researcher – Sasapea's Lab

C/C++のlong double型ってどんな仕様なのかな？と調べてみたら普段使ってるgccだと80bitらしい。でもsizeof()で調べてみるとメモリサイズとしては128bit(16byte)だしなんだか中途半端な感じ。さらに調べてみたら128bit型はIEEE754で仕様が規定されてた。

IEEE754の128bitデータ型であれば必要十分な有効桁数が確保できるけど80bitだと国家予算程度でぎりぎりだしIEEE754の128bitデータ型が使えたとしても標準ライブラリの対応状況がびみょーな気も．．．

それと２進浮動小数点型は小数部を正確に表すことができないことがあり10進浮動小数点型のほうが好ましいことが多いものの10進演算で1桁毎に演算したりすると効率が悪いのが気になってしまう。CPUも高速になってきているのでそう大きな問題でもなさそうではあるが．．．

そこで、ないものは作ればいいといつものいきおい(-_-;)で10進浮動小数点型で128bit以上を扱えて高速な演算が可能なデータ型を考えてみた。

高速な演算を行うには複数桁をまとめて演算する必要があるため32bitデータ型に収まる10億単位で演算する方式にしてみた。64bit-CPUならより大きい単位も考えられるが浮動小数点型となると小数点の位置が一桁単位ではなく複数桁単位になってしまうことから有効桁数が数値により増減してしまうデメリットもあるので10億単位あたりが実用的な上限ではないかと思う。

数値は最上位桁から下位桁に向かって格納する。そうすることで演算時の桁合わせ処理が必要なくなるのと最上位桁だけでゼロ判定ができるので何かと都合が良い。それにともない小数点位置は最上位桁が小数点位置の起点となる。

n乗根(root)が完成した段階でとりあえず公開することにしたが作りたてで試験もあまりしてないためバクバグしてる可能性があることに注意すること。

試しに512bitで2の平方根(sqrt)を求めてみた。ニュートン法で収束或いは反復するまで繰り返して求められた結果は136桁。512bitだと最大144桁であるが小数点位置の関係で144-8=136桁となる。

計算時間は下記の通り。

128bit 28桁 1.9us
256bit 64桁 11.4us
512bit 136桁 170.5us

1.414213562373095048801688724209698078569671875376948073176679737990732478462107038850387534327641572735013846230912297024924836055850737

※　浮動小数点型は表せる数値範囲は広いが有効桁が制限される。より大きな有効桁数を必要とするなら任意精度演算のほうが適している。

【仕様】

/*
 * [specification]
 *
 * 128-bit (fraction: 120-bit, exponent: 7-bit, sign: 1-bit)
 *   e-(64-3)*9 = e-549
 *   e+(63+3)*9 = e+594
 * 256-bit (fraction: 240-bit, exponent: 15-bit, sign: 1-bit)
 *   e-(16384-7)*9 = e-147393
 *   e+(16383+7)*9 = e+147510
 * 512-bit (fraction: 480-bit, exponent: 31-bit, sign: 1-bit)
 *   e-(1073741824-15)*9 = e-1073741767
 *   e+(1073741823+15)*9 = e+1073741958
 */

* [specification]

* 128-bit (fraction: 120-bit, exponent: 7-bit, sign: 1-bit)

* e-(64-3)*9 = e-549

* e+(63+3)*9 = e+594

* 256-bit (fraction: 240-bit, exponent: 15-bit, sign: 1-bit)

* e-(16384-7)*9 = e-147393

* e+(16383+7)*9 = e+147510

* 512-bit (fraction: 480-bit, exponent: 31-bit, sign: 1-bit)

* e-(1073741824-15)*9 = e-1073741767

* e+(1073741823+15)*9 = e+1073741958

しかし、512bitで十億桁って一体どんな世界なんだろう？
想像もつきませーーーーーーーーーーーーーーーん。(笑)

【外部データ形式】

データは32bit値の配列[4/8/16]形式。

32bit[4] 128bit 有効桁数 28-36桁
32bit[8] 256bit 有効桁数 64-72桁
32bit[16] 512bit 有効桁数 136-144桁

[999999999],[999999999],[999999999],[999999999],…

最上位桁:[仮数部(LSB:30bit)+指数部(1bit)+符号部(1bit)]
中下位桁:[仮数部(LSB:30bit)+指数部(2bit)]

桁(32bit)の仮数部には10進９桁の数値をバイナリーで格納。指数部は各桁に分割格納される。この形式は外部保存用の形式。

【内部データ形式】

外部データ形式と同じであるが演算しやすいよう各桁の符号部と指数部は別変数にまとめられ保存時に各桁に振り分けられる。

【オペレータ・オーバー・ロード】

ほとんどのオペレータをオーバーロードしてみたので標準データ型と同じようにコードが書けるし標準データ型との混在(型変換)も可能だがダウンキャスト時の数値のオーバーフローやアンダーフローには注意しなければならない。

【除算のアルゴリズム】

ライブラリを作るにあたって一番悩んだのは除算及び剰余処理だがなんとか完成。アルゴリズムはよく知られている筆算方式で10進一桁あたり０～３回の減算と10倍するための加算で処理している。128bitサイズだと全桁で9×4=36桁格納できるが全桁計算したとしても0回～108回の減算と36回の加算で済む。

小数部を含む剰余は除算と同じ処理で行えるが剰余を求める場合に限り商は整数に制限される。でないと何が余りかわからないので．．．

        ...
        /* optimize of floating point position */
        bool declining = (nod == 0);
        /* division */
        for (i = BDFP_SIZE - 1;; --nod)
        {
          if (nod < 0)
          {
            nod = DIGIT_SIZE - 1;
            if (--i < 0)
              break;
          }
          if (cmp(dividend, divisor[0]) >= 0)
          {
            int32_t n = POW10[nod];
            for (j = sizeof(divisor) / sizeof(divisor[0]) - 1; j >= 0; --j)
            {
              if (cmp(dividend, divisor[j]) >= 0)
              {
                _data.fractions[i].bin += n << j;
                if (sub(dividend, divisor[j]))
                {
                  i = -1;
                  break;
                }
              }
            }
            if (i < 0)
              break;
          }
          else if (declining)
          {
            /* optimize of floating point position */
            ++_data.exponent;
            nod = DIGIT_SIZE;
          }
          declining = false;
          mul10(dividend, dividend);
        }
        ...

