source: webkit/trunk/JavaScriptCore/wtf/dtoa.cpp@ 65588

/****************************************************************
 *
 * 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.
    [email protected]
 */

/* 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 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 <errno.h>
#else
#define NO_ERRNO
#endif
#include <math.h>
#include <stdint.h>
#include <stdio.h>
#include <stdlib.h>
#include <string.h>
#include <wtf/AlwaysInline.h>
#include <wtf/Assertions.h>
#include <wtf/FastMalloc.h>
#include <wtf/MathExtras.h>
#include <wtf/Threading.h>
#include <wtf/Vector.h>

#if COMPILER(MSVC)
#pragma warning(disable: 4244)
#pragma warning(disable: 4245)
#pragma warning(disable: 4554)
#endif

#if CPU(BIG_ENDIAN)
#define IEEE_MC68k
#elif CPU(MIDDLE_ENDIAN)
#define IEEE_ARM
#else
#define IEEE_8087
#endif

#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
 *  *p++ = high << 16 | low & 0xffff;
 */
static ALWAYS_INLINE uint32_t* storeInc(uint32_t* p, uint16_t high, uint16_t low)
{
    uint16_t* p16 = reinterpret_cast<uint16_t*>(p);
#if defined(IEEE_8087) || defined(IEEE_ARM)
    p16[1] = high;
    p16[0] = low;
#else
    p16[0] = high;
    p16[1] = low;
#endif
    return p + 1;
}

#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 CPU(PPC64) || CPU(X86_64)
// FIXME: should we enable this on all 64-bit CPUs?
// 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<uint32_t, 16> 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;
    const uint32_t* xa;
    const uint32_t* xb;
    const uint32_t* xae;
    const uint32_t* xbe;
    uint32_t* xc;
    uint32_t* 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;
                xc = 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;
                xc = 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 : Noncopyable {
    BigInt val;
    P5Node* next;
};
    
static P5Node* p5s;
static int p5sCount;

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;
        p5sCount = 1;
    }

    int p5sCountLocal = p5sCount;
#if ENABLE(JSC_MULTIPLE_THREADS)
    s_dtoaP5Mutex->unlock();
#endif
    int p5sUsed = 0;

    for (;;) {
        if (k & 1)
            mult(b, p5->val);

        if (!(k >>= 1))
            break;

        if (++p5sUsed == p5sCountLocal) {
#if ENABLE(JSC_MULTIPLE_THREADS)
            s_dtoaP5Mutex->lock();
#endif
            if (p5sUsed == p5sCount) {
                ASSERT(!p5->next);
                p5->next = new P5Node;
                p5->next->next = 0;
                p5->next->val = p5->val;
                mult(p5->next->val, p5->next->val);
                ++p5sCount;
            }
            
            p5sCountLocal = p5sCount;
#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]);
    }
#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];
     }
#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;
        xc = 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;
        xc = 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 returnD;
    }
    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 returnD;
    }
    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
returnD:
#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;
    uint32_t 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

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 haveDig;
            }
            goto digDone;
        }
        for (; c >= '0' && c <= '9'; c = *++s) {
haveDig:
            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;
            }
        }
    }
digDone:
    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) {
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 dropDown;
            break;
        }
        if (!i) {
            /* 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)) {
dropDown:
                /* 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)
                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;
            }
            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;
                }
                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) && !word1(&rv))
            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<char*>(s);
    return sign ? -dval(&rv) : dval(&rv);
}

