Otdfbt

This implements 128-bit unsigned integer using C++14.
It works on MSVC and 32-bit architectures being complementary to the unsigned __int128 type provided by GCC and clang on 64-bit architectures.

// intx: extended precision integer library.
// Copyright 2019 Pawel Bylica.
// Licensed under the Apache License, Version 2.0.

#pragma once

#include <algorithm>
#include <climits>
#include <cstdint>
#include <limits>
#include <stdexcept>
#include <string>
#include <type_traits>

#ifdef _MSC_VER
#include <intrin.h>
#endif

namespace intx

template <unsigned N>
struct uint;

/// The 128-bit unsigned integer.
///
/// This type is defined as a specialization of uint<> to easier integration with full intx package,
/// however, uint128 may be used independently.
template <>
struct uint<128>

 uint64_t lo = 0;
 uint64_t hi = 0;

 constexpr uint() noexcept = default;

 constexpr uint(uint64_t high, uint64_t low) noexcept : lolow, hihigh 

 template <typename T,
 typename = typename std::enable_if_t<std::is_convertible<T, uint64_t>::value>>
 constexpr uint(T x) noexcept : lo(static_cast<uint64_t>(x)) // NOLINT
 

#ifdef __SIZEOF_INT128__
#pragma GCC diagnostic push
#pragma GCC diagnostic ignored "-Wpedantic"
 constexpr uint(unsigned __int128 x) noexcept // NOLINT
 : louint64_t(x), hiuint64_t(x >> 64)
 

 constexpr explicit operator unsigned __int128() const noexcept
 lo;
 
#pragma GCC diagnostic pop
#endif

 constexpr explicit operator bool() const noexcept return hi 

 /// Explicit converting operator for all builtin integral types.
 template <typename Int, typename = typename std::enable_if<std::is_integral<Int>::value>::type>
 constexpr explicit operator Int() const noexcept
 
 return static_cast<Int>(lo);
 
;

using uint128 = uint<128>;


/// Linear arithmetic operators.
/// @

constexpr uint128 operator+(uint128 x, uint128 y) noexcept

 const auto lo = x.lo + y.lo;
 const auto carry = x.lo > lo;
 const auto hi = x.hi + y.hi + carry;
 return hi, lo;


constexpr uint128 operator+(uint128 x) noexcept

 return x;


constexpr uint128 operator-(uint128 x, uint128 y) noexcept

 const auto lo = x.lo - y.lo;
 const auto borrow = x.lo < lo;
 const auto hi = x.hi - y.hi - borrow;
 return hi, lo;


constexpr uint128 operator-(uint128 x) noexcept

 // Implementing as subtraction is better than ~x + 1.
 // Clang9: Perfect.
 // GCC8: Does something weird.
 return 0 - x;


inline uint128& operator++(uint128& x) noexcept

 return x = x + 1;


inline uint128& operator--(uint128& x) noexcept

 return x = x - 1;


inline uint128 operator++(uint128& x, int) noexcept

 auto ret = x;
 ++x;
 return ret;


inline uint128 operator--(uint128& x, int) noexcept

 auto ret = x;
 --x;
 return ret;


/// Optimized addition.
///
/// This keeps the multiprecision addition until CodeGen so the pattern is not
/// broken during other optimizations.
constexpr uint128 fast_add(uint128 x, uint128 y) noexcept

#ifdef __SIZEOF_INT128__xxx
 return (unsigned __int128)x + (unsigned __int128)y;
#else
 // Fallback to regular addition.
 return x + y;
#endif


/// @


/// Comparison operators.
///
/// In all implementations bitwise operators are used instead of logical ones
/// to avoid branching.
///
/// @

constexpr bool operator==(uint128 x, uint128 y) noexcept

 // Clang7: generates perfect xor based code,
 // much better than __int128 where it uses vector instructions.
 // GCC8: generates a bit worse cmp based code
 // although it generates the xor based one for __int128.
 return (x.lo == y.lo) & (x.hi == y.hi);


constexpr bool operator!=(uint128 x, uint128 y) noexcept

 // Analogous to ==, but == not used directly, because that confuses GCC8.
 return (x.lo != y.lo) 

constexpr bool operator<(uint128 x, uint128 y) noexcept

 // OPT: This should be implemented by checking the borrow of x - y,
 // but compilers (GCC8, Clang7)
 // have problem with properly optimizing subtraction.
 return (x.hi < y.hi) 

constexpr bool operator<=(uint128 x, uint128 y) noexcept
 (x == y).
 return (x.hi < y.hi) 

constexpr bool operator>(uint128 x, uint128 y) noexcept

 return !(x <= y);


constexpr bool operator>=(uint128 x, uint128 y) noexcept

 return !(x < y);


