source: cpp/PrintFloat/Dragon4.cpp @ 1182

Last change on this file since 1182 was 834, checked in by Maciej Komosinski, 5 years ago

Renamed Math.h to MathDragon4.h because the <math.h> name clash is unfortunate (and caused problems with C++Builder and filename case-insensitive Windows)

File size: 40.9 KB
Line 
1/******************************************************************************
2  Copyright (c) 2014 Ryan Juckett
3  http://www.ryanjuckett.com/
4 
5  This software is provided 'as-is', without any express or implied
6  warranty. In no event will the authors be held liable for any damages
7  arising from the use of this software.
8 
9  Permission is granted to anyone to use this software for any purpose,
10  including commercial applications, and to alter it and redistribute it
11  freely, subject to the following restrictions:
12 
13  1. The origin of this software must not be misrepresented; you must not
14     claim that you wrote the original software. If you use this software
15     in a product, an acknowledgment in the product documentation would be
16     appreciated but is not required.
17 
18  2. Altered source versions must be plainly marked as such, and must not be
19     misrepresented as being the original software.
20 
21  3. This notice may not be removed or altered from any source
22     distribution.
23******************************************************************************/
24
25#include "Dragon4.h"
26#include "MathDragon4.h"
27#include <math.h>
28
29//******************************************************************************
30// Maximum number of 32 bit blocks needed in high precision arithmetic
31// to print out 64 bit IEEE floating point values.
32//******************************************************************************
33const tU32 c_BigInt_MaxBlocks = 35;
34 
35//******************************************************************************
36// This structure stores a high precision unsigned integer. It uses a buffer
37// of 32 bit integer blocks along with a length. The lowest bits of the integer
38// are stored at the start of the buffer and the length is set to the minimum
39// value that contains the integer. Thus, there are never any zero blocks at the
40// end of the buffer.
41//******************************************************************************
42struct tBigInt
43{
44    // Copy integer
45    tBigInt & operator=(const tBigInt &rhs)
46    {
47        tU32 length = rhs.m_length;
48        tU32 * pLhsCur = m_blocks;
49        for (const tU32 *pRhsCur = rhs.m_blocks, *pRhsEnd = pRhsCur + length;
50             pRhsCur != pRhsEnd;
51             ++pLhsCur, ++pRhsCur)
52        {
53            *pLhsCur = *pRhsCur;
54        }
55        m_length = length;
56        return *this;
57    }
58 
59    // Data accessors
60    tU32 GetLength() const        { return m_length; }
61    tU32 GetBlock(tU32 idx) const { return m_blocks[idx]; }
62 
63    // Zero helper functions
64    void    SetZero()       { m_length = 0; }
65    tB      IsZero() const  { return m_length == 0; }
66 
67    // Basic type accessors
68    void SetU64(tU64 val)
69    {
70        if (val > 0xFFFFFFFF)
71        {
72            m_blocks[0] = val & 0xFFFFFFFF;
73            m_blocks[1] = (val >> 32) & 0xFFFFFFFF;
74            m_length    = 2;
75        }
76        else if (val != 0)
77        {
78            m_blocks[0] = val & 0xFFFFFFFF;
79            m_length    = 1;
80        }
81        else
82        {
83            m_length = 0;
84        }
85    }
86 
87    void SetU32(tU32 val)
88    {
89        if (val != 0)
90        {
91            m_blocks[0] = val;
92            m_length    = (val != 0);
93        }
94        else
95        {
96            m_length = 0;
97        }
98    }
99 
100    tU32 GetU32() const { return (m_length == 0) ? 0 : m_blocks[0]; }
101 
102    // Member data
103    tU32 m_length;
104    tU32 m_blocks[c_BigInt_MaxBlocks];
105};
106
107//******************************************************************************       
108// Returns 0 if (lhs = rhs), negative if (lhs < rhs), positive if (lhs > rhs)
109//******************************************************************************       
110static tS32 BigInt_Compare(const tBigInt & lhs, const tBigInt & rhs)
111{
112    // A bigger length implies a bigger number.
113    tS32 lengthDiff = lhs.m_length - rhs.m_length;
114    if (lengthDiff != 0)
115        return lengthDiff;
116     
117    // Compare blocks one by one from high to low.
