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***************************************************************************
*                                                                         * 
* Copyright (c) 1985 Regents of the University of California.             *
*                                                                         * 
* Use and reproduction of this software are granted  in  accordance  with *
* the terms and conditions specified in  the  Berkeley  Software  License *
* Agreement (in particular, this entails acknowledgement of the programs' *
* source, and inclusion of this notice) with the additional understanding *
* that  all  recipients  should regard themselves as participants  in  an *
* ongoing  research  project and hence should  feel  obligated  to report *
* their  experiences (good or bad) with these elementary function  codes, *
* using "sendbug 4bsd-bugs@BERKELEY", to the authors.                     *
*                                                                         *
* 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	1.6 (Berkeley) 9/12/85; 1.3 (ucb.elefunt) %G% */

NB.  The machine-independent Version 7 math library found in 4.2BSD
     is now /usr/lib/libom.a.  To compile with those routines use -lom.

******************************************************************************
*  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.