/// @


/// Bitwise operators.
/// @

constexpr uint128 operator~(uint128 x) noexcept

 return ~x.hi, ~x.lo;


constexpr uint128 operator


/// Multiplication
/// @

/// Portable full unsigned multiplication 64 x 64 -> 128.
constexpr uint128 constexpr_umul(uint64_t x, uint64_t y) noexcept

 uint64_t xl = x & 0xffffffff;
 uint64_t xh = x >> 32;
 uint64_t yl = y & 0xffffffff;
 uint64_t yh = y >> 32;

 uint64_t t0 = xl * yl;
 uint64_t t1 = xh * yl;
 uint64_t t2 = xl * yh;
 uint64_t t3 = xh * yh;

 uint64_t u1 = t1 + (t0 >> 32);
 uint64_t u2 = t2 + (u1 & 0xffffffff);

 uint64_t lo = (u2 << 32) 

/// Full unsigned multiplication 64 x 64 -> 128.
inline uint128 umul(uint64_t x, uint64_t y) noexcept

#if defined(__SIZEOF_INT128__)
#pragma GCC diagnostic push
#pragma GCC diagnostic ignored "-Wpedantic"
 const auto p = static_cast<unsigned __int128>(x) * y;
 return uint64_t(p >> 64), uint64_t(p);
#pragma GCC diagnostic pop
#elif defined(_MSC_VER)
 unsigned __int64 hi;
 const auto lo = _umul128(x, y, &hi);
 return hi, lo;
#else
 return constexpr_umul(x, y);
#endif


inline uint128 operator*(uint128 x, uint128 y) noexcept

 auto p = umul(x.lo, y.lo);
 p.hi += (x.lo * y.hi) + (x.hi * y.lo);
 return p.hi, p.lo;


constexpr uint128 constexpr_mul(uint128 x, uint128 y) noexcept

 auto p = constexpr_umul(x.lo, y.lo);
 p.hi += (x.lo * y.hi) + (x.hi * y.lo);
 return p.hi, p.lo;


/// @


/// Assignment operators.
/// @

constexpr uint128& operator+=(uint128& x, uint128 y) noexcept

 return x = x + y;


constexpr uint128& operator-=(uint128& x, uint128 y) noexcept

 return x = x - y;


inline uint128& operator*=(uint128& x, uint128 y) noexcept

 return x = x * y;


constexpr uint128& operator


inline unsigned clz(uint32_t x) noexcept

#ifdef _MSC_VER
 unsigned long most_significant_bit;
 _BitScanReverse(&most_significant_bit, x);
 return 31 ^ (unsigned)most_significant_bit;
#else
 return unsigned(__builtin_clz(x));
#endif


inline unsigned clz(uint64_t x) noexcept

#ifdef _MSC_VER
 unsigned long most_significant_bit;
 _BitScanReverse64(&most_significant_bit, x);
 return 63 ^ (unsigned)most_significant_bit;
#else
 return unsigned(__builtin_clzll(x));
#endif


inline unsigned clz(uint128 x) noexcept
 64 : clz(x.hi);



inline uint64_t bswap(uint64_t x) noexcept

#ifdef _MSC_VER
 return _byteswap_uint64(x);
#else
 return __builtin_bswap64(x);
#endif


inline uint128 bswap(uint128 x) noexcept

 return bswap(x.lo), bswap(x.hi);



/// Division.
/// @

template <typename T>
struct div_result

 T quot;
 T rem;
;

namespace internal

constexpr uint16_t reciprocal_table_item(uint8_t d9) noexcept

 return uint16_t(0x7fd00 / (0x100 

#define REPEAT4(x) 
 reciprocal_table_item((x) + 0), reciprocal_table_item((x) + 1), 
 reciprocal_table_item((x) + 2), reciprocal_table_item((x) + 3)

#define REPEAT32(x) 
 REPEAT4((x) + 4 * 0), REPEAT4((x) + 4 * 1), REPEAT4((x) + 4 * 2), REPEAT4((x) + 4 * 3), 
 REPEAT4((x) + 4 * 4), REPEAT4((x) + 4 * 5), REPEAT4((x) + 4 * 6), REPEAT4((x) + 4 * 7)

#define REPEAT256() 
 REPEAT32(32 * 0), REPEAT32(32 * 1), REPEAT32(32 * 2), REPEAT32(32 * 3), REPEAT32(32 * 4), 
 REPEAT32(32 * 5), REPEAT32(32 * 6), REPEAT32(32 * 7)

/// Reciprocal lookup table.
constexpr uint16_t reciprocal_table[] = REPEAT256();

