Fixed Math
Volume Number:   9

Issue Number:   10

Column Tag:   C++ Workshop

Related Info: BinaryDec. Pack.
Fast Fixedpoint Math Made Easy
Using C++ to abstract data in an intuitive way
By Brion Sarachan, Albany, New York
Note: Source code files accompanying article are located on MacTech CDROM or source code disks.
With the recent release of Symantec C++, a highquality, lowcost C++ development environment is now available to Macintosh programmers. If used properly, C++ can provide a quantum leap over the representation power of languages like C and Pascal. We often hear about the use of C++ to build objectoriented systems. Even more fundamentally, however, C++ is an excellent language for data abstraction. Using C++, new data types can be defined that can then be used as conveniently as the builtin types like integers, characters, and floatingpoint numbers.
As an example of C++’s ability to support data abstractions, this article will describe a simple C++ class for fixedpoint numbers. Arithmetic operations on fixedpoint numbers are much faster than the corresponding operations on floatingpoint numbers, at the expense of some loss in precision and a smaller range of allowable values. Many computer graphics and animation applications that require highperformance use fixedpoint numbers.
Fixedpoint numbers are not built into C++. The Mac Toolbox provides its own implementation of fixedpoint numbers. However, using the Mac’s fixedpoint numbers can be rather awkward for a couple of reasons:
• Arithmetic operators like * and / cannot be used. Instead, calls must be made to Toolbox routines such as FixMul() and FixDiv(). Therefore, code that uses fixedpoint numbers must be written differently than other arithmetic expressions.
• ANSI library functions such as sqrt() or printf() cannot use fixedpoint numbers directly. A fixedpoint number must be converted to a double by using functions like Fix2X() before using the library functions. This is even more complicated because the manner in which fixedpoint numbers are converted to doubles may differ depending on the type of floatingpoint representation being used by the processor or, if there is one, by the floatingpoint coprocessor.
C++ enables us to solve these problems by allowing us to define a new fixedpoint number class. This new class uses the Mac Toolbox internally, but allows the programmer to declare fixedpoint numbers and use them in expressions just as easily as builtin types like ints or doubles.
The Macintosh Fixed Data Type and its Operations
The Macintosh Fixed data type is described in Inside Macintosh. A Fixed type is actually a long integer:
typedef long Fixed;
A Fixed number therefore consists of 32 bits: a sign bit, a 15bit integer part, and a 16bit fractional part. The value of a Fixed number therefore falls in the range of about + and 32,768.
Fixed numbers can be added and subtracted using the usual + and  operators. However, the * and / operators will not work correctly for multiplication and division. Instead, the Toolbox provides the routines FixMul() and FixDiv(). The Toolbox provides several other routines for conversion to and from other data types and for some math functions. The following Toolbox routines are used in the implementation of the C++ fixedpoint number class:
• FixMul()  multiplies two fixedpoint numbers
• FixDiv()  divides two fixedpoint numbers
• FracSin()  returns the sine of a fixedpoint number as a Fract
• FracCos()  returns the cosine of a fixedpoint number as a Fract
• Frac2Fix()  converts a Fract to a fixedpoint number
• Long2Fix()  converts a long integer to a fixedpoint number
• X2Fix()  converts an extendedformat floatingpoint number to a fixedpoint number
• Fix2X()  converts a fixedpoint number to an extendedformat floatingpoint number
Some of these routines use something called a Fract. A Fract is an alternative fixedpoint type provided by the Toolbox. Like a Fixed number, a Fract also consists of 32 bits. However, for the case of a Fract, there is only one integer bit and a 30bit fractional part.
Normally, one would use fixedpoint numbers by declaring Fixed variables and calling the routines listed above directly. Using our new C++ class, we will be able to use floatingpoint numbers much more easily, while still getting the benefit of fast calculations.
