c,algorithm,math,recursion , Power by squaring for negative exponents


Power by squaring for negative exponents

Question:

Tag: c,algorithm,math,recursion

I am not sure if power by squaring takes care of negative exponent. I implemented the following code which works for only 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 doesn't help as the following code seems 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 fractional exponent need to think though.


Answer:

Integer examples are for 32 bit int arithmetics, DWORD is 32bit unsigned int

  1. floating pow(x,y)=x^y

  2. integer pow(a,b)=a^b where a>=0 , b>=0

  3. integer pow(a,b)=a^b where b>=0

  4. integer pow(a,b)=a^b

  5. integer pow(a,b)=a^b where b is fractional

[edit1] fixed point sqrt example

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

[edit2] 32bit signed fixed point pow C++ example

When you put all the previous steps together you should have something like this:

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

I have tested it like this:

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
    }

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