/**************************************************************** * * The author of this software is David M. Gay. * * Copyright (c) 1991, 2000, 2001 by Lucent Technologies. * Copyright (C) 2002, 2005, 2006, 2007, 2008 Apple Inc. All rights reserved. * * Permission to use, copy, modify, and distribute this software for any * purpose without fee is hereby granted, provided that this entire notice * is included in all copies of any software which is or includes a copy * or modification of this software and in all copies of the supporting * documentation for such software. * * THIS SOFTWARE IS BEING PROVIDED "AS IS", WITHOUT ANY EXPRESS OR IMPLIED * WARRANTY. IN PARTICULAR, NEITHER THE AUTHOR NOR LUCENT MAKES ANY * REPRESENTATION OR WARRANTY OF ANY KIND CONCERNING THE MERCHANTABILITY * OF THIS SOFTWARE OR ITS FITNESS FOR ANY PARTICULAR PURPOSE. * ***************************************************************/ /* Please send bug reports to David M. Gay Bell Laboratories, Room 2C-463 600 Mountain Avenue Murray Hill, NJ 07974-0636 U.S.A. dmg@bell-labs.com */ /* On a machine with IEEE extended-precision registers, it is * necessary to specify double-precision (53-bit) rounding precision * before invoking strtod or dtoa. If the machine uses (the equivalent * of) Intel 80x87 arithmetic, the call * _control87(PC_53, MCW_PC); * does this with many compilers. Whether this or another call is * appropriate depends on the compiler; for this to work, it may be * necessary to #include "float.h" or another system-dependent header * file. */ /* strtod for IEEE-arithmetic machines. * * This strtod returns a nearest machine number to the input decimal * string (or sets errno to ERANGE). With IEEE arithmetic, ties are * broken by the IEEE round-even rule. Otherwise ties are broken by * biased rounding (add half and chop). * * Inspired loosely by William D. Clinger's paper "How to Read Floating * Point Numbers Accurately" [Proc. ACM SIGPLAN '90, pp. 92-101]. * * Modifications: * * 1. We only require IEEE. * 2. We get by with floating-point arithmetic in a case that * Clinger missed -- when we're computing d * 10^n * for a small integer d and the integer n is not too * much larger than 22 (the maximum integer k for which * we can represent 10^k exactly), we may be able to * compute (d*10^k) * 10^(e-k) with just one roundoff. * 3. Rather than a bit-at-a-time adjustment of the binary * result in the hard case, we use floating-point * arithmetic to determine the adjustment to within * one bit; only in really hard cases do we need to * compute a second residual. * 4. Because of 3., we don't need a large table of powers of 10 * for ten-to-e (just some small tables, e.g. of 10^k * for 0 <= k <= 22). */ /* * #define IEEE_8087 for IEEE-arithmetic machines where the least * significant byte has the lowest address. * #define IEEE_MC68k for IEEE-arithmetic machines where the most * significant byte has the lowest address. * #define No_leftright to omit left-right logic in fast floating-point * computation of dtoa. * #define Check_FLT_ROUNDS if FLT_ROUNDS can assume the values 2 or 3 * and Honor_FLT_ROUNDS is not #defined. * #define Inaccurate_Divide for IEEE-format with correctly rounded * products but inaccurate quotients, e.g., for Intel i860. * #define USE_LONG_LONG on machines that have a "long long" * integer type (of >= 64 bits), and performance testing shows that * it is faster than 32-bit fallback (which is often not the case * on 32-bit machines). On such machines, you can #define Just_16 * to store 16 bits per 32-bit int32_t when doing high-precision integer * arithmetic. Whether this speeds things up or slows things down * depends on the machine and the number being converted. * #define Bad_float_h if your system lacks a float.h or if it does not * define some or all of DBL_DIG, DBL_MAX_10_EXP, DBL_MAX_EXP, * FLT_RADIX, FLT_ROUNDS, and DBL_MAX. * #define INFNAN_CHECK on IEEE systems to cause strtod to check for * Infinity and NaN (case insensitively). On some systems (e.g., * some HP systems), it may be necessary to #define NAN_WORD0 * appropriately -- to the most significant word of a quiet NaN. * (On HP Series 700/800 machines, -DNAN_WORD0=0x7ff40000 works.) * When INFNAN_CHECK is #defined and No_Hex_NaN is not #defined, * strtod also accepts (case insensitively) strings of the form * NaN(x), where x is a string of hexadecimal digits and spaces; * if there is only one string of hexadecimal digits, it is taken * for the 52 fraction bits of the resulting NaN; if there are two * or more strings of hex digits, the first is for the high 20 bits, * the second and subsequent for the low 32 bits, with intervening * white space ignored; but if this results in none of the 52 * fraction bits being on (an IEEE Infinity symbol), then NAN_WORD0 * and NAN_WORD1 are used instead. * #define NO_IEEE_Scale to disable new (Feb. 1997) logic in strtod that * avoids underflows on inputs whose result does not underflow. * If you #define NO_IEEE_Scale on a machine that uses IEEE-format * floating-point numbers and flushes underflows to zero rather * than implementing gradual underflow, then you must also #define * Sudden_Underflow. * #define YES_ALIAS to permit aliasing certain double values with * arrays of ULongs. This leads to slightly better code with * some compilers and was always used prior to 19990916, but it * is not strictly legal and can cause trouble with aggressively * optimizing compilers (e.g., gcc 2.95.1 under -O2). * #define SET_INEXACT if IEEE arithmetic is being used and extra * computation should be done to set the inexact flag when the * result is inexact and avoid setting inexact when the result * is exact. In this case, dtoa.c must be compiled in * an environment, perhaps provided by #include "dtoa.c" in a * suitable wrapper, that defines two functions, * int get_inexact(void); * void clear_inexact(void); * such