#undef REPEAT4
#undef REPEAT32
#undef REPEAT256
 // namespace internal

/// Computes the reciprocal (2^128 - 1) / d - 2^64 for normalized d.
///
/// Based on Algorithm 2 from "Improved division by invariant integers".
inline uint64_t reciprocal_2by1(uint64_t d) noexcept

 auto d9 = uint8_t(d >> 55);
 auto v0 = uint64_tinternal::reciprocal_table[d9];

 auto d40 = (d >> 24) + 1;
 auto v1 = (v0 << 11) - (v0 * v0 * d40 >> 40) - 1;

 auto v2 = (v1 << 13) + (v1 * (0x1000000000000000 - v1 * d40) >> 47);

 auto d0 = d % 2;
 auto d63 = d / 2 + d0; // ceil(d/2)
 auto nd0 = uint64_t(-int64_t(d0));
 auto e = ((v2 / 2) & nd0) - v2 * d63;
 auto mh = umul(v2, e).hi;
 auto v3 = (v2 << 31) + (mh >> 1);

 // OPT: The compiler tries a bit too much with 128 + 64 addition and ends up using subtraction.
 // Compare with __int128.
 auto mf = umul(v3, d);
 auto m = fast_add(mf, d);
 auto v3a = m.hi + d;

 auto v4 = v3 - v3a;

 return v4;


inline uint64_t reciprocal_3by2(uint128 d) noexcept

 auto v = reciprocal_2by1(d.hi);
 auto p = d.hi * v;
 p += d.lo;
 if (p < d.lo)
 
 --v;
 if (p >= d.hi)
 
 --v;
 p -= d.hi;
 
 p -= d.hi;
 

 auto t = umul(v, d.lo);

 p += t.hi;
 if (p < t.hi)
 
 --v;
 if (uint128p, t.lo >= d)
 --v;
 
 return v;


inline div_result<uint64_t> udivrem_2by1(uint128 u, uint64_t d, uint64_t v) noexcept

 auto q = umul(v, u.hi);
 q = fast_add(q, u);

 ++q.hi;

 auto r = u.lo - q.hi * d;

 if (r > q.lo)
 
 --q.hi;
 r += d;
 

 if (r >= d)
 
 ++q.hi;
 r -= d;
 

 return q.hi, r;


inline div_result<uint128> udivrem_3by2(
 uint64_t u2, uint64_t u1, uint64_t u0, uint128 d, uint64_t v) noexcept

 auto q = umul(v, u2);
 q = fast_add(q, u2, u1);

 auto r1 = u1 - q.hi * d.hi;

 auto t = umul(d.lo, q.hi);

 auto r = uint128r1, u0 - t - d;
 r1 = r.hi;

 ++q.hi;

 if (r1 >= q.lo)
 
 --q.hi;
 r += d;
 

 if (r >= d)
 
 ++q.hi;
 r -= d;
 

 return q.hi, r;


inline div_result<uint128> udivrem(uint128 x, uint128 y) noexcept

 if (y.hi == 0)
 
 uint64_t xn_ex, xn_hi, xn_lo, yn;

 auto lsh = clz(y.lo);
 if (lsh != 0)
 (x.hi << lsh);
 xn_lo = x.lo << lsh;
 yn = y.lo << lsh;
 
 else
 
 xn_ex = 0;
 xn_hi = x.hi;
 xn_lo = x.lo;
 yn = y.lo;
 

 auto v = reciprocal_2by1(yn);

 auto res = udivrem_2by1(xn_ex, xn_hi, yn, v);
 auto q1 = res.quot;

 res = udivrem_2by1(res.rem, xn_lo, yn, v);

 return q1, res.quot, res.rem >> lsh;
 

 if (y.hi > x.hi)
 return 0, x;

 auto lsh = clz(y.hi);
 if (lsh == 0)
 unsignedy.lo <= x.lo;
 return q, x - (q ? y : 0);
 

 auto rsh = 64 - lsh;

 auto yn_lo = y.lo << lsh;
 auto yn_hi = (y.lo >> rsh) 

inline div_result<uint128> sdivrem(uint128 x, uint128 y) noexcept

 constexpr auto sign_mask = uint1281 << 127;
 const auto x_is_neg = (x & sign_mask) != 0;
 const auto y_is_neg = (y & sign_mask) != 0;

 const auto x_abs = x_is_neg ? -x : x;
 const auto y_abs = y_is_neg ? -y : y;

 const auto q_is_neg = x_is_neg ^ y_is_neg;

 const auto res = udivrem(x_abs, y_abs);

 return q_is_neg ? -res.quot : res.quot, x_is_neg ? -res.rem : res.rem;


inline uint128 operator/(uint128 x, uint128 y) noexcept

 return udivrem(x, y).quot;


inline uint128 operator%(uint128 x, uint128 y) noexcept

 return udivrem(x, y).rem;


inline uint128& operator/=(uint128& x, uint128 y) noexcept

 return x = x / y;


inline uint128& operator%=(uint128& x, uint128 y) noexcept

 return x = x % y;


