xref: /haiku/src/libs/mapm/README (revision 17889a8c70dbb3d59c1412f6431968753c767bab)

**************************************************************************

				   MAPM

			       Version 4.9.5

			     December 10, 2007

			      Michael C. Ring

			    ringx004@tc.umn.edu

		    Latest release will be available at
		        http://tc.umn.edu/~ringx004

***************************************************************************
*									  *
*  Copyright (C) 1999 - 2007   Michael C. Ring                            *
*									  *
*  This software is Freeware.						  *
*									  *
*  Permission to use, copy, and distribute this software and its          *
*  documentation for any purpose with or without fee is hereby granted,   *
*  provided that the above copyright notice appear in all copies and      *
*  that both that copyright notice and this permission notice appear      *
*  in supporting documentation.                                           *
*									  *
*  Permission to modify the software is granted. Permission to distribute *
*  the modified code is granted. Modifications are to be distributed by   *
*  using the file 'license.txt' as a template to modify the file header.  *
*  'license.txt' is available in the official MAPM distribution.          *
*									  *
*  To distribute modified source code, insert the file 'license.txt'      *
*  at the top of all modified source code files and edit accordingly.     *
*									  *
*  This software is provided "as is" without express or implied warranty. *
*									  *
***************************************************************************

		---------------------------------------
		Mike's Arbitrary Precision Math Library
		---------------------------------------

Mike's Arbitrary Precision Math Library is a set of functions that
allow the user to perform math to any level of accuracy that is
desired. The inspiration for this library was Lloyd Zusman's similar
APM package that was released in ~1988. I borrowed some of his ideas
in my implementation, creating a new data type (MAPM) and how the data
type was used by the user. However, there were a few things I wanted
my library to do that the original library did not :

1) Round a value to any desired precision. This is very handy when
   multiplying for many iterations. Since multiplication guarantees an
   exact result, the number of digits will grow without bound. I wanted
   a way to trim the number of significant digits that were retained.

2) A natural support for floating point values. From most of the other
   libraries I looked at, they seem to have a preference for integer
   only type math manipulations. (However, this library will also do
   integer only math if you desire).

   And if a library can only do integers, it can't do ...

3) Trig functions and other common C math library functions. This library
   will perform the following functions to any desired precision level :
   SQRT, CBRT, SIN, COS, TAN, ARC-SIN, ARC-COS, ARC-TAN, ARC-TAN2, LOG,
   LOG10, EXP, POW, SINH, COSH, TANH, ARC-SINH, ARC-COSH, ARC-TANH, and
   also FACTORIAL.  The full 'math.h' is not duplicated, though I think
   these are most of the important ones. My definition of what's important
   is what I've actually used in a real application.

**************************************************************************

NOTE:

There is a COMPILER BUG in Microsoft's Visual C++ 7.x (VS.NET 2003) which
prevents a C++ MAPM application from compiling.

This only affects C++ applications. C applications are OK.

The compiler bug creates an error C2676 similar to this:

<path>...\mapm-49\M_APM.H(###) : error C2676: binary operator '-': 'MAPM'
doesn't define this operator or a conversion to a suitable type for the
predefined operator.

To work around this bug, go to http://www.microsoft.com

In the upper right corner of web page, search for "814455".

The results of the search will point you to an article on how to work
around the problem.

**************************************************************************

NOTE: MAPM Library History can now be found in 'history.txt'

**************************************************************************

ANOTHER NOTE: For the Windows/DOS distribution, the filename convention
will always be in 8.3 format. This is because there are some users who
still use 16 bit DOS ....

(I really wasn't sure what to call this library. 'Arbitrary Precision Math'
is a defacto standard for what this does, but that name was already taken,
so I just put an 'M' in front of it ...)

**************************************************************************

MAPM LIBRARY NUMERICAL LIMITATIONS:

A general floating point number is of the form:

Sn.nnnnnnnnE+yyyy        ex: -4.318384739357974916E+6215
Sn.nnnnnnnnE-yyyy        ex: +8.208237913789131096523645193E-12873

'S' is the sign, + or -.

In MAPM, a number (n.nnnn...) may contain up to ( INT_MAX - 1 ) digits.

For example, an MAPM number with a 16 bit integer may contain 2^15 or 32,767
digits. An MAPM number with a 32 bit integer may contain 2^31 or 2,147,483,647
digits. All MAPM numbers are stored in RAM, there is no "data on disk" option.
So to represent the maximum number of digits on a 32 bit CPU will require
greater than 2 Gig of RAM.

