做定点运算的最好方法是什么?

I need to speed up a program for the Nintendo DS which doesn't have an FPU, so I need to change floating-point math (which is emulated and slow) to fixed-point.

我需要为没有FPU的任天堂DS加速一个程序，所以我需要将浮点数(模拟和缓慢)更改为定点。

How I started was I changed floats to ints and whenever I needed to convert them, I used x>>8 to convert the fixed-point variable x to the actual number and x<<8 to convert to fixed-point. Soon I found out it was impossible to keep track of what needed to be converted and I also realized it would be difficult to change the precision of the numbers (8 in this case.)

我是如何开始的呢，我将浮点数改为ints，每当我需要转换它们时，我使用x>>8将定点变量x转换为实际数字，x< 8转换为定点。很快我发现不可能跟踪需要转换的内容，而且我也意识到很难改变数字的精度(在这种情况下是8)。

My question is, how should I make this easier and still fast? Should I make a FixedPoint class, or just a FixedPoint8 typedef or struct with some functions/macros to convert them, or something else? Should I put something in the variable name to show it's fixed-point?

我的问题是，我应该如何使这变得更容易，而且仍然快速?我是应该创建一个FixedPoint类，还是只创建一个FixedPoint8类型def或具有一些函数/宏来转换它们的结构，还是其他什么?我应该在变量名中加入一些东西来表示它的定点吗?

9 个解决方案

#1

You can try my fixed point class (Latest available @ https://github.com/eteran/cpp-utilities)

您可以尝试我的定点类(最新可用的@ https://github.com/eteran/cpp-utilities)

// From: https://github.com/eteran/cpp-utilities/edit/master/Fixed.h
// See also: http://*.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

It is designed to be a near drop in replacement for floats/doubles and has a choose-able precision. It does make use of boost to add all the necessary math operator overloads, so you will need that as well (I believe for this it is just a header dependency, not a library dependency).

它被设计成浮点/双精度的几乎下降，并且具有可选择的精度。它确实使用boost来添加所有必需的数学运算符重载，所以您也需要它(我认为这只是一个头依赖项，而不是库依赖项)。

BTW, common usage could be something like this:

顺便说一下，通常的用法是这样的:

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

The only real rule is that the number have to add up to a native size of your system such as 8, 16, 32, 64.

唯一真正的规则是，数字必须加到系统的本地大小，比如8,16,32,64。

#2

In modern C++ implementations, there will be no performance penalty for using simple and lean abstractions, such as concrete classes. Fixed-point computation is precisely the place where using a properly engineered class will save you from lots of bugs.

在现代c++实现中，使用简单的和精益的抽象(如具体的类)不会造成性能损失。定点计算正是使用适当的工程类来避免大量错误的地方。

Therefore, you should write a FixedPoint8 class. Test and debug it thoroughly. If you have to convince yourself of its performance as compared to using plain integers, measure it.

因此，您应该编写一个FixedPoint8类。彻底地测试和调试它。如果你必须说服自己与使用普通整数相比，它的性能是怎样的，那就衡量它。

It will save you from many a trouble by moving the complexity of fixed-point calculation to a single place.

通过将定点计算的复杂性转移到一个地方，可以避免许多麻烦。

If you like, you can further increase the utility of your class by making it a template and replacing the old FixedPoint8 with, say, typedef FixedPoint<short, 8> FixedPoint8; But on your target architecture this is not probably necessary, so avoid the complexity of templates at first.

如果您愿意，您可以进一步增加类的效用，将其作为模板，并将旧的FixedPoint8替换为typedef FixedPoint FixedPoint8;但是在您的目标体系结构中，这可能不是必需的，所以首先要避免模板的复杂性。 ,>

There is probably a good fixed point class somewhere in the internet - I'd start looking from the Boost libraries.

在互联网的某个地方可能有一个很好的固定的点类——我将从Boost库开始查找。

#3

Does your floating point code actually make use of the decimal point? If so:

您的浮点代码实际上是否使用了十进制吗?如果是这样:

First you have to read Randy Yates's paper on Intro to Fixed Point Math: http://www.digitalsignallabs.com/fp.pdf

首先，你必须阅读兰迪·耶茨关于定点数学入门的论文:http://www.digitalsignallabs.com/fp.pdf

Then you need to do "profiling" on your floating point code to figure out the appropriate range of fixed-point values required at "critical" points in your code, e.g. U(5,3) = 5 bits to the left, 3 bits to the right, unsigned.

