/* * zmath - declarations for extended precision integer arithmetic * * Copyright (C) 1999-2006 David I. Bell * * Calc is open software; you can redistribute it and/or modify it under * the terms of the version 2.1 of the GNU Lesser General Public License * as published by the Free Software Foundation. * * Calc is distributed in the hope that it will be useful, but WITHOUT * ANY WARRANTY; without even the implied warranty of MERCHANTABILITY * or FITNESS FOR A PARTICULAR PURPOSE. See the GNU Lesser General * Public License for more details. * * A copy of version 2.1 of the GNU Lesser General Public License is * distributed with calc under the filename COPYING-LGPL. You should have * received a copy with calc; if not, write to Free Software Foundation, Inc. * 59 Temple Place, Suite 330, Boston, MA 02111-1307, USA. * * @(#) $Revision: 29.18 $ * @(#) $Id: zmath.h,v 29.18 2006/12/15 16:20:04 chongo Exp $ * @(#) $Source: /usr/local/src/cmd/calc/RCS/zmath.h,v $ * * Under source code control: 1993/07/30 19:42:48 * File existed as early as: 1993 * * Share and enjoy! :-) http://www.isthe.com/chongo/tech/comp/calc/ */ /* * Data structure declarations for extended precision integer arithmetic. * The assumption made is that a long is 32 bits and shorts are 16 bits, * and longs must be addressible on word boundaries. */ #if !defined(__ZMATH_H__) #define __ZMATH_H__ #if defined(CALC_SRC) /* if we are building from the calc source tree */ # include "win32dll.h" # include "alloc.h" # include "endian_calc.h" # include "longbits.h" # include "byteswap.h" # include "have_stdlib.h" #else # include # include # include # include # include # include #endif #ifdef HAVE_STDLIB_H # include #endif #ifndef ALLOCTEST # define freeh(p) { if (((void *)p != (void *)_zeroval_) && \ ((void *)p != (void *)_oneval_)) free((void *)p); } #endif #if !defined(TRUE) #define TRUE ((BOOL) 1) /* booleans */ #endif #if !defined(FALSE) #define FALSE ((BOOL) 0) #endif /* * NOTE: FULL must be twice the storage size of a HALF * HALF must be BASEB bits long */ #if defined(HAVE_B64) #define BASEB 32 /* use base 2^32 */ typedef USB32 HALF; /* unit of number storage */ typedef SB32 SHALF; /* signed HALF */ typedef USB64 FULL; /* double unit of number storage */ typedef SB64 SFULL; /* signed FULL */ #define SWAP_HALF_IN_B64(dest, src) SWAP_B32_IN_B64(dest, src) #define SWAP_HALF_IN_B32(dest, src) (*(dest) = *(src)) #define SWAP_HALF_IN_FULL(dest, src) SWAP_B32_IN_B64(dest, src) #define SWAP_HALF_IN_HASH(dest, src) SWAP_B16_IN_HASH(dest, src) #define SWAP_HALF_IN_FLAG(dest, src) SWAP_B16_IN_FLAG(dest, src) #define SWAP_HALF_IN_BOOL(dest, src) SWAP_B16_IN_BOOL(dest, src) #define SWAP_HALF_IN_LEN(dest, src) SWAP_B16_IN_LEN(dest, src) #define SWAP_B32_IN_FULL(dest, src) SWAP_B32_IN_B64(dest, src) #define SWAP_B16_IN_FULL(dest, src) SWAP_B16_IN_B64(dest, src) #define SWAP_B16_IN_HALF(dest, src) SWAP_B16_IN_B32(dest, src) #define SWAP_B8_IN_FULL(dest, src) SWAP_B8_IN_B64(dest, src) #define SWAP_B8_IN_HALF(dest, src) SWAP_B8_IN_B32(dest, src) #else #define BASEB 16 /* use base 2^16 */ typedef USB16 HALF; /* unit of number storage */ typedef SB16 SHALF; /* signed HALF */ typedef USB32 FULL; /* double unit of number storage */ typedef SB32 SFULL; /* signed FULL */ #define SWAP_HALF_IN_B64(dest, src) SWAP_B16_IN_B64(dest, src) #define