...

/* optimize of floating point position */

bool declining = (nod == 0);

/* division */

for (i = BDFP_SIZE - 1;; --nod)

{

if (nod < 0)

{

nod = DIGIT_SIZE - 1;

if (--i < 0)

break;

}

if (cmp(dividend, divisor[0]) >= 0)

{

int32_t n = POW10[nod];

for (j = sizeof(divisor) / sizeof(divisor[0]) - 1; j >= 0; --j)

{

if (cmp(dividend, divisor[j]) >= 0)

{

_data.fractions[i].bin += n << j;

if (sub(dividend, divisor[j]))

{

i = -1;

break;

}

if (i < 0)

break;

}

else if (declining)

{

/* optimize of floating point position */

++_data.exponent;

nod = DIGIT_SIZE;

}

declining = false;

mul10(dividend, dividend);

}

...

【修正】
2024-12-23
root(),pow()の修正。

2024-12-22
ln()の符号がバグッてたので修正。

2024-12-17
加算と減算が思いっきりバグってたので修正。

2024-12-14
まさかの加算がバグってたので修正。(-_-;)
あとpow()を小数指数に対応＆exp()/ln()/log2()/log10()/log(b)を追加。ほとんど試験してないので全体的にまだまだバグが取り切れてない気がする。

2024-12-12
ceil()を追加。

2024-12-11
floor(負数)とdiv()の剰余処理がまだバグってたので再修正したのとスタティック変数の初期化のためにbdfp.cppを追加。
ついでにround()に丸めモード引数を追加した。ROUND_HALF_EVENモードは累積誤差が一番少なくなるらしく銀行型丸めとも呼ばれることがあるという。初めて知りました。v(-_-;)

typedef enum
{
  ROUND_UP,
  ROUND_DOWN,
  ROUND_CEILING,
  ROUND_FLOOR,
  ROUND_HALF_UP,
  ROUND_HALF_EVEN,
  ROUND_HALF_DOWN,
} ROUND_MODE;

BDFP &round(const BDFP &val = 0, ROUND_MODE mode = ROUND_HALF_EVEN);

typedef enum

{

ROUND_UP,

ROUND_DOWN,

ROUND_CEILING,

ROUND_FLOOR,

ROUND_HALF_UP,

ROUND_HALF_EVEN,

ROUND_HALF_DOWN,

} ROUND_MODE;

BDFP &round(const BDFP &val = 0, ROUND_MODE mode = ROUND_HALF_EVEN);

2024-12-10
丸め処理を行うround()の追加とdiv()の剰余処理がバグってたので修正。あとsqrt()とroot()の符号を無視するよう修正してみた。

2024-12-07
n乗根(root)の最下位桁に誤差が出るため最下位桁を削除。
旧: 9 root 2 -> 3.000………00001
新: 9 root 2 -> 3
除算の10倍処理(mul10)を改良。少しだけ早くなった。

【サンプルの実行形式(sample.exe:win-x64】
sample.zip

【サンプル・プログラム】

#include <windows.h>
#include "bdfp.h"

int main(int argc, char **argv)
{
  int loop = 1;
  const char *a = "";
  const char *c = "";
  const char *b = "";
  for (int i = 1; i < argc; )
  {
    const char *p = argv[i++];
    if ((*a == 0) && (*p == '-'))
    {
      ++p;
      if (strcmp(p, "c") == 0)
      {
        if (i < argc)
          loop = atoi(argv[i++]);
      }
    }
    else if (*a == 0)
      a = p;
    else if (*c == 0)
      c = p;
    else if (*b == 0)
      b = p;
  }
  BDFP dec1(a);
  BDFP dec2(b);
  printf("a = %s\n", dec1.toString().c_str());
  printf("b = %s\n", dec2.toString().c_str());
  BDFP  v = dec1;
  DWORD t = GetTickCount();
  for (int i = 0; i < loop; ++i)
  {
    dec1 = v;
    switch (*c)
    {
      case '+': dec1.add(dec2);   break;
      case '-': dec1.sub(dec2);   break;
      case '*': dec1.mul(dec2);   break;
      case '/': dec1.div(dec2);   break;
      case '%': dec1.mod(dec2);   break;
      case 'r': dec1.root(dec2);  break;
      case 's': dec1.sqrt();      break;
      case 'p': dec1.pow(dec2); 　break;
      case 'o': dec1.round(dec2); break;
    }
  }
  t = GetTickCount() - t;
  if (loop > 1)
    printf("time = %f us\n", (double)t * 1000 / loop);
  printf("result   = %s\n", dec1.toString().c_str());
  printf("exponent = %s\n", dec1.toExponential().c_str());
  return 0;
}

#include <windows.h>

#include "bdfp.h"

int main(int argc, char **argv)

{

int loop = 1;

const char *a = "";

const char *c = "";

const char *b = "";

for (int i = 1; i < argc; )

{

const char *p = argv[i++];

if ((*a == 0) && (*p == '-'))

{

++p;

if (strcmp(p, "c") == 0)

{

if (i < argc)

loop = atoi(argv[i++]);

}

else if (*a == 0)

a = p;

else if (*c == 0)

c = p;

else if (*b == 0)

b = p;

}

BDFP dec1(a);

BDFP dec2(b);

printf("a = %s\n", dec1.toString().c_str());

printf("b = %s\n", dec2.toString().c_str());

BDFP v = dec1;

DWORD t = GetTickCount();

for (int i = 0; i < loop; ++i)

{

dec1 = v;

switch (*c)

{

case '+': dec1.add(dec2); break;

case '-': dec1.sub(dec2); break;

case '*': dec1.mul(dec2); break;

case '/': dec1.div(dec2); break;

case '%': dec1.mod(dec2); break;

case 'r': dec1.root(dec2); break;

case 's': dec1.sqrt(); break;

case 'p': dec1.pow(dec2); 　break;

case 'o': dec1.round(dec2); break;

}

t = GetTickCount() - t;

if (loop > 1)

printf("time = %f us\n", (double)t * 1000 / loop);

printf("result = %s\n", dec1.toString().c_str());

printf("exponent = %s\n", dec1.toExponential().c_str());

return 0;

}

【ライブラリ】

/*
  bdfp.h - Billion Decimal Floating Point Calculation Library

  Copyright (c) 2024 Sasapea's Lab. All right reserved.