C++ Features that Support Data Abstraction
C++ has several features that make it a good language for data abstraction. These include:
• function overloading
• operator overloading
• userdefined type conversions
These features will be briefly described here, and they are discussed in depth in many reference books on C++. (See the References section at the end of this article.)
Function overloading allows multiple functions having the same name to be defined. As long as each instance of the function has a different argument type list, the compiler is able to determine which instance is being called. Function overloading allows us, for example, to define a sin() function, separate from the sin() function found in the C library, that is specialized for our new fixedpoint number class.
Operator overloading allows us to define the meaning of different types of operators as they are applied to userdefined classes. This will allow us, for example, to define the * and / operators for multiplication and division on our new fixedpoint number class.
Userdefined type conversions are a subtle but very powerful feature of C++. In C (or C++), we can write:
double x = 3;
The compiler automatically converts the integer value of 3 to the doubleprecision floatingpoint value of 3.0, which is represented internally in a completely different format. Automatic type conversions save us a lot of work, and eliminate a lot of errors. For example, a library function to calculate a square root may be defined for doubles:
double sqrt(double x);
However, we are generally free to call the sqrt() function with any type of number, such as an integer. The compiler automatically performs any necessary type conversions:
float y = sqrt(3);
The example above performs two automatic conversions: The int 3 is converted to a double before calling sqrt(), and then the result of sqrt() is converted to a float. The effect is the same as if we had written:
float y = (float) sqrt( (double) 3 );
But certainly the simpler form of this expression is easier to write, easier to read, and less subject to errors.
C++ allows us to define the valid set of type conversions that the compiler may apply to the new classes that we write. So, in other words, we can tell the compiler how to convert instances of our new fixedpoint number class to doubles. We can then apply the sqrt() function to a fixedpoint number as easily as any other type of number, and leave the details of the type conversion to the compiler!
Implementation of the C++ FixedPoint Number Class
Let’s take a look at how our new fixedpoint number class is implemented. In order to understand these examples, the reader should know something about C programming. If the reader also has some knowledge of C++, all the better!
Our new C++ class is called simply fix, and recall that the design goal is for fix to be as easily usable as builtin types like float or double. The declaration of fix is structured like this:
/* 1 */
class fix {
private:
// declarations of friend functions
Fixed n;
static fix result;
// declarations of private member functions
public:
// declarations of public member functions
};
The fix class has one data member, n. This member is a number whose data type is the Macintosh Toolbox type Fixed, which was described above. The fix class also has a static data member, result, which is shared among all members of the fix class. It was found during development that using this static data member to hold the results of calculations gave better performance than the alternative of declaring local variables within member functions.
The power of the new class comes from its friend and member functions. Examples of these will be described in detail below.
Arithmetic operators
Friend functions are functions that, although external to the class, may access the class’s private data members, which in this case is the Fixed data member, n. One friend function overloads the + operator for addition. This friend function is declared as:
/* 2 */
friend inline fix operator+(const fix, const fix);
and implemented as:
/* 3 */
inline fix operator+(const fix a, const fix b)
{
fix::result.n = a.n + b.n;
return fix::result;
}
The inline specifier means that the compiler will substitute the code for the function in place, rather than actually generating a stack frame when the function is called. This speeds execution at the expense of program size.
This friend function allows the + operator to be used for the addition of two numbers of type fix. (The minus operator for subtraction is defined similarly.)
From what we have seen so far, there is not yet much benefit to the fix class; the Toolbox Fixed data type also supports the + and  operators. However, let’s take a look at another friend function:
/* 4 */
inline fix operator*(const fix a, const fix b)
{
fix::result.n = FixMul(a.n,b.n);
fix::result.rangecheck();
return fix::result;
}
Now we begin to see some benefit of C++ data abstraction! This function allows the familiar * operator to be used for multiplication rather than the Toolbox FixMul() function. This seems much more natural. (Division is similarly defined.) Also, friend functions of this form play an important role in userdefined data conversions. Either of the function arguments is subject to implicit, userdefined type conversions. This allows, for example, a fix number to multiply a double.
