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1994-12-19
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/*
* Copyright (c) 1985 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 condi-
tions
* are met:
* 1. Redistributions of source code must retain the above copy-
right
* notice, this list of conditions and the following dis-
claimer.
* 2. Redistributions in binary form must reproduce the above
copyright
* notice, this list of conditions and the following dis-
claimer in the
* documentation and/or other materials provided with the dis-
tribution.
* 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 PAR-
TICULAR 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 SUBSTI-
TUTE GOODS
* OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTER-
RUPTION)
* HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CON-
TRACT, STRICT
* LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING
IN ANY WAY
* OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSI-
BILITY 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 5.4 (Berkeley) 10/9/90
*/
******************************************************************************
* 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 overrid-
ing 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 arith-
metic.
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 func-
tions:
the VAX D-format (56-bit precision), and the IEEE double for-
mat
(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, com-
pile 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 hex-
adecimal 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 con-
stant
p1= 2^-6 * .F83ABE67E1066A (Hexadec-
imal)
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 pre-
ferred.
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. Nev-
ertheless,
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 func-
tions:
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}, asin-
cos.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 corre-
sponding ones in 4.2BSD.
In general, to improve performance, all functions in
"IEEE/support.c"
should be written in assembly language and, whenever pos-
sible, 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.