If you have a CPU with 64 bit ints, then the limitation is 2^63 or
9,223,372,036,854,775,807. It should be a very long time before computers
have this much RAM in them.

For the exponent (yyyy), the limitations are also INT_MAX and INT_MIN.

For a 16 bit CPU, the largest number you can represent is approx
0.9999999....E+32767.    (H)

For a 16 bit CPU, the smallest number you can represent (other than 0)
is approx 0.1E-32767.   (L)

For a 32 bit CPU, the largest number you can represent is approx
0.9999999....E+2147483647.   (H)

For a 32 bit CPU, the smallest number you can represent (other than 0)
is approx 0.1E-2147483647.  (L)

The limitations for negative numbers are the same as positive numbers.


                            Real Number Axis

     +------------------------+    ---    +--------------------------+
     |                        |     |     |                          |
    -H                       -L    0.0   +L                         +H



MAPM can represent real numbers from -H to -L, 0.0, +L to +H.

The number of significant digits is typically only limited by available RAM.

In MAPM, numerical overflows and underflows are not handled very well
(actually not at all). There really isn't a clean and portable way to
detect integer overflow and underflow. Per K&R C, the result of integer
overflow/underflow is implementation dependent. In assembly language, when
you add two numbers, you have access to a "carry flag" to see if an overflow
occurred. C has no analogous operator to a carry flag.

It is up to the user to decide if the results are valid for a given operation.
In a 32 bit environment, the limit is so large that this is likely not an
issue for most typical applications. However, it doesn't take much to overflow
a 16 bit int so care should taken in a 16 bit environment.

The reaction to an integer overflow/underflow is unknown at run-time:

    o) adding 2 large positive numbers may silently roll over to a
       negative number.
    o) in some embedded applications an integer overflow/underflow triggers
       a hardware exception.

Since I don't have control over where this library could possibly run,
I chose to ignore the problem for now. If anyone has some suggestions
(that's portable), please let me know.

**************************************************************************

KNOWN BUGS : (Other than integer overflow discussed above....) None

**************************************************************************

IF YOU ARE IN A HURRY ...

UNIX:  (assumes gcc compiler)

run    make          (build library + 4 executables)

run    make -f makefile.osx      (for MAC OSX)

--- OR ---

run:   mklib         (this will create the library, lib_mapm.a)

run:   mkdemo        (this will create 4 executables, 'calc', 'validate',
                      'primenum', and 'cpp_demo')


DOS / Win NT/9x  (in a DOS window for NT/9x):

see the file 'README.DOS' for instructions.


**************************************************************************

calc:  This is a command line version of an RPN calculator. If you are
       familiar with RPN calculators, the use of this program will be
       quite obvious. The default is 30 decimal places but this can be
       changed with the '-d' command line option. This is not an
       interactive program, it just computes an expression from the command
       line. Run 'calc' with no arguments to get a list of the operators.

       compute : (345.2 * 87.33) - (11.88 / 3.21E-2)

       calc 345.2 87.33 x 11.88 3.21E-2 / -
       result: 29776.22254205607476635514018692


       compute PI to 70 decimal places :  (6 * arcsin(0.5))

       calc -d70 6 .5 as x
       result :
3.1415926535897932384626433832795028841971693993751058209749445923078164

       calc -d70 -1 ac             (arccos(-1) for fastest possible way)

validate : This program will compare the MAPM math functions to the C
	   standard library functions, like sqrt, sin, exp, etc.  This
	   should give the user a good idea that the library is operating
	   correctly. The program will also perform high precision math
	   using known quantities like PI, log(2), etc.

	   'validate' also attempts to obtain as much code coverage of the
	   library as is practical. I used 'gcov' (available with the gcc
	   distribution) to test the code coverage. 100% coverage is not
	   obtained, compromises must be made in order to have the program
	   run in a reasonable amount of time.

primenum:  This program will generate the first 10 prime numbers starting
	   with the number entered as an argument.

	   Example:  primenum 1234567890 will output (actually 1 per line):
		     this took ~3 sec on my Linux PC, a 350MHz PII.

	   1234567891, 1234567907, 1234567913, 1234567927, 1234567949,
	   1234567967, 1234567981, 1234568021, 1234568029, 1234568047


**************************************************************************

To use the library, simply include 'm_apm.h' and link your program
with the library, libmapm.a (unix) or mapm.lib (dos).