This function also demonstrates another benefit of C++ classes: that of encapsulation. The function provides a wrapper over fixedpoint number multiplication, so that whenever fix numbers are multiplied, the rangecheck() function will be called automatically to ensure that the result is valid.
sin() and cos() functions
Friend functions are also defined for sin() and cos() of fix numbers. For example:
/* 5 */
fix sin(const fix a)
{
fix result;
Fract temp = FracSin(a.n);
result.n = Frac2Fix(temp);
return result;
}
The calls to the Toolbox functions FracSin() and Frac2Fix() are hidden within the implementation of the friendlier sin() function.
Why is this function defined as a friend function rather than a member function? If it were a member function, it would be called as:
my_fix.sin();
However, as a friend function, it is instead called as:
sin(my_fix);
This matches the normal usage of sin() in the standard ANSI library.
I/O functions
C++ provides the iostream library which may be used in place of the standard C stdio library. Many beginning C++ programmers wonder why an alternative I/O library is provided, and are reluctant to use it since they are already expert in using the stdio library. The benefit of the C++ iostream library is illustrated in the following example.
Suppose we want to print a fix number using stdio. Before doing so, we need to convert the fix to a printable type. (We will assume for now that we may cast a fix to a double. Soon we will take a look at how such userdefined conversions are specified.)
printf("here is the number: %f \n", (double)my_fix);
Converting the fix to a double before printing is at odds with the notion that userdefined types (such as fix) are supposed to behave analogously to the builtin types (like double). The problem is that stdio is not extensible! We cannot add new data types to the stdio mechanism cleanly. New types must be converted to a builtin type before printing.
In contrast, here is how we print the same thing using the iostream library:
cout << "here is the number: " << my_fix << endl;
Note that there is no explicit type conversion specified. We would write essentially the same line of code whether we wanted to print a fix, int, or double variable. As we see, the iostream library can be cleanly extended for new data types.
The following friend function implements the above capability by overloading the << operator:
/* 6 */
ostream& operator<<(ostream& os, const fix& f)
{
os << (double) f;
return os;
}
The explicit conversion from fix to double is now specified once and for all within the implementation of this method.
The assignment operator and copy constructor
The compiler automatically generates a default assignment operator and copy constructor for each userdefined class. Therefore, the following operations are provided automatically by the compiler:
/* 7 */
fix my_fix;
fix your_fix = my_fix;
fix her_fix(my_fix);
Here, your_fix is assigned to have the same value as my_fix, and her_fix is initialized as being a copy of (having the same value as) my_fix. The default assignment operator and copy constructor perform a memberwise copy of all the data members of the class. For the fix class, the value of the fixedpoint number is copied.
Other member functions
A few other member functions are provided, including overloaded operators for increment (++) and decrement (), as well as the +=, =, *=, and /= operators. The implementation of these is fairly straightforward.
The last, and most complex, topic in the implementation of the fix class, is that of userdefined type conversions.
User Defined Type Conversions
In the last section, we had assumed that the compiler knew how to convert fix numbers to and from builtin types such as doubles:
double my_double = (fix) my_fix; // explicit conversion
double another_double = my_fix; // implicit conversion
These type conversions are specified using C++ member functions. In this section we will examine the details of how this is done.
Whenever an instance of a userdefined class is created, a special function called a constructor is called. A class may have multiple constructors having different argument lists. For example, the fix class has several constructors, one of which is:
fix(const long l) { n = Long2Fix(l); };
This constructor allows a fix to be created from a long integer. The implementation of this constructor calls the Toolbox function Long2Fix() in order to convert the integer to a fixedpoint value. The constructor uses the long data type rather than int because the compiler by default casts literal constants to the largest applicable type, so any integer literal has an implicit type of long. Also, the compiler knows how to convert any of the buildin integer types, including int and short, to long.
