数值计算: IEEE754浮点数

文章目录

• IEEE754浮点数
• • • 单精度(float,32bits)
    • 双精度(double,64bits)
    • 浮点数加减法流程逻辑
    • • 工具程序

IEEE754浮点数

What Every Computer Scientist Should Know About Floating-Point Arithmetic

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。数轴上有空隙，多数浮点数是不能被精确表示的，比如是十进制数0.1就不能被IEEE754浮点数精确表示
• 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). 函数atan2(y,x)按IEEE-754标准x取正0和负0,结果对应π \piπ和− π -\pi−π（Intel Fortran和Matlab中的atan2函数对正负零的处理有不同之处)
• 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)（guard bits是三位？）
• 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x 2 + y 2 \sqrt{x^2+y^2}x2+y2​, 和求多维向量长度∑ x i 2 \sqrt{\sum{x_i^2}}∑xi2​​, 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flush 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:

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单精度(float,32bits)

说明
Bias	127
E范围	[1…254], 0和255保留
Range	2 − 126 2^{-126}2−126to2 + 127 2^{+127}2+127

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

浮点数	二进制表示
0	00000000000000000000000000000000
-0	10000000000000000000000000000000
1	00111111100000000000000000000000
-1	10111111100000000000000000000000
eps	00110100000000000000000000000000
1+eps	00111111100000000000000000000001
min	00000000100000000000000000000000
max	01111111011111111111111111111111
denorm_min	00000000000000000000000000000001
infinity	01111111100000000000000000000000
sNaN	01111111101000000000000000000000
qNaN	01111111110000000000000000000000

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双精度(double,64bits)

说明
Bias	1023
E范围	[1…2046], 0和2047保留
Range	2 − 1022 2^{-1022}2−1022to2 + 1023 2^{+1023}2+1023

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

浮点数	二进制表示
0	0000000000000000000000000000000000000000000000000000000000000000
-0	1000000000000000000000000000000000000000000000000000000000000000
1	0011111111110000000000000000000000000000000000000000000000000000
-1	1011111111110000000000000000000000000000000000000000000000000000
eps	0011110010110000000000000000000000000000000000000000000000000000
1+eps	0011111111110000000000000000000000000000000000000000000000000001
min	0000000000010000000000000000000000000000000000000000000000000000
max	0111111111101111111111111111111111111111111111111111111111111111
infinity	0111111111110000000000000000000000000000000000000000000000000000
sNaN	0111111111110100000000000000000000000000000000000000000000000000
qNaN	0111111111111000000000000000000000000000000000000000000000000000

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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(constR x,std::ostream&os=std::cout){uint8_t*u8=(uint8_t*)&x;strings("");//assume little-endianfor(inti=sizeof(R)-1;i>=0;--i){std::bitset<8>b(u8[i]);s+=b.to_string();}os<<s;returnos;}template<typename R>std::ostream&dump_hex(constR x,std::ostream&os=std::cout){os<<std::hexfloat;os<<x;os<<std::defaultfloat;returnos;}template<typename T>voidprint_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>voidprint_bits_and_hex(){static_assert(std::is_same_v<T,float>||std::is_same_v<T,double>||std::is_same_v<T,longdouble>);using flimits=numeric_limits<T>;autodump=[](std::ostream&os,string name,constT&x)->std::ostream&{os<<name<<"\t";dump_bits(x,os);os<<" ";os<<std::hexfloat;os<<x;os<<std::defaultfloat;os<<"\n";returnos;};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());}intmain(intargc,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(longdouble)二进制模式==========================================================================)"<<"\n";print_bits_and_hex<longdouble>();//=======================================================================//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infcout<<numeric_limits<float>::infinity()/2.0f<<"\n";//nan和其他数值比较，结果都为falsecout<<(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