that get_inexact() returns a nonzero value if the * inexact bit is already set, and clear_inexact() sets the * inexact bit to 0. When SET_INEXACT is #defined, strtod * also does extra computations to set the underflow and overflow * flags when appropriate (i.e., when the result is tiny and * inexact or when it is a numeric value rounded to +-infinity). * #define NO_ERRNO if strtod should not assign errno = ERANGE when * the result overflows to +-Infinity or underflows to 0. */ #include "config.h" #include "dtoa.h" #if HAVE(ERRNO_H) #include #else #define NO_ERRNO #endif #include #include #include #include #include #include #include #include #include #include #include #if COMPILER(MSVC) #pragma warning(disable: 4244) #pragma warning(disable: 4245) #pragma warning(disable: 4554) #endif #if PLATFORM(BIG_ENDIAN) #define IEEE_MC68k #elif PLATFORM(MIDDLE_ENDIAN) #define IEEE_ARM #else #define IEEE_8087 #endif #define INFNAN_CHECK #if defined(IEEE_8087) + defined(IEEE_MC68k) + defined(IEEE_ARM) != 1 Exactly one of IEEE_8087, IEEE_ARM or IEEE_MC68k should be defined. #endif namespace WTF { #if ENABLE(JSC_MULTIPLE_THREADS) Mutex* s_dtoaP5Mutex; #endif typedef union { double d; uint32_t L[2]; } U; #ifdef YES_ALIAS #define dval(x) x #ifdef IEEE_8087 #define word0(x) ((uint32_t*)&x)[1] #define word1(x) ((uint32_t*)&x)[0] #else #define word0(x) ((uint32_t*)&x)[0] #define word1(x) ((uint32_t*)&x)[1] #endif #else #ifdef IEEE_8087 #define word0(x) (x)->L[1] #define word1(x) (x)->L[0] #else #define word0(x) (x)->L[0] #define word1(x) (x)->L[1] #endif #define dval(x) (x)->d #endif /* The following definition of Storeinc is appropriate for MIPS processors. * An alternative that might be better on some machines is * #define Storeinc(a,b,c) (*a++ = b << 16 | c & 0xffff) */ #if defined(IEEE_8087) || defined(IEEE_ARM) #define Storeinc(a,b,c) (((unsigned short*)a)[1] = (unsigned short)b, ((unsigned short*)a)[0] = (unsigned short)c, a++) #else #define Storeinc(a,b,c) (((unsigned short*)a)[0] = (unsigned short)b, ((unsigned short*)a)[1] = (unsigned short)c, a++) #endif #define Exp_shift 20 #define Exp_shift1 20 #define Exp_msk1 0x100000 #define Exp_msk11 0x100000 #define Exp_mask 0x7ff00000 #define P 53 #define Bias 1023 #define Emin (-1022) #define Exp_1 0x3ff00000 #define Exp_11 0x3ff00000 #define Ebits 11 #define Frac_mask 0xfffff #define Frac_mask1 0xfffff #define Ten_pmax 22 #define Bletch 0x10 #define Bndry_mask 0xfffff #define Bndry_mask1 0xfffff #define LSB 1 #define Sign_bit 0x80000000 #define Log2P 1 #define Tiny0 0 #define Tiny1 1 #define Quick_max 14 #define Int_max 14 #if !defined(NO_IEEE_Scale) #undef Avoid_Underflow #define Avoid_Underflow #endif #if !defined(Flt_Rounds) #if defined(FLT_ROUNDS) #define Flt_Rounds FLT_ROUNDS #else #define Flt_Rounds 1 #endif #endif /*Flt_Rounds*/ #define rounded_product(a,b) a *= b #define rounded_quotient(a,b) a /= b #define Big0 (Frac_mask1 | Exp_msk1 * (DBL_MAX_EXP + Bias - 1)) #define Big1 0xffffffff // FIXME: we should remove non-Pack_32 mode since it is unused and unmaintained #ifndef Pack_32 #define Pack_32 #endif #if PLATFORM(PPC64) || PLATFORM(X86_64) // 64-bit emulation provided by the compiler is likely to be slower than dtoa own code on 32-bit hardware. #define USE_LONG_LONG #endif #ifndef USE_LONG_LONG #ifdef Just_16 #undef Pack_32 /* When Pack_32 is not defined, we store 16 bits per 32-bit int32_t. * This makes some inner loops simpler and sometimes saves work * during multiplications, but it often seems to make things slightly * slower. Hence the default is now to store 32 bits per int32_t. */ #endif #endif #define Kmax 15 struct BigInt { BigInt() : sign(0) { } int sign; void clear() { sign = 0; m_words.clear(); } size_t size() const { return m_words.size(); } void resize(size_t s) { m_words.resize(s); } uint32_t* words() { return m_words.data(); } const uint32_t* words() const { return m_words.data(); } void append(uint32_t w) { m_words.append(w); } Vector m_words; }; static void multadd(BigInt& b, int m, int a) /* multiply by m and add a */ { #ifdef USE_LONG_LONG unsigned long long carry; #else uint32_t carry; #endif int wds = b.size(); uint32_t* x = b.words(); int i = 0; carry = a; do { #ifdef USE_LONG_LONG unsigned long long y = *x * (unsigned long long)m + carry; carry = y >> 32; *x++ = (uint32_t)y & 0xffffffffUL; #else #ifdef Pack_32 uint32_t xi = *x; uint32_t y = (xi & 0xffff) * m + carry; uint32_t z = (xi >> 16) * m + (y >> 16); carry = z >> 16; *x++ = (z << 16) + (y & 0xffff); #else uint32_t y = *x * m + carry; carry = y >> 16; *x++ = y & 0xffff; #endif #endif } while (++i < wds); if (carry) b.append((uint32_t)carry); } static void s2b(BigInt& b, const char* s, int nd0, int nd, uint32_t y9) { int k; int32_t y; int32_t x = (nd + 8) / 9; for (k = 0, y = 1; x > y; y <<= 1, k++) { } #ifdef Pack_32 b.sign = 0; b.resize(1); b.words()[0] = y9; #else b.sign = 0; b.resize((b->x[1] = y9 >> 16) ? 2 : 1); b.words()[0] = y9 & 0xffff; #endif int i = 9; if (9 < nd0) { s += 9; do { multadd(b, 10, *s++ - '0'); } while (++i < nd0); s++; } else s += 10; for (; i < nd; i++) multadd(b, 10, *s++ - '0'); } static int hi0bits(uint32_t x) { int k = 0; if (!(x & 0xffff0000)) { k = 16; x <<= 16; } if (!(x & 0xff000000)) { k += 8; x <<= 8; } if (!(x & 0xf0000000)) { k += 4; x <<= 4; } if (!(x & 0xc0000000)) { k += 2; x <<= 2; } if (!(x & 0x80000000)) { k++; if (!(x & 0x40000000)) return 32; } return k; } static int lo0bits (uint32_t* y) { int k; uint32_t x = *y; if (x & 7) { if (x & 1) return 0; if (x & 2) { *y = x >> 1; return 1; } *y = x >> 2; return 2; } k = 0; if (!(x & 0xffff)) { k = 16; x >>= 16; } if (!(x & 0xff)) { k += 8; x >>= 8; } if (!(x & 0xf)) { k += 4; x >>= 4; } if (!(x & 0x3)) { k += 2; x >>= 2; } if (!