decimal.InvalidOperation，除法对于非常大的数字来说是不可能的

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：这是我发现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'>]

为什么会Decimal(very_large_int) % Decimal('0.01')出现这个错误？我认为 Decimal 能够处理非常大的数字？

Answer 1

正如L3viathan 回答的那样，问题是结果（不是结果\xe2\x80\x94 这是我在评论中提到的“隐藏部分”）超出了可用精度。

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如果我们使用Python2，隐藏的部分会更明显：

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

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本质上，a % b是通过同时进行除法和取模来实现的（Knuth vol 2 中的算法 D；对于仅限于两个全字的 C 实现，请参阅我在 2000 年代初编写的 qdivrem.c 代码）。因此，库代码需要两个额外的数字（\xe2\x80\x94中小数点右侧的位数，计算Decimal(\'0.01\')实际所需的位数并不像big下面那么简单，因为我们必须查看指数）计算中间商。

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十进制库直接用 C 语言为 Python3 重新实现，隐藏了细节，但两者的解决方法是相同的：扩展精度。下面是一个示例源例程，它捕获异常并重试，但使用了魔术常量 2。

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

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使用Python2：

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$ python2 /tmp/d.py\nTraceback (most recent call last):\n  File "/tmp/d.py", line 24, in <module>\n    print(big % Decimal(\'0.01\'))\n  File "/usr/local/lib/python2.7/decimal.py", line 1460, in __mod__\n    remainder = self._divide(other, context)[1]\n  File "/usr/local/lib/python2.7/decimal.py", line 1381, in _divide\n    \'quotient too large in //, % or divmod\')\n  File "/usr/local/lib/python2.7/decimal.py", line 3873, in _raise_error\n    raise error(explanation)\nInvalidOperation: quotient too large in //, % or divmod\n\nfailed because precision was 28 and big is 1000 digits long\ntrying again with 2 more digits\n0.00\n

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使用Python3：

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$ python3 /tmp/d.py\nTraceback (most recent call last):\n  File "/tmp/d.py", line 24, in <module>\n    print(big % Decimal(\'0.01\'))\ndecimal.InvalidOperation: [<class \'decimal.DivisionImpossible\'>]\n\nfailed because precision was 28 and big is 1000 digits long\ntrying again with 2 more digits\n0.00\n

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请注意，除以一个非常大的数字实际上更容易：正是除以它才0.01导致这里出现问题。如果除数的指数至少为 1000 - 28（1e972 或更大），我们就不会有问题。

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Answer 2

Decimal基于十进制算术规范。你可以在这里看到“除法不可能”的意思是

整数除法或余数运算的整数结果位数过多（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')