1 files changed, 85 insertions, 0 deletions
diff --git a/src/minisketch/doc/log2_factorial.sage b/src/minisketch/doc/log2_factorial.sage
new file mode 100644
index 0000000000..afc6d66c57
--- /dev/null
+++ b/src/minisketch/doc/log2_factorial.sage
@@ -0,0 +1,85 @@
+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)