static ALWAYS_INLINE int quorem(BigInt& b, BigInt& S)
{
    size_t n;
    uint32_t* bx;
    uint32_t* bxe;
    uint32_t q;
    uint32_t* sx;
    uint32_t* 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;
            bx = 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;
            bx = 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(DtoaBuffer 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;
    char* 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");
            if (rve)
                *rve = result + 8;
        } else {
            strcpy(result, "NaN");
            if (rve)
                *rve = result + 3;
        }
        return;
    }
    if (!dval(&u)) {
        *decpt = 1;
        result[0] = '0';
        result[1] = '\0';
        if (rve)
            *rve = result + 1;
        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 fastFailed;
            ilim = ilim1;
            k--;
            dval(&u) *= 10.;
            ieps++;
        }
        dval(&eps) = (ieps * dval(&u)) + 7.;
        word0(&eps) -= (P - 1) * Exp_msk1;
        if (!ilim) {
            S.clear();
            mhi.clear();
            dval(&u) -= 5.;
            if (dval(&u) > dval(&eps))
                goto oneDigit;
            if (dval(&u) < -dval(&eps))
                goto noDigits;
            goto fastFailed;
        }
#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 bumpUp;
                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 bumpUp;
                    if (dval(&u) < 0.5 - dval(&eps)) {
                        while (*--s == '0') { }
                        s++;
                        goto ret;
                    }
                    break;
                }
            }
#ifndef No_leftright
        }
#endif
fastFailed:
        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 noDigits;
            goto oneDigit;
        }
        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))) {
bumpUp:
                    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 && !(word1(&u) & 1)) {
                if (dig == '9')
                    goto round9up;
                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 && !(word1(&u) & 1))) {
                if (!b.words()[0] && b.size() <= 1) {
#ifdef SET_INEXACT
                    inexact = 0;
#endif
                    goto acceptDig;
                }
                if (j1 > 0) {
                    lshift(b, 1);
                    j1 = cmp(b, S);
                    if ((j1 > 0 || (!j1 && (dig & 1))) && dig++ == '9')
                        goto round9up;
                }
acceptDig:
                *s++ = dig;
                goto ret;
            }
            if (j1 > 0) {
                if (dig == '9') { /* possible if i == 1 */
round9up:
                    *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 && (dig & 1))) {
roundoff:
        while (*--s == '9')
            if (s == s0) {
                k++;
                *s++ = '1';
                goto ret;
            }
        ++*s++;
    } else {
        while (*--s == '0') { }
        s++;
    }
    goto ret;
noDigits:
    k = -1 - ndigits;
    goto ret;
oneDigit:
    *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;
}

static ALWAYS_INLINE void append(char*& next, const char* src, unsigned size)
{
    for (unsigned i = 0; i < size; ++i)
        *next++ = *src++;
}

void doubleToStringInJavaScriptFormat(double d, DtoaBuffer buffer, unsigned* resultLength)
{
    ASSERT(buffer);

    // avoid ever printing -NaN, in JS conceptually there is only one NaN value
    if (isnan(d)) {
        append(buffer, "NaN", 3);
        if (resultLength)
            *resultLength = 3;
        return;
    }
    // -0 -> "0"
    if (!d) {
        buffer[0] = '0';
        if (resultLength)
            *resultLength = 1;
        return;
    }

    int decimalPoint;
    int sign;

    DtoaBuffer result;
    char* resultEnd = 0;
    WTF::dtoa(result, d, 0, &decimalPoint, &sign, &resultEnd);
    int length = resultEnd - result;

    char* next = buffer;
    if (sign)
        *next++ = '-';

    if (decimalPoint <= 0 && decimalPoint > -6) {
        *next++ = '0';
        *next++ = '.';
        for (int j = decimalPoint; j < 0; j++)
            *next++ = '0';
        append(next, result, length);
    } else if (decimalPoint <= 21 && decimalPoint > 0) {
        if (length <= decimalPoint) {
            append(next, result, length);
            for (int j = 0; j < decimalPoint - length; j++)
                *next++ = '0';
        } else {
            append(next, result, decimalPoint);
            *next++ = '.';
            append(next, result + decimalPoint, length - decimalPoint);
        }
    } else if (result[0] < '0' || result[0] > '9')
        append(next, result, length);
    else {
        *next++ = result[0];
        if (length > 1) {
            *next++ = '.';
            append(next, result + 1, length - 1);
        }

        *next++ = 'e';
        *next++ = (decimalPoint >= 0) ? '+' : '-';
        // decimalPoint can't be more than 3 digits decimal given the
        // nature of float representation
        int exponential = decimalPoint - 1;
        if (exponential < 0)
            exponential = -exponential;
        if (exponential >= 100)
            *next++ = static_cast<char>('0' + exponential / 100);
        if (exponential >= 10)
            *next++ = static_cast<char>('0' + (exponential % 100) / 10);
        *next++ = static_cast<char>('0' + exponential % 10);
    }
    if (resultLength)
        *resultLength = next - buffer;
}

} // namespace WTF