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1.1 root 1: /*
2: * Copyright (c) 1985 Regents of the University of California.
3: * All rights reserved.
4: *
5: * Redistribution and use in source and binary forms are permitted
6: * provided that the above copyright notice and this paragraph are
7: * duplicated in all such forms and that any documentation,
8: * advertising materials, and other materials related to such
9: * distribution and use acknowledge that the software was developed
10: * by the University of California, Berkeley. The name of the
11: * University may not be used to endorse or promote products derived
12: * from this software without specific prior written permission.
13: * THIS SOFTWARE IS PROVIDED ``AS IS'' AND WITHOUT ANY EXPRESS OR
14: * IMPLIED WARRANTIES, INCLUDING, WITHOUT LIMITATION, THE IMPLIED
15: * WARRANTIES OF MERCHANTIBILITY AND FITNESS FOR A PARTICULAR PURPOSE.
16: *
17: * All recipients should regard themselves as participants in an ongoing
18: * research project and hence should feel obligated to report their
19: * experiences (good or bad) with these elementary function codes, using
20: * the sendbug(8) program, to the authors.
21: *
22: * K.C. Ng, with Z-S. Alex Liu, S. McDonald, P. Tang, W. Kahan.
23: * Revised on 5/10/85, 5/13/85, 6/14/85, 8/20/85, 8/27/85, 9/11/85.
24: *
25: * @(#)README 5.3 (Berkeley) 6/30/88
26: */
27:
28: ******************************************************************************
29: * This is a description of the upgraded elementary functions (listed in 1). *
30: * Bessel functions (j0, j1, jn, y0, y1, yn), floor, and fabs passed over *
31: * from 4.2BSD without change except perhaps for the way floating point *
32: * exception is signaled on a VAX. Three lines that contain "errno" in erf.c*
33: * (error functions erf, erfc) have been deleted to prevent overriding the *
34: * system "errno". *
35: ******************************************************************************
36:
37: 0. Total number of files: 40
38:
39: IEEE/Makefile VAX/Makefile VAX/support.s erf.c lgamma.c
40: IEEE/atan2.c VAX/argred.s VAX/tan.s exp.c log.c
41: IEEE/cabs.c VAX/atan2.s acosh.c exp__E.c log10.c
42: IEEE/cbrt.c VAX/cabs.s asincos.c expm1.c log1p.c
43: IEEE/support.c VAX/cbrt.s asinh.c floor.c log__L.c
44: IEEE/trig.c VAX/infnan.s atan.c j0.c pow.c
45: Makefile VAX/sincos.s atanh.c j1.c sinh.c
46: README VAX/sqrt.s cosh.c jn.c tanh.c
47:
48: 1. Functions implemented :
49: (A). Standard elementary functions (total 22) :
50: acos(x) ...in file asincos.c
51: asin(x) ...in file asincos.c
52: atan(x) ...in file atan.c
53: atan2(x,y) ...in files IEEE/atan2.c, VAX/atan2.s
54: sin(x) ...in files IEEE/trig.c, VAX/sincos.s
55: cos(x) ...in files IEEE/trig.c, VAX/sincos.s
56: tan(x) ...in files IEEE/trig.c, VAX/tan.s
57: cabs(x,y) ...in files IEEE/cabs.c, VAX/cabs.s
58: hypot(x,y) ...in files IEEE/cabs.c, VAX/cabs.s
59: cbrt(x) ...in files IEEE/cbrt.c, VAX/cbrt.s
60: exp(x) ...in file exp.c
61: expm1(x):=exp(x)-1 ...in file expm1.c
62: log(x) ...in file log.c
63: log10(x) ...in file log10.c
64: log1p(x):=log(1+x) ...in file log1p.c
65: pow(x,y) ...in file pow.c
66: sinh(x) ...in file sinh.c
67: cosh(x) ...in file cosh.c
68: tanh(x) ...in file tanh.c
69: asinh(x) ...in file asinh.c
70: acosh(x) ...in file acosh.c
71: atanh(x) ...in file atanh.c
72:
73: (B). Kernel functions :
74: exp__E(x,c) ...in file exp__E.c, used by expm1/exp/pow/cosh
75: log__L(s) ...in file log__L.c, used by log1p/log/pow
76: libm$argred ...in file VAX/argred.s, used by VAX version of sin/cos/tan
77:
78: (C). System supported functions :
79: sqrt() ...in files IEEE/support.c, VAX/sqrt.s
80: drem() ...in files IEEE/support.c, VAX/support.s
81: finite() ...in files IEEE/support.c, VAX/support.s
82: logb() ...in files IEEE/support.c, VAX/support.s
83: scalb() ...in files IEEE/support.c, VAX/support.s
84: copysign() ...in files IEEE/support.c, VAX/support.s
85: rint() ...in file floor.c
86:
87:
88: Notes:
89: i. The codes in files ending with ".s" are written in VAX assembly
90: language. They are intended for VAX computers.
