square and cubic functions in C++

**skibber** · 10-17-2002

It's maybe stupid question, but I can't find any function in C++ for square and higher orders, like x^2, x^3 ....

**Sebastiani** · 10-17-2002

/*whew*/ ... man, I've spent hours perfecting this code for you!

Code:


template <typename number>
number squared( number n ) {
  return (n*n);
 }

template <typename number>
number cubed( number n ) {
  return (n*n*n);
 }

**Sebastiani** · 10-17-2002

For higher orders, use:

Code:


template <typename number>
number exponential( number n, int order ) {
   if( order == 0 ) {
    return 0;
   }
 number value = n;
   while(--order) {
    value *= n;
   }
 return value;
 }

**skibber** · 10-17-2002

Thanx! But that the easiest way??? It seems to me very complicated for simple function like that...

**skibber** · 10-17-2002

That was what I was looking for! It works with float's too ;o)

**Nick** · 10-17-2002

This should be pretty fast for positive integer exponents.
You can make it even faster by taking out the recursion.
It takes advantage of the fact that (a ^ (n/2))^2 = a^n.

Code:

int exp(int a, int n)
{
    int r;

    if (n == 0)
    {
        r = 1;
    }
    else
    {
        if (n % 2 == 0)
        {
            r = exp(a, n/2);
            r *= r;
        }
        else
        {
            r = exp(a, n-1) * a;
        }
    }

    return r;
}

**Sebastiani** · 10-17-2002

Hmm, nice code, Nick. And I wasn't aware of that fact either. I spotted a place where you could speed it up even more, though, by shifting n once to the right instead of dividing by two.

Code:

int exp(int a, int n)
{
    int r;

    if (n == 0)
    {
        r = 1;
    }
    else
    {
        if (n % 2 == 0)
        {
            r = exp(a, (n >> 1) );
            r *= r;
        }
        else
        {
            r = exp(a, n-1) * a;
        }
    }

    return r;
}

And I think making this solution non-recursive would slow down the algorithm due to the additional manipulations, not to mention making it far more unreadable (of course, if you were using it for graphics, readability would be a non-issue...).

**moi** · 10-17-2002

use pow() -- it will be faster and more versatile than anything you can come up with

**Nick** · 10-17-2002

You could also
take out n % 2 == 0 with !(n & 0x00000001). I think
most compilers would see these optimizations if I had used
unsigned int. Another way to speed it up would be
to store maybe the first 200 or so in a table.

**Sebastiani** · 10-17-2002

You could also take out n % 2 == 0 with !(n & 0x00000001).

Ha! We're on a roll now! Ok Nick, here's the challenge:

Let's time the following:

number exponential( number n, int order );

int exp(int a, int n); //..v1.0, v2.0, v.3.0, and the table lookup version.

Sound like fun?

**Sebastiani** · 10-17-2002

The template won!

I didn't use the table lookup, but you can easily add it:

printout:

Looping Template: 8 seconds to run 10, 000, 000 iterations...
Recursive #1: 12 seconds to run 10, 000, 000 iterations...
Recursive #2: 9 seconds to run 10, 000, 000 iterations...
Recursive #3: 10 seconds to run 10, 000, 000 iterations...

**Nick** · 10-17-2002

I think for mine to be faster than yours I think n would
have to be pretty large. In which case
the results would overflow an integer.

You could try this version that uses long double.

Code:

#include <iostream>
using namespace std;

long double exp(long double a, unsigned int n);

int main(void)
{
    long double a = 1.5;
    int n = 300;
    cout << a << "^" <<  n << " = " <<  exp(a, n) << endl;
    return 0;
}

long double exp(long double a, unsigned int n)
{
    unsigned int mask = 0xffffffff;
    unsigned int t = n;
    long double r = a;

    if (n == 0)
        return 1.0;

    // this finds mask such
    // that if n = 0x00001010
    // mask     = 0x00000100
    // there's probably a faster way to do
    // it but it shouldn't matter much if n is large
    while(t != 0)
    {
        mask <<= 1;
        t &= mask;
    }

    mask = ~mask;
    mask ^= (mask >> 1);
    mask >>= 1;

    while(mask != 0)
    {
        r *= r;
        if (n & mask)
        {
            r *= a;
        }
        mask >>= 1;
    }
    return r;
}

**Sang-drax** · 10-17-2002

Originally posted by Sebastiani

Looping Template: 8 seconds to run 10, 000, 000 iterations...
Recursive #1: 12 seconds to run 10, 000, 000 iterations...
Recursive #2: 9 seconds to run 10, 000, 000 iterations...
Recursive #3: 10 seconds to run 10, 000, 000 iterations...

These results seem very strange...
At first glance, I thought the more advanced function would win.
I decided to test it for myself

I got the result:

Looping Template: 0 seconds to run 10,000,000 iterations...
Recursive #1: 0 seconds to run 10,000,000 iterations...
Recursive #2: 0 seconds to run 10,000,000 iterations...
Recursive #3: 0 seconds to run 10,000,000 iterations...

So I had to increase the iterations:

Looping Template: 6 seconds to run 100,000,000 iterations...
Recursive #1: 8 seconds to run 100,000,000 iterations...
Recursive #2: 8 seconds to run 100,000,000 iterations...
Recursive #3: 7 seconds to run 100,000,000 iterations...

Moral: Keep the code simple!

**Sebastiani** · 10-17-2002

So I had to increase the iterations:

My computer may be old, but it's reliable, damn it!

What are you running, anyway - 2 GHZ??

But yeah, I was surprised by the results too, thinking the last recursive would be faster. I think it's wierd that the ratios were so different on our machines, too. Strange...

**Nick** · 10-17-2002

I change the test so that it
works for larger n. This is what
I get
1.5^300 = 6.72013e+052
Looping Template: 46.06 seconds to run 10000000 iterations...
1.5^300 = 6.72013e+052
Recursive #1: 8.111 seconds to run 10000000 iterations...
1.5^300 = 6.72013e+052
Recursive #2: 8.033 seconds to run 10000000 iterations...
1.5^300 = 6.72013e+052
Recursive #3: 11.265 seconds to run 10000000 iterations...
1.5^300 = 6.72013e+052
Iterative #4: 4.414 seconds to run 10000000 iterations...
And the winner is Iterative #4 coming in at 4.414 seconds.

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