a look-up table and call it L1. Eight to 16 bits of precision is often enough depending on the application. 3. Calculate the rst approximation to q1 L1N and the error term e1 L1D (which will approach 1 as the iterations approach in nity).

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