This constructor allows as to declare fix variables such as:
fix x(36); // initializes the fixedpoint number to 36
Less obvious, however, is that this constructor tells the compiler how to implicitly convert between integers and fixedpoint numbers. Having defined this constructor, it is valid to write:
int my_int = 38;
fix my_fix = my_int; // implicit conversion
Of course, now that we have the ability to convert integers to fixedpoint numbers, it would also be desirable to convert floatingpoint numbers to fixedpoint numbers. The implementation of this constructor is similar:
fix(const long double d) { n = X2Fix(d); };
Now that we can convert numbers of different types (integer and floatingpoint) to numbers of our new fixedpoint class, how do we define the inverse conversion? In this case, we overload the fix cast operator:
/* 8*/
fix::operator long double()
{
long double result = Fix2X(n);
return (result);
}
The implementation of this member function is quite analogous to the implementation of the constructor from floatingpoint numbers, except that the conversion is performed in the reverse direction.
The overloaded cast operator allows us to write the following for an explicit type conversion.
/* 9 */
double d;
fix f(3.14);
d = (double) f;
Even more importantly, however, the overloaded cast operator tells the compiler how to convert a fix number to a double implicitly.
At this point, we may wonder whether it is valid to write the following:
/* 10 */
fix my_fix, your_fix;
my_fix = some_value;
your_fix = another_value;
if (my_fix > your_fix) // Can we compare two fix numbers?
{
do something...
}
The answer is: yes! Because we have told the compiler how to implicitly convert fixedpoint numbers to floatingpoint numbers, the compiler actually performs the comparison as:
if ( (long double) my_fix > (long double) your_fix )
The compiler implicitly converts the fixedpoint numbers to floatingpoint numbers in order to use the comparison (>) operator.
Using the FixedPoint Number Class
We’ve already looked at several examples involving the fix class. Let’s study a couple of others in terms of the individual steps  some explicit and some implicit  performed by the compiler.
First, let’s consider:
/* 11 */
const double pi = 3.14159;
fix a(0.75);
cout << "sine of 135 degrees is " << sin(a*pi) << endl;
This calculation is performed by executing these steps:
• The overloaded * operator for fixedpoint numbers computes a*pi.
• The overloaded sin() function for fixedpoint numbers computes the sine of a*pi.
• The overloaded << operator writes the sine value to the output stream.
In contrast, let’s consider this example:
cout << "tangent of 135 degrees is " << tan(a*pi) << endl;
In this case, the calculation consists of these steps:
• The overloaded * operator for fixed numbers computes a*pi.
• The overloaded cast operator implicitly converts a*pi (a fix) to a long double.
• The long double result above is converted implicitly to a double.
• The standard ANSI library tan() function receives the double result from the last step.
• The << operator writes the tangent value to the output stream.
In the first case above, there is a sin() function specialized for fixedpoint numbers, so the entire calculation of sin(a*pi) is performed in the domain of fixedpoint numbers. In contrast, we have not defined a tan() function for fixedpoint numbers. Therefore, the calculation of tan(a*pi) requires an implicit type conversion so that the tangent is actually computed in the domain of floatingpoint numbers. To the programmer using the fix class, the difference is transparent; fixedpoint numbers can be used just like any other numbers. The compiler takes care of the details.
A Benchmark to Measure Performance
In this section we will look at a small sample program that compares the performance of fixedpoint and floatingpoint numbers. To get a fair test, we want to perform the same set of calculations using each type of number. To do this, we will use another C++ feature: templates.
A C++ matrix template
Templates allow functions or classes to be written once for multiple data types. For example, our benchmark program will use a simple template class for matrices. This template class definition defines a simple 4x4 matrix of any type of number:
/* 12 */
template <class Number>
class matrix {
private:
Number m[4][4];
friend ostream&
operator<<(ostream&, const matrix<Number>&);
public:
matrix();
inline Number& operator()(const int,const int);
matrix<Number> operator*(const matrix<Number>);
};
This is not an exhaustive definition for a matrix class by any means. If we were building a matrix class as part of a generalpurpose class library, there are many other matrix functions that we would probably want to provide. However, this definition will suffice for the purpose of benchmarking the fix class.
