1 /* $NetBSD: gdtoa.c,v 1.6 2012/03/13 21:13:33 christos Exp $ */
3 /****************************************************************
5 The author of this software is David M. Gay.
7 Copyright (C) 1998, 1999 by Lucent Technologies
10 Permission to use, copy, modify, and distribute this software and
11 its documentation for any purpose and without fee is hereby
12 granted, provided that the above copyright notice appear in all
13 copies and that both that the copyright notice and this
14 permission notice and warranty disclaimer appear in supporting
15 documentation, and that the name of Lucent or any of its entities
16 not be used in advertising or publicity pertaining to
17 distribution of the software without specific, written prior
20 LUCENT DISCLAIMS ALL WARRANTIES WITH REGARD TO THIS SOFTWARE,
21 INCLUDING ALL IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS.
22 IN NO EVENT SHALL LUCENT OR ANY OF ITS ENTITIES BE LIABLE FOR ANY
23 SPECIAL, INDIRECT OR CONSEQUENTIAL DAMAGES OR ANY DAMAGES
24 WHATSOEVER RESULTING FROM LOSS OF USE, DATA OR PROFITS, WHETHER
25 IN AN ACTION OF CONTRACT, NEGLIGENCE OR OTHER TORTIOUS ACTION,
26 ARISING OUT OF OR IN CONNECTION WITH THE USE OR PERFORMANCE OF
29 ****************************************************************/
31 /* Please send bug reports to David M. Gay (dmg at acm dot org,
32 * with " at " changed at "@" and " dot " changed to "."). */
38 bitstob(bits
, nbits
, bbits
) ULong
*bits
; int nbits
; int *bbits
;
40 bitstob(ULong
*bits
, int nbits
, int *bbits
)
60 be
= bits
+ (((unsigned int)nbits
- 1) >> kshift
);
63 *x
++ = *bits
& ALL_ON
;
65 *x
++ = (*bits
>> 16) & ALL_ON
;
67 } while(++bits
<= be
);
68 ptrdiff_t td
= x
- x0
;
69 _DIAGASSERT(__type_fit(int, td
));
78 *bbits
= i
*ULbits
+ 32 - hi0bits(b
->x
[i
]);
83 /* dtoa for IEEE arithmetic (dmg): convert double to ASCII string.
85 * Inspired by "How to Print Floating-Point Numbers Accurately" by
86 * Guy L. Steele, Jr. and Jon L. White [Proc. ACM SIGPLAN '90, pp. 112-126].
89 * 1. Rather than iterating, we use a simple numeric overestimate
90 * to determine k = floor(log10(d)). We scale relevant
91 * quantities using O(log2(k)) rather than O(k) multiplications.
92 * 2. For some modes > 2 (corresponding to ecvt and fcvt), we don't
93 * try to generate digits strictly left to right. Instead, we
94 * compute with fewer bits and propagate the carry if necessary
95 * when rounding the final digit up. This is often faster.
96 * 3. Under the assumption that input will be rounded nearest,
97 * mode 0 renders 1e23 as 1e23 rather than 9.999999999999999e22.
98 * That is, we allow equality in stopping tests when the
99 * round-nearest rule will give the same floating-point value
100 * as would satisfaction of the stopping test with strict
102 * 4. We remove common factors of powers of 2 from relevant
104 * 5. When converting floating-point integers less than 1e16,
105 * we use floating-point arithmetic rather than resorting
106 * to multiple-precision integers.
107 * 6. When asked to produce fewer than 15 digits, we first try
108 * to get by with floating-point arithmetic; we resort to
109 * multiple-precision integer arithmetic only if we cannot
110 * guarantee that the floating-point calculation has given
111 * the correctly rounded result. For k requested digits and
112 * "uniformly" distributed input, the probability is
113 * something like 10^(k-15) that we must resort to the Long
120 (fpi
, be
, bits
, kindp
, mode
, ndigits
, decpt
, rve
)
121 FPI
*fpi
; int be
; ULong
*bits
;
122 int *kindp
, mode
, ndigits
, *decpt
; char **rve
;
124 (FPI
*fpi
, int be
, ULong
*bits
, int *kindp
, int mode
, int ndigits
, int *decpt
, char **rve
)
127 /* Arguments ndigits and decpt are similar to the second and third
128 arguments of ecvt and fcvt; trailing zeros are suppressed from
129 the returned string. If not null, *rve is set to point
130 to the end of the return value. If d is +-Infinity or NaN,
131 then *decpt is set to 9999.
