xref: /netbsd-src/lib/libc/gdtoa/gdtoa.c (revision ace5b9b5feb0e7608bd2da7a617428d2e1cf8aa3)

/* $NetBSD: gdtoa.c,v 1.11 2024/01/20 14:52:47 christos Exp $ */

/****************************************************************

The author of this software is David M. Gay.

Copyright (C) 1998, 1999 by Lucent Technologies
All Rights Reserved

Permission to use, copy, modify, and distribute this software and
its documentation for any purpose and without fee is hereby
granted, provided that the above copyright notice appear in all
copies and that both that the copyright notice and this
permission notice and warranty disclaimer appear in supporting
documentation, and that the name of Lucent or any of its entities
not be used in advertising or publicity pertaining to
distribution of the software without specific, written prior
permission.

LUCENT DISCLAIMS ALL WARRANTIES WITH REGARD TO THIS SOFTWARE,
INCLUDING ALL IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS.
IN NO EVENT SHALL LUCENT OR ANY OF ITS ENTITIES BE LIABLE FOR ANY
SPECIAL, INDIRECT OR CONSEQUENTIAL DAMAGES OR ANY DAMAGES
WHATSOEVER RESULTING FROM LOSS OF USE, DATA OR PROFITS, WHETHER
IN AN ACTION OF CONTRACT, NEGLIGENCE OR OTHER TORTIOUS ACTION,
ARISING OUT OF OR IN CONNECTION WITH THE USE OR PERFORMANCE OF
THIS SOFTWARE.

****************************************************************/

/* Please send bug reports to David M. Gay (dmg at acm dot org,
 * with " at " changed at "@" and " dot " changed to ".").	*/

#include "gdtoaimp.h"

 static Bigint *
#ifdef KR_headers
bitstob(bits, nbits, bbits) ULong *bits; int nbits; int *bbits;
#else
bitstob(ULong *bits, int nbits, int *bbits)
#endif
{
	int i, k;
	Bigint *b;
	ULong *be, *x, *x0;

	i = ULbits;
	k = 0;
	while(i < nbits) {
		i <<= 1;
		k++;
		}
#ifndef Pack_32
	if (!k)
		k = 1;
#endif
	b = Balloc(k);
	if (b == NULL)
		return NULL;
	be = bits + (((unsigned int)nbits - 1) >> kshift);
	x = x0 = b->x;
	do {
		*x++ = *bits & ALL_ON;
#ifdef Pack_16
		*x++ = (*bits >> 16) & ALL_ON;
#endif
		} while(++bits <= be);
	ptrdiff_t td = x - x0;
	_DIAGASSERT(__type_fit(int, td));
	i = (int)td;
	while(!x0[--i])
		if (!i) {
			b->wds = 0;
			*bbits = 0;
			goto ret;
			}
	b->wds = i + 1;
	*bbits = i*ULbits + 32 - hi0bits(b->x[i]);
 ret:
	return b;
	}

/* dtoa for IEEE arithmetic (dmg): convert double to ASCII string.
 *
 * Inspired by "How to Print Floating-Point Numbers Accurately" by
 * Guy L. Steele, Jr. and Jon L. White [Proc. ACM SIGPLAN '90, pp. 112-126].
 *
 * Modifications:
 *	1. Rather than iterating, we use a simple numeric overestimate
 *	   to determine k = floor(log10(d)).  We scale relevant
 *	   quantities using O(log2(k)) rather than O(k) multiplications.
 *	2. For some modes > 2 (corresponding to ecvt and fcvt), we don't
 *	   try to generate digits strictly left to right.  Instead, we
 *	   compute with fewer bits and propagate the carry if necessary
 *	   when rounding the final digit up.  This is often faster.
 *	3. Under the assumption that input will be rounded nearest,
 *	   mode 0 renders 1e23 as 1e23 rather than 9.999999999999999e22.
 *	   That is, we allow equality in stopping tests when the
 *	   round-nearest rule will give the same floating-point value
 *	   as would satisfaction of the stopping test with strict
 *	   inequality.
 *	4. We remove common factors of powers of 2 from relevant
 *	   quantities.
 *	5. When converting floating-point integers less than 1e16,
 *	   we use floating-point arithmetic rather than resorting
 *	   to multiple-precision integers.
 *	6. When asked to produce fewer than 15 digits, we first try
 *	   to get by with floating-point arithmetic; we resort to
 *	   multiple-precision integer arithmetic only if we cannot
 *	   guarantee that the floating-point calculation has given
 *	   the correctly rounded result.  For k requested digits and
 *	   "uniformly" distributed input, the probability is
 *	   something like 10^(k-15) that we must resort to the Long
 *	   calculation.
 */

