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Sun Oct 6 02:05:52 1996 Ulrich Drepper <drepper@cygnus.com> * posix/getopt.c: Add casts to prevent warnings. * posix/regex.c: Likewise. * math/Makefile (long-m-routines, long-m-yes): Define to correct values to make `long double' functions available. (distribute): Add $(long-m-yes:=.c). * sysdeps/libm-i387/s_ceill.S: Correct loading of return value. Use long double instruction. * sysdeps/libm-ieee754/e_atanhl.c (huge): Correct constant. * sysdeps/libm-ieee754/e_asinhl.c: Likewise. * sysdeps/libm-ieee754/s_tanhl.c: Likewise. * sysdeps/libm-ieee754/e_coshl.c: New file. * sysdeps/libm-ieee754/s_asinh.c [!NO_LONG_DOUBLE]: Define names for `long double' version as alias. * sysdeps/libm-ieee754/s_atan.c: Likewise. * sysdeps/libm-ieee754/s_cbrt.c: Likewise. * sysdeps/libm-ieee754/s_ceil.c: Likewise. * sysdeps/libm-ieee754/s_copysign.c: Likewise. * sysdeps/libm-ieee754/s_cos.c: Likewise. * sysdeps/libm-ieee754/s_erf.c: Likewise. * sysdeps/libm-ieee754/s_expm1.c: Likewise. * sysdeps/libm-ieee754/s_fabs.c: Likewise. * sysdeps/libm-ieee754/s_finite.c: Likewise. * sysdeps/libm-ieee754/s_floor.c: Likewise. * sysdeps/libm-ieee754/s_frexp.c: Likewise. * sysdeps/libm-ieee754/s_ilogb.c: Likewise. * sysdeps/libm-ieee754/s_ldexp.c: Likewise. * sysdeps/libm-ieee754/s_log1p.c: Likewise. * sysdeps/libm-ieee754/s_logb.c: Likewise. * sysdeps/libm-ieee754/s_modf.c: Likewise. * sysdeps/libm-ieee754/s_nextafter.c: Likewise. * sysdeps/libm-ieee754/s_rint.c: Likewise. * sysdeps/libm-ieee754/s_scalbn.c: Likewise. * sysdeps/libm-ieee754/s_significand.c: Likewise. * sysdeps/libm-ieee754/s_sin.c: Likewise. * sysdeps/libm-ieee754/s_tan.c: Likewise. * sysdeps/libm-ieee754/s_tanh.c: Likewise. * sysdeps/libm-ieee754/w_acos.c: Likewise. * sysdeps/libm-ieee754/w_acosh.c: Likewise. * sysdeps/libm-ieee754/w_sin.c: Likewise. * sysdeps/libm-ieee754/w_atan2.c: Likewise. * sysdeps/libm-ieee754/w_atanh.c: Likewise. * sysdeps/libm-ieee754/w_cabs.c: Likewise. * sysdeps/libm-ieee754/w_cosh.c: Likewise. * sysdeps/libm-ieee754/w_drem.c: Likewise. * sysdeps/libm-ieee754/w_exp.c: Likewise. * sysdeps/libm-ieee754/w_fmod.c: Likewise. * sysdeps/libm-ieee754/w_gamma.c: Likewise. * sysdeps/libm-ieee754/w_gamma_r.c: Likewise. * sysdeps/libm-ieee754/w_hypot.c: Likewise. * sysdeps/libm-ieee754/w_j0.c: Likewise. * sysdeps/libm-ieee754/w_j1.c: Likewise. * sysdeps/libm-ieee754/w_jn.c: Likewise. * sysdeps/libm-ieee754/w_lgamma.c: Likewise. * sysdeps/libm-ieee754/w_lgamma_r.c: Likewise. * sysdeps/libm-ieee754/w_log.c: Likewise. * sysdeps/libm-ieee754/w_log10.c: Likewise. * sysdeps/libm-ieee754/w_pow.c: Likewise. * sysdeps/libm-ieee754/w_remainder.c: Likewise. * sysdeps/libm-ieee754/w_scalb.c: Likewise. * sysdeps/libm-ieee754/w_sinh.c: Likewise. * sysdeps/libm-ieee754/w_sqrt.c: Likewise. Stub files for missing long double math functions. * sysdeps/stub/e_acoshl.c: New file. * sysdeps/stub/e_hypotl.c: New file. * sysdeps/stub/e_j0l.c: New file. * sysdeps/stub/e_j1l.c: New file. * sysdeps/stub/e_jnl.c: New file. * sysdeps/stub/e_lgammal_r.c: New file. * sysdeps/stub/e_powl.c: New file. * sysdeps/stub/e_rem_pio2l.c: New file. * sysdeps/stub/e_sinhl.c: New file. * sysdeps/stub/k_cosl.c: New file. * sysdeps/stub/k_rem_pio2l.c: New file. * sysdeps/stub/k_sinl.c: New file. * sysdeps/stub/k_tanl.c: New file. * sysdeps/stub/s_erfl.c: New file. * sysdeps/stub/s_expm1l.c: New file. * sysdeps/i386/__longjmp.S: Use PSEUDO_END macro to provide .size directive. * sysdeps/i386/bsd-_setjmp.S: Likewise. * sysdeps/i386/bsd-setjmp.S: Likewise. * sysdeps/i386/memchr.S: Likewise. * sysdeps/i386/memcmp.S: Likewise. * sysdeps/i386/setjmp.S: Likewise. * sysdeps/i386/stpcpy.S: Likewise. * sysdeps/i386/stpncpy.S: Likewise. * sysdeps/i386/strchr.S: Likewise. * sysdeps/i386/strcspn.S: Likewise. * sysdeps/i386/strpbrk.S: Likewise. * sysdeps/i386/strrchr.S: Likewise. * sysdeps/i386/strspn.S: Likewise. * sysdeps/i386/strtok.S: Likewise. * sysdeps/i386/i486/strcat.S: Likewise. * sysdeps/i386/i486/strlen.S: Likewise. * sysdeps/i386/i586/memset.S: Likewise. * sysdeps/i386/i586/strchr.S: Likewise. * sysdeps/i386/i586/strlen.S: Likewise. * sysdeps/libm-i387/e_acos.S: Likewise. * sysdeps/libm-i387/e_acosl.S: Likewise. * sysdeps/libm-i387/e_asin.S: Likewise. * sysdeps/libm-i387/e_asinl.S: Likewise. * sysdeps/libm-i387/e_atan2.S: Likewise. * sysdeps/libm-i387/e_atan2l.S: Likewise. * sysdeps/libm-i387/e_exp.S: Likewise. * sysdeps/libm-i387/e_expl.S: Likewise. * sysdeps/libm-i387/e_fmod.S: Likewise. * sysdeps/libm-i387/e_fmodl.S: Likewise. * sysdeps/libm-i387/e_log.S: Likewise. * sysdeps/libm-i387/e_log10.S: Likewise. * sysdeps/libm-i387/e_log10l.S: Likewise. * sysdeps/libm-i387/e_logl.S: Likewise. * sysdeps/libm-i387/e_remainder.S: Likewise. * sysdeps/libm-i387/e_remainderf.S: Likewise. * sysdeps/libm-i387/e_remainderl.S: Likewise. * sysdeps/libm-i387/e_scalb.S: Likewise. * sysdeps/libm-i387/e_scalbl.S: Likewise. * sysdeps/libm-i387/e_sqrt.S: Likewise. * sysdeps/libm-i387/e_sqrtf.S: Likewise. * sysdeps/libm-i387/e_sqrtl.S: Likewise. * sysdeps/libm-i387/s_atan.S: Likewise. * sysdeps/libm-i387/s_atanf.S: Likewise. * sysdeps/libm-i387/s_atanl.S: Likewise. * sysdeps/libm-i387/s_ceil.S: Likewise. * sysdeps/libm-i387/s_ceilf.S: Likewise. * sysdeps/libm-i387/s_ceill.S: Likewise. * sysdeps/libm-i387/s_copysign.S: Likewise. * sysdeps/libm-i387/s_copysignf.S: Likewise. * sysdeps/libm-i387/s_copysignl.S: Likewise. * sysdeps/libm-i387/s_cos.S: Likewise. * sysdeps/libm-i387/s_cosf.S: Likewise. * sysdeps/libm-i387/s_cosl.S: Likewise. * sysdeps/libm-i387/s_finite.S: Likewise. * sysdeps/libm-i387/s_finitef.S: Likewise. * sysdeps/libm-i387/s_finitel.S: Likewise. * sysdeps/libm-i387/s_floor.S: Likewise. * sysdeps/libm-i387/s_floorf.S: Likewise. * sysdeps/libm-i387/s_floorl.S: Likewise. * sysdeps/libm-i387/s_ilogb.S: Likewise. * sysdeps/libm-i387/s_ilogbf.S: Likewise. * sysdeps/libm-i387/s_ilogbl.S: Likewise. * sysdeps/libm-i387/s_log1p.S: Likewise. * sysdeps/libm-i387/s_log1pf.S: Likewise. * sysdeps/libm-i387/s_log1pl.S: Likewise. * sysdeps/libm-i387/s_logb.S: Likewise. * sysdeps/libm-i387/s_logbf.S: Likewise. * sysdeps/libm-i387/s_logbl.S: Likewise. * sysdeps/libm-i387/s_rint.S: Likewise. * sysdeps/libm-i387/s_rintf.S: Likewise. * sysdeps/libm-i387/s_rintl.S: Likewise. * sysdeps/libm-i387/s_scalbn.S: Likewise. * sysdeps/libm-i387/s_scalbnf.S: Likewise. * sysdeps/libm-i387/s_scalbnl.S: Likewise. * sysdeps/libm-i387/s_significand.S: Likewise. * sysdeps/libm-i387/s_significandf.S: Likewise. * sysdeps/libm-i387/s_significandl.S: Likewise. * sysdeps/libm-i387/s_sin.S: Likewise. * sysdeps/libm-i387/s_sinf.S: Likewise. * sysdeps/libm-i387/s_sinl.S: Likewise. * sysdeps/libm-i387/s_tan.S: Likewise. * sysdeps/libm-i387/s_tanf.S: Likewise. * sysdeps/libm-i387/s_tanl.S: Likewise. * sysdeps/unix/sysv/linux/i386/clone.S: Add .size directive. * sysdeps/unix/sysv/linux/i386/mmap.S. Likewise. * sysdeps/unix/sysv/linux/i386/socket.S. Likewise. * sysdeps/unix/sysv/linux/i386/sysdep.S. More compact .size line. * sysdeps/i386/sysdep.h (ASM_SIZE_DIRECTIVE): New macro. Used to provide .size directive on ELF systems. * sysdeps/unix/sysv/linux/i386/profil-counter.h: Include <sigcontext.h> and rename parameter type to sigcontext. * sysdeps/unix/sysv/linux/i386/sigcontext.h New file. * sysdeps/unix/sysv/linux/i386/syscall.S: From Linux 2.1 on negative values might occur as positive results. Test against -125 to decide for error or not. * sysdeps/unix/sysv/linux/i386/sysdep.h: Likewise for system calls. Thu Oct 3 21:07:58 1996 Andreas Schwab <schwab@issan.informatik.uni-dortmund.de> * elf/dl-open.c: Use _DYNAMIC instead of _dl_start to decide whether this is a statically linked program. The latter is now always defined. Fri Oct 4 02:08:10 1996 Bang Jun-Young <bangjy@nownuri.nowcom.co.kr> * po/ko.po: Update. Fri Oct 4 02:07:46 1996 Michel Robitaille <robitail@IRO.UMontreal.CA> * po/fr.po: New file. Fri Oct 4 05:04:52 1996 Ulrich Drepper <drepper@cygnus.com> * sysdeps/unix/sysv/linux/i386/sigcontext.h: New file. Add support for MD5 crypt replacement. * Makefile (subdirs): Add crypt. * shlib-versions: Add entry for libcrypt. * crypt/md5-crypt.c: New file. * crypt/md5.c: New file. * crypt/md5.h: New file. * sysdeps/generic/crypt-entry.c: New file. * sysdeps/generic/crypt.h: New file. Thu Oct 3 20:53:23 1996 Andreas Jaeger <aj@arthur.pfalz.de> * dirent/tst-seekdir.c: Provide correct prototypes. * io/test-utime.c: Likewise. * malloc/mallocbug.c: Likewise. * posix/testfnm.c: Likewise. * stdio-common/xbug.c: Likewise. * sysdeps/posix/cuserid.c: Likewise. Wed Oct 2 13:33:48 1996 Richard Henderson <rth@tamu.edu> Provide optimized string functions for Alpha processors. * sysdeps/alpha/bzero.S: New file. * sysdeps/alpha/memset.S: New file. * sysdeps/alpha/stpcpy.S: New file. * sysdeps/alpha/stpncpy.S: New file. * sysdeps/alpha/strcat.S: New file. * sysdeps/alpha/strchr.S: New file. * sysdeps/alpha/strcpy.S: New file. * sysdeps/alpha/strncat.S: New file. * sysdeps/alpha/strncpy.S: New file. * sysdeps/alpha/strrchr.S: New file. * sysdeps/alpha/stxcpy.S: New file. * sysdeps/alpha/stxncpy.S: New file. * sysdeps/alpha/Makefile [$(sudir)==string]: Add stxcpy and stxncpy to sysdep_routines. Minor correction. Wed Oct 2 13:41:48 1996 Ulrich Drepper <drepper@cygnus.com> * sysdeps/generic/strtok.c: Don't set stored pointer to NULL when notheing is found. This guarantees all subsequent calls behave the same. * sysdeps/generic/strtok_r.c: Likewise. Mon Sep 30 22:27:36 1996 Andreas Schwab <schwab@issan.informatik.uni-dortmund.de> * sysdeps/posix/tempname.c (__stdio_gen_tempname): Don't declare `i' as const. Mon Sep 30 22:38:29 1996 Richard Henderson <rth@tamu.edu> * sysdeps/generic/strtok.c: When we hit EOS, don't set olds to NULL immediately; we're going to get called one more time. * sysdeps/generic/strtok_r.c: Likewise. Tue Oct 1 09:12:21 1996 Ulrich Drepper <drepper@cygnus.com> * locale/programs/ld-time.c: Always write some bytes at the end of the file even if no `era's are specified. * stdio-common/tmpname.c (tmpbuf): Don't define as array of pointers. Tue Oct 1 01:43:11 1996 Ulrich Drepper <drepper@cygnus.com> * libio/iofclose.c: Move unlocking to right position. Mon Sep 30 03:08:34 1996 Richard Henderson <rth@tamu.edu> * misc/Makefile: Kill force-wrapper. * misc/force-wrapper.c: Delete. * elf/eval.c: Add <string.h>. * gmon/sys/gmon_out.h: Nonsense like `char foo[sizeof(char*)]' followed by `bcopy(&ptr, &foo, sizeof(foo))' helps portability and binary compatibility not at all. Better to do `char *foo' followed by `foo = ptr' as it is much cleaner. (struct gmon_hdr): Fix version. (struct gmon_hist_hdr): Fix low_pc, high_pc, hist_size, prof_rate. (struct gmon_cg_arc_record): Fix from_pc, self_pc, count. * gmon/gmon.c: De-ansidecl-ify. (write_hist): De-bcopy-fy. (write_call_graph): Likewise. * gmon/mcount.c: Assume _MCOUNT_DECL does the entire declaration. * sysdeps/alpha/machine-gmon.h: Update _MCOUNT_DECL. * sysdeps/generic/machine-gmon.h: Likewise. * resolv/netdb.h: Add __set_h_errno to mimic __set_errno. * nss/getXXent_r.c: Use __set_h_errno to set h_errno in all funcs. * nss/nss_dns/dns-network.c: Likewise. * resolv/getunamaddr.c: Likewise. Also use __set_errno where needed. * resolv/getnetnamadr.c: Likewise. * resolv/res_debug.c: Likewise. * resolv/res_mkquery.c: Likewise. * resolv/res_query.c: Likewise. * libio/clearerr_u.c: It's not necessary to define __ protected function so don't do it. * libio/feof_u.c, libio/ferror_u.c, libio/fputc_u.c, libio/getc_u.c, libio/getchar_u.c, libio/iofflush_u.c: Likewise. * libio/fgetc.c: Avoid a warning by casting _IO_funlockfile for the cleanup registrar. * libio/fputc.c, libio/freopen.c, libio/fseek.c, libio/getc.c, libio/getchar.c, libio/iofclose.c, libio/iofflush.c, libio/iofgetpos.c, libio/iofgets.c, libio/iofputs.c, libio/iofread.c, libio/iofsetpos.c, libio/ioftell.c, libio/iofwrite.c, libio/iogetdelim.c, libio/iogets.c, libio/ioputs.c, libio/iosetbuffer.c, libio/iosetvbuf.c, libio/ioungetc.c, libio/putc.c, libio/putchar.c, libio/rewind.c: Likewise. * locale/programs/ld-ctype.c: Include <alloca.h>. * login/login_tty.c: Get login_tty prototype from <utmp.h>. * posix/sys/types.h: Change #defines to typedefs. This is looking forward to more comprehensive namespace cleanups for C++. * posix/unistd.h: Likewise for ssize_t. * pwd/getpw.c: Prototype and rename getpw -> __getpw and add a weak alias. * resolv/base64.c: Don't do `for (NULL; ...)' as it causes `statement with no effect' warnings. * resolv/inet_neta.c: Include <string.h> for strlen. * stdio-common/getline.c: Undef ssize_t before libio redefinition. * stdio-common/tstgetln.c: Same. De-ansidecl-ify. * stdio-common/vfprintf.c: Same for va_list. * stdio-common/vfscanf.c: Same. * stdio-common/reg-printf.c: Prototype the __ function. * stdio-common/scanf.c [USE_IN_LIBIO]: Include <libioP.h> for _IO_vscanf declaration. * string/tester.c: Tsk, tsk. Don't cast pointers to int, but to unsigned long for the health of 64-bit systems. * sunrpc/svc_udp.c: Define MAX only if we don't have it yet. Many system's <sys/param.h> pulls this in. * sysdeps/generic/dl-cache.c: Kill the bottom half of the patch doubled file. * sysdeps/generic/pty.c (forkpty): Get login_tty decl from <utmp.h> instead of defining it locally. * sysdeps/posix/sigvec.c: Prototype wrapper_handler and convert_mask. * sysdeps/stub/chflags.c: Prototype chflags. * sysdeps/stub/fchflags.c: Prototype fchflags. * sysdeps/stub/sstk.c: Prototype sstk. * sysdeps/unix/alpha/sysdep.S: Add strong alias from errno variable to __errno. Update __syscall_error to store to both the global and per-thread variables. * sysdeps/unix/sysv/linux/init-first.c: Change include of init-first.h from "" to <> as "" does not search the include path. Prototype _dl_start. Sun Sep 29 14:41:17 1996 Andreas Schwab <schwab@issan.informatik.uni-dortmund.de> * sysdeps/unix/Makefile ($(common-objpfx)s-proto.d): Make it a pattern rule to get the right stem. Sat Sep 28 01:30:06 1996 Andreas Schwab <schwab@issan.informatik.uni-dortmund.de> * Makefile ($(objpfx)version-info.h): Fix typo in previous change. Sat Sep 28 00:44:38 1996 Andreas Schwab <schwab@issan.informatik.uni-dortmund.de> * sysdeps/posix/euidaccess.c (__set_errno): Fix typo. Sat Sep 28 00:40:38 1996 Andreas Schwab <schwab@issan.informatik.uni-dortmund.de> * malloc/free.c (_free_internal): Don't call free recursively, use the internal functions instead. Sat Sep 28 00:23:20 1996 Andreas Schwab <schwab@issan.informatik.uni-dortmund.de> * sysdeps/posix/tempname.c (__stdio_gen_tempname) [USE_IN_LIBIO && _IO_MTSAFE_IO]: Allocate and initialize the file lock. Fri Sep 27 23:58:17 1996 Andreas Schwab <schwab@issan.informatik.uni-dortmund.de> * catgets/gencat.c (main): Print version information on stdout, not stderr. * db/makedb.c: Likewise. * locale/programs/locale.c: Likewise. * locale/programs/localedef.c: Likewise. Mon Sep 30 00:14:26 1996 Ulrich Drepper <drepper@cygnus.com> Add implementation of netgroup. * inet/Makefile (distributes): Add netgroup.h. (routines): Add getnetgrent.c and getnetgrent_r.c. * inet/getnetgrent.c: New file. * inet/getnetgrent_r.c: New file. * inet/netgroup.h: New file. * netgroup.h: New file. * nsswitch.h: New file. * nss/Makefile (databases): Add getgrp. * nss/databases.def: Add netgroup. * nss/db-Makefile: Add rules for netgroup. * nss/netgrp-lookup.c: New file. * nss/nss.h: Add new status value NSS_STATUS_RETURN which will always return without a check of the transition table. * nss/nsswitch.h (struct service_user): Extend actions array to five elements. * nss/nsswitch.c (__nss_next): Allow NSS_STATUS_RETURN as legal status value. (nss_parse_service_list): Initialize action for NSS_STATUS_RETURN to NSS_ACTION_RETURN. * nss/nss_db/db-netgrp.c: New file. * nss/nss_files/files-netgrp.c: New file. * nss/nsswitch.conf: Add entry for netgroup. * resolv/netdb.h: Add prototypes for netgroup functions. * nss/nss_db/db-XXX.c: Use `enum nss_status' type when status value is used. Sun Sep 29 12:55:58 1996 Ulrich Drepper <drepper@cygnus.com> * sysdeps/unix/sysv/linux/syscalls.list: Change funciton name for _llseek to _llseek (not llseek). Reported by Matthias Urlichs <smurf@smurf.noris.de>. Sun Sep 29 05:34:32 1996 Ulrich Drepper <drepper@cygnus.com> * inet/rcmd.c: Remove definition of MIN. Sat Sep 28 14:22:51 1996 Ulrich Drepper <drepper@cygnus.com> * dirent/readdir_r.c: Removed. We indeed need a real implementation. * sysdeps/unix/readdir_r.c: New file. * sysdeps/unix/readdir.c: Remove check for correct parameter values. Let them simply die. * manual/filesys.texi: Document readdir_r. * sysdeps/gnu/errlist.awk: Define _sys_errlist as `const char* const'. Proposed by Per Abrahamsen <abraham@dina.kvl.dk>.
321 lines
11 KiB
C
321 lines
11 KiB
C
/* @(#)s_erf.c 5.1 93/09/24 */
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/*
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* ====================================================
