ref: 2459e34d77e2e21ef829c0dfaafd99433899494f
dir: /libmath/dtoa.c/
/* derived from /netlib/fp/dtoa.c assuming IEEE, Standard C */
/* kudos to dmg@bell-labs.com, gripes to ehg@bell-labs.com */
#include "lib9.h"
#ifdef __APPLE__
#pragma clang diagnostic ignored "-Wlogical-op-parentheses"
#pragma clang diagnostic ignored "-Wparentheses"
#endif
#define ACQUIRE_DTOA_LOCK(n) /*nothing*/
#define FREE_DTOA_LOCK(n) /*nothing*/
/* let's provide reasonable defaults for usual implementation of IEEE f.p. */
#ifndef DBL_DIG
#define DBL_DIG 15
#endif
#ifndef DBL_MAX_10_EXP
#define DBL_MAX_10_EXP 308
#endif
#ifndef DBL_MAX_EXP
#define DBL_MAX_EXP 1024
#endif
#ifndef FLT_RADIX
#define FLT_RADIX 2
#endif
#ifndef FLT_ROUNDS
#define FLT_ROUNDS 1
#endif
#ifndef Storeinc
#define Storeinc(a,b,c) (*a++ = b << 16 | c & 0xffff)
#endif
#define Sign_Extend(a,b) if (b < 0) a |= 0xffff0000;
#ifdef USE_FPdbleword
#define word0(x) ((FPdbleword*)&x)->hi
#define word1(x) ((FPdbleword*)&x)->lo
#else
#ifdef __LITTLE_ENDIAN
#define word0(x) ((unsigned long *)&x)[1]
#define word1(x) ((unsigned long *)&x)[0]
#else
#define word0(x) ((unsigned long *)&x)[0]
#define word1(x) ((unsigned long *)&x)[1]
#endif
#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 Bias 1023
#define Emin (-1022)
#define Exp_1 0x3ff00000
#define Exp_11 0x3ff00000
#define Ebits 11
#define Frac_mask 0xfffff
#define Frac_mask1 0xfffff
#define Ten_pmax 22
#define Bletch 0x10
#define Bndry_mask 0xfffff
#define Bndry_mask1 0xfffff
#define LSB 1
#define Sign_bit 0x80000000
#define Log2P 1
#define Tiny0 0
#define Tiny1 1
#define Quick_max 14
#define Int_max 14
#define Infinite(x) (word0(x) == 0x7ff00000) /* sufficient test for here */
#define Avoid_Underflow
#define rounded_product(a,b) a *= b
#define rounded_quotient(a,b) a /= b
#define Big0 (Frac_mask1 | Exp_msk1*(DBL_MAX_EXP+Bias-1))
#define Big1 0xffffffff
#define Kmax 15
struct
Bigint {
struct Bigint *next;
int k, maxwds, sign, wds;
unsigned long x[1];
};
typedef struct Bigint Bigint;
static Bigint *freelist[Kmax+1];
static Bigint *
Balloc(int k)
{
int x;
Bigint * rv;
ACQUIRE_DTOA_LOCK(0);
if (rv = freelist[k]) {
freelist[k] = rv->next;
} else {
x = 1 << k;
rv = (Bigint * )malloc(sizeof(Bigint) + (x - 1) * sizeof(unsigned long));
if(rv == nil)
return nil;
rv->k = k;
rv->maxwds = x;
}
FREE_DTOA_LOCK(0);
rv->sign = rv->wds = 0;
return rv;
}
static void
Bfree(Bigint *v)
{
if (v) {
ACQUIRE_DTOA_LOCK(0);
v->next = freelist[v->k];
freelist[v->k] = v;
FREE_DTOA_LOCK(0);
}
}
#define Bcopy(x,y) memcpy((char *)&x->sign, (char *)&y->sign, \
y->wds*sizeof(long) + 2*sizeof(int))
static Bigint *
multadd(Bigint *b, int m, int a) /* multiply by m and add a */
{
int i, wds;
unsigned long * x, y;
unsigned long xi, z;
Bigint * b1;
wds = b->wds;
x = b->x;
i = 0;
do {
xi = *x;
y = (xi & 0xffff) * m + a;
z = (xi >> 16) * m + (y >> 16);
a = (int)(z >> 16);
*x++ = (z << 16) + (y & 0xffff);
} while (++i < wds);
if (a) {
if (wds >= b->maxwds) {
b1 = Balloc(b->k + 1);
Bcopy(b1, b);
Bfree(b);
b = b1;
}
b->x[wds++] = a;
b->wds = wds;
}
return b;
}
static Bigint *
s2b(const char *s, int nd0, int nd, unsigned long 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);
b->x[0] = y9;
b->wds = 1;
i = 9;
if (9 < nd0) {
s += 9;
do
b = multadd(b, 10, *s++ - '0');
while (++i < nd0);
s++;
} else
s += 10;
for (; i < nd; i++)
b = multadd(b, 10, *s++ - '0');