 char *
gdtoa
#ifdef KR_headers
	(fpi, be, bits, kindp, mode, ndigits, decpt, rve)
	CONST FPI *fpi; int be; ULong *bits;
	int *kindp, mode, ndigits, *decpt; char **rve;
#else
	(CONST FPI *fpi, int be, ULong *bits, int *kindp, int mode, int ndigits, int *decpt, char **rve)
#endif
{
 /*	Arguments ndigits and decpt are similar to the second and third
	arguments 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 -32768.
	be = exponent: value = (integer represented by bits) * (2 to the power of be).

	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, be0, dig, i, ieps, ilim = 0, ilim0, ilim1 = 0, inex;
	int j, jj1, k, k0, k_check, kind, leftright, m2, m5, nbits;
	int rdir, s2, s5, spec_case, try_quick;
	Long L;
	Bigint *b, *b1, *delta, *mlo, *mhi, *mhi1, *S;
	double d2, ds;
	char *s, *s0;
	U d, eps;

#ifndef MULTIPLE_THREADS
	if (dtoa_result) {
		freedtoa(dtoa_result);
		dtoa_result = 0;
		}
#endif
	inex = 0;
	if (*kindp & STRTOG_NoMemory)
		return NULL;
	kind = *kindp &= ~STRTOG_Inexact;
	switch(kind & STRTOG_Retmask) {
	  case STRTOG_Zero:
		goto ret_zero;
	  case STRTOG_Normal:
	  case STRTOG_Denormal:
		break;
	  case STRTOG_Infinite:
		*decpt = -32768;
		return nrv_alloc("Infinity", rve, 8);
	  case STRTOG_NaN:
		*decpt = -32768;
		return nrv_alloc("NaN", rve, 3);
	  default:
		return 0;
	  }
	b = bitstob(bits, nbits = fpi->nbits, &bbits);
	if (b == NULL)
		return NULL;
	be0 = be;
	if ( (i = trailz(b)) !=0) {
		rshift(b, i);
		be += i;
		bbits -= i;
		}
	if (!b->wds) {
		Bfree(b);
 ret_zero:
		*decpt = 1;
		return nrv_alloc("0", rve, 1);
		}

	dval(&d) = b2d(b, &i);
	i = be + bbits - 1;
	word0(&d) &= Frac_mask1;
	word0(&d) |= Exp_11;
#ifdef IBM
	if ( (j = 11 - hi0bits(word0(&d) & Frac_mask)) !=0)
		dval(&d) /= 1 << j;
#endif

	/* 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.)
	 */
#ifdef IBM
	i <<= 2;
	i += j;
#endif
	ds = (dval(&d)-1.5)*0.289529654602168 + 0.1760912590558 + i*0.301029995663981;

	/* correct assumption about exponent range */
	if ((j = i) < 0)
		j = -j;
	if ((j -= 1077) > 0)
		ds += j * 7e-17;

