**ref:** c8b187f9db503465460d8bafa2d7a758bbcf59ca

**dir:** /libmath/fdlibm/readme/

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