dtoa.c

/****************************************************************
 *
 * The author of this software is David M. Gay.
 *
 * Copyright (c) 1991, 2000, 2001 by Lucent Technologies.
 *
 * 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.
 *
 ***************************************************************/

/****************************************************************
 * This is dtoa.c by David M. Gay, downloaded from
 * http://www.netlib.org/fp/dtoa.c on April 15, 2009 and modified for
 * inclusion into the Python core by Mark E. T. Dickinson and Eric V. Smith.
 *
 * Please remember to check http://www.netlib.org/fp regularly (and especially
 * before any Python release) for bugfixes and updates.
 *
 * The major modifications from Gay's original code are as follows:
 *
 *  0. The original code has been specialized to Python's needs by removing
 *     many of the #ifdef'd sections.  In particular, code to support VAX and
 *     IBM floating-point formats, hex NaNs, hex floats, locale-aware
 *     treatment of the decimal point, and setting of the inexact flag have
 *     been removed.
 *
 *  1. We use PyMem_Malloc and PyMem_Free in place of malloc and free.
 *
 *  2. The public functions strtod, dtoa and freedtoa all now have
 *     a _Py_dg_ prefix.
 *
 *  3. Instead of assuming that PyMem_Malloc always succeeds, we thread
 *     PyMem_Malloc failures through the code.  The functions
 *
 *       Balloc, multadd, s2b, i2b, mult, pow5mult, lshift, diff, d2b
 *
 *     of return type *Bigint all return NULL to indicate a malloc failure.
 *     Similarly, rv_alloc and nrv_alloc (return type char *) return NULL on
 *     failure.  bigcomp now has return type int (it used to be void) and
 *     returns -1 on failure and 0 otherwise.  _Py_dg_dtoa returns NULL
 *     on failure.  _Py_dg_strtod indicates failure due to malloc failure
 *     by returning -1.0, setting errno=ENOMEM and *se to s00.
 *
 *  4. The static variable dtoa_result has been removed.  Callers of
 *     _Py_dg_dtoa are expected to call _Py_dg_freedtoa to free
 *     the memory allocated by _Py_dg_dtoa.
 *
 *  5. The code has been reformatted to better fit with Python's
 *     C style guide (PEP 7).
 *
 *  6. A bug in the memory allocation has been fixed: to avoid FREEing memory
 *     that hasn't been MALLOC'ed, private_mem should only be used when k <=
 *     Kmax.
 *
 *  7. _Py_dg_strtod has been modified so that it doesn't accept strings with
 *     leading whitespace.
 *
 ***************************************************************/

/* Please send bug reports for the original dtoa.c code to David M. Gay (dmg
 * at acm dot org, with " at " changed at "@" and " dot " changed to ".").
 * Please report bugs for this modified version using the Python issue tracker
 * (http://bugs.python.org). */

/* 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-, VAX-, and IBM-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, IBM, or VAX double-precision
 *              arithmetic (not IEEE double-extended).
 *      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).
 */

/* Linking of Python's #defines to Gay's #defines starts here. */

#include "Python.h"

/* if PY_NO_SHORT_FLOAT_REPR is defined, then don't even try to compile
   the following code */
#ifndef PY_NO_SHORT_FLOAT_REPR

#include "float.h"

#define MALLOC PyMem_Malloc
#define FREE PyMem_Free

/* This code should also work for ARM mixed-endian format on little-endian
   machines, where doubles have byte order 45670123 (in increasing address
   order, 0 being the least significant byte). */
#ifdef DOUBLE_IS_LITTLE_ENDIAN_IEEE754
#  define IEEE_8087
#endif
#if defined(DOUBLE_IS_BIG_ENDIAN_IEEE754) ||  \
  defined(DOUBLE_IS_ARM_MIXED_ENDIAN_IEEE754)
#  define IEEE_MC68k
#endif
#if defined(IEEE_8087) + defined(IEEE_MC68k) != 1
#error "Exactly one of IEEE_8087 or IEEE_MC68k should be defined."
#endif

/* The code below assumes that the endianness of integers matches the
   endianness of the two 32-bit words of a double.  Check this. */
#if defined(WORDS_BIGENDIAN) && (defined(DOUBLE_IS_LITTLE_ENDIAN_IEEE754) || \
                                 defined(DOUBLE_IS_ARM_MIXED_ENDIAN_IEEE754))
#error "doubles and ints have incompatible endianness"
#endif

#if !defined(WORDS_BIGENDIAN) && defined(DOUBLE_IS_BIG_ENDIAN_IEEE754)
#error "doubles and ints have incompatible endianness"
#endif


#if defined(HAVE_UINT32_T) && defined(HAVE_INT32_T)
typedef PY_UINT32_T ULong;
typedef PY_INT32_T Long;
#else
#error "Failed to find an exact-width 32-bit integer type"
#endif

#if defined(HAVE_UINT64_T)
#define ULLong PY_UINT64_T
#else
#undef ULLong
#endif

#undef DEBUG
#ifdef Py_DEBUG
#define DEBUG
#endif

/* End Python #define linking */

#ifdef DEBUG
#define Bug(x) {fprintf(stderr, "%s\n", x); exit(1);}
#endif

#ifndef PRIVATE_MEM
#define PRIVATE_MEM 2304
#endif
#define PRIVATE_mem ((PRIVATE_MEM+sizeof(double)-1)/sizeof(double))
static double private_mem[PRIVATE_mem], *pmem_next = private_mem;

#ifdef __cplusplus
extern "C" {
#endif

typedef union { double d; ULong L[2]; } U;

#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

#ifndef STRTOD_DIGLIM
#define STRTOD_DIGLIM 40
#endif

/* maximum permitted exponent value for strtod; exponents larger than
   MAX_ABS_EXP in absolute value get truncated to +-MAX_ABS_EXP.  MAX_ABS_EXP
   should fit into an int. */
#ifndef MAX_ABS_EXP
#define MAX_ABS_EXP 1100000000U
#endif
/* Bound on length of pieces of input strings in _Py_dg_strtod; specifically,
   this is used to bound the total number of digits ignoring leading zeros and
   the number of digits that follow the decimal point.  Ideally, MAX_DIGITS
   should satisfy MAX_DIGITS + 400 < MAX_ABS_EXP; that ensures that the
   exponent clipping in _Py_dg_strtod can't affect the value of the output. */
#ifndef MAX_DIGITS
#define MAX_DIGITS 1000000000U
#endif

/* Guard against trying to use the above values on unusual platforms with ints
 * of width less than 32 bits. */
#if MAX_ABS_EXP > INT_MAX
#error "MAX_ABS_EXP should fit in an int"
#endif
#if MAX_DIGITS > INT_MAX
#error "MAX_DIGITS should fit in an int"
#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)
#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 P DBL_MANT_DIG */
/* Ten_pmax = floor(P*log(2)/log(5)) */
/* Bletch = (highest power of 2 < DBL_MAX_10_EXP) / 16 */
/* Quick_max = floor((P-1)*log(FLT_RADIX)/log(10) - 1) */
/* Int_max = floor(P*log(FLT_RADIX)/log(10) - 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 Nbits 53
#define Bias 1023
#define Emax 1023
#define Emin (-1022)
#define Etiny (-1074)  /* smallest denormal is 2**Etiny */
#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 Sign_bit 0x80000000
#define Log2P 1
#define Tiny0 0
#define Tiny1 1
#define Quick_max 14
#define Int_max 14

#ifndef Flt_Rounds
#ifdef FLT_ROUNDS
#define Flt_Rounds FLT_ROUNDS
#else
#define Flt_Rounds 1
#endif
#endif /*Flt_Rounds*/

#define Rounding Flt_Rounds

#define Big0 (Frac_mask1 | Exp_msk1*(DBL_MAX_EXP+Bias-1))
#define Big1 0xffffffff

/* Standard NaN used by _Py_dg_stdnan. */

#define NAN_WORD0 0x7ff80000
#define NAN_WORD1 0

/* Bits of the representation of positive infinity. */

#define POSINF_WORD0 0x7ff00000
#define POSINF_WORD1 0

/* struct BCinfo is used to pass information from _Py_dg_strtod to bigcomp */

typedef struct BCinfo BCinfo;
struct
BCinfo {
    int e0, nd, nd0, scale;
};

#define FFFFFFFF 0xffffffffUL

#define Kmax 7

/* struct Bigint is used to represent arbitrary-precision integers.  These
   integers are stored in sign-magnitude format, with the magnitude stored as
   an array of base 2**32 digits.  Bigints are always normalized: if x is a
   Bigint then x->wds >= 1, and either x->wds == 1 or x[wds-1] is nonzero.