132 be = exponent: value = (integer represented by bits) * (2 to the power of be).
135 0 ==> shortest string that yields d when read in
136 and rounded to nearest.
137 1 ==> like 0, but with Steele & White stopping rule;
138 e.g. with IEEE P754 arithmetic , mode 0 gives
139 1e23 whereas mode 1 gives 9.999999999999999e22.
140 2 ==> max(1,ndigits) significant digits. This gives a
141 return value similar to that of ecvt, except
142 that trailing zeros are suppressed.
143 3 ==> through ndigits past the decimal point. This
144 gives a return value similar to that from fcvt,
145 except that trailing zeros are suppressed, and
146 ndigits can be negative.
147 4-9 should give the same return values as 2-3, i.e.,
148 4 <= mode <= 9 ==> same return as mode
149 2 + (mode & 1). These modes are mainly for
150 debugging; often they run slower but sometimes
151 faster than modes 2-3.
152 4,5,8,9 ==> left-to-right digit generation.
153 6-9 ==> don't try fast floating-point estimate
156 Values of mode other than 0-9 are treated as mode 0.
158 Sufficient space is allocated to the return value
159 to hold the suppressed trailing zeros.
162 int bbits
, b2
, b5
, be0
, dig
, i
, ieps
, ilim
= 0, ilim0
, ilim1
= 0, inex
;
163 int j
, jj1
, k
, k0
, k_check
, kind
, leftright
, m2
, m5
, nbits
;
164 int rdir
, s2
, s5
, spec_case
, try_quick
;
166 Bigint
*b
, *b1
, *delta
, *mlo
, *mhi
, *mhi1
, *S
;
171 #ifndef MULTIPLE_THREADS
173 freedtoa(dtoa_result
);
178 if (*kindp
& STRTOG_NoMemory
)
180 kind
= *kindp
&= ~STRTOG_Inexact
;
181 switch(kind
& STRTOG_Retmask
) {
185 case STRTOG_Denormal
:
187 case STRTOG_Infinite
:
189 return nrv_alloc("Infinity", rve
, 8);
192 return nrv_alloc("NaN", rve
, 3);
196 b
= bitstob(bits
, nbits
= fpi
->nbits
, &bbits
);
200 if ( (i
= trailz(b
)) !=0) {
209 return nrv_alloc("0", rve
, 1);
212 dval(&d
) = b2d(b
, &i
);
214 word0(&d
) &= Frac_mask1
;
217 if ( (j
= 11 - hi0bits(word0(&d
) & Frac_mask
)) !=0)
221 /* log(x) ~=~ log(1.5) + (x-1.5)/1.5
222 * log10(x) = log(x) / log(10)
223 * ~=~ log(1.5)/log(10) + (x-1.5)/(1.5*log(10))
224 * log10(&d) = (i-Bias)*log(2)/log(10) + log10(d2)
226 * This suggests computing an approximation k to log10(&d) by
228 * k = (i - Bias)*0.301029995663981
229 * + ( (d2-1.5)*0.289529654602168 + 0.176091259055681 );
231 * We want k to be too large rather than too small.
232 * The error in the first-order Taylor series approximation
233 * is in our favor, so we just round up the constant enough
234 * to compensate for any error in the multiplication of
235 * (i - Bias) by 0.301029995663981; since |i - Bias| <= 1077,
236 * and 1077 * 0.30103 * 2^-52 ~=~ 7.2e-14,
237 * adding 1e-13 to the constant term more than suffices.
238 * Hence we adjust the constant term to 0.1760912590558.
239 * (We could get a more accurate k by invoking log10,
240 * but this is probably not worthwhile.)
