Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or any later version published by the Free Software Foundation; with no Invariant Sections, no Front-Cover Texts, and no Back-Cover Texts. A copy of the license is included in the section entitled "GNU Free Documentation License".
The High Precision Arithmetic (HPA) library implements a high precision floating point arithmetic together with a comprehensive set of support functions. The general areas covered by these functions include:
The math library support includes evaluation of trigonometric, inverse trigonometric, hyperbolic, logarithm, and exponential functions at the same precision as the floating point math itself. The HPA library also supports high precision complex arithmetic and includes an Extended Precision Complex Math Library.
The HPA library comes from a branch of the source code of the CCMath library, which is a work by Daniel A. Atkinson. Daniel A. Atkinson was very kind to release the code of the CCMath Library under GNU Lesser General Public License. This made possible that Ivano Primi could modify, complete and redistribute this source code under the same terms.
So, the HPA (abbreviation of High Precision Arithmetic) Library is copyrighted by Ivano Primi <ivprimi (a) libero it> and Daniel A. Atkinson (the author of the original code). As it is for the source code of the CCMath Library, the source code of the HPA library is released under the terms of the GNU Lesser General Public License, as published by the Free Software Foundation; either version 2.1 of the License, or (at your option) any later version.
The HPA library is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU Lesser General Public License for more details.
You should have received a copy of the GNU Lesser General Public License along with this library; if not, write to the Free Software Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA 02111-1307 USA
You can contact me by paper mail writing to the next address
Ivano Primi via Colle Cannetacce 50/A C.A.P. 00038 - Valmontone (ROMA) Italy .
If you prefer the electronic mail you can write to the address
<ivprimi (a) libero it> .
The functions forming the HPA library are all implemented in a portable fashion in the C language. The IEEE 754 standard for floating point hardware and software is assumed in the PC/Unix version of this library. The normal configuration of the library employs a floating point mantissa of 112 bits, or approximately 32 decimal digit precision. However, even higher precision is available as an option. An extended floating point number is represented as a combination of the following elements:
sign bit(s): 0 -> positive, 1 -> negative ;
exponent(e): 15-bit biased integer (bias=16383) ;
mantissa(m): 7 words of 16 bit length with the leading 1 explicitly represented .
Thus f = (-1)^s*2^[e-16383] *m , with 1 <= m < 2 . This format supports a dynamic range of:
2^16384 > f > 2^[-16383] or
1.19*10^4932 > f > 1.68*10^-[4932].
Special values of the exponent are:
all ones -> infinity (floating point overflow)
all zeros -> number = zero.
Underflow in operations is handled by a flush to zero. Thus, a number with the exponent zero and nonzero mantissa is invalid (not-a-number). From the point view of the HPA library, a complex number is simply a structure formed by two extended floating point numbers, representing respectively the real and the imaginary part of the complex number.
The HPA library is composed by two modules. The first one is formed
by the functions for the real arithmetic, that is to say by the functions
operating on real arguments. The second one is formed by all the functions
which manipulate complex arguments.
The lists of the functions which compose the HPA library
are in the header files xpre.h and cxpre.h.
xpre.h is the header file for real arithmetic, it contains a definition
of the basic structure of an extended precision real number (struct xpr),
and all the declarations of the functions provided by the library
to manipulate real arguments.
The numeric type struct xpr can be used to declare and define
real variables, just as in
struct xpr q;
The size of a variable of struct xpr type is given by
(2 x XDIM + 2) bytes, where XDIM is a constant defined in the file
xpre.h (actually, in the file hpaconf.h which is included by
xpre.h).
cxpre.h is the header file for complex arithmetic, it contains a definition
of the basic structure of an extended precision complex number
(struct cxpr), and all the declarations of the functions provided
by the library to manipulate complex arguments.
The numeric type struct cxpr can be used to declare and define
complex variables, just as in
struct cxpr q;
The size of a variable of struct cxpr type is given by
(4 x XDIM + 4) bytes, where XDIM is the same constant as above.
Naturally, before declaring or defining variables of struct xpr
type and before using anyone of the functions declared in the header file
xpre.h, you have to put the line
#include <xpre.h>
in your source code file.
Analogously, before declaring or defining variables of struct cxpr
type and before using anyone of the functions declared in the header file
cxpre.h, you have to add the line
#include <cxpre.h>
to your source code file.
