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normalized changes to area, delay, and power with respect to KSA32, using data from Table 2. We assume that the same relative changes would occur to the Bitmain ASIC in terms of area, delay, and power when its adders are changed. For example, to predict the change in profits from adopt- ing a GDA(1,4) design, we first calculate the normalized changes to area, delay, and power between GDA(1,4) and KSA32 based hashing core (Table 2). Next, we scale the Bitmain ASIC’s area, delay, and power by these normal- ized values to predict the modified Bitmain ASIC’s area, delay, and power. Finally, profit is derived from these pre- dicted values, the Bitcoin mining difficulty, exchange rate, and price of electricity. The error rate at each operating point is found through simulation (Section 4.1). Each simulated SHA-256 round has error rate Ei(f), the sum of functional and operational error rates. Bitcoin requires two rounds for each nonce it- eration; hence, we can extrapolate to calculate cumulative error rate E(f), assuming the hash inputs and outputs to be uniform random variables. Ei(f) = Ef +Eo(f) E(f) = 1−[1−Ei(f)]2 (5) The frequency of each design is swept above its nominal value f0 while keeping voltage fixed. Hashing is completely parallel, so the hash rate H(f) ∝ f. The design’s normal- ized operating frequency is Fˆ(f) = f/f0. Combining with Equation 2, we expect the effective hash rate to be: ̃ 1−E(f) ˆ H(f)= Aˆ ·H0·F(f) (6) At 65nm with high duty cycle, dynamic power dominates leakage power in the designs, so P(f) ∝ f, implying: P ̃(f)=P0 ·Fˆ(f) (7) Substituting these expressions into Equation 3, we deter- mine p ̃0(t0,f), the predicted profit at time t0 of the approx- imate Bitmain ASIC. 5. RESULTS We perform the synthesis and simulations discussed above for each adder configuration. Table 1 lists the adders’ delay and area. Each adder was inserted into the hashing core pipelines in the CPA slots indicated in Figure 2. The re- sulting hashing core area and delay are provided in Table 2. Approximate variants are highlighted in gray. Figure 4 shows the error rate-frequency characteristic Ei (f ) of each hashing core for various adders after simulating a full round of SHA-256. The resulting frequency-profit relation is shown in Figure 5. There are several conclusions to be drawn from the re- sults. First, the results show that approximation is feasible in the context of Bitcoin mining since some approximate adder choices raise profits with respect to their exact im- plementation. For example, observing the frequency-error characteristics of Figure 4, the hashing cores corresponding to both approximate adders, GDA(1,4) and KSA16, have neg- ligible error rates at nominal frequency. Also, their nominal operating frequencies are higher than their non-approximate counterparts, CLA and KSA32 respectively. Consequently, Figure 5 shows that profits of both approximate adders at nominal frequency are greater than that of the correspond- ing accurate adders. Figure 4: Frequency/Error Rate Trade-off for Cores 1 0.8 0.6 0.4 0.2 0 0.2 0.3 0.4 frequency (GHz) Figure 5: Frequency/Profit Trade-off for Cores ·10−7 8 RCA CLA GDA(1,4) KSA32 KSA16 6 4 2 0 −2 −4 0.5 0.6 0.7 RCA CLA GDA(1,4) KSA32 KSA16 0.2 0.3 0.4 frequency (GHz) Second, the results show that approximation can increase mining profits significantly. For example, KSA16 performs significantly better than its non-approximate counterpart, producing 15% greater profit at its nominal frequency. A further increase in profit can be gained by operating the design past its nominal frequency. As shown in Figure 5, both KSA designs produce approximately 15% greater profit compared to nominal at their peaks. This indicates KSA16 can raise profits by 30%, 15% from functional approximation and 15% from operational approximation. Third, while mining profit depends on both delay and area of the hashing core, the results show that in a choice between adders with low delay and low area, adders with low delay should be chosen to maximize mining profits. For example, KSA32 generates more profits than both RCA and CLA at all frequency operating points in spite of the fact that the error rate of both RCA and CLA rises more slowly when pushed past nominal frequency. This is not surprising con- 0.5 0.6 0.7 profit (USD/s) error rate (ER)PDF Image | Approximate Bitcoin Mining
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