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8661 bytes, as of the version from 2026-09-23 03:33. Recorded here, not run.
"""Values of xi(u) in the Dickman-de Bruijn estimate -- numberdb.org/T423.
For u > 1, xi(u) is the positive root of
exp(x) - 1 = u*x.
The row u = 1 stores the limiting value 0 exactly. The table stores every
two-decimal argument 1.00 <= u <= 12.00.
Run it with SageMath:
$ sage -pip install numberdb # once
$ sage -python generate.py # check the table against this code
$ sage -python generate.py --publish # fill the draft, with NUMBERDB_API_KEY set
Each non-boundary row is enclosed by bisection on exp(x) - 1 - u*x. The signs
at the final bracket endpoints are checked in ball arithmetic, and the bracket
is the stored enclosure.
"""
import os
import sys
import numberdb.sage as numberdb
from numberdb._write import to_text
from sage.rings.rational_field import QQ
from sage.rings.real_arb import RealBallField
from sage.rings.real_mpfr import RealField
TABLE = os.environ.get("NUMBERDB_TABLE", "T423")
DIGITS = 100
U_MIN_CENTS = 100
U_MAX_CENTS = 1200
# Bits of working precision beyond what the written digits need. Measured at
# 100 digits: the narrowest final sign check occurs near u = 1.01, and still
# has enough margin to decide every bracket endpoint.
WORKING_GUARD = 64
# Decimal places by which the root bracket is narrower than the values written.
BRACKET_GUARD = 8
HYPERGRAPH_MINIMISER_TABLE = (
"Minimisers_in_the_core_threshold_formula_of_random_k-uniform_hypergraphs"
)
def _key_from_stdin():
if os.environ.get("NUMBERDB_KEY_FROM_STDIN") != "1":
return
token = sys.stdin.read().strip()
if "=" in token and token.split("=", 1)[0].isupper():
token = token.split("=", 1)[1].strip().strip("'\"")
if token:
os.environ["NUMBERDB_API_KEY"] = token
def _fields(digits, guard=WORKING_GUARD):
bits = numberdb.bits(digits, losing=guard)
return RealBallField(bits), RealField(bits)
def _root_function(x, u):
return x.exp() - 1 - u * x
def xi(u, digits, guard=WORKING_GUARD):
"""A ball enclosing the positive root of exp(x) - 1 = u*x."""
u = QQ(u)
if u == 1:
return 0
if u < 1:
raise ValueError("this table uses the positive branch for u >= 1")
RBF, RR = _fields(digits, guard)
ub = RBF(u)
ur = RR(u)
delta = RR(10) ** (-(digits + BRACKET_GUARD))
lo = RR(0)
hi = RR(1)
while _root_function(RBF(hi), ub) <= 0:
hi *= 2
# The zero root is always present, so bisection starts with lo = 0 and
# never accepts that endpoint as the root.
while hi - lo > delta / 4:
mid = (lo + hi) / 2
if mid == 0:
mid = delta
value = _root_function(RBF(mid), ub)
if value.contains_zero():
raise ArithmeticError("could not decide sign at x=%s for u=%s" % (mid, u))
if value < 0:
lo = mid
else:
hi = mid
x0 = (lo + hi) / 2
left = _root_function(RBF(x0 - delta), ub)
right = _root_function(RBF(x0 + delta), ub)
if not (left < 0 and right > 0):
raise ArithmeticError("no sign change across xi(u) bracket for u=%s" % u)
root = RBF(x0).add_error(delta)
residual = _root_function(root, ub)
if not residual.contains_zero():
raise ArithmeticError("root enclosure fails defining equation for u=%s" % u)
if not root > 0:
raise ArithmeticError("positive root enclosure is not positive for u=%s" % u)
# Keep `ur` live in this function so accidental float coercions are caught
# by the type checks during development.
assert ur > 1
return root
def _u_values():
for cents in range(U_MIN_CENTS, U_MAX_CENTS + 1):
yield QQ(cents) / QQ(100)
def _integer_u(u):
return u.denominator() == 1 and 2 <= int(u) <= 7
def _integer_entry(u):
k = int(u) + 1
return {
"equals": (
r"HREF{%s#%d,2}[$\lambda_{%d,2}$]"
% (HYPERGRAPH_MINIMISER_TABLE, k, k)
),
"comment": (
r"This is the $r=2$ core-threshold minimiser "
r"$\lambda_{%d,2}$." % k
),
}
class DickmanDeBruijnXi(numberdb.Generator):
table = TABLE
parameters = ("u",)
type = "R"
digits = DIGITS
rigour = "proven"
def enumerate(self):
for u in _u_values():
yield {"u": str(u)}
def value(self, params, digits):
u = QQ(params["u"])
if u == 1:
return {
"number": 0,
"comment": (
r"The positive branch used for $u>1$ has limiting value "
r"$0$ at $u=1$."