return b;
}
static int
hi0bits(register unsigned long x)
{
register int k = 0;
if (!(x & 0xffff0000)) {
k = 16;
x <<= 16;
}
if (!(x & 0xff000000)) {
k += 8;
x <<= 8;
}
if (!(x & 0xf0000000)) {
k += 4;
x <<= 4;
}
if (!(x & 0xc0000000)) {
k += 2;
x <<= 2;
}
if (!(x & 0x80000000)) {
k++;
if (!(x & 0x40000000))
return 32;
}
return k;
}
static int
lo0bits(unsigned long *y)
{
register int k;
register unsigned long x = *y;
if (x & 7) {
if (x & 1)
return 0;
if (x & 2) {
*y = x >> 1;
return 1;
}
*y = x >> 2;
return 2;
}
k = 0;
if (!(x & 0xffff)) {
k = 16;
x >>= 16;
}
if (!(x & 0xff)) {
k += 8;
x >>= 8;
}
if (!(x & 0xf)) {
k += 4;
x >>= 4;
}
if (!(x & 0x3)) {
k += 2;
x >>= 2;
}
if (!(x & 1)) {
k++;
x >>= 1;
if (!x & 1)
return 32;
}
*y = x;
return k;
}
static Bigint *
i2b(int i)
{
Bigint * b;
b = Balloc(1);
b->x[0] = i;
b->wds = 1;
return b;
}
static Bigint *
mult(Bigint *a, Bigint *b)
{
Bigint * c;
int k, wa, wb, wc;
unsigned long carry, y, z;
unsigned long * x, *xa, *xae, *xb, *xbe, *xc, *xc0;
unsigned long z2;
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);
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;
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;
}
}
for (xc0 = c->x, xc = xc0 + wc; wc > 0 && !*--xc; --wc)
;
c->wds = wc;
return c;
}
static Bigint *p5s;
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 (!(k >>= 2))
return b;
if (!(p5 = p5s)) {
/* first time */
ACQUIRE_DTOA_LOCK(1);
if (!(p5 = p5s)) {
p5 = p5s = i2b(625);
p5->next = 0;
}
FREE_DTOA_LOCK(1);
}
for (; ; ) {
if (k & 1) {
b1 = mult(b, p5);
Bfree(b);
b = b1;
}
if (!(k >>= 1))
break;
if (!(p51 = p5->next)) {
ACQUIRE_DTOA_LOCK(1);
if (!(p51 = p5->next)) {
p51 = p5->next = mult(p5, p5);
p51->next = 0;
}
FREE_DTOA_LOCK(1);
}
p5 = p51;
}
return b;
}
static Bigint *
lshift(Bigint *b, int k)
{
int i, k1, n, n1;
Bigint * b1;
unsigned long * x, *x1, *xe, z;
n = k >> 5;
k1 = b->k;
n1 = n + b->wds + 1;
for (i = b->maxwds; n1 > i; i <<= 1)
k1++;
b1 = Balloc(k1);
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;
}
static int
cmp(Bigint *a, Bigint *b)
{
unsigned long * xa, *xa0, *xb, *xb0;
int i, j;
i = a->wds;
j = b->wds;
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;
}
static Bigint *
diff(Bigint *a, Bigint *b)
{
Bigint * c;
int i, wa, wb;
long borrow, y; /* We need signed shifts here. */
unsigned long * xa, *xae, *xb, *xbe, *xc;
long z;
i = cmp(a, b);
if (!i) {
c = Balloc(0);
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);
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;
do {
y = (*xa & 0xffff) - (*xb & 0xffff) + borrow;
borrow = y >> 16;
Sign_Extend(borrow, y);
z = (*xa++ >> 16) - (*xb++ >> 16) + borrow;
borrow = z >> 16;
Sign_Extend(borrow, z);
Storeinc(xc, z, y);
} while (xb < xbe);
while (xa < xae) {
y = (*xa & 0xffff) + borrow;
borrow = y >> 16;
Sign_Extend(borrow, y);
z = (*xa++ >> 16) + borrow;
borrow = z >> 16;
Sign_Extend(borrow, z);
Storeinc(xc, z, y);
}
while (!*--xc)
wa--;
c->wds = wa;
return c;
}
static double
ulp(double x)
{
register long L;
double a;
L = (word0(x) & Exp_mask) - (P - 1) * Exp_msk1;
#ifndef Sudden_Underflow
if (L > 0) {
#endif
word0(a) = L;
word1(a) = 0;
#ifndef Sudden_Underflow
} else {
L = -L >> Exp_shift;
if (L < Exp_shift) {
word0(a) = 0x80000 >> L;
word1(a) = 0;
} else {
word0(a) = 0;
L -= Exp_shift;
word1(a) = L >= 31 ? 1 : 1 << 31 - L;
}
}
#endif
return a;
}
