[C++知识库]IEEE754浮点数

IEEE754浮点数

一个浮点数包括三部分: 符号部分(Sign)、指数部分(Exponent)、分数部分(Fraction)

• IEEE754浮点数不是均匀分布的。仅能代表有限个数的实数
• 对于normal浮点数,1.xxxx中的1是隐含存在的
• normal浮点数0有正负之分(S=0/1, E=0, Fraction=0)
• 有subnormal数(非常小，接近数值0): E=0， Fraction部分不为0
• IEEE754浮点数内部计算寄存器多出两位（保证gaurd & rounding)
• 有四种截断/舍入模式(rounding)
• 有overflow/underflow浮点计算异常。underflow危害不大，overflow需要特殊关注。overflow一个情形是从其他类型转换引起的(e.g.,一个很大的整数转成float, 或double转成float), 另外一个教科书级例子是hypot计算$\sqrt{x^2+y^2}$, 和求多维向量长度$\sqrt{\sum x_i^2}$, 类似计算要时刻避免浮点overflow溢出！
• 比较: inf>1, 返回1; NaN>1、NaN==1、NaN<1都返回0
• 对于subnormal数一般有两种处理方式: flush to zero .vs. gradual underflow。subnormal数对性能影响较大，可以指定编译选项打开或关闭lush to zero；gradual underflow一般对比较精细计算中有帮助，比如求函数数值导数等。

单精度浮点数可表示范围:

单精度(float,32bits)

一些特殊单精度数(调用std::numeric_limits<float>获取)

双精度(double,64bits)

一些特殊双精度数(调用std::numeric_limits<double>获取)

浮点数加减法流程逻辑

工具程序

/************************************
测试IEEE浮点数标准、表示形式等
*************************************/

#include <cmath>
#include <iostream>
#include <bitset>
#include <limits>
#include <type_traits>
#include <cstdint>
#include <sstream>
#include <string>

using namespace std;

template<typename R>
std::ostream &dump_bits(const R x,std::ostream &os=std::cout)
{


    uint8_t *u8=(uint8_t *)&x;
    string s("");

    //assume little-endian 
    for(int i=sizeof(R)-1; i>=0; --i)
    {
        std::bitset<8> b(u8[i]);
        s+=b.to_string();
    }
    os<<s;

    return os;
}

template<typename R>
std::ostream &dump_hex(const R x,std::ostream &os=std::cout)
{
    os<<std::hexfloat;
    os<<x;
    os<<std::defaultfloat;
    return os;
}

template<typename T>
void print_limits()
{
    using flimits=numeric_limits<T>;
    cout<<"radix:\t"<<flimits::radix<<"\n";
    cout<<"min_exponent:\t"<<flimits::min_exponent<<"\n";
    cout<<"max_exponent:\t"<<flimits::max_exponent<<"\n";
    
    cout<<"digits:\t"<<flimits::digits<<"\n";
    cout<<"digits10:\t"<<flimits::digits10<<"\n";

    cout<<"epsilon:\t"<<flimits::epsilon()<<"\n";
    cout<<"inf:\t"<<flimits::infinity()<<"\n";
    cout<<"qNan:\t"<<flimits::quiet_NaN()<<"\n";
    cout<<"sNan:\t"<<flimits::signaling_NaN()<<"\n";

    cout<<"min:\t"<<flimits::min()<<"\n";
    cout<<"max:\t"<<flimits::max()<<"\n";

}

template<typename T>
void print_bits_and_hex()
{
    static_assert(std::is_same_v<T,float> || std::is_same_v<T,double> || std::is_same_v<T,long double>);
    using flimits=numeric_limits<T>;
    
    auto dump=[](std::ostream &os, string name, const T &x)->std::ostream &
    {
        os<<name<<"\t";
        dump_bits(x,os);
        os<<" ";
        os<<std::hexfloat;
        os<<x;
        os<<std::defaultfloat;
        os<<"\n";
        return os;

    };

    dump(cout,"infinity",flimits::infinity());
    dump(cout,"sNaN",flimits::signaling_NaN());
    dump(cout,"qNaN",flimits::quiet_NaN());
    dump(cout,"0",T(0.0));
    dump(cout,"-0",T(-0.0));
    dump(cout,"1",T(1));
    dump(cout,"-1",T(-1));
    dump(cout,"eps", flimits::epsilon());
    dump(cout,"1+eps", flimits::epsilon()+T(1));
    dump(cout,"min",flimits::min());
    dump(cout,"max",flimits::max());
    dump(cout,"denorm_min",flimits::denorm_min());

}

int main(int argc, char **argv)
{

    cout<<R"(
=========================================================================
               单精度浮点数(float) limits
=========================================================================)"<<"\n";
    print_limits<float>();
    cout<<R"(
=========================================================================
               双精度浮点数(double) limits
=========================================================================)"<<"\n";
    print_limits<double>();
    cout<<R"(
=========================================================================
               单精度浮点数(float)二进制模式
=========================================================================)"<<"\n";
    print_bits_and_hex<float>();

    cout<<R"(
==========================================================================
               双精度浮点数(double)二进制模式
==========================================================================)"<<"\n";
    print_bits_and_hex<double>();

    cout<<R"(
==========================================================================
               长精度浮点数(long double)二进制模式
==========================================================================)"<<"\n";
    print_bits_and_hex<long double>();  


    //=======================================================================

    //inf可和其他结果比较
    cout<<((numeric_limits<float>::infinity()>1.0f)?"inf>1":"inf<=1")<<"\n";
    
    //inf参加运算，结果是inf
    cout<<numeric_limits<float>::infinity()/2.0f<<"\n";

    //nan和其他数值比较，结果都为false
    cout<<(numeric_limits<float>::quiet_NaN()>1.0f)<<"\n";
    cout<<(numeric_limits<float>::quiet_NaN()<1.0f)<<"\n";

    //0有特殊运算定义
    cout<<(-0.0f<0.0f)<<"\n";

    return(0);
}

//编译: g++ -std=c++17 cxx_ex3.cpp