  This program is free software: you can redistribute it and/or
  modify it under the terms of the GNU General Public License as
  published by the Free Software Foundation, either version 3 of
  the License, or at your option) any later version.

  This program is distributed in the hope that it will be useful,
  but WITHOUT ANY WARRANTY; without even the implied warranty of
  MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.
  See the GNU General Public License for more details.

  You should have received a copy of the GNU General Public License
  along with this program. If not, see <https://www.gnu.org/licenses/>.
*/
#pragma once

#include <stdint.h>
#include <cstdlib>
#include <cctype>
#include <cstring>
#include <string>
#include <cmath>

/*
 * [specification]
 *
 * 128-bit (fraction: 120-bit, exponent: 7-bit, sign: 1-bit)
 *   e-(64-3)*9 = e-549
 *   e+(63+3)*9 = e+594
 * 256-bit (fraction: 240-bit, exponent: 15-bit, sign: 1-bit)
 *   e-(16384-7)*9 = e-147393
 *   e+(16383+7)*9 = e+147510
 * 512-bit (fraction: 480-bit, exponent: 31-bit, sign: 1-bit)
 *   e-(1073741824-15)*9 = e-1073741767
 *   e+(1073741823+15)*9 = e+1073741958
 */

#ifndef BDFP_SIZE
  #define BDFP_SIZE  4  /*  4 * 4 = 16-bytes (128-bit) */
//#define BDFP_SIZE  8  /*  8 * 4 = 32-bytes (256-bit) */
//#define BDFP_SIZE 16  /* 16 * 4 = 64-bytes (512-bit) */
#endif

#define BDFP_SIZE_MIN 4
#define BDFP_SIZE_MAX 16
#if (BDFP_SIZE < BDFP_SIZE_MIN) || (BDFP_SIZE > BDFP_SIZE_MAX)
  #error "invalid decimal size."
#endif

class BDFP
{
  private:

    static constexpr int32_t  POW10[] = {
      1,
      10,
      100,
      1000,
      10000,
      100000,
      1000000,
      10000000,
      100000000,
      1000000000
    };
    static constexpr int32_t  NEWTON_LOOP   = 1000; /* max loop count (stopper) */
    static constexpr int      DIGIT_SIZE    = 9;
    static constexpr int32_t  DIGIT_BASE    = POW10[DIGIT_SIZE];
    static constexpr uint32_t EXPONENT_MSB  = 1UL << ((BDFP_SIZE << 1) - 1);
    static constexpr uint32_t EXPONENT_MAX  = ((EXPONENT_MSB >> 1) - 1);
    static constexpr uint32_t EXPONENT_MIN  = (~(EXPONENT_MSB - 1) | (EXPONENT_MSB >> 1));

    static const std::string FORMAT1;
    static const std::string FORMAT2;
    static const std::string ZEROSTR;

    typedef union
    {
      struct
      {
#if defined(_BIT_FIELDS_HTOL)
        uint32_t exponent: 2;
        uint32_t fraction: 30;
#else
        uint32_t fraction: 30;
        uint32_t exponent: 2;
#endif
      } bit;
      int32_t bin;
    } fraction_t;

    struct
    {
      fraction_t fractions[BDFP_SIZE];
      int32_t    exponent;
      int8_t     sign;
    } _data;

    int lowest(void) const
    {
      int idx;
      for (idx = 0; idx < BDFP_SIZE; ++idx)
      {
        if (_data.fractions[idx].bin)
          break;
      }
      return idx;
    }

    int highest(void) const
    {
      int idx;
      for (idx = BDFP_SIZE - 1; idx >= 0; --idx)
      {
        if (_data.fractions[idx].bin)
          break;
      }
      return idx;
    }

    int cmp0(const BDFP &val) const
    {
      if (isZero())
        return val.isZero() ? 0 : -1;
      if (val.isZero())
        return 1;
      int32_t off = val._data.exponent - _data.exponent;
      if (off)
        return off < 0 ? -1 : 1;
      for (int i = BDFP_SIZE - 1; i >= 0; --i)
      {
        if ((off = _data.fractions[i].bin - val._data.fractions[i].bin))
          return off < 0 ? -1 : 1;
      }
      return 0;
    }

    BDFP &add0(const BDFP &val)
    {
      if (isZero())
        from(val);
      else if (!val.isZero())
      {
        int32_t off = _data.exponent - val._data.exponent;
        int32_t cy  = 0;
        int i;
        if (off <= 0)
        {
          for (i = 0; i < BDFP_SIZE; ++i)
          {
            cy += _data.fractions[i].bin;
            if (i - off < BDFP_SIZE)
              cy += val._data.fractions[i - off].bin;
            if (cy >= DIGIT_BASE)
            {
              _data.fractions[i].bin = cy - DIGIT_BASE;
              cy = 1;
            }
            else
            {
              _data.fractions[i].bin = cy;
              cy = 0;
            }
          }
        }
        else
        {
          for (i = 0; i < BDFP_SIZE; ++i)
          {
            cy += val._data.fractions[i].bin;
            if (i + off < BDFP_SIZE)
              cy += _data.fractions[i + off].bin;
            if (cy >= DIGIT_BASE)
            {
              _data.fractions[i].bin = cy - DIGIT_BASE;
              cy = 1;
            }
            else
            {
              _data.fractions[i].bin = cy;
             cy = 0;
            }
          }
          _data.exponent = val._data.exponent;
        }
        if (cy)
        {
          for (i = 1; i < BDFP_SIZE; ++i)
            _data.fractions[i - 1].bin = _data.fractions[i].bin;
          _data.fractions[BDFP_SIZE - 1].bin = 1;
          --_data.exponent;
        }
      }
      return *this;
    }