As we can see, the definition of the matrix class includes:
• a 4x4 matrix of numbers
• an overloaded << operator for writing the matrix to an output stream
• a constructor for initializing a matrix
• an overloaded () operator that is used for indexing the elements of the matrix
• an overloaded * operator for multiplying matrices
Let’s take a look at the implementation of the * (matrix multiplication) operator:
/* 13 */
template <class Number>
matrix<Number>
matrix<Number>::operator*(const matrix<Number> mat)
{
matrix<Number> result;
int i,j,k;
for(i=0;i<4;i++)
for(j=0;j<4;j++)
for(k=0;k<4;k++)
result[i][j] += m[i][k] * mat.m[k][j];
return result;
}
This template function implements the traditional algorithm for multiplying two 4x4 matrices. Note that the multiplication of pairs of numbers is performed using the * operator. Of course this is the usual way of multiplying builtin number types like ints and doubles. Because we also defined the fix class to use the * operator for multiplication, this function can multiply matrices of fix numbers, as well.
Matrices for different types of numbers can now be declared as:
/* 14 */
matrix<double> double_matrix;
matrix<fix> fix_matrix;
The benchmark() function
The benchmark() function is itself defined as a template, so that it can be applied to a matrix of any type of numbers. The benchmark() function:
• sets the input matrix to be an identity matrix,
• creates a three dimensional rotation matrix corresponding to a 2 degree rotation about the Z axis,
• premultiplies the input matrix 1000 times by the rotation matrix, and counts the number of seconds that this takes, and finally
• prints the resulting matrix (resulting from the 1000 multiplications).
Matrix operations like those listed above, and their use for threedimensional computer graphics, are described in many standard textbooks, such as those listed in the References section.
It’s interesting to look at the line of code within the benchmark() function that performs the matrix multiplication:
m = rotation * m;
Because we overloaded the * operator for the matrix class to perform matrix multiplication, this line of code can be written in an extremely simple fashion! At this level of detail we do not have to be concerned with the fact that multiplication of two 4x4 matrices actually involves 64 multiplications of numbers, 48 additions, and so forth. The matrix C++ class is a highlevel abstract data type that hides the details of the lowlevel matrix operations.
The main() routine calls the benchmark() function for both a matrix of fix numbers and a matrix of doubles. The results are summarized in the next section.
Benchmark Results
After the initial identity matrix:
10 0 0
01 0 0
00 1 0
00 0 1
was premultiplied 1000 times by the rotation matrix:
cos(Π/90) sin(Π/90) 0 0
sin(Π/90) cos(Π/90) 0 0
0 0 1 0
0 0 0 1
the result, using fix numbers, was the matrix:
0.93808 0.338715 0 0
0.338715 0.93808 0 0
0 0 1 0
0 0 0 1
And using doubles, the result was:
0.939703 0.341992 0 0
0.341992 0.939703 0 0
0 0 1 0
0 0 0 1
It can be seen that the difference in precision between fix numbers and doubles in this case is less than 1% for 1000 calculations. This difference is certainly tolerable for many types of graphics applications where the numbers correspond to screen positions or orientations of geometric objects. For these cases, an infinitesimal difference in some quantity will most likely not even be noticeable.
The benchmark was run on a Powerbook 145 running at 16 MHz. The 1000 matrix multiplications took about 4 seconds for fix numbers and about 13 seconds for doubles. Therefore, the fix numbers executed about 3 times faster!
