38 *
39 * Modifications:
40 * 1. Rather than iterating, we use a simple numeric overestimate
41 * to determine k = floor(log10(d)). We scale relevant
42 * quantities using O(log2(k)) rather than O(k) multiplications.
43 * 2. For some modes > 2 (corresponding to ecvt and fcvt), we don't
44 * try to generate digits strictly left to right. Instead, we
45 * compute with fewer bits and propagate the carry if necessary
46 * when rounding the final digit up. This is often faster.
47 * 3. Under the assumption that input will be rounded nearest,
48 * mode 0 renders 1e23 as 1e23 rather than 9.999999999999999e22.
49 * That is, we allow equality in stopping tests when the
50 * round-nearest rule will give the same floating-point value
51 * as would satisfaction of the stopping test with strict
52 * inequality.
53 * 4. We remove common factors of powers of 2 from relevant
54 * quantities.
55 * 5. When converting floating-point integers less than 1e16,
56 * we use floating-point arithmetic rather than resorting
57 * to multiple-precision integers.
58 * 6. When asked to produce fewer than 15 digits, we first try
59 * to get by with floating-point arithmetic; we resort to
60 * multiple-precision integer arithmetic only if we cannot
61 * guarantee that the floating-point calculation has given
62 * the correctly rounded result. For k requested digits and
63 * "uniformly" distributed input, the probability is
64 * something like 10^(k-15) that we must resort to the Long
65 * calculation.
66 */
67
68#ifdef Honor_FLT_ROUNDS
69#define Rounding rounding
70#undef Check_FLT_ROUNDS
71#define Check_FLT_ROUNDS
72#else
73#define Rounding Flt_Rounds
74#endif
75
76 char *
77dtoa
78#ifdef KR_headers
79 (d, mode, ndigits, decpt, sign, rve)
80 double d; int mode, ndigits, *decpt, *sign; char **rve;
81#else
82 (double d, int mode, int ndigits, int *decpt, int *sign, char **rve)
83#endif
84{
85 /* Arguments ndigits, decpt, sign are similar to those
86 of ecvt and fcvt; trailing zeros are suppressed from
87 the returned string. If not null, *rve is set to point
88 to the end of the return value. If d is +-Infinity or NaN,
89 then *decpt is set to 9999.
90
91 mode:
92 0 ==> shortest string that yields d when read in
93 and rounded to nearest.
94 1 ==> like 0, but with Steele & White stopping rule;
95 e.g. with IEEE P754 arithmetic , mode 0 gives
96 1e23 whereas mode 1 gives 9.999999999999999e22.
97 2 ==> max(1,ndigits) significant digits. This gives a
98 return value similar to that of ecvt, except
99 that trailing zeros are suppressed.
100 3 ==> through ndigits past the decimal point. This
101 gives a return value similar to that from fcvt,
102 except that trailing zeros are suppressed, and
103 ndigits can be negative.
104 4,5 ==> similar to 2 and 3, respectively, but (in
105 round-nearest mode) with the tests of mode 0 to
106 possibly return a shorter string that rounds to d.
107 With IEEE arithmetic and compilation with
108 -DHonor_FLT_ROUNDS, modes 4 and 5 behave the same
109 as modes 2 and 3 when FLT_ROUNDS != 1.
110 6-9 ==> Debugging modes similar to mode - 4: don't try
111 fast floating-point estimate (if applicable).
112
113 Values of mode other than 0-9 are treated as mode 0.
114
115 Sufficient space is allocated to the return value
116 to hold the suppressed trailing zeros.