	k = (int)ds;
	if (ds < 0. && ds != k)
		k--;	/* want k = floor(ds) */
	k_check = 1;
#ifdef IBM
	j = be + bbits - 1;
	if ( (jj1 = j & 3) !=0)
		dval(&d) *= 1 << jj1;
	word0(&d) += j << Exp_shift - 2 & Exp_mask;
#else
	word0(&d) += (be + bbits - 1) << Exp_shift;
#endif
	if (k >= 0 && k <= Ten_pmax) {
		if (dval(&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;
		}
	else if (i >= -4 - Emin || i < Emin)
		try_quick = 0;
	leftright = 1;
	ilim = ilim1 = -1;	/* Values for cases 0 and 1; done here to */
				/* silence erroneous "gcc -Wall" warning. */
	switch(mode) {
		case 0:
		case 1:
			i = (int)(nbits * .30103) + 3;
			ndigits = 0;
			break;
		case 2:
			leftright = 0;
			/*FALLTHROUGH*/
		case 4:
			if (ndigits <= 0)
				ndigits = 1;
			ilim = ilim1 = i = ndigits;
			break;
		case 3:
			leftright = 0;
			/*FALLTHROUGH*/
		case 5:
			i = ndigits + k + 1;
			ilim = i;
			ilim1 = i - 1;
			if (i <= 0)
				i = 1;
		}
	s = s0 = rv_alloc((size_t)i);
	if (s == NULL)
		return NULL;

	if ( (rdir = fpi->rounding - 1) !=0) {
		if (rdir < 0)
			rdir = 2;
		if (kind & STRTOG_Neg)
			rdir = 3 - rdir;
		}

	/* Now rdir = 0 ==> round near, 1 ==> round up, 2 ==> round down. */

	if (ilim >= 0 && ilim <= Quick_max && try_quick && !rdir
#ifndef IMPRECISE_INEXACT
		&& k == 0
#endif
								) {

		/* Try to get by with floating-point arithmetic. */

		i = 0;
		d2 = dval(&d);
#ifdef IBM
		if ( (j = 11 - hi0bits(word0(&d) & Frac_mask)) !=0)
			dval(&d) /= 1 << j;
#endif
		k0 = k;
		ilim0 = ilim;
		ieps = 2; /* conservative */
		if (k > 0) {
			ds = tens[k&0xf];
			j = (unsigned int)k >> 4;
			if (j & Bletch) {
				/* prevent overflows */
				j &= Bletch - 1;
				dval(&d) /= bigtens[n_bigtens-1];
				ieps++;
				}
			for(; j; j /= 2, i++)
				if (j & 1) {
					ieps++;
					ds *= bigtens[i];
					}
			}
		else  {
			ds = 1.;
			if ( (jj1 = -k) !=0) {
				dval(&d) *= tens[jj1 & 0xf];
				for(j = (unsigned int)jj1 >> 4; j; j = (unsigned int)j >> 1, i++)
					if (j & 1) {
						ieps++;
						dval(&d) *= bigtens[i];
						}
				}
			}
		if (k_check && dval(&d) < 1. && ilim > 0) {
			if (ilim1 <= 0)
				goto fast_failed;
			ilim = ilim1;
			k--;
			dval(&d) *= 10.;
			ieps++;
			}
		dval(&eps) = ieps*dval(&d) + 7.;
		word0(&eps) -= (P-1)*Exp_msk1;
		if (ilim == 0) {
			S = mhi = 0;
			dval(&d) -= 5.;
			if (dval(&d) > dval(&eps))
				goto one_digit;
			if (dval(&d) < -dval(&eps))
				goto no_digits;
			goto fast_failed;
			}
#ifndef No_leftright
		if (leftright) {
			/* Use Steele & White method of only
			 * generating digits needed.
			 */
			dval(&eps) = ds*0.5/tens[ilim-1] - dval(&eps);
			for(i = 0;;) {
				L = (Long)(dval(&d)/ds);
				dval(&d) -= L*ds;
				*s++ = '0' + (int)L;
				if (dval(&d) < dval(&eps)) {
					if (dval(&d))
						inex = STRTOG_Inexlo;
					goto ret1;
					}
				if (ds - dval(&d) < dval(&eps))
					goto bump_up;
				if (++i >= ilim)
					break;
				dval(&eps) *= 10.;
				dval(&d) *= 10.;
				}
			}
		else {
#endif
			/* Generate ilim digits, then fix them up. */
			dval(&eps) *= tens[ilim-1];
			for(i = 1;; i++, dval(&d) *= 10.) {
				if ( (L = (Long)(dval(&d)/ds)) !=0)
					dval(&d) -= L*ds;
				*s++ = '0' + (int)L;
				if (i == ilim) {
					ds *= 0.5;
					if (dval(&d) > ds + dval(&eps))
						goto bump_up;
					else if (dval(&d) < ds - dval(&eps)) {
						if (dval(&d))
							inex = STRTOG_Inexlo;
						goto clear_trailing0;
						}
					break;
					}
				}
#ifndef No_leftright
			}
#endif
 fast_failed:
		s = s0;
		dval(&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 || dval(&d) <= 5*ds)
				goto no_digits;
			goto one_digit;
			}
		for(i = 1;; i++, dval(&d) *= 10.) {
			L = dval(&d) / ds;
			dval(&d) -= L*ds;
#ifdef Check_FLT_ROUNDS
			/* If FLT_ROUNDS == 2, L will usually be high by 1 */
			if (dval(&d) < 0) {
				L--;
				dval(&d) += ds;
				}
#endif
			*s++ = '0' + (int)L;
			if (dval(&d) == 0.)
				break;
			if (i == ilim) {
				if (rdir) {
					if (rdir == 1)
						goto bump_up;
					inex = STRTOG_Inexlo;
					goto ret1;
					}
				dval(&d) += dval(&d);
#ifdef ROUND_BIASED
				if (dval(&d) >= ds)
#else
				if (dval(&d) > ds || (dval(&d) == ds && L & 1))
#endif
					{
 bump_up:
					inex = STRTOG_Inexhi;
					while(*--s == '9')
						if (s == s0) {
							k++;
							*s = '0';
							break;
							}
					++*s++;
					}
				else {
					inex = STRTOG_Inexlo;
 clear_trailing0:
					while(*--s == '0'){}
					++s;
					}
				break;
				}
			}
		goto ret1;
		}