static double
b2d(Bigint *a, int *e)
{
unsigned long * xa, *xa0, w, y, z;
int k;
double d;
#define d0 word0(d)
#define d1 word1(d)
xa0 = a->x;
xa = xa0 + a->wds;
y = *--xa;
k = hi0bits(y);
*e = 32 - k;
if (k < Ebits) {
d0 = Exp_1 | y >> Ebits - k;
w = xa > xa0 ? *--xa : 0;
d1 = y << (32 - Ebits) + k | w >> Ebits - k;
goto ret_d;
}
z = xa > xa0 ? *--xa : 0;
if (k -= Ebits) {
d0 = Exp_1 | y << k | z >> 32 - k;
y = xa > xa0 ? *--xa : 0;
d1 = z << k | y >> 32 - k;
} else {
d0 = Exp_1 | y;
d1 = z;
}
ret_d:
#undef d0
#undef d1
return d;
}
static Bigint *
d2b(double d, int *e, int *bits)
{
Bigint * b;
int de, i, k;
unsigned long * x, y, z;
#define d0 word0(d)
#define d1 word1(d)
b = Balloc(1);
x = b->x;
z = d0 & Frac_mask;
d0 &= 0x7fffffff; /* clear sign bit, which we ignore */
#ifdef Sudden_Underflow
de = (int)(d0 >> Exp_shift);
z |= Exp_msk11;
#else
if (de = (int)(d0 >> Exp_shift))
z |= Exp_msk1;
#endif
if (y = d1) {
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;
}
#ifndef Sudden_Underflow
if (de) {
#endif
*e = de - Bias - (P - 1) + k;
*bits = P - k;
#ifndef Sudden_Underflow
} else {
*e = de - Bias - (P - 1) + 1 + k;
*bits = 32 * i - hi0bits(x[i-1]);
}
#endif
return b;
}
#undef d0
#undef d1
static double
ratio(Bigint *a, Bigint *b)
{
double da, db;
int k, ka, kb;
da = b2d(a, &ka);
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 da / 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.e-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 NAN_WORD0 0x7ff80000
#define NAN_WORD1 0
static int
match(const char **sp, char *t)
{
int c, d;
const char * s = *sp;
while (d = *t++) {
if ((c = *++s) >= 'A' && c <= 'Z')
c += 'a' - 'A';
if (c != d)
return 0;
}
*sp = s + 1;
return 1;
}
double
strtod(const char *s00, char **se)
{
int scale;
int bb2, bb5, bbe, bd2, bd5, bbbits, bs2, c, dsign,
e, e1, esign, i, j, k, nd, nd0, nf, nz, nz0, sign;
const char * s, *s0, *s1;
double aadj, aadj1, adj, rv, rv0;
long L;
unsigned long y, z;
Bigint * bb, *bb1, *bd, *bd0, *bs, *delta;
sign = nz0 = nz = 0;
rv = 0.;
for (s = s00; ; s++)
switch (*s) {
case '-':
sign = 1;
/* no break */
case '+':
if (*++s)
goto break2;
/* no break */
case 0:
s = s00;
goto ret;
case '\t':
case '\n':
case '\v':
case '\f':
case '\r':
case ' ':
continue;
default:
goto break2;
}
break2:
if (*s == '0') {
nz0 = 1;
while (*++s == '0')
;
if (!*s)
goto ret;
}
s0 = s;
y = z = 0;
for (nd = nf = 0; (c = *s) >= '0' && c <= '9'; nd++, s++)
if (nd < 9)
y = 10 * y + c - '0';
else if (nd < 16)
z = 10 * z + c - '0';
nd0 = nd;
if (c == '.') {
c = *++s;
if (!nd) {
for (; c == '0'; c = *++s)
nz++;
if (c > '0' && c <= '9') {
s0 = s;
nf += nz;
nz = 0;
goto have_dig;
}
goto dig_done;
}
for (; c >= '0' && c <= '9'; c = *++s) {
have_dig:
nz++;
if (c -= '0') {
nf += nz;
for (i = 1; i < nz; i++)
if (nd++ < 9)
y *= 10;
else if (nd <= DBL_DIG + 1)
z *= 10;
if (nd++ < 9)
y = 10 * y + c;
else if (nd <= DBL_DIG + 1)
z = 10 * z + c;
nz = 0;
}
}
}
dig_done:
e = 0;
if (c == 'e' || c == 'E') {
if (!nd && !nz && !nz0) {
s = s00;
goto ret;
}
s00 = s;
esign = 0;
switch (c = *++s) {
case '-':
esign = 1;
case '+':
c = *++s;
}
if (c >= '0' && c <= '9') {
while (c == '0')
c = *++s;
if (c > '0' && c <= '9') {
L = c - '0';
s1 = s;
while ((c = *++s) >= '0' && c <= '9')
L = 10 * L + c - '0';
if (s - s1 > 8 || L > 19999)
/* Avoid confusion from exponents
* so large that e might overflow.
