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82 lines
2.3 KiB
Python
82 lines
2.3 KiB
Python
#
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# Reference :
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# Noble Mushtak and Daniel Lemire, Fast Number Parsing Without Fallback, Software: Practice and Experience 53 (6), 2023 https://arxiv.org/abs/2212.06644
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#
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all_tqs = []
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# Generates all possible values of T[q]
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# Appendix B of Number parsing at a gigabyte per second.
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# Software: Practice and Experience 2021;51(8):1700–1727.
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for q in range(-342, -27):
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power5 = 5 ** -q
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z = 0
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while (1 << z) < power5:
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z += 1
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b = 2 * z + 2 * 64
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c = 2 ** b // power5 + 1
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while c >= (1 << 128):
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c //= 2
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all_tqs.append(c)
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for q in range(-27, 0):
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power5 = 5 ** -q
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z = 0
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while (1 << z) < power5:
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z += 1
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b = z + 127
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c = 2 ** b // power5 + 1
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all_tqs.append(c)
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for q in range(0, 308 + 1):
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power5 = 5 ** q
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while power5 < (1 << 127):
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power5 *= 2
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while power5 >= (1 << 128):
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power5 //= 2
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all_tqs.append(power5)
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# Returns the continued fraction of numer/denom as a list [a0; a1, a2, ..., an]
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def continued_fraction(numer, denom):
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# (look at page numbers in top-left, not PDF page numbers)
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cf = []
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while denom != 0:
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quot, rem = divmod(numer, denom)
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cf.append(quot)
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numer, denom = denom, rem
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return cf
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# Given a continued fraction [a0; a1, a2, ..., an], returns
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# all the convergents of that continued fraction
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# as pairs of the form (numer, denom), where numer/denom is
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# a convergent of the continued fraction in simple form.
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def convergents(cf):
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p_n_minus_2 = 0
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q_n_minus_2 = 1
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p_n_minus_1 = 1
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q_n_minus_1 = 0
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convergents = []
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for a_n in cf:
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p_n = a_n * p_n_minus_1 + p_n_minus_2
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q_n = a_n * q_n_minus_1 + q_n_minus_2
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convergents.append((p_n, q_n))
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p_n_minus_2, q_n_minus_2, p_n_minus_1, q_n_minus_1 = (
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p_n_minus_1,
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q_n_minus_1,
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p_n,
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q_n,
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)
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return convergents
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# Enumerate through all the convergents of T[q] / 2^137 with denominators < 2^64
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found_solution = False
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for j, tq in enumerate(all_tqs):
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for _, w in convergents(continued_fraction(tq, 2 ** 137)):
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if w >= 2 ** 64:
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break
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if (tq * w) % 2 ** 137 > 2 ** 137 - 2 ** 64:
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print(f"SOLUTION: q={j-342} T[q]={tq} w={w}")
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found_solution = True
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if not found_solution:
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print("No solutions!")
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