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759 lines
16 KiB
759 lines
16 KiB
/**************************************************************** |
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The author of this software is David M. Gay. |
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|
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Copyright (C) 1998, 1999 by Lucent Technologies |
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All Rights Reserved |
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|
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Permission to use, copy, modify, and distribute this software and |
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its documentation for any purpose and without fee is hereby |
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granted, provided that the above copyright notice appear in all |
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copies and that both that the copyright notice and this |
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permission notice and warranty disclaimer appear in supporting |
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documentation, and that the name of Lucent or any of its entities |
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not be used in advertising or publicity pertaining to |
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distribution of the software without specific, written prior |
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permission. |
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LUCENT DISCLAIMS ALL WARRANTIES WITH REGARD TO THIS SOFTWARE, |
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INCLUDING ALL IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS. |
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IN NO EVENT SHALL LUCENT OR ANY OF ITS ENTITIES BE LIABLE FOR ANY |
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SPECIAL, INDIRECT OR CONSEQUENTIAL DAMAGES OR ANY DAMAGES |
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WHATSOEVER RESULTING FROM LOSS OF USE, DATA OR PROFITS, WHETHER |
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IN AN ACTION OF CONTRACT, NEGLIGENCE OR OTHER TORTIOUS ACTION, |
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ARISING OUT OF OR IN CONNECTION WITH THE USE OR PERFORMANCE OF |
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THIS SOFTWARE. |
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|
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****************************************************************/ |
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/* Please send bug reports to |
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David M. Gay |
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Bell Laboratories, Room 2C-463 |
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600 Mountain Avenue |
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Murray Hill, NJ 07974-0636 |
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U.S.A. |
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dmg@bell-labs.com |
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*/ |
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|
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#include "gdtoaimp.h" |
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|
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/* dtoa for IEEE arithmetic (dmg): convert double to ASCII string. |
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* |
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* Inspired by "How to Print Floating-Point Numbers Accurately" by |
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* Guy L. Steele, Jr. and Jon L. White [Proc. ACM SIGPLAN '90, pp. 92-101]. |
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* |
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* Modifications: |
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* 1. Rather than iterating, we use a simple numeric overestimate |
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* to determine k = floor(log10(d)). We scale relevant |
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* quantities using O(log2(k)) rather than O(k) multiplications. |
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* 2. For some modes > 2 (corresponding to ecvt and fcvt), we don't |
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* try to generate digits strictly left to right. Instead, we |
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* compute with fewer bits and propagate the carry if necessary |
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* when rounding the final digit up. This is often faster. |
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* 3. Under the assumption that input will be rounded nearest, |
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* mode 0 renders 1e23 as 1e23 rather than 9.999999999999999e22. |
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* That is, we allow equality in stopping tests when the |
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* round-nearest rule will give the same floating-point value |
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* as would satisfaction of the stopping test with strict |
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* inequality. |
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* 4. We remove common factors of powers of 2 from relevant |
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* quantities. |
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* 5. When converting floating-point integers less than 1e16, |
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* we use floating-point arithmetic rather than resorting |
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* to multiple-precision integers. |
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* 6. When asked to produce fewer than 15 digits, we first try |
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* to get by with floating-point arithmetic; we resort to |
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* multiple-precision integer arithmetic only if we cannot |
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* guarantee that the floating-point calculation has given |
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* the correctly rounded result. For k requested digits and |
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* "uniformly" distributed input, the probability is |
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* something like 10^(k-15) that we must resort to the Long |
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* calculation. |
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*/ |
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|
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#ifdef Honor_FLT_ROUNDS |
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#define Rounding rounding |
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#undef Check_FLT_ROUNDS |
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#define Check_FLT_ROUNDS |
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#else |
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#define Rounding Flt_Rounds |
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#endif |
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|
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char * |
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dtoa |
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#ifdef KR_headers |
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(d, mode, ndigits, decpt, sign, rve) |
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double d; int mode, ndigits, *decpt, *sign; char **rve; |
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#else |
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(double d, int mode, int ndigits, int *decpt, int *sign, char **rve) |
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#endif |
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{ |
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/* Arguments ndigits, decpt, sign are similar to those |
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of ecvt and fcvt; trailing zeros are suppressed from |
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the returned string. If not null, *rve is set to point |
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to the end of the return value. If d is +-Infinity or NaN, |
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then *decpt is set to 9999. |
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|
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mode: |
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0 ==> shortest string that yields d when read in |
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and rounded to nearest. |
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1 ==> like 0, but with Steele & White stopping rule; |
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e.g. with IEEE P754 arithmetic , mode 0 gives |
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1e23 whereas mode 1 gives 9.999999999999999e22. |
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2 ==> max(1,ndigits) significant digits. This gives a |
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return value similar to that of ecvt, except |
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that trailing zeros are suppressed. |