*/
e = 19999; /* safe for 16 bit ints */
else
e = (int)L;
if (esign)
e = -e;
} else
e = 0;
} else
s = s00;
}
if (!nd) {
if (!nz && !nz0) {
/* Check for Nan and Infinity */
switch (c) {
case 'i':
case 'I':
if (match(&s, "nfinity")) {
word0(rv) = 0x7ff00000;
word1(rv) = 0;
goto ret;
}
break;
case 'n':
case 'N':
if (match(&s, "an")) {
word0(rv) = NAN_WORD0;
word1(rv) = NAN_WORD1;
goto ret;
}
}
s = s00;
}
goto ret;
}
e1 = e -= nf;
/* Now we have nd0 digits, starting at s0, followed by a
* decimal point, followed by nd-nd0 digits. The number we're
* after is the integer represented by those digits times
* 10**e */
if (!nd0)
nd0 = nd;
k = nd < DBL_DIG + 1 ? nd : DBL_DIG + 1;
rv = y;
if (k > 9)
rv = tens[k - 9] * rv + z;
bd0 = 0;
if (nd <= DBL_DIG
&& FLT_ROUNDS == 1
) {
if (!e)
goto ret;
if (e > 0) {
if (e <= Ten_pmax) {
/* rv = */ rounded_product(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;
rv *= tens[i];
/* rv = */ rounded_product(rv, tens[e]);
goto ret;
}
} else if (e >= -Ten_pmax) {
/* rv = */ rounded_quotient(rv, tens[-e]);
goto ret;
}
}
e1 += nd - k;
scale = 0;
/* Get starting approximation = rv * 10**e1 */
if (e1 > 0) {
if (i = e1 & 15)
rv *= tens[i];
if (e1 &= ~15) {
if (e1 > DBL_MAX_10_EXP) {
ovfl:
/* Can't trust HUGE_VAL */
word0(rv) = Exp_mask;
word1(rv) = 0;
if (bd0)
goto retfree;
goto ret;
}
if (e1 >>= 4) {
for (j = 0; e1 > 1; j++, e1 >>= 1)
if (e1 & 1)
rv *= bigtens[j];
/* The last multiplication could overflow. */
word0(rv) -= P * Exp_msk1;
rv *= bigtens[j];
if ((z = word0(rv) & Exp_mask)
> Exp_msk1 * (DBL_MAX_EXP + Bias - P))
goto ovfl;
if (z > Exp_msk1 * (DBL_MAX_EXP + Bias - 1 - P)) {
/* set to largest number */
/* (Can't trust DBL_MAX) */
word0(rv) = Big0;
word1(rv) = Big1;
} else
word0(rv) += P * Exp_msk1;
}
}
} else if (e1 < 0) {
e1 = -e1;
if (i = e1 & 15)
rv /= tens[i];
if (e1 &= ~15) {
e1 >>= 4;
if (e1 >= 1 << n_bigtens)
goto undfl;
if (e1 & Scale_Bit)
scale = P;
for (j = 0; e1 > 0; j++, e1 >>= 1)
if (e1 & 1)
rv *= tinytens[j];
if (!rv) {
undfl:
rv = 0.;
if (bd0)
goto retfree;
goto ret;
}
}
}
/* Now the hard part -- adjusting rv to the correct value.*/
/* Put digits into bd: true value = bd * 10^e */
bd0 = s2b(s0, nd0, nd, y);
for (; ; ) {
bd = Balloc(bd0->k);
Bcopy(bd, bd0);
bb = d2b(rv, &bbe, &bbbits); /* rv = bb * 2^bbe */
bs = i2b(1);
if (e >= 0) {
bb2 = bb5 = 0;
bd2 = bd5 = e;
} else {
bb2 = bb5 = -e;
bd2 = bd5 = 0;
}
if (bbe >= 0)
bb2 += bbe;
else
bd2 -= bbe;
bs2 = bb2;
#ifdef Sudden_Underflow
j = P + 1 - bbbits;
#else
i = bbe + bbbits - 1; /* logb(rv) */
if (i < Emin) /* denormal */
j = bbe + (P - Emin);
else
j = P + 1 - bbbits;
#endif
bb2 += j;
bd2 += j;
bd2 += scale;
i = bb2 < bd2 ? bb2 : bd2;
if (i > bs2)
i = bs2;
if (i > 0) {
bb2 -= i;
bd2 -= i;
bs2 -= i;
}
if (bb5 > 0) {
bs = pow5mult(bs, bb5);
bb1 = mult(bs, bb);
Bfree(bb);
bb = bb1;
}
if (bb2 > 0)
bb = lshift(bb, bb2);
if (bd5 > 0)
bd = pow5mult(bd, bd5);
if (bd2 > 0)
bd = lshift(bd, bd2);
if (bs2 > 0)
bs = lshift(bs, bs2);
delta = diff(bb, bd);
dsign = delta->sign;
delta->sign = 0;
i = cmp(delta, bs);
if (i < 0) {
/* Error is less than half an ulp -- check for
* special case of mantissa a power of two.
