进行定点数学运算的最佳方法是什么？ [关闭]

我需要为没有 FPU 的 Nintendo DS 加速一个程序，因此我需要将浮点数学（经过仿真且速度较慢）更改为定点。

我是如何开始的，我将浮点数更改为整数，每当我需要转换它们时，我都会使用x>>8将定点变量 x 转换为实际数字并x转换为定点。很快我发现不可能跟踪需要转换的内容，而且我还意识到很难更改数字的精度（在本例中为 8）。

我的问题是，我应该如何使这变得更容易并且仍然快速？我应该创建一个FixedPoint类，还是只是一个FixedPoint8 typedef或带有一些函数/宏的结构来转换它们，或者其他什么？我应该在变量名中添加一些内容来显示它是定点的吗？

你可以尝试我的定点课程（最新可用@https://github.com/eteran/cpp-utilities)

// From: https://github.com/eteran/cpp-utilities/edit/master/Fixed.h
// See also: http://stackoverflow.com/questions/79677/whats-the-best-way-to-do-fixed-point-math
/*
 * The MIT License (MIT)
 * 
 * Copyright (c) 2015 Evan Teran
 * 
 * Permission is hereby granted, free of charge, to any person obtaining a copy
 * of this software and associated documentation files (the "Software"), to deal
 * in the Software without restriction, including without limitation the rights
 * to use, copy, modify, merge, publish, distribute, sublicense, and/or sell
 * copies of the Software, and to permit persons to whom the Software is
 * furnished to do so, subject to the following conditions:
 * 
 * The above copyright notice and this permission notice shall be included in all
 * copies or substantial portions of the Software.
 * 
 * THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR
 * IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
 * FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE
 * AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER
 * LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM,
 * OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE
 * SOFTWARE.
 */

#ifndef FIXED_H_
#define FIXED_H_

#include <ostream>
#include <exception>
#include <cstddef> // for size_t
#include <cstdint>
#include <type_traits>

#include <boost/operators.hpp>

namespace numeric {

template <size_t I, size_t F>
class Fixed;

namespace detail {

// helper templates to make magic with types :)
// these allow us to determine resonable types from
// a desired size, they also let us infer the next largest type
// from a type which is nice for the division op
template <size_t T>
struct type_from_size {
    static const bool is_specialized = false;
    typedef void      value_type;
};

#if defined(__GNUC__) && defined(__x86_64__)
template <>
struct type_from_size<128> {
    static const bool           is_specialized = true;
    static const size_t         size = 128;
    typedef __int128            value_type;
    typedef unsigned __int128   unsigned_type;
    typedef __int128            signed_type;
    typedef type_from_size<256> next_size;
};
#endif

template <>
struct type_from_size<64> {
    static const bool           is_specialized = true;
    static const size_t         size = 64;
    typedef int64_t             value_type;
    typedef uint64_t            unsigned_type;
    typedef int64_t             signed_type;
    typedef type_from_size<128> next_size;
};

template <>
struct type_from_size<32> {
    static const bool          is_specialized = true;
    static const size_t        size = 32;
    typedef int32_t            value_type;
    typedef uint32_t           unsigned_type;
    typedef int32_t            signed_type;
    typedef type_from_size<64> next_size;
};

template <>
struct type_from_size<16> {
    static const bool          is_specialized = true;
    static const size_t        size = 16;
    typedef int16_t            value_type;
    typedef uint16_t           unsigned_type;
    typedef int16_t            signed_type;
    typedef type_from_size<32> next_size;
};

template <>
struct type_from_size<8> {
    static const bool          is_specialized = true;
    static const size_t        size = 8;
    typedef int8_t             value_type;
    typedef uint8_t            unsigned_type;
    typedef int8_t             signed_type;
    typedef type_from_size<16> next_size;
};