117 */
118
119 int bbits, b2, b5, be, dig, i, ieps, ilim, ilim0, ilim1,
120 j, j1, k, k0, k_check, leftright, m2, m5, s2, s5,
121 spec_case, try_quick;
122 Long L;
123#ifndef Sudden_Underflow
124 int denorm;
125 ULong x;
126#endif
127 Bigint *b, *b1, *delta, *mlo, *mhi, *S;
128 double d2, ds, eps;
129 char *s, *s0;
130#ifdef Honor_FLT_ROUNDS
131 int rounding;
132#endif
133#ifdef SET_INEXACT
134 int inexact, oldinexact;
135#endif
136
137#ifndef MULTIPLE_THREADS
138 if (dtoa_result) {
139 freedtoa(dtoa_result);
140 dtoa_result = 0;
141 }
142#endif
143
144 if (word0(d) & Sign_bit) {
145 /* set sign for everything, including 0's and NaNs */
146 *sign = 1;
147 word0(d) &= ~Sign_bit; /* clear sign bit */
148 }
149 else
150 *sign = 0;
151
152#if defined(IEEE_Arith) + defined(VAX)
153#ifdef IEEE_Arith
154 if ((word0(d) & Exp_mask) == Exp_mask)
155#else
156 if (word0(d) == 0x8000)
157#endif
158 {
159 /* Infinity or NaN */
160 *decpt = 9999;
161#ifdef IEEE_Arith
162 if (!word1(d) && !(word0(d) & 0xfffff))
163 return nrv_alloc("Infinity", rve, 8);
164#endif
165 return nrv_alloc("NaN", rve, 3);
166 }
167#endif
168#ifdef IBM
169 dval(d) += 0; /* normalize */
170#endif
171 if (!dval(d)) {
172 *decpt = 1;
173 return nrv_alloc("0", rve, 1);
174 }
175
176#ifdef SET_INEXACT
177 try_quick = oldinexact = get_inexact();
178 inexact = 1;
179#endif
180#ifdef Honor_FLT_ROUNDS
181 if ((rounding = Flt_Rounds) >= 2) {
182 if (*sign)
183 rounding = rounding == 2 ? 0 : 2;
184 else
185 if (rounding != 2)
186 rounding = 0;
187 }
188#endif
189
190 b = d2b(dval(d), &be, &bbits);
191#ifdef Sudden_Underflow
192 i = (int)(word0(d) >> Exp_shift1 & (Exp_mask>>Exp_shift1));
193#else
194 if (( i = (int)(word0(d) >> Exp_shift1 & (Exp_mask>>Exp_shift1)) )!=0) {
195#endif
196 dval(d2) = dval(d);
197 word0(d2) &= Frac_mask1;
198 word0(d2) |= Exp_11;
199#ifdef IBM
200 if (( j = 11 - hi0bits(word0(d2) & Frac_mask) )!=0)
201 dval(d2) /= 1 << j;
202#endif
203
204 /* log(x) ~=~ log(1.5) + (x-1.5)/1.5
205 * log10(x) = log(x) / log(10)
206 * ~=~ log(1.5)/log(10) + (x-1.5)/(1.5*log(10))
207 * log10(d) = (i-Bias)*log(2)/log(10) + log10(d2)
208 *
209 * This suggests computing an approximation k to log10(d) by
210 *
211 * k = (i - Bias)*0.301029995663981
212 * + ( (d2-1.5)*0.289529654602168 + 0.176091259055681 );
213 *
214 * We want k to be too large rather than too small.
215 * The error in the first-order Taylor series approximation
216 * is in our favor, so we just round up the constant enough
217 * to compensate for any error in the multiplication of
218 * (i - Bias) by 0.301029995663981; since |i - Bias| <= 1077,
219 * and 1077 * 0.30103 * 2^-52 ~=~ 7.2e-14,
220 * adding 1e-13 to the constant term more than suffices.
221 * Hence we adjust the constant term to 0.1760912590558.
222 * (We could get a more accurate k by invoking log10,
223 * but this is probably not worthwhile.)
224 */
225
226 i -= Bias;
227#ifdef IBM
228 i <<= 2;
229 i += j;
230#endif
231#ifndef Sudden_Underflow
232 denorm = 0;
233 }
234 else {
235 /* d is denormalized */
236
237 i = bbits + be + (Bias + (P-1) - 1);
238 x = i > 32 ? word0(d) << 64 - i | word1(d) >> i - 32
239 : word1(d) << 32 - i;
240 dval(d2) = x;
241 word0(d2) -= 31*Exp_msk1; /* adjust exponent */
242 i -= (Bias + (P-1) - 1) + 1;
243 denorm = 1;
244 }
245#endif
246 ds = (dval(d2)-1.5)*0.289529654602168 + 0.1760912590558 + i*0.301029995663981;