We might wonder whether a significant portion of the time for the fix number calculations was due to overhead introduced by the fix number class, as compared to the time required by the fixedpoint math routines themselves. To find out, the profiler was used to take statistics during the 1000 matrix multiplications for fix numbers. The results were:
Function Min Max Avg % Entries
matrix<fix>::operator * 0 3 0 28 1000
operator * 0 1 0 33 64000
fix::operator += 0 1 0 16 64000
fix::rangecheck 0 1 0 8 64000
operator + 0 1 0 11 64000
matrix<fix>::__PT6matrix3fix 0 1 0 0 1000
fix::fix 0 1 0 0 1000
fix::fix 0 1 0 2 16000
As we see, 33% of the execution time was due to fixedpoint number multiplication (performed in the operator *). 28% was due to the matrix multiplication routine, consisting of the nested for loops and assignment statements. Only 11% of the execution time was due to the addition of fixedpoint numbers, although a surprising 16% of time was due to the += operator. We also note that 8% was due to the rangecheck() function. We could of course abstain from calling this function at the expense of some safety. It certainly seems as though the bulk of the execution time was due to the mathematical operations themselves, rather than overhead introduced by the class definition.
Conclusions
We have looked in detail at the design of a new C++ class that allows fixedpoint numbers to be used as easily as floatingpoint numbers, with about a 3 times improvement in performance. The loss in precision was found to be less than 1%, even after a large number of calculations. The values of the fix numbers span the range of about + and  32,768. The fix class should only be used for applications where values will remain within this range.
Modular design and consistent interfaces are extremely important for welldesigned C++ classes. The fix class, although simple, was designed using an extremely powerful idea. By providing the same standard interface as the builtin number types, fix numbers can be used interchangeably with floats or doubles.
We could, for example, develop an application using doubles, and then easily “plug in” fix numbers for increased performance, simply by changing the types of variables. Conversely, if we develop an application using fix numbers, and then find that we need to use doubles after all, we could just “plug in” doubles.
The implementation of the fix class could be easily extended in a modular way. For example, if we want, in the future, to provide a fast square root function optimized for fixedpoint numbers, we could easily “plug in” such a function. In the interim, fixedpoint square roots are easily calculated by the compiler by implicitly converting fix numbers to doubles, and then calling the standard C sqrt() function. A programmer can use the fix class effectively without being concerned with these details.
Looking to the Future
There are many other examples where the power of C++ could greatly simplify Macintosh programming. Large, commercial class libraries such as the THINK Class Library (TCL), MacApp, and, soon, Bedrock, provide us with generalpurpose application frameworks. However, we can still improve the design of our own specialized applications by developing appropriate, applicationspecific class definitions that work along with the larger application frameworks.
For example, suppose we wanted to create a class library to support a variety of threedimensional graphics and animation applications. Using the modest foundation developed in this article, we could:
• extend the simple, initial matrix class introduced here to include other matrix functions,
• derive specialized types of matrices for geometric transformations, perspective projections, and so forth,
• design additional classes for such things as points, lines, and planes, using the fast fix class for calculations, and even
• derive a new type of TCL pane class that uses transformations, projections, and so forth, to support threedimensional geometry.
The only limit is our imagination!
References
C++
The Symantec C++ User’s Guide lists many of the good references on C++. Two others will be mentioned here.
The following book was just recently published and contains a good introduction to C++, the Symantec programming environment, and the THINK Class Library:
Rhodes and McKeehan, Symantec C++ Programming for the Macintosh, Brady, 1993.
The following book has a detailed discussion of the C++ data abstraction capabilities discussed in this article.
K.E. Gorlen, et. al., Data Abstraction and ObjectOriented Programming in C++, Wiley, 1990.
[And don’t forget Dave Mark’s Learn C++ on the Macintosh published by AddisonWesley.  Ed.]
Macintosh Toolbox
The Fixed data type and its operations are documented in Volumes I and IV of Inside Macintosh.
Graphics and Animation
There are many standard textbooks that cover the graphics and animation topics mentioned in this article. Two of the best are:
Foley, et. al., Computer Graphics, Principles and Practice, 2nd ed., AddisonWesley, 1990.
Rogers, Mathematical Elements for Computer Graphics, 2nd ed., McGrawHill, 1990.