After including in your source code the header file xpre.h or,
if you also need functions handling complex arguments, the header file
cxpre.h (which automatically includes xpre.h), you can start
to play with the HPA library, by defining all the variables and recalling
all the functions which are needed to do your computations.
In fact, the HPA library DOES NOT REQUIRE that a special initialization
routine must be called before any other function of the library.
Moreover, variables of struct xpr or struct cxpr type
DO NOT NEED to be initialized before they can be used.
So, with respect to these issues, the HPA library is different
from some other libraries for arbitrary precision computation,
like GNU MP(c) or MAPM(c). Actually, the HPA library is not for
arbitrary precision arithmetic, but only for high precision arithmetic,
if you know the difference.
When I wrote the HPA library, I tried to create a sort of namespace for all the identifiers used by the library. This has been achieved by sticking to the following rules:
xisNaN(). Their names start
by x (if they are defined in xpre.h) or by cx (if they are defined
in cxpre.h) with the only exception of a few functions,
which however have a name ending by tox or tocx: strtox(),
strtocx(), dbltox(), dctocx(), flttox(), fctocx(),
inttox(), ictocx(), uinttox(), uctocx(), atox(),
atocx();
X or CX;
x (real constants) or by cx (complex constants) and the letter
which immediately follows this prefix is always uppercase, just as in
xZero, xPi, cxOne, cxIU (IU stays for imaginary
unit);
int type, whose name is xErrNo.
A trivial program sampling the use of the HPA library, is given by:
#include <stdio.h>
#include <xpre.h>
int main (void)
{
struct xpr s;
int i, n;
do
{
printf ("Give me a number, n = ? \t");
scanf ("%d", &n);
s = xZero;
for (i = 0; i <= n; i++)
s = xadd (s, xpr2(xOne, i), 0);
printf ("The sum 2^0 + 2^1 + ... + 2^n is equal to\n");
xprxpr (s, 30);
putchar ('\n');
} while (n > 0);
return 0;
}
This program takes in input from the user an integer value n
and prints on the screen the sum of the first n powers of 2.
In the program we use the functions xpr2() and xprxpr().
xpr2(x, n), where n is an integer, returns x* 2^n, while
xprxpr(x, m), where m is an integer, prints on the screen
the number x with m decimal digits after the dot (.) .
To know how compiling and linking a program which uses the HPA library,
please refer to the instructions contained in the file README
of the HPA documentation.
A last note: the HPA library is NOT thread safe. Some of the HPA internal data could get corrupted if multiple HPA functions are active at the same time. This is due to the fact that some functions of the HPA library use static variables to store information. So, the user should guarantee that only one thread is performing HPA 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 HPA thread will be active at a time.
During the use of the HPA library it could happen to pass
a function an illegal argument, that is to say an argument whose
value is against the mathematical definition of the function.
For instance, this occurs when a negative value is passed to
the function xsqrt(). This function computes and returns
the square root of its argument, but the square root of a number
is defined only for non-negative numbers.
So, if x is less than zero, then xsqrt(x) can not
be computed and a mathematical error occurs (a so called domain error).
Another type of mathematical error occurs when the second argument
of the division function (xdiv()) is zero: because it is impossible
to divide a number by zero, a division by zero error occurs.
What does it happen when a mathematical error is detected during the
execution of a function of the HPA library ?
Well, it depends upon the way the HPA library was compiled
when it was installed on the system where you are working.
If, during the installation process, the default setting was left
unchanged, then whenever a runtime error occurs within a function of the HPA
library, this function will set an external error indicator to
a suitable value. This value can be looked up later
to know what exactly went wrong.
The name of the variable of int type used as error indicator
is xErrNo. Before any function of the HPA library is
executed, the value of xErrNo is 0.
Then, when the first HPA function is called, if a mathematical error
occurs, then xErrNo is set to a suitable positive value,
which indicates the exact type of the occurred error.
