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Power squared for negative metrics

I'm not sure that squaring takes care of a negative exponent. I have executed the following code, which only works for positive numbers.

    #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);       
    }

Looking at https://en.wikipedia.org/wiki/Exponentiation_by_squaring does not help as the following code seems to be wrong.

    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).

Edit: Thanks to amit, this solution works for both negative and positive numbers:

    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);
    }

For a fractional indicator, we can do below (Spectrum method):

  • Suppose you have x ^ 0.5, then you can easily calculate the square root with this method: start from 0 to x / 2 and continue to check x ^ 2, equal to the result or not to the binary search method .

  • , , x ^ (1/3), if mid*mid <= n if mid*mid*mid <= n, x. x ^ (1/4), x ^ (1/5) .. ^ (2/5) ( ^ (1/5)) ^ 2 5- .

  • , , 1/x. , ? , , .

  • , pow (a, b). , 0,6, (24, 8) (0,6 * 1 < 8) = 153 (10011001). , 153 , (10011001) (2 ^ -1, 0, 0, 2 ^ -3, 2 ^ -4, 0, 0, 2 ^ 7). , pow (a, 0,6), 2,3, 4 7 x . , 1 < 8.

.

:

x ^ y = exp2 (y * log2 (x))

+4
1

32- int, DWORD - 32bit unsigned int

  • pow(x,y)=x^y

    :

    : pow(x,y) = exp2(y*log2(x)). :

  • integer pow(a,b)=a^b a>=0 , b>=0

    ( ), :

        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;
        }

  • integer pow(a,b)=a^b b>=0

    if a

        int powiu(int a,DWORD b)
     {
     int sig=0,c;
     if ((a<0)&&(DWORD(b&1)) { sig=1; a=-a; } // negative output only if a<0 and b is odd
     c=powuu(a,b); if (sig) c=-c;
     return c;
     }

  • pow(a,b)=a^b

    , b<0, 1/powiu(a,-b). , , , ( PI ). :

        float powfii(int a,int b)
     {
     if (b<0) return 1.0/float(powiu(a,-b));
     else return powiu(a,b);
     }

  • pow(a,b)=a^b, b

    - a^(1/bb), bb - . , :

    • a^(1/2) square root(a)
    • a^(1/bb) bb_root(a)

    c MSB LSB , pow(c,bb)<=a, bit, . sqrt:

        int bits(DWORD p) // count how many bits is p
        {
        DWORD m=0x80000000; int b=32;
        for (;m;m>>=1,b--)
         if (p>=m) break;
        return b;
        }

    DWORD sqrt(const DWORD &x)
        {
        DWORD m,a;
        m=(bits(x)>>1);
        if (m) m=1<<m; else m=1;
        for (a=0;m;m>>=1) { a|=m; if (a*a>x) a^=m; }
        return a;
        }

    if (a*a>x) if (pow(a,bb)>x), bb=1/b... b - , , bb - . m - , m=(bits(x)>>1); m=(bits(x)/bb);

[edit1] sqrt

//---------------------------------------------------------------------------
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;
    }
//---------------------------------------------------------------------------

. 16 , 16 .

  • fp → fx conversion: DWORD(float(x)*float(_fx32_one))
  • fp < - fx: float(DWORD(x))/float(_fx32_one))
  • fx32_mul(x,y) x*y 80386 + 32 ( karatsuba - , )

  • fx32_sqrt(x) sqrt(x)

    : (a<<16)*(b<<16)=(a*b<<32) >>16, (a*b<<16). 32, 64 .

[edit2] 32- pow ++ example

, - :

//---------------------------------------------------------------------------
//--- 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
    }
  • a,b
  • x,y a,b
  • c0 - .
  • c1 - fx32_pow

  • d

    - , , . , , ...

P.S. :

+3

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