246 ds
= (dval(&d
)-1.5)*0.289529654602168 + 0.1760912590558 + i
*0.301029995663981;
248 /* correct assumption about exponent range */
255 if (ds
< 0. && ds
!= k
)
256 k
--; /* want k = floor(ds) */
260 if ( (jj1
= j
& 3) !=0)
261 dval(&d
) *= 1 << jj1
;
262 word0(&d
) += j
<< Exp_shift
- 2 & Exp_mask
;
264 word0(&d
) += (be
+ bbits
- 1) << Exp_shift
;
266 if (k
>= 0 && k
<= Ten_pmax
) {
267 if (dval(&d
) < tens
[k
])
290 if (mode
< 0 || mode
> 9)
297 else if (i
>= -4 - Emin
|| i
< Emin
)
300 ilim
= ilim1
= -1; /* Values for cases 0 and 1; done here to */
301 /* silence erroneous "gcc -Wall" warning. */
305 i
= (int)(nbits
* .30103) + 3;
314 ilim
= ilim1
= i
= ndigits
;
326 s
= s0
= rv_alloc((size_t)i
);
330 if ( (rdir
= fpi
->rounding
- 1) !=0) {
333 if (kind
& STRTOG_Neg
)
337 /* Now rdir = 0 ==> round near, 1 ==> round up, 2 ==> round down. */
339 if (ilim
>= 0 && ilim
<= Quick_max
&& try_quick
&& !rdir
340 #ifndef IMPRECISE_INEXACT
345 /* Try to get by with floating-point arithmetic. */
350 if ( (j
= 11 - hi0bits(word0(&d
) & Frac_mask
)) !=0)
355 ieps
= 2; /* conservative */
358 j
= (unsigned int)k
>> 4;
360 /* prevent overflows */
362 dval(&d
) /= bigtens
[n_bigtens
-1];
365 for(; j
; j
/= 2, i
++)
373 if ( (jj1
= -k
) !=0) {
374 dval(&d
) *= tens
[jj1
& 0xf];
375 for(j
= jj1
>> 4; j
; j
>>= 1, i
++)
378 dval(&d
) *= bigtens
[i
];
382 if (k_check
&& dval(&d
) < 1. && ilim
> 0) {
390 dval(&eps
) = ieps
*dval(&d
) + 7.;
391 word0(&eps
) -= (P
-1)*Exp_msk1
;
395 if (dval(&d
) > dval(&eps
))
397 if (dval(&d
) < -dval(&eps
))
403 /* Use Steele & White method of only
404 * generating digits needed.
406 dval(&eps
) = ds
*0.5/tens
[ilim
-1] - dval(&eps
);
408 L
= (Long
)(dval(&d
)/ds
);
411 if (dval(&d
) < dval(&eps
)) {
413 inex
= STRTOG_Inexlo
;
416 if (ds
- dval(&d
) < dval(&eps
))
426 /* Generate ilim digits, then fix them up. */
427 dval(&eps
) *= tens
[ilim
-1];
428 for(i
= 1;; i
++, dval(&d
) *= 10.) {
429 if ( (L
= (Long
)(dval(&d
)/ds
)) !=0)
434 if (dval(&d
) > ds
+ dval(&eps
))
436 else if (dval(&d
) < ds
- dval(&eps
)) {
438 inex
= STRTOG_Inexlo
;
439 goto clear_trailing0
;
454 /* Do we have a "small" integer? */
456 if (be
>= 0 && k
<= Int_max
) {
459 if (ndigits
< 0 && ilim
<= 0) {
461 if (ilim
< 0 || dval(&d
) <= 5*ds
)
465 for(i
= 1;; i
++, dval(&d
) *= 10.) {
468 #ifdef Check_FLT_ROUNDS
469 /* If FLT_ROUNDS == 2, L will usually be high by 1 */
482 inex
= STRTOG_Inexlo
;
485 dval(&d
) += dval(&d
);
489 if (dval(&d
) > ds
|| (dval(&d
) == ds
&& L
& 1))
493 inex
= STRTOG_Inexhi
;
503 inex
= STRTOG_Inexlo
;
519 if (be
- i
++ < fpi
->emin
&& mode
!= 3 && mode
!= 5) {
521 i
= be
- fpi
->emin
+ 1;
522 if (mode
>= 2 && ilim
> 0 && ilim
< i
)
525 else if (mode
>= 2) {
535 if ((i
= ilim
) < 0) {
544 if (m2
> 0 && s2
> 0) {
545 i
= m2
< s2
? m2
: s2
;
553 mhi
= pow5mult(mhi
, m5
);
562 if ( (j
= b5
- m5
) !=0) {
583 /* Check for special case that d is a normalized power of 2. */
587 if (bbits
== 1 && be0
> fpi
->emin
+ 1) {
588 /* The special case */
595 /* Arrange for convenient computation of quotients:
596 * shift left if necessary so divisor has 4 leading 0 bits.
