## Arbitrary-precision integer arithmetic in C++

This was mostly just for fun, motivated by questions about a student programming problem.  There are plenty of arbitrary- and extended-precision arithmetic libraries out there, and there are plenty of great references describing the algorithms involved.  But there can be a lot of added complexity in the translation from pseudo-code to actual lines of working code.  My goal here was to provide the latter, while maintaining the readability of the former.  I’m not sure I succeeded.  But it was still fun.

The end result is about 500 lines of code in a single C++ header file, defining an unsigned (i.e., non-negative) integer type with overloading of all sensible arithmetic, bitwise, and relational operators (basically everything except bitwise one’s complement), and implicit conversions to/from a decimal string representation.  The code follows the conventions and variable names of Knuth’s TAOCP Volume 2 very closely throughout, although the division algorithm is slightly simplified.

[Edit 2018-09-02: See this more recent post extending this implementation to also support signed integer and rational arithmetic.]

Following is the source code, which is free, unencumbered, released into the public domain; do what you want with it.

```#ifndef MATH_UNSIGNED_H
#define MATH_UNSIGNED_H

#include <cstdint>
#include <vector>
#include <iosfwd>
#include <stdexcept>
#include <algorithm>
#include <sstream>
#include <cctype>
using std::size_t;

namespace math
{
class Unsigned
{
public:
typedef std::uint32_t Digit;
typedef std::uint64_t Wigit;
static const unsigned BITS = 32;

Unsigned(Digit u = 0) :
digits(1, u)
{
// empty
}

Unsigned(const std::string& s) :
digits(1, 0)
{
std::istringstream iss(s);
iss >> *this;
if (iss.fail() || !iss.eof())
{
throw std::runtime_error("Error: Unsigned::string");
}
}

Unsigned(const Unsigned& copy) :
digits(copy.digits)
{
// empty
}

Unsigned& operator= (const Unsigned& rhs)
{
digits = rhs.digits;
return *this;
}

Unsigned operator++ (int)
{
Unsigned w(*this);
++(*this);
return w;
}

Unsigned& operator++ ()
{
for (size_t j = 0; j < digits.size() && ++digits[j] == 0; ++j);
if (digits.back() == 0)
{
digits.push_back(1);
}
return *this;
}

Unsigned operator-- (int)
{
Unsigned w(*this);
--(*this);
return w;
}

Unsigned& operator-- ()
{
if (digits.back() == 0)
{
throw std::underflow_error("Error: Unsigned::underflow");
}
for (size_t j = 0; j < digits.size() && digits[j]-- == 0; ++j);
trim();
return *this;
}

friend Unsigned operator+ (Unsigned u, const Unsigned& v)
{
u += v;
return u;
}

Unsigned& operator+= (const Unsigned& rhs)
{
const size_t n = rhs.digits.size();
if (digits.size() < n)
{
digits.resize(n, 0);
}
size_t j = 0;
Wigit k = 0;
for (; j < n; ++j)
{
k = k + digits[j] + rhs.digits[j];
digits[j] = static_cast<Digit>(k);
k >>= BITS;
}
for (; k != 0 && j < digits.size(); ++j)
{
k += digits[j];
digits[j] = static_cast<Digit>(k);
k >>= BITS;
}
if (k != 0)
{
digits.push_back(1);
}
return *this;
}

friend Unsigned operator- (Unsigned u, const Unsigned& v)
{
u -= v;
return u;
}

Unsigned& operator-= (const Unsigned& rhs)
{
if ((*this) < rhs)
{
throw std::underflow_error("Error: Unsigned::underflow");
}
size_t j = 0;
Wigit k = 0;
for (; j < rhs.digits.size(); ++j)
{
k = k + digits[j] - rhs.digits[j];
digits[j] = static_cast<Digit>(k);
k = ((k >> BITS) ? -1 : 0);
}
for (; k != 0 && j < digits.size(); ++j)
{
k += digits[j];
digits[j] = static_cast<Digit>(k);
k = ((k >> BITS) ? -1 : 0);
}
trim();
return *this;
}

friend Unsigned operator* (const Unsigned& u, const Unsigned& v)
{
const size_t m = u.digits.size();
const size_t n = v.digits.size();
Unsigned w;
w.digits.resize(m + n, 0);
for (size_t j = 0; j < n; ++j)
{
Wigit k = 0;
for (size_t i = 0; i < m; ++i)
{
k += static_cast<Wigit>(u.digits[i]) * v.digits[j] +
w.digits[i + j];
w.digits[i + j] = static_cast<Digit>(k);
k >>= BITS;
}
w.digits[j + m] = static_cast<Digit>(k);
}
w.trim();
return w;
}

Unsigned& operator*= (const Unsigned& rhs)
{
*this = (*this) * rhs;
return *this;
}

friend Unsigned operator/ (const Unsigned& u, const Unsigned& v)
{
Unsigned q, r;
u.divide(v, q, r);
return q;
}

Unsigned& operator/= (const Unsigned& rhs)
{
Unsigned r;
divide(rhs, *this, r);
return *this;
}

friend Unsigned operator% (const Unsigned& u, const Unsigned& v)
{
Unsigned q, r;
u.divide(v, q, r);
return r;
}

Unsigned& operator%= (const Unsigned& rhs)
{
Unsigned q;
divide(rhs, q, *this);
return *this;
}

void divide(Unsigned v, Unsigned& q, Unsigned& r) const
{
// Handle special cases (m < n).
if (v.digits.back() == 0)
{
throw std::overflow_error("Error: Unsigned::overflow");
}
r.digits = digits;
const size_t n = v.digits.size();
if (digits.size() < n)
{
q.digits.assign(1, 0);
return;
}

// Normalize divisor (v[n-1] >= BASE/2).
unsigned d = BITS;
for (Digit vn = v.digits.back(); vn != 0; vn >>= 1, --d);
v <<= d;
r <<= d;
const Digit vn = v.digits.back();

// Ensure first single-digit quotient (u[m-1] < v[n-1]).
r.digits.push_back(0);
const size_t m = r.digits.size();