SWAP_HALF_IN_B32(dest, src) SWAP_B16_IN_B32(dest, src) #define SWAP_HALF_IN_FULL(dest, src) SWAP_B16_IN_B32(dest, src) #define SWAP_HALF_IN_HASH(dest, src) SWAP_B16_IN_HASH(dest, src) #define SWAP_HALF_IN_FLAG(dest, src) SWAP_B16_IN_FLAG(dest, src) #define SWAP_HALF_IN_BOOL(dest, src) SWAP_B16_IN_BOOL(dest, src) #define SWAP_HALF_IN_LEN(dest, src) SWAP_B16_IN_LEN(dest, src) #define SWAP_B32_IN_FULL(dest, src) (*(dest) = *(src)) #define SWAP_B16_IN_FULL(dest, src) SWAP_B16_IN_B32(dest, src) #define SWAP_B16_IN_HALF(dest, src) (*(dest) = *(src)) #define SWAP_B8_IN_FULL(dest, src) SWAP_B8_IN_B32(dest, src) #define SWAP_B8_IN_HALF(dest, src) SWAP_B8_IN_B16(dest, src) #endif #define BASE ((FULL)1< /\../\ * * Note that the \'s above are not back-slashing escape characters. * They are literal ASCII backslash 0x5c characters. * * The effect of this virgin initial value is the same as starting * with 0 and pre-pending those 32 characters onto the data being * hashed. * * Yes, even with this non-zero virgin value there is a set of data * that will result in a zero hash value. Worse, appending any * about of zero bytes will continue to produce a zero hash value. * But that would happen with any initial value so long as the * hash of the initial was the `inverse' of the virgin prefix string. * * But then again for any hash function, there exists sets of data * which that the hash of every member is the same value. That is * life with many to few mapping functions. All we do here is to * prevent sets whose members consist of 0 or more bytes of 0's from * being such an awkward set. * * And yes, someone can figure out what the magic 'inverse' of the * 32 ASCII character are ... but this hash function is NOT intended * to be a cryptographic hash function, just a fast and reasonably * good hash function. */ #define FNV1_32_BASIS ((QCKHASH)(0x811c9dc5)) /* * The largest power of 10 we will compute for our decimal conversion * internal constants is: 10^(2^TEN_MAX). */ #define TEN_MAX 31 /* 10^2^31 requires about 1.66 * 2^29 bytes */ /* * LEN storage size must be <= FULL storage size */ #define MAXLEN ((LEN) 0x7fffffff >> 3) /* longest value allowed */ #define MAXREDC 64 /* number of entries in REDC cache */ #define SQ_ALG2 3388 /* size for alternative squaring */ #define MUL_ALG2 1780 /* size for alternative multiply */ #define POW_ALG2 176 /* size for using REDC for powers */ /* old REDC_ALG2 was 5/4 of POW_ALG2, so we will keep the same ratio */ #define REDC_ALG2 220 /* size for using alternative REDC */ typedef union { FULL ivalue; struct { HALF Svalue1; HALF Svalue2; } sis; } SIUNION; #if !defined(LITTLE_ENDIAN) #define LITTLE_ENDIAN 1234 /* Least Significant Byte first */ #endif #if !defined(BIG_ENDIAN) #define BIG_ENDIAN 4321 /* Most Significant Byte first */ #endif /* PDP_ENDIAN - LSB in word, MSW in long is not supported */ #if CALC_BYTE_ORDER == LITTLE_ENDIAN # define silow sis.Svalue1 /* low order half of full value */ # define sihigh sis.Svalue2 /* high order half of full value */ #else # if CALC_BYTE_ORDER == BIG_ENDIAN # define silow sis.Svalue2 /* low order half of full value */ # define sihigh sis.Svalue1 /* high order half of full value */ # else /\oo/\ CALC_BYTE_ORDER must be BIG_ENDIAN or LITTLE_ENDIAN /\oo/\ !!! # endif #endif typedef struct { HALF *v; /* pointer