Boost Algorithms & Mathematics

Multiprecision: Arbitrary-Precision Arithmetic

Core design: frontend/backend separation. The frontend (number<Backend>) provides a unified operator interface, while the backend determines storage and algorithms.

template<typename Backend>
class number {
    Backend backend_;
public:
    number& operator*=(number const& other) {
        backend_.multiply(backend_, other.backend_);
        return *this;
    }
};

Backend	Storage	Precision	Use Case
cpp_int	Dynamic vector<limb_type>	Runtime arbitrary precision	General-purpose big integers
int128_backend	Stack-allocated 128-bit	Fixed 128-bit	Beyond uint64_t
gmp_backend	GMP's mpz_t	Arbitrary precision	Maximum performance

Small Value Optimization and Multiplication Strategy

void multiply(cpp_int_backend& result, ...) {
    if (a.size() <= 2 && b.size() <= 2) {
        // Small: single mul instruction
    } else if (a.size() < 30 || b.size() < 30) {
        // Medium: naive O(n*m) multiplication
    } else {
        // Large: Karatsuba O(n^1.585)
        // Even larger: Toom-Cook-3 or FFT O(n log n)
    }
}

Expression Templates

a = b * c + d * e uses expression templates to avoid 3 temporary objects — the entire expression tree is computed into a's backend in a single assignment.

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Math: Mathematical Functions

Boost.Math provides extensions to standard mathematical functions:

• Special functions: Bessel functions, elliptic integrals, Gamma/Beta functions
• Statistical distributions: PDF/CDF/quantiles for normal, Poisson, etc.
• Numerical integration: adaptive integration, Gaussian quadrature
• Constants: compile-time mathematical constants (π, e, √2, etc.)

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Geometry: Computational Geometry

namespace bg = boost::geometry;
using Point = bg::model::d2::point_xy<double>;
using Polygon = bg::model::polygon<Point>;

Polygon poly;
bg::read_wkt("POLYGON((0 0, 0 5, 5 5, 5 0, 0 0))", poly);
std::cout << bg::area(poly);             // 25
std::cout << bg::within(Point(2.5, 2.5), poly);  // 1

The strategy pattern allows switching implementations: distance calculations can choose Euclidean/Manhattan distance, point-in-polygon tests can choose different precision strategies.

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Range: Range Algorithms

Boost.Range is the predecessor of C++20 Ranges, providing pipe-based range operations:

auto result = data | boost::adaptors::filtered(pred)
                   | boost::adaptors::transformed(fn);

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Other Algorithm Libraries

Library	Description
Algorithm	String algorithms (trim, split, join, replace)
CRC	CRC-16/32/64 checksums
Conversion	Type conversion (lexical_cast, numeric_cast)
Endian	Byte order conversion (big_to_native, little_to_native)