   The Bigint fields are as follows:

     - next is a header used by Balloc and Bfree to keep track of lists
         of freed Bigints;  it's also used for the linked list of
         powers of 5 of the form 5**2**i used by pow5mult.
     - k indicates which pool this Bigint was allocated from
     - maxwds is the maximum number of words space was allocated for
       (usually maxwds == 2**k)
     - sign is 1 for negative Bigints, 0 for positive.  The sign is unused
       (ignored on inputs, set to 0 on outputs) in almost all operations
       involving Bigints: a notable exception is the diff function, which
       ignores signs on inputs but sets the sign of the output correctly.
     - wds is the actual number of significant words
     - x contains the vector of words (digits) for this Bigint, from least
       significant (x[0]) to most significant (x[wds-1]).
*/

struct
Bigint {
    struct Bigint *next;
    int k, maxwds, sign, wds;
    ULong x[1];
};

typedef struct Bigint Bigint;

#ifndef Py_USING_MEMORY_DEBUGGER

/* Memory management: memory is allocated from, and returned to, Kmax+1 pools
   of memory, where pool k (0 <= k <= Kmax) is for Bigints b with b->maxwds ==
   1 << k.  These pools are maintained as linked lists, with freelist[k]
   pointing to the head of the list for pool k.

   On allocation, if there's no free slot in the appropriate pool, MALLOC is
   called to get more memory.  This memory is not returned to the system until
   Python quits.  There's also a private memory pool that's allocated from
   in preference to using MALLOC.

   For Bigints with more than (1 << Kmax) digits (which implies at least 1233
   decimal digits), memory is directly allocated using MALLOC, and freed using
   FREE.

   XXX: it would be easy to bypass this memory-management system and
   translate each call to Balloc into a call to PyMem_Malloc, and each
   Bfree to PyMem_Free.  Investigate whether this has any significant
   performance on impact. */

static Bigint *freelist[Kmax+1];

/* Allocate space for a Bigint with up to 1<<k digits */

static Bigint *
Balloc(int k)
{
    int x;
    Bigint *rv;
    unsigned int len;

    if (k <= Kmax && (rv = freelist[k]))
        freelist[k] = rv->next;
    else {
        x = 1 << k;
        len = (sizeof(Bigint) + (x-1)*sizeof(ULong) + sizeof(double) - 1)
            /sizeof(double);
        if (k <= Kmax && pmem_next - private_mem + len <= (Py_ssize_t)PRIVATE_mem) {
            rv = (Bigint*)pmem_next;
            pmem_next += len;
        }
        else {
            rv = (Bigint*)MALLOC(len*sizeof(double));
            if (rv == NULL)
                return NULL;
        }
        rv->k = k;
        rv->maxwds = x;
    }
    rv->sign = rv->wds = 0;
    return rv;
}

/* Free a Bigint allocated with Balloc */

static void
Bfree(Bigint *v)
{
    if (v) {
        if (v->k > Kmax)
            FREE((void*)v);
        else {
            v->next = freelist[v->k];
            freelist[v->k] = v;
        }
    }
}

#else

/* Alternative versions of Balloc and Bfree that use PyMem_Malloc and
   PyMem_Free directly in place of the custom memory allocation scheme above.
   These are provided for the benefit of memory debugging tools like
   Valgrind. */

/* Allocate space for a Bigint with up to 1<<k digits */

static Bigint *
Balloc(int k)
{
    int x;
    Bigint *rv;
    unsigned int len;

    x = 1 << k;
    len = (sizeof(Bigint) + (x-1)*sizeof(ULong) + sizeof(double) - 1)
        /sizeof(double);

    rv = (Bigint*)MALLOC(len*sizeof(double));
    if (rv == NULL)
        return NULL;

    rv->k = k;
    rv->maxwds = x;
    rv->sign = rv->wds = 0;
    return rv;
}

/* Free a Bigint allocated with Balloc */

static void
Bfree(Bigint *v)
{
    if (v) {
        FREE((void*)v);
    }
}

#endif /* Py_USING_MEMORY_DEBUGGER */

#define Bcopy(x,y) memcpy((char *)&x->sign, (char *)&y->sign,   \
                          y->wds*sizeof(Long) + 2*sizeof(int))

/* Multiply a Bigint b by m and add a.  Either modifies b in place and returns
   a pointer to the modified b, or Bfrees b and returns a pointer to a copy.
   On failure, return NULL.  In this case, b will have been already freed. */

static Bigint *
multadd(Bigint *b, int m, int a)       /* multiply by m and add a */
{
    int i, wds;
#ifdef ULLong
    ULong *x;
    ULLong carry, y;
#else
    ULong carry, *x, y;
    ULong xi, z;
#endif
    Bigint *b1;

    wds = b->wds;
    x = b->x;
    i = 0;
    carry = a;
    do {
#ifdef ULLong
        y = *x * (ULLong)m + carry;
        carry = y >> 32;
        *x++ = (ULong)(y & FFFFFFFF);
#else
        xi = *x;
        y = (xi & 0xffff) * m + carry;
        z = (xi >> 16) * m + (y >> 16);
        carry = z >> 16;
        *x++ = (z << 16) + (y & 0xffff);
#endif
    }
    while(++i < wds);
    if (carry) {
        if (wds >= b->maxwds) {
            b1 = Balloc(b->k+1);
            if (b1 == NULL){
                Bfree(b);
                return NULL;
            }
            Bcopy(b1, b);
            Bfree(b);
            b = b1;
        }
        b->x[wds++] = (ULong)carry;
        b->wds = wds;
    }
    return b;
}

/* convert a string s containing nd decimal digits (possibly containing a
   decimal separator at position nd0, which is ignored) to a Bigint.  This
   function carries on where the parsing code in _Py_dg_strtod leaves off: on
   entry, y9 contains the result of converting the first 9 digits.  Returns
   NULL on failure. */

static Bigint *
s2b(const char *s, int nd0, int nd, ULong y9)
{
    Bigint *b;
    int i, k;
    Long x, y;

    x = (nd + 8) / 9;
    for(k = 0, y = 1; x > y; y <<= 1, k++) ;
    b = Balloc(k);
    if (b == NULL)
        return NULL;
    b->x[0] = y9;
    b->wds = 1;

    if (nd <= 9)
      return b;

    s += 9;
    for (i = 9; i < nd0; i++) {
        b = multadd(b, 10, *s++ - '0');
        if (b == NULL)
            return NULL;
    }
    s++;
    for(; i < nd; i++) {
        b = multadd(b, 10, *s++ - '0');
        if (b == NULL)
            return NULL;
    }
    return b;
}

/* count leading 0 bits in the 32-bit integer x. */

static int
hi0bits(ULong 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;
}

/* count trailing 0 bits in the 32-bit integer y, and shift y right by that
   number of bits. */

static int
lo0bits(ULong *y)
{
    int k;
    ULong 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)
            return 32;
    }
    *y = x;
    return k;
}

/* convert a small nonnegative integer to a Bigint */

static Bigint *
i2b(int i)
{
    Bigint *b;

    b = Balloc(1);
    if (b == NULL)
        return NULL;
    b->x[0] = i;
    b->wds = 1;
    return b;
}

/* multiply two Bigints.  Returns a new Bigint, or NULL on failure.  Ignores
   the signs of a and b. */

static Bigint *
mult(Bigint *a, Bigint *b)
{
    Bigint *c;
    int k, wa, wb, wc;
    ULong *x, *xa, *xae, *xb, *xbe, *xc, *xc0;
    ULong y;
#ifdef ULLong
    ULLong carry, z;
#else
    ULong carry, z;
    ULong z2;
#endif

    if ((!a->x[0] && a->wds == 1) || (!b->x[0] && b->wds == 1)) {
        c = Balloc(0);
        if (c == NULL)
            return NULL;
        c->wds = 1;
        c->x[0] = 0;
        return c;
    }

    if (a->wds < b->wds) {
        c = a;
        a = b;
        b = c;
    }
    k = a->k;
    wa = a->wds;
    wb = b->wds;
    wc = wa + wb;
    if (wc > a->maxwds)
        k++;
    c = Balloc(k);
    if (c == NULL)
        return NULL;
    for(x = c->x, xa = x + wc; x < xa; x++)
        *x = 0;
    xa = a->x;
    xae = xa + wa;
    xb = b->x;
    xbe = xb + wb;
    xc0 = c->x;
#ifdef ULLong
    for(; xb < xbe; xc0++) {
        if ((y = *xb++)) {
            x = xa;
            xc = xc0;
            carry = 0;
            do {
                z = *x++ * (ULLong)y + *xc + carry;
                carry = z >> 32;
                *xc++ = (ULong)(z & FFFFFFFF);
            }
            while(x < xae);
            *xc = (ULong)carry;
        }
    }
#else
    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;
                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;
            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;
        }
    }
#endif
    for(xc0 = c->x, xc = xc0 + wc; wc > 0 && !*--xc; --wc) ;
    c->wds = wc;
    return c;
}

