C 负指数平方幂
我不确定平方幂是否考虑负指数。我实现了以下代码,它只适用于正数C 负指数平方幂,c,algorithm,math,recursion,C,Algorithm,Math,Recursion,我不确定平方幂是否考虑负指数。我实现了以下代码,它只适用于正数 #include <stdio.h> int powe(int x, int exp) { if (x == 0) return 1; if (x == 1) return x; if (x&1) return powe(x*x, exp/2);
#include <stdio.h>
int powe(int x, int exp)
{
if (x == 0)
return 1;
if (x == 1)
return x;
if (x&1)
return powe(x*x, exp/2);
else
return x*powe(x*x, (exp-1)/2);
}
#包括
整数功率(整数x,整数exp)
{
如果(x==0)
返回1;
如果(x==1)
返回x;
如果(x&1)
返回功率(x*x,exp/2);
其他的
返回x*功率(x*x,(exp-1)/2);
}
查看没有帮助,因为下面的代码似乎是错误的
Function exp-by-squaring(x, n )
if n < 0 then return exp-by-squaring(1 / x, - n );
else if n = 0 then return 1;
else if n = 1 then return x ;
else if n is even then return exp-by-squaring(x * x, n / 2);
else if n is odd then return x * exp-by-squaring(x * x, (n - 1) / 2).
平方函数exp(x,n)
如果n<0,则通过平方(1/x,-n)返回exp;
否则,如果n=0,则返回1;
否则,如果n=1,则返回x;
否则,如果n为偶数,则通过平方(x*x,n/2)返回exp;
否则,如果n是奇数,则通过平方(x*x,(n-1)/2)返回x*exp。
编辑:
由于amit,此解决方案适用于负数和正数:
float powe(float x, int exp)
{
if (exp < 0)
return powe(1/x, -exp);
if (exp == 0)
return 1;
if (exp == 1)
return x;
if (((int)exp)%2==0)
return powe(x*x, exp/2);
else
return x*powe(x*x, (exp-1)/2);
}
float功率(float x,int exp)
{
如果(exp<0)
返回功率(1/x,-exp);
如果(exp==0)
返回1;
如果(exp==1)
返回x;
如果((int)exp)%2==0)
返回功率(x*x,exp/2);
其他的
返回x*功率(x*x,(exp-1)/2);
}
对于分数指数,我们可以执行以下操作(Spektre方法):
,如果mid*mid整数示例用于32位int
算术,DWORD
是32位无符号int
浮动pow(x,y)=x^y
通常是这样评估的:
//---------------------------------------------------------------------------
//--- 32bit signed fixed point format (2os complement)
//---------------------------------------------------------------------------
// |MSB LSB|
// |integer|.|fractional|
//---------------------------------------------------------------------------
const int _fx32_bits=32; // all bits count
const int _fx32_fract_bits=16; // fractional bits count
const int _fx32_integ_bits=_fx32_bits-_fx32_fract_bits; // integer bits count
//---------------------------------------------------------------------------
const int _fx32_one =1<<_fx32_fract_bits; // constant=1.0 (fixed point)
const float _fx32_onef =_fx32_one; // constant=1.0 (floating point)
const int _fx32_fract_mask=_fx32_one-1; // fractional bits mask
const int _fx32_integ_mask=0xFFFFFFFF-_fx32_fract_mask; // integer bits mask
const int _fx32_sMSB_mask =1<<(_fx32_bits-1); // max signed bit mask
const int _fx32_uMSB_mask =1<<(_fx32_bits-2); // max unsigned bit mask
//---------------------------------------------------------------------------
float fx32_get(int x) { return float(x)/_fx32_onef; }
int fx32_set(float x) { return int(float(x*_fx32_onef)); }
//---------------------------------------------------------------------------
int fx32_mul(const int &x,const int &y) // x*y
{
int a=x,b=y; // asm has access only to local variables
asm { // compute (a*b)>>_fx32_fract
mov eax,a
mov ebx,b
mul eax,ebx // (edx,eax)=a*b
mov ebx,_fx32_one
div ebx // eax=(a*b)>>_fx32_fract
mov a,eax;
}
return a;
}