247 k = (int)ds;
248 if (ds < 0. && ds != k)
249 k--; /* want k = floor(ds) */
250 k_check = 1;
251 if (k >= 0 && k <= Ten_pmax) {
252 if (dval(d) < tens[k])
253 k--;
254 k_check = 0;
255 }
256 j = bbits - i - 1;
257 if (j >= 0) {
258 b2 = 0;
259 s2 = j;
260 }
261 else {
262 b2 = -j;
263 s2 = 0;
264 }
265 if (k >= 0) {
266 b5 = 0;
267 s5 = k;
268 s2 += k;
269 }
270 else {
271 b2 -= k;
272 b5 = -k;
273 s5 = 0;
274 }
275 if (mode < 0 || mode > 9)
276 mode = 0;
277
278#ifndef SET_INEXACT
279#ifdef Check_FLT_ROUNDS
280 try_quick = Rounding == 1;
281#else
282 try_quick = 1;
283#endif
284#endif /*SET_INEXACT*/
285
286 if (mode > 5) {
287 mode -= 4;
288 try_quick = 0;
289 }
290 leftright = 1;
291 switch(mode) {
292 case 0:
293 case 1:
294 ilim = ilim1 = -1;
295 i = 18;
296 ndigits = 0;
297 break;
298 case 2:
299 leftright = 0;
300 /* no break */
301 case 4:
302 if (ndigits <= 0)
303 ndigits = 1;
304 ilim = ilim1 = i = ndigits;
305 break;
306 case 3:
307 leftright = 0;
308 /* no break */
309 case 5:
310 i = ndigits + k + 1;
311 ilim = i;
312 ilim1 = i - 1;
313 if (i <= 0)
314 i = 1;
315 }
316 s = s0 = rv_alloc(i);
317
318#ifdef Honor_FLT_ROUNDS
319 if (mode > 1 && rounding != 1)
320 leftright = 0;
321#endif
322
323 if (ilim >= 0 && ilim <= Quick_max && try_quick) {
324
325 /* Try to get by with floating-point arithmetic. */
326
327 i = 0;
328 dval(d2) = dval(d);
329 k0 = k;
330 ilim0 = ilim;
331 ieps = 2; /* conservative */
332 if (k > 0) {
333 ds = tens[k&0xf];
334 j = k >> 4;
335 if (j & Bletch) {
336 /* prevent overflows */
337 j &= Bletch - 1;
338 dval(d) /= bigtens[n_bigtens-1];
339 ieps++;
340 }
341 for(; j; j >>= 1, i++)
342 if (j & 1) {
343 ieps++;
344 ds *= bigtens[i];
345 }
346 dval(d) /= ds;
347 }
348 else if (( j1 = -k )!=0) {
349 dval(d) *= tens[j1 & 0xf];
350 for(j = j1 >> 4; j; j >>= 1, i++)
351 if (j & 1) {
352 ieps++;
353 dval(d) *= bigtens[i];
354 }
355 }
356 if (k_check && dval(d) < 1. && ilim > 0) {
357 if (ilim1 <= 0)
358 goto fast_failed;
359 ilim = ilim1;
360 k--;
361 dval(d) *= 10.;
362 ieps++;
363 }
364 dval(eps) = ieps*dval(d) + 7.;
365 word0(eps) -= (P-1)*Exp_msk1;
366 if (ilim == 0) {
367 S = mhi = 0;
368 dval(d) -= 5.;
369 if (dval(d) > dval(eps))
370 goto one_digit;
371 if (dval(d) < -dval(eps))
372 goto no_digits;
373 goto fast_failed;
374 }
375#ifndef No_leftright
376 if (leftright) {
377 /* Use Steele & White method of only
378 * generating digits needed.
379 */
380 dval(eps) = 0.5/tens[ilim-1] - dval(eps);
381 for(i = 0;;) {
382 L = dval(d);
383 dval(d) -= L;
384 *s++ = '0' + (int)L;
385 if (dval(d) < dval(eps))
386 goto ret1;
387 if (1. - dval(d) < dval(eps))
388 goto bump_up;
389 if (++i >= ilim)
390 break;
391 dval(eps) *= 10.;
392 dval(d) *= 10.;
393 }
394 }
395 else {
396#endif
397 /* Generate ilim digits, then fix them up. */
398 dval(eps) *= tens[ilim-1];
399 for(i = 1;; i++, dval(d) *= 10.) {
400 L = (Long)(dval(d));
401 if (!(dval(d) -= L))
402 ilim = i;
403 *s++ = '0' + (int)L;
404 if (i == ilim) {
405 if (dval(d) > 0.5 + dval(eps))
406 goto bump_up;
407 else if (dval(d) < 0.5 - dval(eps)) {
408 while(*--s == '0');
409 s++;
410 goto ret1;
411 }
412 break;
413 }
414 }
415#ifndef No_leftright
416 }
417#endif
418 fast_failed:
419 s = s0;
420 dval(d) = dval(d2);
421 k = k0;
422 ilim = ilim0;
423 }
424
425 /* Do we have a "small" integer? */
426
427 if (be >= 0 && k <= Int_max) {
428 /* Yes. */