(x & 1)) { k++; x >>= 1; if (!x & 1) return 32; } *y = x; return k; } static void i2b(BigInt& b, int i) { b.sign = 0; b.resize(1); b.words()[0] = i; } static void mult(BigInt& aRef, const BigInt& bRef) { const BigInt* a = &aRef; const BigInt* b = &bRef; BigInt c; int wa, wb, wc; const uint32_t *x = 0, *xa, *xb, *xae, *xbe; uint32_t *xc, *xc0; uint32_t y; #ifdef USE_LONG_LONG unsigned long long carry, z; #else uint32_t carry, z; #endif if (a->size() < b->size()) { const BigInt* tmp = a; a = b; b = tmp; } wa = a->size(); wb = b->size(); wc = wa + wb; c.resize(wc); for (xc = c.words(), xa = xc + wc; xc < xa; xc++) *xc = 0; xa = a->words(); xae = xa + wa; xb = b->words(); xbe = xb + wb; xc0 = c.words(); #ifdef USE_LONG_LONG for (; xb < xbe; xc0++) { if ((y = *xb++)) { x = xa; xc = xc0; carry = 0; do { z = *x++ * (unsigned long long)y + *xc + carry; carry = z >> 32; *xc++ = (uint32_t)z & 0xffffffffUL; } while (x < xae); *xc = (uint32_t)carry; } } #else #ifdef Pack_32 for (; xb < xbe; xb++, xc0++) { if ((y = *xb & 0xffff)) { x = xa; xc = xc0; carry = 0; do { z = (*x & 0xffff) * y + (*xc & 0xffff) + carry; carry = z >> 16; uint32_t z2 = (*x++ >> 16) * y + (*xc >> 16) + carry; carry = z2 >> 16; Storeinc(xc, z2, z); } while (x < xae); *xc = carry; } if ((y = *xb >> 16)) { x = xa; xc = xc0; carry = 0; uint32_t z2 = *xc; do { z = (*x & 0xffff) * y + (*xc >> 16) + carry; carry = z >> 16; Storeinc(xc, z, z2); z2 = (*x++ >> 16) * y + (*xc & 0xffff) + carry; carry = z2 >> 16; } while (x < xae); *xc = z2; } } #else for(; xb < xbe; xc0++) { if ((y = *xb++)) { x = xa; xc = xc0; carry = 0; do { z = *x++ * y + *xc + carry; carry = z >> 16; *xc++ = z & 0xffff; } while (x < xae); *xc = carry; } } #endif #endif for (xc0 = c.words(), xc = xc0 + wc; wc > 0 && !*--xc; --wc) { } c.resize(wc); aRef = c; } struct P5Node { BigInt val; P5Node* next; }; static P5Node* p5s; static int p5s_count; static ALWAYS_INLINE void pow5mult(BigInt& b, int k) { static int p05[3] = { 5, 25, 125 }; if (int i = k & 3) multadd(b, p05[i - 1], 0); if (!(k >>= 2)) return; #if ENABLE(JSC_MULTIPLE_THREADS) s_dtoaP5Mutex->lock(); #endif P5Node* p5 = p5s; if (!p5) { /* first time */ p5 = new P5Node; i2b(p5->val, 625); p5->next = 0; p5s = p5; p5s_count = 1; } int p5s_count_local = p5s_count; #if ENABLE(JSC_MULTIPLE_THREADS) s_dtoaP5Mutex->unlock(); #endif int p5s_used = 0; for (;;) { if (k & 1) mult(b, p5->val); if (!(k >>= 1)) break; if (++p5s_used == p5s_count_local) { #if ENABLE(JSC_MULTIPLE_THREADS) s_dtoaP5Mutex->lock(); #endif if (p5s_used == p5s_count) { ASSERT(!p5->next); p5->next = new P5Node; p5->next->next = 0; p5->next->val = p5->val; mult(p5->next->val, p5->next->val); ++p5s_count; } p5s_count_local = p5s_count; #if ENABLE(JSC_MULTIPLE_THREADS) s_dtoaP5Mutex->unlock(); #endif } p5 = p5->next; } } static ALWAYS_INLINE void lshift(BigInt& b, int k) { #ifdef Pack_32 int n = k >> 5; #else int n = k >> 4; #endif int origSize = b.size(); int n1 = n + origSize + 1; if (k &= 0x1f) b.resize(b.size() + n + 1); else b.resize(b.size() + n); const uint32_t* srcStart = b.words(); uint32_t* dstStart = b.words(); const uint32_t* src = srcStart + origSize - 1; uint32_t* dst = dstStart + n1 - 1; #ifdef Pack_32 if (k) { uint32_t hiSubword = 0; int s = 32 - k; for (; src >= srcStart; --src) { *dst-- = hiSubword | *src >> s; hiSubword = *src << k; } *dst = hiSubword; ASSERT(dst == dstStart + n); b.resize(origSize + n + (b.words()[n1 - 1] != 0)); } #else if (k &= 0xf) { uint32_t hiSubword = 0; int s = 16 - k; for (; src >= srcStart; --src) { *dst-- = hiSubword | *src >> s; hiSubword = (*src << k) & 0xffff; } *dst = hiSubword; ASSERT(dst == dstStart + n); result->wds = b->wds + n + (result->x[n1 - 1] != 0); } #endif else { do { *--dst = *src--; } while (src >= srcStart); } for (dst = dstStart + n; dst != dstStart; ) *--dst = 0; ASSERT(b.size() <= 1 || b.words()[b.size() - 1]); } static int cmp(const BigInt& a, const BigInt& b) { const uint32_t *xa, *xa0, *xb, *xb0; int i, j; i = a.size(); j = b.size(); ASSERT(i <= 1 || a.words()[i - 1]); ASSERT(j <= 1 || b.words()[j - 1]); if (i -= j) return i; xa0 = a.words(); xa = xa0 + j; xb0 = b.words(); xb = xb0 + j; for (;;) { if (*--xa != *--xb) return *xa < *xb ? -1 : 1; if (xa <= xa0) break; } return 0; } static ALWAYS_INLINE void diff(BigInt& c, const BigInt& aRef, const BigInt& bRef) { const BigInt* a = &aRef; const BigInt* b = &bRef; int i, wa, wb; uint32_t *xc; i = cmp(*a, *b); if (!i) { c.sign = 0; c.resize(1); c.words()[0] = 0; return; } if (i < 0) { const BigInt* tmp = a; a = b; b = tmp; i = 1; } else i = 0; wa = a->size(); const uint32_t* xa = a->words(); const uint32_t* xae = xa + wa; wb = b->size(); const uint32_t* xb = b->words(); const uint32_t* xbe = xb + wb; c.resize(wa); c.sign = i; xc = c.words(); #ifdef USE_LONG_LONG unsigned long long borrow = 0; do { unsigned long long y = (unsigned long long)*xa++ - *xb++ - borrow; borrow = y >> 32 & (uint32_t)1; *xc++ = (uint32_t)y & 0xffffffffUL; } while (xb < xbe); while (xa < xae) { unsigned long long y = *xa++ - borrow; borrow = y >> 32 & (uint32_t)1; *xc++ = (uint32_t)y & 0xffffffffUL; } #else uint32_t borrow = 0; #ifdef Pack_32 do { uint32_t y = (*xa & 0xffff) - (*xb & 0xffff) - borrow; borrow = (y & 0x10000) >> 16; uint32_t z = (*xa++ >> 16) - (*xb++ >> 16) - borrow; borrow = (z & 0x10000) >> 16; Storeinc(xc, z, y); } while (xb < xbe); while (xa < xae) { uint32_t y = (*xa & 0xffff) - borrow; borrow = (y & 0x10000) >> 16; uint32_t z = (*xa++ >> 16) - borrow; borrow = (z & 0x10000) >> 16; Storeinc(xc, z, y); } #else do { uint32_t y = *xa++ - *xb++ - borrow; borrow = (y & 0x10000) >> 16; *xc++ = y & 0xffff; } while (xb < xbe); while (xa < xae) { uint32_t y = *xa++ - borrow; borrow = (y & 0x10000) >> 16; *xc++ = y & 0xffff; } #endif #endif while (!*--xc) wa--; c.resize(wa); } static double ulp(U *x) { register int32_t L; U u; L = (word0(x) & Exp_mask) - (P - 1) * Exp_msk1; #ifndef Avoid_Underflow #ifndef Sudden_Underflow if (L > 0) { #endif #endif word0(&u) = L; word1(&u) = 0; #ifndef Avoid_Underflow #ifndef Sudden_Underflow } else { L = -L >> Exp_shift; if (L < Exp_shift) { word0(&u) = 0x80000 >> L; word1(&u) = 0; } else { word0(&u) = 0; L -= Exp_shift; word1(&u) = L >= 31 ? 