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* Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
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*
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* Developed at SunPro, a Sun Microsystems, Inc. business.
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* Permission to use, copy, modify, and distribute this
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* software is freely granted, provided that this notice
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* is preserved.
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* ====================================================
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*/
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#if defined(LIBM_SCCS) && !defined(lint)
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static char rcsid[] = "$NetBSD: s_erf.c,v 1.8 1995/05/10 20:47:05 jtc Exp $";
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#endif
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/* double erf(double x)
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* double erfc(double x)
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* x
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* 2 |\
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* erf(x) = --------- | exp(-t*t)dt
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* sqrt(pi) \|
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* 0
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*
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* erfc(x) = 1-erf(x)
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* Note that
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* erf(-x) = -erf(x)
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* erfc(-x) = 2 - erfc(x)
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*
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* Method:
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* 1. For |x| in [0, 0.84375]
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* erf(x) = x + x*R(x^2)
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* erfc(x) = 1 - erf(x) if x in [-.84375,0.25]
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* = 0.5 + ((0.5-x)-x*R) if x in [0.25,0.84375]
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* where R = P/Q where P is an odd poly of degree 8 and
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* Q is an odd poly of degree 10.
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* -57.90
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* | R - (erf(x)-x)/x | <= 2
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*
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*
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* Remark. The formula is derived by noting
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* erf(x) = (2/sqrt(pi))*(x - x^3/3 + x^5/10 - x^7/42 + ....)
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* and that
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* 2/sqrt(pi) = 1.128379167095512573896158903121545171688
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* is close to one. The interval is chosen because the fix
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* point of erf(x) is near 0.6174 (i.e., erf(x)=x when x is
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* near 0.6174), and by some experiment, 0.84375 is chosen to
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* guarantee the error is less than one ulp for erf.
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*
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* 2. For |x| in [0.84375,1.25], let s = |x| - 1, and
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* c = 0.84506291151 rounded to single (24 bits)
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* erf(x) = sign(x) * (c + P1(s)/Q1(s))
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* erfc(x) = (1-c) - P1(s)/Q1(s) if x > 0
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* 1+(c+P1(s)/Q1(s)) if x < 0
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* |P1/Q1 - (erf(|x|)-c)| <= 2**-59.06
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* Remark: here we use the taylor series expansion at x=1.