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3 ==> through ndigits past the decimal point. This |
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gives a return value similar to that from fcvt, |
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except that trailing zeros are suppressed, and |
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ndigits can be negative. |
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4,5 ==> similar to 2 and 3, respectively, but (in |
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round-nearest mode) with the tests of mode 0 to |
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possibly return a shorter string that rounds to d. |
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With IEEE arithmetic and compilation with |
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-DHonor_FLT_ROUNDS, modes 4 and 5 behave the same |
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as modes 2 and 3 when FLT_ROUNDS != 1. |
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6-9 ==> Debugging modes similar to mode - 4: don't try |
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fast floating-point estimate (if applicable). |
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|
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Values of mode other than 0-9 are treated as mode 0. |
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|
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Sufficient space is allocated to the return value |
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to hold the suppressed trailing zeros. |
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*/ |
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int bbits, b2, b5, be, dig, i, ieps, ilim, ilim0, ilim1, |
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j, j1, k, k0, k_check, leftright, m2, m5, s2, s5, |
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spec_case, try_quick; |
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Long L; |
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#ifndef Sudden_Underflow |
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int denorm; |
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ULong x; |
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#endif |
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Bigint *b, *b1, *delta, *mlo, *mhi, *S; |
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double d2, ds, eps; |
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char *s, *s0; |
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#ifdef Honor_FLT_ROUNDS |
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int rounding; |
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#endif |
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#ifdef SET_INEXACT |
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int inexact, oldinexact; |
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#endif |
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|
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#ifndef MULTIPLE_THREADS |
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if (dtoa_result) { |
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freedtoa(dtoa_result); |
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dtoa_result = 0; |
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} |
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#endif |
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|
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if (word0(d) & Sign_bit) { |
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/* set sign for everything, including 0's and NaNs */ |
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*sign = 1; |
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word0(d) &= ~Sign_bit; /* clear sign bit */ |
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} |
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else |
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*sign = 0; |
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|
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#if defined(IEEE_Arith) + defined(VAX) |
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#ifdef IEEE_Arith |
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if ((word0(d) & Exp_mask) == Exp_mask) |
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#else |
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if (word0(d) == 0x8000) |
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#endif |
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{ |
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/* Infinity or NaN */ |
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*decpt = 9999; |
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#ifdef IEEE_Arith |
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if (!word1(d) && !(word0(d) & 0xfffff)) |
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return nrv_alloc("Infinity", rve, 8); |
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#endif |
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return nrv_alloc("NaN", rve, 3); |
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} |
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#endif |
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#ifdef IBM |
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dval(d) += 0; /* normalize */ |
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#endif |
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if (!dval(d)) { |
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*decpt = 1; |
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return nrv_alloc("0", rve, 1); |
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} |
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|
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#ifdef SET_INEXACT |
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try_quick = oldinexact = get_inexact(); |
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inexact = 1; |
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#endif |
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#ifdef Honor_FLT_ROUNDS |
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if ((rounding = Flt_Rounds) >= 2) { |
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if (*sign) |
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rounding = rounding == 2 ? 0 : 2; |
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else |
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if (rounding != 2) |
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rounding = 0; |
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} |
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#endif |
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|
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b = d2b(dval(d), &be, &bbits); |
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#ifdef Sudden_Underflow |
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i = (int)(word0(d) >> Exp_shift1 & (Exp_mask>>Exp_shift1)); |
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#else |
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if (( i = (int)(word0(d) >> Exp_shift1 & (Exp_mask>>Exp_shift1)) )!=0) { |
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#endif |
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dval(d2) = dval(d); |
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word0(d2) &= Frac_mask1; |
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word0(d2) |= Exp_11; |
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#ifdef IBM |
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if (( j = 11 - hi0bits(word0(d2) & Frac_mask) )!=0) |
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dval(d2) /= 1 << j; |
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#endif |
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|
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/* log(x) ~=~ log(1.5) + (x-1.5)/1.5 |
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* log10(x) = log(x) / log(10) |
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* ~=~ log(1.5)/log(10) + (x-1.5)/(1.5*log(10)) |
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* log10(d) = (i-Bias)*log(2)/log(10) + log10(d2) |
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* |
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* This suggests computing an approximation k to log10(d) by |
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* |
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* k = (i - Bias)*0.301029995663981 |
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* + ( (d2-1.5)*0.289529654602168 + 0.176091259055681 ); |
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* |
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* We want k to be too large rather than too small. |
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* The error in the first-order Taylor series approximation |
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* is in our favor, so we just round up the constant enough |
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* to compensate for any error in the multiplication of |
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* (i - Bias) by 0.301029995663981; since |i - Bias| <= 1077, |
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* and 1077 * 0.30103 * 2^-52 ~=~ 7.2e-14, |