// this is to assist in adding support for non-native base
// types (for adding big-int support), this should be fine
// unless your bit-int class doesn't nicely support casting
template <class B, class N>
B next_to_base(const N& rhs) {
    return static_cast<B>(rhs);
}

struct divide_by_zero : std::exception {
};

template <size_t I, size_t F>
Fixed<I,F> divide(const Fixed<I,F> &numerator, const Fixed<I,F> &denominator, Fixed<I,F> &remainder, typename std::enable_if<type_from_size<I+F>::next_size::is_specialized>::type* = 0) {

    typedef typename Fixed<I,F>::next_type next_type;
    typedef typename Fixed<I,F>::base_type base_type;
    static const size_t fractional_bits = Fixed<I,F>::fractional_bits;

    next_type t(numerator.to_raw());
    t <<= fractional_bits;

    Fixed<I,F> quotient;

    quotient  = Fixed<I,F>::from_base(next_to_base<base_type>(t / denominator.to_raw()));
    remainder = Fixed<I,F>::from_base(next_to_base<base_type>(t % denominator.to_raw()));

    return quotient;
}

template <size_t I, size_t F>
Fixed<I,F> divide(Fixed<I,F> numerator, Fixed<I,F> denominator, Fixed<I,F> &remainder, typename std::enable_if<!type_from_size<I+F>::next_size::is_specialized>::type* = 0) {

    // NOTE(eteran): division is broken for large types :-(
    // especially when dealing with negative quantities

    typedef typename Fixed<I,F>::base_type     base_type;
    typedef typename Fixed<I,F>::unsigned_type unsigned_type;

    static const int bits = Fixed<I,F>::total_bits;

    if(denominator == 0) {
        throw divide_by_zero();
    } else {

        int sign = 0;

        Fixed<I,F> quotient;

        if(numerator < 0) {
            sign ^= 1;
            numerator = -numerator;
        }

        if(denominator < 0) {
            sign ^= 1;
            denominator = -denominator;
        }

            base_type n      = numerator.to_raw();
            base_type d      = denominator.to_raw();
            base_type x      = 1;
            base_type answer = 0;

            // egyptian division algorithm
            while((n >= d) && (((d >> (bits - 1)) & 1) == 0)) {
                x <<= 1;
                d <<= 1;
            }

            while(x != 0) {
                if(n >= d) {
                    n      -= d;
                    answer += x;
                }

                x >>= 1;
                d >>= 1;
            }

            unsigned_type l1 = n;
            unsigned_type l2 = denominator.to_raw();

            // calculate the lower bits (needs to be unsigned)
            // unfortunately for many fractions this overflows the type still :-/
            const unsigned_type lo = (static_cast<unsigned_type>(n) << F) / denominator.to_raw();

            quotient  = Fixed<I,F>::from_base((answer << F) | lo);
            remainder = n;

        if(sign) {
            quotient = -quotient;
        }

        return quotient;
    }
}

// this is the usual implementation of multiplication
template <size_t I, size_t F>
void multiply(const Fixed<I,F> &lhs, const Fixed<I,F> &rhs, Fixed<I,F> &result, typename std::enable_if<type_from_size<I+F>::next_size::is_specialized>::type* = 0) {

    typedef typename Fixed<I,F>::next_type next_type;
    typedef typename Fixed<I,F>::base_type base_type;

    static const size_t fractional_bits = Fixed<I,F>::fractional_bits;

    next_type t(static_cast<next_type>(lhs.to_raw()) * static_cast<next_type>(rhs.to_raw()));
    t >>= fractional_bits;
    result = Fixed<I,F>::from_base(next_to_base<base_type>(t));
}

// this is the fall back version we use when we don't have a next size
// it is slightly slower, but is more robust since it doesn't
// require and upgraded type
template <size_t I, size_t F>
void multiply(const Fixed<I,F> &lhs, const Fixed<I,F> &rhs, Fixed<I,F> &result, typename std::enable_if<!type_from_size<I+F>::next_size::is_specialized>::type* = 0) {

    typedef typename Fixed<I,F>::base_type base_type;

    static const size_t fractional_bits = Fixed<I,F>::fractional_bits;
    static const size_t integer_mask    = Fixed<I,F>::integer_mask;
    static const size_t fractional_mask = Fixed<I,F>::fractional_mask;