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* erf(1+s) = erf(1) + s*Poly(s)
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* = 0.845.. + P1(s)/Q1(s)
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* That is, we use rational approximation to approximate
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* erf(1+s) - (c = (single)0.84506291151)
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* Note that |P1/Q1|< 0.078 for x in [0.84375,1.25]
|
|
* where
|
|
* P1(s) = degree 6 poly in s
|
|
* Q1(s) = degree 6 poly in s
|
|
*
|
|
* 3. For x in [1.25,1/0.35(~2.857143)],
|
|
* erfc(x) = (1/x)*exp(-x*x-0.5625+R1/S1)
|
|
* erf(x) = 1 - erfc(x)
|
|
* where
|
|
* R1(z) = degree 7 poly in z, (z=1/x^2)
|
|
* S1(z) = degree 8 poly in z
|
|
*
|
|
* 4. For x in [1/0.35,28]
|
|
* erfc(x) = (1/x)*exp(-x*x-0.5625+R2/S2) if x > 0
|
|
* = 2.0 - (1/x)*exp(-x*x-0.5625+R2/S2) if -6<x<0
|
|
* = 2.0 - tiny (if x <= -6)
|
|
* erf(x) = sign(x)*(1.0 - erfc(x)) if x < 6, else
|
|
* erf(x) = sign(x)*(1.0 - tiny)
|
|
* where
|
|
* R2(z) = degree 6 poly in z, (z=1/x^2)
|
|
* S2(z) = degree 7 poly in z
|
|
*
|
|
* Note1:
|
|
* To compute exp(-x*x-0.5625+R/S), let s be a single
|
|
* precision number and s := x; then
|
|
* -x*x = -s*s + (s-x)*(s+x)
|
|
* exp(-x*x-0.5626+R/S) =
|
|
* exp(-s*s-0.5625)*exp((s-x)*(s+x)+R/S);
|
|
* Note2:
|
|
* Here 4 and 5 make use of the asymptotic series
|
|
* exp(-x*x)
|
|
* erfc(x) ~ ---------- * ( 1 + Poly(1/x^2) )
|
|
* x*sqrt(pi)
|
|
* We use rational approximation to approximate
|
|
* g(s)=f(1/x^2) = log(erfc(x)*x) - x*x + 0.5625
|
|
* Here is the error bound for R1/S1 and R2/S2
|
|
* |R1/S1 - f(x)| < 2**(-62.57)
|
|
* |R2/S2 - f(x)| < 2**(-61.52)
|
|
*
|
|
* 5. For inf > x >= 28
|
|
* erf(x) = sign(x) *(1 - tiny) (raise inexact)
|
|
* erfc(x) = tiny*tiny (raise underflow) if x > 0
|
|
* = 2 - tiny if x<0
|
|
*
|
|
* 7. Special case:
|
|
* erf(0) = 0, erf(inf) = 1, erf(-inf) = -1,
|
|
* erfc(0) = 1, erfc(inf) = 0, erfc(-inf) = 2,
|
|
* erfc/erf(NaN) is NaN
|
|
*/
|
|
|
|
|
|
#include "math.h"
|
|
#include "math_private.h"
|
|
|
|
#ifdef __STDC__
|
|
static const double
|
|
#else
|
|
static double
|
|
#endif
|
|
tiny = 1e-300,
|
|
half= 5.00000000000000000000e-01, /* 0x3FE00000, 0x00000000 */
|
|
one = 1.00000000000000000000e+00, /* 0x3FF00000, 0x00000000 */
|
|
two = 2.00000000000000000000e+00, /* 0x40000000, 0x00000000 */
|
|
/* c = (float)0.84506291151 */
|
|
erx = 8.45062911510467529297e-01, /* 0x3FEB0AC1, 0x60000000 */
|
|
/*
|
|
* Coefficients for approximation to erf on [0,0.84375]
|
|
*/
|
|
efx = 1.28379167095512586316e-01, /* 0x3FC06EBA, 0x8214DB69 */
|
|
efx8= 1.02703333676410069053e+00, /* 0x3FF06EBA, 0x8214DB69 */
|
|
pp0 = 1.28379167095512558561e-01, /* 0x3FC06EBA, 0x8214DB68 */
|
|
pp1 = -3.25042107247001499370e-01, /* 0xBFD4CD7D, 0x691CB913 */
|
|
pp2 = -2.84817495755985104766e-02, /* 0xBF9D2A51, 0xDBD7194F */
|
|
pp3 = -5.77027029648944159157e-03, /* 0xBF77A291, 0x236668E4 */
|
|
pp4 = -2.37630166566501626084e-05, /* 0xBEF8EAD6, 0x120016AC */
|
|
qq1 = 3.97917223959155352819e-01, /* 0x3FD97779, 0xCDDADC09 */
|
|
qq2 = 6.50222499887672944485e-02, /* 0x3FB0A54C, 0x5536CEBA */
|
|
qq3 = 5.08130628187576562776e-03, /* 0x3F74D022, 0xC4D36B0F */
|
|
qq4 = 1.32494738004321644526e-04, /* 0x3F215DC9, 0x221C1A10 */
|
|