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* adding 1e-13 to the constant term more than suffices. |
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* Hence we adjust the constant term to 0.1760912590558. |
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* (We could get a more accurate k by invoking log10, |
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* but this is probably not worthwhile.) |
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*/ |
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|
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i -= Bias; |
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#ifdef IBM |
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i <<= 2; |
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i += j; |
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#endif |
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#ifndef Sudden_Underflow |
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denorm = 0; |
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} |
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else { |
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/* d is denormalized */ |
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|
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i = bbits + be + (Bias + (P-1) - 1); |
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x = i > 32 ? word0(d) << 64 - i | word1(d) >> i - 32 |
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: word1(d) << 32 - i; |
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dval(d2) = x; |
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word0(d2) -= 31*Exp_msk1; /* adjust exponent */ |
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i -= (Bias + (P-1) - 1) + 1; |
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denorm = 1; |
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} |
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#endif |
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ds = (dval(d2)-1.5)*0.289529654602168 + 0.1760912590558 + i*0.301029995663981; |
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k = (int)ds; |
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if (ds < 0. && ds != k) |
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k--; /* want k = floor(ds) */ |
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k_check = 1; |
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if (k >= 0 && k <= Ten_pmax) { |
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if (dval(d) < tens[k]) |
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k--; |
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k_check = 0; |
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} |
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j = bbits - i - 1; |
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if (j >= 0) { |
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b2 = 0; |
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s2 = j; |
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} |
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else { |
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b2 = -j; |
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s2 = 0; |
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} |
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if (k >= 0) { |
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b5 = 0; |
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s5 = k; |
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s2 += k; |
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} |
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else { |
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b2 -= k; |
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b5 = -k; |
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s5 = 0; |
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} |
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if (mode < 0 || mode > 9) |
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mode = 0; |
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|
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#ifndef SET_INEXACT |
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#ifdef Check_FLT_ROUNDS |
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try_quick = Rounding == 1; |
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#else |
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try_quick = 1; |
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#endif |
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#endif /*SET_INEXACT*/ |
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|
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if (mode > 5) { |
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mode -= 4; |
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try_quick = 0; |
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} |
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leftright = 1; |
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switch(mode) { |
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case 0: |
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case 1: |
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ilim = ilim1 = -1; |
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i = 18; |
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ndigits = 0; |
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break; |
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case 2: |
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leftright = 0; |
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/* no break */ |
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case 4: |
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if (ndigits <= 0) |
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ndigits = 1; |
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ilim = ilim1 = i = ndigits; |
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break; |
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case 3: |
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leftright = 0; |
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/* no break */ |
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case 5: |
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i = ndigits + k + 1; |
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ilim = i; |
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ilim1 = i - 1; |
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if (i <= 0) |
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i = 1; |
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} |
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s = s0 = rv_alloc(i); |
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|
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#ifdef Honor_FLT_ROUNDS |
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if (mode > 1 && rounding != 1) |
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leftright = 0; |
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#endif |
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|
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if (ilim >= 0 && ilim <= Quick_max && try_quick) { |
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|
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/* Try to get by with floating-point arithmetic. */ |
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|
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i = 0; |
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dval(d2) = dval(d); |
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k0 = k; |
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ilim0 = ilim; |
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ieps = 2; /* conservative */ |
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if (k > 0) { |
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ds = tens[k&0xf]; |
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j = k >> 4; |
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if (j & Bletch) { |
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/* prevent overflows */ |
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j &= Bletch - 1; |
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dval(d) /= bigtens[n_bigtens-1]; |
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ieps++; |
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} |
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for(; j; j >>= 1, i++) |
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if (j & 1) { |
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ieps++; |
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ds *= bigtens[i]; |
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} |
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dval(d) /= ds; |
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} |
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else if (( j1 = -k )!=0) { |
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dval(d) *= tens[j1 & 0xf]; |
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for(j = j1 >> 4; j; j >>= 1, i++) |
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if (j & 1) { |
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ieps++; |
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dval(d) *= bigtens[i]; |
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} |
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} |
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if (k_check && dval(d) < 1. && ilim > 0) { |