118    for (tS32 i = lhs.m_length - 1; i >= 0; --i)
119    {
120        if (lhs.m_blocks[i] == rhs.m_blocks[i])
121            continue;
122        else if (lhs.m_blocks[i] > rhs.m_blocks[i])
123            return 1;
124        else
125            return -1;
126    }
127 
128    // no blocks differed
129    return 0;
130}
131
132//******************************************************************************
133// result = lhs + rhs
134//******************************************************************************
135static void BigInt_Add(tBigInt * pResult, const tBigInt & lhs, const tBigInt & rhs)
136{
137    // determine which operand has the smaller length
138    const tBigInt * pLarge;
139    const tBigInt * pSmall;
140    if (lhs.m_length < rhs.m_length)
141    {
142        pSmall = &lhs;
143        pLarge = &rhs;
144    }
145    else
146    {
147        pSmall = &rhs;
148        pLarge = &lhs;
149    }
150 
151    const tU32 largeLen = pLarge->m_length;
152    const tU32 smallLen = pSmall->m_length;
153 
154    // The output will be at least as long as the largest input
155    pResult->m_length = largeLen;
156 
157    // Add each block and add carry the overflow to the next block
158    tU64 carry = 0;
159    const tU32 * pLargeCur  = pLarge->m_blocks;
160    const tU32 * pLargeEnd  = pLargeCur + largeLen;
161    const tU32 * pSmallCur  = pSmall->m_blocks;
162    const tU32 * pSmallEnd  = pSmallCur + smallLen;
163    tU32 *       pResultCur = pResult->m_blocks;
164    while (pSmallCur != pSmallEnd)
165    {
166        tU64 sum = carry + (tU64)(*pLargeCur) + (tU64)(*pSmallCur);
167        carry = sum >> 32;
168        (*pResultCur) = sum & 0xFFFFFFFF;
169        ++pLargeCur;
170        ++pSmallCur;
171        ++pResultCur;
172    }
173     
174    // Add the carry to any blocks that only exist in the large operand
175    while (pLargeCur != pLargeEnd)
176    {
177        tU64 sum = carry + (tU64)(*pLargeCur);
178        carry = sum >> 32;
179        (*pResultCur) = sum & 0xFFFFFFFF;
180        ++pLargeCur;
181        ++pResultCur;
182    }
183     
184    // If there's still a carry, append a new block
185    if (carry != 0)
186    {
187        RJ_ASSERT(carry == 1);
188        RJ_ASSERT((tU32)(pResultCur - pResult->m_blocks) == largeLen && (largeLen < c_BigInt_MaxBlocks));
189        *pResultCur = 1;
190        pResult->m_length = largeLen + 1;
191    }
192    else
193    {
194        pResult->m_length = largeLen;
195    }
196}
197
198//******************************************************************************
199// result = lhs * rhs
200//******************************************************************************
201static void BigInt_Multiply(tBigInt * pResult, const tBigInt &lhs, const tBigInt &rhs)
202{
203    RJ_ASSERT( pResult != &lhs && pResult != &rhs );
204     
205    // determine which operand has the smaller length
206    const tBigInt * pLarge;
207    const tBigInt * pSmall;
208    if (lhs.m_length < rhs.m_length)
209    {
210        pSmall = &lhs;
211        pLarge = &rhs;
212    }
213    else
214    {
215        pSmall = &rhs;
216        pLarge = &lhs;
217    }
218 
219    // set the maximum possible result length
220    tU32 maxResultLen = pLarge->m_length + pSmall->m_length;     
221    RJ_ASSERT( maxResultLen <= c_BigInt_MaxBlocks );
222 
223    // clear the result data
224    for(tU32 * pCur = pResult->m_blocks, *pEnd = pCur + maxResultLen; pCur != pEnd; ++pCur)
225        *pCur = 0;
226 
227    // perform standard long multiplication
228    const tU32 *pLargeBeg = pLarge->m_blocks;
229    const tU32 *pLargeEnd = pLargeBeg + pLarge->m_length;
230 
231    // for each small block
232    tU32 *pResultStart = pResult->m_blocks;
233    for(const tU32 *pSmallCur = pSmall->m_blocks, *pSmallEnd = pSmallCur + pSmall->m_length;
234        pSmallCur != pSmallEnd;
235        ++pSmallCur, ++pResultStart)
236    {
237        // if non-zero, multiply against all the large blocks and add into the result
238        const tU32 multiplier = *pSmallCur;
239        if (multiplier != 0)
240        {
241            const tU32 *pLargeCur = pLargeBeg;
242            tU32 *pResultCur = pResultStart;
243            tU64 carry = 0;
244            do
245            {
246                tU64 product = (*pResultCur) + (*pLargeCur)*(tU64)multiplier + carry;
247                carry = product >> 32;
248                *pResultCur = product & 0xFFFFFFFF;
249                ++pLargeCur;
250                ++pResultCur;
251            } while(pLargeCur != pLargeEnd);
252 
253            RJ_ASSERT(pResultCur < pResult->m_blocks + maxResultLen);
254            *pResultCur = (tU32)(carry & 0xFFFFFFFF);
255        }
256    }
257 
258    // check if the terminating block has no set bits
259    if (maxResultLen > 0 && pResult->m_blocks[maxResultLen - 1] == 0)
260        pResult->m_length = maxResultLen-1;
261    else
262        pResult->m_length = maxResultLen;
263}
264 
265//******************************************************************************
266// result = lhs * rhs
267//******************************************************************************
268static void BigInt_Multiply(tBigInt * pResult, const tBigInt & lhs, tU32 rhs)
269{
270    // perform long multiplication
271    tU32 carry = 0;         
272    tU32 *pResultCur = pResult->m_blocks;
273    const tU32 *pLhsCur = lhs.m_blocks;
274    const tU32 *pLhsEnd = lhs.m_blocks + lhs.m_length;
275    for ( ; pLhsCur != pLhsEnd; ++pLhsCur, ++pResultCur )
276    {
277        tU64 product = (tU64)(*pLhsCur) * rhs + carry;