    // more costly but doesn't need a larger type
    const base_type a_hi = (lhs.to_raw() & integer_mask) >> fractional_bits;
    const base_type b_hi = (rhs.to_raw() & integer_mask) >> fractional_bits;
    const base_type a_lo = (lhs.to_raw() & fractional_mask);
    const base_type b_lo = (rhs.to_raw() & fractional_mask);

    const base_type x1 = a_hi * b_hi;
    const base_type x2 = a_hi * b_lo;
    const base_type x3 = a_lo * b_hi;
    const base_type x4 = a_lo * b_lo;

    result = Fixed<I,F>::from_base((x1 << fractional_bits) + (x3 + x2) + (x4 >> fractional_bits));

}
}

/*
 * inheriting from boost::operators enables us to be a drop in replacement for base types
 * without having to specify all the different versions of operators manually
 */
template <size_t I, size_t F>
class Fixed : boost::operators<Fixed<I,F>> {
    static_assert(detail::type_from_size<I + F>::is_specialized, "invalid combination of sizes");

public:
    static const size_t fractional_bits = F;
    static const size_t integer_bits    = I;
    static const size_t total_bits      = I + F;

    typedef detail::type_from_size<total_bits>             base_type_info;

    typedef typename base_type_info::value_type            base_type;
    typedef typename base_type_info::next_size::value_type next_type;
    typedef typename base_type_info::unsigned_type         unsigned_type;

public:
    static const size_t base_size          = base_type_info::size;
    static const base_type fractional_mask = ~((~base_type(0)) << fractional_bits);
    static const base_type integer_mask    = ~fractional_mask;

public:
    static const base_type one = base_type(1) << fractional_bits;

public: // constructors
    Fixed() : data_(0) {
    }

    Fixed(long n) : data_(base_type(n) << fractional_bits) {
        // TODO(eteran): assert in range!
    }

    Fixed(unsigned long n) : data_(base_type(n) << fractional_bits) {
        // TODO(eteran): assert in range!
    }

    Fixed(int n) : data_(base_type(n) << fractional_bits) {
        // TODO(eteran): assert in range!
    }

    Fixed(unsigned int n) : data_(base_type(n) << fractional_bits) {
        // TODO(eteran): assert in range!
    }

    Fixed(float n) : data_(static_cast<base_type>(n * one)) {
        // TODO(eteran): assert in range!
    }

    Fixed(double n) : data_(static_cast<base_type>(n * one))  {
        // TODO(eteran): assert in range!
    }

    Fixed(const Fixed &o) : data_(o.data_) {
    }

    Fixed& operator=(const Fixed &o) {
        data_ = o.data_;
        return *this;
    }

private:
    // this makes it simpler to create a fixed point object from
    // a native type without scaling
    // use "Fixed::from_base" in order to perform this.
    struct NoScale {};

    Fixed(base_type n, const NoScale &) : data_(n) {
    }

public:
    static Fixed from_base(base_type n) {
        return Fixed(n, NoScale());
    }

public: // comparison operators
    bool operator==(const Fixed &o) const {
        return data_ == o.data_;
    }

    bool operator<(const Fixed &o) const {
        return data_ < o.data_;
    }

public: // unary operators
    bool operator!() const {
        return !data_;
    }

    Fixed operator~() const {
        Fixed t(*this);
        t.data_ = ~t.data_;
        return t;
    }

    Fixed operator-() const {
        Fixed t(*this);
        t.data_ = -t.data_;
        return t;
    }

    Fixed operator+() const {
        return *this;
    }

    Fixed& operator++() {
        data_ += one;
        return *this;
    }

    Fixed& operator--() {
        data_ -= one;
        return *this;
    }

public: // basic math operators
    Fixed& operator+=(const Fixed &n) {
        data_ += n.data_;
        return *this;
    }

    Fixed& operator-=(const Fixed &n) {
        data_ -= n.data_;
        return *this;
    }

    Fixed& operator&=(const Fixed &n) {
        data_ &= n.data_;
        return *this;
    }

    Fixed& operator|=(const Fixed &n) {
        data_ |= n.data_;
        return *this;
    }

    Fixed& operator^=(const Fixed &n) {
        data_ ^= n.data_;
        return *this;
    }