429 ds = tens[k];
430 if (ndigits < 0 && ilim <= 0) {
431 S = mhi = 0;
432 if (ilim < 0 || dval(d) <= 5*ds)
433 goto no_digits;
434 goto one_digit;
435 }
436 for(i = 1;; i++, dval(d) *= 10.) {
437 L = (Long)(dval(d) / ds);
438 dval(d) -= L*ds;
439#ifdef Check_FLT_ROUNDS
440 /* If FLT_ROUNDS == 2, L will usually be high by 1 */
441 if (dval(d) < 0) {
442 L--;
443 dval(d) += ds;
444 }
445#endif
446 *s++ = '0' + (int)L;
447 if (!dval(d)) {
448#ifdef SET_INEXACT
449 inexact = 0;
450#endif
451 break;
452 }
453 if (i == ilim) {
454#ifdef Honor_FLT_ROUNDS
455 if (mode > 1)
456 switch(rounding) {
457 case 0: goto ret1;
458 case 2: goto bump_up;
459 }
460#endif
461 dval(d) += dval(d);
462 if (dval(d) > ds || dval(d) == ds && L & 1) {
463 bump_up:
464 while(*--s == '9')
465 if (s == s0) {
466 k++;
467 *s = '0';
468 break;
469 }
470 ++*s++;
471 }
472 break;
473 }
474 }
475 goto ret1;
476 }
477
478 m2 = b2;
479 m5 = b5;
480 mhi = mlo = 0;
481 if (leftright) {
482 i =
483#ifndef Sudden_Underflow
484 denorm ? be + (Bias + (P-1) - 1 + 1) :
485#endif
486#ifdef IBM
487 1 + 4*P - 3 - bbits + ((bbits + be - 1) & 3);
488#else
489 1 + P - bbits;
490#endif
491 b2 += i;
492 s2 += i;
493 mhi = i2b(1);
494 }
495 if (m2 > 0 && s2 > 0) {
496 i = m2 < s2 ? m2 : s2;
497 b2 -= i;
498 m2 -= i;
499 s2 -= i;
500 }
501 if (b5 > 0) {
502 if (leftright) {
503 if (m5 > 0) {
504 mhi = pow5mult(mhi, m5);
505 b1 = mult(mhi, b);
506 Bfree(b);
507 b = b1;
508 }
509 if (( j = b5 - m5 )!=0)
510 b = pow5mult(b, j);
511 }
512 else
513 b = pow5mult(b, b5);
514 }
515 S = i2b(1);
516 if (s5 > 0)
517 S = pow5mult(S, s5);
518
519 /* Check for special case that d is a normalized power of 2. */
520
521 spec_case = 0;
522 if ((mode < 2 || leftright)
523#ifdef Honor_FLT_ROUNDS
524 && rounding == 1
525#endif
526 ) {
527 if (!word1(d) && !(word0(d) & Bndry_mask)
528#ifndef Sudden_Underflow
529 && word0(d) & (Exp_mask & ~Exp_msk1)
530#endif
531 ) {
532 /* The special case */
533 b2 += Log2P;
534 s2 += Log2P;
535 spec_case = 1;
536 }
537 }
538
539 /* Arrange for convenient computation of quotients:
540 * shift left if necessary so divisor has 4 leading 0 bits.
541 *
542 * Perhaps we should just compute leading 28 bits of S once
543 * and for all and pass them and a shift to quorem, so it
544 * can do shifts and ors to compute the numerator for q.
545 */
546#ifdef Pack_32
547 if (( i = ((s5 ? 32 - hi0bits(S->x[S->wds-1]) : 1) + s2) & 0x1f )!=0)
548 i = 32 - i;
549#else
550 if (( i = ((s5 ? 32 - hi0bits(S->x[S->wds-1]) : 1) + s2) & 0xf )!=0)
551 i = 16 - i;
552#endif
553 if (i > 4) {
554 i -= 4;
555 b2 += i;
556 m2 += i;
557 s2 += i;
558 }
559 else if (i < 4) {
560 i += 28;
561 b2 += i;
562 m2 += i;
563 s2 += i;
564 }
565 if (b2 > 0)
566 b = lshift(b, b2);
567 if (s2 > 0)
568 S = lshift(S, s2);
569 if (k_check) {
570 if (cmp(b,S) < 0) {
571 k--;
572 b = multadd(b, 10, 0); /* we botched the k estimate */
573 if (leftright)
574 mhi = multadd(mhi, 10, 0);
575 ilim = ilim1;
576 }
577 }
578 if (ilim <= 0 && (mode == 3 || mode == 5)) {
579 if (ilim < 0 || cmp(b,S = multadd(S,5,0)) <= 0) {
580 /* no digits, fcvt style */
581 no_digits:
582 k = -1 - ndigits;
583 goto ret;
584 }
585 one_digit:
586 *s++ = '1';
587 k++;
588 goto ret;
589 }
590 if (leftright) {
591 if (m2 > 0)
592 mhi = lshift(mhi, m2);
593
594 /* Compute mlo -- check for special case
595 * that d is a normalized power of 2.
