inferno-os/libmath/fdlibm

********************************
* Announcing FDLIBM Version 5 *
********************************
============================================================
FDLIBM
============================================================
(developed at SunSoft, a Sun Microsystems, Inc. business.)

What's new in FDLIBM 5.2?
BUGS FIXED
1. Little endian bug in frexp (affect only little endian machine):
in file s_frexp.c, last line of program frexp before exit
*(int*)&x = hx;
should read
*(n0+(int*)&x) = hx;

2. jn(-1,x) is three times larger that the actual answer:
in file e_jn.c, the line
sign = 1 - ((n&1)<<2);
should read
sign = 1 - ((n&1)<<1);

3. Compiler failure on non-standard code
J.T. Conklin found that gcc optimizing out the manipulation of doubles
via pointer bashing of the form
double x = 0;
*(((int*)&x)+n0)=0x7fff0000;
foo(x);
C experts confirmed that the behavior of *(((int*)&x)+n0)=0x7fff0000
is undefined. By replacing n0 with a constant 0 or 1, the GCC "knows"
that the assignment is modifying the double, and "does the right thing."
Thus, in FDLIBM 5.2, we replace n0 with a constant and use a macro
__HI() and __LO() with #ifdef __LITTLE_ENDIAN to avoid the above problem.

4. Performance issue on rem_pio2
An attempt to speed up the argument reduction in the trig function is the
consider pi/4 < x < 3pi/4 a special case. This was done in the file
e_rem_pio2.c


FDLIBM (Freely Distributable LIBM) is a C math library
for machines that support IEEE 754 floating-point arithmetic.
In this release, only double precision is supported.

FDLIBM is intended to provide a reasonably portable (see
assumptions below), reference quality (below one ulp for
major functions like sin,cos,exp,log) math library
(libm.a). For a copy of FDLIBM, please send a message "send index from fdlibm"
to netlib@research.att.com.

--------------
1. ASSUMPTIONS
--------------
FDLIBM (double precision version) assumes:
a. IEEE 754 style (if not precise compliance) arithmetic;
b. 32 bit 2's complement integer arithmetic;
c. Each double precision floating-point number must be in IEEE 754
double format, and that each number can be retrieved as two 32-bit
integers through the using of pointer bashing as in the example
below:

Example: let y = 2.0
double fp number y: 2.0
IEEE double format: 0x4000000000000000

Referencing y as two integers:
*(int*)&y,*(1+(int*)&y) = {0x40000000,0x0} (on sparc)
{0x0,0x40000000} (on 386)

Note: Four macros are defined in fdlibm.h to handle this kind of
retrieving:

__HI(x) the high part of a double x
(sign,exponent,the first 21 significant bits)
__LO(x) the least 32 significant bits of x
__HIp(x) same as __HI except that the argument is a pointer
to a double
__LOp(x) same as __LO except that the argument is a pointer
to a double

To ensure obtaining correct ordering, one must define __LITTLE_ENDIAN
during compilation for little endian machine (like 386,486). The
default is big endian.

If the behavior of pointer bashing is undefined, one may hack on the
macro in fdlibm.h.

d. IEEE exceptions may trigger "signals" as is common in Unix
implementations.

-------------------
2. EXCEPTION CASES
-------------------
All exception cases in the FDLIBM functions will be mapped
to one of the following four exceptions:

+-huge*huge, +-tiny*tiny, +-1.0/0.0, +-0.0/0.0
(overflow) (underflow) (divided-by-zero) (invalid)

For example, log(0) is a singularity and is thus mapped to
-1.0/0.0 = -infinity.
That is, FDLIBM's log will compute -one/zero and return the
computed value. On an IEEE machine, this will trigger the
divided-by-zero exception and a negative infinity is returned by
default.

Similarly, exp(-huge) will be mapped to tiny*tiny to generate
an underflow signal.


--------------------------------
3. STANDARD CONFORMANCE WRAPPER
--------------------------------
The default FDLIBM functions (compiled with -D_IEEE_LIBM flag)
are in "IEEE spirit" (i.e., return the most reasonable result in
floating-point arithmetic). If one wants FDLIBM to comply with
standards like SVID, X/OPEN, or POSIX/ANSI, then one can
create a multi-standard compliant FDLIBM. In this case, each
function in FDLIBM is actually a standard compliant wrapper
function.