278        *pResultCur = (tU32)(product & 0xFFFFFFFF);
279        carry = product >> 32;
280    }
281 
282    // if there is a remaining carry, grow the array
283    if (carry != 0)
284    {
285        // grow the array
286        RJ_ASSERT(lhs.m_length + 1 <= c_BigInt_MaxBlocks);
287        *pResultCur = (tU32)carry;
288        pResult->m_length = lhs.m_length + 1;
289    }
290    else
291    {
292        pResult->m_length = lhs.m_length;
293    }
294}
295 
296//******************************************************************************
297// result = in * 2
298//******************************************************************************
299static void BigInt_Multiply2(tBigInt * pResult, const tBigInt &in)
300{
301    // shift all the blocks by one
302    tU32 carry = 0;
303     
304    tU32 *pResultCur = pResult->m_blocks;
305    const tU32 *pLhsCur = in.m_blocks;
306    const tU32 *pLhsEnd = in.m_blocks + in.m_length;
307    for ( ; pLhsCur != pLhsEnd; ++pLhsCur, ++pResultCur )
308    {
309        tU32 cur = *pLhsCur;
310        *pResultCur = (cur << 1) | carry;
311        carry = cur >> 31;
312    }
313 
314    if (carry != 0)
315    {
316        // grow the array
317        RJ_ASSERT(in.m_length + 1 <= c_BigInt_MaxBlocks);
318        *pResultCur = carry;
319        pResult->m_length = in.m_length + 1;
320    }
321    else
322    {
323        pResult->m_length = in.m_length;
324    }
325}
326 
327//******************************************************************************
328// result = result * 2
329//******************************************************************************
330static void BigInt_Multiply2(tBigInt * pResult)
331{
332    // shift all the blocks by one
333    tU32 carry = 0;
334     
335    tU32 *pCur = pResult->m_blocks;
336    tU32 *pEnd = pResult->m_blocks + pResult->m_length;
337    for ( ; pCur != pEnd; ++pCur )
338    {
339        tU32 cur = *pCur;
340        *pCur = (cur << 1) | carry;
341        carry = cur >> 31;
342    }
343 
344    if (carry != 0)
345    {
346        // grow the array
347        RJ_ASSERT(pResult->m_length + 1 <= c_BigInt_MaxBlocks);
348        *pCur = carry;
349        ++pResult->m_length;
350    }
351}
352 
353//******************************************************************************
354// result = result * 10
355//******************************************************************************
356static void BigInt_Multiply10(tBigInt * pResult)
357{
358    // multiply all the blocks
359    tU64 carry = 0;
360     
361    tU32 *pCur = pResult->m_blocks;
362    tU32 *pEnd = pResult->m_blocks + pResult->m_length;
363    for ( ; pCur != pEnd; ++pCur )
364    {
365        tU64 product = (tU64)(*pCur) * 10ull + carry;
366        (*pCur) = (tU32)(product & 0xFFFFFFFF);
367        carry = product >> 32;
368    }
369 
370    if (carry != 0)
371    {
372        // grow the array
373        RJ_ASSERT(pResult->m_length + 1 <= c_BigInt_MaxBlocks);
374        *pCur = (tU32)carry;
375        ++pResult->m_length;
376    }
377}
378
379//******************************************************************************
380//******************************************************************************
381static tU32 g_PowerOf10_U32[] =
382{
383    1,          // 10 ^ 0
384    10,         // 10 ^ 1
385    100,        // 10 ^ 2
386    1000,       // 10 ^ 3
387    10000,      // 10 ^ 4
388    100000,     // 10 ^ 5
389    1000000,    // 10 ^ 6
390    10000000,   // 10 ^ 7
391};
392 
393//******************************************************************************
394// Note: This has a lot of wasted space in the big integer structures of the
395//       early table entries. It wouldn't be terribly hard to make the multiply
396//       function work on integer pointers with an array length instead of
397//       the tBigInt struct which would allow us to store a minimal amount of
398//       data here.
399//******************************************************************************
400static tBigInt g_PowerOf10_Big[] =
401{
402    // 10 ^ 8
403    { 1, { 100000000 } },
404    // 10 ^ 16
405    { 2, { 0x6fc10000, 0x002386f2 } },
406    // 10 ^ 32
407    { 4, { 0x00000000, 0x85acef81, 0x2d6d415b, 0x000004ee, } },
408    // 10 ^ 64
409    { 7, { 0x00000000, 0x00000000, 0xbf6a1f01, 0x6e38ed64, 0xdaa797ed, 0xe93ff9f4, 0x00184f03, } },
410    // 10 ^ 128
411    { 14, { 0x00000000, 0x00000000, 0x00000000, 0x00000000, 0x2e953e01, 0x03df9909, 0x0f1538fd,
412            0x2374e42f, 0xd3cff5ec, 0xc404dc08, 0xbccdb0da, 0xa6337f19, 0xe91f2603, 0x0000024e, } },
413    // 10 ^ 256
414    { 27, { 0x00000000, 0x00000000, 0x00000000, 0x00000000, 0x00000000, 0x00000000, 0x00000000,
415            0x00000000, 0x982e7c01, 0xbed3875b, 0xd8d99f72, 0x12152f87, 0x6bde50c6, 0xcf4a6e70,
416            0xd595d80f, 0x26b2716e, 0xadc666b0, 0x1d153624, 0x3c42d35a, 0x63ff540e, 0xcc5573c0,
417            0x65f9ef17, 0x55bc28f2, 0x80dcc7f7, 0xf46eeddc, 0x5fdcefce, 0x000553f7, } }
418};
419 
420//******************************************************************************
421// result = 10^exponent
422//******************************************************************************
423static void BigInt_Pow10(tBigInt * pResult, tU32 exponent)
424{
425    // make sure the exponent is within the bounds of the lookup table data
426    RJ_ASSERT(exponent < 512);
427                 
428    // create two temporary values to reduce large integer copy operations
429    tBigInt temp1;
430    tBigInt temp2;
431    tBigInt *pCurTemp = &temp1;
432    tBigInt *pNextTemp = &temp2;
433 