    BDFP &sub0(const BDFP &val)
    {
      if (isZero())
        from(val);
      else if (!val.isZero())
      {
        int i = cmp0(val);
        if (i == 0)
          zero();
        else
        {
          int32_t off = _data.exponent - val._data.exponent;
          int32_t cy  = 0;
          if ((off < 0) || ((off == 0) && (i >= 0)))
          {
            for (i = 0; i < BDFP_SIZE; ++i)
            {
              cy += _data.fractions[i].bin;
              if (i - off < BDFP_SIZE)
                cy -= val._data.fractions[i - off].bin;
              if (cy < 0)
              {
                _data.fractions[i].bin = cy + DIGIT_BASE;
                cy = -1;
              }
              else
              {
                _data.fractions[i].bin = cy;
                cy = 0;
              }
            }
          }
          else
          {
            for (i = 0; i < BDFP_SIZE; ++i)
            {
              cy += val._data.fractions[i].bin;
              if (i + off < BDFP_SIZE)
                cy -= _data.fractions[i + off].bin;
              if (cy < 0)
              {
                _data.fractions[i].bin = cy + DIGIT_BASE;
                cy = -1;
              }
              else
              {
                _data.fractions[i].bin = cy;
                cy = 0;
              }
            }
            _data.exponent = val._data.exponent;
          }
          off = highest();
          if (off < 0)
            zero();
          else
          {
            off = BDFP_SIZE - 1 - off;
            for (i = BDFP_SIZE - 1; i >= 0; --i)
              _data.fractions[i].bin = (i - off >= 0 ? _data.fractions[i - off].bin : 0);
            _data.exponent += off;
          }
        }
      }
      return *this;
    }

    BDFP &mul0(const BDFP &val)
    {
      if (isZero() || val.isZero())
        zero();
      else
      {
        fraction_t result[BDFP_SIZE * 2] = {};
        uint32_t part1, part2;
        uint64_t cy;
        int i, j = 0;
        for (i = 0; i < BDFP_SIZE; ++i)
        {
          if ((part1 = _data.fractions[i].bin))
          {
            cy = 0;
            for (j = i; j < BDFP_SIZE * 2; ++j)
            {
              if ((j - i) < BDFP_SIZE)
              {
                cy += result[j].bin;
                if ((part2 = val._data.fractions[j - i].bin))
                  cy += (uint64_t)part1 * part2;
              }
              else if (cy == 0)
                break;
              if (cy >= DIGIT_BASE)
              {
                std::lldiv_t x = std::lldiv(cy, DIGIT_BASE);
                result[j].bin = x.rem;
                cy = x.quot;
              }
              else
              {
                result[j].bin = cy;
                cy = 0;
              }
            }
          }
        }
        _data.sign ^= val._data.sign;
        _data.exponent = (_data.exponent + val._data.exponent) - (BDFP_SIZE - 1) - (j == BDFP_SIZE * 2);
        for (i = BDFP_SIZE; i > 0; )
          _data.fractions[--i].bin = result[--j].bin;
      }
      return *this;
    }

    int mul2(fraction_t *dst, fraction_t *src) const
    {
      int32_t cy = 0;
      for (int i = 0; i < BDFP_SIZE + 1; ++i)
      {
        if ((cy += (src[i].bin << 1)) >= DIGIT_BASE)
        {
          dst[i].bin = cy - DIGIT_BASE;
          cy = 1;
        }
        else
        {
          dst[i].bin = cy;
          cy = 0;
        }
      }
      return cy;
    }

    int mul10(fraction_t *dst, fraction_t *src) const
    {
      int cy = 0;
      for (int i = 0; i < BDFP_SIZE + 1; ++i)
      {
        int64_t d = src[i].bin << 1;
        d += (d << 2) + cy;
        for (cy = 0; d >= DIGIT_BASE; d -= DIGIT_BASE, ++cy)
          continue;
        dst[i].bin = d;
      }
      return cy;
    }

    bool sub(fraction_t *a, fraction_t *b) const
    {
      int cy, zc, i;
      for (cy = zc = i = 0; i < BDFP_SIZE + 1; ++i)
      {
        if ((a[i].bin -= b[i].bin + cy) < 0)
        {
          a[i].bin += DIGIT_BASE;
          cy = 1;
        }
        else
        {
          zc += a[i].bin == 0;
          cy = 0;
        }
      }
      return zc == BDFP_SIZE + 1;
    }

    int32_t cmp(fraction_t *a, fraction_t *b) const
    {
      int32_t rv;
      for (int i = BDFP_SIZE + 1 - 1; i >= 0; --i)
      {
        if ((rv = a[i].bin - b[i].bin))
          break;
      }
      return rv;
    }

    int msd(int32_t val) const
    {
      int i;
      for (i = DIGIT_SIZE - 1; i >= 0; --i)
      {
        if (POW10[i] <= val)
          break;
      }
      return i + 1;
    }

    BDFP &div0(const BDFP &val, BDFP *rem = nullptr)
    {
      int i = msd(_data.fractions[BDFP_SIZE - 1].bin);
      int j = msd(val._data.fractions[BDFP_SIZE - 1].bin);
      int k, r = 0;
      if (rem)
        rem->zero();
      if ((i == 0) || (j == 0))
        zero();
      else
      {
        /* sign and exponent */
        _data.sign ^= val._data.sign;
        int32_t e = (_data.exponent - val._data.exponent) + (BDFP_SIZE - 1) + (i < j);
        if (rem)
        {
          if (e >= BDFP_SIZE)
          {
            rem->from(*this);
            zero();
            return *this;
          }
          rem->_data.sign = _data.sign;
          rem->_data.exponent = (_data.exponent < val._data.exponent) || (i > j)
            ? val._data.exponent : _data.exponent;
          r = ((_data.exponent == val._data.exponent) || (i > j) ? (i - j) : 0);
        }
        _data.exponent = e;
        /* copy data to working variables */
        fraction_t dividend[BDFP_SIZE + 1];
        fraction_t divisor[4][BDFP_SIZE + 1];
        for (k = 0; k < BDFP_SIZE; ++k)
        {
          divisor[0][k] = val._data.fractions[k];
          dividend[k] = _data.fractions[k];
          _data.fractions[k].bin = 0;
        }
        dividend[BDFP_SIZE].bin = 0;
        divisor[0][BDFP_SIZE].bin = 0;
        /* Align the most significant digit */
        for (k = j; i > k; ++k)
          mul10(divisor[0], divisor[0]);
        for (k = i; k < j; ++k)
          mul10(dividend, dividend);
        /* multiply the divisor by 2, 4, or 8 */
        for (k = 1; k < (int)(sizeof(divisor) / sizeof(divisor[0])); ++k)
          mul2(divisor[k], divisor[k - 1]);
        /* digit position */
        int nod = i - j;
        if (nod < 0)
          nod += DIGIT_SIZE;
        /* optimize of floating point position */
        bool declining = (nod == 0);
        /* division */
        for (i = BDFP_SIZE - 1;; --nod)
        {
          if (nod < 0)
          {
            nod = DIGIT_SIZE - 1;
            if (--i < 0)
              break;
          }
          if (cmp(dividend, divisor[0]) >= 0)
          {
            int32_t n = POW10[nod];
            for (j = sizeof(divisor) / sizeof(divisor[0]) - 1; j >= 0; --j)
            {
              if (cmp(dividend, divisor[j]) >= 0)
              {
                _data.fractions[i].bin += n << j;
                if (sub(dividend, divisor[j]))
                {
                  i = -1;
                  break;
                }
              }
            }
            if (i < 0)
              break;
          }
          else if (declining)
          {
            /* optimize of floating point position */
            ++_data.exponent;
            nod = DIGIT_SIZE;
          }
          declining = false;
          if (rem && ((nod == 0) || (nod == DIGIT_SIZE)) && (i == _data.exponent))
          {
            for (i = 0; (i < BDFP_SIZE) && (dividend[BDFP_SIZE - 1 - i].bin == 0); ++i)
              continue;
            if (i >= BDFP_SIZE)
              rem->zero();
            else
            {
              rem->_data.exponent += i;
              for (j = 0; j < BDFP_SIZE; ++j)
                rem->_data.fractions[j].bin = (j - i >= 0 ? dividend[j - i].bin : 0);
              if (r)
                rem->div0(BDFP(POW10[r]));
            }
            break;
          }
          mul10(dividend, dividend);
        }
      }
      return *this;
    }

