dtoa.c

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

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"

/* 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.
 */

#ifdef Honor_FLT_ROUNDS
#define Rounding rounding
#undef Check_FLT_ROUNDS
#define Check_FLT_ROUNDS
#else
#define Rounding Flt_Rounds
#endif

 char *
dtoa
#ifdef KR_headers
	(d, mode, ndigits, decpt, sign, rve)
	double d; int mode, ndigits, *decpt, *sign; char **rve;
#else
	(double d, int mode, int ndigits, int *decpt, int *sign, char **rve)
#endif
{
 /*	Arguments ndigits, decpt, sign are similar to those
	of ecvt and fcvt; trailing zeros are suppressed from
	the returned string.  If not null, *rve is set to point
	to the end of the return value.  If d is +-Infinity or NaN,
	then *decpt is set to 9999.

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

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

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

	int bbits, b2, b5, be, dig, i, ieps, ilim, ilim0, ilim1,
		j, j1, k, k0, k_check, leftright, m2, m5, s2, s5,
		spec_case, try_quick;
	Long L;
#ifndef Sudden_Underflow
	int denorm;
	ULong x;
#endif
	Bigint *b, *b1, *delta, *mlo, *mhi, *S;
	double d2, ds, eps;
	char *s, *s0;
#ifdef Honor_FLT_ROUNDS
	int rounding;
#endif
#ifdef SET_INEXACT
	int inexact, oldinexact;
#endif

#ifndef MULTIPLE_THREADS
	if (dtoa_result) {
		freedtoa(dtoa_result);
		dtoa_result = 0;
		}
#endif

	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 defined(IEEE_Arith) + defined(VAX)
#ifdef IEEE_Arith
	if ((word0(d) & Exp_mask) == Exp_mask)
#else
	if (word0(d)  == 0x8000)
#endif
		{
		/* Infinity or NaN */
		*decpt = 9999;
#ifdef IEEE_Arith
		if (!word1(d) && !(word0(d) & 0xfffff))
			return nrv_alloc("Infinity", rve, 8);
#endif
		return nrv_alloc("NaN", rve, 3);
		}
#endif
#ifdef IBM
	dval(d) += 0; /* normalize */
#endif
	if (!dval(d)) {
		*decpt = 1;
		return nrv_alloc("0", rve, 1);
		}

#ifdef SET_INEXACT
	try_quick = oldinexact = get_inexact();
	inexact = 1;
#endif
#ifdef Honor_FLT_ROUNDS
	if ((rounding = Flt_Rounds) >= 2) {
		if (*sign)
			rounding = rounding == 2 ? 0 : 2;
		else
			if (rounding != 2)
				rounding = 0;
		}
#endif

	b = d2b(dval(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)) )!=0) {
#endif
		dval(d2) = dval(d);
		word0(d2) &= Frac_mask1;
		word0(d2) |= Exp_11;
#ifdef IBM
		if (( j = 11 - hi0bits(word0(d2) & Frac_mask) )!=0)
			dval(d2) /= 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.)
		 */

		i -= Bias;
#ifdef IBM
		i <<= 2;
		i += j;
#endif
#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;
		dval(d2) = x;
		word0(d2) -= 31*Exp_msk1; /* adjust exponent */
		i -= (Bias + (P-1) - 1) + 1;
		denorm = 1;
		}
#endif
	ds = (dval(d2)-1.5)*0.289529654602168 + 0.1760912590558 + i*0.301029995663981;
	k = (int)ds;
	if (ds < 0. && ds != k)
		k--;	/* want k = floor(ds) */
	k_check = 1;
	if (k >= 0 && k <= Ten_pmax) {
		if (dval(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;

#ifndef SET_INEXACT
#ifdef Check_FLT_ROUNDS
	try_quick = Rounding == 1;
#else
	try_quick = 1;
#endif
#endif /*SET_INEXACT*/

	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);

#ifdef Honor_FLT_ROUNDS
	if (mode > 1 && rounding != 1)
		leftright = 0;
#endif

	if (ilim >= 0 && ilim <= Quick_max && try_quick) {

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

		i = 0;
		dval(d2) = dval(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;
				dval(d) /= bigtens[n_bigtens-1];
				ieps++;
				}
			for(; j; j >>= 1, i++)
				if (j & 1) {
					ieps++;
					ds *= bigtens[i];
					}
			dval(d) /= ds;
			}
		else if (( j1 = -k )!=0) {
			dval(d) *= tens[j1 & 0xf];
			for(j = j1 >> 4; j; 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) = 0.5/tens[ilim-1] - dval(eps);
			for(i = 0;;) {
				L = dval(d);
				dval(d) -= L;
				*s++ = '0' + (int)L;
				if (dval(d) < dval(eps))
					goto ret1;
				if (1. - 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.) {
				L = (Long)(dval(d));
				if (!(dval(d) -= L))
					ilim = i;
				*s++ = '0' + (int)L;
				if (i == ilim) {
					if (dval(d) > 0.5 + dval(eps))
						goto bump_up;
					else if (dval(d) < 0.5 - dval(eps)) {
						while(*--s == '0');
						s++;
						goto ret1;
						}
					break;
					}
				}
#ifndef No_leftright
			}
#endif
 fast_failed:
		s = s0;
		dval(d) = dval(d2);
		k = k0;
		ilim = ilim0;
		}