1 : 1 << 31 - L; } } #endif #endif return dval(&u); } static double b2d(const BigInt& a, int* e) { const uint32_t* xa; const uint32_t* xa0; uint32_t w; uint32_t y; uint32_t z; int k; U d; #define d0 word0(&d) #define d1 word1(&d) xa0 = a.words(); xa = xa0 + a.size(); y = *--xa; ASSERT(y); k = hi0bits(y); *e = 32 - k; #ifdef Pack_32 if (k < Ebits) { d0 = Exp_1 | (y >> (Ebits - k)); w = xa > xa0 ? *--xa : 0; d1 = (y << (32 - Ebits + k)) | (w >> (Ebits - k)); goto ret_d; } z = xa > xa0 ? *--xa : 0; if (k -= Ebits) { d0 = Exp_1 | (y << k) | (z >> (32 - k)); y = xa > xa0 ? *--xa : 0; d1 = (z << k) | (y >> (32 - k)); } else { d0 = Exp_1 | y; d1 = z; } #else if (k < Ebits + 16) { z = xa > xa0 ? *--xa : 0; d0 = Exp_1 | y << k - Ebits | z >> Ebits + 16 - k; w = xa > xa0 ? *--xa : 0; y = xa > xa0 ? *--xa : 0; d1 = z << k + 16 - Ebits | w << k - Ebits | y >> 16 + Ebits - k; goto ret_d; } z = xa > xa0 ? *--xa : 0; w = xa > xa0 ? *--xa : 0; k -= Ebits + 16; d0 = Exp_1 | y << k + 16 | z << k | w >> 16 - k; y = xa > xa0 ? *--xa : 0; d1 = w << k + 16 | y << k; #endif ret_d: #undef d0 #undef d1 return dval(&d); } static ALWAYS_INLINE void d2b(BigInt& b, U* d, int* e, int* bits) { int de, k; uint32_t *x, y, z; #ifndef Sudden_Underflow int i; #endif #define d0 word0(d) #define d1 word1(d) b.sign = 0; #ifdef Pack_32 b.resize(1); #else b.resize(2); #endif x = b.words(); z = d0 & Frac_mask; d0 &= 0x7fffffff; /* clear sign bit, which we ignore */ #ifdef Sudden_Underflow de = (int)(d0 >> Exp_shift); #else if ((de = (int)(d0 >> Exp_shift))) z |= Exp_msk1; #endif #ifdef Pack_32 if ((y = d1)) { if ((k = lo0bits(&y))) { x[0] = y | (z << (32 - k)); z >>= k; } else x[0] = y; if (z) { b.resize(2); x[1] = z; } #ifndef Sudden_Underflow i = b.size(); #endif } else { k = lo0bits(&z); x[0] = z; #ifndef Sudden_Underflow i = 1; #endif b.resize(1); k += 32; } #else if ((y = d1)) { if ((k = lo0bits(&y))) { if (k >= 16) { x[0] = y | z << 32 - k & 0xffff; x[1] = z >> k - 16 & 0xffff; x[2] = z >> k; i = 2; } else { x[0] = y & 0xffff; x[1] = y >> 16 | z << 16 - k & 0xffff; x[2] = z >> k & 0xffff; x[3] = z >> k + 16; i = 3; } } else { x[0] = y & 0xffff; x[1] = y >> 16; x[2] = z & 0xffff; x[3] = z >> 16; i = 3; } } else { k = lo0bits(&z); if (k >= 16) { x[0] = z; i = 0; } else { x[0] = z & 0xffff; x[1] = z >> 16; i = 1; } k += 32; } while (!x[i]) --i; b->resize(i + 1); #endif #ifndef Sudden_Underflow if (de) { #endif *e = de - Bias - (P - 1) + k; *bits = P - k; #ifndef Sudden_Underflow } else { *e = de - Bias - (P - 1) + 1 + k; #ifdef Pack_32 *bits = (32 * i) - hi0bits(x[i - 1]); #else *bits = (i + 2) * 16 - hi0bits(x[i]); #endif } #endif } #undef d0 #undef d1 static double ratio(const BigInt& a, const BigInt& b) { U da, db; int k, ka, kb; dval(&da) = b2d(a, &ka); dval(&db) = b2d(b, &kb); #ifdef Pack_32 k = ka - kb + 32 * (a.size() - b.size()); #else k = ka - kb + 16 * (a.size() - b.size()); #endif if (k > 0) word0(&da) += k * Exp_msk1; else { k = -k; word0(&db) += k * Exp_msk1; } return dval(&da) / dval(&db); } static const double tens[] = { 1e0, 1e1, 1e2, 1e3, 1e4, 1e5, 1e6, 1e7, 1e8, 1e9, 1e10, 1e11, 1e12, 1e13, 1e14, 1e15, 1e16, 1e17, 1e18, 1e19, 1e20, 1e21, 1e22 }; static const double bigtens[] = { 1e16, 1e32, 1e64, 1e128, 1e256 }; static const double tinytens[] = { 1e-16, 1e-32, 1e-64, 1e-128, #ifdef Avoid_Underflow 9007199254740992. * 9007199254740992.e-256 /* = 2^106 * 1e-53 */ #else 1e-256 #endif }; /* The factor of 2^53 in tinytens[4] helps us avoid setting the underflow */ /* flag unnecessarily. It leads to a song and dance at the end of strtod. */ #define Scale_Bit 0x10 #define n_bigtens 5 #if defined(INFNAN_CHECK) #ifndef NAN_WORD0 #define NAN_WORD0 0x7ff80000 #endif #ifndef NAN_WORD1 #define NAN_WORD1 0 #endif static int match(const char** sp, const char* t) { int c, d; const char* s = *sp; while ((d = *t++)) { if ((c = *++s) >= 'A' && c <= 'Z') c += 'a' - 'A'; if (c != d) return 0; } *sp = s + 1; return 1; } #ifndef No_Hex_NaN static void hexnan(U* rvp, const char** sp) { uint32_t c, x[2]; const char* s; int havedig, udx0, xshift; x[0] = x[1] = 0; havedig = xshift = 0; udx0 = 1; s = *sp; while ((c = *(const unsigned char*)++s)) { if (c >= '0' && c <= '9') c -= '0'; else if (c >= 'a' && c <= 'f') c += 10 - 'a'; else if (c >= 'A' && c <= 'F') c += 10 - 'A'; else if (c <= ' ') { if (udx0 && havedig) { udx0 = 0; xshift = 1; } continue; } else if (/*(*/ c == ')' && havedig) { *sp = s + 1; break; } else return; /* invalid form: don't change *sp */ havedig = 1; if (xshift) { xshift = 0; x[0] = x[1]; x[1] = 0; } if (udx0) x[0] = (x[0] << 4) | (x[1] >> 28); x[1] = (x[1] << 4) | c; } if ((x[0] &= 0xfffff) || x[1]) { word0(rvp) = Exp_mask | x[0]; word1(rvp) = x[1]; } } #endif /*No_Hex_NaN*/ #endif /* INFNAN_CHECK */ double strtod(const char* s00, char** se) { #ifdef Avoid_Underflow int scale; #endif int bb2, bb5, bbe, bd2, bd5, bbbits, bs2, c, dsign, e, e1, esign, i, j, k, nd, nd0, nf, nz, nz0, sign; const char *s, *s0, *s1; double aadj, aadj1; U aadj2, adj, rv, rv0; int32_t L; uint32_t y, z; BigInt bb, bb1, bd, bd0, bs, delta; #ifdef SET_INEXACT int inexact, oldinexact; #endif sign = nz0 = nz = 0; dval(&rv) = 0; for (s = s00; ; s++) switch (*s) { case '-': sign = 1; /* no break */ case '+': if (*++s) goto break2; /* no break */ case 0: goto ret0; case '\t': case '\n': case '\v': case '\f': case '\r': case ' ': continue; default: goto break2; } break2: if (*s == '0') { nz0 = 1; while (*++s == '0') { } if (!