Note: If your system creates libraries with a '.a' extension, then the
library will be named libmapm.a. (This conforms to typical unix naming
conventions).

Note: If your system creates libraries with a '.lib' extension, then the
library will be named mapm.lib.

For unix builds, you also may need to specify the math library (-lm) when
linking. The reason is some of the MAPM functions use an iterative algorithm.
When you use an iterative solution, you have to supply an initial guess. I
use the standard math library to generate this initial guess. I debated
whether this library should be stand-alone, i.e. generate it's own initial
guesses with some algorithm. In the end, I decided using the standard math
library would not be a big inconvienence and also it was just too tempting
having an immediate 15 digits of precision. When you prime the iterative
routines with 15 accurate digits, the MAPM functions converge faster.

See the file 'algorithms.used' to see a description of the algorithms I
used in the library. Some I derived on my own, others I borrowed from people
smarter than me. Version 2 of the library supports a 'fast' multiplication
algorithm. The algorithm used is described in the algorithms.used file. A
considerable amount of time went into finding fast algorithms for the
library. However, some could possibly be even better. If anyone has a
more efficient algorithm for any of these functions, I would like to here
from you.

See the file 'function.ref' to see a description of all the functions in
the library and the calling conventions, prototypes, etc.

See the file 'struct.ref' which documents how I store the numbers internally
in the MAPM data structure. This will not be needed for normal use, but it
will be very useful if you need to change/add to the existing library.

**************************************************************************

USING MAPM IN A MULTI-THREADED APPLICATION :

Note that the default MAPM library is NOT thread safe. MAPM internal data
structures could get corrupted if multiple MAPM functions are active at the
same time. The user should guarantee that only one thread is performing
MAPM functions. This can usually be achieved by a call to the operating
system to obtain a 'semaphore', 'mutex', or 'critical code section' so the
operating system will guarantee that only one MAPM thread will be active
at a time.

The necessary function wrappers for thread safe operation can be found in
the sub-directory 'multi_thread' (unix) or 'multithd' (Win/Dos). For now,
only Microsoft Visual C++ 6.0 is supported.

**************************************************************************

QUICK TEMPLATE FOR NORMAL USE :

The MAPM math is done on a new data type called "M_APM".  This is
actually a pointer to a structure, but the contents of the structure
should never be manipulated: all operations on MAPM entities are done
through a functional interface.

The MAPM routines will automatically allocate enough space in their
results to hold the proper number of digits.

The caller must initialize all MAPM values before the routines can
operate on them (including the values intended to contain results of
calculations).  Once this initialization is done, the user never needs
to worry about resizing the MAPM values, as this is handled inside the
MAPM routines and is totally invisible to the user.

The result of a MAPM operation cannot be one of the other MAPM operands.
If you want this to be the case, you must put the result into a
temporary variable and then assign (m_apm_copy) it to the appropriate operand.

All of the MAPM math functions begin with m_apm_*.

There are some predefined constants for your use. In case it's not obvious,
these should never appear as a 'result' parameter in a function call.

The following constants are available : (declared in m_apm.h)

MM_Zero             MM_One              MM_Two             MM_Three
MM_Four             MM_Five             MM_Ten
MM_PI               MM_HALF_PI          MM_2_PI            MM_E
MM_LOG_E_BASE_10    MM_LOG_10_BASE_E    MM_LOG_2_BASE_E    MM_LOG_3_BASE_E


The non-integer constants above (PI, log(2), etc) are accurate to 128
decimal places. The file mapmcnst.c contains these constants. I've
included 512 digit constants in this file also that are commented out.
If you need more than 512 digits, you can simply use the 'calc' program
to generate more precise constants (or create more precise constants at
run-time in your app).  The number of significant digits in the constants
should be 6-8 more than the value specified in the #define.


Basic plan of attack:

(1)  get your 'numbers' into M_APM format.
(2)  do your high precision math.
(3)  get the M_APM numbers back into a format you can use.