After, xErrNo is modified if and only if, during the execution of
an HPA function, another mathematical error occurs. xErrNo is never
reset to 0 by the HPA library, so one has to
zero xErrNo before calling a function of the HPA library
in order to detect possible errors. A sample of it is given by:
#include <stdio.h>
#include <xpre.h>
extern int xErrNo;
int main (void)
{
int n;
struct xpr sr;
do
{
printf ("Give me a number, n = ? \t");
scanf ("%d", &n);
xErrNo = 0;
sr = xsqrt (inttox (n));
if (xErrNo == 0)
{
printf ("The square root of %d is\n", n);
xprxpr (sr, 30);
putchar ('\n');
}
else
{
fprintf (stderr, "*** Error: Out of domain\n");
}
} while (n != 0);
return 0;
}
In this sample xErrNo is reset to zero at each execution of the
do {...} while(); loop before the call to the xsqrt() function.
However, the HPA library could be compiled to deal differently with
runtime errors.
For instance, in case of error a suitable message could be
printed onto stderr and the library could also cause the
termination of the calling program via EXIT(1).
At last, the library could also be compiled to ignore
at all any mathematical error (sigh !). To know how the routines of
the HPA library deal with errors is sufficient to examine the file
hpaconf.h (which is automatically included by xpre.h and
cxpre.h). This file defines the macro
XERR_DFL to mean that, in case of error, 'xErrNo' is suitably set;
XERR_WARN ... ... ... ... ... ... ... a suitable message is printed
on stderr;
XERR_EXIT ... ... ... ... ... ... ... the calling program is terminated
through a call to exit(1) after
printing a message on stderr;
XERR_IGN ... ... ... ... ... ... ... nothing is done or signaled.
When the macro XERR_DFL is defined, the header file xpre.h
also defines the macros XENONE, XEDIV, XEDOM, XEBADEXP, XFPOFLOW and
XNERR:
#define XENONE 0 /* No error */ #define XEDIV 1 /* Division by zero */ #define XEDOM 2 /* Out of domain */ #define XEBADEXP 3 /* Bad exponent */ #define XFPOFLOW 4 /* Floating point overflow */ #define XNERR 4 /* Number of the non-null error codes */
that can be used, together with xErrNo, to recover the exact type of
the error occurred during a call to a routine of the HPA library.
The first module of the HPA library is made of functions for
Extended Precision Floating Point Arithmetic, functions of the
Extended Precision Math Library and applications of the Extended
Precision Arithmetic. They are all declared in the file xreal.h
together with some macros and numerical constants.
The header file xpre.h also defines the structure xoutflags:
struct xoutflags
{
short fmt, notat, sf, mfwd, lim;
signed char padding, ldel, rdel;
};
A structure of such kind is used by the output functions
xfout(), xout() and xsout() to know how they have to print the
numbers.
The field notat refers to the notation: it can
be equal to XOUT_SCIENTIFIC (scientific notation) or to XOUT_FIXED
(floating point notation). Both XOUT_SCIENTIFIC and XOUT_FIXED are
macros defined inside xpre.h.
The field sf refers to the sign: when sf is not zero every non-negative
number is printed with a plus sign (+) ahead.
The field mfwd indicates the minimum field width to use in
printing numbers. When mfwd is zero no minimum field width is used.
When mfwd is negative, then the actual minimum field width is given
by -mfwd and the printed number is left adjusted on the
field boundary. (The default is right justification).
lim has a different meaning depending on the actual notation in use.
Together with the scientific notation, lim gives the number of
decimal digits to the right of the decimal point
(lim+1 = total digits displayed). Otherwise, lim + 1 is the
number of significant digits displayed. When lim is negative,
the default value (6) is used.
At last, padding defines the padding character to use together
with a non-zero minimum field width.
If padding is negative, then the default padding char (i.e. the
blank character) is used.
The fields fmt, ldel and rdel are ignored by the functions
xfout(), xout() and xsout(). They are only used by the complex
printing functions cxfout(), cxout() and cxsout().
fmt specifies the format to use in printing complex numbers.
The possible values for fmt are XFMT_STD, XFMT_RAW and XFMT_ALT
(these macros are declared inside cxpre.h !).
If fmt == XFMT_STD, then the complex number (a, b) is printed
using the notation a+bi or a-bi (depending on the sign of b).
Naturally, a and b are printed under the rules above
exposed.
If fmt == XFMT_RAW, then (a, b) is printed in the form
a<two blank spaces>b
just as in
1.0 2.5
under the hypothesis that a = 1.0 and b = 2.5.