*s) goto ret; } s0 = s; y = z = 0; for (nd = nf = 0; (c = *s) >= '0' && c <= '9'; nd++, s++) if (nd < 9) y = (10 * y) + c - '0'; else if (nd < 16) z = (10 * z) + c - '0'; nd0 = nd; if (c == '.') { c = *++s; if (!nd) { for (; c == '0'; c = *++s) nz++; if (c > '0' && c <= '9') { s0 = s; nf += nz; nz = 0; goto have_dig; } goto dig_done; } for (; c >= '0' && c <= '9'; c = *++s) { have_dig: nz++; if (c -= '0') { nf += nz; for (i = 1; i < nz; i++) if (nd++ < 9) y *= 10; else if (nd <= DBL_DIG + 1) z *= 10; if (nd++ < 9) y = (10 * y) + c; else if (nd <= DBL_DIG + 1) z = (10 * z) + c; nz = 0; } } } dig_done: e = 0; if (c == 'e' || c == 'E') { if (!nd && !nz && !nz0) { goto ret0; } s00 = s; esign = 0; switch (c = *++s) { case '-': esign = 1; case '+': c = *++s; } if (c >= '0' && c <= '9') { while (c == '0') c = *++s; if (c > '0' && c <= '9') { L = c - '0'; s1 = s; while ((c = *++s) >= '0' && c <= '9') L = (10 * L) + c - '0'; if (s - s1 > 8 || L > 19999) /* Avoid confusion from exponents * so large that e might overflow. */ e = 19999; /* safe for 16 bit ints */ else e = (int)L; if (esign) e = -e; } else e = 0; } else s = s00; } if (!nd) { if (!nz && !nz0) { #ifdef INFNAN_CHECK /* Check for Nan and Infinity */ switch(c) { case 'i': case 'I': if (match(&s,"nf")) { --s; if (!match(&s,"inity")) ++s; word0(&rv) = 0x7ff00000; word1(&rv) = 0; goto ret; } break; case 'n': case 'N': if (match(&s, "an")) { word0(&rv) = NAN_WORD0; word1(&rv) = NAN_WORD1; #ifndef No_Hex_NaN if (*s == '(') /*)*/ hexnan(&rv, &s); #endif goto ret; } } #endif /* INFNAN_CHECK */ ret0: s = s00; sign = 0; } goto ret; } e1 = e -= nf; /* Now we have nd0 digits, starting at s0, followed by a * decimal point, followed by nd-nd0 digits. The number we're * after is the integer represented by those digits times * 10**e */ if (!nd0) nd0 = nd; k = nd < DBL_DIG + 1 ? nd : DBL_DIG + 1; dval(&rv) = y; if (k > 9) { #ifdef SET_INEXACT if (k > DBL_DIG) oldinexact = get_inexact(); #endif dval(&rv) = tens[k - 9] * dval(&rv) + z; } if (nd <= DBL_DIG && Flt_Rounds == 1) { if (!e) goto ret; if (e > 0) { if (e <= Ten_pmax) { /* rv = */ rounded_product(dval(&rv), tens[e]); goto ret; } i = DBL_DIG - nd; if (e <= Ten_pmax + i) { /* A fancier test would sometimes let us do * this for larger i values. */ e -= i; dval(&rv) *= tens[i]; /* rv = */ rounded_product(dval(&rv), tens[e]); goto ret; } } #ifndef Inaccurate_Divide else if (e >= -Ten_pmax) { /* rv = */ rounded_quotient(dval(&rv), tens[-e]); goto ret; } #endif } e1 += nd - k; #ifdef SET_INEXACT inexact = 1; if (k <= DBL_DIG) oldinexact = get_inexact(); #endif #ifdef Avoid_Underflow scale = 0; #endif /* Get starting approximation = rv * 10**e1 */ if (e1 > 0) { if ((i = e1 & 15)) dval(&rv) *= tens[i]; if (e1 &= ~15) { if (e1 > DBL_MAX_10_EXP) { ovfl: #ifndef NO_ERRNO errno = ERANGE; #endif /* Can't trust HUGE_VAL */ word0(&rv) = Exp_mask; word1(&rv) = 0; #ifdef SET_INEXACT /* set overflow bit */ dval(&rv0) = 1e300; dval(&rv0) *= dval(&rv0); #endif goto ret; } e1 >>= 4; for (j = 0; e1 > 1; j++, e1 >>= 1) if (e1 & 1) dval(&rv) *= bigtens[j]; /* The last multiplication could overflow. */ word0(&rv) -= P * Exp_msk1; dval(&rv) *= bigtens[j]; if ((z = word0(&rv) & Exp_mask) > Exp_msk1 * (DBL_MAX_EXP + Bias - P)) goto ovfl; if (z > Exp_msk1 * (DBL_MAX_EXP + Bias - 1 - P)) { /* set to largest number */ /* (Can't trust DBL_MAX) */ word0(&rv) = Big0; word1(&rv) = Big1; } else word0(&rv) += P * Exp_msk1; } } else if (e1 < 0) { e1 = -e1; if ((i = e1 & 15)) dval(&rv) /= tens[i]; if (e1 >>= 4) { if (e1 >= 1 << n_bigtens) goto undfl; #ifdef Avoid_Underflow if (e1 & Scale_Bit) scale = 2 * P; for (j = 0; e1 > 0; j++, e1 >>= 1) if (e1 & 1) dval(&rv) *= tinytens[j]; if (scale && (j = (2 * P) + 1 - ((word0(&rv) & Exp_mask) >> Exp_shift)) > 0) { /* scaled rv is denormal; zap j low bits */ if (j >= 32) { word1(&rv) = 0; if (j >= 53) word0(&rv) = (P + 2) * Exp_msk1; else word0(&rv) &= 0xffffffff << (j - 32); } else word1(&rv) &= 0xffffffff << j; } #else for (j = 0; e1 > 1; j++, e1 >>= 1) if (e1 & 1) dval(&rv) *= tinytens[j]; /* The last multiplication could underflow. */ dval(&rv0) = dval(&rv); dval(&rv) *= tinytens[j]; if (!dval(&rv)) { dval(&rv) = 2. * dval(&rv0); dval(&rv) *= tinytens[j]; #endif if (!dval(&rv)) { undfl: dval(&rv) = 0.; #ifndef NO_ERRNO errno = ERANGE; #endif goto ret; } #ifndef Avoid_Underflow word0(&rv) = Tiny0; word1(&rv) = Tiny1; /* The refinement below will clean * this approximation up. */ } #endif } } /* Now the hard part -- adjusting rv to the correct value.*/ /* Put digits into bd: true value = bd * 10^e */ s2b(bd0, s0, nd0, nd, y); for (;;) { bd = bd0; d2b(bb, &rv, &bbe, &bbbits); /* rv = bb * 2^bbe */ i2b(bs, 1); if (e >= 0) { bb2 = bb5 = 0; bd2 = bd5 = e; } else { bb2 = bb5 = -e; bd2 = bd5 = 0; } if (bbe >= 0) bb2 += bbe; else bd2 -= bbe; bs2 = bb2; #ifdef Avoid_Underflow j = bbe - scale; i = j + bbbits - 1; /* logb(rv) */ if (i < Emin) /* denormal */ j += P - Emin; else j = P + 1 - bbbits; #else /*Avoid_Underflow*/ #ifdef Sudden_Underflow j = P + 1 - bbbits; #else /*Sudden_Underflow*/ j = bbe; i = j + bbbits - 1; /* logb(rv) */ if (i < Emin) /* denormal */ j += P - Emin; else j = P + 1 - bbbits; #endif /*Sudden_Underflow*/ #endif /*Avoid_Underflow*/ bb2 += j; bd2 += j; #ifdef Avoid_Underflow bd2 += scale; #endif i = bb2 < bd2 ? bb2 : bd2; if (i > bs2) i = bs2; if (i > 0) { bb2 -= i; bd2 -= i; bs2 -= i; } if (bb5 > 0) { pow5mult(bs, bb5); mult(bb, bs); } if (bb2 > 0) lshift(bb, bb2); if (bd5 > 0) pow5mult(bd, bd5); if (bd2 > 0) lshift(bd, bd2); if (bs2 > 0) lshift(bs, bs2); diff(delta, bb, bd); dsign = delta.sign; delta.sign = 0; i = cmp(delta, bs); if (i < 0) { /* Error is less than half an ulp -- check for * special case of mantissa a power of two. */ if (dsign || word1(&rv) || word0(&rv) & Bndry_mask #ifdef Avoid_Underflow || (word0(&rv) & Exp_mask) <= (2 * P + 1) * Exp_msk1 #else || (word0(&rv) & Exp_mask) <= Exp_msk1 #endif ) { #ifdef SET_INEXACT if (!delta->words()[0] && delta->size() <= 1) inexact = 0; #endif break; } if (!delta.words()[0] && delta.size() <= 1) { /* exact result */ #ifdef SET_INEXACT inexact = 0; #endif break; } lshift(delta, Log2P); if (cmp(delta, bs) > 0) goto drop_down; break; } if (i == 0) { /* exactly half-way between */ if (dsign) { if ((word0(&rv) & Bndry_mask1) == Bndry_mask1 && word1(&rv) == ( #ifdef Avoid_Underflow (scale && (y = word0(&rv) & Exp_mask) <= 2 * P * Exp_msk1) ? (0xffffffff & (0xffffffff << (2 * P + 1 - (y >> Exp_shift)))) : #endif 0xffffffff)) { /*boundary case -- increment exponent*/ word0(&rv) = (word0(&rv) & Exp_mask) + Exp_msk1; word1(&rv) = 0; #ifdef Avoid_Underflow dsign = 0; #endif break; } } else if (!