    void dump(fraction_t *val, const char *msg = nullptr) const
    {
      printf(msg ? "%s [" : "[", msg);
      for (int i = 0; i < BDFP_SIZE + 1; ++i)
        printf(i ? ", %09d" : " %09d", val[BDFP_SIZE - i].bin);
      printf(" ]\n");
    }

  public:

    static const BDFP TEN;
    static const BDFP FIVE;
    static const BDFP TWO;
    static const BDFP ONE;
    static const BDFP ZERO;
    static const BDFP E;
    static const BDFP LOG2;
    static const BDFP LOG10;

    typedef struct
    {
      fraction_t fractions[BDFP_SIZE];
    } storage_t;

    bool isZero(void) const
    {
      return _data.fractions[BDFP_SIZE - 1].bin == 0;
    }

    BDFP &zero(void)
    {
      _data = { {}, BDFP_SIZE };
      return *this;
    }

    bool sign(void) const
    {
      return _data.sign;
    }

    BDFP &neg(void)
    {
      _data.sign = isZero() ? 0 : _data.sign ^ 1;
      return *this;
    }

    BDFP &abs(void)
    {
      _data.sign = 0;
      return *this;
    }

    bool isOdd(void) const
    {
      return (_data.exponent >= 0) && (_data.exponent < BDFP_SIZE) && (_data.fractions[_data.exponent].bin & 1);
    }

    int cmp(const BDFP &val) const
    {
      int sign = val._data.sign - _data.sign;
      if (sign)
        return sign;
      sign = cmp0(val);
      return sign == 0 ? 0 : ((sign < 0) ^ _data.sign ? -1 : 1);
    }

    BDFP &add(const BDFP &val)
    {
      if (isZero())
        from(val);
      else if (sign() == val.sign())
        add0(val);
      else
      {
        _data.sign = sign() ? cmp0(val) >= 0 : cmp0(val) < 0;
        sub0(val);
      }
      return *this;
    }

    BDFP &sub(const BDFP &val)
    {
      if (isZero())
        from(val).neg();
      else if (sign() != val.sign())
        add0(val);
      else
      {
        _data.sign = sign() ? cmp0(val) >= 0 : cmp0(val) < 0;
        sub0(val);
      }
      return *this;
    }

    BDFP &mul(const BDFP &val)
    {
      return mul0(val);
    }

    BDFP &div(const BDFP &val, BDFP *rem = nullptr)
    {
      return div0(val, rem);
    }

    BDFP &mod(const BDFP &val)
    {
      BDFP rem;
      div0(val, &rem);
      return from(rem);
    }

    BDFP &modf(BDFP *iptr = nullptr)
    {
      int msd = -1;
      if (iptr)
        iptr->from(*this);
      for (int i = 0; i < BDFP_SIZE; ++i)
      {
        if (i >= _data.exponent)
          _data.fractions[i].bin = 0;
        else
        {
          if (_data.fractions[i].bin)
            msd = i;
          if (iptr)
            iptr->_data.fractions[i].bin = 0;
        }
      }
      if (iptr)
      {
        if (iptr->highest() < 0)
          iptr->zero();
      }
      if (msd < 0)
        zero();
      else if (BDFP_SIZE - 1 > msd)
      {
        for (int i = 0; i < BDFP_SIZE; ++i)
          _data.fractions[(BDFP_SIZE - 1) - i].bin = (msd - i >= 0 ? _data.fractions[msd - i].bin : 0);
        _data.exponent += (BDFP_SIZE - 1) - msd;
      }
      return *this;
    }

    BDFP &floor(void)
    {
      BDFP i;
      if (!modf(&i).isZero() && sign())
        i.add0(ONE);
      return from(i);
    }

    BDFP &ceil(void)
    {
      BDFP i;
      if (!modf(&i).isZero() && !sign())
        i.add0(ONE);
      return from(i);
    }

    BDFP &trunc(void)
    {
      for (int i = 0; i < BDFP_SIZE; ++i)
      {
        if (i < _data.exponent)
          _data.fractions[i].bin = 0;
      }
      return *this;
    }

    typedef enum
    {
      ROUND_UP,
      ROUND_DOWN,
      ROUND_CEILING,
      ROUND_FLOOR,
      ROUND_HALF_UP,
      ROUND_HALF_EVEN,
      ROUND_HALF_DOWN,
    } ROUND_MODE;

