172 lines
12 KiB
Plaintext
172 lines
12 KiB
Plaintext
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Copyright 2011 - 2020 John Maddock.
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Copyright 2013 - 2019 Paul A. Bristow.
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Copyright 2013 Christopher Kormanyos.
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Distributed under the Boost Software License, Version 1.0.
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(See accompanying file LICENSE_1_0.txt or copy at
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http://www.boost.org/LICENSE_1_0.txt).
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]
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[section:cpp_int cpp_int]
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`#include <boost/multiprecision/cpp_int.hpp>`
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namespace boost{ namespace multiprecision{
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typedef unspecified-type limb_type;
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enum cpp_integer_type { signed_magnitude, unsigned_magnitude };
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enum cpp_int_check_type { checked, unchecked };
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template <unsigned MinBits = 0,
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unsigned MaxBits = 0,
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cpp_integer_type SignType = signed_magnitude,
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cpp_int_check_type Checked = unchecked,
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class Allocator = std::allocator<limb_type> >
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class cpp_int_backend;
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//
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// Expression templates default to et_off if there is no allocator:
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//
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template <unsigned MinBits, unsigned MaxBits, cpp_integer_type SignType, cpp_int_check_type Checked>
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struct expression_template_default<cpp_int_backend<MinBits, MaxBits, SignType, Checked, void> >
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{ static const expression_template_option value = et_off; };
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typedef number<cpp_int_backend<> > cpp_int; // arbitrary precision integer
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typedef rational_adaptor<cpp_int_backend<> > cpp_rational_backend;
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typedef number<cpp_rational_backend> cpp_rational; // arbitrary precision rational number
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// Fixed precision unsigned types:
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typedef number<cpp_int_backend<128, 128, unsigned_magnitude, unchecked, void> > uint128_t;
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typedef number<cpp_int_backend<256, 256, unsigned_magnitude, unchecked, void> > uint256_t;
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typedef number<cpp_int_backend<512, 512, unsigned_magnitude, unchecked, void> > uint512_t;
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typedef number<cpp_int_backend<1024, 1024, unsigned_magnitude, unchecked, void> > uint1024_t;
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// Fixed precision signed types:
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typedef number<cpp_int_backend<128, 128, signed_magnitude, unchecked, void> > int128_t;
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typedef number<cpp_int_backend<256, 256, signed_magnitude, unchecked, void> > int256_t;
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typedef number<cpp_int_backend<512, 512, signed_magnitude, unchecked, void> > int512_t;
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typedef number<cpp_int_backend<1024, 1024, signed_magnitude, unchecked, void> > int1024_t;
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// Over again, but with checking enabled this time:
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typedef number<cpp_int_backend<0, 0, signed_magnitude, checked> > checked_cpp_int;
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typedef rational_adaptor<cpp_int_backend<0, 0, signed_magnitude, checked> > checked_cpp_rational_backend;
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typedef number<cpp_rational_backend> checked_cpp_rational;
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// Checked fixed precision unsigned types:
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typedef number<cpp_int_backend<128, 128, unsigned_magnitude, checked, void> > checked_uint128_t;
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typedef number<cpp_int_backend<256, 256, unsigned_magnitude, checked, void> > checked_uint256_t;
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typedef number<cpp_int_backend<512, 512, unsigned_magnitude, checked, void> > checked_uint512_t;
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typedef number<cpp_int_backend<1024, 1024, unsigned_magnitude, checked, void> > checked_uint1024_t;
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// Fixed precision signed types:
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typedef number<cpp_int_backend<128, 128, signed_magnitude, checked, void> > checked_int128_t;
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typedef number<cpp_int_backend<256, 256, signed_magnitude, checked, void> > checked_int256_t;
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typedef number<cpp_int_backend<512, 512, signed_magnitude, checked, void> > checked_int512_t;
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typedef number<cpp_int_backend<1024, 1024, signed_magnitude, checked, void> > checked_int1024_t;
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}} // namespaces
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The `cpp_int_backend` type is normally used via one of the convenience typedefs given above.
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This back-end is the "Swiss Army Knife" of integer types as it can represent both fixed and
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[@http://en.wikipedia.org/wiki/Arbitrary-precision_arithmetic arbitrary precision]
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integer types, and both signed and unsigned types. There are five template arguments:
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[variablelist
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[[MinBits][Determines the number of Bits to store directly within the object before resorting to dynamic memory
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allocation. When zero, this field is determined automatically based on how many bits can be stored
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in union with the dynamic storage header: setting a larger value may improve performance as larger integer
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values will be stored internally before memory allocation is required.]]
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[[MaxBits][Determines the maximum number of bits to be stored in the type: resulting in a fixed precision type.
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When this value is the same as MinBits, then the Allocator parameter is ignored, as no dynamic
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memory allocation will ever be performed: in this situation the Allocator parameter should be set to
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type `void`. Note that this parameter should not be used simply to prevent large memory
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allocations, not only is that role better performed by the allocator, but fixed precision
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integers have a tendency to allocate all of MaxBits of storage more often than one would expect.]]
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[[SignType][Determines whether the resulting type is signed or not. Note that for
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[@http://en.wikipedia.org/wiki/Arbitrary-precision_arithmetic arbitrary precision] types
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this parameter must be `signed_magnitude`. For fixed precision
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types then this type may be either `signed_magnitude` or `unsigned_magnitude`.]]
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[[Checked][This parameter has two values: `checked` or `unchecked`. See below.]]
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[[Allocator][The allocator to use for dynamic memory allocation, or type `void` if MaxBits == MinBits.]]