434    // initialize the result by looking up a 32-bit power of 10 corresponding to the first 3 bits
435    tU32 smallExponent = exponent & 0x7;
436    pCurTemp->SetU32(g_PowerOf10_U32[smallExponent]);
437     
438    // remove the low bits that we used for the 32-bit lookup table
439    exponent >>= 3;
440    tU32 tableIdx = 0;
441 
442    // while there are remaining bits in the exponent to be processed
443    while (exponent != 0)
444    {
445        // if the current bit is set, multiply it with the corresponding power of 10
446        if(exponent & 1)
447        {
448            // multiply into the next temporary
449            BigInt_Multiply( pNextTemp, *pCurTemp, g_PowerOf10_Big[tableIdx] );
450             
451            // swap to the next temporary
452            tBigInt * pSwap = pCurTemp;
453            pCurTemp = pNextTemp;
454            pNextTemp = pSwap;
455        }
456 
457        // advance to the next bit
458        ++tableIdx;
459        exponent >>= 1;
460    }
461 
462    // output the result
463    *pResult = *pCurTemp;
464}
465 
466//******************************************************************************
467// result = in * 10^exponent
468//******************************************************************************
469static void BigInt_MultiplyPow10(tBigInt * pResult, const tBigInt & in, tU32 exponent)
470{
471    // make sure the exponent is within the bounds of the lookup table data
472    RJ_ASSERT(exponent < 512);
473                 
474    // create two temporary values to reduce large integer copy operations
475    tBigInt temp1;
476    tBigInt temp2;
477    tBigInt *pCurTemp = &temp1;
478    tBigInt *pNextTemp = &temp2;
479 
480    // initialize the result by looking up a 32-bit power of 10 corresponding to the first 3 bits
481    tU32 smallExponent = exponent & 0x7;
482    if (smallExponent != 0)
483    {
484        BigInt_Multiply( pCurTemp, in, g_PowerOf10_U32[smallExponent] );
485    }
486    else
487    {
488        *pCurTemp = in;
489    }
490     
491    // remove the low bits that we used for the 32-bit lookup table
492    exponent >>= 3;
493    tU32 tableIdx = 0;
494 
495    // while there are remaining bits in the exponent to be processed
496    while (exponent != 0)
497    {
498        // if the current bit is set, multiply it with the corresponding power of 10
499        if(exponent & 1)
500        {                   
501            // multiply into the next temporary
502            BigInt_Multiply( pNextTemp, *pCurTemp, g_PowerOf10_Big[tableIdx] );
503             
504            // swap to the next temporary
505            tBigInt * pSwap = pCurTemp;
506            pCurTemp = pNextTemp;
507            pNextTemp = pSwap;
508        }
509         
510        // advance to the next bit
511        ++tableIdx;
512        exponent >>= 1;
513    }
514 
515    // output the result
516    *pResult = *pCurTemp;
517}
518 
519//******************************************************************************   
520// result = 2^exponent
521//******************************************************************************       
522static inline void BigInt_Pow2(tBigInt * pResult, tU32 exponent)
523{
524    tU32 blockIdx = exponent / 32;
525    RJ_ASSERT( blockIdx < c_BigInt_MaxBlocks );
526 
527    for ( tU32 i = 0; i <= blockIdx; ++i)
528        pResult->m_blocks[i] = 0;
529 
530    pResult->m_length = blockIdx + 1;
531     
532    tU32 bitIdx = (exponent % 32);
533    pResult->m_blocks[blockIdx] |= (1 << bitIdx);
534}
535
536//******************************************************************************
537// This function will divide two large numbers under the assumption that the
538// result is within the range [0,10) and the input numbers have been shifted
539// to satisfy:
540// - The highest block of the divisor is greater than or equal to 8 such that
541//   there is enough precision to make an accurate first guess at the quotient.
542// - The highest block of the divisor is less than the maximum value on an
543//   unsigned 32-bit integer such that we can safely increment without overflow.
544// - The dividend does not contain more blocks than the divisor such that we
545//   can estimate the quotient by dividing the equivalently placed high blocks.
546//
547// quotient  = floor(dividend / divisor)
548// remainder = dividend - quotient*divisor
549//
550// pDividend is updated to be the remainder and the quotient is returned.
551//******************************************************************************
552static tU32 BigInt_DivideWithRemainder_MaxQuotient9(tBigInt * pDividend, const tBigInt & divisor)
553{
554    // Check that the divisor has been correctly shifted into range and that it is not
555    // smaller than the dividend in length.
556    RJ_ASSERT(  !divisor.IsZero() &&
557                divisor.m_blocks[divisor.m_length-1] >= 8 &&
558                divisor.m_blocks[divisor.m_length-1] < 0xFFFFFFFF &&
559                pDividend->m_length <= divisor.m_length );
560 
561    // If the dividend is smaller than the divisor, the quotient is zero and the divisor is already
562    // the remainder.
563    tU32 length = divisor.m_length;
564    if (pDividend->m_length < divisor.m_length)
565        return 0;
566 
567    const tU32 * pFinalDivisorBlock  = divisor.m_blocks + length - 1;           
568    tU32 *       pFinalDividendBlock = pDividend->m_blocks + length - 1;
569 
570    // Compute an estimated quotient based on the high block value. This will either match the actual quotient or
571    // undershoot by one.