Listing: fix.h
/***************************************
* fix.h
* fixedpoint number class interface
***************************************/
#pragma once
#include <FixMath.h>
#include <iostream.h>
#include <assert.h>
const Fixed fixed_zero = Long2Fix((long) 0);
class fix {
private:
// arithmetic operators
friend inline fix operator+(const fix, const fix);
friend inline fix operator(const fix, const fix);
friend inline fix operator*(const fix, const fix);
friend inline fix operator/(const fix, const fix);
// I/O operators
friend ostream& operator<<(ostream&, const fix&);
friend istream& operator>>(istream&, fix&);
// transcendental functions that are optimized
// for fixedpoint numbers
friend fix sin(const fix);
friend fix cos(const fix);
// data value
Fixed n;
// static data member for computation results
static fix result;
// private range check member function
inline void rangecheck();
public:
// constructors
fix() { n = fixed_zero; };
fix(const long l) { n = Long2Fix(l); };
fix(const long double d) { n = X2Fix(d); };
// incremant and decrement operators
fix operator++() { n += Long2Fix(1); return *this; };
// prefix
fix operator() { n = Long2Fix(1); return *this; };
fix operator++(int) { n += Long2Fix(1); return *this; };
// postfix
fix operator(int) { n = Long2Fix(1); return *this; };
// cast operators
operator long double();
// other operators...
fix operator+=(const fix f)
{ *this = *this + f; return *this; };
fix operator=(const fix f)
{ *this = *this  f; return *this; };
fix operator*=(const fix f)
{ *this = *this * f; return *this; };
fix operator/=(const fix f)
{ *this = *this / f; return *this; };
};
inline void fix::rangecheck()
{
assert((n != 0x7FFFFFFF) && (n != 0x80000000));
}
inline fix operator+(const fix a, const fix b)
{
fix::result.n = a.n + b.n;
return fix::result;
}
inline fix operator(const fix a, const fix b)
{
fix::result.n = a.n  b.n;
return fix::result;
}
inline fix operator*(const fix a, const fix b)
{
fix::result.n = FixMul(a.n,b.n);
fix::result.rangecheck();
return fix::result;
}
inline fix operator/(const fix a, const fix b)
{
fix::result.n = FixDiv(a.n,b.n);
fix::result.rangecheck();
return fix::result;
}
Listing: fix.cp
/***************************************
* fix.cp
* fixedpoint number class implementation
***************************************/
#include "fix.h"
#include <math.h>
#include <SANE.h>
fix fix::result; // initialize static data member
fix::operator long double()
{
long double result = Fix2X(n);
return (result);
}
ostream& operator<<(ostream& os, const fix& f)
{
os << (double) f;
return os;
}
istream& operator>>(istream& is, fix& f)
{
double temp;
is >> temp;
// Since this routine reads a floatingpoint number,
// and then converts it, errors are handled in the same
// way as for reading floatingpoint numbers. The
// iostream library includes functions for testing
// and reseting the error state.
