# Blog Archives

## Generating Random Numbers

To be precise, random number generation functions, including rand, return pseudo-random numbers as opposed to truly random numbers, so whenever I say random, I actually mean pseudo-random. Before using the rand function you need to seed (i.e., initialize) the random number generator with a call to srand. This assures that subsequent calls to rand won’t produce the same sequence of numbers each time the program is run. The simplest way to seed the random number generator is to pass the result from a call to clock from the header as an unsigned int. Reseeding a random number generator causes number generation to be less random.

The rand function is limited in many ways. To begin with, it only generates integers, and only does so using a uniform distribution. Furthermore, the specific random number generation algorithm used is implementation specific and, thus, random number sequences are not reproducible from system to system given the same seed. This is a problem for certain kinds of applications, as well as when testing and debugging.

If you want to generate some random floating-point numbers in the interval of [0.0, 1.0) with a uniform distribution. The C++ standard provides the C runtime function rand in the header that returns a random number in the range of 0 to RAND_MAX inclusive. The RAND_MAX macro represents the highest value returnable by the rand function. A demonstration of

using rand to generate random floating-point numbers below.

#include <cstdlib> #include <ctime> #include <iostream> using namespace std; double doubleRand( ) { return double(rand( )) / (double(RAND_MAX) + 1.0); } int main( ) { srand(static_cast(clock( ))); cout << "expect 5 numbers within the interval [0.0, 1.0)" << endl; for (int i=0; i < 5; i++) { cout << doubleRand( ) << "\n"; } cout << endl; }

The program above should produce output similar to:

expect 5 numbers within the interval [0.0, 1.0)

0.010437

0.740997

0.34906

0.369293

0.544373

A much more sophisticated alternative to rand is the Boost Random library by Jens Maurer. The Boost Random library provides several high-quality random number generation functions for both integer and floating-point types, and support for numerous kinds

of distributions. Below code demonstrates how you can produce random floating- point numbers in the interval [0,1).

#include <boost/random.hpp> #include <iostream> #include <cstdlib> using namespace std; using namespace boost; typedef boost::mt19937 BaseGenerator; typedef boost::uniform_real Distribution; typedef boost::variate_generator<basegenerator, distribution=""> Generator; double boostDoubleRand( ) { static BaseGenerator base; static Distribution dist; static Generator rng(base, dist); return rng( ); } int main( ) { cout << "expect 5 numbers within the interval [0,1)" << endl; for (int i=0; i < 5; i++) { cout << boostDoubleRand( ) << "\n"; } cout << endl; }

The main advantage of the Boost Random library, is that the pseudo-random number generation algorithm has guaranteed and reproducible randomness properties based on the precise algorithm chosen. In above code I use the Mersenne Twister generator (mt19937) because it offers a good blend of performance and randomness.

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## Implementing Fixed-Point Numbers in C++

A fixed-point number, like a floating-point number, is an approximate representation of a real number. A floating-point number is stored as a mantissa (m), and an exponent (e), to form the equation m * be, where b is some constant. A fixed-point number is almost the same but the exponent is also a constant. This constant is passed to the basic_fixed_real in below class template as a template parameter.

By representing e as a constant, it allows fixed-point numbers to be represented internally as integers and for the arithmetic operations on them to be performed using integer arithmetic. This can often improve the speed of basic arithmetic operations

especially addition and subtraction. Fixed-point representations are less flexible than floating-point numbers, as they can only represent a narrow range of values. The fixed_real type in below function has a range that can only represent values from –2,097,151 to +2,097,151 with a precision of 1/1,024.

Implementing addition and subtraction of fixed-point numbers is straightforward enough: I simply add or subtract the underlying representation. To perform division and multiplication, I need an extra step of shifting the mantissa left or right to adjust for the binary point.

Below class template provides the implementation of a fixed-point real number, where the number of places to the right of the binary point is a template parameter. For instance basic_fixed_real<10> has 10 binary digits to the right of the binary point, allowing it to represent numbers up to a precision of 1/1,024.

#include <iostream> using namespace std; template<int E> struct BasicFixedReal { typedef BasicFixedReal self; static const int factor = 1 << (E - 1); BasicFixedReal( ) : m(0) { } BasicFixedReal(double d) : m(static_cast(d * factor)) { } self& operator+=(const self& x) { m += x.m; return *this; } self& operator-=(const self& x) { m -= x.m; return *this; } self& operator*=(const self& x) { m *= x.m; m >>= E; return *this; } self& operator/=(const self& x) { m /= x.m; m *= factor; return *this; } self& operator*=(int x) { m *= x; return *this; } self& operator/=(int x) { m /= x; return *this; } self operator-( ) { return self(-m); } double toDouble( ) const { return double(m) / factor; } // friend functions friend self operator+(self x, const self& y) { return x += y; } friend self operator-(self x, const self& y) { return x -= y; } friend self operator*(self x, const self& y) { return x *= y; } friend self operator/(self x, const self& y) { return x /= y; } // comparison operators friend bool operator==(const self& x, const self& y) { return x.m == y.m; } friend bool operator!=(const self& x, const self& y) { return x.m != y.m; } friend bool operator>(const self& x, const self& y) { return x.m > y.m; } friend bool operator<(const self& x, const self& y) { return x.m < y.m; } friend bool operator>=(const self& x, const self& y) { return x.m >= y.m; } friend bool operator<=(const self& x, const self& y) { return x.m <= y.m; } private: int m; }; typedef BasicFixedReal<10> FixedReal; int main( ) { FixedReal x(0); for (int i=0; i < 100; ++i) { x += FixedReal(0.0625); } cout << x.toDouble( ) << endl; }

The program outputs:

6.25

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