At last, if fmt == XFMT_ALT, then (a,b) is printed
as
<left delimiter>a, b<right delimiter>
where <left_delimiter> and <right_delimiter> are the characters
given by the fields ldel and rdel respectively.
If ldel < 0 (rdel < 0), then ( (or ) ) is used as
default <left_delimiter> (<right_delimiter>).
Be careful ! None of the functions xfout(), xout(), xsout(),
cxfout(), cxout() or cxsout() adds a newline at the end
of the printed number !
The header file xpre.h defines several constants. Between the
constants defined in xpre.h there are those ones
corresponding to particular mathematical values:
extern const struct xpr xZero, xOne, xTwo, xTen;
extern const struct xpr xPinf, xMinf, xNaN;
extern const struct xpr xPi, xPi2, xPi4, xEe, xSqrt2;
extern const struct xpr xLn2, xLn10, xLog2_e, xLog2_10, xLog10_e;
xZero (= 0), xOne (= 1), xTwo (= 2)and xTen (= 10) do not need a comment. xPi, xPi2, xPi4, xEe, xSqrt2, xLn2, xLn10, xLog2_e, xLog2_10, xLog10_e represent respectively the values PI (= 3.14159...), PI/2, PI/4, e (= 2.7182818...), square root of 2 (= 1.4142135...), natural logarithm of 2 and 10, base-2 logarithm of e and 10, 10-base logarithm of e.
xPinf, xMinf and xNan are special values: xPinf represents the value
+oo (plus infty), xMinf the value -oo (minus infty) and xNaN
is used to mean an invalid number (NaN stays for Not a Number).
xPinf and xMinf are usually returned by a function to signal a
floating point overflow (positive and negative respectively), while
xNaN is returned by the functions converting ASCII strings to floating
point numbers to indicate that the string given to them as argument
did not contain any valid number.
xPinf, xMinf and xNaN should never be used as arguments for functions,
since a such use has unpredictable results.
The arithmetic functions support the basic computation and input/output
operations needed for extended precision floating point mathematics. Some
of the operations supply capabilities designed to enhance the computational
efficiency of this arithmetic (e.g., xpwr).
What follows is their complete list including the synopsis for each of them.
xadd
Add (subtract) two extended precision numbers.
struct xpr xadd(struct xpr s,struct xpr t,int f)
s = structure containing first number;
t = structure containing second number;
f = control flag: if 0, then s and t are added,
else they are subtracted (s-t).
The value returned by xadd() is a structure containing the result
of the addition (subtraction). xadd() can return xPinf or xMinf to
signal a floating point overflow.
xmul
Multiply two extended precision numbers.
struct xpr xmul(struct xpr s,struct xpr t)
s = structure containing first number;
t = structure containing second number.
The value returned by xmul() is a structure containing the
product s*t. It can be xPinf or xMinf in case of overflow.
The manual of the HPA library is still to be completed.
Refer to the header files xpre.h and cxpre.h for the
numeric types, the definitions of macros, the
prototypes of the functions and their use.
Actually, the functions of the real module of the HPA library
are fully listed and commented inside the document manual,
which is a simple ASCII text-file contained in the doc
directory of the HPA library.
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Front-Cover Text and one of Back-Cover Text may be added by (or
through arrangements made by) any one entity. If the Document already
includes a cover text for the same cover, previously added by you or
by arrangement made by the same entity you are acting on behalf of,
you may not add another; but you may replace the old one, on explicit
permission from the previous publisher that added the old one.
The author(s) and publisher(s) of the Document do not by this License
give permission to use their names for publicity for or to assert or
imply endorsement of any Modified Version.
5. COMBINING DOCUMENTS
You may combine the Document with other documents released under this
License, under the terms defined in section 4 above for modified
versions, provided that you include in the combination all of the
Invariant Sections of all of the original documents, unmodified, and
list them all as Invariant Sections of your combined work in its
license notice, and that you preserve all their Warranty Disclaimers.
The combined work need only contain one copy of this License, and
multiple identical Invariant Sections may be replaced with a single
copy. If there are multiple Invariant Sections with the same name but
different contents, make the title of each such section unique by
adding at the end of it, in parentheses, the name of the original
author or publisher of that section if known, or else a unique number.