if (is.good())
f = (fix) temp;
return is;
}
fix sin(const fix a)
{
fix result;
Fract temp = FracSin(a.n);
result.n = Frac2Fix(temp);
return result;
}
fix cos(const fix a)
{
fix result;
Fract temp = FracCos(a.n);
result.n = Frac2Fix(temp);
return result;
}
Listing: matrix.h
/***************************************
* matrix.h
* matrix template class interface
***************************************/
#pragma once
#include <iostream.h>
#include <assert.h>
template <class Number>
class matrix {
private:
Number m[4][4];
friend ostream& operator<<(ostream&, const matrix<Number>&);
public:
matrix();
inline Number& operator()(const int,const int);
// for matrix indexing
matrix<Number> operator*(const matrix<Number>);
};
template <class Number>
inline Number& matrix<Number>::operator()(const int row,
const int col)
{
assert(row >=0 && row <= 3 && col >=0 && col <= 3);
return m[row][col];
}
Listing: matrix.cp
/***************************************
* matrix.cp
* matrix template class implementations
***************************************/
#include "matrix.h"
#include <iomanip.h>
template <class Number>
matrix<Number>::matrix()
{
int i,j;
Number zero(0); // initialize any type of Number to zero
for(i=0;i<4;i++)
for(j=0;j<4;j++)
m[i][j] = zero;
}
template <class Number>
matrix<Number> matrix<Number>::operator*(
const matrix<Number> mat)
{
matrix<Number> result;
int i,j,k;
for(i=0;i<4;i++)
for(j=0;j<4;j++)
for(k=0;k<4;k++)
result.m[i][j] += m[i][k] * mat.m[k][j];
return result;
}
template <class Number>
ostream& operator<<(ostream& os, const matrix<Number>& mat)
{
int i,j;
for(i=0;i<4;i++)
{
for(j=0;j<4;j++)
os << setw(10) << mat(i,j);
os << endl;
}
return os;
}
Listing: matrix<double>.cp
/***************************************
* matrix<double>.cp
* pragmas to create public double matrix
***************************************/
#include "matrix.cp"
#pragma template matrix<double>
#pragma template_access public
Listing: matrix<fix>.cp
/***************************************
* matrix<fix>.cp
* pragmas to create public fix matrix
***************************************/
#include "fix.h"
#include "matrix.cp"
#pragma template matrix<fix>
#pragma template_access public
Listing: benchmark.cp
/***************************************
* benchmark.cp
* benchmark template function and main routine
***************************************/
#include "fix.h"
#include "matrix.h"
#include <math.h>
#include <iostream.h>
#include <console.h>
#include <profile.h>
// Change the 0 below to a 1 to use the profiler.
// Also turn on the following C++ compiler options:
// "Generate profiler calls"
// "Always generate stack frames"
// "Use function call for inlines"
const int use_profiler = 0;
const double pi = 3.14159;
const int num_iterations = 1000;
template <class Number>
unsigned long benchmark(matrix<Number> m,const int use_prof=0)
{
unsigned long time1, time2, total;
// Make the matrix an identity matrix, and print.
m(0,0) = m(1,1) = m(2,2) = m(3,3) = 1;
cout << "identity matrix:" << endl << m;
// Create a rotation matrix, and print.
matrix<Number> rotation;
rotation(0,0) = rotation(1,1) = cos(pi/90);
// 2 degree rotation
rotation(0,1) = rotation(1,0) = sin(pi/90);
rotation(0,1) *= 1;
rotation(2,2) = rotation(3,3) = 1;
cout << "rotation matrix:" << endl << rotation;
// Premultiply the matrix by the rotation
// matrix 1000 times.
GetDateTime(&time1);
if (use_prof) InitProfile(200,200);
for (int i=0; i<num_iterations; i++)
m = rotation * m;
if (use_prof) DumpProfile();
GetDateTime(&time2);
// Print results.
cout << "result of " << num_iterations <<
" multiplications:" << endl << m;
total = time2time1;
cout << "duration: " << total << " seconds" << endl;
return total;
}
void main(int argc, char *argv[])
{
unsigned long double_time, fix_time;
double ratio;
// The ccommand() function allows results to be
// easily saved to a file.
argc = ccommand(&argv);
cout <<
"calling the benchmark function on a double matrix..."
<< endl;
matrix<double> double_matrix;
double_time = benchmark(double_matrix);
cout << "calling the benchmark function on a fix matrix..."
<< endl;
matrix<fix> fix_matrix;
fix_time = benchmark(fix_matrix,use_profiler);
if (!use_profiler)
if (fix_time > 0)
{
ratio = (double)double_time/(double)fix_time;
cout << "fixedpoint calculations were about "
<< ratio << " times faster!" << endl;
}
else
cout <<
"fixedpoint calculations completed in less than 1 second!"
<< endl;
}