--------  (1)  --------

#include "m_apm.h"        /* must be included before any M_APM 'declares' */

M_APM    area_mapm;                /* declare your variables */
M_APM    tmp_mapm;
M_APM    array_mapm[10];           /* can use normal array notation */
M_APM    array2d_mapm[10][10];

area_mapm = m_apm_init()           /* init your variables */
tmp_mapm  = m_apm_init();

for (i=0; i < M; i++)              /* must init every element of the array */
  array_mapm[i] = m_apm_init();

for (i=0; i < M; i++)
   for (j=0; j < N; j++)
     array2d_mapm[i][j] = m_apm_init();

/*
 *   there are 3 ways to convert your number into an M_APM number
 *   (see the file function.ref)
 *
 *   a) literal string   (exponential notation OK)
 *   b) long variable
 *   c) double variable
 */

m_apm_set_string(tmp_mapm, "5.3286E-7");
m_apm_set_long(array_mapm[6], -872253L);
m_apm_set_double(array2d_mapm[3][7], -529.4486711);


--------  (2)  --------

do your math ...

m_apm_add(cc_mapm, aa_mapm, MM_PI);
m_apm_divide(bb_mapm, DECIMAL_PLACES, aa_mapm, MM_LOG_2_BASE_E);
m_apm_sin(bb_mapm, DECIMAL_PLACES, aa_mapm);
whatever ...


--------  (3)  --------

There are 5 total functions for converting an M_APM number into something
useful.  (See the file 'function.ref' for full function descriptions)

For these 5 functions, M_APM number -> string is the conversion

===================
====  METHOD 1 ==== : floating point values	(m_apm_to_string)
===================

the format will be in scientific (exponential) notation

output string = [-]n.nnnnnE+x   or ...E-x

where 'n' are digits and the exponent will be always be present,
including E+0

it's easy to convert this to a double:

double dtmp = atof(out_buffer);


===================
====  METHOD 2 ==== : floating point values	(m_apm_to_fixpt_string)
===================				(m_apm_to_fixpt_stringex)
						(m_apm_to_fixpt_stringexp)
the format will be in fixed point notation

output string = [-]mmm.nnnnnn

where 'm' & 'n' are digits.


===================
====  METHOD 3 ==== : integer values		(m_apm_to_integer_string)
===================

the format will simply be digits with a possible leading '-' sign.

output string = [-]nnnnnn

where 'n' are digits.

it's easy to convert this to a long :
long mtmp = atol(out_buffer);

... or an int :
int itmp = atoi(out_buffer);

Note that if the M_APM number has a fractional portion, the fraction
will be truncated and only the integer portion will be output.


char  out_buffer[1024];

m_apm_to_string(out_buffer, DECIMAL_PLACES, mapm_number);

m_apm_to_fixpt_string(out_buffer, DECIMAL_PLACES, mapm_number);

m_apm_to_integer_string(out_buffer, mapm_number);


**************************************************************************

*********************************************************
****  NOTES on the fixed point formatting functions  ****
*********************************************************

Assume you have the following code:


-->   m_apm_set_string(aa_mapm, "2.0E18");
-->   m_apm_sqrt(bb_mapm, 40, aa_mapm);

-->   m_apm_to_string(buffer, 40, bb_mapm);
-->   fprintf(stdout,"[%s]\n",buffer);

-->   m_apm_to_fixpt_string(buffer, 40, bb_mapm);
-->   fprintf(stdout,"[%s]\n",buffer);


It is desired to compute the sqrt(2.0E+18) to
40 significant digits. You then want the result
output with 40 decimal places. But the output
from above is :

[1.4142135623730950488016887242096980785697E+9]
[1414213562.3730950488016887242096980785697000000000]


Why are there 9 '0' in the fixed point formatted string??

The sqrt calculation computed 40 significant digits relative
to the number in EXPONENTIAL format. When the number is
output in exponential format, the 40 digits are as expected
with an exponent of 'E+9'.

The same number formatted as fixed point appears to be an
error.  Remember, we computed 40 significant digits.  However,
the result has an exponent of '+9'. So, 9 of the digits are
needed *before* the decimal point. In our calculation, only
31 digits of precision remain from our original 40. We then
asked the fixed point formatting function to format 40 digits.
Only 31 are left so 9 zeros are used as pad at the end to
fulfill the 40 places asked for.

Keep this in mind if you truly desire more accurate results
in fixed point formatting and your result contains a large
positive exponent.

**************************************************************************

MAPM C++ WRAPPER CLASS:

Orion Sky Lawlor (olawlor@acm.org) has added a very nice C++ wrapper
class to m_apm.h. This C++ class will have no effect if you just use
a normal C compiler. The library will operate as before with no user
impacts.

For now, I recommend compiling the library as 'C'. The library will
compile as a C++ library, but then it is likely that you would not
be able to use the library in a straight C application.  Since the C++
wrapper class works very nicely as is, there is no pressing need to compile
the library as C++.