	m2 = b2;
	m5 = b5;
	mhi = mlo = 0;
	if (leftright) {
		i = nbits - bbits;
		if (be - i++ < fpi->emin && mode != 3 && mode != 5) {
			/* denormal */
			i = be - fpi->emin + 1;
			if (mode >= 2 && ilim > 0 && ilim < i)
				goto small_ilim;
			}
		else if (mode >= 2) {
 small_ilim:
			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);
				if (mhi == NULL)
					return NULL;
				b1 = mult(mhi, b);
				if (b1 == NULL)
					return NULL;
				Bfree(b);
				b = b1;
				}
			if ( (j = b5 - m5) !=0) {
				b = pow5mult(b, j);
				if (b == NULL)
					return NULL;
				}
			}
		else {
			b = pow5mult(b, b5);
			if (b == NULL)
				return NULL;
			}
		}
	S = i2b(1);
	if (S == NULL)
		return NULL;
	if (s5 > 0) {
		S = pow5mult(S, s5);
		if (S == NULL)
			return NULL;
		}

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

	spec_case = 0;
	if (mode < 2) {
		if (bbits == 1 && be0 > fpi->emin + 1) {
			/* The special case */
			b2++;
			s2++;
			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.
	 */
	i = ((s5 ? hi0bits(S->x[S->wds-1]) : ULbits - 1) - s2 - 4) & kmask;
	m2 += i;
	if ((b2 += i) > 0) {
		b = lshift(b, b2);
		if (b == NULL)
			return NULL;
		}
	if ((s2 += i) > 0) {
		S = lshift(S, s2);
		if (S == NULL)
			return NULL;
		}
	if (k_check) {
		if (cmp(b,S) < 0) {
			k--;
			b = multadd(b, 10, 0);	/* we botched the k estimate */
			if (b == NULL)
				return NULL;
			if (leftright) {
				mhi = multadd(mhi, 10, 0);
				if (mhi == NULL)
					return NULL;
				}
			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;
			inex = STRTOG_Inexlo;
			goto ret;
			}
 one_digit:
		inex = STRTOG_Inexhi;
		*s++ = '1';
		k++;
		goto ret;
		}
	if (leftright) {
		if (m2 > 0) {
			mhi = lshift(mhi, m2);
			if (mhi == NULL)
				return NULL;
			}

