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`

**floating**`pow(x,y)=x^y`

- is usually evaluated like this: How Math.Pow (and so on) actualy works
- so the fractional exponent can be evaluated:
`pow(x,y) = exp2(y*log2(x))`

- this can be done also on fixed point fixed point bignum pow

**integer**`pow(a,b)=a^b`

where`a>=0 , b>=0`

this is easy (you already have that) done by squaring

`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`

where`b>=0`

just add few ifs to handle the negative 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; }`

**integer**`pow(a,b)=a^b`

- so if
`b<0`

then it means`1/powiu(a,-b)`

- as you can see the result is not integer at all
- so either ignore this case
- or return floating value
- or add a multiplier variable (so you can evaluate PI equations on pure Integer arithmetics)
this is float result:

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

- so if
**integer**`pow(a,b)=a^b`

where`b`

is fractional- you can do something like this
`a^(1/bb)`

where`bb`

is integer - in reality this is rooting
- so you can use binary search to evaluate
`a^(1/2)`

is`square root(a)`

`a^(1/bb)`

is`bb_root(a)`

- so do a binary search for c from MSB to LSB
- and evaluate if
`pow(c,bb)<=a`

then leave the bit as is else clear it this is sqrt example:

`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; }`

so now just change the

`if (a*a>x)`

with`if (pow(a,bb)>x)`

- where
`bb=1/b`

... so b is fractional exponent you looking for and bb is integer - also m is the number of bits of the result so change
`m=(bits(x)>>1);`

to`m=(bits(x)/bb);`

- you can do something like this

**[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;
}
//---------------------------------------------------------------------------
```

- so this is unsigned fixed point
- high 16 bits are integer
- low 16 bits are fractional part
- this is fp -> fx conversion:
`DWORD(float(x)*float(_fx32_one))`

- this is fp <- fx conversion:
`float(DWORD(x))/float(_fx32_one))`

`fx32_mul(x,y)`

is`x*y`

it uses assembler of 80386+ 32bit architecture (you can rewrite it to karatsuba or whatever else to be platform independent)`fx32_sqrt(x)`

is`sqrt(x)`

- in fixed point you should be avare of the fractional bit shift
- for multiplication:
`(a<<16)*(b<<16)=(a*b<<32)`

- you need to shift back by
`>>16`

to get result`(a*b<<16)`

- also the result can overflow 32 bit therefore I use 64 bit result in assembly

**[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
}
```

`a,b`

are floating point`x,y`

are closest fixed point representations of`a,b`

`c0`

is math pow result`c1`

is fx32_pow result`d`

is difference- hope did not forget something trivial but it seems like it works properly
- do not forget that fixed point has very limited precision
- so the results will differ a bit ...

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Below is a C program i have written to print different combination of characters in a string. This is not an efficient way as this algorithm creats a lot of extra strings. However my question is NOT about how to solve this problem more efficiently. The program works(inefficiently though) and...

algorithm

You are managing a software project that involves building a computer-assisted instrument for medical surgery. The exact placement of the surgical knife is dependent on a number of different parameters, usually at least 25, sometimes more. Your programmer has developed two algorithms for positioning the cutting tool, and is seeking...