#ifndef Py_USING_MEMORY_DEBUGGER

/* p5s is a linked list of powers of 5 of the form 5**(2**i), i >= 2 */

static Bigint *p5s;

/* multiply the Bigint b by 5**k.  Returns a pointer to the result, or NULL on
   failure; if the returned pointer is distinct from b then the original
   Bigint b will have been Bfree'd.   Ignores the sign of b. */

static Bigint *
pow5mult(Bigint *b, int k)
{
    Bigint *b1, *p5, *p51;
    int i;
    static int p05[3] = { 5, 25, 125 };

    if ((i = k & 3)) {
        b = multadd(b, p05[i-1], 0);
        if (b == NULL)
            return NULL;
    }

    if (!(k >>= 2))
        return b;
    p5 = p5s;
    if (!p5) {
        /* first time */
        p5 = i2b(625);
        if (p5 == NULL) {
            Bfree(b);
            return NULL;
        }
        p5s = p5;
        p5->next = 0;
    }
    for(;;) {
        if (k & 1) {
            b1 = mult(b, p5);
            Bfree(b);
            b = b1;
            if (b == NULL)
                return NULL;
        }
        if (!(k >>= 1))
            break;
        p51 = p5->next;
        if (!p51) {
            p51 = mult(p5,p5);
            if (p51 == NULL) {
                Bfree(b);
                return NULL;
            }
            p51->next = 0;
            p5->next = p51;
        }
        p5 = p51;
    }
    return b;
}

#else

/* Version of pow5mult that doesn't cache powers of 5. Provided for
   the benefit of memory debugging tools like Valgrind. */

static Bigint *
pow5mult(Bigint *b, int k)
{
    Bigint *b1, *p5, *p51;
    int i;
    static int p05[3] = { 5, 25, 125 };

    if ((i = k & 3)) {
        b = multadd(b, p05[i-1], 0);
        if (b == NULL)
            return NULL;
    }

    if (!(k >>= 2))
        return b;
    p5 = i2b(625);
    if (p5 == NULL) {
        Bfree(b);
        return NULL;
    }

    for(;;) {
        if (k & 1) {
            b1 = mult(b, p5);
            Bfree(b);
            b = b1;
            if (b == NULL) {
                Bfree(p5);
                return NULL;
            }
        }
        if (!(k >>= 1))
            break;
        p51 = mult(p5, p5);
        Bfree(p5);
        p5 = p51;
        if (p5 == NULL) {
            Bfree(b);
            return NULL;
        }
    }
    Bfree(p5);
    return b;
}

#endif /* Py_USING_MEMORY_DEBUGGER */

/* shift a Bigint b left by k bits.  Return a pointer to the shifted result,
   or NULL on failure.  If the returned pointer is distinct from b then the
   original b will have been Bfree'd.   Ignores the sign of b. */

static Bigint *
lshift(Bigint *b, int k)
{
    int i, k1, n, n1;
    Bigint *b1;
    ULong *x, *x1, *xe, z;

    if (!k || (!b->x[0] && b->wds == 1))
        return b;

    n = k >> 5;
    k1 = b->k;
    n1 = n + b->wds + 1;
    for(i = b->maxwds; n1 > i; i <<= 1)
        k1++;
    b1 = Balloc(k1);
    if (b1 == NULL) {
        Bfree(b);
        return NULL;
    }
    x1 = b1->x;
    for(i = 0; i < n; i++)
        *x1++ = 0;
    x = b->x;
    xe = x + b->wds;
    if (k &= 0x1f) {
        k1 = 32 - k;
        z = 0;
        do {
            *x1++ = *x << k | z;
            z = *x++ >> k1;
        }
        while(x < xe);
        if ((*x1 = z))
            ++n1;
    }
    else do
             *x1++ = *x++;
        while(x < xe);
    b1->wds = n1 - 1;
    Bfree(b);
    return b1;
}

/* Do a three-way compare of a and b, returning -1 if a < b, 0 if a == b and
   1 if a > b.  Ignores signs of a and b. */

static int
cmp(Bigint *a, Bigint *b)
{
    ULong *xa, *xa0, *xb, *xb0;
    int i, j;

    i = a->wds;
    j = b->wds;
#ifdef DEBUG
    if (i > 1 && !a->x[i-1])
        Bug("cmp called with a->x[a->wds-1] == 0");
    if (j > 1 && !b->x[j-1])
        Bug("cmp called with b->x[b->wds-1] == 0");
#endif
    if (i -= j)
        return i;
    xa0 = a->x;
    xa = xa0 + j;
    xb0 = b->x;
    xb = xb0 + j;
    for(;;) {
        if (*--xa != *--xb)
            return *xa < *xb ? -1 : 1;
        if (xa <= xa0)
            break;
    }
    return 0;
}

/* Take the difference of Bigints a and b, returning a new Bigint.  Returns
   NULL on failure.  The signs of a and b are ignored, but the sign of the
   result is set appropriately. */

static Bigint *
diff(Bigint *a, Bigint *b)
{
    Bigint *c;
    int i, wa, wb;
    ULong *xa, *xae, *xb, *xbe, *xc;
#ifdef ULLong
    ULLong borrow, y;
#else
    ULong borrow, y;
    ULong z;
#endif

    i = cmp(a,b);
    if (!i) {
        c = Balloc(0);
        if (c == NULL)
            return NULL;
        c->wds = 1;
        c->x[0] = 0;
        return c;
    }
    if (i < 0) {
        c = a;
        a = b;
        b = c;
        i = 1;
    }
    else
        i = 0;
    c = Balloc(a->k);
    if (c == NULL)
        return NULL;
    c->sign = i;
    wa = a->wds;
    xa = a->x;
    xae = xa + wa;
    wb = b->wds;
    xb = b->x;
    xbe = xb + wb;
    xc = c->x;
    borrow = 0;
#ifdef ULLong
    do {
        y = (ULLong)*xa++ - *xb++ - borrow;
        borrow = y >> 32 & (ULong)1;
        *xc++ = (ULong)(y & FFFFFFFF);
    }
    while(xb < xbe);
    while(xa < xae) {
        y = *xa++ - borrow;
        borrow = y >> 32 & (ULong)1;
        *xc++ = (ULong)(y & FFFFFFFF);
    }
#else
    do {
        y = (*xa & 0xffff) - (*xb & 0xffff) - borrow;
        borrow = (y & 0x10000) >> 16;
        z = (*xa++ >> 16) - (*xb++ >> 16) - borrow;
        borrow = (z & 0x10000) >> 16;
        Storeinc(xc, z, y);
    }
    while(xb < xbe);
    while(xa < xae) {
        y = (*xa & 0xffff) - borrow;
        borrow = (y & 0x10000) >> 16;
        z = (*xa++ >> 16) - borrow;
        borrow = (z & 0x10000) >> 16;
        Storeinc(xc, z, y);
    }
#endif
    while(!*--xc)
        wa--;
    c->wds = wa;
    return c;
}

/* Given a positive normal double x, return the difference between x and the
   next double up.  Doesn't give correct results for subnormals. */

static double
ulp(U *x)
{
    Long L;
    U u;

    L = (word0(x) & Exp_mask) - (P-1)*Exp_msk1;
    word0(&u) = L;
    word1(&u) = 0;
    return dval(&u);
}

/* Convert a Bigint to a double plus an exponent */

static double
b2d(Bigint *a, int *e)
{
    ULong *xa, *xa0, w, y, z;
    int k;
    U d;

    xa0 = a->x;
    xa = xa0 + a->wds;
    y = *--xa;
#ifdef DEBUG
    if (!y) Bug("zero y in b2d");
#endif
    k = hi0bits(y);
    *e = 32 - k;
    if (k < Ebits) {
        word0(&d) = Exp_1 | y >> (Ebits - k);
        w = xa > xa0 ? *--xa : 0;
        word1(&d) = y << ((32-Ebits) + k) | w >> (Ebits - k);
        goto ret_d;
    }
    z = xa > xa0 ? *--xa : 0;
    if (k -= Ebits) {
        word0(&d) = Exp_1 | y << k | z >> (32 - k);
        y = xa > xa0 ? *--xa : 0;
        word1(&d) = z << k | y >> (32 - k);
    }
    else {
        word0(&d) = Exp_1 | y;
        word1(&d) = z;
    }
  ret_d:
    return dval(&d);
}

/* Convert a scaled double to a Bigint plus an exponent.  Similar to d2b,
   except that it accepts the scale parameter used in _Py_dg_strtod (which
   should be either 0 or 2*P), and the normalization for the return value is
   different (see below).  On input, d should be finite and nonnegative, and d
   / 2**scale should be exactly representable as an IEEE 754 double.

   Returns a Bigint b and an integer e such that

     dval(d) / 2**scale = b * 2**e.