File organization:
1. For FDLIBM's kernel (internal) function,
File name Entry point
---------------------------
k_sin.c __kernel_sin
k_tan.c __kernel_tan
---------------------------
2. For functions that have no standards conflict
File name Entry point
---------------------------
s_sin.c sin
s_erf.c erf
---------------------------
3. Ieee754 core functions
File name Entry point
---------------------------
e_exp.c __ieee754_exp
e_sinh.c __ieee754_sinh
---------------------------
4. Wrapper functions
File name Entry point
---------------------------
w_exp.c exp
w_sinh.c sinh
---------------------------

Wrapper functions will twist the result of the ieee754
function to comply to the standard specified by the value
of _LIB_VERSION
if _LIB_VERSION = _IEEE_, return the ieee754 result;
if _LIB_VERSION = _SVID_, return SVID result;
if _LIB_VERSION = _XOPEN_, return XOPEN result;
if _LIB_VERSION = _POSIX_, return POSIX/ANSI result.
(These are macros, see fdlibm.h for their definition.)


--------------------------------
4. HOW TO CREATE FDLIBM's libm.a
--------------------------------
There are two types of libm.a. One is IEEE only, and the other is
multi-standard compliant (supports IEEE,XOPEN,POSIX/ANSI,SVID).

To create the IEEE only libm.a, use
make "CFLAGS = -D_IEEE_LIBM"
This will create an IEEE libm.a, which is smaller in size, and
somewhat faster.

To create a multi-standard compliant libm, use
make "CFLAGS = -D_IEEE_MODE" --- multi-standard fdlibm: default
to IEEE
make "CFLAGS = -D_XOPEN_MODE" --- multi-standard fdlibm: default
to X/OPEN
make "CFLAGS = -D_POSIX_MODE" --- multi-standard fdlibm: default
to POSIX/ANSI
make "CFLAGS = -D_SVID3_MODE" --- multi-standard fdlibm: default
to SVID


Here is how one makes a SVID compliant libm.
Make the library by
make "CFLAGS = -D_SVID3_MODE".
The libm.a of FDLIBM will be multi-standard compliant and
_LIB_VERSION is initialized to the value _SVID_ .

example1:
---------
main()
{
double y0();
printf("y0(1e300) = %1.20e\n",y0(1e300));
exit(0);
}

% cc example1.c libm.a
% a.out
y0: TLOSS error
y0(1e300) = 0.00000000000000000000e+00


It is possible to change the default standard in multi-standard
fdlibm. Here is an example of how to do it:
example2:
---------
#include "fdlibm.h" /* must include FDLIBM's fdlibm.h */
main()
{
double y0();
_LIB_VERSION = _IEEE_;
printf("IEEE: y0(1e300) = %1.20e\n",y0(1e300));
_LIB_VERSION = _XOPEN_;
printf("XOPEN y0(1e300) = %1.20e\n",y0(1e300));
_LIB_VERSION = _POSIX_;
printf("POSIX y0(1e300) = %1.20e\n",y0(1e300));
_LIB_VERSION = _SVID_;
printf("SVID y0(1e300) = %1.20e\n",y0(1e300));
exit(0);
}

% cc example2.c libm.a
% a.out
IEEE: y0(1e300) = -1.36813604503424810557e-151
XOPEN y0(1e300) = 0.00000000000000000000e+00
POSIX y0(1e300) = 0.00000000000000000000e+00
y0: TLOSS error
SVID y0(1e300) = 0.00000000000000000000e+00

Note: Here _LIB_VERSION is a global variable. If global variables
are forbidden, then one should modify fdlibm.h to change
_LIB_VERSION to be a global constant. In this case, one
may not change the value of _LIB_VERSION as in example2.

---------------------------
5. NOTES ON PORTING FDLIBM
---------------------------
Care must be taken when installing FDLIBM over existing
libm.a.
All co-existing function prototypes must agree, otherwise
users will encounter mysterious failures.

So far, the only known likely conflict is the declaration
of the IEEE recommended function scalb:

double scalb(double,double) (1) SVID3 defined
double scalb(double,int) (2) IBM,DEC,...

FDLIBM follows Sun definition and use (1) as default.
If one's existing libm.a uses (2), then one may raise
the flags _SCALB_INT during the compilation of FDLIBM
to get the correct function prototype.
(E.g., make "CFLAGS = -D_IEEE_LIBM -D_SCALB_INT".)
NOTE that if -D_SCALB_INT is raised, it won't be SVID3
conformant.

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6. PROBLEMS ?
--------------
Please send comments and bug report to:
fdlibm-comments@sunpro.eng.sun.com