//---------------------------------------------------------------------------
int fx32_div(const int &x,const int &y) // x/y
{
int a=x,b=y; // asm has access only to local variables
asm { // compute (a*b)>>_fx32_fract
mov eax,a
mov ebx,_fx32_one
mul eax,ebx // (edx,eax)=a<<_fx32_fract
mov ebx,b
div ebx // eax=(a<<_fx32_fract)/b
mov a,eax;
}
return a;
}
//---------------------------------------------------------------------------
int fx32_abs_sqrt(int x) // |x|^(0.5)
{
int m,a;
if (!x) return 0;
if (x<0) x=-x;
m=bits(x); // integer bits
for (a=x,m=0;a;a>>=1,m++); // count all bits
m-=_fx32_fract_bits; // compute result integer bits (half of x integer bits)
if (m<0) m=0; m>>=1;
m=_fx32_one<<m; // MSB of result mask
for (a=0;m;m>>=1) // test bits from MSB to 0
{
a|=m; // bit set
if (fx32_mul(a,a)>x) // if result is too big
a^=m; // bit clear
}
return a;
}
//---------------------------------------------------------------------------
int fx32_pow(int x,int y) // x^y
{
// handle special cases
if (!y) return _fx32_one; // x^0 = 1
if (!x) return 0; // 0^y = 0 if y!=0
if (y==-_fx32_one) return fx32_div(_fx32_one,x); // x^-1 = 1/x
if (y==+_fx32_one) return x; // x^+1 = x
int m,a,b,_y; int sx,sy;
// handle the signs
sx=0; if (x<0) { sx=1; x=-x; }
sy=0; if (y<0) { sy=1; y=-y; }
_y=y&_fx32_fract_mask; // _y fractional part of exponent
y=y&_fx32_integ_mask; // y integer part of exponent
a=_fx32_one; // ini result
// powering by squaring x^y
if (y)
{
for (m=_fx32_uMSB_mask;(m>_fx32_one)&&(m>y);m>>=1); // find mask of highest bit of exponent
for (;m>=_fx32_one;m>>=1)
{
a=fx32_mul(a,a);
if (int(y&m)) a=fx32_mul(a,x);
}
}
// powering by rooting x^_y
if (_y)
{
for (b=x,m=_fx32_one>>1;m;m>>=1) // use only fractional part
{
b=fx32_abs_sqrt(b);
if (int(_y&m)) a=fx32_mul(a,b);
}
}
// handle signs
if (sy) { if (a) a=fx32_div(_fx32_one,a); else a=0; /*Error*/ } // underflow
if (sx) { if (_y) a=0; /*Error*/ else if(int(y&_fx32_one)) a=-a; } // negative number ^ non integer exponent, here could add test if 1/_y is integer instead
return a;
}
//---------------------------------------------------------------------------
float a,b,c0,c1,d;
int x,y;
for (a=0.0,x=fx32_set(a);a<=10.0;a+=0.1,x=fx32_set(a))
for (b=-2.5,y=fx32_set(b);b<=2.5;b+=0.1,y=fx32_set(b))
{
if (!x) continue; // math pow has problems with this
if (!y) continue; // math pow has problems with this
c0=pow(a,b);
c1=fx32_get(fx32_pow(x,y));
d=0.0;
if (fabs(c1)<1e-3) d=c1-c0; else d=(c0/c1)-1.0;
if (fabs(d)>0.1)
d=d; // here add breakpoint to check inconsistencies with math pow
}
因此可以计算分数指数:pow(x,y)=exp2(y*log2(x))
。这也可以在固定点上完成:
整数pow(a,b)=a^b
其中a>=0,b>=0
这很容易(您已经做到了)通过平方:
DWORD powuu(DWORD a,DWORD b)
{
int i,bits=32;
DWORD d=1;
for (i=0;i<bits;i++)
{
d*=d;
if (DWORD(b&0x80000000)) d*=a;
b<<=1;
}
return d;
}
整数pow(a,b)=a^b
所以如果bx)
与if(pow(a,bb)>x)
其中bb=1/b
。。。所以b
是您要查找的分数指数,bb
是整数。另外m
是结果的位数,因此改变m=(位(x)>>1)代码>至m=(位(x)/bb)代码>
[edit1]定点sqrt示例
//---------------------------------------------------------------------------
