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/*
* Copyright (c) 1985, 1993
* The Regents of the University of California. All rights reserved.
*
* Redistribution and use in source and binary forms, with or without
* modification, are permitted provided that the following conditions
* are met:
* 1. Redistributions of source code must retain the above copyright
* notice, this list of conditions and the following disclaimer.
* 2. Redistributions in binary form must reproduce the above copyright
* notice, this list of conditions and the following disclaimer in the
* documentation and/or other materials provided with the distribution.
* 3. All advertising materials mentioning features or use of this software
* must display the following acknowledgement:
* This product includes software developed by the University of
* California, Berkeley and its contributors.
* 4. Neither the name of the University nor the names of its contributors
* may be used to endorse or promote products derived from this software
* without specific prior written permission.
*
* THIS SOFTWARE IS PROVIDED BY THE REGENTS AND CONTRIBUTORS ``AS IS'' AND
* ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
* IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE
* ARE DISCLAIMED. IN NO EVENT SHALL THE REGENTS OR CONTRIBUTORS BE LIABLE
* FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL
* DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS
* OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION)
* HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT
* LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY
* OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF
* SUCH DAMAGE.
*
* K.C. Ng, with Z-S. Alex Liu, S. McDonald, P. Tang, W. Kahan.
* Revised on 5/10/85, 5/13/85, 6/14/85, 8/20/85, 8/27/85, 9/11/85.
*
* @(#)README 8.1 (Berkeley) 6/4/93
*/
******************************************************************************
* This is a description of the upgraded elementary functions (listed in 1). *
* Bessel functions (j0, j1, jn, y0, y1, yn), floor, and fabs passed over *
* from 4.2BSD without change except perhaps for the way floating point *
* exception is signaled on a VAX. Three lines that contain "errno" in erf.c*
* (error functions erf, erfc) have been deleted to prevent overriding the *
* system "errno". *
******************************************************************************
0. Total number of files: 40
IEEE/Makefile VAX/Makefile VAX/support.s erf.c lgamma.c
IEEE/atan2.c VAX/argred.s VAX/tan.s exp.c log.c
IEEE/cabs.c VAX/atan2.s acosh.c exp__E.c log10.c
IEEE/cbrt.c VAX/cabs.s asincos.c expm1.c log1p.c
IEEE/support.c VAX/cbrt.s asinh.c floor.c log__L.c
IEEE/trig.c VAX/infnan.s atan.c j0.c pow.c
Makefile VAX/sincos.s atanh.c j1.c sinh.c
README VAX/sqrt.s cosh.c jn.c tanh.c
1. Functions implemented :
(A). Standard elementary functions (total 22) :
acos(x) ...in file asincos.c
asin(x) ...in file asincos.c
atan(x) ...in file atan.c
atan2(x,y) ...in files IEEE/atan2.c, VAX/atan2.s
sin(x) ...in files IEEE/trig.c, VAX/sincos.s
cos(x) ...in files IEEE/trig.c, VAX/sincos.s
tan(x) ...in files IEEE/trig.c, VAX/tan.s
cabs(x,y) ...in files IEEE/cabs.c, VAX/cabs.s
hypot(x,y) ...in files IEEE/cabs.c, VAX/cabs.s
cbrt(x) ...in files IEEE/cbrt.c, VAX/cbrt.s
exp(x) ...in file exp.c
expm1(x):=exp(x)-1 ...in file expm1.c
log(x) ...in file log.c
log10(x) ...in file log10.c
log1p(x):=log(1+x) ...in file log1p.c
pow(x,y) ...in file pow.c
sinh(x) ...in file sinh.c
cosh(x) ...in file cosh.c
tanh(x) ...in file tanh.c
asinh(x) ...in file asinh.c
acosh(x) ...in file acosh.c
atanh(x) ...in file atanh.c
(B). Kernel functions :
exp__E(x,c) ...in file exp__E.c, used by expm1/exp/pow/cosh
log__L(s) ...in file log__L.c, used by log1p/log/pow
libm$argred ...in file VAX/argred.s, used by VAX version of sin/cos/tan
(C). System supported functions :
sqrt() ...in files IEEE/support.c, VAX/sqrt.s
drem() ...in files IEEE/support.c, VAX/support.s
finite() ...in files IEEE/support.c, VAX/support.s
logb() ...in files IEEE/support.c, VAX/support.s
scalb() ...in files IEEE/support.c, VAX/support.s
copysign() ...in files IEEE/support.c, VAX/support.s
rint() ...in file floor.c
Notes:
i. The codes in files ending with ".s" are written in VAX assembly
language. They are intended for VAX computers.
