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git-subtree-dir: src/minisketch git-subtree-split: 89629eb2c7e262b39ba489b93b111760baded4b3
85 lines
2.5 KiB
Python
85 lines
2.5 KiB
Python
import bisect
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INPUT_BITS = 32
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TABLE_BITS = 5
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INT_BITS = 64
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EXACT_FPBITS = 256
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F = RealField(100) # overkill
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def BestOverApproxInvLog2(mulof, maxd):
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"""
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Compute denominator of an approximation of 1/log(2).
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Specifically, find the value of d (<= maxd, and a multiple of mulof)
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such that ceil(d/log(2))/d is the best approximation of 1/log(2).
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"""
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dist=1
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best=0
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# Precomputed denominators that lead to good approximations of 1/log(2)
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for d in [1, 2, 9, 70, 131, 192, 445, 1588, 4319, 11369, 18419, 25469, 287209, 836158, 3057423, 8336111, 21950910, 35565709, 49180508, 161156323, 273132138, 385107953, 882191721]:
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kd = lcm(mulof, d)
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if kd <= maxd:
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n = ceil(kd / log(2))
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dis = F((n / kd) - 1 / log(2))
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if dis < dist:
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dist = dis
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best = kd
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return best
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LOG2_TABLE = []
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A = 0
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B = 0
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C = 0
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D = 0
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K = 0
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def Setup(k):
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global LOG2_TABLE, A, B, C, D, K
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K = k
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LOG2_TABLE = []
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for i in range(2 ** TABLE_BITS):
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LOG2_TABLE.append(int(floor(F(K * log(1 + i / 2**TABLE_BITS, 2)))))
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# Maximum for (2*x+1)*LogK2(x)
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max_T = (2^(INPUT_BITS + 1) - 1) * (INPUT_BITS*K - 1)
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# Maximum for A
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max_A = (2^INT_BITS - 1) // max_T
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D = BestOverApproxInvLog2(2 * K, max_A * 2 * K)
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A = D // (2 * K)
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B = int(ceil(F(D/log(2))))
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C = int(floor(F(D*log(2*pi,2)/2)))
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def LogK2(n):
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assert(n >= 1 and n < (1 << INPUT_BITS))
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bits = Integer(n).nbits()
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return K * (bits - 1) + LOG2_TABLE[((n << (INPUT_BITS - bits)) >> (INPUT_BITS - TABLE_BITS - 1)) - 2**TABLE_BITS]
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def Log2Fact(n):
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# Use formula (A*(2*x+1)*LogK2(x) - B*x + C) / D
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return (A*(2*n+1)*LogK2(n) - B*n + C) // D + (n < 3)
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RES = [int(F(log(factorial(i),2))) for i in range(EXACT_FPBITS * 10)]
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best_worst_ratio = 0
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for K in range(1, 10000):
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Setup(K)
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assert(LogK2(1) == 0)
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assert(LogK2(2) == K)
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assert(LogK2(4) == 2 * K)
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good = True
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worst_ratio = 1
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for i in range(1, EXACT_FPBITS * 10):
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exact = RES[i]
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approx = Log2Fact(i)
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if not (approx <= exact and ((approx == exact) or (approx >= EXACT_FPBITS and exact >= EXACT_FPBITS))):
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good = False
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break
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if worst_ratio * exact > approx:
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worst_ratio = approx / exact
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if good and worst_ratio > best_worst_ratio:
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best_worst_ratio = worst_ratio
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print("Formula: (%i*(2*x+1)*floor(%i*log2(x)) - %i*x + %i) / %i; log(max_ratio)=%f" % (A, K, B, C, D, RR(-log(worst_ratio))))
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print("LOG2K_TABLE: %r" % LOG2_TABLE)
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