const int _fx32_fract=16; // fractional bits count
const int _fx32_one =1<<_fx32_fract;
DWORD fx32_mul(const DWORD &x,const DWORD &y) // unsigned fixed point mul
{
DWORD a=x,b=y; // asm has access only to local variables
asm { // compute (a*b)>>_fx32_fract
mov eax,a // eax=a
mov ebx,b // ebx=b
mul eax,ebx // (edx,eax)=eax*ebx
mov ebx,_fx32_one
div ebx // eax=(edx,eax)>>_fx32_fract
mov a,eax;
}
return a;
}
DWORD fx32_sqrt(const DWORD &x) // unsigned fixed point sqrt
{
DWORD m,a;
if (!x) return 0;
m=bits(x); // integer bits
if (m>_fx32_fract) m-=_fx32_fract; else m=0;
m>>=1; // sqrt integer result is half of x integer bits
m=_fx32_one<<m; // MSB of result mask
for (a=0;m;m>>=1) // test bits from MSB to 0
{
a|=m; // bit set
if (fx32_mul(a,a)>x) // if result is too big
a^=m; // bit clear
}
return a;
}
//---------------------------------------------------------------------------
我对它进行了如下测试:
//---------------------------------------------------------------------------
//--- 32bit signed fixed point format (2os complement)
//---------------------------------------------------------------------------
// |MSB LSB|
// |integer|.|fractional|
//---------------------------------------------------------------------------
const int _fx32_bits=32; // all bits count
const int _fx32_fract_bits=16; // fractional bits count
const int _fx32_integ_bits=_fx32_bits-_fx32_fract_bits; // integer bits count
//---------------------------------------------------------------------------
const int _fx32_one =1<<_fx32_fract_bits; // constant=1.0 (fixed point)
const float _fx32_onef =_fx32_one; // constant=1.0 (floating point)
const int _fx32_fract_mask=_fx32_one-1; // fractional bits mask
const int _fx32_integ_mask=0xFFFFFFFF-_fx32_fract_mask; // integer bits mask
const int _fx32_sMSB_mask =1<<(_fx32_bits-1); // max signed bit mask
const int _fx32_uMSB_mask =1<<(_fx32_bits-2); // max unsigned bit mask
//---------------------------------------------------------------------------
float fx32_get(int x) { return float(x)/_fx32_onef; }
int fx32_set(float x) { return int(float(x*_fx32_onef)); }
//---------------------------------------------------------------------------
int fx32_mul(const int &x,const int &y) // x*y
{
int a=x,b=y; // asm has access only to local variables
asm { // compute (a*b)>>_fx32_fract
mov eax,a
mov ebx,b
mul eax,ebx // (edx,eax)=a*b
mov ebx,_fx32_one
div ebx // eax=(a*b)>>_fx32_fract
mov a,eax;
}
return a;
}
//---------------------------------------------------------------------------
int fx32_div(const int &x,const int &y) // x/y
{
int a=x,b=y; // asm has access only to local variables
asm { // compute (a*b)>>_fx32_fract
mov eax,a
mov ebx,_fx32_one
mul eax,ebx // (edx,eax)=a<<_fx32_fract
mov ebx,b
div ebx // eax=(a<<_fx32_fract)/b
mov a,eax;
}
return a;
}
//---------------------------------------------------------------------------
int fx32_abs_sqrt(int x) // |x|^(0.5)
{
int m,a;
if (!x) return 0;
if (x<0) x=-x;
m=bits(x); // integer bits