Files that end with ".c" are written in C. They are intended
for either a VAX or a machine that conforms to the IEEE
standard 754 for double precision floating-point arithmetic.
ii. On other than VAX or IEEE machines, run the original math
library, formerly "/usr/lib/libm.a", now "/usr/lib/libom.a", if
nothing better is available.
iii. The trigonometric functions sin/cos/tan/atan2 in files "VAX/sincos.s",
"VAX/tan.s" and "VAX/atan2.s" are different from those in
"IEEE/trig.c" and "IEEE/atan2.c". The VAX assembler code uses the
true value of pi to perform argument reduction, while the C code uses
a machine value of PI (see "IEEE/trig.c").
2. A computer system that conforms to IEEE standard 754 should provide
sqrt(x),
drem(x,p), (double precision remainder function)
copysign(x,y),
finite(x),
scalb(x,N),
logb(x) and
rint(x).
These functions are either required or recommended by the standard.
For convenience, a (slow) C implementation of these functions is
provided in the file "IEEE/support.c".
Warning: The functions in IEEE/support.c are somewhat machine dependent.
Some modifications may be necessary to run them on a different machine.
Currently, if compiled with a suitable flag, "IEEE/support.c" will work
on a National 32000, a Zilog 8000, a VAX, and a SUN (cf. the "Makefile"
in this directory). Invoke the C compiler thus:
cc -c -DVAX IEEE/support.c ... on a VAX, D-format
cc -c -DNATIONAL IEEE/support.c ... on a National 32000
cc -c IEEE/support.c ... on other IEEE machines,
we hope.
Notes:
1. Faster versions of "drem" and "sqrt" for IEEE double precision
(coded in C but intended for assembly language) are given at the
end of "IEEE/support.c" but commented out since they require certain
machine-dependent functions.
2. A fast VAX assembler version of the system supported functions
copysign(), logb(), scalb(), finite(), and drem() appears in file
"VAX/support.s". A fast VAX assembler version of sqrt() is in
file "VAX/sqrt.s".
3. Two formats are supported by all the standard elementary functions:
the VAX D-format (56-bit precision), and the IEEE double format
(53-bit precision). The cbrt() in "IEEE/cbrt.c" is for IEEE machines
only. The functions in files that end with ".s" are for VAX computers
only. The functions in files that end with ".c" (except "IEEE/cbrt.c")
are for VAX and IEEE machines. To use the VAX D-format, compile the code
with -DVAX; to use IEEE double format on various IEEE machines, see
"Makefile" in this directory).
Example:
cc -c -DVAX sin.c ... for VAX D-format
Warning: The values of floating-point constants used in the code are
given in both hexadecimal and decimal. The hexadecimal values
are the intended ones. The decimal values may be used provided
that the compiler converts from decimal to binary accurately
enough to produce the hexadecimal values shown. If the
conversion is inaccurate, then one must know the exact machine
representation of the constants and alter the assembly
language output from the compiler, or play tricks like
the following in a C program.
Example: to store the floating-point constant
p1= 2^-6 * .F83ABE67E1066A (Hexadecimal)
on a VAX in C, we use two longwords to store its
machine value and define p1 to be the double constant
at the location of these two longwords:
static long p1x[] = { 0x3abe3d78, 0x066a67e1};
#define p1 (*(double*)p1x)
Note: On a VAX, some functions have two codes. For example, cabs() has
one implementation in "IEEE/cabs.c", and another in "VAX/cabs.s".
In this case, the assembly language version is preferred.