572    tU32  quotient = *pFinalDividendBlock / (*pFinalDivisorBlock + 1);
573    RJ_ASSERT(quotient <= 9);
574 
575    // Divide out the estimated quotient
576    if (quotient != 0)
577    {
578        // dividend = dividend - divisor*quotient
579        const tU32 *pDivisorCur = divisor.m_blocks;
580        tU32 *pDividendCur      = pDividend->m_blocks;
581 
582        tU64 borrow = 0;
583        tU64 carry = 0;
584        do
585        {
586            tU64 product = (tU64)*pDivisorCur * (tU64)quotient + carry;
587            carry = product >> 32;
588 
589            tU64 difference = (tU64)*pDividendCur - (product & 0xFFFFFFFF) - borrow;
590            borrow = (difference >> 32) & 1;
591             
592            *pDividendCur = difference & 0xFFFFFFFF;
593 
594            ++pDivisorCur;
595            ++pDividendCur;
596        } while(pDivisorCur <= pFinalDivisorBlock);
597 
598        // remove all leading zero blocks from dividend
599        while (length > 0 && pDividend->m_blocks[length - 1] == 0)
600            --length;
601         
602        pDividend->m_length = length;
603    }
604 
605    // If the dividend is still larger than the divisor, we overshot our estimate quotient. To correct,
606    // we increment the quotient and subtract one more divisor from the dividend.
607    if ( BigInt_Compare(*pDividend, divisor) >= 0 )
608    {
609        ++quotient;
610 
611        // dividend = dividend - divisor
612        const tU32 *pDivisorCur = divisor.m_blocks;
613        tU32 *pDividendCur      = pDividend->m_blocks;
614 
615        tU64 borrow = 0;
616        do
617        {
618            tU64 difference = (tU64)*pDividendCur - (tU64)*pDivisorCur - borrow;
619            borrow = (difference >> 32) & 1;
620 
621            *pDividendCur = difference & 0xFFFFFFFF;
622 
623            ++pDivisorCur;
624            ++pDividendCur;
625        } while(pDivisorCur <= pFinalDivisorBlock);
626 
627        // remove all leading zero blocks from dividend
628        while (length > 0 && pDividend->m_blocks[length - 1] == 0)
629            --length;
630         
631        pDividend->m_length = length;
632    }
633 
634    return quotient;
635}
636
637//******************************************************************************
638// result = result << shift
639//******************************************************************************
640static void BigInt_ShiftLeft(tBigInt * pResult, tU32 shift)
641{
642    RJ_ASSERT( shift != 0 );
643 
644    tU32 shiftBlocks = shift / 32;
645    tU32 shiftBits = shift % 32;
646     
647    // process blocks high to low so that we can safely process in place
648    const tU32 *    pInBlocks   = pResult->m_blocks;
649    tS32            inLength    = pResult->m_length;
650    RJ_ASSERT( inLength + shiftBlocks <= c_BigInt_MaxBlocks );
651 
652    // check if the shift is block aligned
653    if (shiftBits == 0)
654    {
655        // copy blocks from high to low
656        for (tU32 * pInCur = pResult->m_blocks + inLength - 1, *pOutCur = pInCur + shiftBlocks;
657             pInCur >= pInBlocks;
658             --pInCur, --pOutCur)
659        {
660            *pOutCur = *pInCur;
661        }
662 
663        // zero the remaining low blocks
664        for ( tU32 i = 0; i < shiftBlocks; ++i)
665            pResult->m_blocks[i] = 0;
666 
667        pResult->m_length += shiftBlocks;
668    }
669    // else we need to shift partial blocks
670    else
671    {
672        tS32 inBlockIdx  = inLength - 1;
673        tU32 outBlockIdx = inLength + shiftBlocks;
674     
675        // set the length to hold the shifted blocks
676        RJ_ASSERT( outBlockIdx < c_BigInt_MaxBlocks );
677        pResult->m_length = outBlockIdx + 1;
678     
679        // output the initial blocks
680        const tU32 lowBitsShift = (32 - shiftBits);
681        tU32 highBits = 0;
682        tU32 block = pResult->m_blocks[inBlockIdx];             
683        tU32 lowBits = block >> lowBitsShift;
684        while ( inBlockIdx > 0 )
685        {
686            pResult->m_blocks[outBlockIdx] = highBits | lowBits;
687            highBits = block << shiftBits;
688 
689            --inBlockIdx;
690            --outBlockIdx;
691 
692            block = pResult->m_blocks[inBlockIdx];
693            lowBits = block >> lowBitsShift;
694        }
695 
696        // output the final blocks
697        RJ_ASSERT( outBlockIdx == shiftBlocks + 1 );
698        pResult->m_blocks[outBlockIdx] = highBits | lowBits;
699        pResult->m_blocks[outBlockIdx-1] = block << shiftBits;
700     
701        // zero the remaining low blocks
702        for ( tU32 i = 0; i < shiftBlocks; ++i)
703            pResult->m_blocks[i] = 0;
704     
705        // check if the terminating block has no set bits
706        if (pResult->m_blocks[pResult->m_length - 1] == 0)
707            --pResult->m_length;
708    }
709}
710
711//******************************************************************************
712// This is an implementation the Dragon4 algorithm to convert a binary number
713// in floating point format to a decimal number in string format. The function
714// returns the number of digits written to the output buffer and the output is
715// not NUL terminated.
716//
717// The floating point input value is (mantissa * 2^exponent).
