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Issue #12080: Fix a performance issue in Decimal._power_exact that causes some corner-case Decimal.__pow__ calls to take an unreasonably long time.

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This commit is contained in:
Mark Dickinson 2011-06-04 18:14:23 +01:00
parent 4dfcb1a00d
commit 7ce0fa8775
3 changed files with 96 additions and 37 deletions
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117
Lib/decimal.py
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@ -2001,9 +2001,9 @@ def _power_exact(self, other, p):
nonzero. For efficiency, other._exp should not be too large,
so that 10**abs(other._exp) is a feasible calculation."""
# In the comments below, we write x for the value of self and
# y for the value of other. Write x = xc*10**xe and y =
# yc*10**ye.
# In the comments below, we write x for the value of self and y for the
# value of other. Write x = xc*10**xe and abs(y) = yc*10**ye, with xc
# and yc positive integers not divisible by 10.
# The main purpose of this method is to identify the *failure*
# of x**y to be exactly representable with as little effort as
@ -2011,13 +2011,12 @@ def _power_exact(self, other, p):
# eliminate the possibility of x**y being exact. Only if all
# these tests are passed do we go on to actually compute x**y.
# Here's the main idea. First normalize both x and y. We
# express y as a rational m/n, with m and n relatively prime
# and n>0. Then for x**y to be exactly representable (at
# *any* precision), xc must be the nth power of a positive
# integer and xe must be divisible by n. If m is negative
# then additionally xc must be a power of either 2 or 5, hence
# a power of 2**n or 5**n.
# Here's the main idea. Express y as a rational number m/n, with m and
# n relatively prime and n>0. Then for x**y to be exactly
# representable (at *any* precision), xc must be the nth power of a
# positive integer and xe must be divisible by n. If y is negative
# then additionally xc must be a power of either 2 or 5, hence a power
# of 2**n or 5**n.
#
# There's a limit to how small |y| can be: if y=m/n as above
# then:
@ -2089,21 +2088,43 @@ def _power_exact(self, other, p):
return None
# now xc is a power of 2; e is its exponent
e = _nbits(xc)-1
# find e*y and xe*y; both must be integers
if ye >= 0:
y_as_int = yc*10**ye
e = e*y_as_int
xe = xe*y_as_int
else:
ten_pow = 10**-ye
e, remainder = divmod(e*yc, ten_pow)
if remainder:
return None
xe, remainder = divmod(xe*yc, ten_pow)
if remainder:
return None
if e*65 >= p*93: # 93/65 > log(10)/log(5)
# We now have:
#
# x = 2**e * 10**xe, e > 0, and y < 0.
#
# The exact result is:
#
# x**y = 5**(-e*y) * 10**(e*y + xe*y)
#
# provided that both e*y and xe*y are integers. Note that if
# 5**(-e*y) >= 10**p, then the result can't be expressed
# exactly with p digits of precision.
#
# Using the above, we can guard against large values of ye.
# 93/65 is an upper bound for log(10)/log(5), so if
#
# ye >= len(str(93*p//65))
#
# then
#
# -e*y >= -y >= 10**ye > 93*p/65 > p*log(10)/log(5),
#
# so 5**(-e*y) >= 10**p, and the coefficient of the result
# can't be expressed in p digits.
# emax >= largest e such that 5**e < 10**p.
emax = p*93//65
if ye >= len(str(emax)):
return None
# Find -e*y and -xe*y; both must be integers
e = _decimal_lshift_exact(e * yc, ye)
xe = _decimal_lshift_exact(xe * yc, ye)
if e is None or xe is None:
return None
if e > emax:
return None
xc = 5**e
@ -2117,19 +2138,20 @@ def _power_exact(self, other, p):
while xc % 5 == 0:
xc //= 5
e -= 1
if ye >= 0:
y_as_integer = yc*10**ye
e = e*y_as_integer
xe = xe*y_as_integer
else:
ten_pow = 10**-ye
e, remainder = divmod(e*yc, ten_pow)
if remainder:
return None
xe, remainder = divmod(xe*yc, ten_pow)
if remainder:
return None
if e*3 >= p*10: # 10/3 > log(10)/log(2)
# Guard against large values of ye, using the same logic as in
# the 'xc is a power of 2' branch. 10/3 is an upper bound for
# log(10)/log(2).
emax = p*10//3
if ye >= len(str(emax)):
return None
e = _decimal_lshift_exact(e * yc, ye)
xe = _decimal_lshift_exact(xe * yc, ye)
if e is None or xe is None:
return None
if e > emax:
return None
xc = 2**e
else:
@ -5529,6 +5551,27 @@ def _normalize(op1, op2, prec = 0):
_nbits = int.bit_length
def _decimal_lshift_exact(n, e):
""" Given integers n and e, return n * 10**e if it's an integer, else None.
The computation is designed to avoid computing large powers of 10
unnecessarily.
>>> _decimal_lshift_exact(3, 4)
30000
>>> _decimal_lshift_exact(300, -999999999) # returns None
"""
if n == 0:
return 0
elif e >= 0:
return n * 10**e
else:
# val_n = largest power of 10 dividing n.
str_n = str(abs(n))
val_n = len(str_n) - len(str_n.rstrip('0'))
return None if val_n < -e else n // 10**-e
def _sqrt_nearest(n, a):
"""Closest integer to the square root of the positive integer n. a is
an initial approximation to the square root. Any positive integer