*/
if (dsign || word1(rv) || word0(rv) & Bndry_mask
|| (word0(rv) & Exp_mask) <= Exp_msk1
) {
if (!delta->x[0] && delta->wds == 1)
dsign = 2;
break;
}
delta = lshift(delta, Log2P);
if (cmp(delta, bs) > 0)
goto drop_down;
break;
}
if (i == 0) {
/* exactly half-way between */
if (dsign) {
if ((word0(rv) & Bndry_mask1) == Bndry_mask1
&& word1(rv) == 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)) {
dsign = 2;
drop_down:
/* boundary case -- decrement exponent */
#ifdef Sudden_Underflow
L = word0(rv) & Exp_mask;
if (L <= Exp_msk1)
goto undfl;
L -= Exp_msk1;
#else
L = (word0(rv) & Exp_mask) - Exp_msk1;
#endif
word0(rv) = L | Bndry_mask1;
word1(rv) = 0xffffffff;
break;
}
if (!(word1(rv) & LSB))
break;
if (dsign)
rv += ulp(rv);
else {
rv -= ulp(rv);
#ifndef Sudden_Underflow
if (!rv)
goto undfl;
#endif
}
dsign = 1 - dsign;
break;
}
if ((aadj = ratio(delta, bs)) <= 2.) {
if (dsign)
aadj = aadj1 = 1.;
else if (word1(rv) || word0(rv) & Bndry_mask) {
#ifndef Sudden_Underflow
if (word1(rv) == Tiny1 && !word0(rv))
goto undfl;
#endif
aadj = 1.;
aadj1 = -1.;
} else {
/* special case -- power of FLT_RADIX to be */
/* rounded down... */
if (aadj < 2. / FLT_RADIX)
aadj = 1. / FLT_RADIX;
else
aadj *= 0.5;
aadj1 = -aadj;
}
} else {
aadj *= 0.5;
aadj1 = dsign ? aadj : -aadj;
if (FLT_ROUNDS == 0)
aadj1 += 0.5;
}
y = word0(rv) & Exp_mask;
/* Check for overflow */
if (y == Exp_msk1 * (DBL_MAX_EXP + Bias - 1)) {
rv0 = rv;
word0(rv) -= P * Exp_msk1;
adj = aadj1 * ulp(rv);
rv += adj;
if ((word0(rv) & Exp_mask) >=
Exp_msk1 * (DBL_MAX_EXP + Bias - P)) {
if (word0(rv0) == Big0 && word1(rv0) == Big1)
goto ovfl;
word0(rv) = Big0;
word1(rv) = Big1;
goto cont;
} else
word0(rv) += P * Exp_msk1;
} else {
#ifdef Sudden_Underflow
if ((word0(rv) & Exp_mask) <= P * Exp_msk1) {
rv0 = rv;
word0(rv) += P * Exp_msk1;
adj = aadj1 * ulp(rv);
rv += adj;
if ((word0(rv) & Exp_mask) <= P * Exp_msk1) {
if (word0(rv0) == Tiny0
&& word1(rv0) == Tiny1)
goto undfl;
word0(rv) = Tiny0;
word1(rv) = Tiny1;
goto cont;
} else
word0(rv) -= P * Exp_msk1;
} else {
adj = aadj1 * ulp(rv);
rv += adj;
}
#else
/* Compute adj so that the IEEE rounding rules will
* correctly round rv + adj in some half-way cases.
* If rv * ulp(rv) is denormalized (i.e.,
* y <= (P-1)*Exp_msk1), we must adjust aadj to avoid
* trouble from bits lost to denormalization;
* example: 1.2e-307 .