	/* Do we have a "small" integer? */

	if (be >= 0 && k <= Int_max) {
		/* Yes. */
		ds = tens[k];
		if (ndigits < 0 && ilim <= 0) {
			S = mhi = 0;
			if (ilim < 0 || dval(d) <= 5*ds)
				goto no_digits;
			goto one_digit;
			}
		for(i = 1;; i++, dval(d) *= 10.) {
			L = (Long)(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)) {
#ifdef SET_INEXACT
				inexact = 0;
#endif
				break;
				}
			if (i == ilim) {
#ifdef Honor_FLT_ROUNDS
				if (mode > 1)
				switch(rounding) {
				  case 0: goto ret1;
				  case 2: goto bump_up;
				  }
#endif
				dval(d) += dval(d);
				if (dval(d) > ds || dval(d) == ds && L & 1) {
 bump_up:
					while(*--s == '9')
						if (s == s0) {
							k++;
							*s = '0';
							break;
							}
					++*s++;
					}
				break;
				}
			}
		goto ret1;
		}

	m2 = b2;
	m5 = b5;
	mhi = mlo = 0;
	if (leftright) {
		i =
#ifndef Sudden_Underflow
			denorm ? be + (Bias + (P-1) - 1 + 1) :
#endif
#ifdef IBM
			1 + 4*P - 3 - bbits + ((bbits + be - 1) & 3);
#else
			1 + P - bbits;
#endif
		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 )!=0)
				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 || leftright)
#ifdef Honor_FLT_ROUNDS
			&& rounding == 1
#endif
				) {
		if (!word1(d) && !(word0(d) & Bndry_mask)
#ifndef Sudden_Underflow
		 && word0(d) & (Exp_mask & ~Exp_msk1)
#endif
				) {
			/* The special case */
			b2 += Log2P;
			s2 += Log2P;
			spec_case = 1;
			}
		}

	/* Arrange for convenient computation of quotients:
	 * shift left if necessary so divisor has 4 leading 0 bits.
	 *
	 * Perhaps we should just compute leading 28 bits of S once
	 * and for all and pass them and a shift to quorem, so it
	 * can do shifts and ors to compute the numerator for q.
	 */
#ifdef Pack_32
	if (( i = ((s5 ? 32 - hi0bits(S->x[S->wds-1]) : 1) + s2) & 0x1f )!=0)
		i = 32 - i;
#else
	if (( i = ((s5 ? 32 - hi0bits(S->x[S->wds-1]) : 1) + s2) & 0xf )!=0)
		i = 16 - i;
#endif
	if (i > 4) {
		i -= 4;
		b2 += i;
		m2 += i;
		s2 += i;
		}
	else if (i < 4) {
		i += 28;
		b2 += i;
		m2 += i;
		s2 += i;
		}
	if (b2 > 0)
		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 == 3 || mode == 5)) {
		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);
#ifndef ROUND_BIASED
			if (j1 == 0 && mode != 1 && !(word1(d) & 1)
#ifdef Honor_FLT_ROUNDS
				&& rounding >= 1
#endif
								   ) {
				if (dig == '9')
					goto round_9_up;
				if (j > 0)
					dig++;
#ifdef SET_INEXACT
				else if (!b->x[0] && b->wds <= 1)
					inexact = 0;
#endif
				*s++ = dig;
				goto ret;
				}
#endif
			if (j < 0 || j == 0 && mode != 1
#ifndef ROUND_BIASED
							&& !(word1(d) & 1)
#endif
					) {
				if (!b->x[0] && b->wds <= 1) {
#ifdef SET_INEXACT
					inexact = 0;
#endif
					goto accept_dig;
					}
#ifdef Honor_FLT_ROUNDS
				if (mode > 1)
				 switch(rounding) {
				  case 0: goto accept_dig;
				  case 2: goto keep_dig;
				  }
#endif /*Honor_FLT_ROUNDS*/
				if (j1 > 0) {
					b = lshift(b, 1);
					j1 = cmp(b, S);
					if ((j1 > 0 || j1 == 0 && dig & 1)
					&& dig++ == '9')
						goto round_9_up;
					}
 accept_dig:
				*s++ = dig;
				goto ret;
				}
			if (j1 > 0) {
#ifdef Honor_FLT_ROUNDS
				if (!rounding)
					goto accept_dig;
#endif
				if (dig == '9') { /* possible if i == 1 */
 round_9_up:
					*s++ = '9';
					goto roundoff;
					}
				*s++ = dig + 1;
				goto ret;
				}
#ifdef Honor_FLT_ROUNDS
 keep_dig:
#endif
			*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 (!b->x[0] && b->wds <= 1) {
#ifdef SET_INEXACT
				inexact = 0;
#endif
				goto ret;
				}
			if (i >= ilim)
				break;
			b = multadd(b, 10, 0);
			}

	/* Round off last digit */

#ifdef Honor_FLT_ROUNDS
	switch(rounding) {
	  case 0: goto trimzeros;
	  case 2: goto roundoff;
	  }
#endif
	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 {
 trimzeros:
		while(*--s == '0');
		s++;
		}
 ret:
	Bfree(S);
	if (mhi) {
		if (mlo && mlo != mhi)
			Bfree(mlo);
		Bfree(mhi);
		}
 ret1:
#ifdef SET_INEXACT
	if (inexact) {
		if (!oldinexact) {
			word0(d) = Exp_1 + (70 << Exp_shift);
			word1(d) = 0;
			dval(d) += 1.;
			}
		}
	else if (!oldinexact)
		clear_inexact();
#endif
	Bfree(b);
	*s = 0;
	*decpt = k + 1;
	if (rve)
		*rve = s;
	return s0;
	}