See the file 'cpp_function.ref' to see a description of how to use
the MAPM class.

To compile and link the C++ demo program: (assuming the library is
already built)

UNIX:

g++ cpp_demo.cpp lib_mapm.a -s -o cpp_demo -lm

GCC for DOS: (gxx (or g++) is the C++ compiler)

gxx cpp_demo.cpp lib_mapm.a -s -o cpp_demo.exe -lm


Using the C++ wrapper allows you to do things like:

// Compute the factorial of the integer n

MAPM factorial(MAPM n)
{
	MAPM i;
	MAPM product = 1;
	for (i=2; i <= n; i++)
		product *= i;
	return product;
}


The syntax is the same as if you were just writing normal code, but all
the computations will be performed with the high precision math library,
using the new 'datatype' MAPM.

The default precision of the computations is as follows:

Addition, subtraction, and multiplication will maintain ALL significant
digits.

All other operations (divide, sin, etc) will use the following rules:

1) if the operation uses only one input value [y = sin(x)], the result 'y'
   will be the same precision as 'x', with a minimum of 30 digits if 'x' is
   less than 30 digits.

2) if the operation uses two input values [z = atan2(y,x)], the result 'z'
   will be the max digits of 'x' or 'y' with a minimum of 30.

The default precision is 30 digits. You can change the precision at
any time with the function 'm_apm_cpp_precision'. (See function.ref)

---->        m_apm_cpp_precision(80);

will result in all operations being accurate to a minimum of 80 significant
digits. If any operand contains more than the minimum number of digits, then
the result will contain the max number of digits of the operands.


NOTE!: Some real life use with the C++ wrapper has revealed a certain
       tendency for a program to become quite slow after many iterations
       (like in a for/while loop).  After a little debug, the reason
       became clear. Remember that multiplication will maintain ALL
       significant digits :

       20 digit number x 20 digit number =  40 digits
       40 digit number x 40 digit number =  80 digits
       80 digit number x 80 digit number = 160 digits
       etc.

       So after numerous iterations, the number of significant digits
       was growing without bound. The easy way to fix the problem is
       to simply *round* your result after a multiply or some other
       complex operation. For example:

       #define MAX_DIGITS 256

       p1 = (p0 * sin(b1) * exp(1 + u1)) / sqrt(1 + b1);
       p1 = p1.round(MAX_DIGITS);

       If you 'round' as shown here, your program will likely be
       nearly as fast as a straight 'C' version.


NOTE #2!

       Reference the following code snippet:

       ...

       MAPM  pi1, pi2;
       char  obuf[256];

       m_apm_cpp_precision(62);

       pi1 = 2 * asin("1");
       pi2 = 2 * asin(1.0);

       pi1.toString(obuf, 60);   printf("PI1 = [%s] \n",obuf);
       pi2.toString(obuf, 60);   printf("PI2 = [%s] \n",obuf);

       ...

       On my system, the output is :

PI1 = [3.141592653589793238462643383279502884197169399375105820974945E+0]
PI2 = [3.141592653589790000000000000000000000000000000000000000000000E+0]


       PI2 only has 15 significant digits! This is due to how the second
       asin is called. It is called with a 'double' as the argument, hence
       the compiler will use the default double asin function from the
       standard library. This is likely not the intent but this would be
       easy to miss if this was a complex calculation and we didn't know
       the 'right' answer.

       In order to force the use of the overloaded MAPM functions, call the
       MAPM functions with a quoted string as the argument (if the argument
       is a constant and not a variable).

       This would also work (though it seems less elegant ...) :

       MAPM t = 1;
       pi2 = 2 * asin(t);

-----------

If you have any questions or problems with the C++ wrapper, please let
me know. I am not very C++ proficient, but I'd still like to know about any
problems.  Orion Sky Lawlor (olawlor@acm.org) is the one who implemented
the MAPM class, so he'll have to resolve any real hardcore problems, if you
have any.

**************************************************************************

TESTING :

Testing the library was interesting.  How do I know it's right?  Since I
test the library against the standard C runtime math library (see below)
I have high confidence that the MAPM library is giving correct results.
The logic here is the basic algorithms are independent of the number of
digits calculated, more digits just takes longer.