598 * Perhaps we should just compute leading 28 bits of S once
599 * and for all and pass them and a shift to quorem, so it
600 * can do shifts and ors to compute the numerator for q.
602 i
= ((s5
? hi0bits(S
->x
[S
->wds
-1]) : ULbits
- 1) - s2
- 4) & kmask
;
611 b
= multadd(b
, 10, 0); /* we botched the k estimate */
615 mhi
= multadd(mhi
, 10, 0);
622 if (ilim
<= 0 && mode
> 2) {
623 if (ilim
< 0 || cmp(b
,S
= multadd(S
,5,0)) <= 0) {
624 /* no digits, fcvt style */
627 inex
= STRTOG_Inexlo
;
631 inex
= STRTOG_Inexhi
;
638 mhi
= lshift(mhi
, m2
);
643 /* Compute mlo -- check for special case
644 * that d is a normalized power of 2.
649 mhi
= Balloc(mhi
->k
);
653 mhi
= lshift(mhi
, 1);
659 dig
= quorem(b
,S
) + '0';
660 /* Do we yet have the shortest decimal string
661 * that will round to d?
664 delta
= diff(S
, mhi
);
667 jj1
= delta
->sign
? 1 : cmp(b
, delta
);
670 if (jj1
== 0 && !mode
&& !(bits
[0] & 1) && !rdir
) {
674 if (b
->wds
> 1 || b
->x
[0])
675 inex
= STRTOG_Inexlo
;
679 inex
= STRTOG_Inexhi
;
685 if (j
< 0 || (j
== 0 && !mode
690 if (rdir
&& (b
->wds
> 1 || b
->x
[0])) {
692 inex
= STRTOG_Inexlo
;
695 while (cmp(S
,mhi
) > 0) {
697 mhi1
= multadd(mhi
, 10, 0);
703 b
= multadd(b
, 10, 0);
706 dig
= quorem(b
,S
) + '0';
710 inex
= STRTOG_Inexhi
;
721 if ((jj1
> 0 || (jj1
== 0 && dig
& 1))
725 inex
= STRTOG_Inexhi
;
727 if (b
->wds
> 1 || b
->x
[0])
728 inex
= STRTOG_Inexlo
;
733 if (jj1
> 0 && rdir
!= 2) {
734 if (dig
== '9') { /* possible if i == 1 */
737 inex
= STRTOG_Inexhi
;
740 inex
= STRTOG_Inexhi
;
747 b
= multadd(b
, 10, 0);
751 mlo
= mhi
= multadd(mhi
, 10, 0);
756 mlo
= multadd(mlo
, 10, 0);
759 mhi
= multadd(mhi
, 10, 0);
767 *s
++ = dig
= quorem(b
,S
) + '0';
770 b
= multadd(b
, 10, 0);
775 /* Round off last digit */
778 if (rdir
== 2 || (b
->wds
<= 1 && !b
->x
[0]))
789 if (j
> 0 || (j
== 0 && dig
& 1))
793 inex
= STRTOG_Inexhi
;
804 if (b
->wds
> 1 || b
->x
[0])
805 inex
= STRTOG_Inexlo
;
812 if (mlo
&& mlo
!= mhi
)