4. Accuracy.
The errors in expm1(), log1p(), exp(), log(), cabs(), hypot()
and cbrt() are below 1 ULP (Unit in the Last Place).
The error in pow(x,y) grows with the size of y. Nevertheless,
for integers x and y, pow(x,y) returns the correct integer value
on all tested machines (VAX, SUN, NATIONAL, ZILOG), provided that
x to the power of y is representable exactly.
cosh, sinh, acosh, asinh, tanh, atanh and log10 have errors below
about 3 ULPs.
For trigonometric and inverse trigonometric functions:
Let [trig(x)] denote the value actually computed for trig(x),
1) Those codes using the machine's value PI (true pi rounded):
(source codes: IEEE/{trig.c,atan2.c}, asincos.c and atan.c)
The errors in [sin(x)], [cos(x)], and [atan(x)] are below
1 ULP compared with sin(x*pi/PI), cos(x*pi/PI), and
atan(x)*PI/pi respectively, where PI is the machine's
value of pi rounded. [tan(x)] returns tan(x*pi/PI) within
about 2 ULPs; [acos(x)], [asin(x)], and [atan2(y,x)]
return acos(x)*PI/pi, asin(x)*PI/pi, and atan2(y,x)*PI/pi
respectively to similar accuracy.
2) Those using true pi (for VAX D-format only):
(source codes: VAX/{sincos.s,tan.s,atan2.s}, asincos.c and
atan.c)
The errors in [sin(x)], [cos(x)], and [atan(x)] are below
1 ULP. [tan(x)], [atan2(y,x)], [acos(x)], and [asin(x)]
have errors below about 2 ULPs.
Here are the results of some test runs to find worst errors on
the VAX :
tan : 2.09 ULPs ...1,024,000 random arguments (machine PI)
sin : .861 ULPs ...1,024,000 random arguments (machine PI)
cos : .857 ULPs ...1,024,000 random arguments (machine PI)
(compared with tan, sin, cos of (x*pi/PI))
acos : 2.07 ULPs .....200,000 random arguments (machine PI)
asin : 2.06 ULPs .....200,000 random arguments (machine PI)
atan2 : 1.41 ULPs .....356,000 random arguments (machine PI)
atan : 0.86 ULPs ...1,536,000 random arguments (machine PI)
(compared with (PI/pi)*(atan, asin, acos, atan2 of x))
tan : 2.15 ULPs ...1,024,000 random arguments (true pi)
sin : .814 ULPs ...1,024,000 random arguments (true pi)
cos : .792 ULPs ...1,024,000 random arguments (true pi)
acos : 2.15 ULPs ...1,024,000 random arguments (true pi)
asin : 1.99 ULPs ...1,024,000 random arguments (true pi)
atan2 : 1.48 ULPs ...1,024,000 random arguments (true pi)
atan : .850 ULPs ...1,024,000 random arguments (true pi)
acosh : 3.30 ULPs .....512,000 random arguments
asinh : 1.58 ULPs .....512,000 random arguments
atanh : 1.71 ULPs .....512,000 random arguments
cosh : 1.23 ULPs .....768,000 random arguments
sinh : 1.93 ULPs ...1,024,000 random arguments
tanh : 2.22 ULPs ...1,024,000 random arguments
log10 : 1.74 ULPs ...1,536,000 random arguments
pow : 1.79 ULPs .....100,000 random arguments, 0 < x, y < 20.
exp : .768 ULPs ...1,156,000 random arguments
expm1 : .844 ULPs ...1,166,000 random arguments
log1p : .846 ULPs ...1,536,000 random arguments
log : .826 ULPs ...1,536,000 random arguments
cabs : .959 ULPs .....500,000 random arguments
cbrt : .666 ULPs ...5,120,000 random arguments
5. Speed.
Some functions coded in VAX assembly language (cabs(), hypot() and
sqrt()) are significantly faster than the corresponding ones in 4.2BSD.
In general, to improve performance, all functions in "IEEE/support.c"
should be written in assembly language and, whenever possible, should
be called via short subroutine calls.
6. j0, j1, jn.
The modifications to these routines were only in how an invalid
floating point operations is signaled.