   Unlike d2b, b is not necessarily odd: b and e are normalized so
   that either 2**(P-1) <= b < 2**P and e >= Etiny, or b < 2**P
   and e == Etiny.  This applies equally to an input of 0.0: in that
   case the return values are b = 0 and e = Etiny.

   The above normalization ensures that for all possible inputs d,
   2**e gives ulp(d/2**scale).

   Returns NULL on failure.
*/

static Bigint *
sd2b(U *d, int scale, int *e)
{
    Bigint *b;

    b = Balloc(1);
    if (b == NULL)
        return NULL;

    /* First construct b and e assuming that scale == 0. */
    b->wds = 2;
    b->x[0] = word1(d);
    b->x[1] = word0(d) & Frac_mask;
    *e = Etiny - 1 + (int)((word0(d) & Exp_mask) >> Exp_shift);
    if (*e < Etiny)
        *e = Etiny;
    else
        b->x[1] |= Exp_msk1;

    /* Now adjust for scale, provided that b != 0. */
    if (scale && (b->x[0] || b->x[1])) {
        *e -= scale;
        if (*e < Etiny) {
            scale = Etiny - *e;
            *e = Etiny;
            /* We can't shift more than P-1 bits without shifting out a 1. */
            assert(0 < scale && scale <= P - 1);
            if (scale >= 32) {
                /* The bits shifted out should all be zero. */
                assert(b->x[0] == 0);
                b->x[0] = b->x[1];
                b->x[1] = 0;
                scale -= 32;
            }
            if (scale) {
                /* The bits shifted out should all be zero. */
                assert(b->x[0] << (32 - scale) == 0);
                b->x[0] = (b->x[0] >> scale) | (b->x[1] << (32 - scale));
                b->x[1] >>= scale;
            }
        }
    }
    /* Ensure b is normalized. */
    if (!b->x[1])
        b->wds = 1;

    return b;
}

/* Convert a double to a Bigint plus an exponent.  Return NULL on failure.

   Given a finite nonzero double d, return an odd Bigint b and exponent *e
   such that fabs(d) = b * 2**e.  On return, *bbits gives the number of
   significant bits of b; that is, 2**(*bbits-1) <= b < 2**(*bbits).

   If d is zero, then b == 0, *e == -1010, *bbits = 0.
 */

static Bigint *
d2b(U *d, int *e, int *bits)
{
    Bigint *b;
    int de, k;
    ULong *x, y, z;
    int i;

    b = Balloc(1);
    if (b == NULL)
        return NULL;
    x = b->x;

    z = word0(d) & Frac_mask;
    word0(d) &= 0x7fffffff;   /* clear sign bit, which we ignore */
    if ((de = (int)(word0(d) >> Exp_shift)))
        z |= Exp_msk1;
    if ((y = word1(d))) {
        if ((k = lo0bits(&y))) {
            x[0] = y | z << (32 - k);
            z >>= k;
        }
        else
            x[0] = y;
        i =
            b->wds = (x[1] = z) ? 2 : 1;
    }
    else {
        k = lo0bits(&z);
        x[0] = z;
        i =
            b->wds = 1;
        k += 32;
    }
    if (de) {
        *e = de - Bias - (P-1) + k;
        *bits = P - k;
    }
    else {
        *e = de - Bias - (P-1) + 1 + k;
        *bits = 32*i - hi0bits(x[i-1]);
    }
    return b;
}

/* Compute the ratio of two Bigints, as a double.  The result may have an
   error of up to 2.5 ulps. */

static double
ratio(Bigint *a, Bigint *b)
{
    U da, db;
    int k, ka, kb;

    dval(&da) = b2d(a, &ka);
    dval(&db) = b2d(b, &kb);
    k = ka - kb + 32*(a->wds - b->wds);
    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,
                                   9007199254740992.*9007199254740992.e-256
                                   /* = 2^106 * 1e-256 */
};
/* 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

#define ULbits 32
#define kshift 5
#define kmask 31


static int
dshift(Bigint *b, int p2)
{
    int rv = hi0bits(b->x[b->wds-1]) - 4;
    if (p2 > 0)
        rv -= p2;
    return rv & kmask;
}

/* special case of Bigint division.  The quotient is always in the range 0 <=
   quotient < 10, and on entry the divisor S is normalized so that its top 4
   bits (28--31) are zero and bit 27 is set. */

static int
quorem(Bigint *b, Bigint *S)
{
    int n;
    ULong *bx, *bxe, q, *sx, *sxe;
#ifdef ULLong
    ULLong borrow, carry, y, ys;
#else
    ULong borrow, carry, y, ys;
    ULong si, z, zs;
#endif

    n = S->wds;
#ifdef DEBUG
    /*debug*/ if (b->wds > n)
        /*debug*/       Bug("oversize b in quorem");
#endif
    if (b->wds < n)
        return 0;
    sx = S->x;
    sxe = sx + --n;
    bx = b->x;
    bxe = bx + n;
    q = *bxe / (*sxe + 1);      /* ensure q <= true quotient */
#ifdef DEBUG
    /*debug*/ if (q > 9)
        /*debug*/       Bug("oversized quotient in quorem");
#endif
    if (q) {
        borrow = 0;
        carry = 0;
        do {
#ifdef ULLong
            ys = *sx++ * (ULLong)q + carry;
            carry = ys >> 32;
            y = *bx - (ys & FFFFFFFF) - borrow;
            borrow = y >> 32 & (ULong)1;
            *bx++ = (ULong)(y & FFFFFFFF);
#else
            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);
#endif
        }
        while(sx <= sxe);
        if (!*bxe) {
            bx = b->x;
            while(--bxe > bx && !*bxe)
                --n;
            b->wds = n;
        }
    }
    if (cmp(b, S) >= 0) {
        q++;
        borrow = 0;
        carry = 0;
        bx = b->x;
        sx = S->x;
        do {
#ifdef ULLong
            ys = *sx++ + carry;
            carry = ys >> 32;
            y = *bx - (ys & FFFFFFFF) - borrow;
            borrow = y >> 32 & (ULong)1;
            *bx++ = (ULong)(y & FFFFFFFF);
#else
            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);
#endif
        }
        while(sx <= sxe);
        bx = b->x;
        bxe = bx + n;
        if (!*bxe) {
            while(--bxe > bx && !*bxe)
                --n;
            b->wds = n;
        }
    }
    return q;
}

/* sulp(x) is a version of ulp(x) that takes bc.scale into account.

   Assuming that x is finite and nonnegative (positive zero is fine
   here) and x / 2^bc.scale is exactly representable as a double,
   sulp(x) is equivalent to 2^bc.scale * ulp(x / 2^bc.scale). */

static double
sulp(U *x, BCinfo *bc)
{
    U u;

    if (bc->scale && 2*P + 1 > (int)((word0(x) & Exp_mask) >> Exp_shift)) {
        /* rv/2^bc->scale is subnormal */
        word0(&u) = (P+2)*Exp_msk1;
        word1(&u) = 0;
        return u.d;
    }
    else {
        assert(word0(x) || word1(x)); /* x != 0.0 */
        return ulp(x);
    }
}

/* The bigcomp function handles some hard cases for strtod, for inputs
   with more than STRTOD_DIGLIM digits.  It's called once an initial
   estimate for the double corresponding to the input string has
   already been obtained by the code in _Py_dg_strtod.

   The bigcomp function is only called after _Py_dg_strtod has found a
   double value rv such that either rv or rv + 1ulp represents the
   correctly rounded value corresponding to the original string.  It
   determines which of these two values is the correct one by
   computing the decimal digits of rv + 0.5ulp and comparing them with
   the corresponding digits of s0.

   In the following, write dv for the absolute value of the number represented
   by the input string.

   Inputs:

     s0 points to the first significant digit of the input string.

     rv is a (possibly scaled) estimate for the closest double value to the
        value represented by the original input to _Py_dg_strtod.  If
        bc->scale is nonzero, then rv/2^(bc->scale) is the approximation to
        the input value.

     bc is a struct containing information gathered during the parsing and
        estimation steps of _Py_dg_strtod.  Description of fields follows:

        bc->e0 gives the exponent of the input value, such that dv = (integer
           given by the bd->nd digits of s0) * 10**e0

        bc->nd gives the total number of significant digits of s0.  It will
           be at least 1.

        bc->nd0 gives the number of significant digits of s0 before the
           decimal separator.  If there's no decimal separator, bc->nd0 ==
           bc->nd.

        bc->scale is the value used to scale rv to avoid doing arithmetic with
           subnormal values.  It's either 0 or 2*P (=106).

   Outputs:

     On successful exit, rv/2^(bc->scale) is the closest double to dv.

     Returns 0 on success, -1 on failure (e.g., due to a failed malloc call). */

static int
bigcomp(U *rv, const char *s0, BCinfo *bc)
{
    Bigint *b, *d;
    int b2, d2, dd, i, nd, nd0, odd, p2, p5;

    nd = bc->nd;
    nd0 = bc->nd0;
    p5 = nd + bc->e0;
    b = sd2b(rv, bc->scale, &p2);
    if (b == NULL)
        return -1;

    /* record whether the lsb of rv/2^(bc->scale) is odd:  in the exact halfway
       case, this is used for round to even. */
    odd = b->x[0] & 1;

    /* left shift b by 1 bit and or a 1 into the least significant bit;
       this gives us b * 2**p2 = rv/2^(bc->scale) + 0.5 ulp. */
    b = lshift(b, 1);
    if (b == NULL)
        return -1;
    b->x[0] |= 1;
    p2--;

    p2 -= p5;
    d = i2b(1);
    if (d == NULL) {
        Bfree(b);
        return -1;
    }
    /* Arrange for convenient computation of quotients:
     * shift left if necessary so divisor has 4 leading 0 bits.
     */
    if (p5 > 0) {
        d = pow5mult(d, p5);
        if (d == NULL) {
            Bfree(b);
            return -1;
        }
    }
    else if (p5 < 0) {
        b = pow5mult(b, -p5);
        if (b == NULL) {
            Bfree(d);
            return -1;
        }
    }
    if (p2 > 0) {
        b2 = p2;
        d2 = 0;
    }
    else {
        b2 = 0;
        d2 = -p2;
    }
    i = dshift(d, d2);
    if ((b2 += i) > 0) {
        b = lshift(b, b2);
        if (b == NULL) {
            Bfree(d);
            return -1;
        }
    }
    if ((d2 += i) > 0) {
        d = lshift(d, d2);
        if (d == NULL) {
            Bfree(b);
            return -1;
        }
    }

    /* Compare s0 with b/d: set dd to -1, 0, or 1 according as s0 < b/d, s0 ==
     * b/d, or s0 > b/d.  Here the digits of s0 are thought of as representing
     * a number in the range [0.1, 1). */
    if (cmp(b, d) >= 0)
        /* b/d >= 1 */
        dd = -1;
    else {
        i = 0;
        for(;;) {
            b = multadd(b, 10, 0);
            if (b == NULL) {
                Bfree(d);
                return -1;
            }
            dd = s0[i < nd0 ? i : i+1] - '0' - quorem(b, d);
            i++;

            if (dd)
                break;
            if (!b->x[0] && b->wds == 1) {
                /* b/d == 0 */
                dd = i < nd;
                break;
            }
            if (!(i < nd)) {
                /* b/d != 0, but digits of s0 exhausted */
                dd = -1;
                break;
            }
        }
    }
    Bfree(b);
    Bfree(d);
    if (dd > 0 || (dd == 0 && odd))
        dval(rv) += sulp(rv, bc);
    return 0;
}

/* Return a 'standard' NaN value.

   There are exactly two quiet NaNs that don't arise by 'quieting' signaling
   NaNs (see IEEE 754-2008, section 6.2.1).  If sign == 0, return the one whose
   sign bit is cleared.  Otherwise, return the one whose sign bit is set.
*/

double
_Py_dg_stdnan(int sign)
{
    U rv;
    word0(&rv) = NAN_WORD0;
    word1(&rv) = NAN_WORD1;
    if (sign)
        word0(&rv) |= Sign_bit;
    return dval(&rv);
}

/* Return positive or negative infinity, according to the given sign (0 for
 * positive infinity, 1 for negative infinity). */

double
_Py_dg_infinity(int sign)
{
    U rv;
    word0(&rv) = POSINF_WORD0;
    word1(&rv) = POSINF_WORD1;
    return sign ? -dval(&rv) : dval(&rv);
}

double
_Py_dg_strtod(const char *s00, char **se)
{
    int bb2, bb5, bbe, bd2, bd5, bs2, c, dsign, e, e1, error;
    int esign, i, j, k, lz, nd, nd0, odd, sign;
    const char *s, *s0, *s1;
    double aadj, aadj1;
    U aadj2, adj, rv, rv0;
    ULong y, z, abs_exp;
    Long L;
    BCinfo bc;
    Bigint *bb, *bb1, *bd, *bd0, *bs, *delta;
    size_t ndigits, fraclen;

    dval(&rv) = 0.;

    /* Start parsing. */
    c = *(s = s00);

    /* Parse optional sign, if present. */
    sign = 0;
    switch (c) {
    case '-':
        sign = 1;
        /* no break */
    case '+':
        c = *++s;
    }

    /* Skip leading zeros: lz is true iff there were leading zeros. */
    s1 = s;
    while (c == '0')
        c = *++s;
    lz = s != s1;

    /* Point s0 at the first nonzero digit (if any).  fraclen will be the
       number of digits between the decimal point and the end of the
       digit string.  ndigits will be the total number of digits ignoring
       leading zeros. */
    s0 = s1 = s;
    while ('0' <= c && c <= '9')
        c = *++s;
    ndigits = s - s1;
    fraclen = 0;

    /* Parse decimal point and following digits. */
    if (c == '.') {
        c = *++s;
        if (!ndigits) {
            s1 = s;
            while (c == '0')
                c = *++s;
            lz = lz || s != s1;
            fraclen += (s - s1);
            s0 = s;
        }
        s1 = s;
        while ('0' <= c && c <= '9')
            c = *++s;
        ndigits += s - s1;
        fraclen += s - s1;
    }

    /* Now lz is true if and only if there were leading zero digits, and
       ndigits gives the total number of digits ignoring leading zeros.  A
       valid input must have at least one digit. */
    if (!ndigits && !lz) {
        if (se)
            *se = (char *)s00;
        goto parse_error;
    }

    /* Range check ndigits and fraclen to make sure that they, and values
       computed with them, can safely fit in an int. */
    if (ndigits > MAX_DIGITS || fraclen > MAX_DIGITS) {
        if (se)
            *se = (char *)s00;
        goto parse_error;
    }
    nd = (int)ndigits;
    nd0 = (int)ndigits - (int)fraclen;

    /* Parse exponent. */
    e = 0;
    if (c == 'e' || c == 'E') {
        s00 = s;
        c = *++s;

        /* Exponent sign. */
        esign = 0;
        switch (c) {
        case '-':
            esign = 1;
            /* no break */
        case '+':
            c = *++s;
        }

        /* Skip zeros.  lz is true iff there are leading zeros. */
        s1 = s;
        while (c == '0')
            c = *++s;
        lz = s != s1;

        /* Get absolute value of the exponent. */
        s1 = s;
        abs_exp = 0;
        while ('0' <= c && c <= '9') {
            abs_exp = 10*abs_exp + (c - '0');
            c = *++s;
        }

        /* abs_exp will be correct modulo 2**32.  But 10**9 < 2**32, so if
           there are at most 9 significant exponent digits then overflow is
           impossible. */
        if (s - s1 > 9 || abs_exp > MAX_ABS_EXP)
            e = (int)MAX_ABS_EXP;
        else
            e = (int)abs_exp;
        if (esign)
            e = -e;

        /* A valid exponent must have at least one digit. */
        if (s == s1 && !lz)
            s = s00;
    }

    /* Adjust exponent to take into account position of the point. */
    e -= nd - nd0;
    if (nd0 <= 0)
        nd0 = nd;

    /* Finished parsing.  Set se to indicate how far we parsed */
    if (se)
        *se = (char *)s;

    /* If all digits were zero, exit with return value +-0.0.  Otherwise,
       strip trailing zeros: scan back until we hit a nonzero digit. */
    if (!nd)
        goto ret;
    for (i = nd; i > 0; ) {
        --i;
        if (s0[i < nd0 ? i : i+1] != '0') {
            ++i;
            break;
        }
    }
    e += nd - i;
    nd = i;
    if (nd0 > nd)
        nd0 = nd;

    /* Summary of parsing results.  After parsing, and dealing with zero
     * inputs, we have values s0, nd0, nd, e, sign, where:
     *
     *  - s0 points to the first significant digit of the input string
     *
     *  - nd is the total number of significant digits (here, and
     *    below, 'significant digits' means the set of digits of the
     *    significand of the input that remain after ignoring leading
     *    and trailing zeros).
     *
     *  - nd0 indicates the position of the decimal point, if present; it
     *    satisfies 1 <= nd0 <= nd.  The nd significant digits are in
     *    s0[0:nd0] and s0[nd0+1:nd+1] using the usual Python half-open slice
     *    notation.  (If nd0 < nd, then s0[nd0] contains a '.'  character; if
     *    nd0 == nd, then s0[nd0] could be any non-digit character.)
     *
     *  - e is the adjusted exponent: the absolute value of the number
     *    represented by the original input string is n * 10**e, where
     *    n is the integer represented by the concatenation of
     *    s0[0:nd0] and s0[nd0+1:nd+1]
     *
     *  - sign gives the sign of the input:  1 for negative, 0 for positive
     *
     *  - the first and last significant digits are nonzero
     */

    /* put first DBL_DIG+1 digits into integer y and z.
     *
     *  - y contains the value represented by the first min(9, nd)
     *    significant digits
     *
     *  - if nd > 9, z contains the value represented by significant digits
     *    with indices in [9, min(16, nd)).  So y * 10**(min(16, nd) - 9) + z
     *    gives the value represented by the first min(16, nd) sig. digits.
     */

    bc.e0 = e1 = e;
    y = z = 0;
    for (i = 0; i < nd; i++) {
        if (i < 9)
            y = 10*y + s0[i < nd0 ? i : i+1] - '0';
        else if (i < DBL_DIG+1)
            z = 10*z + s0[i < nd0 ? i : i+1] - '0';
        else
            break;
    }

    k = nd < DBL_DIG + 1 ? nd : DBL_DIG + 1;
    dval(&rv) = y;
    if (k > 9) {
        dval(&rv) = tens[k - 9] * dval(&rv) + z;
    }
    bd0 = 0;
    if (nd <= DBL_DIG
        && Flt_Rounds == 1
        ) {
        if (!e)
            goto ret;
        if (e > 0) {
            if (e <= Ten_pmax) {
                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];
                dval(&rv) *= tens[e];
                goto ret;
            }
        }
        else if (e >= -Ten_pmax) {
            dval(&rv) /= tens[-e];
            goto ret;
        }
    }
    e1 += nd - k;

    bc.scale = 0;

    /* 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)
                goto ovfl;
            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) {
        /* The input decimal value lies in [10**e1, 10**(e1+16)).

           If e1 <= -512, underflow immediately.
           If e1 <= -256, set bc.scale to 2*P.

           So for input value < 1e-256, bc.scale is always set;
           for input value >= 1e-240, bc.scale is never set.
           For input values in [1e-256, 1e-240), bc.scale may or may
           not be set. */

        e1 = -e1;
        if ((i = e1 & 15))
            dval(&rv) /= tens[i];
        if (e1 >>= 4) {
            if (e1 >= 1 << n_bigtens)
                goto undfl;
            if (e1 & Scale_Bit)
                bc.scale = 2*P;
            for(j = 0; e1 > 0; j++, e1 >>= 1)
                if (e1 & 1)
                    dval(&rv) *= tinytens[j];
            if (bc.scale && (j = 2*P + 1 - ((word0(&rv) & Exp_mask)
                                            >> Exp_shift)) > 0) {
                /* scaled rv is denormal; clear 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;
            }
            if (!dval(&rv))
                goto undfl;
        }
    }

    /* Now the hard part -- adjusting rv to the correct value.*/

    /* Put digits into bd: true value = bd * 10^e */

    bc.nd = nd;
    bc.nd0 = nd0;       /* Only needed if nd > STRTOD_DIGLIM, but done here */
                        /* to silence an erroneous warning about bc.nd0 */
                        /* possibly not being initialized. */
    if (nd > STRTOD_DIGLIM) {
        /* ASSERT(STRTOD_DIGLIM >= 18); 18 == one more than the */
        /* minimum number of decimal digits to distinguish double values */
        /* in IEEE arithmetic. */

        /* Truncate input to 18 significant digits, then discard any trailing
           zeros on the result by updating nd, nd0, e and y suitably. (There's
           no need to update z; it's not reused beyond this point.) */
        for (i = 18; i > 0; ) {
            /* scan back until we hit a nonzero digit.  significant digit 'i'
            is s0[i] if i < nd0, s0[i+1] if i >= nd0. */
            --i;
            if (s0[i < nd0 ? i : i+1] != '0') {
                ++i;
                break;
            }
        }
        e += nd - i;
        nd = i;
        if (nd0 > nd)
            nd0 = nd;
        if (nd < 9) { /* must recompute y */
            y = 0;
            for(i = 0; i < nd0; ++i)
                y = 10*y + s0[i] - '0';
            for(; i < nd; ++i)
                y = 10*y + s0[i+1] - '0';
        }
    }
    bd0 = s2b(s0, nd0, nd, y);
    if (bd0 == NULL)
        goto failed_malloc;

    /* Notation for the comments below.  Write:

         - dv for the absolute value of the number represented by the original
           decimal input string.

         - if we've truncated dv, write tdv for the truncated value.
           Otherwise, set tdv == dv.

         - srv for the quantity rv/2^bc.scale; so srv is the current binary
           approximation to tdv (and dv).  It should be exactly representable
           in an IEEE 754 double.
    */

    for(;;) {

        /* This is the main correction loop for _Py_dg_strtod.

           We've got a decimal value tdv, and a floating-point approximation
           srv=rv/2^bc.scale to tdv.  The aim is to determine whether srv is
           close enough (i.e., within 0.5 ulps) to tdv, and to compute a new
           approximation if not.

           To determine whether srv is close enough to tdv, compute integers
           bd, bb and bs proportional to tdv, srv and 0.5 ulp(srv)
           respectively, and then use integer arithmetic to determine whether
           |tdv - srv| is less than, equal to, or greater than 0.5 ulp(srv).
        */

        bd = Balloc(bd0->k);
        if (bd == NULL) {
            Bfree(bd0);
            goto failed_malloc;
        }
        Bcopy(bd, bd0);
        bb = sd2b(&rv, bc.scale, &bbe);   /* srv = bb * 2^bbe */
        if (bb == NULL) {
            Bfree(bd);
            Bfree(bd0);
            goto failed_malloc;
        }
        /* Record whether lsb of bb is odd, in case we need this
           for the round-to-even step later. */
        odd = bb->x[0] & 1;

        /* tdv = bd * 10**e;  srv = bb * 2**bbe */
        bs = i2b(1);
        if (bs == NULL) {
            Bfree(bb);
            Bfree(bd);
            Bfree(bd0);
            goto failed_malloc;
        }

        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;
        bb2++;
        bd2++;

        /* At this stage bd5 - bb5 == e == bd2 - bb2 + bbe, bb2 - bs2 == 1,
           and bs == 1, so:

              tdv == bd * 10**e = bd * 2**(bbe - bb2 + bd2) * 5**(bd5 - bb5)
              srv == bb * 2**bbe = bb * 2**(bbe - bb2 + bb2)
              0.5 ulp(srv) == 2**(bbe-1) = bs * 2**(bbe - bb2 + bs2)

           It follows that:

              M * tdv = bd * 2**bd2 * 5**bd5
              M * srv = bb * 2**bb2 * 5**bb5
              M * 0.5 ulp(srv) = bs * 2**bs2 * 5**bb5

           for some constant M.  (Actually, M == 2**(bb2 - bbe) * 5**bb5, but
           this fact is not needed below.)
        */

        /* Remove factor of 2**i, where i = min(bb2, bd2, bs2). */
        i = bb2 < bd2 ? bb2 : bd2;
        if (i > bs2)
            i = bs2;
        if (i > 0) {
            bb2 -= i;
            bd2 -= i;
            bs2 -= i;
        }

        /* Scale bb, bd, bs by the appropriate powers of 2 and 5. */
        if (bb5 > 0) {
            bs = pow5mult(bs, bb5);
            if (bs == NULL) {
                Bfree(bb);
                Bfree(bd);
                Bfree(bd0);
                goto failed_malloc;
            }
            bb1 = mult(bs, bb);
            Bfree(bb);
            bb = bb1;
            if (bb == NULL) {
                Bfree(bs);
                Bfree(bd);
                Bfree(bd0);
                goto failed_malloc;
            }
        }
        if (bb2 > 0) {
            bb = lshift(bb, bb2);
            if (bb == NULL) {
                Bfree(bs);
                Bfree(bd);
                Bfree(bd0);
                goto failed_malloc;
            }
        }
        if (bd5 > 0) {
            bd = pow5mult(bd, bd5);
            if (bd == NULL) {
                Bfree(bb);
                Bfree(bs);
                Bfree(bd0);
                goto failed_malloc;
            }
        }
        if (bd2 > 0) {
            bd = lshift(bd, bd2);
            if (bd == NULL) {
                Bfree(bb);
                Bfree(bs);
                Bfree(bd0);
                goto failed_malloc;
            }
        }
        if (bs2 > 0) {
            bs = lshift(bs, bs2);
            if (bs == NULL) {
                Bfree(bb);
                Bfree(bd);
                Bfree(bd0);
                goto failed_malloc;
            }
        }

        /* Now bd, bb and bs are scaled versions of tdv, srv and 0.5 ulp(srv),
           respectively.  Compute the difference |tdv - srv|, and compare
           with 0.5 ulp(srv). */

        delta = diff(bb, bd);
        if (delta == NULL) {
            Bfree(bb);
            Bfree(bs);
            Bfree(bd);
            Bfree(bd0);
            goto failed_malloc;
        }
        dsign = delta->sign;
        delta->sign = 0;
        i = cmp(delta, bs);
        if (bc.nd > nd && i <= 0) {
            if (dsign)
                break;  /* Must use bigcomp(). */

            /* Here rv overestimates the truncated decimal value by at most
               0.5 ulp(rv).  Hence rv either overestimates the true decimal
               value by <= 0.5 ulp(rv), or underestimates it by some small
               amount (< 0.1 ulp(rv)); either way, rv is within 0.5 ulps of
               the true decimal value, so it's possible to exit.

               Exception: if scaled rv is a normal exact power of 2, but not
               DBL_MIN, then rv - 0.5 ulp(rv) takes us all the way down to the
               next double, so the correctly rounded result is either rv - 0.5
               ulp(rv) or rv; in this case, use bigcomp to distinguish. */

            if (!word1(&rv) && !(word0(&rv) & Bndry_mask)) {
                /* rv can't be 0, since it's an overestimate for some
                   nonzero value.  So rv is a normal power of 2. */
                j = (int)(word0(&rv) & Exp_mask) >> Exp_shift;
                /* rv / 2^bc.scale = 2^(j - 1023 - bc.scale); use bigcomp if
                   rv / 2^bc.scale >= 2^-1021. */
                if (j - bc.scale >= 2) {
                    dval(&rv) -= 0.5 * sulp(&rv, &bc);
                    break; /* Use bigcomp. */
                }
            }

            {
                bc.nd = nd;
                i = -1; /* Discarded digits make delta smaller. */
            }
        }

        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
                || (word0(&rv) & Exp_mask) <= (2*P+1)*Exp_msk1
                ) {
                break;
            }
            if (!delta->x[0] && delta->wds <= 1) {
                /* exact result */
                break;
            }
            delta = lshift(delta,Log2P);
            if (delta == NULL) {
                Bfree(bb);
                Bfree(bs);
                Bfree(bd);
                Bfree(bd0);
                goto failed_malloc;
            }
            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) == (
                        (bc.scale &&
                         (y = word0(&rv) & Exp_mask) <= 2*P*Exp_msk1) ?
                        (0xffffffff & (0xffffffff << (2*P+1-(y>>Exp_shift)))) :
                        0xffffffff)) {
                    /*boundary case -- increment exponent*/
                    word0(&rv) = (word0(&rv) & Exp_mask)
                        + Exp_msk1
                        ;
                    word1(&rv) = 0;
                    /* dsign = 0; */
                    break;
                }
            }
            else if (!(word0(&rv) & Bndry_mask) && !word1(&rv)) {
              drop_down:
                /* boundary case -- decrement exponent */
                if (bc.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 */
                        if (bc.nd > nd)
                            break;
                        goto undfl;
                    }
                }
                L = (word0(&rv) & Exp_mask) - Exp_msk1;
                word0(&rv) = L | Bndry_mask1;
                word1(&rv) = 0xffffffff;
                break;
            }
            if (!odd)
                break;
            if (dsign)
                dval(&rv) += sulp(&rv, &bc);
            else {
                dval(&rv) -= sulp(&rv, &bc);
                if (!dval(&rv)) {
                    if (bc.nd >nd)
                        break;
                    goto undfl;
                }
            }
            /* dsign = 1 - dsign; */
            break;
        }
        if ((aadj = ratio(delta, bs)) <= 2.) {
            if (dsign)
                aadj = aadj1 = 1.;
            else if (word1(&rv) || word0(&rv) & Bndry_mask) {
                if (word1(&rv) == Tiny1 && !word0(&rv)) {
                    if (bc.nd >nd)
                        break;
                    goto undfl;
                }
                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;
            if (Flt_Rounds == 0)
                aadj1 += 0.5;
        }
        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) {
                    Bfree(bb);
                    Bfree(bd);
                    Bfree(bs);
                    Bfree(bd0);
                    Bfree(delta);
                    goto ovfl;
                }
                word0(&rv) = Big0;
                word1(&rv) = Big1;
                goto cont;
            }
            else
                word0(&rv) += P*Exp_msk1;
        }
        else {
            if (bc.scale && y <= 2*P*Exp_msk1) {
                if (aadj <= 0x7fffffff) {
                    if ((z = (ULong)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;
        }
        z = word0(&rv) & Exp_mask;
        if (bc.nd == nd) {
            if (!bc.scale)
                if (y == z) {
                    /* Can we stop now? */
                    L = (Long)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;
                }
        }
      cont:
        Bfree(bb);
        Bfree(bd);
        Bfree(bs);
        Bfree(delta);
    }
    Bfree(bb);
    Bfree(bd);
    Bfree(bs);
    Bfree(bd0);
    Bfree(delta);
    if (bc.nd > nd) {
        error = bigcomp(&rv, s0, &bc);
        if (error)
            goto failed_malloc;
    }

    if (bc.scale) {
        word0(&rv0) = Exp_1 - 2*P*Exp_msk1;
        word1(&rv0) = 0;
        dval(&rv) *= dval(&rv0);
    }

  ret:
    return sign ? -dval(&rv) : dval(&rv);

  parse_error:
    return 0.0;

  failed_malloc:
    errno = ENOMEM;
    return -1.0;

  undfl:
    return sign ? -0.0 : 0.0;

  ovfl:
    errno = ERANGE;
    /* Can't trust HUGE_VAL */
    word0(&rv) = Exp_mask;
    word1(&rv) = 0;
    return sign ? -dval(&rv) : dval(&rv);

}

static char *
rv_alloc(int i)
{
    int j, k, *r;

    j = sizeof(ULong);
    for(k = 0;
        sizeof(Bigint) - sizeof(ULong) - sizeof(int) + j <= (unsigned)i;
        j <<= 1)
        k++;
    r = (int*)Balloc(k);
    if (r == NULL)
        return NULL;
    *r = k;
    return (char *)(r+1);
}

static char *
nrv_alloc(char *s, char **rve, int n)
{
    char *rv, *t;

    rv = rv_alloc(n);
    if (rv == NULL)
        return NULL;
    t = rv;
    while((*t = *s++)) t++;
    if (rve)
        *rve = t;
    return rv;
}

/* freedtoa(s) must be used to free values s returned by dtoa
 * when MULTIPLE_THREADS is #defined.  It should be used in all cases,
 * but for consistency with earlier versions of dtoa, it is optional
 * when MULTIPLE_THREADS is not defined.
 */

void
_Py_dg_freedtoa(char *s)
{
    Bigint *b = (Bigint *)((int *)s - 1);
    b->maxwds = 1 << (b->k = *(int*)b);
    Bfree(b);
}

/* 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. 112-126].
 *
 * 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 Long
 *         calculation.
 */

/* Additional notes (METD): (1) returns NULL on failure.  (2) to avoid memory
   leakage, a successful call to _Py_dg_dtoa should always be matched by a
   call to _Py_dg_freedtoa. */

char *
_Py_dg_dtoa(double dd, int mode, 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.

        mode:
        0 ==> shortest string that yields d when read in
        and rounded to nearest.
        1 ==> like 0, but with Steele & White stopping rule;
        e.g. with IEEE P754 arithmetic , mode 0 gives
        1e23 whereas mode 1 gives 9.999999999999999e22.
        2 ==> max(1,ndigits) significant digits.  This gives a
        return value similar to that of ecvt, except
        that trailing zeros are suppressed.
        3 ==> through ndigits past the decimal point.  This
        gives a return value similar to that from fcvt,
        except that trailing zeros are suppressed, and
        ndigits can be negative.
        4,5 ==> similar to 2 and 3, respectively, but (in
        round-nearest mode) with the tests of mode 0 to
        possibly return a shorter string that rounds to d.
        With IEEE arithmetic and compilation with
        -DHonor_FLT_ROUNDS, modes 4 and 5 behave the same
        as modes 2 and 3 when FLT_ROUNDS != 1.
        6-9 ==> Debugging modes similar to mode - 4:  don't try
        fast floating-point estimate (if applicable).

        Values of mode other than 0-9 are treated as mode 0.

        Sufficient space is allocated to the return value
        to hold the suppressed trailing zeros.
    */

    int bbits, b2, b5, be, dig, i, ieps, ilim, ilim0, ilim1,
        j, j1, k, k0, k_check, leftright, m2, m5, s2, s5,
        spec_case, try_quick;
    Long L;
    int denorm;
    ULong x;
    Bigint *b, *b1, *delta, *mlo, *mhi, *S;
    U d2, eps, u;
    double ds;
    char *s, *s0;

    /* set pointers to NULL, to silence gcc compiler warnings and make
       cleanup easier on error */
    mlo = mhi = S = 0;
    s0 = 0;

    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;

    /* quick return for Infinities, NaNs and zeros */
    if ((word0(&u) & Exp_mask) == Exp_mask)
    {
        /* Infinity or NaN */
        *decpt = 9999;
        if (!word1(&u) && !(word0(&u) & 0xfffff))
            return nrv_alloc("Infinity", rve, 8);
        return nrv_alloc("NaN", rve, 3);
    }
    if (!dval(&u)) {
        *decpt = 1;
        return nrv_alloc("0", rve, 1);
    }

    /* compute k = floor(log10(d)).  The computation may leave k
       one too large, but should never leave k too small. */
    b = d2b(&u, &be, &bbits);
    if (b == NULL)
        goto failed_malloc;
    if ((i = (int)(word0(&u) >> Exp_shift1 & (Exp_mask>>Exp_shift1)))) {
        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;
        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;
    }
    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;
    }
    if (mode < 0 || mode > 9)
        mode = 0;

    try_quick = 1;

    if (mode > 5) {
        mode -= 4;
        try_quick = 0;
    }
    leftright = 1;
    ilim = ilim1 = -1;  /* Values for cases 0 and 1; done here to */
    /* silence erroneous "gcc -Wall" warning. */
    switch(mode) {
    case 0:
    case 1:
        i = 18;
        ndigits = 0;
        break;
    case 2:
        leftright = 0;
        /* no break */
    case 4:
        if (ndigits <= 0)
            ndigits = 1;
        ilim = ilim1 = i = ndigits;
        break;
    case 3:
        leftright = 0;
        /* no break */
    case 5:
        i = ndigits + k + 1;
        ilim = i;
        ilim1 = i - 1;
        if (i <= 0)
            i = 1;
    }
    s0 = rv_alloc(i);
    if (s0 == NULL)
        goto failed_malloc;
    s = s0;


    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 = mhi = 0;
            dval(&u) -= 5.;
            if (dval(&u) > dval(&eps))
                goto one_digit;
            if (dval(&u) < -dval(&eps))
                goto no_digits;
            goto fast_failed;
        }
        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)dval(&u);
                dval(&u) -= L;
                *s++ = '0' + (int)L;
                if (dval(&u) < dval(&eps))
                    goto ret1;
                if (1. - dval(&u) < dval(&eps))
                    goto bump_up;
                if (++i >= ilim)
                    break;
                dval(&eps) *= 10.;
                dval(&u) *= 10.;
            }
        }
        else {
            /* Generate ilim digits, then fix them up. */
            dval(&eps) *= tens[ilim-1];
            for(i = 1;; i++, dval(&u) *= 10.) {
                L = (Long)(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 ret1;
                    }
                    break;
                }
            }
        }
      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 = mhi = 0;
            if (ilim < 0 || dval(&u) <= 5*ds)
                goto no_digits;
            goto one_digit;
        }
        for(i = 1;; i++, dval(&u) *= 10.) {
            L = (Long)(dval(&u) / ds);
            dval(&u) -= L*ds;
            *s++ = '0' + (int)L;
            if (!dval(&u)) {
                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 ret1;
    }

    m2 = b2;
    m5 = b5;
    if (leftright) {
        i =
            denorm ? be + (Bias + (P-1) - 1 + 1) :
            1 + P - bbits;
        b2 += i;
        s2 += i;
        mhi = i2b(1);
        if (mhi == NULL)
            goto failed_malloc;
    }
    if (m2 > 0 && s2 > 0) {
        i = m2 < s2 ? m2 : s2;
        b2 -= i;
        m2 -= i;
        s2 -= i;
    }
    if (b5 > 0) {
        if (leftright) {
            if (m5 > 0) {
                mhi = pow5mult(mhi, m5);
                if (mhi == NULL)
                    goto failed_malloc;
                b1 = mult(mhi, b);
                Bfree(b);
                b = b1;
                if (b == NULL)
                    goto failed_malloc;
            }
            if ((j = b5 - m5)) {
                b = pow5mult(b, j);
                if (b == NULL)
                    goto failed_malloc;
            }
        }
        else {
            b = pow5mult(b, b5);
            if (b == NULL)
                goto failed_malloc;
        }
    }
    S = i2b(1);
    if (S == NULL)
        goto failed_malloc;
    if (s5 > 0) {
        S = pow5mult(S, s5);
        if (S == NULL)
            goto failed_malloc;
    }

    /* Check for special case that d is a normalized power of 2. */

    spec_case = 0;
    if ((mode < 2 || leftright)
        ) {
        if (!word1(&u) && !(word0(&u) & Bndry_mask)
            && word0(&u) & (Exp_mask & ~Exp_msk1)
            ) {
            /* 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.
     */
#define iInc 28
    i = dshift(S, s2);
    b2 += i;
    m2 += i;
    s2 += i;
    if (b2 > 0) {
        b = lshift(b, b2);
        if (b == NULL)
            goto failed_malloc;
    }
    if (s2 > 0) {
        S = lshift(S, s2);
        if (S == NULL)
            goto failed_malloc;
    }
    if (k_check) {
        if (cmp(b,S) < 0) {
            k--;
            b = multadd(b, 10, 0);      /* we botched the k estimate */
            if (b == NULL)
                goto failed_malloc;
            if (leftright) {
                mhi = multadd(mhi, 10, 0);
                if (mhi == NULL)
                    goto failed_malloc;
            }
            ilim = ilim1;
        }
    }
    if (ilim <= 0 && (mode == 3 || mode == 5)) {
        if (ilim < 0) {
            /* no digits, fcvt style */
          no_digits:
            k = -1 - ndigits;
            goto ret;
        }
        else {
            S = multadd(S, 5, 0);
            if (S == NULL)
                goto failed_malloc;
            if (cmp(b, S) <= 0)
                goto no_digits;
        }
      one_digit:
        *s++ = '1';
        k++;
        goto ret;
    }
    if (leftright) {
        if (m2 > 0) {
            mhi = lshift(mhi, m2);
            if (mhi == NULL)
                goto failed_malloc;
        }

        /* Compute mlo -- check for special case
         * that d is a normalized power of 2.
         */

        mlo = mhi;
        if (spec_case) {
            mhi = Balloc(mhi->k);
            if (mhi == NULL)
                goto failed_malloc;
            Bcopy(mhi, mlo);
            mhi = lshift(mhi, Log2P);
            if (mhi == NULL)
                goto failed_malloc;
        }

        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);
            delta = diff(S, mhi);
            if (delta == NULL)
                goto failed_malloc;
            j1 = delta->sign ? 1 : cmp(b, delta);
            Bfree(delta);
            if (j1 == 0 && mode != 1 && !(word1(&u) & 1)
                ) {
                if (dig == '9')
                    goto round_9_up;
                if (j > 0)
                    dig++;
                *s++ = dig;
                goto ret;
            }
            if (j < 0 || (j == 0 && mode != 1
                          && !(word1(&u) & 1)
                    )) {
                if (!b->x[0] && b->wds <= 1) {
                    goto accept_dig;
                }
                if (j1 > 0) {
                    b = lshift(b, 1);
                    if (b == NULL)
                        goto failed_malloc;
                    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;
            b = multadd(b, 10, 0);
            if (b == NULL)
                goto failed_malloc;
            if (mlo == mhi) {
                mlo = mhi = multadd(mhi, 10, 0);
                if (mlo == NULL)
                    goto failed_malloc;
            }
            else {
                mlo = multadd(mlo, 10, 0);
                if (mlo == NULL)
                    goto failed_malloc;
                mhi = multadd(mhi, 10, 0);
                if (mhi == NULL)
                    goto failed_malloc;
            }
        }
    }
    else
        for(i = 1;; i++) {
            *s++ = dig = quorem(b,S) + '0';
            if (!b->x[0] && b->wds <= 1) {
                goto ret;
            }
            if (i >= ilim)
                break;
            b = multadd(b, 10, 0);
            if (b == NULL)
                goto failed_malloc;
        }

    /* Round off last digit */

    b = lshift(b, 1);
    if (b == NULL)
        goto failed_malloc;
    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++;
    }
  ret:
    Bfree(S);
    if (mhi) {
        if (mlo && mlo != mhi)
            Bfree(mlo);
        Bfree(mhi);
    }
  ret1:
    Bfree(b);
    *s = 0;
    *decpt = k + 1;
    if (rve)
        *rve = s;
    return s0;
  failed_malloc:
    if (S)
        Bfree(S);
    if (mlo && mlo != mhi)
        Bfree(mlo);
    if (mhi)
        Bfree(mhi);
    if (b)
        Bfree(b);
    if (s0)
        _Py_dg_freedtoa(s0);
    return NULL;
}
#ifdef __cplusplus
}
#endif

#endif  /* PY_NO_SHORT_FLOAT_REPR */