qq5 = -3.96022827877536812320e-06, /* 0xBED09C43, 0x42A26120 */
|
|
/*
|
|
* Coefficients for approximation to erf in [0.84375,1.25]
|
|
*/
|
|
pa0 = -2.36211856075265944077e-03, /* 0xBF6359B8, 0xBEF77538 */
|
|
pa1 = 4.14856118683748331666e-01, /* 0x3FDA8D00, 0xAD92B34D */
|
|
pa2 = -3.72207876035701323847e-01, /* 0xBFD7D240, 0xFBB8C3F1 */
|
|
pa3 = 3.18346619901161753674e-01, /* 0x3FD45FCA, 0x805120E4 */
|
|
pa4 = -1.10894694282396677476e-01, /* 0xBFBC6398, 0x3D3E28EC */
|
|
pa5 = 3.54783043256182359371e-02, /* 0x3FA22A36, 0x599795EB */
|
|
pa6 = -2.16637559486879084300e-03, /* 0xBF61BF38, 0x0A96073F */
|
|
qa1 = 1.06420880400844228286e-01, /* 0x3FBB3E66, 0x18EEE323 */
|
|
qa2 = 5.40397917702171048937e-01, /* 0x3FE14AF0, 0x92EB6F33 */
|
|
qa3 = 7.18286544141962662868e-02, /* 0x3FB2635C, 0xD99FE9A7 */
|
|
qa4 = 1.26171219808761642112e-01, /* 0x3FC02660, 0xE763351F */
|
|
qa5 = 1.36370839120290507362e-02, /* 0x3F8BEDC2, 0x6B51DD1C */
|
|
qa6 = 1.19844998467991074170e-02, /* 0x3F888B54, 0x5735151D */
|
|
/*
|
|
* Coefficients for approximation to erfc in [1.25,1/0.35]
|
|
*/
|
|
ra0 = -9.86494403484714822705e-03, /* 0xBF843412, 0x600D6435 */
|
|
ra1 = -6.93858572707181764372e-01, /* 0xBFE63416, 0xE4BA7360 */
|
|
ra2 = -1.05586262253232909814e+01, /* 0xC0251E04, 0x41B0E726 */
|
|
ra3 = -6.23753324503260060396e+01, /* 0xC04F300A, 0xE4CBA38D */
|
|
ra4 = -1.62396669462573470355e+02, /* 0xC0644CB1, 0x84282266 */
|
|
ra5 = -1.84605092906711035994e+02, /* 0xC067135C, 0xEBCCABB2 */
|
|
ra6 = -8.12874355063065934246e+01, /* 0xC0545265, 0x57E4D2F2 */
|
|
ra7 = -9.81432934416914548592e+00, /* 0xC023A0EF, 0xC69AC25C */
|
|
sa1 = 1.96512716674392571292e+01, /* 0x4033A6B9, 0xBD707687 */
|
|
sa2 = 1.37657754143519042600e+02, /* 0x4061350C, 0x526AE721 */
|
|
sa3 = 4.34565877475229228821e+02, /* 0x407B290D, 0xD58A1A71 */
|
|
sa4 = 6.45387271733267880336e+02, /* 0x40842B19, 0x21EC2868 */
|
|
sa5 = 4.29008140027567833386e+02, /* 0x407AD021, 0x57700314 */
|
|
sa6 = 1.08635005541779435134e+02, /* 0x405B28A3, 0xEE48AE2C */
|
|
sa7 = 6.57024977031928170135e+00, /* 0x401A47EF, 0x8E484A93 */
|
|
sa8 = -6.04244152148580987438e-02, /* 0xBFAEEFF2, 0xEE749A62 */
|
|
/*
|
|
* Coefficients for approximation to erfc in [1/.35,28]
|
|
*/
|
|
rb0 = -9.86494292470009928597e-03, /* 0xBF843412, 0x39E86F4A */
|
|
rb1 = -7.99283237680523006574e-01, /* 0xBFE993BA, 0x70C285DE */
|
|
rb2 = -1.77579549177547519889e+01, /* 0xC031C209, 0x555F995A */
|
|
rb3 = -1.60636384855821916062e+02, /* 0xC064145D, 0x43C5ED98 */
|
|
rb4 = -6.37566443368389627722e+02, /* 0xC083EC88, 0x1375F228 */
|
|
rb5 = -1.02509513161107724954e+03, /* 0xC0900461, 0x6A2E5992 */
|
|
rb6 = -4.83519191608651397019e+02, /* 0xC07E384E, 0x9BDC383F */
|
|
sb1 = 3.03380607434824582924e+01, /* 0x403E568B, 0x261D5190 */
|
|
sb2 = 3.25792512996573918826e+02, /* 0x40745CAE, 0x221B9F0A */
|
|
sb3 = 1.53672958608443695994e+03, /* 0x409802EB, 0x189D5118 */
|
|
sb4 = 3.19985821950859553908e+03, /* 0x40A8FFB7, 0x688C246A */
|
|
sb5 = 2.55305040643316442583e+03, /* 0x40A3F219, 0xCEDF3BE6 */
|
|
sb6 = 4.74528541206955367215e+02, /* 0x407DA874, 0xE79FE763 */
|
|
sb7 = -2.24409524465858183362e+01; /* 0xC03670E2, 0x42712D62 */
|
|
|
|
#ifdef __STDC__
|
|
double __erf(double x)
|
|
#else
|
|
double __erf(x)
|
|
double x;
|
|
#endif
|
|
{
|
|
int32_t hx,ix,i;
|
|
double R,S,P,Q,s,y,z,r;
|
|
GET_HIGH_WORD(hx,x);