    BDFP &round(const BDFP &val = 0, ROUND_MODE mode = ROUND_HALF_EVEN)
    {
      BDFP d, r, q;
      d.from(TEN).pow(BDFP(val).trunc().neg());
      switch (mode)
      {
        case ROUND_UP:
          add0(BDFP(d).sub0(BDFP(d).div(TEN)));
          BDFP(*this).div(d, &r);
          sub0(r);
          break;
        case ROUND_DOWN:
          BDFP(*this).div(d, &r);
          sub0(r);
          break;
        case ROUND_CEILING:
          if (!sign())
            add0(BDFP(d).sub0(BDFP(d).div(TEN)));
          BDFP(*this).div(d, &r);
          sub0(r);
          break;
        case ROUND_FLOOR:
          if (sign())
            add0(BDFP(d).sub0(BDFP(d).div(TEN)));
          BDFP(*this).div(d, &r);
          sub0(r);
          break;
        case ROUND_HALF_UP:
          add0(BDFP(d).div(TWO));
          BDFP(*this).div(d, &r);
          sub0(r);
          break;
        case ROUND_HALF_DOWN:
          add0(BDFP(d).div(TWO).sub0(BDFP(d).div(TEN)));
          BDFP(*this).div(d, &r);
          sub0(r);
          break;
        case ROUND_HALF_EVEN:
        default:
          q.from(*this).div(d, &r);
          if (r.isZero())
            break;
          sub0(r);
          switch (r.div(BDFP(d).div(TEN)).trunc().cmp0(FIVE))
          {
            case 0:
              if (q.isOdd())
                add0(d);
              break;
            case 1:
              add0(d);
              break;
          }
          break;
      }
      return *this;
    }

    BDFP &root(const BDFP &val)
    {
      bool s = sign();
      BDFP a, b(ONE), c(val), d, e;
      if (val.sign() || (!val.isOdd() && s))
        zero();
      else
      {
        a.from(abs());
        e.from(c.abs().trunc()).sub(ONE);
        for (int i = 0; i < NEWTON_LOOP; ++i)
        {
          from(BDFP(b).sub(BDFP(b).powi(c).sub(a).div(BDFP(b).powi(e).mul(c))));
          if (b.sub(*this).isZero() || (b.cmp0(d) == 0))
            break;
          d.from(b.from(*this));
        }
        if (s)
          neg();
        /* delete the error */
        if (_data.fractions[0].bin)
          _data.fractions[0].bin -= _data.fractions[0].bin % 10;
      }
      return *this;
    }

    BDFP &sqrt(void)
    {
      BDFP a(*this), b, c, d;
      if (a.sign())
        zero();
      else
      {
        for (int i = 0; i < NEWTON_LOOP; i++)
        {
          mul(b.from(c.from(*this))).add(a).div(c.add(c));
          if (b.sub(*this).isZero() || (b.cmp0(d) == 0))
            break;
          d.from(b);
        }
      }
      return *this;
    }

    BDFP &exp(void)
    {
      static const BDFP PREC("1e-" + std::to_string((BDFP_SIZE - 1) * DIGIT_SIZE));
      BDFP a(*this), b, c, d;
      from(ONE);
      if (!a.isZero())
      {
        bool sign = a.sign();
        for (b.from(a.abs()).modf(&a); !a.isZero(); a.sub0(ONE))
          mul(E);
        if (!b.isZero())
        {
          d.from(c.from(ONE));
          for (a.from(ONE); c.cmp0(PREC) > 0; a.add0(ONE))
            d.add0(c.mul(b).div(a));
          mul(d);
        }
        if (sign)
          from(BDFP(ONE).div(*this));
      }
      return *this;
    }

    BDFP &exp2(void)
    {
      return mul(LOG2).exp();
    }

    BDFP &exp10(void)
    {
      return mul(LOG10).exp();
    }

    BDFP &exp(const BDFP &val)
    {
      return mul(log(val)).exp();
    }

    BDFP &expm1(void)
    {
      return exp().sub(ONE);
    }

    BDFP &ln(void)
    {
      static const BDFP PREC(1e16);
      if (cmp(ZERO) <= 0)
        zero();
      else
        pow(BDFP(ONE).div(PREC)).sub(ONE).mul(PREC);
      return *this;
    }

    BDFP &log2(void)
    {
      return ln().div(LOG2);
    }

    BDFP &log10(void)
    {
      return ln().div(LOG10);
    }

    BDFP &log(const BDFP &val)
    {
      return ln().div(BDFP(val).ln());
    }

    BDFP &log1p(void)
    {
      return add(ONE).ln();
    }

    BDFP &pow(const BDFP &val)
    {
      BDFP d(val), n;
      d.fraction(n);
      return root(d).powi(n);
    }

    BDFP &powi(const BDFP &val)
    {
      BDFP f(1);
      if (!val.isZero())
      {
        if (isZero())
          f.zero();
        else
        {
          uint32_t n[BDFP_SIZE];
          BDFP(val).abs().to(n, BDFP_SIZE);
          int i, j;
          for (j = BDFP_SIZE; j > 0; --j)
          {
            if (n[j - 1])
              break;
          }
          for (j <<= 5, i = 0; i < j; ++i)
          {
            if (n[i >> 5] & (1 << (i & 31)))
            {
              if (val._data.sign)
                f.div0(*this);
              else
                f.mul0(*this);
            }
            mul0(*this);
          }
        }
      }
      return from(f);
    }

    BDFP &fraction(BDFP &val)
    {
      BDFP i, n(modf(&i));
      zero();
      for (int j = 0; j < BDFP_SIZE; ++j)
      {
        int32_t d = n._data.fractions[j].bin;
        if (d == 0)
          continue;
        if (n._data.exponent > j)
        {
          from((n._data.exponent - j) * DIGIT_SIZE + 1);
          for (j = 0; j < DIGIT_SIZE; ++j)
          {
            if (d % POW10[j])
              break;
            sub0(ONE);
          }
        }
        break;
      }
      from(BDFP(TEN).powi(*this));
      if (i.from(BDFP(*this).gcd(val.from(i.mul(*this)).add0(n.mul(*this)))).cmp0(ONE))
      {
        val.div(i);
        div(i);
      }
      return *this;
    }