for (a=x,m=0;a;a>>=1,m++); // count all bits
m-=_fx32_fract_bits; // compute result integer bits (half of x integer bits)
if (m<0) m=0; m>>=1;
m=_fx32_one<<m; // MSB of result mask
for (a=0;m;m>>=1) // test bits from MSB to 0
{
a|=m; // bit set
if (fx32_mul(a,a)>x) // if result is too big
a^=m; // bit clear
}
return a;
}
//---------------------------------------------------------------------------
int fx32_pow(int x,int y) // x^y
{
// handle special cases
if (!y) return _fx32_one; // x^0 = 1
if (!x) return 0; // 0^y = 0 if y!=0
if (y==-_fx32_one) return fx32_div(_fx32_one,x); // x^-1 = 1/x
if (y==+_fx32_one) return x; // x^+1 = x
int m,a,b,_y; int sx,sy;
// handle the signs
sx=0; if (x<0) { sx=1; x=-x; }
sy=0; if (y<0) { sy=1; y=-y; }
_y=y&_fx32_fract_mask; // _y fractional part of exponent
y=y&_fx32_integ_mask; // y integer part of exponent
a=_fx32_one; // ini result
// powering by squaring x^y
if (y)
{
for (m=_fx32_uMSB_mask;(m>_fx32_one)&&(m>y);m>>=1); // find mask of highest bit of exponent
for (;m>=_fx32_one;m>>=1)
{
a=fx32_mul(a,a);
if (int(y&m)) a=fx32_mul(a,x);
}
}
// powering by rooting x^_y
if (_y)
{
for (b=x,m=_fx32_one>>1;m;m>>=1) // use only fractional part
{
b=fx32_abs_sqrt(b);
if (int(_y&m)) a=fx32_mul(a,b);
}
}
// handle signs
if (sy) { if (a) a=fx32_div(_fx32_one,a); else a=0; /*Error*/ } // underflow
if (sx) { if (_y) a=0; /*Error*/ else if(int(y&_fx32_one)) a=-a; } // negative number ^ non integer exponent, here could add test if 1/_y is integer instead
return a;
}
//---------------------------------------------------------------------------
float a,b,c0,c1,d;
int x,y;
for (a=0.0,x=fx32_set(a);a<=10.0;a+=0.1,x=fx32_set(a))
for (b=-2.5,y=fx32_set(b);b<=2.5;b+=0.1,y=fx32_set(b))
{
if (!x) continue; // math pow has problems with this
if (!y) continue; // math pow has problems with this
c0=pow(a,b);
c1=fx32_get(fx32_pow(x,y));
d=0.0;
if (fabs(c1)<1e-3) d=c1-c0; else d=(c0/c1)-1.0;
if (fabs(d)>0.1)
d=d; // here add breakpoint to check inconsistencies with math pow
}
浮点a、b、c0、c1、d;
int x,y;
对于(a=0.0,x=fx32_set(a);a您认为这是错误的吗?第二个代码实际上处理负指数问题,因为x^n=(x^-1)^(-n)
-@amit:终止条件是什么?我对这个解决方案进行了编码,它在负指数的无限循环中运行。第一个代码确实没有考虑负指数的行为。但是第二个代码是正确的,尽管你说它“似乎错了”我的问题是:为什么第二个代码对你来说似乎是错误的?我之所以提到它是因为x^n=(x^-1)^(-n)
@amit:你能澄清一下哪个是“第一个代码”,哪个是“第二个代码”吗你指的是这里?@FalconUA:是的,愚蠢的错误,我想做exp&1来检查奇数。s:请解释更多分数?@newbie\u old你知道二进制搜索是如何工作的吗(例如sqrt?)@新手添加了sqrt二进制搜索的旧代码,并提示要更改的内容…如果不够,请评论我,我将编写一些代码,但不是现在,我需要工作几小时Spektre:是,从0开始到x/2,并不断检查x^2是否等于二进制搜索中的数字method@newbie_old因此,您不必检查x^bb
或x^(1/b)
其中bb=1/b
因此使用整数指数。但从MSB位开始,逐位移位,因此只检查位(x)/bb次(二进制搜索,而不是线性搜索)
float a,b,c0,c1,d;
int x,y;
for (a=0.0,x=fx32_set(a);a<=10.0;a+=0.1,x=fx32_set(a))
for (b=-2.5,y=fx32_set(b);b<=2.5;b+=0.1,y=fx32_set(b))
{
if (!x) continue; // math pow has problems with this
if (!y) continue; // math pow has problems with this
c0=pow(a,b);
c1=fx32_get(fx32_pow(x,y));
d=0.0;
if (fabs(c1)<1e-3) d=c1-c0; else d=(c0/c1)-1.0;
if (fabs(d)>0.1)
d=d; // here add breakpoint to check inconsistencies with math pow
}