91:
92: Files that end with ".c" are written in C. They are intended
93: for either a VAX or a machine that conforms to the IEEE
94: standard 754 for double precision floating-point arithmetic.
95:
96: ii. On other than VAX or IEEE machines, run the original math
97: library, formerly "/usr/lib/libm.a", now "/usr/lib/libom.a", if
98: nothing better is available.
99:
100: iii. The trigonometric functions sin/cos/tan/atan2 in files "VAX/sincos.s",
101: "VAX/tan.s" and "VAX/atan2.s" are different from those in
102: "IEEE/trig.c" and "IEEE/atan2.c". The VAX assembler code uses the
103: true value of pi to perform argument reduction, while the C code uses
104: a machine value of PI (see "IEEE/trig.c").
105:
106:
107: 2. A computer system that conforms to IEEE standard 754 should provide
108: sqrt(x),
109: drem(x,p), (double precision remainder function)
110: copysign(x,y),
111: finite(x),
112: scalb(x,N),
113: logb(x) and
114: rint(x).
115: These functions are either required or recommended by the standard.
116: For convenience, a (slow) C implementation of these functions is
117: provided in the file "IEEE/support.c".
118:
119: Warning: The functions in IEEE/support.c are somewhat machine dependent.
120: Some modifications may be necessary to run them on a different machine.
121: Currently, if compiled with a suitable flag, "IEEE/support.c" will work
122: on a National 32000, a Zilog 8000, a VAX, and a SUN (cf. the "Makefile"
123: in this directory). Invoke the C compiler thus:
124:
125: cc -c -DVAX IEEE/support.c ... on a VAX, D-format
126: cc -c -DNATIONAL IEEE/support.c ... on a National 32000
127: cc -c IEEE/support.c ... on other IEEE machines,
128: we hope.
129:
130: Notes:
131: 1. Faster versions of "drem" and "sqrt" for IEEE double precision
132: (coded in C but intended for assembly language) are given at the
133: end of "IEEE/support.c" but commented out since they require certain
134: machine-dependent functions.
135:
136: 2. A fast VAX assembler version of the system supported functions
137: copysign(), logb(), scalb(), finite(), and drem() appears in file
138: "VAX/support.s". A fast VAX assembler version of sqrt() is in
139: file "VAX/sqrt.s".
140:
141: 3. Two formats are supported by all the standard elementary functions:
142: the VAX D-format (56-bit precision), and the IEEE double format
143: (53-bit precision). The cbrt() in "IEEE/cbrt.c" is for IEEE machines
144: only. The functions in files that end with ".s" are for VAX computers
145: only. The functions in files that end with ".c" (except "IEEE/cbrt.c")
146: are for VAX and IEEE machines. To use the VAX D-format, compile the code
147: with -DVAX; to use IEEE double format on various IEEE machines, see
148: "Makefile" in this directory).
149:
150: Example:
151: cc -c -DVAX sin.c ... for VAX D-format
152:
153: Warning: The values of floating-point constants used in the code are
154: given in both hexadecimal and decimal. The hexadecimal values
155: are the intended ones. The decimal values may be used provided
156: that the compiler converts from decimal to binary accurately
157: enough to produce the hexadecimal values shown. If the
158: conversion is inaccurate, then one must know the exact machine
159: representation of the constants and alter the assembly
160: language output from the compiler, or play tricks like
161: the following in a C program.
162:
163: Example: to store the floating-point constant
164:
165: p1= 2^-6 * .F83ABE67E1066A (Hexadecimal)
166:
167: on a VAX in C, we use two longwords to store its
168: machine value and define p1 to be the double constant
169: at the location of these two longwords:
170:
171: static long p1x[] = { 0x3abe3d78, 0x066a67e1};
172: #define p1 (*(double*)p1x)
173:
174: Note: On a VAX, some functions have two codes. For example, cabs() has
175: one implementation in "IEEE/cabs.c", and another in "VAX/cabs.s".
176: In this case, the assembly language version is preferred.
177:
178:
179: 4. Accuracy.
180:
181: The errors in expm1(), log1p(), exp(), log(), cabs(), hypot()
182: and cbrt() are below 1 ULP (Unit in the Last Place).