    BDFP &gcd(const BDFP &val)
    {
      BDFP a(*this), b(val), r;
      if (a.trunc().isZero() || b.trunc().isZero())
        zero();
      else
      {
        int i = cmp0(val);
        if (i == 0)
          from(ONE);
        else
        {
          if (i > 0)
            from(a);
          else
          {
            from(b);
            b.from(a);
          }
          while (!b.isZero())
          {
            r.from(mod(b));
            from(b);
            b.from(r);
          }
        }
      }
      return *this;
    }

    void to(storage_t &val) const
    {
      uint32_t exponent = (_data.sign ? EXPONENT_MSB : 0) | (_data.exponent & (EXPONENT_MSB - 1));
      for (int i = 0; i < BDFP_SIZE; ++i)
      {
        val.fractions[i] = _data.fractions[i];
        val.fractions[i].bit.exponent = exponent >> (i << 1);
      }
    }

    void to(long double &val) const
    {
      val = std::stod(toExponential());
    }

    bool to(uint32_t *buf, size_t len) const
    {
      if (buf)
      {
        size_t  i;
        int32_t j;
        uint64_t cy;
        for (i = 0; i < len; ++i)
          buf[i] = 0;
        for (j = BDFP_SIZE - 1; j >= _data.exponent; --j)
        {
          cy = (j >= 0 ? _data.fractions[j].bin : 0);
          for (i = 0; i < len; ++i)
          {
            if ((cy += (uint64_t)buf[i] * DIGIT_BASE) == 0)
              break;
            buf[i] = cy;
            cy >>= 32;
          }
          if (cy)
            return false; /* overflow */
        }
        if (_data.sign)
        {
          cy = 1;
          for (i = 0; i < len; ++i)
          {
            cy += ~buf[i];
            if (cy > (uint64_t)UINT32_MAX)
            {
              buf[i] = 0;
              cy = 1;
            }
            else
            {
              buf[i] = cy;
              cy = 0;
            }
          }
        }
        return true;
      }
      return false;
    }

    BDFP &from(const BDFP &val)
    {
      _data = val._data;
      return *this;
    }

    BDFP &from(const storage_t &val)
    {
      zero();
      _data.exponent = 0;
      for (int i = 0; i < BDFP_SIZE; ++i)
      {
        fraction_t part = val.fractions[i];
        _data.fractions[i].bin = part.bit.fraction;
        _data.exponent |= part.bit.exponent << (i << 1);
      }
      _data.sign = _data.exponent & EXPONENT_MSB ? 1 : 0;
      if (_data.exponent & (EXPONENT_MSB >> 1))
        _data.exponent |= ~(EXPONENT_MSB - 1);
      return *this;
    }

    BDFP &fromI64(const int64_t &val)
    {
      zero();
      _data.sign = val < 0;
      if (!_data.sign && (val <= DIGIT_BASE))
      {
        _data.fractions[BDFP_SIZE - 1].bin = val;
        _data.exponent = BDFP_SIZE - 1;
      }
      else
      {
        std::lldiv_t x = std::lldiv(val, DIGIT_BASE);
        if (_data.sign)
        {
          x.quot = std::abs(x.quot);
          x.rem  = std::abs(x.rem);
        }
        if (x.quot >= DIGIT_BASE)
        {
          std::lldiv_t y = std::lldiv(x.quot, DIGIT_BASE);
          _data.fractions[BDFP_SIZE - 3].bin = x.rem;
          _data.fractions[BDFP_SIZE - 2].bin = y.rem;
          _data.fractions[BDFP_SIZE - 1].bin = y.quot;
          _data.exponent = BDFP_SIZE - 3;
        }
        else if (x.quot)
        {
          _data.fractions[BDFP_SIZE - 2].bin = x.rem;
          _data.fractions[BDFP_SIZE - 1].bin = x.quot;
          _data.exponent = BDFP_SIZE - 2;
        }
        else
        {
          _data.fractions[BDFP_SIZE - 1].bin = x.rem;
          _data.exponent = BDFP_SIZE - 1;
        }
      }
      return *this;
    }

    BDFP &fromU64(const uint64_t &val)
    {
      if (val <= INT64_MAX)
        fromI64(val);
      else
      {
        fromI64(INT64_MIN);
        _data.sign = 0;
        add(BDFP(val + INT64_MIN));
      }
      return *this;
    }

    BDFP &fromDBL(const double &val, int len = __DBL_DIG__)
    {
      return fromLDBL(val, len);
    }

    BDFP &fromLDBL(const long double &val, int len = __LDBL_DIG__)
    {
      char buf[64];
      snprintf(buf, sizeof(buf), "%*Lg", len, val);
      return fromStr(std::string(buf));
    }

    BDFP &fromStr(const std::string &str)
    {
      std::string num, dec;
      std::string::size_type len, pos, off = 0;
      int32_t exp = 0;
      int i;
      char c;
      zero();
      do
        c = str[off++];
      while (c == ' ');
      if ((c == '-') || (c == '+'))
      {
        if (c == '-')
          _data.sign = 1;
        c = str[off++];
      }
      while ((c == ' ') || (c == '0'))
        c = str[off++];
      while (std::isdigit(c))
      {
        num += c;
        c = str[off++];
      }
      if (c == '.')
      {
        len = 0;
        c = str[off++];
        while (std::isdigit(c))
        {
          dec += c;
          if (c != '0')
            len = dec.length();
          c = str[off++];
        }
        dec.resize(len);
      }
      if ((c == 'E') || (c == 'e'))
        exp = std::strtol(&str[off], nullptr, 10);
      if (num.length() + dec.length() == 0)
        zero();
      else
      {
        if (exp < 0)
        {
          std::string::size_type len = num.length();
          if ((exp += len) <= 0)
          {
            dec.insert(0, num);
            num.clear();
          }
          else
          {
            dec.insert(0, num.substr(exp));
            num.resize(exp);
            exp = 0;
          }
        }
        else if (exp > 0)
        {
          std::string::size_type len = dec.length();
          if ((exp -= len) >= 0)
          {
            num += dec;
            dec.clear();
          }
          else
          {
            num += dec.substr(0, len + exp);
            dec = dec.substr(len + exp);
            exp = 0;
          }
        }
        _data.exponent = i = BDFP_SIZE;
        if (num.length())
        {
          if ((len = (num.length() + exp) % DIGIT_SIZE) == 0)
            len = DIGIT_SIZE;
          if (num.length() < len)
          {
            exp -= (off = len - num.length());
            num += ZEROSTR.substr(0, off);
          }
          else if (num.length() > len)
          {
            if ((off = (num.length() - len) % DIGIT_SIZE))
            {
              exp -= (off = DIGIT_SIZE - off);
              num += ZEROSTR.substr(0, off);
            }
          }
          for (off = 0; i > 0; off += pos)
          {
            pos = num.length() - off;
            if (pos == 0)
              break;
            if (pos > len)
              pos = len;
            len = DIGIT_SIZE;
            _data.fractions[--i].bin = std::stoul(num.substr(off, pos));
          }
          _data.exponent = i - (num.length() - off + exp) / DIGIT_SIZE;
        }
        if (dec.length() && (i > 0))
        {
          exp = -exp;
          if ((off = exp % DIGIT_SIZE))
          {
            exp -= off;
            dec.insert(0, ZEROSTR.substr(0, off));
          }
          for (off = 0; i > 0; off += DIGIT_SIZE)
          {
            if ((len = dec.length() - off) == 0)
              break;
            if (len < DIGIT_SIZE)
              dec += ZEROSTR.substr(0, DIGIT_SIZE - len);
            int32_t part = std::stoul(dec.substr(off, DIGIT_SIZE));
            if ((i == BDFP_SIZE) && (part == 0))
              ++_data.exponent;
            else
              _data.fractions[--i].bin = part;
          }
          _data.exponent += exp / DIGIT_SIZE;
        }
      }
      return *this;
    }