(word0(&rv) & Bndry_mask) && !word1(&rv)) { drop_down: /* boundary case -- decrement exponent */ #ifdef Sudden_Underflow /*{{*/ L = word0(&rv) & Exp_mask; #ifdef Avoid_Underflow if (L <= (scale ? (2 * P + 1) * Exp_msk1 : Exp_msk1)) #else if (L <= Exp_msk1) #endif /*Avoid_Underflow*/ goto undfl; L -= Exp_msk1; #else /*Sudden_Underflow}{*/ #ifdef Avoid_Underflow if (scale) { L = word0(&rv) & Exp_mask; if (L <= (2 * P + 1) * Exp_msk1) { if (L > (P + 2) * Exp_msk1) /* round even ==> */ /* accept rv */ break; /* rv = smallest denormal */ goto undfl; } } #endif /*Avoid_Underflow*/ L = (word0(&rv) & Exp_mask) - Exp_msk1; #endif /*Sudden_Underflow}}*/ word0(&rv) = L | Bndry_mask1; word1(&rv) = 0xffffffff; break; } if (!(word1(&rv) & LSB)) break; if (dsign) dval(&rv) += ulp(&rv); else { dval(&rv) -= ulp(&rv); #ifndef Sudden_Underflow if (!dval(&rv)) goto undfl; #endif } #ifdef Avoid_Underflow dsign = 1 - dsign; #endif break; } if ((aadj = ratio(delta, bs)) <= 2.) { if (dsign) aadj = aadj1 = 1.; else if (word1(&rv) || word0(&rv) & Bndry_mask) { #ifndef Sudden_Underflow if (word1(&rv) == Tiny1 && !word0(&rv)) goto undfl; #endif aadj = 1.; aadj1 = -1.; } else { /* special case -- power of FLT_RADIX to be */ /* rounded down... */ if (aadj < 2. / FLT_RADIX) aadj = 1. / FLT_RADIX; else aadj *= 0.5; aadj1 = -aadj; } } else { aadj *= 0.5; aadj1 = dsign ? aadj : -aadj; #ifdef Check_FLT_ROUNDS switch (Rounding) { case 2: /* towards +infinity */ aadj1 -= 0.5; break; case 0: /* towards 0 */ case 3: /* towards -infinity */ aadj1 += 0.5; } #else if (Flt_Rounds == 0) aadj1 += 0.5; #endif /*Check_FLT_ROUNDS*/ } y = word0(&rv) & Exp_mask; /* Check for overflow */ if (y == Exp_msk1 * (DBL_MAX_EXP + Bias - 1)) { dval(&rv0) = dval(&rv); word0(&rv) -= P * Exp_msk1; adj.d = aadj1 * ulp(&rv); dval(&rv) += adj.d; if ((word0(&rv) & Exp_mask) >= Exp_msk1 * (DBL_MAX_EXP + Bias - P)) { if (word0(&rv0) == Big0 && word1(&rv0) == Big1) goto ovfl; word0(&rv) = Big0; word1(&rv) = Big1; goto cont; } else word0(&rv) += P * Exp_msk1; } else { #ifdef Avoid_Underflow if (scale && y <= 2 * P * Exp_msk1) { if (aadj <= 0x7fffffff) { if ((z = (uint32_t)aadj) <= 0) z = 1; aadj = z; aadj1 = dsign ? aadj : -aadj; } dval(&aadj2) = aadj1; word0(&aadj2) += (2 * P + 1) * Exp_msk1 - y; aadj1 = dval(&aadj2); } adj.d = aadj1 * ulp(&rv); dval(&rv) += adj.d; #else #ifdef Sudden_Underflow if ((word0(&rv) & Exp_mask) <= P * Exp_msk1) { dval(&rv0) = dval(&rv); word0(&rv) += P * Exp_msk1; adj.d = aadj1 * ulp(&rv); dval(&rv) += adj.d; if ((word0(&rv) & Exp_mask) <= P * Exp_msk1) { if (word0(&rv0) == Tiny0 && word1(&rv0) == Tiny1) goto undfl; word0(&rv) = Tiny0; word1(&rv) = Tiny1; goto cont; } else word0(&rv) -= P * Exp_msk1; } else { adj.d = aadj1 * ulp(&rv); dval(&rv) += adj.d; } #else /*Sudden_Underflow*/ /* Compute adj so that the IEEE rounding rules will * correctly round rv + adj in some half-way cases. * If rv * ulp(rv) is denormalized (i.e., * y <= (P - 1) * Exp_msk1), we must adjust aadj to avoid * trouble from bits lost to denormalization; * example: 1.2e-307 . */ if (y <= (P - 1) * Exp_msk1 && aadj > 1.) { aadj1 = (double)(int)(aadj + 0.5); if (!dsign) aadj1 = -aadj1; } adj.d = aadj1 * ulp(&rv); dval(&rv) += adj.d; #endif /*Sudden_Underflow*/ #endif /*Avoid_Underflow*/ } z = word0(&rv) & Exp_mask; #ifndef SET_INEXACT #ifdef Avoid_Underflow if (!scale) #endif if (y == z) { /* Can we stop now? */ L = (int32_t)aadj; aadj -= L; /* The tolerances below are conservative. */ if (dsign || word1(&rv) || word0(&rv) & Bndry_mask) { if (aadj < .4999999 || aadj > .5000001) break; } else if (aadj < .4999999 / FLT_RADIX) break; } #endif cont: ; } #ifdef SET_INEXACT if (inexact) { if (!oldinexact) { word0(&rv0) = Exp_1 + (70 << Exp_shift); word1(&rv0) = 0; dval(&rv0) += 1.; } } else if (!oldinexact) clear_inexact(); #endif #ifdef Avoid_Underflow if (scale) { word0(&rv0) = Exp_1 - 2 * P * Exp_msk1; word1(&rv0) = 0; dval(&rv) *= dval(&rv0); #ifndef NO_ERRNO /* try to avoid the bug of testing an 8087 register value */ if (word0(&rv) == 0 && word1(&rv) == 0) errno = ERANGE; #endif } #endif /* Avoid_Underflow */ #ifdef SET_INEXACT if (inexact && !(word0(&rv) & Exp_mask)) { /* set underflow bit */ dval(&rv0) = 1e-300; dval(&rv0) *= dval(&rv0); } #endif ret: if (se) *se = const_cast(s); return sign ? -dval(&rv) : dval(&rv); } static ALWAYS_INLINE int quorem(BigInt& b, BigInt& S) { size_t n; uint32_t *bx, *bxe, q, *sx, *sxe; #ifdef USE_LONG_LONG unsigned long long borrow, carry, y, ys; #else uint32_t borrow, carry, y, ys; #ifdef Pack_32 uint32_t si, z, zs; #endif #endif ASSERT(b.size() <= 1 || b.words()[b.size() - 1]); ASSERT(S.size() <= 1 || S.words()[S.size() - 1]); n = S.size(); ASSERT_WITH_MESSAGE(b.size() <= n, "oversize b in quorem"); if (b.size() < n) return 0; sx = S.words(); sxe = sx + --n; bx = b.words(); bxe = bx + n; q = *bxe / (*sxe + 1); /* ensure q <= true quotient */ ASSERT_WITH_MESSAGE(q <= 9, "oversized quotient in quorem"); if (q) { borrow = 0; carry = 0; do { #ifdef USE_LONG_LONG ys = *sx++ * (unsigned long long)q + carry; carry = ys >> 32; y = *bx - (ys & 0xffffffffUL) - borrow; borrow = y >> 32 & (uint32_t)1; *bx++ = (uint32_t)y & 0xffffffffUL; #else #ifdef Pack_32 si = *sx++; ys = (si & 0xffff) * q + carry; zs = (si >> 16) * q + (ys >> 16); carry = zs >> 16; y = (*bx & 0xffff) - (ys & 0xffff) - borrow; borrow = (y & 0x10000) >> 16; z = (*bx >> 16) - (zs & 0xffff) - borrow; borrow = (z & 0x10000) >> 16; Storeinc(bx, z, y); #else ys = *sx++ * q + carry; carry = ys >> 16; y = *bx - (ys & 0xffff) - borrow; borrow = (y & 0x10000) >> 16; *bx++ = y & 0xffff; #endif #endif } while (sx <= sxe); if (!