The MAPM library has been tested in the following environments :

Linux i486 / gcc 2.7.2.3, gcc 2.95.2
Linux i686 / gcc 2.91.66, gcc 2.95.2, gcc 2.95.3, gcc 3.0.4, gcc 3.2.3
Linux i686 / Intel Linux C/C++ Compiler Verison 7.0 / 8.1
FreeBSD 4.1 / gcc 2.95.1
FreeBSD 4.8 / gcc 2.95.4
FreeBSD 5.x / gcc 2.95.2, gcc 2.95.3
Redhat Linux 8.2 / gcc 3.2
HP-UX 9.x /gcc 2.5.8
HP-UX 10.x / gcc 2.95.2
Sun 5.5.1  (output from uname), gcc 3.1.1
Sun Solaris 2.6 / gcc 2.95.1, gcc 2.95.3, gcc 3.2.3
MAC OSX / gcc ?
DOS 5.0 using GCC 2.8.1 for DOS
DOS 5.0 using GCC 2.95.2 for DOS
DOS ??? using Borland Turbo C++ 3.0
WIN NT+SP5 using Borland C++ 5.02 IDE, 5.2 & 5.5 command line.
WIN 2000 using National Instruments LabWindows CVI 6.0
WIN98 using Borland C++ 5.5 command line.
WIN98 & NT & 2000 & XP using LCC-WIN32 Ver 3.2, 3.3
WIN98 & NT using Watcom C 11.x
WIN95 & NT using Open Watcom 1.0
WIN95 & WIN98 using MINGW-32 version mingw-1.0.1-20010726
WIN95 & WIN2000 using DEV-CPP 5.0 Beta 8, 4.9.8.0
MINGW-32 with gcc 3.2   (mingw special 20020817-1)
MINGW-32 with gcc 3.2.3 (mingw special 20030504-1)
WINXP & MINGW-32 with gcc 3.4.5
WINXP & Digital Mars Compiler 8.49
WIN?? using Metrowerks CodeWarrior Pro 7.0 for Windows
DOS 5.0 using Microsoft C 5.1 and 8.00c  (16 bit EXEs)
WIN98 & NT using Microsoft Visual C++ 6.0
WIN98 & NT using Microsoft Visual C++ 7.x (VS.NET 2003 except for
                                           known compiler bug C2676)



As a general rule, when calculating a quantity to a given number of decimal
places, I calculated 4-6 extra digits and then rounded the result to what was
asked for. I decided to be conservative and give a correct answer rather than
to be faster and possibly have the last 2-3 digits in error. Also, some of
the functions call other functions (calculating arc-cos will call cos, log
will call exp, etc.) so I had to calculate a few extra digits in each iteration
to guarantee the loops terminated correctly.

1)  I debugged the 4 basic math operations. I threw numerous test cases at
    each of the operations until I knew they were correct.

    Also note that the math.h type functions all call the 4 basic operations
    numerous times. So if all the math.h functions work, it is highly
    probable the 4 basic math operations work also.

2)  'math.h' type functions.

     SQRT:     Not real hard to check. Single stepping through the iterative
	       loop showed it was always converging to the sqrt.

     CBRT:     Similar to sqrt, single stepping through the iterative loop
	       showed it was always converging to the cube root.

     EXP:      I wrote a separate algorithm which expanded the Taylor series
	       manually and compared the results against the library.

     POW:      Straightforward since this just calls 'EXP'.

     LOG:      I wrote a separate algorithm which expanded the Taylor series
	       manually and compared the results against the library. This
	       took a long time to execute since the normal series converges
	       VERY slowly for the log function. This is why the LOG function
	       uses an iterative algorithm.

     LOG10:    Straightforward since this just calls 'LOG'.

     SIN/COS:  I wrote a separate algorithm which expanded the Taylor series
	       manually and compared the results against the library.

     TAN:      Straightforward since this just calls 'SIN' and 'COS'.

     ARC-x:    Single stepping through the iterative loop showed the arc
	       family of functions were always converging. Also used these
	       to compute PI. The value of PI is now known to many, many
	       digits. I computed PI to 1000+ digits by computing:

	       6 * arcsin(0.5)  and  4 * arctan(1)  and  3 * arccos(0.5)

	       and compared the output to the published 'real' values of PI.

	       The arc family of functions exercise considerable portions
	       of the library.

  HYPERBOLIC:  The hyperbolic functions just use exp, log, and the 4 basic
  	       math operations. All of these functions simply use other
	       existing functions in the library.

     FINALLY:  Run the program 'validate'. This program compares the first
	       13-14 significant digits of the standard C library against
	       the MAPM library. If this program passes, you can feel
	       confident that the MAPM library is giving correct results.

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