to array of values */ LEN len; /* number of values in array */ BOOL sign; /* sign, nonzero is negative */ } ZVALUE; /* * Function prototypes for integer math routines. */ extern DLL HALF * alloc(LEN len); #ifdef ALLOCTEST extern DLL void freeh(HALF *); #endif /* * Input, output, and conversion routines. */ extern DLL void zcopy(ZVALUE z, ZVALUE *res); extern DLL void itoz(long i, ZVALUE *res); extern DLL void utoz(FULL i, ZVALUE *res); extern DLL void stoz(SFULL i, ZVALUE *res); extern DLL void str2z(char *s, ZVALUE *res); extern DLL long ztoi(ZVALUE z); extern DLL FULL ztou(ZVALUE z); extern DLL SFULL ztos(ZVALUE z); extern DLL void zprintval(ZVALUE z, long decimals, long width); extern DLL void zprintx(ZVALUE z, long width); extern DLL void zprintb(ZVALUE z, long width); extern DLL void zprinto(ZVALUE z, long width); extern DLL void fitzprint(ZVALUE, long, long); /* * Basic numeric routines. */ extern DLL void zmuli(ZVALUE z, long n, ZVALUE *res); extern DLL long zdivi(ZVALUE z, long n, ZVALUE *res); extern DLL long zmodi(ZVALUE z, long n); extern DLL void zadd(ZVALUE z1, ZVALUE z2, ZVALUE *res); extern DLL void zsub(ZVALUE z1, ZVALUE z2, ZVALUE *res); extern DLL void zmul(ZVALUE z1, ZVALUE z2, ZVALUE *res); extern DLL long zdiv(ZVALUE z1, ZVALUE z2, ZVALUE *res, ZVALUE *rem, long R); extern DLL long zquo(ZVALUE z1, ZVALUE z2, ZVALUE *res, long R); extern DLL long zmod(ZVALUE z1, ZVALUE z2, ZVALUE *rem, long R); extern DLL void zequo(ZVALUE z1, ZVALUE z2, ZVALUE *res); extern DLL BOOL zdivides(ZVALUE z1, ZVALUE z2); extern DLL void zor(ZVALUE z1, ZVALUE z2, ZVALUE *res); extern DLL void zand(ZVALUE z1, ZVALUE z2, ZVALUE *res); extern DLL void zxor(ZVALUE z1, ZVALUE z2, ZVALUE *res); extern DLL void zandnot(ZVALUE z1, ZVALUE z2, ZVALUE *res); extern DLL long zpopcnt(ZVALUE z, int bitval); extern DLL void zshift(ZVALUE z, long n, ZVALUE *res); extern DLL void zsquare(ZVALUE z, ZVALUE *res); extern DLL long zlowbit(ZVALUE z); extern DLL LEN zhighbit(ZVALUE z); extern DLL void zbitvalue(long n, ZVALUE *res); extern DLL BOOL zisset(ZVALUE z, long n); extern DLL BOOL zisonebit(ZVALUE z); extern DLL BOOL zisallbits(ZVALUE z); extern DLL FLAG ztest(ZVALUE z); extern DLL FLAG zrel(ZVALUE z1, ZVALUE z2); extern DLL FLAG zabsrel(ZVALUE z1, ZVALUE z2); extern DLL BOOL zcmp(ZVALUE z1, ZVALUE z2); /* * More complicated numeric functions. */ extern DLL FULL uugcd(FULL i1, FULL i2); extern DLL long iigcd(long i1, long i2); extern DLL void zgcd(ZVALUE z1, ZVALUE z2, ZVALUE *res); extern DLL void zlcm(ZVALUE z1, ZVALUE z2, ZVALUE *res); extern DLL void zreduce(ZVALUE z1, ZVALUE z2, ZVALUE *z1res, ZVALUE *z2res); extern DLL void zfact(ZVALUE z, ZVALUE *dest); extern DLL void zperm(ZVALUE z1, ZVALUE z2, ZVALUE *res); extern DLL int zcomb(ZVALUE z1, ZVALUE z2, ZVALUE *res); extern DLL FLAG zjacobi(ZVALUE z1, ZVALUE z2); extern DLL void zfib(ZVALUE z, ZVALUE *res); extern DLL void zpowi(ZVALUE z1, ZVALUE z2, ZVALUE *res); extern DLL void ztenpow(long power, ZVALUE *res); extern DLL void zpowermod(ZVALUE z1, ZVALUE z2, ZVALUE z3, ZVALUE *res); extern DLL BOOL zmodinv(ZVALUE z1, ZVALUE z2, ZVALUE *res); extern DLL BOOL zrelprime(ZVALUE z1, ZVALUE z2); extern DLL long zlog(ZVALUE z1, ZVALUE z2); extern DLL long zlog10(ZVALUE z, BOOL *was_10_power); extern DLL long zdivcount(ZVALUE