/// @

 // namespace intx


namespace std

template <unsigned N>
struct numeric_limits<intx::uint<N>>

 using type = intx::uint<N>;

 static constexpr bool is_specialized = true;
 static constexpr bool is_integer = true;
 static constexpr bool is_signed = false;
 static constexpr bool is_exact = true;
 static constexpr bool has_infinity = false;
 static constexpr bool has_quiet_NaN = false;
 static constexpr bool has_signaling_NaN = false;
 static constexpr float_denorm_style has_denorm = denorm_absent;
 static constexpr bool has_denorm_loss = false;
 static constexpr float_round_style round_style = round_toward_zero;
 static constexpr bool is_iec559 = false;
 static constexpr bool is_bounded = true;
 static constexpr bool is_modulo = true;
 static constexpr int digits = CHAR_BIT * sizeof(type);
 static constexpr int digits10 = int(0.3010299956639812 * digits);
 static constexpr int max_digits10 = 0;
 static constexpr int radix = 2;
 static constexpr int min_exponent = 0;
 static constexpr int min_exponent10 = 0;
 static constexpr int max_exponent = 0;
 static constexpr int max_exponent10 = 0;
 static constexpr bool traps = std::numeric_limits<unsigned>::traps;
 static constexpr bool tinyness_before = false;

 static constexpr type min() noexcept return 0; 
 static constexpr type lowest() noexcept return min(); 
 static constexpr type max() noexcept return ~type0; 
 static constexpr type epsilon() noexcept return 0; 
 static constexpr type round_error() noexcept return 0; 
 static constexpr type infinity() noexcept return 0; 
 static constexpr type quiet_NaN() noexcept return 0; 
 static constexpr type signaling_NaN() noexcept return 0; 
 static constexpr type denorm_min() noexcept return 0; 
;
 // namespace std

namespace intx

template <typename Int>
constexpr Int from_string(const char* s)

 using namespace std::literals;

 auto x = Int;
 int num_digits = 0;

 if (s[0] == '0' && s[1] == 'x')
 
 s += 2;
 while (auto d = *s++)
 = d;
 
 return x;
 

 while (auto d = *s++)
 
 if (num_digits++ > std::numeric_limits<Int>::digits10)
 throw std::overflow_error"Integer overflow";

 x = constexpr_mul(x, Int10);
 if (d >= '0' && d <= '9')
 d -= '0';
 else
 throw std::invalid_argument"Invalid literal character: "s + d;
 x += d;
 if (x < d)
 throw std::overflow_error"Integer overflow";
 
 return x;


template <typename Int>
constexpr Int from_string(const std::string& s)

 return from_string<Int>(s.c_str());


constexpr uint128 operator""_u128(const char* s)

 return from_string<uint128>(s);


template <unsigned N>
inline std::string to_string(uint<N> x, int base = 10)


template <unsigned N>
inline std::string hex(uint<N> x)

 return to_string(x, 16);

 // namespace intx

std::uint64_t is consistently misspelt throughout the code, and may be a poor choice anyway (since an exact 64-bit type need not be provided). It's better to use one of std::uint_fast64_t or std::uint_least64_t instead, for better portability. (Edit: as suggested by supercat in a comment, only the exact-width type will work correctly here, given the implicit masking we rely on for detecting carry and elsewhere, so disregard this recommendation: we want to fail to compile where this won't hold.)

I'd expect constructors from other intx::uint<> instantiations (obviously, the narrowing conversions should be explicit).

The ++ and -- operators could be implemented much more efficiently by using the members rather than converting and adding/subtracting. Unary - might also be better implemented element-wise.

It's good to see that you've included a specialization of std::numeric_limits for this type. A minor nitpick: I think that digits10 needs to round up rather than down. It's unfortunate that we're not allowed to specialize std::is_integral too.

A few problems in from_string():

• Style (minor): writing sizeof(x) instead of simply sizeof x makes it look like x is a type name.


• Actually, there's a more serious error in that line, in assuming that char is 8 bits (2 hex digits), rather than using CHAR_BIT - I'd write sizeof x * CHAR_BIT / 4 there for full portability.


• Alternatively, allow redundant leading zeros by simply checking clz(x) >= 4 before shifting.


• Also, why are octal inputs to from_string() treated as decimal? That's surprising to me.


• It might be useful to have (private) multiply/divide routines for the small multiplicands in from_string() and to_string().


• Perhaps we could skip over any leading whitespace and/or +, like the standard conversion functions (std::stoi(), std::from_chars(), etc) do?

These string conversions look like they could easily be adapted to become streaming operators << and >>.

Finally, it's a great shame that you didn't include the unit tests in the review; that would have greatly aided reviewers (especially when making suggestions for improvement).