    Fixed& operator*=(const Fixed &n) {
        detail::multiply(*this, n, *this);
        return *this;
    }

    Fixed& operator/=(const Fixed &n) {
        Fixed temp;
        *this = detail::divide(*this, n, temp);
        return *this;
    }

    Fixed& operator>>=(const Fixed &n) {
        data_ >>= n.to_int();
        return *this;
    }

    Fixed& operator<<=(const Fixed &n) {
        data_ <<= n.to_int();
        return *this;
    }

public: // conversion to basic types
    int to_int() const {
        return (data_ & integer_mask) >> fractional_bits;
    }

    unsigned int to_uint() const {
        return (data_ & integer_mask) >> fractional_bits;
    }

    float to_float() const {
        return static_cast<float>(data_) / Fixed::one;
    }

    double to_double() const        {
        return static_cast<double>(data_) / Fixed::one;
    }

    base_type to_raw() const {
        return data_;
    }

public:
    void swap(Fixed &rhs) {
        using std::swap;
        swap(data_, rhs.data_);
    }

public:
    base_type data_;
};

// if we have the same fractional portion, but differing integer portions, we trivially upgrade the smaller type
template <size_t I1, size_t I2, size_t F>
typename std::conditional<I1 >= I2, Fixed<I1,F>, Fixed<I2,F>>::type operator+(const Fixed<I1,F> &lhs, const Fixed<I2,F> &rhs) {

    typedef typename std::conditional<
        I1 >= I2,
        Fixed<I1,F>,
        Fixed<I2,F>
    >::type T;

    const T l = T::from_base(lhs.to_raw());
    const T r = T::from_base(rhs.to_raw());
    return l + r;
}

template <size_t I1, size_t I2, size_t F>
typename std::conditional<I1 >= I2, Fixed<I1,F>, Fixed<I2,F>>::type operator-(const Fixed<I1,F> &lhs, const Fixed<I2,F> &rhs) {

    typedef typename std::conditional<
        I1 >= I2,
        Fixed<I1,F>,
        Fixed<I2,F>
    >::type T;

    const T l = T::from_base(lhs.to_raw());
    const T r = T::from_base(rhs.to_raw());
    return l - r;
}

template <size_t I1, size_t I2, size_t F>
typename std::conditional<I1 >= I2, Fixed<I1,F>, Fixed<I2,F>>::type operator*(const Fixed<I1,F> &lhs, const Fixed<I2,F> &rhs) {

    typedef typename std::conditional<
        I1 >= I2,
        Fixed<I1,F>,
        Fixed<I2,F>
    >::type T;

    const T l = T::from_base(lhs.to_raw());
    const T r = T::from_base(rhs.to_raw());
    return l * r;
}

template <size_t I1, size_t I2, size_t F>
typename std::conditional<I1 >= I2, Fixed<I1,F>, Fixed<I2,F>>::type operator/(const Fixed<I1,F> &lhs, const Fixed<I2,F> &rhs) {

    typedef typename std::conditional<
        I1 >= I2,
        Fixed<I1,F>,
        Fixed<I2,F>
    >::type T;

    const T l = T::from_base(lhs.to_raw());
    const T r = T::from_base(rhs.to_raw());
    return l / r;
}

template <size_t I, size_t F>
std::ostream &operator<<(std::ostream &os, const Fixed<I,F> &f) {
    os << f.to_double();
    return os;
}

template <size_t I, size_t F>
const size_t Fixed<I,F>::fractional_bits;

template <size_t I, size_t F>
const size_t Fixed<I,F>::integer_bits;

template <size_t I, size_t F>
const size_t Fixed<I,F>::total_bits;

}

#endif

它被设计为几乎可以替代浮点数/双精度数，并且具有可选择的精度。它确实利用 boost 来添加所有必要的数学运算符重载，因此您也需要它（我相信为此它只是一个标头依赖项，而不是库依赖项）。

顺便说一句，常见用法可能是这样的：

using namespace numeric;
typedef Fixed<16, 16> fixed;
fixed f;

唯一真正的规则是该数字必须加起来等于系统的本机大小，例如 8、16、32、64。