z1, ZVALUE z2); extern DLL long zfacrem(ZVALUE z1, ZVALUE z2, ZVALUE *rem); extern DLL long zgcdrem(ZVALUE z1, ZVALUE z2, ZVALUE *res); extern DLL long zdigits(ZVALUE z1); extern DLL long zdigit(ZVALUE z1, long n); extern DLL FLAG zsqrt(ZVALUE z1, ZVALUE *dest, long R); extern DLL void zroot(ZVALUE z1, ZVALUE z2, ZVALUE *dest); extern DLL BOOL zissquare(ZVALUE z); extern DLL void zhnrmod(ZVALUE v, ZVALUE h, ZVALUE zn, ZVALUE zr, ZVALUE *res); /* * Prime related functions. */ extern DLL FLAG zisprime(ZVALUE z); extern DLL FULL znprime(ZVALUE z); extern DLL FULL next_prime(FULL v); extern DLL FULL zpprime(ZVALUE z); extern DLL void zpfact(ZVALUE z, ZVALUE *dest); extern DLL BOOL zprimetest(ZVALUE z, long count, ZVALUE skip); extern DLL BOOL zredcprimetest(ZVALUE z, long count, ZVALUE skip); extern DLL BOOL znextcand(ZVALUE z1, long count, ZVALUE skip, ZVALUE res, ZVALUE mod, ZVALUE *cand); extern DLL BOOL zprevcand(ZVALUE z1, long count, ZVALUE skip, ZVALUE res, ZVALUE mod, ZVALUE *cand); extern DLL FULL zlowfactor(ZVALUE z, long count); extern DLL FLAG zfactor(ZVALUE z1, ZVALUE z2, ZVALUE *res); extern DLL long zpix(ZVALUE z1); extern DLL void zlcmfact(ZVALUE z, ZVALUE *dest); /* * Misc misc functions. :-) */ extern DLL void zsquaremod(ZVALUE z1, ZVALUE z2, ZVALUE *res); extern DLL void zminmod(ZVALUE z1, ZVALUE z2, ZVALUE *res); extern DLL BOOL zcmpmod(ZVALUE z1, ZVALUE z2, ZVALUE z3); extern DLL void zio_init(void); /* * These functions are for internal use only. */ extern DLL void ztrim(ZVALUE *z); extern DLL void zshiftr(ZVALUE z, long n); extern DLL void zshiftl(ZVALUE z, long n); extern DLL HALF *zalloctemp(LEN len); /* * Modulo arithmetic definitions. * Structure holding state of REDC initialization. * Multiple instances of this structure can be used allowing * calculations with more than one modulus at the same time. * Len of zero means the structure is not initialized. */ typedef struct { LEN len; /* number of words in binary modulus */ ZVALUE mod; /* modulus REDC is computing with */ ZVALUE inv; /* inverse of modulus in binary modulus */ ZVALUE one; /* REDC format for the number 1 */ } REDC; extern DLL REDC *zredcalloc(ZVALUE z1); extern DLL void zredcfree(REDC *rp); extern DLL void zredcencode(REDC *rp, ZVALUE z1, ZVALUE *res); extern DLL void zredcdecode(REDC *rp, ZVALUE z1, ZVALUE *res); extern DLL void zredcmul(REDC *rp, ZVALUE z1, ZVALUE z2, ZVALUE *res); extern DLL void zredcsquare(REDC *rp, ZVALUE z1, ZVALUE *res); extern DLL void zredcpower(REDC *rp, ZVALUE z1, ZVALUE z2, ZVALUE *res); /* * macro expansions to speed this thing up */ #define ziseven(z) (!(*(z).v & 0x1)) #define zisodd(z) (*(z).v & 0x1) #define ziszero(z) ((*(z).v == 0) && ((z).len == 1)) #define zisneg(z) ((z).sign) #define zispos(z) (((z).sign == 0) && (*(z).v || ((z).len > 1))) #define zisunit(z) ((*(z).v == 1) && ((z).len == 1)) #define zisone(z) ((*(z).v == 1) && ((z).len == 1) && !(z).sign) #define zisnegone(z) ((*(z).v == 1) && ((z).len == 1) && (z).sign) #define zltnegone(z) (zisneg(z) && (((z).len > 1) || *(z).v > 1)) #define zistwo(z) ((*(z).v == 2) && ((z).len == 1) && !