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import bisect
INPUT_BITS = 32
TABLE_BITS = 5
INT_BITS = 64
EXACT_FPBITS = 256
F = RealField(100) # overkill
def BestOverApproxInvLog2(mulof, maxd):
"""
Compute denominator of an approximation of 1/log(2).
Specifically, find the value of d (<= maxd, and a multiple of mulof)
such that ceil(d/log(2))/d is the best approximation of 1/log(2).
"""
dist=1
best=0
# Precomputed denominators that lead to good approximations of 1/log(2)
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]:
kd = lcm(mulof, d)
if kd <= maxd:
n = ceil(kd / log(2))
dis = F((n / kd) - 1 / log(2))
if dis < dist:
dist = dis
best = kd
return best
LOG2_TABLE = []
A = 0
B = 0
C = 0
D = 0
K = 0
def Setup(k):
global LOG2_TABLE, A, B, C, D, K
K = k
LOG2_TABLE = []
for i in range(2 ** TABLE_BITS):
LOG2_TABLE.append(int(floor(F(K * log(1 + i / 2**TABLE_BITS, 2)))))
# Maximum for (2*x+1)*LogK2(x)
max_T = (2^(INPUT_BITS + 1) - 1) * (INPUT_BITS*K - 1)
# Maximum for A
max_A = (2^INT_BITS - 1) // max_T
D = BestOverApproxInvLog2(2 * K, max_A * 2 * K)
A = D // (2 * K)
B = int(ceil(F(D/log(2))))
C = int(floor(F(D*log(2*pi,2)/2)))
def LogK2(n):
assert(n >= 1 and n < (1 << INPUT_BITS))
bits = Integer(n).nbits()
return K * (bits - 1) + LOG2_TABLE[((n << (INPUT_BITS - bits)) >> (INPUT_BITS - TABLE_BITS - 1)) - 2**TABLE_BITS]
def Log2Fact(n):
# Use formula (A*(2*x+1)*LogK2(x) - B*x + C) / D
return (A*(2*n+1)*LogK2(n) - B*n + C) // D + (n < 3)
RES = [int(F(log(factorial(i),2))) for i in range(EXACT_FPBITS * 10)]
best_worst_ratio = 0
for K in range(1, 10000):
Setup(K)
assert(LogK2(1) == 0)
assert(LogK2(2) == K)
assert(LogK2(4) == 2 * K)
good = True
worst_ratio = 1
for i in range(1, EXACT_FPBITS * 10):
exact = RES[i]
approx = Log2Fact(i)
if not (approx <= exact and ((approx == exact) or (approx >= EXACT_FPBITS and exact >= EXACT_FPBITS))):
good = False
break
if worst_ratio * exact > approx:
worst_ratio = approx / exact
if good and worst_ratio > best_worst_ratio:
best_worst_ratio = worst_ratio
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))))
print("LOG2K_TABLE: %r" % LOG2_TABLE)
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