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if (ilim1 <= 0) |
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goto fast_failed; |
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ilim = ilim1; |
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k--; |
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dval(d) *= 10.; |
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ieps++; |
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} |
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dval(eps) = ieps*dval(d) + 7.; |
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word0(eps) -= (P-1)*Exp_msk1; |
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if (ilim == 0) { |
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S = mhi = 0; |
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dval(d) -= 5.; |
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if (dval(d) > dval(eps)) |
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goto one_digit; |
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if (dval(d) < -dval(eps)) |
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goto no_digits; |
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goto fast_failed; |
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} |
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#ifndef No_leftright |
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if (leftright) { |
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/* Use Steele & White method of only |
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* generating digits needed. |
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*/ |
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dval(eps) = 0.5/tens[ilim-1] - dval(eps); |
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for(i = 0;;) { |
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L = dval(d); |
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dval(d) -= L; |
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*s++ = '0' + (int)L; |
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if (dval(d) < dval(eps)) |
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goto ret1; |
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if (1. - dval(d) < dval(eps)) |
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goto bump_up; |
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if (++i >= ilim) |
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break; |
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dval(eps) *= 10.; |
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dval(d) *= 10.; |
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} |
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} |
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else { |
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#endif |
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/* Generate ilim digits, then fix them up. */ |
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dval(eps) *= tens[ilim-1]; |
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for(i = 1;; i++, dval(d) *= 10.) { |
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L = (Long)(dval(d)); |
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if (!(dval(d) -= L)) |
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ilim = i; |
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*s++ = '0' + (int)L; |
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if (i == ilim) { |
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if (dval(d) > 0.5 + dval(eps)) |
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goto bump_up; |
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else if (dval(d) < 0.5 - dval(eps)) { |
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while(*--s == '0'); |
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s++; |
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goto ret1; |
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} |
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break; |
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} |
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} |
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#ifndef No_leftright |
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} |
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#endif |
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fast_failed: |
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s = s0; |
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dval(d) = dval(d2); |
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k = k0; |
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ilim = ilim0; |
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} |
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|
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/* Do we have a "small" integer? */ |
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|
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if (be >= 0 && k <= Int_max) { |
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/* Yes. */ |
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ds = tens[k]; |
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if (ndigits < 0 && ilim <= 0) { |
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S = mhi = 0; |
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if (ilim < 0 || dval(d) <= 5*ds) |
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goto no_digits; |
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goto one_digit; |
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} |
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for(i = 1;; i++, dval(d) *= 10.) { |
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L = (Long)(dval(d) / ds); |
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dval(d) -= L*ds; |
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#ifdef Check_FLT_ROUNDS |
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/* If FLT_ROUNDS == 2, L will usually be high by 1 */ |
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if (dval(d) < 0) { |
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L--; |
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dval(d) += ds; |
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} |
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#endif |
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*s++ = '0' + (int)L; |
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if (!dval(d)) { |
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#ifdef SET_INEXACT |
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inexact = 0; |
|
#endif |
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break; |
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} |
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if (i == ilim) { |
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#ifdef Honor_FLT_ROUNDS |
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if (mode > 1) |
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switch(rounding) { |
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case 0: goto ret1; |
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case 2: goto bump_up; |
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} |
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#endif |
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dval(d) += dval(d); |
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if (dval(d) > ds || dval(d) == ds && L & 1) { |
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bump_up: |
|
while(*--s == '9') |
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if (s == s0) { |
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k++; |
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*s = '0'; |
|
break; |
|
} |
|
++*s++; |
|
} |
|
break; |
|
} |
|
} |
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goto ret1; |
|
} |
|
|
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m2 = b2; |
|
m5 = b5; |
|
mhi = mlo = 0; |
|
if (leftright) { |
|
i = |
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#ifndef Sudden_Underflow |
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denorm ? be + (Bias + (P-1) - 1 + 1) : |
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#endif |
|
#ifdef IBM |
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1 + 4*P - 3 - bbits + ((bbits + be - 1) & 3); |
|
#else |
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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; |
|
} |
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if (( j = b5 - m5 )!=0) |
|
b = pow5mult(b, j); |
|
} |
|
else |
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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) |
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#ifdef Honor_FLT_ROUNDS |
|
&& rounding == 1 |
|
#endif |
|
) { |
|
if (!word1(d) && !(word0(d) & Bndry_mask) |
|
#ifndef Sudden_Underflow |
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&& 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. |
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*/ |
|
#ifdef Pack_32 |
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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) { |
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i -= 4; |
|
b2 += i; |
|
m2 += i; |
|
s2 += i; |
|
} |
|
else if (i < 4) { |
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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; |
|
}
|
|
|