(z).sign) #define zisabstwo(z) ((*(z).v == 2) && ((z).len == 1)) #define zisabsleone(z) ((*(z).v <= 1) && ((z).len == 1)) #define zislezero(z) (zisneg(z) || ziszero(z)) #define zisleone(z) (zisneg(z) || zisabsleone(z)) #define zistiny(z) ((z).len == 1) /* * zgtmaxfull(z) TRUE if abs(z) > MAXFULL */ #define zgtmaxfull(z) (((z).len > 2) || (((z).len == 2) && (((SHALF)(z).v[1]) < 0))) /* * zgtmaxufull(z) TRUE if abs(z) will not fit into a FULL (> MAXUFULL) */ #define zgtmaxufull(z) ((z).len > 2) /* * zgtmaxulong(z) TRUE if abs(z) > MAXULONG */ #if BASEB >= LONG_BITS #define zgtmaxulong(z) ((z).len > 1) #else #define zgtmaxulong(z) zgtmaxufull(z) #endif /* * zgtmaxlong(z) TRUE if abs(z) > MAXLONG */ #if BASEB >= LONG_BITS #define zgtmaxlong(z) (((z).len > 1) || (((z).len == 1) && (((SHALF)(z).v[0]) < 0))) #else #define zgtmaxlong(z) zgtmaxfull(z) #endif /* * Some algorithms testing for values of a certain length. Macros such as * zistiny() do this well. In other cases algorthms require tests for values * in comparison to a given power of 2. In the later case, zistiny() compares * against a different power of 2 on a 64 bit machine. The macros below * provide a tests against powers of 2 that are independent of the work size. * * zge16b(z) TRUE if abs(z) >= 2^16 * zge24b(z) TRUE if abs(z) >= 2^24 * zge31b(z) TRUE if abs(z) >= 2^31 * zge32b(z) TRUE if abs(z) >= 2^32 * zge64b(z) TRUE if abs(z) >= 2^64 * zge128b(z) TRUE if abs(z) >= 2^128 * zge256b(z) TRUE if abs(z) >= 2^256 * zge512b(z) TRUE if abs(z) >= 2^512 * zge1024b(z) TRUE if abs(z) >= 2^1024 * zge2048b(z) TRUE if abs(z) >= 2^2048 * zge4096b(z) TRUE if abs(z) >= 2^4096 * zge8192b(z) TRUE if abs(z) >= 2^8192 */ #if BASEB == 32 #define zge16b(z) (!zistiny(z) || ((z).v[0] >= (HALF)0x10000)) #define zge24b(z) (!zistiny(z) || ((z).v[0] >= (HALF)0x1000000)) #define zge31b(z) (!zistiny(z) || (((SHALF)(z).v[0]) < 0)) #define zge32b(z) (!zistiny(z)) #define zge64b(z) ((z).len > 2) #define zge128b(z) ((z).len > 4) #define zge256b(z) ((z).len > 8) #define zge512b(z) ((z).len > 16) #define zge1024b(z) ((z).len > 32) #define zge2048b(z) ((z).len > 64) #define zge4096b(z) ((z).len > 128) #define zge8192b(z) ((z).len > 256) #else #define zge16b(z) (!zistiny(z)) #define zge24b(z) (((z).len > 2) || (((z).len == 2) && ((z).v[1] >= (HALF)0x100))) #define zge31b(z) (((z).len > 2) || (((z).len == 2) && (((SHALF)(z).v[1]) < 0))) #define zge32b(z) ((z).len > 2) #define zge64b(z) ((z).len > 4) #define zge128b(z) ((z).len > 8) #define zge256b(z) ((z).len > 16) #define zge512b(z) ((z).len > 32) #define zge1024b(z) ((z).len > 64) #define zge2048b(z) ((z).len > 128) #define zge4096b(z) ((z).len > 256) #define zge8192b(z) ((z).len > 512) #endif /* * ztofull - convert an absolute value of a ZVALUE to a FULL if possible * * If the value is too large, only the low order bits that are able to * be converted into a FULL will be used. */ #define ztofull(z) (zistiny(z) ? ((FULL)((z).v[0])) : \ ((FULL)((z).v[0]) + \ ((FULL)((z).v[1]) << BASEB))) #define z1tol(z) ((long)((z).v[0])) #define z2tol(z) ((long)(((z).v[0]) + \ (((z).v[1] & MAXHALF) << BASEB))) /* * ztoulong - convert an absolute value of a ZVALUE to an unsigned long * * If the value is too large, only the low order bits that are able to * be converted into a long will be used. */ #if BASEB >= LONG_BITS # define ztoulong(z) ((unsigned long)z1tol(z)) #else # define ztoulong(z) ((unsigned long)ztofull(z)) #endif /* * ztolong - convert an absolute value of a ZVALUE to a long * * If the value is too