1  import bisect
 2  
 3  INPUT_BITS = 32
 4  TABLE_BITS = 5
 5  INT_BITS = 64
 6  EXACT_FPBITS = 256
 7  
 8  F = RealField(100) # overkill
 9  
10  def BestOverApproxInvLog2(mulof, maxd):
11      """
12      Compute denominator of an approximation of 1/log(2).
13  
14      Specifically, find the value of d (<= maxd, and a multiple of mulof)
15      such that ceil(d/log(2))/d is the best approximation of 1/log(2).
16      """
17      dist=1
18      best=0
19      # Precomputed denominators that lead to good approximations of 1/log(2)
20      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]:
21          kd = lcm(mulof, d)
22          if kd <= maxd:
23              n = ceil(kd / log(2))
24              dis = F((n / kd) - 1 / log(2))
25              if dis < dist:
26                  dist = dis
27                  best = kd
28      return best
29  
30  
31  LOG2_TABLE = []
32  A = 0
33  B = 0
34  C = 0
35  D = 0
36  K = 0
37  
38  def Setup(k):
39      global LOG2_TABLE, A, B, C, D, K
40      K = k
41      LOG2_TABLE = []
42      for i in range(2 ** TABLE_BITS):
43          LOG2_TABLE.append(int(floor(F(K * log(1 + i / 2**TABLE_BITS, 2)))))
44  
45      # Maximum for (2*x+1)*LogK2(x)
46      max_T = (2^(INPUT_BITS + 1) - 1) * (INPUT_BITS*K - 1)
47      # Maximum for A
48      max_A = (2^INT_BITS - 1) // max_T
49      D = BestOverApproxInvLog2(2 * K, max_A * 2 * K)
50      A = D // (2 * K)
51      B = int(ceil(F(D/log(2))))
52      C = int(floor(F(D*log(2*pi,2)/2)))
53  
54  def LogK2(n):
55      assert(n >= 1 and n < (1 << INPUT_BITS))
56      bits = Integer(n).nbits()
57      return K * (bits - 1) + LOG2_TABLE[((n << (INPUT_BITS - bits)) >> (INPUT_BITS - TABLE_BITS - 1)) - 2**TABLE_BITS]
58  
59  def Log2Fact(n):
60      # Use formula (A*(2*x+1)*LogK2(x) - B*x + C) / D
61      return (A*(2*n+1)*LogK2(n) - B*n + C) // D + (n < 3)
62  
63  RES = [int(F(log(factorial(i),2))) for i in range(EXACT_FPBITS * 10)]
64  
65  best_worst_ratio = 0
66  
67  for K in range(1, 10000):
68      Setup(K)
69      assert(LogK2(1) == 0)
70      assert(LogK2(2) == K)
71      assert(LogK2(4) == 2 * K)
72      good = True
73      worst_ratio = 1
74      for i in range(1, EXACT_FPBITS * 10):
75          exact = RES[i]
76          approx = Log2Fact(i)
77          if not (approx <= exact and ((approx == exact) or (approx >= EXACT_FPBITS and exact >= EXACT_FPBITS))):
78              good = False
79              break
80          if worst_ratio * exact > approx:
81              worst_ratio = approx / exact
82      if good and worst_ratio > best_worst_ratio:
83          best_worst_ratio = worst_ratio
84          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))))
85          print("LOG2K_TABLE: %r" % LOG2_TABLE)