q.digits.resize(m - n);
Unsigned w;
w.digits.resize(n + 1);
const Wigit MAX_DIGIT = (static_cast<Wigit>(1) << BITS) - 1;
for (size_t j = m - n; j-- != 0;)
{
// Estimate quotient digit.
Wigit qhat = std::min(MAX_DIGIT,
(static_cast<Wigit>(r.digits[j + n]) << BITS |
r.digits[j + n - 1]) / vn);

// Compute partial product (w = qhat * v).
Wigit k = 0;
for (size_t i = 0; i < n; ++i)
{
k += qhat * v.digits[i];
w.digits[i] = static_cast<Digit>(k);
k >>= BITS;
}
w.digits[n] = static_cast<Digit>(k);

// Check if qhat is too large (u - w < 0).
bool is_trial = true;
while (is_trial)
{
size_t i = n;
for (; i != 0 && r.digits[j + i] == w.digits[i]; --i);
if ((is_trial = (r.digits[j + i] < w.digits[i])))
{
// Adjust partial product (w -= v).
--qhat;
k = 0;
for (size_t i = 0; i < n; ++i)
{
k = k + w.digits[i] - v.digits[i];
w.digits[i] = static_cast<Digit>(k);
k = ((k >> BITS) ? -1 : 0);
}
w.digits[n] = static_cast<Digit>(k + w.digits[n]);
}
}
q.digits[j] = static_cast<Digit>(qhat);

// Compute partial remainder (u -= w).
k = 0;
for (size_t i = 0; i < n; ++i)
{
k = k + r.digits[j + i] - w.digits[i];
r.digits[j + i] = static_cast<Digit>(k);
k = ((k >> BITS) ? -1 : 0);
}
}

// Denormalize remainder.
q.trim();
r.digits.resize(n);
r >>= d;
}

friend Unsigned operator<< (Unsigned u, size_t v)
{
u <<= v;
return u;
}

Unsigned& operator<<= (size_t rhs)
{
if (digits.back() != 0 && rhs != 0)
{
const size_t n = rhs / BITS;
digits.insert(digits.begin(), n, 0);
rhs -= n * BITS;
Wigit k = 0;
for (size_t j = n; j < digits.size(); ++j)
{
k |= static_cast<Wigit>(digits[j]) << rhs;
digits[j] = static_cast<Digit>(k);
k >>= BITS;
}
if (k != 0)
{
digits.push_back(static_cast<Digit>(k));
}
}
return *this;
}

friend Unsigned operator>> (Unsigned u, size_t v)
{
u >>= v;
return u;
}

Unsigned& operator>>= (size_t rhs)
{
const size_t n = rhs / BITS;
if (n >= digits.size())
{
digits.assign(1, 0);
}
else
{
digits.erase(digits.begin(), digits.begin() + n);
rhs -= n * BITS;
Wigit k = 0;
for (size_t j = digits.size(); j-- != 0;)
{
k = k << BITS | digits[j];
digits[j] = static_cast<Digit>(k >> rhs);
k = static_cast<Digit>(k);
}
trim();
}
return *this;
}

friend Unsigned operator& (Unsigned u, const Unsigned& v)
{
u &= v;
return u;
}

Unsigned& operator&= (const Unsigned& rhs)
{
const size_t n = rhs.digits.size();
if (digits.size() > n)
{
digits.resize(n);
}
for (size_t j = 0; j < digits.size(); ++j)
{
digits[j] &= rhs.digits[j];
}
trim();
return *this;
}

Unsigned and_not(const Unsigned& v) const
{
Unsigned u(*this);
const size_t n = v.digits.size();
if (u.digits.size() > n)
{
u.digits.resize(n);
}
for (size_t j = 0; j < u.digits.size(); ++j)
{
u.digits[j] &= ~v.digits[j];
}
u.trim();
return u;
}

friend Unsigned operator^ (Unsigned u, const Unsigned& v)
{
u ^= v;
return u;
}

Unsigned& operator^= (const Unsigned& rhs)
{
const size_t n = rhs.digits.size();
if (digits.size() < n)
{
digits.resize(n, 0);
}
for (size_t j = 0; j < n; ++j)
{
digits[j] ^= rhs.digits[j];
}
trim();
return *this;
}

friend Unsigned operator| (Unsigned u, const Unsigned& v)
{
u |= v;
return u;
}

Unsigned& operator|= (const Unsigned& rhs)
{
const size_t n = rhs.digits.size();
if (digits.size() < n)
{
digits.resize(n, 0);
}
for (size_t j = 0; j < n; ++j)
{
digits[j] |= rhs.digits[j];
}
return *this;
}

friend bool operator< (const Unsigned& u, const Unsigned& v)
{
const size_t m = u.digits.size();
size_t n = v.digits.size();
if (m != n)
{
return (m < n);
}
for (--n; n != 0 && u.digits[n] == v.digits[n]; --n);
return (u.digits[n] < v.digits[n]);
}

friend bool operator> (const Unsigned& u, const Unsigned& v)
{
return (v < u);
}

friend bool operator<= (const Unsigned& u, const Unsigned& v)
{
return !(v < u);
}

friend bool operator>= (const Unsigned& u, const Unsigned& v)
{
return !(u < v);
}

friend bool operator== (const Unsigned& u, const Unsigned& v)
{
return (u.digits == v.digits);
}

friend bool operator!= (const Unsigned& u, const Unsigned& v)
{
return (u.digits != v.digits);
}

// Return 1 + floor(log2(u)), or 0 for u == 0.
int bits() const
{
size_t count = (digits.size() - 1) * BITS;
for (Digit u = digits.back(); u != 0; u >>= 1, ++count);
return static_cast<int>(count);
}

Digit to_uint() const
{
return digits[0];
}

std::string to_string() const
{
std::ostringstream oss;
Unsigned q(*this), r;
do
{
q.divide(10, q, r);
oss << r.digits[0];
} while (q.digits.back() != 0);
std::string s(oss.str());
std::reverse(s.begin(), s.end());
return s;
}

friend std::ostream& operator<< (std::ostream& os, const Unsigned& u)
{
os << u.to_string();
return os;
}

friend std::istream& operator>> (std::istream& is, Unsigned& u)
{
char digit = '\0';
is >> digit;
if (is.good() && std::isdigit(digit))
{
u = digit - '0';
while (std::isdigit(is.peek()))
{
is >> digit;
u = 10 * u + (digit - '0');
}
}
else
{
is.setstate(std::ios_base::failbit);
}
return is;
}

private:
std::vector<Digit> digits;

void trim()
{
while (digits.back() == 0 && digits.size() > 1)
{
digits.pop_back();
}
}
};
} // namespace math

#endif // MATH_UNSIGNED_H
```