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

		mlo = mhi;
		if (spec_case) {
			mhi = Balloc(mhi->k);
			if (mhi == NULL)
				return NULL;
			Bcopy(mhi, mlo);
			mhi = lshift(mhi, 1);
			if (mhi == NULL)
				return NULL;
			}

		for(i = 1;;i++) {
			dig = quorem(b,S) + '0';
			/* Do we yet have the shortest decimal string
			 * that will round to d?
			 */
			j = cmp(b, mlo);
			delta = diff(S, mhi);
			if (delta == NULL)
				return NULL;
			jj1 = delta->sign ? 1 : cmp(b, delta);
			Bfree(delta);
#ifndef ROUND_BIASED
			if (jj1 == 0 && !mode && !(bits[0] & 1) && !rdir) {
				if (dig == '9')
					goto round_9_up;
				if (j <= 0) {
					if (b->wds > 1 || b->x[0])
						inex = STRTOG_Inexlo;
					}
				else {
					dig++;
					inex = STRTOG_Inexhi;
					}
				*s++ = dig;
				goto ret;
				}
#endif
			if (j < 0 || (j == 0 && !mode
#ifndef ROUND_BIASED
							&& !(bits[0] & 1)
#endif
					)) {
				if (rdir && (b->wds > 1 || b->x[0])) {
					if (rdir == 2) {
						inex = STRTOG_Inexlo;
						goto accept;
						}
					while (cmp(S,mhi) > 0) {
						*s++ = dig;
						mhi1 = multadd(mhi, 10, 0);
						if (mhi1 == NULL)
							return NULL;
						if (mlo == mhi)
							mlo = mhi1;
						mhi = mhi1;
						b = multadd(b, 10, 0);
						if (b == NULL)
							return NULL;
						dig = quorem(b,S) + '0';
						}
					if (dig++ == '9')
						goto round_9_up;
					inex = STRTOG_Inexhi;
					goto accept;
					}
				if (jj1 > 0) {
					b = lshift(b, 1);
					if (b == NULL)
						return NULL;
					jj1 = cmp(b, S);
#ifdef ROUND_BIASED
					if (jj1 >= 0 /*)*/
#else
					if ((jj1 > 0 || (jj1 == 0 && dig & 1))
#endif
					&& dig++ == '9')
						goto round_9_up;
					inex = STRTOG_Inexhi;
					}
				if (b->wds > 1 || b->x[0])
					inex = STRTOG_Inexlo;
 accept:
				*s++ = dig;
				goto ret;
				}
			if (jj1 > 0 && rdir != 2) {
				if (dig == '9') { /* possible if i == 1 */
 round_9_up:
					*s++ = '9';
					inex = STRTOG_Inexhi;
					goto roundoff;
					}
				inex = STRTOG_Inexhi;
				*s++ = dig + 1;
				goto ret;
				}
			*s++ = dig;
			if (i == ilim)
				break;
			b = multadd(b, 10, 0);
			if (b == NULL)
				return NULL;
			if (mlo == mhi) {
				mlo = mhi = multadd(mhi, 10, 0);
				if (mlo == NULL)
					return NULL;
				}
			else {
				mlo = multadd(mlo, 10, 0);
				if (mlo == NULL)
					return NULL;
				mhi = multadd(mhi, 10, 0);
				if (mhi == NULL)
					return NULL;
				}
			}
		}
	else
		for(i = 1;; i++) {
			*s++ = dig = quorem(b,S) + '0';
			if (i >= ilim)
				break;
			b = multadd(b, 10, 0);
			if (b == NULL)
				return NULL;
			}

	/* Round off last digit */

	if (rdir) {
		if (rdir == 2 || (b->wds <= 1 && !b->x[0]))
			goto chopzeros;
		goto roundoff;
		}
	b = lshift(b, 1);
	if (b == NULL)
		return NULL;
	j = cmp(b, S);
#ifdef ROUND_BIASED
	if (j >= 0)
#else
	if (j > 0 || (j == 0 && dig & 1))
#endif
		{
 roundoff:
		inex = STRTOG_Inexhi;
		while(*--s == '9')
			if (s == s0) {
				k++;
				*s++ = '1';
				goto ret;
				}
		++*s++;
		}
	else {
 chopzeros:
		if (b->wds > 1 || b->x[0])
			inex = STRTOG_Inexlo;
		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;
	*kindp |= inex;
	return s0;
	}