*/
if (y <= (P - 1) * Exp_msk1 && aadj >= 1.) {
aadj1 = (double)(int)(aadj + 0.5);
if (!dsign)
aadj1 = -aadj1;
}
adj = aadj1 * ulp(rv);
rv += adj;
#endif
}
z = word0(rv) & Exp_mask;
if (!scale)
if (y == z) {
/* Can we stop now? */
L = 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);
}
if (scale) {
if ((word0(rv) & Exp_mask) <= P * Exp_msk1
&& word1(rv) & 1
&& dsign != 2)
if (dsign)
rv += ulp(rv);
else
word1(rv) &= ~1;
word0(rv0) = Exp_1 - P * Exp_msk1;
word1(rv0) = 0;
rv *= rv0;
}
retfree:
Bfree(bb);
Bfree(bd);
Bfree(bs);
Bfree(bd0);
Bfree(delta);
ret:
if (se)
*se = (char *)s;
return sign ? -rv : rv;
}
static int
quorem(Bigint *b, Bigint *S)
{
int n;
long borrow, y;
unsigned long carry, q, ys;
unsigned long * bx, *bxe, *sx, *sxe;
long z;
unsigned long si, zs;
n = S->wds;
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 */
if (q) {
borrow = 0;
carry = 0;
do {
si = *sx++;
ys = (si & 0xffff) * q + carry;
zs = (si >> 16) * q + (ys >> 16);
carry = zs >> 16;
y = (*bx & 0xffff) - (ys & 0xffff) + borrow;
borrow = y >> 16;
Sign_Extend(borrow, y);
z = (*bx >> 16) - (zs & 0xffff) + borrow;
borrow = z >> 16;
Sign_Extend(borrow, z);
Storeinc(bx, z, y);
} 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 {
si = *sx++;
ys = (si & 0xffff) + carry;
zs = (si >> 16) + (ys >> 16);
carry = zs >> 16;
y = (*bx & 0xffff) - (ys & 0xffff) + borrow;
borrow = y >> 16;
Sign_Extend(borrow, y);
z = (*bx >> 16) - (zs & 0xffff) + borrow;
borrow = z >> 16;
Sign_Extend(borrow, z);
Storeinc(bx, z, y);
} while (sx <= sxe);
bx = b->x;
bxe = bx + n;
if (!*bxe) {
while (--bxe > bx && !*bxe)
--n;
b->wds = n;
}
}
return q;
}
static char *
rv_alloc(int i)
{
int j, k, *r;
j = sizeof(unsigned long);
for (k = 0;
sizeof(Bigint) - sizeof(unsigned long) - sizeof(int) + j <= i;
j <<= 1)
k++;
r = (int * )Balloc(k);
*r = k;
return
(char *)(r + 1);
}
static char *
nrv_alloc(char *s, char **rve, int n)
{
char *rv, *t;
t = rv = rv_alloc(n);
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
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. 92-101].
*
* Modifications:
* 1. Rather than iterating, we use a simple numeric overestimate
* to determine k = floor(log10(d)). We scale relevant
* quantities using O(log2(k)) rather than O(k) multiplications.
* 2. For some modes > 2 (corresponding to ecvt and fcvt), we don't
* try to generate digits strictly left to right. Instead, we
* compute with fewer bits and propagate the carry if necessary
* when rounding the final digit up. This is often faster.
* 3. Under the assumption that input will be rounded nearest,
* mode 0 renders 1e23 as 1e23 rather than 9.999999999999999e22.
* That is, we allow equality in stopping tests when the
* round-nearest rule will give the same floating-point value
* as would satisfaction of the stopping test with strict
* inequality.
* 4. We remove common factors of powers of 2 from relevant
* quantities.
* 5. When converting floating-point integers less than 1e16,
* we use floating-point arithmetic rather than resorting
* to multiple-precision integers.
* 6. When asked to produce fewer than 15 digits, we first try
* to get by with floating-point arithmetic; we resort to
* multiple-precision integer arithmetic only if we cannot
* guarantee that the floating-point calculation has given
* the correctly rounded result. For k requested digits and
* "uniformly" distributed input, the probability is
* something like 10^(k-15) that we must resort to the long
* calculation.
*/
char *
dtoa(double d, 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.
4-9 should give the same return values as 2-3, i.e.,
4 <= mode <= 9 ==> same return as mode
2 + (mode & 1). These modes are mainly for
debugging; often they run slower but sometimes
faster than modes 2-3.
4,5,8,9 ==> left-to-right digit generation.