),
}
entry = {"number": xi(u, digits)}
if _integer_u(u):
entry.update(_integer_entry(u))
return entry
def _mpmath_lambert_xi(u, digits=140):
"""Independent computation from the Lambert W formula."""
from mpmath import mp
mp.dps = digits + 30
u = QQ(u)
u = mp.mpf(int(u.numerator())) / mp.mpf(int(u.denominator()))
value = -mp.lambertw(-mp.e ** (-1 / u) / u, -1) - 1 / u
if abs(mp.im(value)) > mp.mpf(10) ** (-(digits + 10)):
raise ArithmeticError("Lambert W check produced a non-real value")
return mp.re(value)
def _check_lambert_formula():
RBF, _RR = _fields(120, guard=WORKING_GUARD)
tolerance = RBF("1e-95")
for u in _u_values():
if u == 1:
continue
computed = xi(u, DIGITS)
independent = RBF(str(_mpmath_lambert_xi(u)))
if not (abs(computed - independent) < tolerance):
raise ArithmeticError(
"Lambert W comparison failed for u=%s: %s vs %s"
% (u, computed, independent)
)
def _check_hypergraph_minimisers():
table = numberdb.table("T289")
numbers = table["Numbers"]
RBF, _RR = _fields(120, guard=WORKING_GUARD)
tolerance = RBF("1e-98")
for u in range(2, 8):
ours = xi(QQ(u), DIGITS)
k = str(u + 1)
theirs = numbers[k]["2"]
if isinstance(theirs, dict):
theirs = theirs["number"]
other = RBF(theirs)
if not (abs(ours - other) < tolerance):
raise ArithmeticError(
"T289 comparison failed for u=%d: %s vs %s" % (u, ours, other)
)
def run_integrity_checks():
"""Independent checks for the identities stated in the table."""
_check_lambert_formula()
_check_hypergraph_minimisers()
def fill_draft_once(generator, message):
"""Fill a fresh draft without the client's empty upsert probe."""
from numberdb._generate import (
_check_precision,
_check_rigour,
_producer,
_run_name,
_source_files,
)
from numberdb._write import Entries, attach, submit_entries
table = generator.table
run = _run_name(generator)
entries = Entries(*generator.parameters)
for params in generator.enumerate():
params = dict(params)
wanted = generator.digits_for(params)
entry = generator._entry(params, wanted)
value = entry["number"]
identity = ",".join(str(params[name]) for name in generator.parameters)
_check_rigour(generator, table, identity, value)
written = to_text(value, wanted, generator.format)
_check_precision(table, identity, written, wanted, lowering=False)
record = dict(entry)
record.pop("digits", None)
entries.add(**params, **record, digits=wanted)
answer = submit_entries(
table,
entries,
message=message,
produced_by=_producer(
generator,
assisted_by=os.environ.get("NUMBERDB_ASSISTED_BY", "codex"),
),
upsert=False,
run=run,
rigour=generator.rigour,
)
stored = []
for name, body in sorted(_source_files(generator).items()):
attach(table, name, body, run=run, message=message,
rigour=generator.rigour)
stored.append(name)
return {
"tid": answer.get("tid", table),
"revision": answer.get("revision"),
"entries": len(entries),
"files": stored,
}
if __name__ == "__main__":
_key_from_stdin()
generator = DickmanDeBruijnXi()
run_integrity_checks()
if os.environ.get("NUMBERDB_PUBLISH") == "1" or "--publish" in sys.argv:
print(fill_draft_once(
generator,
message=(
"xi(u) in the Dickman-de Bruijn estimate on the two-decimal "
"grid 1 <= u <= 12"
),
))
else:
report = generator.verify(sample=None)
print(report)
sys.exit(0 if report.ok else 1)