然后，您需要对浮点代码进行“分析”，以确定在代码的“关键”点所需的固定点值的适当范围，例如，U(5,3) =左5位，右3位，无符号。

At this point, you can apply the arithmetic rules in the paper mentioned above; the rules specify how to interpret the bits which result from arithmetic operations. You can write macros or functions to perform the operations.

此时，你可以应用上述论文中的算术规则;规则指定如何解释算术运算产生的位元。您可以编写宏或函数来执行操作。

It's handy to keep the floating point version around, in order to compare the floating point vs fixed point results.

为了比较浮点数和定点数的结果，将浮点数版本保留在周围是很方便的。

#4

I wouldn't use floating point at all on a CPU without special hardware for handling it. My advice is to treat ALL numbers as integers scaled to a specific factor. For example, all monetary values are in cents as integers rather than dollars as floats. For example, 0.72 is represented as the integer 72.

如果没有特殊的硬件，我不会在CPU上使用浮点数。我的建议是把所有的数字都看成是按特定因子缩放的整数。例如，所有的货币值都是以美分为整数，而不是以美元为浮点数。例如，0.72表示为整数72。

Addition and subtraction are then a very simple integer operation such as (0.72 + 1 becomes 72 + 100 becomes 172 becomes 1.72).

加法和减法是一个非常简单的整数运算，比如(0.72 + 1变成72 + 100 = 172)。

Multiplication is slightly more complex as it needs an integer multiply followed by a scale back such as (0.72 * 2 becomes 72 * 200 becomes 14400 becomes 144 (scaleback) becomes 1.44).

乘法运算要稍微复杂一些，因为它需要一个整数乘以，然后进行一个回调(0.72 * 2变成72 * 200变成14400变成144(回调)变成1.44)。

That may require special functions for performing more complex math (sine, cosine, etc) but even those can be sped up by using lookup tables. Example: since you're using fixed-2 representation, there's only 100 values in the range (0.0,1] (0-99) and sin/cos repeat outside this range so you only need a 100-integer lookup table.

这可能需要特殊的函数来执行更复杂的数学运算(正弦、余弦等)，但即使是这些函数也可以通过使用查找表来加速。示例:由于您使用的是固定2表示，所以在这个范围内只有100个值(0.0,1)(0-99)和sin/cos在这个范围之外重复，所以您只需要一个100整数的查找表。

Cheers, Pax.

干杯,罗马帝国。

#5

Changing fixed point representations is commonly called 'scaling'.

改变定点表示法通常被称为“缩放”。

If you can do this with a class with no performance penalty, then that's the way to go. It depends heavily on the compiler and how it inlines. If there is a performance penalty using classes, then you need a more traditional C-style approach. The OOP approach will give you compiler-enforced type safety which the traditional implementation only approximates.

如果您可以在不影响性能的情况下对类执行此操作，那么就应该这样做。这在很大程度上取决于编译器和它如何内联。如果有使用类的性能惩罚，那么您需要更传统的c风格方法。OOP方法将为您提供编译器强制的类型安全性，而传统的实现只接近于这种安全性。

@cibyr has a good OOP implementation. Now for the more traditional one.

@cibyr有一个很好的OOP实现。现在是更传统的。

To keep track of which variables are scaled, you need to use a consistent convention. Make a notation at the end of each variable name to indicate whether the value is scaled or not, and write macros SCALE() and UNSCALE() that expand to x>>8 and x<<8.

要跟踪哪些变量被缩放，需要使用一致的约定。在每个变量名的末尾做一个标记，以指示值是否缩放，并编写扩展到x>>8和x< 8的宏SCALE()和UNSCALE()。

#define SCALE(x) (x>>8)
#define UNSCALE(x) (x<<8)

xPositionUnscaled = UNSCALE(10);
xPositionScaled = SCALE(xPositionUnscaled);

It may seem like extra work to use so much notation, but notice how you can tell at a glance that any line is correct without looking at other lines. For example:

使用如此多的符号似乎是额外的工作，但是请注意，您如何能够一眼看出任何一行都是正确的，而不需要查看其他行。例如:

xPositionScaled = SCALE(xPositionScaled);

is obviously wrong, by inspection.

经检查，显然是错的。

This is a variation of the Apps Hungarian idea that Joel mentions in this post.

这是Joel在这篇文章中提到的匈牙利应用的变体。

#6

When I first encountered fixed point numbers I found Joe Lemieux's article, Fixed-point Math in C, very helpful, and it does suggest one way of representing fixed-point values.