*bxe) { bx = b.words(); while (--bxe > bx && !*bxe) --n; b.resize(n); } } if (cmp(b, S) >= 0) { q++; borrow = 0; carry = 0; bx = b.words(); sx = S.words(); do { #ifdef USE_LONG_LONG ys = *sx++ + carry; carry = ys >> 32; y = *bx - (ys & 0xffffffffUL) - borrow; borrow = y >> 32 & (uint32_t)1; *bx++ = (uint32_t)y & 0xffffffffUL; #else #ifdef Pack_32 si = *sx++; ys = (si & 0xffff) + carry; zs = (si >> 16) + (ys >> 16); carry = zs >> 16; y = (*bx & 0xffff) - (ys & 0xffff) - borrow; borrow = (y & 0x10000) >> 16; z = (*bx >> 16) - (zs & 0xffff) - borrow; borrow = (z & 0x10000) >> 16; Storeinc(bx, z, y); #else ys = *sx++ + carry; carry = ys >> 16; y = *bx - (ys & 0xffff) - borrow; borrow = (y & 0x10000) >> 16; *bx++ = y & 0xffff; #endif #endif } while (sx <= sxe); bx = b.words(); bxe = bx + n; if (!*bxe) { while (--bxe > bx && !*bxe) --n; b.resize(n); } } return q; } /* dtoa for IEEE arithmetic (dmg): convert double to ASCII string. * * Inspired by "How to Print Floating-Point Numbers Accurately" by * Guy L. Steele, Jr. and Jon L. White [Proc. ACM SIGPLAN '90, pp. 92-101]. * * Modifications: * 1. Rather than iterating, we use a simple numeric overestimate * to determine k = floor(log10(d)). We scale relevant * quantities using O(log2(k)) rather than O(k) multiplications. * 2. For some modes > 2 (corresponding to ecvt and fcvt), we don't * try to generate digits strictly left to right. Instead, we * compute with fewer bits and propagate the carry if necessary * when rounding the final digit up. This is often faster. * 3. Under the assumption that input will be rounded nearest, * mode 0 renders 1e23 as 1e23 rather than 9.999999999999999e22. * That is, we allow equality in stopping tests when the * round-nearest rule will give the same floating-point value * as would satisfaction of the stopping test with strict * inequality. * 4. We remove common factors of powers of 2 from relevant * quantities. * 5. When converting floating-point integers less than 1e16, * we use floating-point arithmetic rather than resorting * to multiple-precision integers. * 6. When asked to produce fewer than 15 digits, we first try * to get by with floating-point arithmetic; we resort to * multiple-precision integer arithmetic only if we cannot * guarantee that the floating-point calculation has given * the correctly rounded result. For k requested digits and * "uniformly" distributed input, the probability is * something like 10^(k-15) that we must resort to the int32_t * calculation. */ void dtoa(char* result, double dd, int ndigits, int* decpt, int* sign, char** rve) { /* Arguments ndigits, decpt, sign are similar to those of ecvt and fcvt; trailing zeros are suppressed from the returned string. If not null, *rve is set to point to the end of the return value. If d is +-Infinity or NaN, then *decpt is set to 9999. */ int bbits, b2, b5, be, dig, i, ieps, ilim = 0, ilim0, ilim1 = 0, j, j1, k, k0, k_check, leftright, m2, m5, s2, s5, spec_case, try_quick; int32_t L; #ifndef Sudden_Underflow int denorm; uint32_t x; #endif BigInt b, b1, delta, mlo, mhi, S; U d2, eps, u; double ds; char *s, *s0; #ifdef SET_INEXACT int inexact, oldinexact; #endif u.d = dd; if (word0(&u) & Sign_bit) { /* set sign for everything, including 0's and NaNs */ *sign = 1; word0(&u) &= ~Sign_bit; /* clear sign bit */ } else *sign = 0; if ((word0(&u) & Exp_mask) == Exp_mask) { /* Infinity or NaN */ *decpt = 9999; if (!word1(&u) && !(word0(&u) & 0xfffff)) strcpy(result, "Infinity"); else strcpy(result, "NaN"); return; } if (!dval(&u)) { *decpt = 1; result[0] = '0'; result[1] = '\0'; return; } #ifdef SET_INEXACT try_quick = oldinexact = get_inexact(); inexact = 1; #endif d2b(b, &u, &be, &bbits); #ifdef Sudden_Underflow i = (int)(word0(&u) >> Exp_shift1 & (Exp_mask >> Exp_shift1)); #else if ((i = (int)(word0(&u) >> Exp_shift1 & (Exp_mask >> Exp_shift1)))) { #endif dval(&d2) = dval(&u); word0(&d2) &= Frac_mask1; word0(&d2) |= Exp_11; /* log(x) ~=~ log(1.5) + (x-1.5)/1.5 * log10(x) = log(x) / log(10) * ~=~ log(1.5)/log(10) + (x-1.5)/(1.5*log(10)) * log10(d) = (i-Bias)*log(2)/log(10) + log10(d2) * * This suggests computing an approximation k to log10(d) by * * k = (i - Bias)*0.301029995663981 * + ( (d2-1.5)*0.289529654602168 + 0.176091259055681 ); * * We want k to be too large rather than too small. * The error in the first-order Taylor series approximation * is in our favor, so we just round up the constant enough * to compensate for any error in the multiplication of * (i - Bias) by 0.301029995663981; since |i - Bias| <= 1077, * and 1077 * 0.30103 * 2^-52 ~=~ 7.2e-14, * adding 1e-13 to the constant term more than suffices. * Hence we adjust the constant term to 0.1760912590558. * (We could get a more accurate k by invoking log10, * but this is probably not worthwhile.) */ i -= Bias; #ifndef Sudden_Underflow denorm = 0; } else { /* d is denormalized */ i = bbits + be + (Bias + (P - 1) - 1); x = (i > 32) ? (word0(&u) << (64 - i)) | (word1(&u) >> (i - 32)) : word1(&u) << (32 - i); dval(&d2) = x; word0(&d2) -= 31 * Exp_msk1; /* adjust exponent */ i -= (Bias + (P - 1) - 1) + 1; denorm = 1; } #endif ds = (dval(&d2) - 1.5) * 0.289529654602168 + 0.1760912590558 + (i * 0.301029995663981); k = (int)ds; if (ds < 0. && ds != k) k--; /* want k = floor(ds) */ k_check = 1; if (k >= 0 && k <= Ten_pmax) { if (dval(&u) < tens[k]) k--; k_check = 0; } j = bbits - i - 1; if (j >= 0) { b2 = 0; s2 = j; } else { b2 = -j; s2 = 0; } if (k >= 0) { b5 = 0; s5 = k; s2 += k; } else { b2 -= k; b5 = -k; s5 = 0; } #ifndef SET_INEXACT #ifdef Check_FLT_ROUNDS try_quick = Rounding == 1; #else try_quick = 1; #endif #endif /*SET_INEXACT*/ leftright = 1; ilim = ilim1 = -1; i = 18; ndigits = 0; s = s0 = result; if (ilim >= 0 && ilim <= Quick_max && try_quick) { /* Try to get by with floating-point arithmetic. */ i = 0; dval(&d2) = dval(&u); k0 = k; ilim0 = ilim; ieps = 2; /* conservative */ if (k > 0) { ds = tens[k & 0xf]; j = k >> 4; if (j & Bletch) { /* prevent overflows */ j &= Bletch - 1; dval(&u) /= bigtens[n_bigtens - 1]; ieps++; } for (; j; j >>= 1, i++) { if (j & 1) { ieps++; ds *= bigtens[i]; } } dval(&u) /= ds; } else if ((j1 = -k)) { dval(&u) *= tens[j1 & 0xf]; for (j = j1 >> 4; j; j >>= 1, i++) { if (j & 1) { ieps++; dval(&u) *= bigtens[i]; } } } if (k_check && dval(&u) < 1. && ilim > 0) { if (ilim1 <= 0) goto fast_failed; ilim = ilim1; k--; dval(&u) *= 10.; ieps++; } dval(&eps) = (ieps * dval(&u)) + 7.; word0(&eps) -= (P - 1) * Exp_msk1; if (ilim == 0) { S.clear(); mhi.clear(); dval(&u) -= 5.; if (dval(&u) > dval(&eps)) goto one_digit; if (dval(&u) < -dval(&eps)) goto no_digits; goto fast_failed; } #ifndef No_leftright if (leftright) { /* Use Steele & White method of only * generating digits needed. */ dval(&eps) = (0.5 / tens[ilim - 1]) - dval(&eps); for (i = 0;;) { L = (long int)dval(&u); dval(&u) -= L; *s++ = '0' + (int)L; if (dval(&u) < dval(&eps)) goto ret; if (1. - dval(&u) < dval(&eps)) goto bump_up; if (++i >= ilim) break; dval(&eps) *= 10.; dval(&u) *= 10.; } } else { #endif /* Generate ilim digits, then fix them up. */ dval(&eps) *= tens[ilim - 1]; for (i = 1;; i++, dval(&u) *= 10.) { L = (int32_t)(dval(&u)); if (!(dval(&u) -= L)) ilim = i; *s++ = '0' + (int)L; if (i == ilim) { if (dval(&u) > 0.5 + dval(&eps)) goto bump_up; else if (dval(&u) < 0.5 - dval(&eps)) { while (*--s == '0') { } s++; goto ret; } break; } } #ifndef No_leftright } #endif fast_failed: s = s0; dval(&u) = dval(&d2); k = k0; ilim = ilim0; } /* Do we have a "small" integer? */ if (be >= 0 && k <= Int_max) { /* Yes. */ ds = tens[k]; if (ndigits < 0 && ilim <= 0) { S.clear(); mhi.clear(); if (ilim < 0 || dval(&u) <= 5 * ds) goto no_digits; goto one_digit; } for (i = 1;; i++, dval(&u) *= 10.) { L = (int32_t)(dval(&u) / ds); dval(&u) -= L * ds; #ifdef Check_FLT_ROUNDS /* If FLT_ROUNDS == 2, L will usually be high by 1 */ if (dval(&u) < 0) { L--; dval(&u) += ds; } #endif *s++ = '0' + (int)L; if (!dval(&u)) { #ifdef SET_INEXACT inexact = 0; #endif break; } if (i == ilim) { dval(&u) += dval(&u); if (dval(&u) > ds || (dval(&u) == ds && (L & 1))) { bump_up: while (*--s == '9') if (s == s0) { k++; *s = '0'; break; } ++*s++; } break; } } goto ret; } m2 = b2; m5 = b5; mhi.clear(); mlo.clear(); if (leftright) { i = #ifndef Sudden_Underflow denorm ? be + (Bias + (P - 1) - 1 + 1) : #endif 1 + P - bbits; b2 += i; s2 += i; i2b(mhi, 1); } if (m2 > 0 && s2 > 0) { i = m2 < s2 ? m2 : s2; b2 -= i; m2 -= i; s2 -= i; } if (b5 > 0) { if (leftright) { if (m5 > 0) { pow5mult(mhi, m5); mult(b, mhi); } if ((j = b5 - m5)) pow5mult(b, j); } else pow5mult(b, b5); } i2b(S, 1); if (s5 > 0) pow5mult(S, s5); /* Check for special case that d is a normalized power of 2. */ spec_case = 0; if (!word1(&u) && !(word0(&u) & Bndry_mask) #ifndef Sudden_Underflow && word0(&u) & (Exp_mask & ~Exp_msk1) #endif ) { /* The special case */ b2 += Log2P; s2 += Log2P; spec_case = 1; } /* Arrange for convenient computation of quotients: * shift left if necessary so divisor has 4 leading 0 bits. * * Perhaps we should just compute leading 28 bits of S once * and for all and pass them and a shift to quorem, so it * can do shifts and ors to compute the numerator for q. */ #ifdef Pack_32 if ((i = ((s5 ? 32 - hi0bits(S.words()[S.size() - 1]) : 1) + s2) & 0x1f)) i = 32 - i; #else if ((i = ((s5 ? 32 - hi0bits(S.words()[S.size() - 1]) : 1) + s2) & 0xf)) i = 16 - i; #endif if (i > 4) { i -= 4; b2 += i; m2 += i; s2 += i; } else if (i < 4) { i += 28; b2 += i; m2 += i; s2 += i; } if (b2 > 0) lshift(b, b2); if (s2 > 0) lshift(S, s2); if (k_check) { if (cmp(b,S) < 0) { k--; multadd(b, 10, 0); /* we botched the k estimate */ if (leftright) multadd(mhi, 10, 0); ilim = ilim1; } } if (leftright) { if (m2 > 0) lshift(mhi, m2); /* Compute mlo -- check for special case * that d is a normalized power of 2. */ mlo = mhi; if (spec_case) { mhi = mlo; lshift(mhi, Log2P); } for (i = 1;;i++) { dig = quorem(b,S) + '0'; /* Do we yet have the shortest decimal string * that will round to d? */ j = cmp(b, mlo); diff(delta, S, mhi); j1 = delta.sign ? 1 : cmp(b, delta); if (j1 == 0 && !(word1(&u) & 1)) { if (dig == '9') goto round_9_up; if (j > 0) dig++; #ifdef SET_INEXACT else if (!b->x[0] && b->wds <= 1) inexact = 0; #endif *s++ = dig; goto ret; } if (j < 0 || (j == 0 && !(word1(&u) & 1))) { if (!b.words()[0] && b.size() <= 1) { #ifdef SET_INEXACT inexact = 0; #endif goto accept_dig; } if (j1 > 0) { lshift(b, 1); j1 = cmp(b, S); if ((j1 > 0 || (j1 == 0 && (dig & 1))) && dig++ == '9') goto round_9_up; } accept_dig: *s++ = dig; goto ret; } if (j1 > 0) { if (dig == '9') { /* possible if i == 1 */ round_9_up: *s++ = '9'; goto roundoff; } *s++ = dig + 1; goto ret; } *s++ = dig; if (i == ilim) break; multadd(b, 10, 0); multadd(mlo, 10, 0); multadd(mhi, 10, 0); } } else for (i = 1;; i++) { *s++ = dig = quorem(b,S) + '0'; if (!b.words()[0] && b.size() <= 1) { #ifdef SET_INEXACT inexact = 0; #endif goto ret; } if (i >= ilim) break; multadd(b, 10, 0); } /* Round off last digit */ lshift(b, 1); j = cmp(b, S); if (j > 0 || (j == 0 && (dig & 1))) { roundoff: while (*--s == '9') if (s == s0) { k++; *s++ = '1'; goto ret; } ++*s++; } else { while (*--s == '0') { } s++; } goto ret; no_digits: k = -1 - ndigits; goto ret; one_digit: *s++ = '1'; k++; goto ret; ret: #ifdef SET_INEXACT if (inexact) { if (!oldinexact) { word0(&u) = Exp_1 + (70 << Exp_shift); word1(&u) = 0; dval(&u) += 1.; } } else if (!oldinexact) clear_inexact(); #endif *s = 0; *decpt = k + 1; if (rve) *rve = s; } } // namespace WTF