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]
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When the template parameter Checked is set to `checked` then the result is a ['checked-integer], checked
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and unchecked integers have the following properties:
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[table
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[[Condition][Checked-Integer][Unchecked-Integer]]
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[[Numeric overflow in fixed precision arithmetic][Throws a `std::overflow_error`.][Performs arithmetic modulo 2[super MaxBits]]]
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[[Constructing an integer from a value that can not be represented in the target type][Throws a `std::range_error`.]
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[Converts the value modulo 2[super MaxBits], signed to unsigned conversions extract the last MaxBits bits of the
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2's complement representation of the input value.]]
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[[Unsigned subtraction yielding a negative value.][Throws a `std::range_error`.][Yields the value that would
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result from treating the unsigned type as a 2's complement signed type.]]
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[[Attempting a bitwise operation on a negative value.][Throws a `std::range_error`][Yields the value, but not the bit pattern,
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that would result from performing the operation on a 2's complement integer type.]]
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]
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Things you should know when using this type:
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* Default constructed `cpp_int_backend`s have the value zero.
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* Division by zero results in a `std::overflow_error` being thrown.
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* Construction from a string that contains invalid non-numeric characters results in a `std::runtime_error` being thrown.
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* Since the precision of `cpp_int_backend` is necessarily limited when the allocator parameter is void,
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care should be taken to avoid numeric overflow when using this type
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unless you actually want modulo-arithmetic behavior.
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* The type uses a sign-magnitude representation internally, so type `int128_t` has 128-bits of precision plus an extra sign bit.
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In this respect the behaviour of these types differs from __fundamental 2's complement types. In might be tempting to use a
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127-bit type instead, and indeed this does work, but behaviour is still slightly different from a 2's complement __fundamental type
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as the minimum and maximum values are identical (apart from the sign), where as they differ by one for a true 2's complement type.
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That said it should be noted that there's no requirement for fundamental_types to be 2's complement either - it's simply that this
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is the most common format by far.
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* Attempting to print negative values as either an Octal or Hexadecimal string results in a `std::runtime_error` being thrown,
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this is a direct consequence of the sign-magnitude representation.
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* The fixed precision types `[checked_][u]intXXX_t` have expression template support turned off - it seems to make little
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difference to the performance of these types either way - so we may as well have the faster compile times by turning
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the feature off.
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* Unsigned types support subtraction - the result is "as if" a 2's complement operation had been performed as long as they are not
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['checked-integers] (see above).
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In other words they behave pretty much as a __fundamental integer type would in this situation. So for example if we were using
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`uint128_t` then `uint128_t(1)-4` would result in the value `0xFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFD`
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of type `uint128_t`. However, had this operation been performed on `checked_uint128_t` then a `std::range_error` would have
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been thrown.
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* Unary negation of unsigned types results in a compiler error (static assertion).
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* This backend supports rvalue-references and is move-aware, making instantiations of `number` on this backend move aware.
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* When used at fixed precision, the size of this type is always one machine word (plus any compiler-applied alignment padding)
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larger than you would expect for an N-bit integer:
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the extra word stores both the sign, and how many machine words in the integer are actually in use.
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The latter is an optimisation for larger fixed precision integers, so that a 1024-bit integer has almost the same performance
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characteristics as a 128-bit integer, rather than being 4 times slower for addition and 16 times slower for multiplication
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(assuming the values involved would always fit in 128 bits).
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Typically this means you can use
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an integer type wide enough for the "worst case scenario" with only minor performance degradation even if most of the time
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the arithmetic could in fact be done with a narrower type.
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Also note that unsigned fixed precision types small enough to fit inside the largest native integer become a simple wrapper around that type,
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this includes the "checked" variants. Small signed types will always have an extra sign word and so be larger than their native equivalent.
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* When used at fixed precision and MaxBits is smaller than the number of bits in the largest native integer type, then
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internally `cpp_int_backend` switches to a "trivial" implementation where it is just a thin wrapper around a single
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integer. Note that it will still be slightly slower than a bare native integer, as it emulates a
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signed-magnitude representation rather than simply using the platforms native sign representation: this ensures
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there is no step change in behavior as a cpp_int grows in size.
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* Fixed precision `cpp_int`'s have some support for `constexpr` values and user-defined literals, see
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[link boost_multiprecision.tut.lits here] for the full description. For example `0xfffff_cppi1024`
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specifies a 1024-bit integer with the value 0xffff. This can be used to generate compile-time constants that are
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too large to fit into any __fundamental number type.
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* The __cpp_int types support constexpr arithmetic, provided it is a fixed-precision type with no allocator. It may also
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be a checked integer: in which case a compiler error will be generated on overflow or undefined behaviour. In addition
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the free functions `abs`, `swap`, `multiply`, `add`, `subtract`, `divide_qr`, `integer_modulus`, `powm`, `lsb`, `msb`,
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`bit_test`, `bit_set`, `bit_unset`, `bit_flip`, `sqrt`, `gcd`, `lcm` are all supported. Use of __cpp_int in this way
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requires either a C++2a compiler (one which supports `std::is_constant_evaluated()`), or GCC-6 or later in C++14 mode.
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Compilers other than GCC and without `std::is_constant_evaluated()` will support a very limited set of operations:
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expect to hit roadblocks rather easily.
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* You can import/export the raw bits of a __cpp_int to and from external storage via the `import_bits` and `export_bits`
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functions. More information is in the [link boost_multiprecision.tut.import_export section on import/export].
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[h5:cpp_int_eg Example:]
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[cpp_int_eg]
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[endsect] [/section:cpp_int cpp_int]
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