183:
184: The error in pow(x,y) grows with the size of y. Nevertheless,
185: for integers x and y, pow(x,y) returns the correct integer value
186: on all tested machines (VAX, SUN, NATIONAL, ZILOG), provided that
187: x to the power of y is representable exactly.
188:
189: cosh, sinh, acosh, asinh, tanh, atanh and log10 have errors below
190: about 3 ULPs.
191:
192: For trigonometric and inverse trigonometric functions:
193:
194: Let [trig(x)] denote the value actually computed for trig(x),
195:
196: 1) Those codes using the machine's value PI (true pi rounded):
197: (source codes: IEEE/{trig.c,atan2.c}, asincos.c and atan.c)
198:
199: The errors in [sin(x)], [cos(x)], and [atan(x)] are below
200: 1 ULP compared with sin(x*pi/PI), cos(x*pi/PI), and
201: atan(x)*PI/pi respectively, where PI is the machine's
202: value of pi rounded. [tan(x)] returns tan(x*pi/PI) within
203: about 2 ULPs; [acos(x)], [asin(x)], and [atan2(y,x)]
204: return acos(x)*PI/pi, asin(x)*PI/pi, and atan2(y,x)*PI/pi
205: respectively to similar accuracy.
206:
207:
208: 2) Those using true pi (for VAX D-format only):
209: (source codes: VAX/{sincos.s,tan.s,atan2.s}, asincos.c and
210: atan.c)
211:
212: The errors in [sin(x)], [cos(x)], and [atan(x)] are below
213: 1 ULP. [tan(x)], [atan2(y,x)], [acos(x)], and [asin(x)]
214: have errors below about 2 ULPs.
215:
216:
217: Here are the results of some test runs to find worst errors on
218: the VAX :
219:
220: tan : 2.09 ULPs ...1,024,000 random arguments (machine PI)
221: sin : .861 ULPs ...1,024,000 random arguments (machine PI)
222: cos : .857 ULPs ...1,024,000 random arguments (machine PI)
223: (compared with tan, sin, cos of (x*pi/PI))
224:
225: acos : 2.07 ULPs .....200,000 random arguments (machine PI)
226: asin : 2.06 ULPs .....200,000 random arguments (machine PI)
227: atan2 : 1.41 ULPs .....356,000 random arguments (machine PI)
228: atan : 0.86 ULPs ...1,536,000 random arguments (machine PI)
229: (compared with (PI/pi)*(atan, asin, acos, atan2 of x))
230:
231: tan : 2.15 ULPs ...1,024,000 random arguments (true pi)
232: sin : .814 ULPs ...1,024,000 random arguments (true pi)
233: cos : .792 ULPs ...1,024,000 random arguments (true pi)
234: acos : 2.15 ULPs ...1,024,000 random arguments (true pi)
235: asin : 1.99 ULPs ...1,024,000 random arguments (true pi)
236: atan2 : 1.48 ULPs ...1,024,000 random arguments (true pi)
237: atan : .850 ULPs ...1,024,000 random arguments (true pi)
238:
239: acosh : 3.30 ULPs .....512,000 random arguments
240: asinh : 1.58 ULPs .....512,000 random arguments
241: atanh : 1.71 ULPs .....512,000 random arguments
242: cosh : 1.23 ULPs .....768,000 random arguments
243: sinh : 1.93 ULPs ...1,024,000 random arguments
244: tanh : 2.22 ULPs ...1,024,000 random arguments
245: log10 : 1.74 ULPs ...1,536,000 random arguments
246: pow : 1.79 ULPs .....100,000 random arguments, 0 < x, y < 20.
247:
248: exp : .768 ULPs ...1,156,000 random arguments
249: expm1 : .844 ULPs ...1,166,000 random arguments
250: log1p : .846 ULPs ...1,536,000 random arguments
251: log : .826 ULPs ...1,536,000 random arguments
252: cabs : .959 ULPs .....500,000 random arguments
253: cbrt : .666 ULPs ...5,120,000 random arguments
254:
255:
256: 5. Speed.
257:
258: Some functions coded in VAX assembly language (cabs(), hypot() and
259: sqrt()) are significantly faster than the corresponding ones in 4.2BSD.
260: In general, to improve performance, all functions in "IEEE/support.c"
261: should be written in assembly language and, whenever possible, should
262: be called via short subroutine calls.
263:
264:
265: 6. j0, j1, jn.
266:
267: The modifications to these routines were only in how an invalid
268: floating point operations is signaled.
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