large, only the low order bits that are able to * be converted into a long will be used. */ #define ztolong(z) ((long)(ztoulong(z) & MAXLONG)) #define zclearval(z) memset((z).v, 0, (z).len * sizeof(HALF)) #define zcopyval(z1,z2) memcpy((z2).v, (z1).v, (z1).len * sizeof(HALF)) #define zquicktrim(z) {if (((z).len > 1) && ((z).v[(z).len-1] == 0)) \ (z).len--;} #define zfree(z) freeh((z).v) /* * Output modes for numeric displays. */ #define MODE_DEFAULT 0 #define MODE_FRAC 1 #define MODE_INT 2 #define MODE_REAL 3 #define MODE_EXP 4 #define MODE_HEX 5 #define MODE_OCTAL 6 #define MODE_BINARY 7 #define MODE_MAX 7 #define MODE2_OFF (MODE_MAX+1) #define MODE_INITIAL MODE_REAL #define MODE2_INITIAL MODE2_OFF /* * Output routines for either FILE handles or strings. */ extern DLL void math_chr(int ch); extern DLL void math_str(char *str); extern DLL void math_fill(char *str, long width); extern DLL void math_flush(void); extern DLL void math_divertio(void); extern DLL void math_cleardiversions(void); extern DLL char *math_getdivertedio(void); extern DLL int math_setmode(int mode); extern DLL int math_setmode2(int mode); extern DLL LEN math_setdigits(LEN digits); extern DLL void math_fmt(char *, ...); /* * The error routine. */ extern DLL void math_error(char *, ...); /* * external swap functions */ extern DLL HALF *swap_b8_in_HALFs(HALF *dest, HALF *src, LEN len); extern DLL ZVALUE *swap_b8_in_ZVALUE(ZVALUE *dest, ZVALUE *src, BOOL all); extern DLL HALF *swap_b16_in_HALFs(HALF *dest, HALF *src, LEN len); extern DLL ZVALUE *swap_b16_in_ZVALUE(ZVALUE *dest, ZVALUE *src, BOOL all); extern DLL ZVALUE *swap_HALF_in_ZVALUE(ZVALUE *dest, ZVALUE *src, BOOL all); /* * constants used often by the arithmetic routines */ extern HALF _zeroval_[], _oneval_[], _twoval_[], _threeval_[], _fourval_[]; extern HALF _fiveval_[], _sixval_[], _sevenval_[], _eightval_[], _nineval_[]; extern HALF _tenval_[], _elevenval_[], _twelveval_[], _thirteenval_[]; extern HALF _fourteenval_[], _fifteenval_[]; extern HALF _sqbaseval_[]; extern HALF _fourthbaseval_[]; extern ZVALUE zconst[]; /* ZVALUE integers from 0 thru 15 */ extern ZVALUE _zero_, _one_, _two_, _ten_, _neg_one_; extern ZVALUE _sqbase_, _pow4base_, _pow8base_; extern ZVALUE _b32_, _b64_; extern DLL BOOL _math_abort_; /* nonzero to abort calculations */ extern ZVALUE _tenpowers_[]; /* table of 10^2^n */ /* * Bit fiddeling functions and types */ extern HALF bitmask[]; /* bit rotation, norm 0 */ extern HALF lowhalf[]; /* bit masks from low end of HALF */ extern HALF rlowhalf[]; /* reversed bit masks from low end of HALF */ extern HALF highhalf[]; /* bit masks from high end of HALF */ extern HALF rhighhalf[]; /* reversed bit masks from high end of HALF */ #define HAVE_REVERSED_MASKS /* allows old code to know libcalc.a has them */ /* * BITSTR - string of bits within an array of HALFs * * This typedef records a location of a bit in an array of HALFs. * Bit 0 in a HALF is assumed to be the least significant bit in that HALF. * * The most significant bit is found at (loc,bit). Bits of lesser * significance may be found in previous bits and HALFs. */ typedef struct { HALF *loc; /* half address of most significant bit */ int bit; /* bit position within half of most significant bit */ int len; /* length of string in bits */ } BITSTR; #endif /* !__ZMATH_H__*/