718//
719// See the following papers for more information on the algorithm:
720//  "How to Print Floating-Point Numbers Accurately"
721//    Steele and White
722//    http://kurtstephens.com/files/p372-steele.pdf
723//  "Printing Floating-Point Numbers Quickly and Accurately"
724//    Burger and Dybvig
725//    http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.72.4656&rep=rep1&type=pdf
726//******************************************************************************
727tU32 Dragon4
728(
729    const tU64          mantissa,           // value significand
730    const tS32          exponent,           // value exponent in base 2
731    const tU32          mantissaHighBitIdx, // index of the highest set mantissa bit
732    const tB            hasUnequalMargins,  // is the high margin twice as large as the low margin
733    const tCutoffMode   cutoffMode,         // how to determine output length
734    tU32                cutoffNumber,       // parameter to the selected cutoffMode
735    tC8 *               pOutBuffer,         // buffer to output into
736    tU32                bufferSize,         // maximum characters that can be printed to pOutBuffer
737    tS32 *              pOutExponent        // the base 10 exponent of the first digit
738)
739{
740    tC8 * pCurDigit = pOutBuffer;
741 
742    RJ_ASSERT( bufferSize > 0 );
743 
744    // if the mantissa is zero, the value is zero regardless of the exponent
745    if (mantissa == 0)
746    {
747        *pCurDigit = '0';
748        *pOutExponent = 0;
749        return 1;
750    }
751 
752    // compute the initial state in integral form such that
753    //  value     = scaledValue / scale
754    //  marginLow = scaledMarginLow / scale
755    tBigInt scale;              // positive scale applied to value and margin such that they can be
756                                //  represented as whole numbers
757    tBigInt scaledValue;        // scale * mantissa
758    tBigInt scaledMarginLow;    // scale * 0.5 * (distance between this floating-point number and its
759                                //  immediate lower value)
760 
761    // For normalized IEEE floating point values, each time the exponent is incremented the margin also
762    // doubles. That creates a subset of transition numbers where the high margin is twice the size of
763    // the low margin.
764    tBigInt * pScaledMarginHigh;
765    tBigInt optionalMarginHigh;
766 
767    if ( hasUnequalMargins )
768    {
769        // if we have no fractional component
770        if (exponent > 0)
771        {
772            // 1) Expand the input value by multiplying out the mantissa and exponent. This represents
773            //    the input value in its whole number representation.
774            // 2) Apply an additional scale of 2 such that later comparisons against the margin values
775            //    are simplified.
776            // 3) Set the margin value to the lowest mantissa bit's scale.
777             
778            // scaledValue      = 2 * 2 * mantissa*2^exponent
779            scaledValue.SetU64( 4 * mantissa );
780            BigInt_ShiftLeft( &scaledValue, exponent );
781             
782            // scale            = 2 * 2 * 1
783            scale.SetU32( 4 );
784             
785            // scaledMarginLow  = 2 * 2^(exponent-1)
786            BigInt_Pow2( &scaledMarginLow, exponent );
787 
788            // scaledMarginHigh = 2 * 2 * 2^(exponent-1)
789            BigInt_Pow2( &optionalMarginHigh, exponent + 1 );
790        }
791        // else we have a fractional exponent
792        else
793        {
794            // In order to track the mantissa data as an integer, we store it as is with a large scale
795             
796            // scaledValue      = 2 * 2 * mantissa
797            scaledValue.SetU64( 4 * mantissa );
798             
799            // scale            = 2 * 2 * 2^(-exponent)
800            BigInt_Pow2(&scale, -exponent + 2 );
801             
802            // scaledMarginLow  = 2 * 2^(-1)
803            scaledMarginLow.SetU32( 1 );
804             
805            // scaledMarginHigh = 2 * 2 * 2^(-1)
806            optionalMarginHigh.SetU32( 2 );
807        }
808 
809        // the high and low margins are different
810        pScaledMarginHigh = &optionalMarginHigh;
811    }
812    else
813    {
814        // if we have no fractional component
815        if (exponent > 0)
816        {
817            // 1) Expand the input value by multiplying out the mantissa and exponent. This represents
818            //    the input value in its whole number representation.
819            // 2) Apply an additional scale of 2 such that later comparisons against the margin values
820            //    are simplified.
821            // 3) Set the margin value to the lowest mantissa bit's scale.
822             
823            // scaledValue     = 2 * mantissa*2^exponent
824            scaledValue.SetU64( 2 * mantissa );
825            BigInt_ShiftLeft( &scaledValue, exponent );
826             
827            // scale           = 2 * 1
828            scale.SetU32( 2 );
829 
830            // scaledMarginLow = 2 * 2^(exponent-1)
831            BigInt_Pow2( &scaledMarginLow, exponent );
832        }
833        // else we have a fractional exponent
834        else
835        {
836            // In order to track the mantissa data as an integer, we store it as is with a large scale
837 
838            // scaledValue     = 2 * mantissa
839            scaledValue.SetU64( 2 * mantissa );
840             
841            // scale           = 2 * 2^(-exponent)
842            BigInt_Pow2(&scale, -exponent + 1 );
843 
844            // scaledMarginLow = 2 * 2^(-1)
845            scaledMarginLow.SetU32( 1 );
846        }
847     
848        // the high and low margins are equal
849        pScaledMarginHigh = &scaledMarginLow;
850    }
851 
852    // Compute an estimate for digitExponent that will be correct or undershoot by one.
853    // This optimization is based on the paper "Printing Floating-Point Numbers Quickly and Accurately"
854    // by Burger and Dybvig http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.72.4656&rep=rep1&type=pdf
855    // We perform an additional subtraction of 0.69 to increase the frequency of a failed estimate
856    // because that lets us take a faster branch in the code. 0.69 is chosen because 0.69 + log10(2) is
857    // less than one by a reasonable epsilon that will account for any floating point error.