6-9 ==> 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;
#ifndef Sudden_Underflow
int denorm;
unsigned long x;
#endif
Bigint * b, *b1, *delta, *mlo, *mhi, *S;
double d2, ds, eps;
char *s, *s0;
if (word0(d) & Sign_bit) {
/* set sign for everything, including 0's and NaNs */
*sign = 1;
word0(d) &= ~Sign_bit; /* clear sign bit */
} else
*sign = 0;
if ((word0(d) & Exp_mask) == Exp_mask) {
/* Infinity or NaN */
*decpt = 9999;
if (!word1(d) && !(word0(d) & 0xfffff))
return nrv_alloc("Infinity", rve, 8);
return nrv_alloc("NaN", rve, 3);
}
if (!d) {
*decpt = 1;
return nrv_alloc("0", rve, 1);
}
b = d2b(d, &be, &bbits);
#ifdef Sudden_Underflow
i = (int)(word0(d) >> Exp_shift1 & (Exp_mask >> Exp_shift1));
#else
if (i = (int)(word0(d) >> Exp_shift1 & (Exp_mask >> Exp_shift1))) {
#endif
word0(d2) = (word0(d) & Frac_mask1) | Exp_11;
word1(d2) = word1(d);
/* log(x) ~=~ log(1.5) + (x-1.5)/1.5
* log10(x) = log(x) / log(10)
* ~=~ log(1.5)/log(10) + (x-1.5)/(1.5*log(10))
* log10(d) = (i-Bias)*log(2)/log(10) + log10(d2)
*
* This suggests computing an approximation k to log10(d) by
*
* k = (i - Bias)*0.301029995663981
* + ( (d2-1.5)*0.289529654602168 + 0.176091259055681 );
*
* We want k to be too large rather than too small.
* The error in the first-order Taylor series approximation
* is in our favor, so we just round up the constant enough
* to compensate for any error in the multiplication of
* (i - Bias) by 0.301029995663981; since |i - Bias| <= 1077,
* and 1077 * 0.30103 * 2^-52 ~=~ 7.2e-14,
* adding 1e-13 to the constant term more than suffices.
* Hence we adjust the constant term to 0.1760912590558.
* (We could get a more accurate k by invoking log10,
* but this is probably not worthwhile.)
*/
i -= Bias;
#ifndef Sudden_Underflow
denorm = 0;
} else {
/* d is denormalized */
i = bbits + be + (Bias + (P - 1) - 1);
x = i > 32 ? word0(d) << 64 - i | word1(d) >> i - 32
: word1(d) << 32 - i;
d2 = x;
word0(d2) -= 31 * Exp_msk1; /* adjust exponent */
i -= (Bias + (P - 1) - 1) + 1;
denorm = 1;
}
#endif
ds = (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 (d < 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;
switch (mode) {
case 0:
case 1:
ilim = ilim1 = -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;
}
s = s0 = rv_alloc(i);
if (ilim >= 0 && ilim <= Quick_max && try_quick) {
/* Try to get by with floating-point arithmetic. */
i = 0;
d2 = d;
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;
d /= bigtens[n_bigtens-1];
ieps++;
}
for (; j; j >>= 1, i++)
if (j & 1) {
ieps++;
ds *= bigtens[i];
}
d /= ds;
} else if (j1 = -k) {
d *= tens[j1 & 0xf];
for (j = j1 >> 4; j; j >>= 1, i++)
if (j & 1) {
ieps++;
d *= bigtens[i];
}
}
if (k_check && d < 1. && ilim > 0) {
if (ilim1 <= 0)
goto fast_failed;
ilim = ilim1;
k--;
d *= 10.;
ieps++;
}
eps = ieps * d + 7.;
word0(eps) -= (P - 1) * Exp_msk1;
if (ilim == 0) {
S = mhi = 0;
d -= 5.;
if (d > eps)
goto one_digit;
if (d < -eps)
goto no_digits;
goto fast_failed;
}
/* Generate ilim digits, then fix them up. */
eps *= tens[ilim-1];
for (i = 1; ; i++, d *= 10.) {
L = d;
d -= L;
*s++ = '0' + (int)L;
if (i == ilim) {
if (d > 0.5 + eps)
goto bump_up;
else if (d < 0.5 - eps) {
while (*--s == '0')
;
s++;
goto ret1;
}
break;
}
}
fast_failed:
s = s0;
d = 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 || d <= 5 * ds)
goto no_digits;
goto one_digit;
}
for (i = 1; ; i++) {
L = d / ds;
d -= L * ds;
*s++ = '0' + (int)L;
if (i == ilim) {
d += d;
if (d > ds || d == ds && L & 1) {
bump_up:
while (*--s == '9')
if (s == s0) {
k++;
*s = '0';
break;
}
++ * s++;
}
break;
}
if (!(d *= 10.))