596 */
597
598 mlo = mhi;
599 if (spec_case) {
600 mhi = Balloc(mhi->k);
601 Bcopy(mhi, mlo);
602 mhi = lshift(mhi, Log2P);
603 }
604
605 for(i = 1;;i++) {
606 dig = quorem(b,S) + '0';
607 /* Do we yet have the shortest decimal string
608 * that will round to d?
609 */
610 j = cmp(b, mlo);
611 delta = diff(S, mhi);
612 j1 = delta->sign ? 1 : cmp(b, delta);
613 Bfree(delta);
614#ifndef ROUND_BIASED
615 if (j1 == 0 && mode != 1 && !(word1(d) & 1)
616#ifdef Honor_FLT_ROUNDS
617 && rounding >= 1
618#endif
619 ) {
620 if (dig == '9')
621 goto round_9_up;
622 if (j > 0)
623 dig++;
624#ifdef SET_INEXACT
625 else if (!b->x[0] && b->wds <= 1)
626 inexact = 0;
627#endif
628 *s++ = dig;
629 goto ret;
630 }
631#endif
632 if (j < 0 || j == 0 && mode != 1
633#ifndef ROUND_BIASED
634 && !(word1(d) & 1)
635#endif
636 ) {
637 if (!b->x[0] && b->wds <= 1) {
638#ifdef SET_INEXACT
639 inexact = 0;
640#endif
641 goto accept_dig;
642 }
643#ifdef Honor_FLT_ROUNDS
644 if (mode > 1)
645 switch(rounding) {
646 case 0: goto accept_dig;
647 case 2: goto keep_dig;
648 }
649#endif /*Honor_FLT_ROUNDS*/
650 if (j1 > 0) {
651 b = lshift(b, 1);
652 j1 = cmp(b, S);
653 if ((j1 > 0 || j1 == 0 && dig & 1)
654 && dig++ == '9')
655 goto round_9_up;
656 }
657 accept_dig:
658 *s++ = dig;
659 goto ret;
660 }
661 if (j1 > 0) {
662#ifdef Honor_FLT_ROUNDS
663 if (!rounding)
664 goto accept_dig;
665#endif
666 if (dig == '9') { /* possible if i == 1 */
667 round_9_up:
668 *s++ = '9';
669 goto roundoff;
670 }
671 *s++ = dig + 1;
672 goto ret;
673 }
674#ifdef Honor_FLT_ROUNDS
675 keep_dig:
676#endif
677 *s++ = dig;
678 if (i == ilim)
679 break;
680 b = multadd(b, 10, 0);
681 if (mlo == mhi)
682 mlo = mhi = multadd(mhi, 10, 0);
683 else {
684 mlo = multadd(mlo, 10, 0);
685 mhi = multadd(mhi, 10, 0);
686 }
687 }
688 }
689 else
690 for(i = 1;; i++) {
691 *s++ = dig = quorem(b,S) + '0';
692 if (!b->x[0] && b->wds <= 1) {
693#ifdef SET_INEXACT
694 inexact = 0;
695#endif
696 goto ret;
697 }
698 if (i >= ilim)
699 break;
700 b = multadd(b, 10, 0);
701 }
702
703 /* Round off last digit */
704
705#ifdef Honor_FLT_ROUNDS
706 switch(rounding) {
707 case 0: goto trimzeros;
708 case 2: goto roundoff;
709 }
710#endif
711 b = lshift(b, 1);
712 j = cmp(b, S);
713 if (j > 0 || j == 0 && dig & 1) {
714 roundoff:
715 while(*--s == '9')
716 if (s == s0) {
717 k++;
718 *s++ = '1';
719 goto ret;
720 }
721 ++*s++;
722 }
723 else {
724 trimzeros:
725 while(*--s == '0');
726 s++;
727 }
728 ret:
729 Bfree(S);
730 if (mhi) {
731 if (mlo && mlo != mhi)
732 Bfree(mlo);
733 Bfree(mhi);
734 }
735 ret1:
736#ifdef SET_INEXACT
737 if (inexact) {
738 if (!oldinexact) {
739 word0(d) = Exp_1 + (70 << Exp_shift);
740 word1(d) = 0;
741 dval(d) += 1.;
742 }
743 }
744 else if (!oldinexact)
745 clear_inexact();
746#endif
747 Bfree(b);
748 *s = 0;
749 *decpt = k + 1;
750 if (rve)
751 *rve = s;
752 return s0;
753 }
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