858    //
859    // We want to set digitExponent to floor(log10(v)) + 1
860    //  v = mantissa*2^exponent
861    //  log2(v) = log2(mantissa) + exponent;
862    //  log10(v) = log2(v) * log10(2)
863    //  floor(log2(v)) = mantissaHighBitIdx + exponent;
864    //  log10(v) - log10(2) < (mantissaHighBitIdx + exponent) * log10(2) <= log10(v)
865    //  log10(v) < (mantissaHighBitIdx + exponent) * log10(2) + log10(2) <= log10(v) + log10(2)
866    //  floor( log10(v) ) < ceil( (mantissaHighBitIdx + exponent) * log10(2) ) <= floor( log10(v) ) + 1
867    const tF64 log10_2 = 0.30102999566398119521373889472449;
868        tS32 digitExponent = (tS32)(ceil(tF64((tS32)mantissaHighBitIdx + exponent) * log10_2 - 0.69));
869 
870    // if the digit exponent is smaller than the smallest desired digit for fractional cutoff,
871    // pull the digit back into legal range at which point we will round to the appropriate value.
872    // Note that while our value for digitExponent is still an estimate, this is safe because it
873    // only increases the number. This will either correct digitExponent to an accurate value or it
874    // will clamp it above the accurate value.
875    if (cutoffMode == CutoffMode_FractionLength && digitExponent <= -(tS32)cutoffNumber)
876    {
877        digitExponent = -(tS32)cutoffNumber + 1;
878    }
879 
880    // Divide value by 10^digitExponent.
881    if (digitExponent > 0)
882    {
883        // The exponent is positive creating a division so we multiply up the scale.
884        tBigInt temp;
885        BigInt_MultiplyPow10( &temp, scale, digitExponent );
886        scale = temp;
887    }
888    else if (digitExponent < 0)
889    {
890        // The exponent is negative creating a multiplication so we multiply up the scaledValue,
891        // scaledMarginLow and scaledMarginHigh.
892        tBigInt pow10;
893        BigInt_Pow10( &pow10, -digitExponent);
894 
895        tBigInt temp;
896        BigInt_Multiply( &temp, scaledValue, pow10);
897        scaledValue = temp;
898 
899        BigInt_Multiply( &temp, scaledMarginLow, pow10);
900        scaledMarginLow = temp;
901         
902        if (pScaledMarginHigh != &scaledMarginLow)
903            BigInt_Multiply2( pScaledMarginHigh, scaledMarginLow );
904    }
905 
906    // If (value >= 1), our estimate for digitExponent was too low
907    if( BigInt_Compare(scaledValue,scale) >= 0 )
908    {
909        // The exponent estimate was incorrect.
910        // Increment the exponent and don't perform the premultiply needed
911        // for the first loop iteration.
912        digitExponent = digitExponent + 1;
913    }
914    else
915    {
916        // The exponent estimate was correct.
917        // Multiply larger by the output base to prepare for the first loop iteration.
918        BigInt_Multiply10( &scaledValue );
919        BigInt_Multiply10( &scaledMarginLow );
920        if (pScaledMarginHigh != &scaledMarginLow)
921            BigInt_Multiply2( pScaledMarginHigh, scaledMarginLow );
922    }
923     
924    // Compute the cutoff exponent (the exponent of the final digit to print).
925    // Default to the maximum size of the output buffer.
926    tS32 cutoffExponent = digitExponent - bufferSize;
927    switch(cutoffMode)
928    {
929    // print digits until we pass the accuracy margin limits or buffer size
930    case CutoffMode_Unique:
931        break;
932 
933    // print cutoffNumber of digits or until we reach the buffer size
934    case CutoffMode_TotalLength:
935        {
936            tS32 desiredCutoffExponent = digitExponent - (tS32)cutoffNumber;
937            if (desiredCutoffExponent > cutoffExponent)
938                cutoffExponent = desiredCutoffExponent;
939        }
940        break;
941 
942    // print cutoffNumber digits past the decimal point or until we reach the buffer size
943    case CutoffMode_FractionLength:
944        {
945            tS32 desiredCutoffExponent = -(tS32)cutoffNumber;
946            if (desiredCutoffExponent > cutoffExponent)
947                cutoffExponent = desiredCutoffExponent;
948        }
949        break;
950    }
951 
952    // Output the exponent of the first digit we will print
953    *pOutExponent = digitExponent-1;
954 
955    // In preparation for calling BigInt_DivideWithRemainder_MaxQuotient9(),
956    // we need to scale up our values such that the highest block of the denominator
957    // is greater than or equal to 8. We also need to guarantee that the numerator
958    // can never have a length greater than the denominator after each loop iteration.
959    // This requires the highest block of the denominator to be less than or equal to
960    // 429496729 which is the highest number that can be multiplied by 10 without
961    // overflowing to a new block.
962    RJ_ASSERT( scale.GetLength() > 0 );
963    tU32 hiBlock = scale.GetBlock( scale.GetLength() - 1 );
964    if (hiBlock < 8 || hiBlock > 429496729)
965    {
966        // Perform a bit shift on all values to get the highest block of the denominator into
967        // the range [8,429496729]. We are more likely to make accurate quotient estimations
968        // in BigInt_DivideWithRemainder_MaxQuotient9() with higher denominator values so
969        // we shift the denominator to place the highest bit at index 27 of the highest block.
970        // This is safe because (2^28 - 1) = 268435455 which is less than 429496729. This means
971        // that all values with a highest bit at index 27 are within range.         