当我第一次遇到定点数时，我发现了Joe Lemieux的文章，C中的定点数学，非常有用，它确实提出了一种表示定点值的方法。

I didn't wind up using his union representation for fixed-point numbers though. I mostly have experience with fixed-point in C, so I haven't had the option to use a class either. For the most part though, I think that defining your number of fraction bits in a macro and using descriptive variable names makes this fairly easy to work with. Also, I've found that it is best to have macros or functions for multiplication and especially division, or you quickly get unreadable code.

不过，我并没有使用他的工会代表来使用定点数。我在C语言中大多有固定点的经验，所以我也没有选择使用类。在大多数情况下，我认为在宏中定义分数位的数量并使用描述性变量名使这一工作变得相当容易。此外，我发现最好使用宏或函数进行乘法运算，尤其是除法运算，否则很快就会出现不可读的代码。

For example, with 24.8 values:

例如，有24.8个值:

 #include "stdio.h"

/* Declarations for fixed point stuff */

typedef int int_fixed;

#define FRACT_BITS 8
#define FIXED_POINT_ONE (1 << FRACT_BITS)
#define MAKE_INT_FIXED(x) ((x) << FRACT_BITS)
#define MAKE_FLOAT_FIXED(x) ((int_fixed)((x) * FIXED_POINT_ONE))
#define MAKE_FIXED_INT(x) ((x) >> FRACT_BITS)
#define MAKE_FIXED_FLOAT(x) (((float)(x)) / FIXED_POINT_ONE)

#define FIXED_MULT(x, y) ((x)*(y) >> FRACT_BITS)
#define FIXED_DIV(x, y) (((x)<<FRACT_BITS) / (y))

/* tests */
int main()
{
    int_fixed fixed_x = MAKE_FLOAT_FIXED( 4.5f );
    int_fixed fixed_y = MAKE_INT_FIXED( 2 );

    int_fixed fixed_result = FIXED_MULT( fixed_x, fixed_y );
    printf( "%.1f\n", MAKE_FIXED_FLOAT( fixed_result ) );

    fixed_result = FIXED_DIV( fixed_result, fixed_y );
    printf( "%.1f\n", MAKE_FIXED_FLOAT( fixed_result ) );

    return 0;
}

Which writes out

这写

9.0
4.5

Note that there are all kinds of integer overflow issues with those macros, I just wanted to keep the macros simple. This is just a quick and dirty example of how I've done this in C. In C++ you could make something a lot cleaner using operator overloading. Actually, you could easily make that C code a lot prettier too...

注意，在这些宏中有各种各样的整数溢出问题，我只是想让宏保持简单。这只是我在C语言中做的一个简单而肮脏的例子。实际上，你也可以让C代码更漂亮……

I guess this is a long-winded way of saying: I think it's OK to use a typedef and macro approach. So long as you're clear about what variables contain fixed point values it isn't too hard to maintain, but it probably won't be as pretty as a C++ class.

我想这是一种长篇大论的说法:我认为使用类型定义和宏方法是可以的。只要您清楚哪些变量包含固定点值，就不难维护，但它可能不会像c++类那样漂亮。

If I was in your position, I would try to get some profiling numbers to show where the bottlenecks are. If there are relatively few of them then go with a typedef and macros. If you decide that you need a global replacement of all floats with fixed-point math though, then you'll probably be better off with a class.

如果我处在您的位置，我将尝试获取一些分析数据，以显示瓶颈在哪里。如果它们中相对较少，则使用typedef和宏。如果您决定需要一个全局变量替换所有浮点数的浮点数，那么您可能会更好地使用一个类。

#7

The original version of Tricks of the Game Programming Gurus has an entire chapter on implementing fixed-point math.

游戏编程大师的原来版本有一整章是关于实现定点数学的。

#8

Whichever way you decide to go (I'd lean toward a typedef and some CPP macros for converting), you will need to be careful to convert back and forth with some discipline.

无论您选择哪种方式(我倾向于使用typedef和一些CPP宏进行转换)，您都需要小心地使用一些规则进行来回转换。

You might find that you never need to convert back and forth. Just imagine everything in the whole system is x256.

你可能会发现你不需要来回转换。假设整个系统中的所有东西都是x256。

#9

template <int precision = 8> class FixedPoint {
private:
    int val_;
public:
    inline FixedPoint(int val) : val_ (val << precision) {};
    inline operator int() { return val_ >> precision; }
    // Other operators...
};