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### 5 Responses to Arbitrary-precision integer arithmetic in C++

1. Nice work! I really like that you put it in the public domain. One of my favorite excuses for writing my own library when an open source library already exists is that I’m writing the world’s first public domain implementation. Public domain is pretty rare, even in open source, so it means my implementation has an automatic feature above every other implementation, which are each burdened with one license or another.

Is “widgit” your term or Knuth’s? I’ve never seen it before but I like it. It reminds me of byte/nibble.

I’ve rolled my own unsigned bigint library a few times as needed (dailyprogrammer, etc.), too, implementing some subset of these operations each time — except division. That one never seems like it would be fun or interesting to implement, just tedious bookkeeping. It really shows here, where it’s your largest method.

• Thanks! I haven’t seen “wigit” used before, but I liked it as a shortened “wide digit” better than my first idea of Digit2 :).

Actually, the division is what snagged my interest here, *because* it’s the hard part. I wanted to demonstrate that everything in the implementation is really just “how we learned in school” … and that mechanizing how we learned division is actually pretty tricky. (At least for large bases; you can do it in just 10-ish lines of code with shift/compare/subtract, but only one quotient bit at a time.) That’s why I left out Knuth’s more complex initial quotient digit estimate, that only saves one of three possible adjustments.

(The divide() method *could* be simplified somewhat; my initial code actually used the pseudo-code in the comments for computing and adjusting the partial product, turning 16 lines into just 3. But that involves extra unnecessary trim()ing and resize()ing of leading zeros in those intermediate calculations.)

2. interested_programmer says:

It would be really nice to have an implementation of this in C with strings using functions like add(), subtract(), multiply(), cos(), sin(), tan(), pow() and so on and so forth. What you wrote is impressive and interesting, but manipulates the language at an abstracted level that is not really useful for my needs. Thanks

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