972                tU32 hiBlockLog2 = LogBase2(hiBlock);
973        RJ_ASSERT(hiBlockLog2 < 3 || hiBlockLog2 > 27);
974        tU32 shift = (32 + 27 - hiBlockLog2) % 32;
975 
976        BigInt_ShiftLeft( &scale, shift );
977        BigInt_ShiftLeft( &scaledValue, shift);
978        BigInt_ShiftLeft( &scaledMarginLow, shift);
979        if (pScaledMarginHigh != &scaledMarginLow)
980            BigInt_Multiply2( pScaledMarginHigh, scaledMarginLow );
981    }
982 
983    // These values are used to inspect why the print loop terminated so we can properly
984    // round the final digit.
985    tB      low;            // did the value get within marginLow distance from zero
986    tB      high;           // did the value get within marginHigh distance from one
987    tU32    outputDigit;    // current digit being output
988     
989    if (cutoffMode == CutoffMode_Unique)
990    {
991        // For the unique cutoff mode, we will try to print until we have reached a level of
992        // precision that uniquely distinguishes this value from its neighbors. If we run
993        // out of space in the output buffer, we terminate early.
994        for (;;)
995        {
996            digitExponent = digitExponent-1;
997 
998            // divide out the scale to extract the digit
999            outputDigit = BigInt_DivideWithRemainder_MaxQuotient9(&scaledValue, scale);
1000            RJ_ASSERT( outputDigit < 10 );
1001 
1002            // update the high end of the value
1003            tBigInt scaledValueHigh;
1004            BigInt_Add( &scaledValueHigh, scaledValue, *pScaledMarginHigh );
1005 
1006            // stop looping if we are far enough away from our neighboring values
1007            // or if we have reached the cutoff digit
1008            low = BigInt_Compare(scaledValue, scaledMarginLow) < 0;
1009            high = BigInt_Compare(scaledValueHigh, scale) > 0;
1010            if (low | high | (digitExponent == cutoffExponent))
1011                break;
1012             
1013            // store the output digit
1014            *pCurDigit = (tC8)('0' + outputDigit);
1015            ++pCurDigit;
1016 
1017            // multiply larger by the output base
1018            BigInt_Multiply10( &scaledValue );
1019            BigInt_Multiply10( &scaledMarginLow );
1020            if (pScaledMarginHigh != &scaledMarginLow)
1021                BigInt_Multiply2( pScaledMarginHigh, scaledMarginLow );                 
1022        }
1023    }
1024    else
1025    {
1026        // For length based cutoff modes, we will try to print until we
1027        // have exhausted all precision (i.e. all remaining digits are zeros) or
1028        // until we reach the desired cutoff digit.
1029        low = false;
1030        high = false;
1031 
1032        for (;;)
1033        {
1034            digitExponent = digitExponent-1;
1035 
1036            // divide out the scale to extract the digit
1037            outputDigit = BigInt_DivideWithRemainder_MaxQuotient9(&scaledValue, scale);
1038            RJ_ASSERT( outputDigit < 10 );
1039 
1040            if ( scaledValue.IsZero() | (digitExponent == cutoffExponent) )
1041                break;
1042 
1043            // store the output digit
1044            *pCurDigit = (tC8)('0' + outputDigit);
1045            ++pCurDigit;
1046 
1047            // multiply larger by the output base
1048            BigInt_Multiply10(&scaledValue);
1049        }
1050    }
1051 
1052    // round off the final digit
1053    // default to rounding down if value got too close to 0
1054        tB roundDown = low;
1055     
1056    // if it is legal to round up and down
1057    if (low == high)
1058    {
1059        // round to the closest digit by comparing value with 0.5. To do this we need to convert
1060        // the inequality to large integer values.
1061        //  compare( value, 0.5 )
1062        //  compare( scale * value, scale * 0.5 )
1063        //  compare( 2 * scale * value, scale )
1064        BigInt_Multiply2(&scaledValue);
1065        tS32 compare = BigInt_Compare(scaledValue, scale);
1066        roundDown = compare < 0;
1067         
1068        // if we are directly in the middle, round towards the even digit (i.e. IEEE rouding rules)
1069        if (compare == 0)
1070            roundDown = (outputDigit & 1) == 0;             
1071    }
1072 
1073    // print the rounded digit
1074    if (roundDown)
1075    {
1076        *pCurDigit = (tC8)('0' + outputDigit);
1077        ++pCurDigit;
1078    }
1079    else
1080    {
1081        // handle rounding up
1082        if (outputDigit == 9)
1083        {
1084            // find the first non-nine prior digit
1085            for (;;)
1086            {
1087                // if we are at the first digit
1088                if (pCurDigit == pOutBuffer)
1089                {
1090                    // output 1 at the next highest exponent
1091                    *pCurDigit = '1';
1092                    ++pCurDigit;
1093                    *pOutExponent += 1;
1094                    break;
1095                }
1096
1097                --pCurDigit;
1098                if (*pCurDigit != '9')
1099                {
1100                    // increment the digit
1101                    *pCurDigit += 1;
1102                    ++pCurDigit;
1103                    break;
1104                }
1105            }
1106        }
1107        else
1108        {
1109            // values in the range [0,8] can perform a simple round up
1110            *pCurDigit = (tC8)('0' + outputDigit + 1);
1111            ++pCurDigit;
1112        }
1113    }
1114 
1115    // return the number of digits output
1116    tU32 outputLen = (tU32)(pCurDigit - pOutBuffer);
1117    RJ_ASSERT(outputLen <= bufferSize);
1118    return outputLen;
1119}
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