xref: /plan9/sys/src/ape/lib/ap/stdio/_dtoa.c (revision 219b2ee8daee37f4aad58d63f21287faa8e4ffdc)

#include "fconv.h"

static int quorem(Bigint *, Bigint *);

/* 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 darg, 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 ds;
	Dul d2, eps;
	char *s, *s0;
	static Bigint *result;
	static int result_k;
	Dul d;

	d.d = darg;
	if (result) {
		result->k = result_k;
		result->maxwds = 1 << result_k;
		Bfree(result);
		result = 0;
		}

	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;
		s =
#ifdef IEEE_Arith
			!word1(d) && !(word0(d) & 0xfffff) ? "Infinity" :
#endif
				"NaN";
		if (rve)
			*rve =
#ifdef IEEE_Arith
				s[3] ? s + 8 :
#endif
						s + 3;
		return s;
		}
#endif
#ifdef IBM
	d.d += 0; /* normalize */
#endif
	if (!d.d) {
		*decpt = 1;
		s = "0";
		if (rve)
			*rve = s + 1;
		return s;
		}

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

		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;
		d2.d = x;
		word0(d2) -= 31*Exp_msk1; /* adjust exponent */
		i -= (Bias + (P-1) - 1) + 1;
		denorm = 1;
		}
#endif
	ds = (d2.d-1.5)*0.289529654602168 + 0.1760912590558 + i*0.301029995663981;
	k = floor(ds);
	k_check = 1;
	if (k >= 0 && k <= Ten_pmax) {
		if (d.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;
		}
	j = sizeof(unsigned long);
	for(result_k = 0; sizeof(Bigint) - sizeof(unsigned long) + j <= i;
		j <<= 1) result_k++;
	result = Balloc(result_k);
	s = s0 = (char *)result;

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

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

		i = 0;
		d2.d = d.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.d /= bigtens[n_bigtens-1];
				ieps++;
				}
			for(; j; j >>= 1, i++)
				if (j & 1) {
					ieps++;
					ds *= bigtens[i];
					}
			d.d /= ds;
			}
		else if (j1 = -k) {
			d.d *= tens[j1 & 0xf];
			for(j = j1 >> 4; j; j >>= 1, i++)
				if (j & 1) {
					ieps++;
					d.d *= bigtens[i];
					}
			}
		if (k_check && d.d < 1. && ilim > 0) {
			if (ilim1 <= 0)
				goto fast_failed;
			ilim = ilim1;
			k--;
			d.d *= 10.;
			ieps++;
			}
		eps.d = ieps*d.d + 7.;
		word0(eps) -= (P-1)*Exp_msk1;
		if (ilim == 0) {
			S = mhi = 0;
			d.d -= 5.;
			if (d.d > eps.d)
				goto one_digit;
			if (d.d < -eps.d)
				goto no_digits;
			goto fast_failed;
			}
#ifndef No_leftright
		if (leftright) {
			/* Use Steele & White method of only
			 * generating digits needed.
			 */
			eps.d = 0.5/tens[ilim-1] - eps.d;
			for(i = 0;;) {
				L = floor(d.d);
				d.d -= L;
				*s++ = '0' + (int)L;
				if (d.d < eps.d)
					goto ret1;
				if (1. - d.d < eps.d)
					goto bump_up;
				if (++i >= ilim)
					break;
				eps.d *= 10.;
				d.d *= 10.;
				}
			}
		else {
#endif
			/* Generate ilim digits, then fix them up. */
			eps.d *= tens[ilim-1];
			for(i = 1;; i++, d.d *= 10.) {
				L = floor(d.d);
				d.d -= L;
				*s++ = '0' + (int)L;
				if (i == ilim) {
					if (d.d > 0.5 + eps.d)
						goto bump_up;
					else if (d.d < 0.5 - eps.d) {
						while(*--s == '0');
						s++;
						goto ret1;
						}
					break;
					}
				}
#ifndef No_leftright
			}
#endif
 fast_failed:
		s = s0;
		d.d = d2.d;
		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.d <= 5*ds)
				goto no_digits;
			goto one_digit;
			}
		for(i = 1;; i++) {
			L = floor(d.d / ds);
			d.d -= L*ds;
#ifdef Check_FLT_ROUNDS
			/* If FLT_ROUNDS == 2, L will usually be high by 1 */
			if (d.d < 0) {
				L--;
				d.d += ds;
				}
#endif
			*s++ = '0' + (int)L;
			if (i == ilim) {
				d.d += d.d;
				if (d.d > ds || d.d == ds && L & 1) {
 bump_up:
					while(*--s == '9')
						if (s == s0) {
							k++;
							*s = '0';
							break;
							}
					++*s++;
					}
				break;
				}
			d.d *= 10.;
			if (d.d == 0.)
				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
#ifdef IBM
				1 + 4*P - 3 - bbits + ((bbits + be - 1) & 3);
#else
				1 + P - bbits;
#endif
			}
		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. */

	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;
			}
		else
			spec_case = 0;
		}

	/* 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)
		i = 32 - i;
#else
	if (i = ((s5 ? 32 - hi0bits(S->x[S->wds-1]) : 1) + s2) & 0xf)
		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 > 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);
#ifndef ROUND_BIASED
			if (j1 == 0 && !mode && !(word1(d) & 1)) {
				if (dig == '9')
					goto round_9_up;
				if (j > 0)
					dig++;
				*s++ = dig;
				goto ret;
				}
#endif
			if (j < 0 || j == 0 && !mode
#ifndef ROUND_BIASED
							&& !(word1(d) & 1)
#endif
					) {
				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;
	}

 static int
quorem(Bigint *b, Bigint *S)
{
	int n;
	long borrow, y;
	unsigned long carry, q, ys;
	unsigned long *bx, *bxe, *sx, *sxe;
#ifdef Pack_32
	long z;
	unsigned long si, zs;
#endif

	n = S->wds;
#ifdef DEBUG
	/*debug*/ if (b->wds > n)
	/*debug*/	Bug("oversize b in quorem");
#endif
	if (b->wds < n)
		return 0;
	sx = S->x;
	sxe = sx + --n;
	bx = b->x;
	bxe = bx + n;
	q = *bxe / (*sxe + 1);	/* ensure q <= true quotient */
#ifdef DEBUG
	/*debug*/ if (q > 9)
	/*debug*/	Bug("oversized quotient in quorem");
#endif
	if (q) {
		borrow = 0;
		carry = 0;
		do {
#ifdef Pack_32
			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);
#else
			ys = *sx++ * q + carry;
			carry = ys >> 16;
			y = *bx - (ys & 0xffff) + borrow;
			borrow = y >> 16;
			Sign_Extend(borrow, y);
			*bx++ = y & 0xffff;
#endif
			}
			while(sx <= sxe);
		if (!*bxe) {
			bx = b->x;
			while(--bxe > bx && !*bxe)
				--n;
			b->wds = n;
			}
		}
	if (cmp(b, S) >= 0) {
		q++;
		borrow = 0;
		carry = 0;
		bx = b->x;
		sx = S->x;
		do {
#ifdef Pack_32
			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);
#else
			ys = *sx++ + carry;
			carry = ys >> 16;
			y = *bx - (ys & 0xffff) + borrow;
			borrow = y >> 16;
			Sign_Extend(borrow, y);
			*bx++ = y & 0xffff;
#endif
			}
			while(sx <= sxe);
		bx = b->x;
		bxe = bx + n;
		if (!*bxe) {
			while(--bxe > bx && !*bxe)
				--n;
			b->wds = n;
			}
		}
	return q;
	}