|
|
ix = hx&0x7fffffff;
|
|
if(ix>=0x7ff00000) { /* erf(nan)=nan */
|
|
i = ((u_int32_t)hx>>31)<<1;
|
|
return (double)(1-i)+one/x; /* erf(+-inf)=+-1 */
|
|
}
|
|
|
|
if(ix < 0x3feb0000) { /* |x|<0.84375 */
|
|
if(ix < 0x3e300000) { /* |x|<2**-28 */
|
|
if (ix < 0x00800000)
|
|
return 0.125*(8.0*x+efx8*x); /*avoid underflow */
|
|
return x + efx*x;
|
|
}
|
|
z = x*x;
|
|
r = pp0+z*(pp1+z*(pp2+z*(pp3+z*pp4)));
|
|
s = one+z*(qq1+z*(qq2+z*(qq3+z*(qq4+z*qq5))));
|
|
y = r/s;
|
|
return x + x*y;
|
|
}
|
|
if(ix < 0x3ff40000) { /* 0.84375 <= |x| < 1.25 */
|
|
s = fabs(x)-one;
|
|
P = pa0+s*(pa1+s*(pa2+s*(pa3+s*(pa4+s*(pa5+s*pa6)))));
|
|
Q = one+s*(qa1+s*(qa2+s*(qa3+s*(qa4+s*(qa5+s*qa6)))));
|
|
if(hx>=0) return erx + P/Q; else return -erx - P/Q;
|
|
}
|
|
if (ix >= 0x40180000) { /* inf>|x|>=6 */
|
|
if(hx>=0) return one-tiny; else return tiny-one;
|
|
}
|
|
x = fabs(x);
|
|
s = one/(x*x);
|
|
if(ix< 0x4006DB6E) { /* |x| < 1/0.35 */
|
|
R=ra0+s*(ra1+s*(ra2+s*(ra3+s*(ra4+s*(
|
|
ra5+s*(ra6+s*ra7))))));
|
|
S=one+s*(sa1+s*(sa2+s*(sa3+s*(sa4+s*(
|
|
sa5+s*(sa6+s*(sa7+s*sa8)))))));
|
|
} else { /* |x| >= 1/0.35 */
|
|
R=rb0+s*(rb1+s*(rb2+s*(rb3+s*(rb4+s*(
|
|
rb5+s*rb6)))));
|
|
S=one+s*(sb1+s*(sb2+s*(sb3+s*(sb4+s*(
|
|
sb5+s*(sb6+s*sb7))))));
|
|
}
|
|
z = x;
|
|
SET_LOW_WORD(z,0);
|
|
r = __ieee754_exp(-z*z-0.5625)*__ieee754_exp((z-x)*(z+x)+R/S);
|
|
if(hx>=0) return one-r/x; else return r/x-one;
|
|
}
|
|
weak_alias (__erf, erf)
|
|
|
|
#ifdef __STDC__
|
|
double __erfc(double x)
|
|
#else
|
|
double __erfc(x)
|
|
double x;
|
|
#endif
|
|
{
|
|
int32_t hx,ix;
|
|
double R,S,P,Q,s,y,z,r;
|
|
GET_HIGH_WORD(hx,x);
|
|
ix = hx&0x7fffffff;
|
|
if(ix>=0x7ff00000) { /* erfc(nan)=nan */
|
|
/* erfc(+-inf)=0,2 */
|
|
return (double)(((u_int32_t)hx>>31)<<1)+one/x;
|
|
}
|
|
|
|
if(ix < 0x3feb0000) { /* |x|<0.84375 */
|
|
if(ix < 0x3c700000) /* |x|<2**-56 */
|
|
return one-x;
|
|
z = x*x;
|
|
r = pp0+z*(pp1+z*(pp2+z*(pp3+z*pp4)));
|
|
s = one+z*(qq1+z*(qq2+z*(qq3+z*(qq4+z*qq5))));
|
|
y = r/s;
|
|
if(hx < 0x3fd00000) { /* x<1/4 */
|
|
return one-(x+x*y);
|
|
} else {
|
|
r = x*y;
|
|
r += (x-half);
|
|
return half - r ;
|
|
}
|
|
}
|
|
if(ix < 0x3ff40000) { /* 0.84375 <= |x| < 1.25 */
|
|
s = fabs(x)-one;
|
|
P = pa0+s*(pa1+s*(pa2+s*(pa3+s*(pa4+s*(pa5+s*pa6)))));
|
|
Q = one+s*(qa1+s*(qa2+s*(qa3+s*(qa4+s*(qa5+s*qa6)))));
|
|
if(hx>=0) {
|
|
z = one-erx; return z - P/Q;
|
|
} else {
|
|
z = erx+P/Q; return one+z;
|
|
}
|
|
}
|
|
if (ix < 0x403c0000) { /* |x|<28 */
|
|
x = fabs(x);
|
|
s = one/(x*x);
|
|
if(ix< 0x4006DB6D) { /* |x| < 1/.35 ~ 2.857143*/
|
|
R=ra0+s*(ra1+s*(ra2+s*(ra3+s*(ra4+s*(
|
|
ra5+s*(ra6+s*ra7))))));
|
|
S=one+s*(sa1+s*(sa2+s*(sa3+s*(sa4+s*(
|
|
sa5+s*(sa6+s*(sa7+s*sa8)))))));
|
|
} else { /* |x| >= 1/.35 ~ 2.857143 */
|
|
if(hx<0&&ix>=0x40180000) return two-tiny;/* x < -6 */
|
|
R=rb0+s*(rb1+s*(rb2+s*(rb3+s*(rb4+s*(
|
|
rb5+s*rb6)))));
|
|
S=one+s*(sb1+s*(sb2+s*(sb3+s*(sb4+s*(
|
|
sb5+s*(sb6+s*sb7))))));
|
|
}
|
|
z = x;
|
|
SET_LOW_WORD(z,0);
|
|
r = __ieee754_exp(-z*z-0.5625)*
|
|
__ieee754_exp((z-x)*(z+x)+R/S);
|
|
if(hx>0) return r/x; else return two-r/x;
|
|
} else {
|
|
if(hx>0) return tiny*tiny; else return two-tiny;
|
|
}
|
|
}
|
|
weak_alias (__erfc, erfc)
|
|
#ifdef NO_LONG_DOUBLE
|
|
strong_alias (__erfc, __erfcl)
|
|
weak_alias (__erfc, erfcl)
|
|
#endif
|