Make the same adjustment to the section titles in the list of
Invariant Sections in the license notice of the combined work.
In the combination, you must combine any sections Entitled "History"
in the various original documents, forming one section Entitled
"History"; likewise combine any sections Entitled "Acknowledgements",
and any sections Entitled "Dedications". You must delete all sections
Entitled "Endorsements".
6. COLLECTIONS OF DOCUMENTS
You may make a collection consisting of the Document and other documents
released under this License, and replace the individual copies of this
License in the various documents with a single copy that is included in
the collection, provided that you follow the rules of this License for
verbatim copying of each of the documents in all other respects.
You may extract a single document from such a collection, and distribute
it individually under this License, provided you insert a copy of this
License into the extracted document, and follow this License in all
other respects regarding verbatim copying of that document.
7. AGGREGATION WITH INDEPENDENT WORKS
A compilation of the Document or its derivatives with other separate
and independent documents or works, in or on a volume of a storage or
distribution medium, is called an "aggregate" if the copyright
resulting from the compilation is not used to limit the legal rights
of the compilation's users beyond what the individual works permit.
When the Document is included in an aggregate, this License does not
apply to the other works in the aggregate which are not themselves
derivative works of the Document.
If the Cover Text requirement of section 3 is applicable to these
copies of the Document, then if the Document is less than one half of
the entire aggregate, the Document's Cover Texts may be placed on
covers that bracket the Document within the aggregate, or the
electronic equivalent of covers if the Document is in electronic form.
Otherwise they must appear on printed covers that bracket the whole
aggregate.
8. TRANSLATION
Translation is considered a kind of modification, so you may
distribute translations of the Document under the terms of section 4.
Replacing Invariant Sections with translations requires special
permission from their copyright holders, but you may include
translations of some or all Invariant Sections in addition to the
original versions of these Invariant Sections. You may include a
translation of this License, and all the license notices in the
Document, and any Warranty Disclaimers, provided that you also include
the original English version of this License and the original versions
of those notices and disclaimers. In case of a disagreement between
the translation and the original version of this License or a notice
or disclaimer, the original version will prevail.
If a section in the Document is Entitled "Acknowledgements",
"Dedications", or "History", the requirement (section 4) to Preserve
its Title (section 1) will typically require changing the actual
title.
9. TERMINATION
You may not copy, modify, sublicense, or distribute the Document except
as expressly provided for under this License. Any other attempt to
copy, modify, sublicense or distribute the Document is void, and will
automatically terminate your rights under this License. However,
parties who have received copies, or rights, from you under this
License will not have their licenses terminated so long as such
parties remain in full compliance.
10. FUTURE REVISIONS OF THIS LICENSE
The Free Software Foundation may publish new, revised versions
of the GNU Free Documentation License from time to time. Such new
versions will be similar in spirit to the present version, but may
differ in detail to address new problems or concerns. See
http://www.gnu.org/copyleft/.
Each version of the License is given a distinguishing version number.
If the Document specifies that a particular numbered version of this
License "or any later version" applies to it, you have the option of
following the terms and conditions either of that specified version or
of any later version that has been published (not as a draft) by the
Free Software Foundation. If the Document does not specify a version
number of this License, you may choose any version ever published (not
as a draft) by the Free Software Foundation.
ADDENDUM: How to use this License for your documents
To use this License in a document you have written, include a copy of
the License in the document and put the following copyright and
license notices just after the title page:
Copyright (c) YEAR YOUR NAME.
Permission is granted to copy, distribute and/or modify this document
under the terms of the GNU Free Documentation License, Version 1.2
or any later version published by the Free Software Foundation;
with no Invariant Sections, no Front-Cover Texts, and no Back-Cover Texts.
A copy of the license is included in the section entitled "GNU
Free Documentation License".
If you have Invariant Sections, Front-Cover Texts and Back-Cover Texts,
replace the "with...Texts." line with this:
with the Invariant Sections being LIST THEIR TITLES, with the
Front-Cover Texts being LIST, and with the Back-Cover Texts being LIST.
If you have Invariant Sections without Cover Texts, or some other
combination of the three, merge those two alternatives to suit the
situation.
If your document contains nontrivial examples of program code, we
recommend releasing these examples in parallel under your choice of
free software license, such as the GNU General Public License,
to permit their use in free software.