Question

使用python 3.5.2

>>> from decimal import Decimal
>>> Decimal('12') % Decimal('0.01')
Decimal('0.00')
>>> Decimal('234567') % Decimal('0.01')
Decimal('0.00')

按预期工作。但...

>>> Decimal('7316717653133062491922511967442657474235534919493496983520312774506326239578318016984801869478851843858615607891129494954595017379583319528532088055111254069874715852386305071569329096329522744304355766896648950445244523161731856403098711121722383113622298934233803081353362766142828064444866452387493035890729629049156044077239071381051585930796086670172427121883998797908792274921901699720888093776657273330010533678812202354218097512545405947522435258490771167055601360483958644670632441572215539753697817977846174064955149290862569321978468622482839722413756570560574902614079729686524145351004748216637048440319989000889524345065854122758866688116427171479924442928230863465674813919123162824586178664583591245665294765456828489128831426076900422421902267105562632111110937054421750694165896040807198403850962455444362981230987879927244284909188845801561660979191338754992005240636899125607176060588611646710940507754100225698315520005593572972571636269561882670428252483600823257530420752963450') % Decimal('0.01')
Traceback (most recent call last):
File "<stdin>", line 1, in <module>
decimal.InvalidOperation: [<class 'decimal.DivisionImpossible'>]

编辑：这是我找到的可能导致此错误的最小数字：

>>> Decimal(10**26) % Decimal('0.01') 
Traceback (most recent call last):
  File "<stdin>", line 1, in <module>
decimal.InvalidOperation: [<class 'decimal.DivisionImpossible'>]

为什么Decimal(very_large_int) % Decimal('0.01')会出现此错误？我认为Decimal能够处理非常大的数字吗？

Answer 1

作为L3viathan answered，问题是结果（不是结果 - 这是＆＃34;隐藏的部分＆＃34;我在评论中提及）已超出可用范围精度。

如果我们使用Python2，隐藏部分会更明显：

Traceback (most recent call last):
  File "/tmp/d.py", line 24, in <module>
    print(big % Decimal('0.01'))
  File "/usr/local/lib/python2.7/decimal.py", line 1460, in __mod__
    remainder = self._divide(other, context)[1]
  File "/usr/local/lib/python2.7/decimal.py", line 1381, in _divide
    'quotient too large in //, % or divmod')
  File "/usr/local/lib/python2.7/decimal.py", line 3873, in _raise_error
    raise error(explanation)
InvalidOperation: quotient too large in //, % or divmod

基本上，a % b是通过同时进行除法和模数来实现的（在Knuth第2卷中使用算法D;对于限制为两个全字的C实现，请参见qdivrem.c code I wrote in the early 2000s）。因此，库代码需要两个额外的数字（Decimal('0.01')中小数点右边的位数 - 计算所需的实际位数并不像下面的big那么简单，因为我们必须看看指数）来计算中间商。

十进制库直接在C中为Python3重新实现，它隐藏了细节，但两种方法的解决方法是相同的：扩展精度。这是一个捕获异常并再次尝试的示例源例程，但是使用魔法常量2。

from __future__ import print_function
import decimal
Decimal = decimal.Decimal
import traceback
big = Decimal(
    '731671765313306249192251196744265747423553491949349698352031277'
    '4506326239578318016984801869478851843858615607891129494954595017379'
    '5833195285320880551112540698747158523863050715693290963295227443043'
    '5576689664895044524452316173185640309871112172238311362229893423380'
    '3081353362766142828064444866452387493035890729629049156044077239071'
    '3810515859307960866701724271218839987979087922749219016997208880937'
    '7665727333001053367881220235421809751254540594752243525849077116705'
    '5601360483958644670632441572215539753697817977846174064955149290862'
    '5693219784686224828397224137565705605749026140797296865241453510047'
    '4821663704844031998900088952434506585412275886668811642717147992444'
    '2928230863465674813919123162824586178664583591245665294765456828489'
    '1288314260769004224219022671055626321111109370544217506941658960408'
    '0719840385096245544436298123098787992724428490918884580156166097919'
    '1338754992005240636899125607176060588611646710940507754100225698315'
    '520005593572972571636269561882670428252483600823257530420752963450')
try:
    print(big % Decimal('0.01'))
except decimal.DecimalException:
    traceback.print_exc()
    print('')
    ctx = decimal.getcontext()
    print('failed because precision was', ctx.prec, 'and big is',
        len(big.as_tuple().digits), 'digits long')
    print('trying again with 2 more digits')
    with decimal.localcontext() as ctx:
        ctx.prec = len(big.as_tuple().digits) + 2
        try:
            print(big % Decimal('0.01'))
        except decimal.DecimalException:
            traceback.print_exc()

使用Python2：

$ python2 /tmp/d.py
Traceback (most recent call last):
  File "/tmp/d.py", line 24, in <module>
    print(big % Decimal('0.01'))
  File "/usr/local/lib/python2.7/decimal.py", line 1460, in __mod__
    remainder = self._divide(other, context)[1]
  File "/usr/local/lib/python2.7/decimal.py", line 1381, in _divide
    'quotient too large in //, % or divmod')
  File "/usr/local/lib/python2.7/decimal.py", line 3873, in _raise_error
    raise error(explanation)
InvalidOperation: quotient too large in //, % or divmod

failed because precision was 28 and big is 1000 digits long
trying again with 2 more digits
0.00

使用Python3：

$ python3 /tmp/d.py
Traceback (most recent call last):
  File "/tmp/d.py", line 24, in <module>
    print(big % Decimal('0.01'))
decimal.InvalidOperation: [<class 'decimal.DivisionImpossible'>]

failed because precision was 28 and big is 1000 digits long
trying again with 2 more digits
0.00

请注意，除以非常大的数字实际上更容易：它是0.01的除法，这会导致问题。如果除数上的指数至少为1000 - 28（1e972或更大），我们就不会有问题。

Answer 2

Decimal基于十进制算术规范。您可以看到here＆＃34;分区不可能＆＃34;意味着

除数整数或余数运算的整数结果有太多数字（比精度长）。

您可以调整此精度：

>>> decimal.getcontext().prec=10000
>>> Decimal('7316717653133062491922511967442657474235534919493496983520312774506326239578318016984801869478851843858615607891129494954595017379583319528532088
... 0551112540698747158523863050715693290963295227443043557668966489504452445231617318564030987111217223831136222989342338030813533627661428280644448664523874
... 9303589072962904915604407723907138105158593079608667017242712188399879790879227492190169972088809377665727333001053367881220235421809751254540594752243525
... 8490771167055601360483958644670632441572215539753697817977846174064955149290862569321978468622482839722413756570560574902614079729686524145351004748216637
... 0484403199890008895243450658541227588666881164271714799244429282308634656748139191231628245861786645835912456652947654568284891288314260769004224219022671
... 0556263211111093705442175069416589604080719840385096245544436298123098787992724428490918884580156166097919133875499200524063689912560717606058861164671094
... 0507754100225698315520005593572972571636269561882670428252483600823257530420752963450') % Decimal('0.01')
Decimal('0.00')

decimal.InvalidOperation，Division不可能为非常大的数字

2 个答案: