duktape-1.5.2/src-separate/duk_numconv.c

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/*
 *  Number-to-string and string-to-number conversions.
 *
 *  Slow path number-to-string and string-to-number conversion is based on
 *  a Dragon4 variant, with fast paths for small integers.  Big integer
 *  arithmetic is needed for guaranteeing that the conversion is correct
 *  and uses a minimum number of digits.  The big number arithmetic has a
 *  fixed maximum size and does not require dynamic allocations.
 *
 *  See: doc/number-conversion.rst.
 */
 
#include "duk_internal.h"
 
#define DUK__IEEE_DOUBLE_EXP_BIAS  1023
#define DUK__IEEE_DOUBLE_EXP_MIN   (-1022)   /* biased exp == 0 -> denormal, exp -1022 */
 
#define DUK__DIGITCHAR(x)  duk_lc_digits[(x)]
 
/*
 *  Tables generated with src/gennumdigits.py.
 *
 *  duk__str2num_digits_for_radix indicates, for each radix, how many input
 *  digits should be considered significant for string-to-number conversion.
 *  The input is also padded to this many digits to give the Dragon4
 *  conversion enough (apparent) precision to work with.
 *
 *  duk__str2num_exp_limits indicates, for each radix, the radix-specific
 *  minimum/maximum exponent values (for a Dragon4 integer mantissa)
 *  below and above which the number is guaranteed to underflow to zero
 *  or overflow to Infinity.  This allows parsing to keep bigint values
 *  bounded.
 */
 
DUK_LOCAL const duk_uint8_t duk__str2num_digits_for_radix[] = {
    69, 44, 35, 30, 27, 25, 23, 22, 20, 20,    /* 2 to 11 */
    20, 19, 19, 18, 18, 17, 17, 17, 16, 16,    /* 12 to 21 */
    16, 16, 16, 15, 15, 15, 15, 15, 15, 14,    /* 22 to 31 */
    14, 14, 14, 14, 14                         /* 31 to 36 */
};
 
typedef struct {
    duk_int16_t upper;
    duk_int16_t lower;
} duk__exp_limits;
 
DUK_LOCAL const duk__exp_limits duk__str2num_exp_limits[] = {
    { 957, -1147 }, { 605, -725 },  { 479, -575 },  { 414, -496 },
    { 372, -446 },  { 342, -411 },  { 321, -384 },  { 304, -364 },
    { 291, -346 },  { 279, -334 },  { 268, -323 },  { 260, -312 },
    { 252, -304 },  { 247, -296 },  { 240, -289 },  { 236, -283 },
    { 231, -278 },  { 227, -273 },  { 223, -267 },  { 220, -263 },
    { 216, -260 },  { 213, -256 },  { 210, -253 },  { 208, -249 },
    { 205, -246 },  { 203, -244 },  { 201, -241 },  { 198, -239 },
    { 196, -237 },  { 195, -234 },  { 193, -232 },  { 191, -230 },
    { 190, -228 },  { 188, -226 },  { 187, -225 },
};
 
/*
 *  Limited functionality bigint implementation.
 *
 *  Restricted to non-negative numbers with less than 32 * DUK__BI_MAX_PARTS bits,
 *  with the caller responsible for ensuring this is never exceeded.  No memory
 *  allocation (except stack) is needed for bigint computation.  Operations
 *  have been tailored for number conversion needs.
 *
 *  Argument order is "assignment order", i.e. target first, then arguments:
 *  x <- y * z  -->  duk__bi_mul(x, y, z);
 */
 
/* This upper value has been experimentally determined; debug build will check
 * bigint size with assertions.
 */
#define DUK__BI_MAX_PARTS  37  /* 37x32 = 1184 bits */
 
#ifdef DUK_USE_DDDPRINT
#define DUK__BI_PRINT(name,x)  duk__bi_print((name),(x))
#else
#define DUK__BI_PRINT(name,x)
#endif
 
/* Current size is about 152 bytes. */
typedef struct {
    duk_small_int_t n;
    duk_uint32_t v[DUK__BI_MAX_PARTS];  /* low to high */
} duk__bigint;
 
#ifdef DUK_USE_DDDPRINT
DUK_LOCAL void duk__bi_print(const char *name, duk__bigint *x) {
    /* Overestimate required size; debug code so not critical to be tight. */
    char buf[DUK__BI_MAX_PARTS * 9 + 64];
    char *p = buf;
    duk_small_int_t i;
 
    /* No NUL term checks in this debug code. */
    p += DUK_SPRINTF(p, "%p n=%ld", (void *) x, (long) x->n);
    if (x->n == 0) {
        p += DUK_SPRINTF(p, " 0");
    }
    for (i = x->n - 1; i >= 0; i--) {
        p += DUK_SPRINTF(p, " %08lx", (unsigned long) x->v[i]);
    }
 
    DUK_DDD(DUK_DDDPRINT("%s: %s", (const char *) name, (const char *) buf));
}
#endif
 
#ifdef DUK_USE_ASSERTIONS
DUK_LOCAL duk_small_int_t duk__bi_is_valid(duk__bigint *x) {
    return (duk_small_int_t)
           ( ((x->n >= 0) && (x->n <= DUK__BI_MAX_PARTS)) /* is valid size */ &&
             ((x->n == 0) || (x->v[x->n - 1] != 0)) /* is normalized */ );
}
#endif
 
DUK_LOCAL void duk__bi_normalize(duk__bigint *x) {
    duk_small_int_t i;
 
    for (i = x->n - 1; i >= 0; i--) {
        if (x->v[i] != 0) {
            break;
        }
    }
 
    /* Note: if 'x' is zero, x->n becomes 0 here */
    x->n = i + 1;
    DUK_ASSERT(duk__bi_is_valid(x));
}
 
/* x <- y */
DUK_LOCAL void duk__bi_copy(duk__bigint *x, duk__bigint *y) {
    duk_small_int_t n;
 
    n = y->n;
    x->n = n;
    if (n == 0) {
        return;
    }
    DUK_MEMCPY((void *) x->v, (const void *) y->v, (size_t) (sizeof(duk_uint32_t) * n));
}
 
DUK_LOCAL void duk__bi_set_small(duk__bigint *x, duk_uint32_t v) {
    if (v == 0U) {
        x->n = 0;
    } else {
        x->n = 1;
        x->v[0] = v;
    }
    DUK_ASSERT(duk__bi_is_valid(x));
}
 
/* Return value: <0  <=>  x < y
 *                0  <=>  x == y
 *               >0  <=>  x > y
 */
DUK_LOCAL int duk__bi_compare(duk__bigint *x, duk__bigint *y) {
    duk_small_int_t i, nx, ny;
    duk_uint32_t tx, ty;
 
    DUK_ASSERT(duk__bi_is_valid(x));
    DUK_ASSERT(duk__bi_is_valid(y));
 
    nx = x->n;
    ny = y->n;
    if (nx > ny) {
        goto ret_gt;
    }
    if (nx < ny) {
        goto ret_lt;
    }
    for (i = nx - 1; i >= 0; i--) {
        tx = x->v[i];
        ty = y->v[i];
 
        if (tx > ty) {
            goto ret_gt;
        }
        if (tx < ty) {
            goto ret_lt;
        }
    }
 
    return 0;
 
 ret_gt:
    return 1;
 
 ret_lt:
    return -1;
}
 
/* x <- y + z */
#ifdef DUK_USE_64BIT_OPS
DUK_LOCAL void duk__bi_add(duk__bigint *x, duk__bigint *y, duk__bigint *z) {
    duk_uint64_t tmp;
    duk_small_int_t i, ny, nz;
 
    DUK_ASSERT(duk__bi_is_valid(y));
    DUK_ASSERT(duk__bi_is_valid(z));
 
    if (z->n > y->n) {
        duk__bigint *t;
        t = y; y = z; z = t;
    }
    DUK_ASSERT(y->n >= z->n);
 
    ny = y->n; nz = z->n;
    tmp = 0U;
    for (i = 0; i < ny; i++) {
        DUK_ASSERT(i < DUK__BI_MAX_PARTS);
        tmp += y->v[i];
        if (i < nz) {
            tmp += z->v[i];
        }
        x->v[i] = (duk_uint32_t) (tmp & 0xffffffffUL);
        tmp = tmp >> 32;
    }
    if (tmp != 0U) {
        DUK_ASSERT(i < DUK__BI_MAX_PARTS);
        x->v[i++] = (duk_uint32_t) tmp;
    }
    x->n = i;
    DUK_ASSERT(x->n <= DUK__BI_MAX_PARTS);
 
    /* no need to normalize */
    DUK_ASSERT(duk__bi_is_valid(x));
}
#else  /* DUK_USE_64BIT_OPS */
DUK_LOCAL void duk__bi_add(duk__bigint *x, duk__bigint *y, duk__bigint *z) {
    duk_uint32_t carry, tmp1, tmp2;
    duk_small_int_t i, ny, nz;
 
    DUK_ASSERT(duk__bi_is_valid(y));
    DUK_ASSERT(duk__bi_is_valid(z));
 
    if (z->n > y->n) {
        duk__bigint *t;
        t = y; y = z; z = t;
    }
    DUK_ASSERT(y->n >= z->n);
 
    ny = y->n; nz = z->n;
    carry = 0U;
    for (i = 0; i < ny; i++) {
        /* Carry is detected based on wrapping which relies on exact 32-bit
         * types.
         */
        DUK_ASSERT(i < DUK__BI_MAX_PARTS);
        tmp1 = y->v[i];
        tmp2 = tmp1;
        if (i < nz) {
            tmp2 += z->v[i];
        }
 
        /* Careful with carry condition:
         *  - If carry not added: 0x12345678 + 0 + 0xffffffff = 0x12345677 (< 0x12345678)
         *  - If carry added:     0x12345678 + 1 + 0xffffffff = 0x12345678 (== 0x12345678)
         */
        if (carry) {
            tmp2++;
            carry = (tmp2 <= tmp1 ? 1U : 0U);
        } else {
            carry = (tmp2 < tmp1 ? 1U : 0U);
        }
 
        x->v[i] = tmp2;
    }
    if (carry) {
        DUK_ASSERT(i < DUK__BI_MAX_PARTS);
        DUK_ASSERT(carry == 1U);
        x->v[i++] = carry;
    }
    x->n = i;
    DUK_ASSERT(x->n <= DUK__BI_MAX_PARTS);
 
    /* no need to normalize */
    DUK_ASSERT(duk__bi_is_valid(x));
}
#endif  /* DUK_USE_64BIT_OPS */
 
/* x <- y + z */
DUK_LOCAL void duk__bi_add_small(duk__bigint *x, duk__bigint *y, duk_uint32_t z) {
    duk__bigint tmp;
 
    DUK_ASSERT(duk__bi_is_valid(y));
 
    /* XXX: this could be optimized; there is only one call site now though */
    duk__bi_set_small(&tmp, z);
    duk__bi_add(x, y, &tmp);
 
    DUK_ASSERT(duk__bi_is_valid(x));
}
 
#if 0  /* unused */
/* x <- x + y, use t as temp */
DUK_LOCAL void duk__bi_add_copy(duk__bigint *x, duk__bigint *y, duk__bigint *t) {
    duk__bi_add(t, x, y);
    duk__bi_copy(x, t);
}
#endif
 
/* x <- y - z, require x >= y => z >= 0, i.e. y >= z */
#ifdef DUK_USE_64BIT_OPS
DUK_LOCAL void duk__bi_sub(duk__bigint *x, duk__bigint *y, duk__bigint *z) {
    duk_small_int_t i, ny, nz;
    duk_uint32_t ty, tz;
    duk_int64_t tmp;
 
    DUK_ASSERT(duk__bi_is_valid(y));
    DUK_ASSERT(duk__bi_is_valid(z));
    DUK_ASSERT(duk__bi_compare(y, z) >= 0);
    DUK_ASSERT(y->n >= z->n);
 
    ny = y->n; nz = z->n;
    tmp = 0;
    for (i = 0; i < ny; i++) {
        ty = y->v[i];
        if (i < nz) {
            tz = z->v[i];
        } else {
            tz = 0;
        }
        tmp = (duk_int64_t) ty - (duk_int64_t) tz + tmp;
        x->v[i] = (duk_uint32_t) (tmp & 0xffffffffUL);
        tmp = tmp >> 32;  /* 0 or -1 */
    }
    DUK_ASSERT(tmp == 0);
 
    x->n = i;
    duk__bi_normalize(x);  /* need to normalize, may even cancel to 0 */
    DUK_ASSERT(duk__bi_is_valid(x));
}
#else
DUK_LOCAL void duk__bi_sub(duk__bigint *x, duk__bigint *y, duk__bigint *z) {
    duk_small_int_t i, ny, nz;
    duk_uint32_t tmp1, tmp2, borrow;
 
    DUK_ASSERT(duk__bi_is_valid(y));
    DUK_ASSERT(duk__bi_is_valid(z));
    DUK_ASSERT(duk__bi_compare(y, z) >= 0);
    DUK_ASSERT(y->n >= z->n);
 
    ny = y->n; nz = z->n;
    borrow = 0U;
    for (i = 0; i < ny; i++) {
        /* Borrow is detected based on wrapping which relies on exact 32-bit
         * types.
         */
        tmp1 = y->v[i];
        tmp2 = tmp1;
        if (i < nz) {
            tmp2 -= z->v[i];
        }
 
        /* Careful with borrow condition:
         *  - If borrow not subtracted: 0x12345678 - 0 - 0xffffffff = 0x12345679 (> 0x12345678)
         *  - If borrow subtracted:     0x12345678 - 1 - 0xffffffff = 0x12345678 (== 0x12345678)
         */
        if (borrow) {
            tmp2--;
            borrow = (tmp2 >= tmp1 ? 1U : 0U);
        } else {
            borrow = (tmp2 > tmp1 ? 1U : 0U);
        }
 
        x->v[i] = tmp2;
    }
    DUK_ASSERT(borrow == 0U);
 
    x->n = i;
    duk__bi_normalize(x);  /* need to normalize, may even cancel to 0 */
    DUK_ASSERT(duk__bi_is_valid(x));
}
#endif
 
#if 0  /* unused */
/* x <- y - z */
DUK_LOCAL void duk__bi_sub_small(duk__bigint *x, duk__bigint *y, duk_uint32_t z) {
    duk__bigint tmp;
 
    DUK_ASSERT(duk__bi_is_valid(y));
 
    /* XXX: this could be optimized */
    duk__bi_set_small(&tmp, z);
    duk__bi_sub(x, y, &tmp);
 
    DUK_ASSERT(duk__bi_is_valid(x));
}
#endif
 
/* x <- x - y, use t as temp */
DUK_LOCAL void duk__bi_sub_copy(duk__bigint *x, duk__bigint *y, duk__bigint *t) {
    duk__bi_sub(t, x, y);
    duk__bi_copy(x, t);
}
 
/* x <- y * z */
DUK_LOCAL void duk__bi_mul(duk__bigint *x, duk__bigint *y, duk__bigint *z) {
    duk_small_int_t i, j, nx, nz;
 
    DUK_ASSERT(duk__bi_is_valid(y));
    DUK_ASSERT(duk__bi_is_valid(z));
 
    nx = y->n + z->n;  /* max possible */
    DUK_ASSERT(nx <= DUK__BI_MAX_PARTS);
 
    if (nx == 0) {
        /* Both inputs are zero; cases where only one is zero can go
         * through main algorithm.
         */
        x->n = 0;
        return;
    }
 
    DUK_MEMZERO((void *) x->v, (size_t) (sizeof(duk_uint32_t) * nx));
    x->n = nx;
 
    nz = z->n;
    for (i = 0; i < y->n; i++) {
#ifdef DUK_USE_64BIT_OPS
        duk_uint64_t tmp = 0U;
        for (j = 0; j < nz; j++) {
            tmp += (duk_uint64_t) y->v[i] * (duk_uint64_t) z->v[j] + x->v[i+j];
            x->v[i+j] = (duk_uint32_t) (tmp & 0xffffffffUL);
            tmp = tmp >> 32;
        }
        if (tmp > 0) {
            DUK_ASSERT(i + j < nx);
            DUK_ASSERT(i + j < DUK__BI_MAX_PARTS);
            DUK_ASSERT(x->v[i+j] == 0U);
            x->v[i+j] = (duk_uint32_t) tmp;
        }
#else
        /*
         *  Multiply + add + carry for 32-bit components using only 16x16->32
         *  multiplies and carry detection based on unsigned overflow.
         *
         *    1st mult, 32-bit: (A*2^16 + B)
         *    2nd mult, 32-bit: (C*2^16 + D)
         *    3rd add, 32-bit: E
         *    4th add, 32-bit: F
         *
         *      (AC*2^16 + B) * (C*2^16 + D) + E + F
         *    = AC*2^32 + AD*2^16 + BC*2^16 + BD + E + F
         *    = AC*2^32 + (AD + BC)*2^16 + (BD + E + F)
         *    = AC*2^32 + AD*2^16 + BC*2^16 + (BD + E + F)
         */
        duk_uint32_t a, b, c, d, e, f;
        duk_uint32_t r, s, t;
 
        a = y->v[i]; b = a & 0xffffUL; a = a >> 16;
 
        f = 0;
        for (j = 0; j < nz; j++) {
            c = z->v[j]; d = c & 0xffffUL; c = c >> 16;
            e = x->v[i+j];
 
            /* build result as: (r << 32) + s: start with (BD + E + F) */
            r = 0;
            s = b * d;
 
            /* add E */
            t = s + e;
            if (t < s) { r++; }  /* carry */
            s = t;
 
            /* add F */
            t = s + f;
            if (t < s) { r++; }  /* carry */
            s = t;
 
            /* add BC*2^16 */
            t = b * c;
            r += (t >> 16);
            t = s + ((t & 0xffffUL) << 16);
            if (t < s) { r++; }  /* carry */
            s = t;
 
            /* add AD*2^16 */
            t = a * d;
            r += (t >> 16);
            t = s + ((t & 0xffffUL) << 16);
            if (t < s) { r++; }  /* carry */
            s = t;
 
            /* add AC*2^32 */
            t = a * c;
            r += t;
 
            DUK_DDD(DUK_DDDPRINT("ab=%08lx cd=%08lx ef=%08lx -> rs=%08lx %08lx",
                                 (unsigned long) y->v[i], (unsigned long) z->v[j],
                                 (unsigned long) x->v[i+j], (unsigned long) r,
                                 (unsigned long) s));
 
            x->v[i+j] = s;
            f = r;
        }
        if (f > 0U) {
            DUK_ASSERT(i + j < nx);
            DUK_ASSERT(i + j < DUK__BI_MAX_PARTS);
            DUK_ASSERT(x->v[i+j] == 0U);
            x->v[i+j] = (duk_uint32_t) f;
        }
#endif  /* DUK_USE_64BIT_OPS */
    }
 
    duk__bi_normalize(x);
    DUK_ASSERT(duk__bi_is_valid(x));
}
 
/* x <- y * z */
DUK_LOCAL void duk__bi_mul_small(duk__bigint *x, duk__bigint *y, duk_uint32_t z) {
    duk__bigint tmp;
 
    DUK_ASSERT(duk__bi_is_valid(y));
 
    /* XXX: this could be optimized */
    duk__bi_set_small(&tmp, z);
    duk__bi_mul(x, y, &tmp);
 
    DUK_ASSERT(duk__bi_is_valid(x));
}
 
/* x <- x * y, use t as temp */
DUK_LOCAL void duk__bi_mul_copy(duk__bigint *x, duk__bigint *y, duk__bigint *t) {
    duk__bi_mul(t, x, y);
    duk__bi_copy(x, t);
}
 
/* x <- x * y, use t as temp */
DUK_LOCAL void duk__bi_mul_small_copy(duk__bigint *x, duk_uint32_t y, duk__bigint *t) {
    duk__bi_mul_small(t, x, y);
    duk__bi_copy(x, t);
}
 
DUK_LOCAL int duk__bi_is_even(duk__bigint *x) {
    DUK_ASSERT(duk__bi_is_valid(x));
    return (x->n == 0) || ((x->v[0] & 0x01) == 0);
}
 
DUK_LOCAL int duk__bi_is_zero(duk__bigint *x) {
    DUK_ASSERT(duk__bi_is_valid(x));
    return (x->n == 0);  /* this is the case for normalized numbers */
}
 
/* Bigint is 2^52.  Used to detect normalized IEEE double mantissa values
 * which are at the lowest edge (next floating point value downwards has
 * a different exponent).  The lowest mantissa has the form:
 *
 *     1000........000    (52 zeroes; only "hidden bit" is set)
 */
DUK_LOCAL duk_small_int_t duk__bi_is_2to52(duk__bigint *x) {
    DUK_ASSERT(duk__bi_is_valid(x));
    return (duk_small_int_t)
            (x->n == 2) && (x->v[0] == 0U) && (x->v[1] == (1U << (52-32)));
}
 
/* x <- (1<<y) */
DUK_LOCAL void duk__bi_twoexp(duk__bigint *x, duk_small_int_t y) {
    duk_small_int_t n, r;
 
    n = (y / 32) + 1;
    DUK_ASSERT(n > 0);
    r = y % 32;
    DUK_MEMZERO((void *) x->v, sizeof(duk_uint32_t) * n);
    x->n = n;
    x->v[n - 1] = (((duk_uint32_t) 1) << r);
}
 
/* x <- b^y; use t1 and t2 as temps */
DUK_LOCAL void duk__bi_exp_small(duk__bigint *x, duk_small_int_t b, duk_small_int_t y, duk__bigint *t1, duk__bigint *t2) {
    /* Fast path the binary case */
 
    DUK_ASSERT(x != t1 && x != t2 && t1 != t2);  /* distinct bignums, easy mistake to make */
    DUK_ASSERT(b >= 0);
    DUK_ASSERT(y >= 0);
 
    if (b == 2) {
        duk__bi_twoexp(x, y);
        return;
    }
 
    /* http://en.wikipedia.org/wiki/Exponentiation_by_squaring */
 
    DUK_DDD(DUK_DDDPRINT("exp_small: b=%ld, y=%ld", (long) b, (long) y));
 
    duk__bi_set_small(x, 1);
    duk__bi_set_small(t1, b);
    for (;;) {
        /* Loop structure ensures that we don't compute t1^2 unnecessarily
         * on the final round, as that might create a bignum exceeding the
         * current DUK__BI_MAX_PARTS limit.
         */
        if (y & 0x01) {
            duk__bi_mul_copy(x, t1, t2);
        }
        y = y >> 1;
        if (y == 0) {
            break;
        }
        duk__bi_mul_copy(t1, t1, t2);
    }
 
    DUK__BI_PRINT("exp_small result", x);
}
 
/*
 *  A Dragon4 number-to-string variant, based on:
 *
 *    Guy L. Steele Jr., Jon L. White: "How to Print Floating-Point Numbers
 *    Accurately"
 *
 *    Robert G. Burger, R. Kent Dybvig: "Printing Floating-Point Numbers
 *    Quickly and Accurately"
 *
 *  The current algorithm is based on Figure 1 of the Burger-Dybvig paper,
 *  i.e. the base implementation without logarithm estimation speedups
 *  (these would increase code footprint considerably).  Fixed-format output
 *  does not follow the suggestions in the paper; instead, we generate an
 *  extra digit and round-with-carry.
 *
 *  The same algorithm is used for number parsing (with b=10 and B=2)
 *  by generating one extra digit and doing rounding manually.
 *
 *  See doc/number-conversion.rst for limitations.
 */
 
/* Maximum number of digits generated. */
#define DUK__MAX_OUTPUT_DIGITS          1040  /* (Number.MAX_VALUE).toString(2).length == 1024, + spare */
 
/* Maximum number of characters in formatted value. */
#define DUK__MAX_FORMATTED_LENGTH       1040  /* (-Number.MAX_VALUE).toString(2).length == 1025, + spare */
 
/* Number and (minimum) size of bigints in the nc_ctx structure. */
#define DUK__NUMCONV_CTX_NUM_BIGINTS    7
#define DUK__NUMCONV_CTX_BIGINTS_SIZE   (sizeof(duk__bigint) * DUK__NUMCONV_CTX_NUM_BIGINTS)
 
typedef struct {
    /* Currently about 7*152 = 1064 bytes.  The space for these
     * duk__bigints is used also as a temporary buffer for generating
     * the final string.  This is a bit awkard; a union would be
     * more correct.
     */
    duk__bigint f, r, s, mp, mm, t1, t2;
 
    duk_small_int_t is_s2n;        /* if 1, doing a string-to-number; else doing a number-to-string */
    duk_small_int_t is_fixed;      /* if 1, doing a fixed format output (not free format) */
    duk_small_int_t req_digits;    /* requested number of output digits; 0 = free-format */
    duk_small_int_t abs_pos;       /* digit position is absolute, not relative */
    duk_small_int_t e;             /* exponent for 'f' */
    duk_small_int_t b;             /* input radix */
    duk_small_int_t B;             /* output radix */
    duk_small_int_t k;             /* see algorithm */
    duk_small_int_t low_ok;        /* see algorithm */
    duk_small_int_t high_ok;       /* see algorithm */
    duk_small_int_t unequal_gaps;  /* m+ != m- (very rarely) */
 
    /* Buffer used for generated digits, values are in the range [0,B-1]. */
    duk_uint8_t digits[DUK__MAX_OUTPUT_DIGITS];
    duk_small_int_t count;  /* digit count */
} duk__numconv_stringify_ctx;
 
/* Note: computes with 'idx' in assertions, so caller beware.
 * 'idx' is preincremented, i.e. '1' on first call, because it
 * is more convenient for the caller.
 */
#define DUK__DRAGON4_OUTPUT_PREINC(nc_ctx,preinc_idx,x)  do { \
        DUK_ASSERT((preinc_idx) - 1 >= 0); \
        DUK_ASSERT((preinc_idx) - 1 < DUK__MAX_OUTPUT_DIGITS); \
        ((nc_ctx)->digits[(preinc_idx) - 1]) = (duk_uint8_t) (x); \
    } while (0)
 
DUK_LOCAL duk_size_t duk__dragon4_format_uint32(duk_uint8_t *buf, duk_uint32_t x, duk_small_int_t radix) {
    duk_uint8_t *p;
    duk_size_t len;
    duk_small_int_t dig;
    duk_small_int_t t;
 
    DUK_ASSERT(radix >= 2 && radix <= 36);
 
    /* A 32-bit unsigned integer formats to at most 32 digits (the
     * worst case happens with radix == 2).  Output the digits backwards,
     * and use a memmove() to get them in the right place.
     */
 
    p = buf + 32;
    for (;;) {
        t = x / radix;
        dig = x - t * radix;
        x = t;
 
        DUK_ASSERT(dig >= 0 && dig < 36);
        *(--p) = DUK__DIGITCHAR(dig);
 
        if (x == 0) {
            break;
        }
    }
    len = (duk_size_t) ((buf + 32) - p);
 
    DUK_MEMMOVE((void *) buf, (const void *) p, (size_t) len);
 
    return len;
}
 
DUK_LOCAL void duk__dragon4_prepare(duk__numconv_stringify_ctx *nc_ctx) {
    duk_small_int_t lowest_mantissa;
 
#if 1
    /* Assume IEEE round-to-even, so that shorter encoding can be used
     * when round-to-even would produce correct result.  By removing
     * this check (and having low_ok == high_ok == 0) the results would
     * still be accurate but in some cases longer than necessary.
     */
    if (duk__bi_is_even(&nc_ctx->f)) {
        DUK_DDD(DUK_DDDPRINT("f is even"));
        nc_ctx->low_ok = 1;
        nc_ctx->high_ok = 1;
    } else {
        DUK_DDD(DUK_DDDPRINT("f is odd"));
        nc_ctx->low_ok = 0;
        nc_ctx->high_ok = 0;
    }
#else
    /* Note: not honoring round-to-even should work but now generates incorrect
     * results.  For instance, 1e23 serializes to "a000...", i.e. the first digit
     * equals the radix (10).  Scaling stops one step too early in this case.
     * Don't know why this is the case, but since this code path is unused, it
     * doesn't matter.
     */
    nc_ctx->low_ok = 0;
    nc_ctx->high_ok = 0;
#endif
 
    /* For string-to-number, pretend we never have the lowest mantissa as there
     * is no natural "precision" for inputs.  Having lowest_mantissa == 0, we'll
     * fall into the base cases for both e >= 0 and e < 0.
     */
    if (nc_ctx->is_s2n) {
        lowest_mantissa = 0;
    } else {
        lowest_mantissa = duk__bi_is_2to52(&nc_ctx->f);
    }
 
    nc_ctx->unequal_gaps = 0;
    if (nc_ctx->e >= 0) {
        /* exponent non-negative (and thus not minimum exponent) */
 
        if (lowest_mantissa) {
            /* (>= e 0) AND (= f (expt b (- p 1)))
             *
             * be <- (expt b e) == b^e
             * be1 <- (* be b) == (expt b (+ e 1)) == b^(e+1)
             * r <- (* f be1 2) == 2 * f * b^(e+1)    [if b==2 -> f * b^(e+2)]
             * s <- (* b 2)                           [if b==2 -> 4]
             * m+ <- be1 == b^(e+1)
             * m- <- be == b^e
             * k <- 0
             * B <- B
             * low_ok <- round
             * high_ok <- round
             */
 
            DUK_DDD(DUK_DDDPRINT("non-negative exponent (not smallest exponent); "
                                 "lowest mantissa value for this exponent -> "
                                 "unequal gaps"));
 
            duk__bi_exp_small(&nc_ctx->mm, nc_ctx->b, nc_ctx->e, &nc_ctx->t1, &nc_ctx->t2);  /* mm <- b^e */
            duk__bi_mul_small(&nc_ctx->mp, &nc_ctx->mm, nc_ctx->b);  /* mp <- b^(e+1) */
            duk__bi_mul_small(&nc_ctx->t1, &nc_ctx->f, 2);
            duk__bi_mul(&nc_ctx->r, &nc_ctx->t1, &nc_ctx->mp);       /* r <- (2 * f) * b^(e+1) */
            duk__bi_set_small(&nc_ctx->s, nc_ctx->b * 2);            /* s <- 2 * b */
            nc_ctx->unequal_gaps = 1;
        } else {
            /* (>= e 0) AND (not (= f (expt b (- p 1))))
             *
             * be <- (expt b e) == b^e
             * r <- (* f be 2) == 2 * f * b^e    [if b==2 -> f * b^(e+1)]
             * s <- 2
             * m+ <- be == b^e
             * m- <- be == b^e
             * k <- 0
             * B <- B
             * low_ok <- round
             * high_ok <- round
             */
 
            DUK_DDD(DUK_DDDPRINT("non-negative exponent (not smallest exponent); "
                                 "not lowest mantissa for this exponent -> "
                                 "equal gaps"));
 
            duk__bi_exp_small(&nc_ctx->mm, nc_ctx->b, nc_ctx->e, &nc_ctx->t1, &nc_ctx->t2);  /* mm <- b^e */
            duk__bi_copy(&nc_ctx->mp, &nc_ctx->mm);                /* mp <- b^e */
            duk__bi_mul_small(&nc_ctx->t1, &nc_ctx->f, 2);
            duk__bi_mul(&nc_ctx->r, &nc_ctx->t1, &nc_ctx->mp);     /* r <- (2 * f) * b^e */
            duk__bi_set_small(&nc_ctx->s, 2);                      /* s <- 2 */
        }
    } else {
        /* When doing string-to-number, lowest_mantissa is always 0 so
         * the exponent check, while incorrect, won't matter.
         */
        if (nc_ctx->e > DUK__IEEE_DOUBLE_EXP_MIN /*not minimum exponent*/ &&
            lowest_mantissa /* lowest mantissa for this exponent*/) {
            /* r <- (* f b 2)                                [if b==2 -> (* f 4)]
             * s <- (* (expt b (- 1 e)) 2) == b^(1-e) * 2    [if b==2 -> b^(2-e)]
             * m+ <- b == 2
             * m- <- 1
             * k <- 0
             * B <- B
             * low_ok <- round
             * high_ok <- round
             */
 
            DUK_DDD(DUK_DDDPRINT("negative exponent; not minimum exponent and "
                                 "lowest mantissa for this exponent -> "
                                 "unequal gaps"));
 
            duk__bi_mul_small(&nc_ctx->r, &nc_ctx->f, nc_ctx->b * 2);  /* r <- (2 * b) * f */
            duk__bi_exp_small(&nc_ctx->t1, nc_ctx->b, 1 - nc_ctx->e, &nc_ctx->s, &nc_ctx->t2);  /* NB: use 's' as temp on purpose */
            duk__bi_mul_small(&nc_ctx->s, &nc_ctx->t1, 2);             /* s <- b^(1-e) * 2 */
            duk__bi_set_small(&nc_ctx->mp, 2);
            duk__bi_set_small(&nc_ctx->mm, 1);
            nc_ctx->unequal_gaps = 1;
        } else {
            /* r <- (* f 2)
             * s <- (* (expt b (- e)) 2) == b^(-e) * 2    [if b==2 -> b^(1-e)]
             * m+ <- 1
             * m- <- 1
             * k <- 0
             * B <- B
             * low_ok <- round
             * high_ok <- round
             */
 
            DUK_DDD(DUK_DDDPRINT("negative exponent; minimum exponent or not "
                                 "lowest mantissa for this exponent -> "
                                 "equal gaps"));
 
            duk__bi_mul_small(&nc_ctx->r, &nc_ctx->f, 2);            /* r <- 2 * f */
            duk__bi_exp_small(&nc_ctx->t1, nc_ctx->b, -nc_ctx->e, &nc_ctx->s, &nc_ctx->t2);  /* NB: use 's' as temp on purpose */
            duk__bi_mul_small(&nc_ctx->s, &nc_ctx->t1, 2);           /* s <- b^(-e) * 2 */
            duk__bi_set_small(&nc_ctx->mp, 1);
            duk__bi_set_small(&nc_ctx->mm, 1);
        }
    }
}
 
DUK_LOCAL void duk__dragon4_scale(duk__numconv_stringify_ctx *nc_ctx) {
    duk_small_int_t k = 0;
 
    /* This is essentially the 'scale' algorithm, with recursion removed.
     * Note that 'k' is either correct immediately, or will move in one
     * direction in the loop.  There's no need to do the low/high checks
     * on every round (like the Scheme algorithm does).
     *
     * The scheme algorithm finds 'k' and updates 's' simultaneously,
     * while the logical algorithm finds 'k' with 's' having its initial
     * value, after which 's' is updated separately (see the Burger-Dybvig
     * paper, Section 3.1, steps 2 and 3).
     *
     * The case where m+ == m- (almost always) is optimized for, because
     * it reduces the bigint operations considerably and almost always
     * applies.  The scale loop only needs to work with m+, so this works.
     */
 
    /* XXX: this algorithm could be optimized quite a lot by using e.g.
     * a logarithm based estimator for 'k' and performing B^n multiplication
     * using a lookup table or using some bit-representation based exp
     * algorithm.  Currently we just loop, with significant performance
     * impact for very large and very small numbers.
     */
 
    DUK_DDD(DUK_DDDPRINT("scale: B=%ld, low_ok=%ld, high_ok=%ld",
                         (long) nc_ctx->B, (long) nc_ctx->low_ok, (long) nc_ctx->high_ok));
    DUK__BI_PRINT("r(init)", &nc_ctx->r);
    DUK__BI_PRINT("s(init)", &nc_ctx->s);
    DUK__BI_PRINT("mp(init)", &nc_ctx->mp);
    DUK__BI_PRINT("mm(init)", &nc_ctx->mm);
 
    for (;;) {
        DUK_DDD(DUK_DDDPRINT("scale loop (inc k), k=%ld", (long) k));
        DUK__BI_PRINT("r", &nc_ctx->r);
        DUK__BI_PRINT("s", &nc_ctx->s);
        DUK__BI_PRINT("m+", &nc_ctx->mp);
        DUK__BI_PRINT("m-", &nc_ctx->mm);
 
        duk__bi_add(&nc_ctx->t1, &nc_ctx->r, &nc_ctx->mp);  /* t1 = (+ r m+) */
        if (duk__bi_compare(&nc_ctx->t1, &nc_ctx->s) >= (nc_ctx->high_ok ? 0 : 1)) {
            DUK_DDD(DUK_DDDPRINT("k is too low"));
            /* r <- r
             * s <- (* s B)
             * m+ <- m+
             * m- <- m-
             * k <- (+ k 1)
             */
 
            duk__bi_mul_small_copy(&nc_ctx->s, nc_ctx->B, &nc_ctx->t1);
            k++;
        } else {
            break;
        }
    }
 
    /* k > 0 -> k was too low, and cannot be too high */
    if (k > 0) {
        goto skip_dec_k;
    }
 
    for (;;) {
        DUK_DDD(DUK_DDDPRINT("scale loop (dec k), k=%ld", (long) k));
        DUK__BI_PRINT("r", &nc_ctx->r);
        DUK__BI_PRINT("s", &nc_ctx->s);
        DUK__BI_PRINT("m+", &nc_ctx->mp);
        DUK__BI_PRINT("m-", &nc_ctx->mm);
 
        duk__bi_add(&nc_ctx->t1, &nc_ctx->r, &nc_ctx->mp);  /* t1 = (+ r m+) */
        duk__bi_mul_small(&nc_ctx->t2, &nc_ctx->t1, nc_ctx->B);   /* t2 = (* (+ r m+) B) */
        if (duk__bi_compare(&nc_ctx->t2, &nc_ctx->s) <= (nc_ctx->high_ok ? -1 : 0)) {
            DUK_DDD(DUK_DDDPRINT("k is too high"));
            /* r <- (* r B)
             * s <- s
             * m+ <- (* m+ B)
             * m- <- (* m- B)
             * k <- (- k 1)
             */
            duk__bi_mul_small_copy(&nc_ctx->r, nc_ctx->B, &nc_ctx->t1);
            duk__bi_mul_small_copy(&nc_ctx->mp, nc_ctx->B, &nc_ctx->t1);
            if (nc_ctx->unequal_gaps) {
                DUK_DDD(DUK_DDDPRINT("m+ != m- -> need to update m- too"));
                duk__bi_mul_small_copy(&nc_ctx->mm, nc_ctx->B, &nc_ctx->t1);
            }
            k--;
        } else {
            break;
        }
    }
 
 skip_dec_k:
 
    if (!nc_ctx->unequal_gaps) {
        DUK_DDD(DUK_DDDPRINT("equal gaps, copy m- from m+"));
        duk__bi_copy(&nc_ctx->mm, &nc_ctx->mp);  /* mm <- mp */
    }
    nc_ctx->k = k;
 
    DUK_DDD(DUK_DDDPRINT("final k: %ld", (long) k));
    DUK__BI_PRINT("r(final)", &nc_ctx->r);
    DUK__BI_PRINT("s(final)", &nc_ctx->s);
    DUK__BI_PRINT("mp(final)", &nc_ctx->mp);
    DUK__BI_PRINT("mm(final)", &nc_ctx->mm);
}
 
DUK_LOCAL void duk__dragon4_generate(duk__numconv_stringify_ctx *nc_ctx) {
    duk_small_int_t tc1, tc2;    /* terminating conditions */
    duk_small_int_t d;           /* current digit */
    duk_small_int_t count = 0;   /* digit count */
 
    /*
     *  Digit generation loop.
     *
     *  Different termination conditions:
     *
     *    1. Free format output.  Terminate when shortest accurate
     *       representation found.
     *
     *    2. Fixed format output, with specific number of digits.
     *       Ignore termination conditions, terminate when digits
     *       generated.  Caller requests an extra digit and rounds.
     *
     *    3. Fixed format output, with a specific absolute cut-off
     *       position (e.g. 10 digits after decimal point).  Note
     *       that we always generate at least one digit, even if
     *       the digit is below the cut-off point already.
     */
 
    for (;;) {
        DUK_DDD(DUK_DDDPRINT("generate loop, count=%ld, k=%ld, B=%ld, low_ok=%ld, high_ok=%ld",
                             (long) count, (long) nc_ctx->k, (long) nc_ctx->B,
                             (long) nc_ctx->low_ok, (long) nc_ctx->high_ok));
        DUK__BI_PRINT("r", &nc_ctx->r);
        DUK__BI_PRINT("s", &nc_ctx->s);
        DUK__BI_PRINT("m+", &nc_ctx->mp);
        DUK__BI_PRINT("m-", &nc_ctx->mm);
 
        /* (quotient-remainder (* r B) s) using a dummy subtraction loop */
        duk__bi_mul_small(&nc_ctx->t1, &nc_ctx->r, nc_ctx->B);       /* t1 <- (* r B) */
        d = 0;
        for (;;) {
            if (duk__bi_compare(&nc_ctx->t1, &nc_ctx->s) < 0) {
                break;
            }
            duk__bi_sub_copy(&nc_ctx->t1, &nc_ctx->s, &nc_ctx->t2);  /* t1 <- t1 - s */
            d++;
        }
        duk__bi_copy(&nc_ctx->r, &nc_ctx->t1);  /* r <- (remainder (* r B) s) */
                                                /* d <- (quotient (* r B) s)   (in range 0...B-1) */
        DUK_DDD(DUK_DDDPRINT("-> d(quot)=%ld", (long) d));
        DUK__BI_PRINT("r(rem)", &nc_ctx->r);
 
        duk__bi_mul_small_copy(&nc_ctx->mp, nc_ctx->B, &nc_ctx->t2); /* m+ <- (* m+ B) */
        duk__bi_mul_small_copy(&nc_ctx->mm, nc_ctx->B, &nc_ctx->t2); /* m- <- (* m- B) */
        DUK__BI_PRINT("mp(upd)", &nc_ctx->mp);
        DUK__BI_PRINT("mm(upd)", &nc_ctx->mm);
 
        /* Terminating conditions.  For fixed width output, we just ignore the
         * terminating conditions (and pretend that tc1 == tc2 == false).  The
         * the current shortcut for fixed-format output is to generate a few
         * extra digits and use rounding (with carry) to finish the output.
         */
 
        if (nc_ctx->is_fixed == 0) {
            /* free-form */
            tc1 = (duk__bi_compare(&nc_ctx->r, &nc_ctx->mm) <= (nc_ctx->low_ok ? 0 : -1));
 
            duk__bi_add(&nc_ctx->t1, &nc_ctx->r, &nc_ctx->mp);  /* t1 <- (+ r m+) */
            tc2 = (duk__bi_compare(&nc_ctx->t1, &nc_ctx->s) >= (nc_ctx->high_ok ? 0 : 1));
 
            DUK_DDD(DUK_DDDPRINT("tc1=%ld, tc2=%ld", (long) tc1, (long) tc2));
        } else {
            /* fixed-format */
            tc1 = 0;
            tc2 = 0;
        }
 
        /* Count is incremented before DUK__DRAGON4_OUTPUT_PREINC() call
         * on purpose, which is taken into account by the macro.
         */
        count++;
 
        if (tc1) {
            if (tc2) {
                /* tc1 = true, tc2 = true */
                duk__bi_mul_small(&nc_ctx->t1, &nc_ctx->r, 2);
                if (duk__bi_compare(&nc_ctx->t1, &nc_ctx->s) < 0) {  /* (< (* r 2) s) */
                    DUK_DDD(DUK_DDDPRINT("tc1=true, tc2=true, 2r > s: output d --> %ld (k=%ld)",
                                         (long) d, (long) nc_ctx->k));
                    DUK__DRAGON4_OUTPUT_PREINC(nc_ctx, count, d);
                } else {
                    DUK_DDD(DUK_DDDPRINT("tc1=true, tc2=true, 2r <= s: output d+1 --> %ld (k=%ld)",
                                         (long) (d + 1), (long) nc_ctx->k));
                    DUK__DRAGON4_OUTPUT_PREINC(nc_ctx, count, d + 1);
                }
                break;
            } else {
                /* tc1 = true, tc2 = false */
                DUK_DDD(DUK_DDDPRINT("tc1=true, tc2=false: output d --> %ld (k=%ld)",
                                     (long) d, (long) nc_ctx->k));
                DUK__DRAGON4_OUTPUT_PREINC(nc_ctx, count, d);
                break;
            }
        } else {
            if (tc2) {
                /* tc1 = false, tc2 = true */
                DUK_DDD(DUK_DDDPRINT("tc1=false, tc2=true: output d+1 --> %ld (k=%ld)",
                                     (long) (d + 1), (long) nc_ctx->k));
                DUK__DRAGON4_OUTPUT_PREINC(nc_ctx, count, d + 1);
                break;
            } else {
                /* tc1 = false, tc2 = false */
                DUK_DDD(DUK_DDDPRINT("tc1=false, tc2=false: output d --> %ld (k=%ld)",
                                     (long) d, (long) nc_ctx->k));
                DUK__DRAGON4_OUTPUT_PREINC(nc_ctx, count, d);
 
                /* r <- r    (updated above: r <- (remainder (* r B) s)
                 * s <- s
                 * m+ <- m+  (updated above: m+ <- (* m+ B)
                 * m- <- m-  (updated above: m- <- (* m- B)
                 * B, low_ok, high_ok are fixed
                 */
 
                /* fall through and continue for-loop */
            }
        }
 
        /* fixed-format termination conditions */
        if (nc_ctx->is_fixed) {
            if (nc_ctx->abs_pos) {
                int pos = nc_ctx->k - count + 1;  /* count is already incremented, take into account */
                DUK_DDD(DUK_DDDPRINT("fixed format, absolute: abs pos=%ld, k=%ld, count=%ld, req=%ld",
                                     (long) pos, (long) nc_ctx->k, (long) count, (long) nc_ctx->req_digits));
                if (pos <= nc_ctx->req_digits) {
                    DUK_DDD(DUK_DDDPRINT("digit position reached req_digits, end generate loop"));
                    break;
                }
            } else {
                DUK_DDD(DUK_DDDPRINT("fixed format, relative: k=%ld, count=%ld, req=%ld",
                                     (long) nc_ctx->k, (long) count, (long) nc_ctx->req_digits));
                if (count >= nc_ctx->req_digits) {
                    DUK_DDD(DUK_DDDPRINT("digit count reached req_digits, end generate loop"));
                    break;
                }
            }
        }
    }  /* for */
 
    nc_ctx->count = count;
 
    DUK_DDD(DUK_DDDPRINT("generate finished"));
 
#ifdef DUK_USE_DDDPRINT
    {
        duk_uint8_t buf[2048];
        duk_small_int_t i, t;
        DUK_MEMZERO(buf, sizeof(buf));
        for (i = 0; i < nc_ctx->count; i++) {
            t = nc_ctx->digits[i];
            if (t < 0 || t > 36) {
                buf[i] = (duk_uint8_t) '?';
            } else {
                buf[i] = (duk_uint8_t) DUK__DIGITCHAR(t);
            }
        }
        DUK_DDD(DUK_DDDPRINT("-> generated digits; k=%ld, digits='%s'",
                             (long) nc_ctx->k, (const char *) buf));
    }
#endif
}
 
/* Round up digits to a given position.  If position is out-of-bounds,
 * does nothing.  If carry propagates over the first digit, a '1' is
 * prepended to digits and 'k' will be updated.  Return value indicates
 * whether carry propagated over the first digit.
 *
 * Note that nc_ctx->count is NOT updated based on the rounding position
 * (it is updated only if carry overflows over the first digit and an
 * extra digit is prepended).
 */
DUK_LOCAL duk_small_int_t duk__dragon4_fixed_format_round(duk__numconv_stringify_ctx *nc_ctx, duk_small_int_t round_idx) {
    duk_small_int_t t;
    duk_uint8_t *p;
    duk_uint8_t roundup_limit;
    duk_small_int_t ret = 0;
 
    /*
     *  round_idx points to the digit which is considered for rounding; the
     *  digit to its left is the final digit of the rounded value.  If round_idx
     *  is zero, rounding will be performed; the result will either be an empty
     *  rounded value or if carry happens a '1' digit is generated.
     */
 
    if (round_idx >= nc_ctx->count) {
        DUK_DDD(DUK_DDDPRINT("round_idx out of bounds (%ld >= %ld (count)) -> no rounding",
                             (long) round_idx, (long) nc_ctx->count));
        return 0;
    } else if (round_idx < 0) {
        DUK_DDD(DUK_DDDPRINT("round_idx out of bounds (%ld < 0) -> no rounding",
                             (long) round_idx));
        return 0;
    }
 
    /*
     *  Round-up limit.
     *
     *  For even values, divides evenly, e.g. 10 -> roundup_limit=5.
     *
     *  For odd values, rounds up, e.g. 3 -> roundup_limit=2.
     *  If radix is 3, 0/3 -> down, 1/3 -> down, 2/3 -> up.
     */
    roundup_limit = (duk_uint8_t) ((nc_ctx->B + 1) / 2);
 
    p = &nc_ctx->digits[round_idx];
    if (*p >= roundup_limit) {
        DUK_DDD(DUK_DDDPRINT("fixed-format rounding carry required"));
        /* carry */
        for (;;) {
            *p = 0;
            if (p == &nc_ctx->digits[0]) {
                DUK_DDD(DUK_DDDPRINT("carry propagated to first digit -> special case handling"));
                DUK_MEMMOVE((void *) (&nc_ctx->digits[1]),
                            (const void *) (&nc_ctx->digits[0]),
                            (size_t) (sizeof(char) * nc_ctx->count));
                nc_ctx->digits[0] = 1;  /* don't increase 'count' */
                nc_ctx->k++;  /* position of highest digit changed */
                nc_ctx->count++;  /* number of digits changed */
                ret = 1;
                break;
            }
 
            DUK_DDD(DUK_DDDPRINT("fixed-format rounding carry: B=%ld, roundup_limit=%ld, p=%p, digits=%p",
                                 (long) nc_ctx->B, (long) roundup_limit, (void *) p, (void *) nc_ctx->digits));
            p--;
            t = *p;
            DUK_DDD(DUK_DDDPRINT("digit before carry: %ld", (long) t));
            if (++t < nc_ctx->B) {
                DUK_DDD(DUK_DDDPRINT("rounding carry terminated"));
                *p = (duk_uint8_t) t;
                break;
            }
 
            DUK_DDD(DUK_DDDPRINT("wraps, carry to next digit"));
        }
    }
 
    return ret;
}
 
#define DUK__NO_EXP  (65536)  /* arbitrary marker, outside valid exp range */
 
DUK_LOCAL void duk__dragon4_convert_and_push(duk__numconv_stringify_ctx *nc_ctx,
                                          duk_context *ctx,
                                          duk_small_int_t radix,
                                          duk_small_int_t digits,
                                          duk_small_uint_t flags,
                                          duk_small_int_t neg) {
    duk_small_int_t k;
    duk_small_int_t pos, pos_end;
    duk_small_int_t expt;
    duk_small_int_t dig;
    duk_uint8_t *q;
    duk_uint8_t *buf;
 
    /*
     *  The string conversion here incorporates all the necessary Ecmascript
     *  semantics without attempting to be generic.  nc_ctx->digits contains
     *  nc_ctx->count digits (>= 1), with the topmost digit's 'position'
     *  indicated by nc_ctx->k as follows:
     *
     *    digits="123" count=3 k=0   -->   0.123
     *    digits="123" count=3 k=1   -->   1.23
     *    digits="123" count=3 k=5   -->   12300
     *    digits="123" count=3 k=-1  -->   0.0123
     *
     *  Note that the identifier names used for format selection are different
     *  in Burger-Dybvig paper and Ecmascript specification (quite confusingly
     *  so, because e.g. 'k' has a totally different meaning in each).  See
     *  documentation for discussion.
     *
     *  Ecmascript doesn't specify any specific behavior for format selection
     *  (e.g. when to use exponent notation) for non-base-10 numbers.
     *
     *  The bigint space in the context is reused for string output, as there
     *  is more than enough space for that (>1kB at the moment), and we avoid
     *  allocating even more stack.
     */
 
    DUK_ASSERT(DUK__NUMCONV_CTX_BIGINTS_SIZE >= DUK__MAX_FORMATTED_LENGTH);
    DUK_ASSERT(nc_ctx->count >= 1);
 
    k = nc_ctx->k;
    buf = (duk_uint8_t *) &nc_ctx->f;  /* XXX: union would be more correct */
    q = buf;
 
    /* Exponent handling: if exponent format is used, record exponent value and
     * fake k such that one leading digit is generated (e.g. digits=123 -> "1.23").
     *
     * toFixed() prevents exponent use; otherwise apply a set of criteria to
     * match the other API calls (toString(), toPrecision, etc).
     */
 
    expt = DUK__NO_EXP;
    if (!nc_ctx->abs_pos /* toFixed() */) {
        if ((flags & DUK_N2S_FLAG_FORCE_EXP) ||             /* exponential notation forced */
            ((flags & DUK_N2S_FLAG_NO_ZERO_PAD) &&          /* fixed precision and zero padding would be required */
                 (k - digits >= 1)) ||                          /* (e.g. k=3, digits=2 -> "12X") */
            ((k > 21 || k <= -6) && (radix == 10))) {       /* toString() conditions */
            DUK_DDD(DUK_DDDPRINT("use exponential notation: k=%ld -> expt=%ld",
                                 (long) k, (long) (k - 1)));
            expt = k - 1;  /* e.g. 12.3 -> digits="123" k=2 -> 1.23e1 */
            k = 1;  /* generate mantissa with a single leading whole number digit */
        }
    }
 
    if (neg) {
        *q++ = '-';
    }
 
    /* Start position (inclusive) and end position (exclusive) */
    pos = (k >= 1 ? k : 1);
    if (nc_ctx->is_fixed) {
        if (nc_ctx->abs_pos) {
            /* toFixed() */
            pos_end = -digits;
        } else {
            pos_end = k - digits;
        }
    } else {
        pos_end = k - nc_ctx->count;
    }
    if (pos_end > 0) {
        pos_end = 0;
    }
 
    DUK_DDD(DUK_DDDPRINT("expt=%ld, k=%ld, count=%ld, pos=%ld, pos_end=%ld, is_fixed=%ld, "
                         "digits=%ld, abs_pos=%ld",
                         (long) expt, (long) k, (long) nc_ctx->count, (long) pos, (long) pos_end,
                         (long) nc_ctx->is_fixed, (long) digits, (long) nc_ctx->abs_pos));
 
    /* Digit generation */
    while (pos > pos_end) {
        DUK_DDD(DUK_DDDPRINT("digit generation: pos=%ld, pos_end=%ld",
                             (long) pos, (long) pos_end));
        if (pos == 0) {
            *q++ = (duk_uint8_t) '.';
        }
        if (pos > k) {
            *q++ = (duk_uint8_t) '0';
        } else if (pos <= k - nc_ctx->count) {
            *q++ = (duk_uint8_t) '0';
        } else {
            dig = nc_ctx->digits[k - pos];
            DUK_ASSERT(dig >= 0 && dig < nc_ctx->B);
            *q++ = (duk_uint8_t) DUK__DIGITCHAR(dig);
        }
 
        pos--;
    }
    DUK_ASSERT(pos <= 1);
 
    /* Exponent */
    if (expt != DUK__NO_EXP) {
        /*
         *  Exponent notation for non-base-10 numbers isn't specified in Ecmascript
         *  specification, as it never explicitly turns up: non-decimal numbers can
         *  only be formatted with Number.prototype.toString([radix]) and for that,
         *  behavior is not explicitly specified.
         *
         *  Logical choices include formatting the exponent as decimal (e.g. binary
         *  100000 as 1e+5) or in current radix (e.g. binary 100000 as 1e+101).
         *  The Dragon4 algorithm (in the original paper) prints the exponent value
         *  in the target radix B.  However, for radix values 15 and above, the
         *  exponent separator 'e' is no longer easily parseable.  Consider, for
         *  instance, the number "1.faecee+1c".
         */
 
        duk_size_t len;
        char expt_sign;
 
        *q++ = 'e';
        if (expt >= 0) {
            expt_sign = '+';
        } else {
            expt_sign = '-';
            expt = -expt;
        }
        *q++ = (duk_uint8_t) expt_sign;
        len = duk__dragon4_format_uint32(q, (duk_uint32_t) expt, radix);
        q += len;
    }
 
    duk_push_lstring(ctx, (const char *) buf, (size_t) (q - buf));
}
 
/*
 *  Conversion helpers
 */
 
DUK_LOCAL void duk__dragon4_double_to_ctx(duk__numconv_stringify_ctx *nc_ctx, duk_double_t x) {
    duk_double_union u;
    duk_uint32_t tmp;
    duk_small_int_t expt;
 
    /*
     *    seeeeeee eeeeffff ffffffff ffffffff ffffffff ffffffff ffffffff ffffffff
     *       A        B        C        D        E        F        G        H
     *
     *    s       sign bit
     *    eee...  exponent field
     *    fff...  fraction
     *
     *    ieee value = 1.ffff... * 2^(e - 1023)  (normal)
     *               = 0.ffff... * 2^(-1022)     (denormal)
     *
     *    algorithm v = f * b^e
     */
 
    DUK_DBLUNION_SET_DOUBLE(&u, x);
 
    nc_ctx->f.n = 2;
 
    tmp = DUK_DBLUNION_GET_LOW32(&u);
    nc_ctx->f.v[0] = tmp;
    tmp = DUK_DBLUNION_GET_HIGH32(&u);
    nc_ctx->f.v[1] = tmp & 0x000fffffUL;
    expt = (duk_small_int_t) ((tmp >> 20) & 0x07ffUL);
 
    if (expt == 0) {
        /* denormal */
        expt = DUK__IEEE_DOUBLE_EXP_MIN - 52;
        duk__bi_normalize(&nc_ctx->f);
    } else {
        /* normal: implicit leading 1-bit */
        nc_ctx->f.v[1] |= 0x00100000UL;
        expt = expt - DUK__IEEE_DOUBLE_EXP_BIAS - 52;
        DUK_ASSERT(duk__bi_is_valid(&nc_ctx->f));  /* true, because v[1] has at least one bit set */
    }
 
    DUK_ASSERT(duk__bi_is_valid(&nc_ctx->f));
 
    nc_ctx->e = expt;
}
 
DUK_LOCAL void duk__dragon4_ctx_to_double(duk__numconv_stringify_ctx *nc_ctx, duk_double_t *x) {
    duk_double_union u;
    duk_small_int_t expt;
    duk_small_int_t i;
    duk_small_int_t bitstart;
    duk_small_int_t bitround;
    duk_small_int_t bitidx;
    duk_small_int_t skip_round;
    duk_uint32_t t, v;
 
    DUK_ASSERT(nc_ctx->count == 53 + 1);
 
    /* Sometimes this assert is not true right now; it will be true after
     * rounding.  See: test-bug-numconv-mantissa-assert.js.
     */
    DUK_ASSERT_DISABLE(nc_ctx->digits[0] == 1);  /* zero handled by caller */
 
    /* Should not be required because the code below always sets both high
     * and low parts, but at least gcc-4.4.5 fails to deduce this correctly
     * (perhaps because the low part is set (seemingly) conditionally in a
     * loop), so this is here to avoid the bogus warning.
     */
    DUK_MEMZERO((void *) &u, sizeof(u));
 
    /*
     *  Figure out how generated digits match up with the mantissa,
     *  and then perform rounding.  If mantissa overflows, need to
     *  recompute the exponent (it is bumped and may overflow to
     *  infinity).
     *
     *  For normal numbers the leading '1' is hidden and ignored,
     *  and the last bit is used for rounding:
     *
     *                          rounding pt
     *       <--------52------->|
     *     1 x x x x ... x x x x|y  ==>  x x x x ... x x x x
     *
     *  For denormals, the leading '1' is included in the number,
     *  and the rounding point is different:
     *
     *                      rounding pt
     *     <--52 or less--->|
     *     1 x x x x ... x x|x x y  ==>  0 0 ... 1 x x ... x x
     *
     *  The largest denormals will have a mantissa beginning with
     *  a '1' (the explicit leading bit); smaller denormals will
     *  have leading zero bits.
     *
     *  If the exponent would become too high, the result becomes
     *  Infinity.  If the exponent is so small that the entire
     *  mantissa becomes zero, the result becomes zero.
     *
     *  Note: the Dragon4 'k' is off-by-one with respect to the IEEE
     *  exponent.  For instance, k==0 indicates that the leading '1'
     *  digit is at the first binary fraction position (0.1xxx...);
     *  the corresponding IEEE exponent would be -1.
     */
 
    skip_round = 0;
 
 recheck_exp:
 
    expt = nc_ctx->k - 1;   /* IEEE exp without bias */
    if (expt > 1023) {
        /* Infinity */
        bitstart = -255;  /* needed for inf: causes mantissa to become zero,
                           * and rounding to be skipped.
                           */
        expt = 2047;
    } else if (expt >= -1022) {
        /* normal */
        bitstart = 1;  /* skip leading digit */
        expt += DUK__IEEE_DOUBLE_EXP_BIAS;
        DUK_ASSERT(expt >= 1 && expt <= 2046);
    } else {
        /* denormal or zero */
        bitstart = 1023 + expt;  /* expt==-1023 -> bitstart=0 (leading 1);
                                  * expt==-1024 -> bitstart=-1 (one left of leading 1), etc
                                  */
        expt = 0;
    }
    bitround = bitstart + 52;
 
    DUK_DDD(DUK_DDDPRINT("ieee expt=%ld, bitstart=%ld, bitround=%ld",
                         (long) expt, (long) bitstart, (long) bitround));
 
    if (!skip_round) {
        if (duk__dragon4_fixed_format_round(nc_ctx, bitround)) {
            /* Corner case: see test-numconv-parse-mant-carry.js.  We could
             * just bump the exponent and update bitstart, but it's more robust
             * to recompute (but avoid rounding twice).
             */
            DUK_DDD(DUK_DDDPRINT("rounding caused exponent to be bumped, recheck exponent"));
            skip_round = 1;
            goto recheck_exp;
        }
    }
 
    /*
     *  Create mantissa
     */
 
    t = 0;
    for (i = 0; i < 52; i++) {
        bitidx = bitstart + 52 - 1 - i;
        if (bitidx >= nc_ctx->count) {
            v = 0;
        } else if (bitidx < 0) {
            v = 0;
        } else {
            v = nc_ctx->digits[bitidx];
        }
        DUK_ASSERT(v == 0 || v == 1);
        t += v << (i % 32);
        if (i == 31) {
            /* low 32 bits is complete */
            DUK_DBLUNION_SET_LOW32(&u, t);
            t = 0;
        }
    }
    /* t has high mantissa */
 
    DUK_DDD(DUK_DDDPRINT("mantissa is complete: %08lx %08lx",
                         (unsigned long) t,
                         (unsigned long) DUK_DBLUNION_GET_LOW32(&u)));
 
    DUK_ASSERT(expt >= 0 && expt <= 0x7ffL);
    t += expt << 20;
#if 0  /* caller handles sign change */
    if (negative) {
        t |= 0x80000000U;
    }
#endif
    DUK_DBLUNION_SET_HIGH32(&u, t);
 
    DUK_DDD(DUK_DDDPRINT("number is complete: %08lx %08lx",
                         (unsigned long) DUK_DBLUNION_GET_HIGH32(&u),
                         (unsigned long) DUK_DBLUNION_GET_LOW32(&u)));
 
    *x = DUK_DBLUNION_GET_DOUBLE(&u);
}
 
/*
 *  Exposed number-to-string API
 *
 *  Input: [ number ]
 *  Output: [ string ]
 */
 
DUK_INTERNAL void duk_numconv_stringify(duk_context *ctx, duk_small_int_t radix, duk_small_int_t digits, duk_small_uint_t flags) {
    duk_double_t x;
    duk_small_int_t c;
    duk_small_int_t neg;
    duk_uint32_t uval;
    duk__numconv_stringify_ctx nc_ctx_alloc;  /* large context; around 2kB now */
    duk__numconv_stringify_ctx *nc_ctx = &nc_ctx_alloc;
 
    x = (duk_double_t) duk_require_number(ctx, -1);
    duk_pop(ctx);
 
    /*
     *  Handle special cases (NaN, infinity, zero).
     */
 
    c = (duk_small_int_t) DUK_FPCLASSIFY(x);
    if (DUK_SIGNBIT((double) x)) {
        x = -x;
        neg = 1;
    } else {
        neg = 0;
    }
 
    /* NaN sign bit is platform specific with unpacked, un-normalized NaNs */
    DUK_ASSERT(c == DUK_FP_NAN || DUK_SIGNBIT((double) x) == 0);
 
    if (c == DUK_FP_NAN) {
        duk_push_hstring_stridx(ctx, DUK_STRIDX_NAN);
        return;
    } else if (c == DUK_FP_INFINITE) {
        if (neg) {
            /* -Infinity */
            duk_push_hstring_stridx(ctx, DUK_STRIDX_MINUS_INFINITY);
        } else {
            /* Infinity */
            duk_push_hstring_stridx(ctx, DUK_STRIDX_INFINITY);
        }
        return;
    } else if (c == DUK_FP_ZERO) {
        /* We can't shortcut zero here if it goes through special formatting
         * (such as forced exponential notation).
         */
        ;
    }
 
    /*
     *  Handle integers in 32-bit range (that is, [-(2**32-1),2**32-1])
     *  specially, as they're very likely for embedded programs.  This
     *  is now done for all radix values.  We must be careful not to use
     *  the fast path when special formatting (e.g. forced exponential)
     *  is in force.
     *
     *  XXX: could save space by supporting radix 10 only and using
     *  sprintf "%lu" for the fast path and for exponent formatting.
     */
 
    uval = (unsigned int) x;
    if (((double) uval) == x &&  /* integer number in range */
        flags == 0) {            /* no special formatting */
        /* use bigint area as a temp */
        duk_uint8_t *buf = (duk_uint8_t *) (&nc_ctx->f);
        duk_uint8_t *p = buf;
 
        DUK_ASSERT(DUK__NUMCONV_CTX_BIGINTS_SIZE >= 32 + 1);  /* max size: radix=2 + sign */
        if (neg && uval != 0) {
            /* no negative sign for zero */
            *p++ = (duk_uint8_t) '-';
        }
        p += duk__dragon4_format_uint32(p, uval, radix);
        duk_push_lstring(ctx, (const char *) buf, (duk_size_t) (p - buf));
        return;
    }
 
    /*
     *  Dragon4 setup.
     *
     *  Convert double from IEEE representation for conversion;
     *  normal finite values have an implicit leading 1-bit.  The
     *  slow path algorithm doesn't handle zero, so zero is special
     *  cased here but still creates a valid nc_ctx, and goes
     *  through normal formatting in case special formatting has
     *  been requested (e.g. forced exponential format: 0 -> "0e+0").
     */
 
    /* Would be nice to bulk clear the allocation, but the context
     * is 1-2 kilobytes and nothing should rely on it being zeroed.
     */
#if 0
    DUK_MEMZERO((void *) nc_ctx, sizeof(*nc_ctx));  /* slow init, do only for slow path cases */
#endif
 
    nc_ctx->is_s2n = 0;
    nc_ctx->b = 2;
    nc_ctx->B = radix;
    nc_ctx->abs_pos = 0;
    if (flags & DUK_N2S_FLAG_FIXED_FORMAT) {
        nc_ctx->is_fixed = 1;
        if (flags & DUK_N2S_FLAG_FRACTION_DIGITS) {
            /* absolute req_digits; e.g. digits = 1 -> last digit is 0,
             * but add an extra digit for rounding.
             */
            nc_ctx->abs_pos = 1;
            nc_ctx->req_digits = (-digits + 1) - 1;
        } else {
            nc_ctx->req_digits = digits + 1;
        }
    } else {
        nc_ctx->is_fixed = 0;
        nc_ctx->req_digits = 0;
    }
 
    if (c == DUK_FP_ZERO) {
        /* Zero special case: fake requested number of zero digits; ensure
         * no sign bit is printed.  Relative and absolute fixed format
         * require separate handling.
         */
        duk_small_int_t count;
        if (nc_ctx->is_fixed) {
            if (nc_ctx->abs_pos) {
                count = digits + 2;  /* lead zero + 'digits' fractions + 1 for rounding */
            } else {
                count = digits + 1;  /* + 1 for rounding */
            }
        } else {
            count = 1;
        }
        DUK_DDD(DUK_DDDPRINT("count=%ld", (long) count));
        DUK_ASSERT(count >= 1);
        DUK_MEMZERO((void *) nc_ctx->digits, count);
        nc_ctx->count = count;
        nc_ctx->k = 1;  /* 0.000... */
        neg = 0;
        goto zero_skip;
    }
 
    duk__dragon4_double_to_ctx(nc_ctx, x);   /* -> sets 'f' and 'e' */
    DUK__BI_PRINT("f", &nc_ctx->f);
    DUK_DDD(DUK_DDDPRINT("e=%ld", (long) nc_ctx->e));
 
    /*
     *  Dragon4 slow path digit generation.
     */
 
    duk__dragon4_prepare(nc_ctx);  /* setup many variables in nc_ctx */
 
    DUK_DDD(DUK_DDDPRINT("after prepare:"));
    DUK__BI_PRINT("r", &nc_ctx->r);
    DUK__BI_PRINT("s", &nc_ctx->s);
    DUK__BI_PRINT("mp", &nc_ctx->mp);
    DUK__BI_PRINT("mm", &nc_ctx->mm);
 
    duk__dragon4_scale(nc_ctx);
 
    DUK_DDD(DUK_DDDPRINT("after scale; k=%ld", (long) nc_ctx->k));
    DUK__BI_PRINT("r", &nc_ctx->r);
    DUK__BI_PRINT("s", &nc_ctx->s);
    DUK__BI_PRINT("mp", &nc_ctx->mp);
    DUK__BI_PRINT("mm", &nc_ctx->mm);
 
    duk__dragon4_generate(nc_ctx);
 
    /*
     *  Convert and push final string.
     */
 
 zero_skip:
 
    if (flags & DUK_N2S_FLAG_FIXED_FORMAT) {
        /* Perform fixed-format rounding. */
        duk_small_int_t roundpos;
        if (flags & DUK_N2S_FLAG_FRACTION_DIGITS) {
            /* 'roundpos' is relative to nc_ctx->k and increases to the right
             * (opposite of how 'k' changes).
             */
            roundpos = -digits;  /* absolute position for digit considered for rounding */
            roundpos = nc_ctx->k - roundpos;
        } else {
            roundpos = digits;
        }
        DUK_DDD(DUK_DDDPRINT("rounding: k=%ld, count=%ld, digits=%ld, roundpos=%ld",
                             (long) nc_ctx->k, (long) nc_ctx->count, (long) digits, (long) roundpos));
        (void) duk__dragon4_fixed_format_round(nc_ctx, roundpos);
 
        /* Note: 'count' is currently not adjusted by rounding (i.e. the
         * digits are not "chopped off".  That shouldn't matter because
         * the digit position (absolute or relative) is passed on to the
         * convert-and-push function.
         */
    }
 
    duk__dragon4_convert_and_push(nc_ctx, ctx, radix, digits, flags, neg);
}
 
/*
 *  Exposed string-to-number API
 *
 *  Input: [ string ]
 *  Output: [ number ]
 *
 *  If number parsing fails, a NaN is pushed as the result.  If number parsing
 *  fails due to an internal error, an InternalError is thrown.
 */
 
DUK_INTERNAL void duk_numconv_parse(duk_context *ctx, duk_small_int_t radix, duk_small_uint_t flags) {
    duk_hthread *thr = (duk_hthread *) ctx;
    duk__numconv_stringify_ctx nc_ctx_alloc;  /* large context; around 2kB now */
    duk__numconv_stringify_ctx *nc_ctx = &nc_ctx_alloc;
    duk_double_t res;
    duk_hstring *h_str;
    duk_small_int_t expt;
    duk_small_int_t expt_neg;
    duk_small_int_t expt_adj;
    duk_small_int_t neg;
    duk_small_int_t dig;
    duk_small_int_t dig_whole;
    duk_small_int_t dig_lzero;
    duk_small_int_t dig_frac;
    duk_small_int_t dig_expt;
    duk_small_int_t dig_prec;
    const duk__exp_limits *explim;
    const duk_uint8_t *p;
    duk_small_int_t ch;
 
    /* This seems to waste a lot of stack frame entries, but good compilers
     * will compute these as needed below.  Some of these initial flags are
     * also modified in the code below, so they can't all be removed.
     */
    duk_small_int_t trim_white = (flags & DUK_S2N_FLAG_TRIM_WHITE);
    duk_small_int_t allow_expt = (flags & DUK_S2N_FLAG_ALLOW_EXP);
    duk_small_int_t allow_garbage = (flags & DUK_S2N_FLAG_ALLOW_GARBAGE);
    duk_small_int_t allow_plus = (flags & DUK_S2N_FLAG_ALLOW_PLUS);
    duk_small_int_t allow_minus = (flags & DUK_S2N_FLAG_ALLOW_MINUS);
    duk_small_int_t allow_infinity = (flags & DUK_S2N_FLAG_ALLOW_INF);
    duk_small_int_t allow_frac = (flags & DUK_S2N_FLAG_ALLOW_FRAC);
    duk_small_int_t allow_naked_frac = (flags & DUK_S2N_FLAG_ALLOW_NAKED_FRAC);
    duk_small_int_t allow_empty_frac = (flags & DUK_S2N_FLAG_ALLOW_EMPTY_FRAC);
    duk_small_int_t allow_empty = (flags & DUK_S2N_FLAG_ALLOW_EMPTY_AS_ZERO);
    duk_small_int_t allow_leading_zero = (flags & DUK_S2N_FLAG_ALLOW_LEADING_ZERO);
    duk_small_int_t allow_auto_hex_int = (flags & DUK_S2N_FLAG_ALLOW_AUTO_HEX_INT);
    duk_small_int_t allow_auto_oct_int = (flags & DUK_S2N_FLAG_ALLOW_AUTO_OCT_INT);
 
    DUK_DDD(DUK_DDDPRINT("parse number: %!T, radix=%ld, flags=0x%08lx",
                         (duk_tval *) duk_get_tval(ctx, -1),
                         (long) radix, (unsigned long) flags));
 
    DUK_ASSERT(radix >= 2 && radix <= 36);
    DUK_ASSERT(radix - 2 < (duk_small_int_t) sizeof(duk__str2num_digits_for_radix));
 
    /*
     *  Preliminaries: trim, sign, Infinity check
     *
     *  We rely on the interned string having a NUL terminator, which will
     *  cause a parse failure wherever it is encountered.  As a result, we
     *  don't need separate pointer checks.
     *
     *  There is no special parsing for 'NaN' in the specification although
     *  'Infinity' (with an optional sign) is allowed in some contexts.
     *  Some contexts allow plus/minus sign, while others only allow the
     *  minus sign (like JSON.parse()).
     *
     *  Automatic hex number detection (leading '0x' or '0X') and octal
     *  number detection (leading '0' followed by at least one octal digit)
     *  is done here too.
     */
 
    if (trim_white) {
        /* Leading / trailing whitespace is sometimes accepted and
         * sometimes not.  After white space trimming, all valid input
         * characters are pure ASCII.
         */
        duk_trim(ctx, -1);
    }
    h_str = duk_require_hstring(ctx, -1);
    DUK_ASSERT(h_str != NULL);
    p = (const duk_uint8_t *) DUK_HSTRING_GET_DATA(h_str);
 
    neg = 0;
    ch = *p;
    if (ch == (duk_small_int_t) '+') {
        if (!allow_plus) {
            DUK_DDD(DUK_DDDPRINT("parse failed: leading plus sign not allowed"));
            goto parse_fail;
        }
        p++;
    } else if (ch == (duk_small_int_t) '-') {
        if (!allow_minus) {
            DUK_DDD(DUK_DDDPRINT("parse failed: leading minus sign not allowed"));
            goto parse_fail;
        }
        p++;
        neg = 1;
    }
 
    ch = *p;
    if (allow_infinity && ch == (duk_small_int_t) 'I') {
        /* Don't check for Infinity unless the context allows it.
         * 'Infinity' is a valid integer literal in e.g. base-36:
         *
         *   parseInt('Infinity', 36)
         *   1461559270678
         */
 
        const duk_uint8_t *q;
 
        /* borrow literal Infinity from builtin string */
        q = (const duk_uint8_t *) DUK_HSTRING_GET_DATA(DUK_HTHREAD_STRING_INFINITY(thr));
        if (DUK_STRNCMP((const char *) p, (const char *) q, 8) == 0) {
            if (!allow_garbage && (p[8] != (duk_uint8_t) 0)) {
                DUK_DDD(DUK_DDDPRINT("parse failed: trailing garbage after matching 'Infinity' not allowed"));
                goto parse_fail;
            } else {
                res = DUK_DOUBLE_INFINITY;
                goto negcheck_and_ret;
            }
        }
    }
    if (ch == (duk_small_int_t) '0') {
        duk_small_int_t detect_radix = 0;
        ch = p[1];
        if (allow_auto_hex_int && (ch == (duk_small_int_t) 'x' || ch == (duk_small_int_t) 'X')) {
            DUK_DDD(DUK_DDDPRINT("detected 0x/0X hex prefix, changing radix and preventing fractions and exponent"));
            detect_radix = 16;
            allow_empty = 0;  /* interpret e.g. '0x' and '0xg' as a NaN (= parse error) */
            p += 2;
        } else if (allow_auto_oct_int && (ch >= (duk_small_int_t) '0' && ch <= (duk_small_int_t) '9')) {
            DUK_DDD(DUK_DDDPRINT("detected 0n oct prefix, changing radix and preventing fractions and exponent"));
            detect_radix = 8;
            allow_empty = 1;  /* interpret e.g. '09' as '0', not NaN */
            p += 1;
        }
        if (detect_radix > 0) {
            radix = detect_radix;
            allow_expt = 0;
            allow_frac = 0;
            allow_naked_frac = 0;
            allow_empty_frac = 0;
            allow_leading_zero = 1;  /* allow e.g. '0x0009' and '00077' */
        }
    }
 
    /*
     *  Scan number and setup for Dragon4.
     *
     *  The fast path case is detected during setup: an integer which
     *  can be converted without rounding, no net exponent.  The fast
     *  path could be implemented as a separate scan, but may not really
     *  be worth it: the multiplications for building 'f' are not
     *  expensive when 'f' is small.
     *
     *  The significand ('f') must contain enough bits of (apparent)
     *  accuracy, so that Dragon4 will generate enough binary output digits.
     *  For decimal numbers, this means generating a 20-digit significand,
     *  which should yield enough practical accuracy to parse IEEE doubles.
     *  In fact, the Ecmascript specification explicitly allows an
     *  implementation to treat digits beyond 20 as zeroes (and even
     *  to round the 20th digit upwards).  For non-decimal numbers, the
     *  appropriate number of digits has been precomputed for comparable
     *  accuracy.
     *
     *  Digit counts:
     *
     *    [ dig_lzero ]
     *      |
     *     .+-..---[ dig_prec ]----.
     *     |  ||                   |
     *     0000123.456789012345678901234567890e+123456
     *     |     | |                         |  |    |
     *     `--+--' `------[ dig_frac ]-------'  `-+--'
     *        |                                   |
     *    [ dig_whole ]                       [ dig_expt ]
     *
     *    dig_frac and dig_expt are -1 if not present
     *    dig_lzero is only computed for whole number part
     *
     *  Parsing state
     *
     *     Parsing whole part      dig_frac < 0 AND dig_expt < 0
     *     Parsing fraction part   dig_frac >= 0 AND dig_expt < 0
     *     Parsing exponent part   dig_expt >= 0   (dig_frac may be < 0 or >= 0)
     *
     *  Note: in case we hit an implementation limit (like exponent range),
     *  we should throw an error, NOT return NaN or Infinity.  Even with
     *  very large exponent (or significand) values the final result may be
     *  finite, so NaN/Infinity would be incorrect.
     */
 
    duk__bi_set_small(&nc_ctx->f, 0);
    dig_prec = 0;
    dig_lzero = 0;
    dig_whole = 0;
    dig_frac = -1;
    dig_expt = -1;
    expt = 0;
    expt_adj = 0;  /* essentially tracks digit position of lowest 'f' digit */
    expt_neg = 0;
    for (;;) {
        ch = *p++;
 
        DUK_DDD(DUK_DDDPRINT("parse digits: p=%p, ch='%c' (%ld), expt=%ld, expt_adj=%ld, "
                             "dig_whole=%ld, dig_frac=%ld, dig_expt=%ld, dig_lzero=%ld, dig_prec=%ld",
                             (const void *) p, (int) ((ch >= 0x20 && ch <= 0x7e) ? ch : '?'), (long) ch,
                             (long) expt, (long) expt_adj, (long) dig_whole, (long) dig_frac,
                             (long) dig_expt, (long) dig_lzero, (long) dig_prec));
        DUK__BI_PRINT("f", &nc_ctx->f);
 
        /* Most common cases first. */
        if (ch >= (duk_small_int_t) '0' && ch <= (duk_small_int_t) '9') {
            dig = (int) ch - '0' + 0;
        } else if (ch == (duk_small_int_t) '.') {
            /* A leading digit is not required in some cases, e.g. accept ".123".
             * In other cases (JSON.parse()) a leading digit is required.  This
             * is checked for after the loop.
             */
            if (dig_frac >= 0 || dig_expt >= 0) {
                if (allow_garbage) {
                    DUK_DDD(DUK_DDDPRINT("garbage termination (invalid period)"));
                    break;
                } else {
                    DUK_DDD(DUK_DDDPRINT("parse failed: period not allowed"));
                    goto parse_fail;
                }
            }
 
            if (!allow_frac) {
                /* Some contexts don't allow fractions at all; this can't be a
                 * post-check because the state ('f' and expt) would be incorrect.
                 */
                if (allow_garbage) {
                    DUK_DDD(DUK_DDDPRINT("garbage termination (invalid first period)"));
                    break;
                } else {
                    DUK_DDD(DUK_DDDPRINT("parse failed: fraction part not allowed"));
                }
            }
 
            DUK_DDD(DUK_DDDPRINT("start fraction part"));
            dig_frac = 0;
            continue;
        } else if (ch == (duk_small_int_t) 0) {
            DUK_DDD(DUK_DDDPRINT("NUL termination"));
            break;
        } else if (allow_expt && dig_expt < 0 && (ch == (duk_small_int_t) 'e' || ch == (duk_small_int_t) 'E')) {
            /* Note: we don't parse back exponent notation for anything else
             * than radix 10, so this is not an ambiguous check (e.g. hex
             * exponent values may have 'e' either as a significand digit
             * or as an exponent separator).
             *
             * If the exponent separator occurs twice, 'e' will be interpreted
             * as a digit (= 14) and will be rejected as an invalid decimal
             * digit.
             */
 
            DUK_DDD(DUK_DDDPRINT("start exponent part"));
 
            /* Exponent without a sign or with a +/- sign is accepted
             * by all call sites (even JSON.parse()).
             */
            ch = *p;
            if (ch == (duk_small_int_t) '-') {
                expt_neg = 1;
                p++;
            } else if (ch == (duk_small_int_t) '+') {
                p++;
            }
            dig_expt = 0;
            continue;
        } else if (ch >= (duk_small_int_t) 'a' && ch <= (duk_small_int_t) 'z') {
            dig = (duk_small_int_t) (ch - (duk_small_int_t) 'a' + 0x0a);
        } else if (ch >= (duk_small_int_t) 'A' && ch <= (duk_small_int_t) 'Z') {
            dig = (duk_small_int_t) (ch - (duk_small_int_t) 'A' + 0x0a);
        } else {
            dig = 255;  /* triggers garbage digit check below */
        }
        DUK_ASSERT((dig >= 0 && dig <= 35) || dig == 255);
 
        if (dig >= radix) {
            if (allow_garbage) {
                DUK_DDD(DUK_DDDPRINT("garbage termination"));
                break;
            } else {
                DUK_DDD(DUK_DDDPRINT("parse failed: trailing garbage or invalid digit"));
                goto parse_fail;
            }
        }
 
        if (dig_expt < 0) {
            /* whole or fraction digit */
 
            if (dig_prec < duk__str2num_digits_for_radix[radix - 2]) {
                /* significant from precision perspective */
 
                duk_small_int_t f_zero = duk__bi_is_zero(&nc_ctx->f);
                if (f_zero && dig == 0) {
                    /* Leading zero is not counted towards precision digits; not
                     * in the integer part, nor in the fraction part.
                     */
                    if (dig_frac < 0) {
                        dig_lzero++;
                    }
                } else {
                    /* XXX: join these ops (multiply-accumulate), but only if
                     * code footprint decreases.
                     */
                    duk__bi_mul_small(&nc_ctx->t1, &nc_ctx->f, radix);
                    duk__bi_add_small(&nc_ctx->f, &nc_ctx->t1, dig);
                    dig_prec++;
                }
            } else {
                /* Ignore digits beyond a radix-specific limit, but note them
                 * in expt_adj.
                 */
                expt_adj++;
            }
 
            if (dig_frac >= 0) {
                dig_frac++;
                expt_adj--;
            } else {
                dig_whole++;
            }
        } else {
            /* exponent digit */
 
            expt = expt * radix + dig;
            if (expt > DUK_S2N_MAX_EXPONENT) {
                /* impose a reasonable exponent limit, so that exp
                 * doesn't need to get tracked using a bigint.
                 */
                DUK_DDD(DUK_DDDPRINT("parse failed: exponent too large"));
                goto parse_explimit_error;
            }
            dig_expt++;
        }
    }
 
    /* Leading zero. */
 
    if (dig_lzero > 0 && dig_whole > 1) {
        if (!allow_leading_zero) {
            DUK_DDD(DUK_DDDPRINT("parse failed: leading zeroes not allowed in integer part"));
            goto parse_fail;
        }
    }
 
    /* Validity checks for various fraction formats ("0.1", ".1", "1.", "."). */
 
    if (dig_whole == 0) {
        if (dig_frac == 0) {
            /* "." is not accepted in any format */
            DUK_DDD(DUK_DDDPRINT("parse failed: plain period without leading or trailing digits"));
            goto parse_fail;
        } else if (dig_frac > 0) {
            /* ".123" */
            if (!allow_naked_frac) {
                DUK_DDD(DUK_DDDPRINT("parse failed: fraction part not allowed without "
                                     "leading integer digit(s)"));
                goto parse_fail;
            }
        } else {
            /* empty ("") is allowed in some formats (e.g. Number(''), as zero */
            if (!allow_empty) {
                DUK_DDD(DUK_DDDPRINT("parse failed: empty string not allowed (as zero)"));
                goto parse_fail;
            }
        }
    } else {
        if (dig_frac == 0) {
            /* "123." is allowed in some formats */
            if (!allow_empty_frac) {
                DUK_DDD(DUK_DDDPRINT("parse failed: empty fractions"));
                goto parse_fail;
            }
        } else if (dig_frac > 0) {
            /* "123.456" */
            ;
        } else {
            /* "123" */
            ;
        }
    }
 
    /* Exponent without digits (e.g. "1e" or "1e+").  If trailing garbage is
     * allowed, ignore exponent part as garbage (= parse as "1", i.e. exp 0).
     */
 
    if (dig_expt == 0) {
        if (!allow_garbage) {
            DUK_DDD(DUK_DDDPRINT("parse failed: empty exponent"));
            goto parse_fail;
        }
        DUK_ASSERT(expt == 0);
    }
 
    if (expt_neg) {
        expt = -expt;
    }
    DUK_DDD(DUK_DDDPRINT("expt=%ld, expt_adj=%ld, net exponent -> %ld",
                         (long) expt, (long) expt_adj, (long) (expt + expt_adj)));
    expt += expt_adj;
 
    /* Fast path check. */
 
    if (nc_ctx->f.n <= 1 &&   /* 32-bit value */
        expt == 0    /* no net exponent */) {
        /* Fast path is triggered for no exponent and also for balanced exponent
         * and fraction parts, e.g. for "1.23e2" == "123".  Remember to respect
         * zero sign.
         */
 
        /* XXX: could accept numbers larger than 32 bits, e.g. up to 53 bits? */
        DUK_DDD(DUK_DDDPRINT("fast path number parse"));
        if (nc_ctx->f.n == 1) {
            res = (double) nc_ctx->f.v[0];
        } else {
            res = 0.0;
        }
        goto negcheck_and_ret;
    }
 
    /* Significand ('f') padding. */
 
    while (dig_prec < duk__str2num_digits_for_radix[radix - 2]) {
        /* Pad significand with "virtual" zero digits so that Dragon4 will
         * have enough (apparent) precision to work with.
         */
        DUK_DDD(DUK_DDDPRINT("dig_prec=%ld, pad significand with zero", (long) dig_prec));
        duk__bi_mul_small_copy(&nc_ctx->f, radix, &nc_ctx->t1);
        DUK__BI_PRINT("f", &nc_ctx->f);
        expt--;
        dig_prec++;
    }
 
    DUK_DDD(DUK_DDDPRINT("final exponent: %ld", (long) expt));
 
    /* Detect zero special case. */
 
    if (nc_ctx->f.n == 0) {
        /* This may happen even after the fast path check, if exponent is
         * not balanced (e.g. "0e1").  Remember to respect zero sign.
         */
        DUK_DDD(DUK_DDDPRINT("significand is zero"));
        res = 0.0;
        goto negcheck_and_ret;
    }
 
 
    /* Quick reject of too large or too small exponents.  This check
     * would be incorrect for zero (e.g. "0e1000" is zero, not Infinity)
     * so zero check must be above.
     */
 
    explim = &duk__str2num_exp_limits[radix - 2];
    if (expt > explim->upper) {
        DUK_DDD(DUK_DDDPRINT("exponent too large -> infinite"));
        res = (duk_double_t) DUK_DOUBLE_INFINITY;
        goto negcheck_and_ret;
    } else if (expt < explim->lower) {
        DUK_DDD(DUK_DDDPRINT("exponent too small -> zero"));
        res = (duk_double_t) 0.0;
        goto negcheck_and_ret;
    }
 
    nc_ctx->is_s2n = 1;
    nc_ctx->e = expt;
    nc_ctx->b = radix;
    nc_ctx->B = 2;
    nc_ctx->is_fixed = 1;
    nc_ctx->abs_pos = 0;
    nc_ctx->req_digits = 53 + 1;
 
    DUK__BI_PRINT("f", &nc_ctx->f);
    DUK_DDD(DUK_DDDPRINT("e=%ld", (long) nc_ctx->e));
 
    /*
     *  Dragon4 slow path (binary) digit generation.
     *  An extra digit is generated for rounding.
     */
 
    duk__dragon4_prepare(nc_ctx);  /* setup many variables in nc_ctx */
 
    DUK_DDD(DUK_DDDPRINT("after prepare:"));
    DUK__BI_PRINT("r", &nc_ctx->r);
    DUK__BI_PRINT("s", &nc_ctx->s);
    DUK__BI_PRINT("mp", &nc_ctx->mp);
    DUK__BI_PRINT("mm", &nc_ctx->mm);
 
    duk__dragon4_scale(nc_ctx);
 
    DUK_DDD(DUK_DDDPRINT("after scale; k=%ld", (long) nc_ctx->k));
    DUK__BI_PRINT("r", &nc_ctx->r);
    DUK__BI_PRINT("s", &nc_ctx->s);
    DUK__BI_PRINT("mp", &nc_ctx->mp);
    DUK__BI_PRINT("mm", &nc_ctx->mm);
 
    duk__dragon4_generate(nc_ctx);
 
    DUK_ASSERT(nc_ctx->count == 53 + 1);
 
    /*
     *  Convert binary digits into an IEEE double.  Need to handle
     *  denormals and rounding correctly.
     */
 
    duk__dragon4_ctx_to_double(nc_ctx, &res);
    goto negcheck_and_ret;
 
 negcheck_and_ret:
    if (neg) {
        res = -res;
    }
    duk_pop(ctx);
    duk_push_number(ctx, (double) res);
    DUK_DDD(DUK_DDDPRINT("result: %!T", (duk_tval *) duk_get_tval(ctx, -1)));
    return;
 
 parse_fail:
    DUK_DDD(DUK_DDDPRINT("parse failed"));
    duk_pop(ctx);
    duk_push_nan(ctx);
    return;
 
 parse_explimit_error:
    DUK_DDD(DUK_DDDPRINT("parse failed, internal error, can't return a value"));
    DUK_ERROR_RANGE(thr, "exponent too large");
    return;
}