break;
}
goto ret1;
}
m2 = b2;
m5 = b5;
mhi = mlo = 0;
if (leftright) {
if (mode < 2) {
i =
#ifndef Sudden_Underflow
denorm ? be + (Bias + (P - 1) - 1 + 1) :
#endif
1 + P - bbits;
} else {
j = ilim - 1;
if (m5 >= j)
m5 -= j;
else {
s5 += j -= m5;
b5 += j;
m5 = 0;
}
if ((i = ilim) < 0) {
m2 -= i;
i = 0;
}
}
b2 += i;
s2 += i;
mhi = i2b(1);
}
if (m2 > 0 && s2 > 0) {
i = m2 < s2 ? m2 : s2;
b2 -= i;
m2 -= i;
s2 -= i;
}
if (b5 > 0) {
if (leftright) {
if (m5 > 0) {
mhi = pow5mult(mhi, m5);
b1 = mult(mhi, b);
Bfree(b);
b = b1;
}
if (j = b5 - m5)
b = pow5mult(b, j);
} else
b = pow5mult(b, b5);
}
S = i2b(1);
if (s5 > 0)
S = pow5mult(S, s5);
/* Check for special case that d is a normalized power of 2. */
spec_case = 0;
if (mode < 2) {
if (!word1(d) && !(word0(d) & Bndry_mask)
#ifndef Sudden_Underflow
&& word0(d) & Exp_mask
#endif
) {
/* The special case */
b2 += Log2P;
s2 += Log2P;
spec_case = 1;
}
}
/* Arrange for convenient computation of quotients:
* shift left if necessary so divisor has 4 leading 0 bits.
*
* Perhaps we should just compute leading 28 bits of S once
* and for all and pass them and a shift to quorem, so it
* can do shifts and ors to compute the numerator for q.
*/
if (i = ((s5 ? 32 - hi0bits(S->x[S->wds-1]) : 1) + s2) & 0x1f)
i = 32 - i;
if (i > 4) {
i -= 4;
b2 += i;
m2 += i;
s2 += i;
} else if (i < 4) {
i += 28;
b2 += i;
m2 += i;
s2 += i;
}
if (b2 > 0)
b = lshift(b, b2);
if (s2 > 0)
S = lshift(S, s2);
if (k_check) {
if (cmp(b, S) < 0) {
k--;
b = multadd(b, 10, 0); /* we botched the k estimate */
if (leftright)
mhi = multadd(mhi, 10, 0);
ilim = ilim1;
}
}
if (ilim <= 0 && mode > 2) {
if (ilim < 0 || cmp(b, S = multadd(S, 5, 0)) <= 0) {
/* no digits, fcvt style */
no_digits:
k = -1 - ndigits;
goto ret;
}
one_digit:
*s++ = '1';
k++;
goto ret;
}
if (leftright) {
if (m2 > 0)
mhi = lshift(mhi, m2);
/* Compute mlo -- check for special case
* that d is a normalized power of 2.
*/
mlo = mhi;
if (spec_case) {
mhi = Balloc(mhi->k);
Bcopy(mhi, mlo);
mhi = lshift(mhi, Log2P);
}
for (i = 1; ; i++) {
dig = quorem(b, S) + '0';
/* Do we yet have the shortest decimal string
* that will round to d?
*/
j = cmp(b, mlo);
delta = diff(S, mhi);
j1 = delta->sign ? 1 : cmp(b, delta);
Bfree(delta);
if (j1 == 0 && !mode && !(word1(d) & 1)) {
if (dig == '9')
goto round_9_up;
if (j > 0)
dig++;
*s++ = dig;
goto ret;
}
if (j < 0 || j == 0 && !mode
&& !(word1(d) & 1)
) {
if (j1 > 0) {
b = lshift(b, 1);
j1 = cmp(b, S);
if ((j1 > 0 || j1 == 0 && dig & 1)
&& dig++ == '9')
goto round_9_up;
}
*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 (mlo == mhi)
mlo = mhi = multadd(mhi, 10, 0);
else {
mlo = multadd(mlo, 10, 0);
mhi = multadd(mhi, 10, 0);
}
}
} else
for (i = 1; ; i++) {
*s++ = dig = quorem(b, S) + '0';
if (i >= ilim)
break;
b = multadd(b, 10, 0);
}
/* Round off last digit */
b = lshift(b, 1);
j = cmp(b, S);
if (j > 0 || j == 0 && dig & 1) {
roundoff:
while (*--s == '9')
if (s == s0) {
k++;
*s++ = '1';
goto ret;
}
++ * s++;
} else {
while (*--s == '0')
;
s++;
}
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;
}