    std::string toString(void) const
    {
      std::string str;
      if (!isZero())
      {
        const char *fmt  = FORMAT1.c_str();
        const char *fmt2 = FORMAT2.c_str();
        int32_t sign, dot, cnt, idx;
        sign = _data.sign;
        dot  = 0;
        cnt  = -1;
        if (_data.exponent >= BDFP_SIZE)
        {
          dot = 1;
          cnt = _data.exponent - BDFP_SIZE;
          fmt = fmt2;
        }
        for (idx = BDFP_SIZE - 1; idx >= 0; --idx)
        {
          int32_t part = _data.fractions[idx].bin;
          if (part)
          {
            if (sign)
              str += '-';
            if (dot > 0)
            {
              if (str.length() == (std::string::size_type)sign)
                str += '0';
              str += '.';
              dot = -1;
            }
            sign = 0;
            for (; cnt > 0; --cnt)
              str += ZEROSTR;
            char buf[32];
            snprintf(buf, sizeof(buf), fmt, part);
            str += buf;
            fmt = fmt2;
          }
          else if (cnt >= 0)
            ++cnt;
          else if (fmt == fmt2)
            str += ZEROSTR;
          if (idx == _data.exponent)
          {
            dot = 1;
            cnt = 0;
            fmt = fmt2;
          }
        }
        if (dot < 0)
        {
          std::string::size_type len = str.length();
          char c;
          do
            c = str[--len];
          while (c == '0');
          str.resize(c == '.' ? len : len + 1);
        }
        if (fmt == fmt2)
        {
          for (idx = 0; idx > _data.exponent; --idx)
            str += ZEROSTR;
        }
      }
      return str.length() ? str : "0";
    }

    std::string toExponential(void) const
    {
      std::string str;
      if (!isZero())
      {
        const char *fmt  = FORMAT1.c_str();
        const char *fmt2 = FORMAT2.c_str();
        int32_t sign, dot, cnt, idx;
        sign = _data.sign;
        dot  = 0;
        cnt  = -1;
        for (idx = BDFP_SIZE - 1; idx >= 0; --idx)
        {
          int32_t part = _data.fractions[idx].bin;
          if (part)
          {
            if (sign)
              str += '-';
            sign = 0;
            for (; cnt > 0; --cnt)
              str += ZEROSTR;
            char buf[32];
            snprintf(buf, sizeof(buf), fmt, part);
            fmt = fmt2;
            if (dot == 0)
            {
              str += buf[0];
              str += '.';
              str += buf + 1;
              dot += std::strlen(buf);
            }
            else
              str += buf;
            cnt = 0;
          }
          else if (cnt >= 0)
            ++cnt;
        }
        std::string::size_type len = str.length();
        char c;
        do
          c = str[--len];
        while (c == '0');
        str.resize(c == '.' ? len : len + 1);
        if ((dot = ((BDFP_SIZE - 1) - _data.exponent) * DIGIT_SIZE + dot - 1))
        {
          str += 'e';
          str += std::to_string(dot);
        }
      }
      return str.length() ? str : "0";
    }

    std::string dump(void) const
    {
      char buf[64];
      std::string str;
      str += "fractions: [";
      for (int32_t i = 0; i < BDFP_SIZE; ++i)
      {
        str += i ? ", " : " ";
        snprintf(buf, sizeof(buf), FORMAT2.c_str(), _data.fractions[(BDFP_SIZE - 1) - i].bin);
        str += buf;
      }
      snprintf(buf, sizeof(buf), " ], exponent: %d, sign: %d", _data.exponent, _data.sign);
      str += buf;
      return str;
    }

    /* constructor's */

    BDFP(void)
    {
      zero();
    }

    BDFP(const BDFP &val)
    {
      from(val);
    }

    BDFP(const storage_t &val)
    {
      from(val);
    }

    BDFP(const std::string &val)
    {
      fromStr(val);
    }

    BDFP(int32_t val)
    {
      fromI64(val);
    }

    BDFP(uint32_t val)
    {
      fromI64(val);
    }

    BDFP(const int64_t &val)
    {
      fromI64(val);
    }

    BDFP(const uint64_t &val)
    {
      fromU64(val);
    }

    BDFP(const double &val)
    {
      fromLDBL(val);
    }

    BDFP(const long double &val)
    {
      fromLDBL(val);
    }

    /* destructor */

    virtual ~BDFP(void)
    {
    }

    /* operator's */

    BDFP &operator=(const storage_t &val)
    {
      return from(val);
    }

    BDFP &operator=(const std::string &val)
    {
      return fromStr(val);
    }

    BDFP &operator=(const BDFP &val)
    {
      from(val);
      return *this;
    }

    BDFP &operator=(int32_t val)
    {
      return fromI64(val);
    }

    BDFP &operator=(uint32_t val)
    {
      return fromI64(val);
    }

    BDFP &operator=(const int64_t &val)
    {
      return fromI64(val);
    }

    BDFP &operator=(const uint64_t &val)
    {
      return fromU64(val);
    }

    BDFP &operator=(const double &val)
    {
      return fromLDBL(val);
    }

    BDFP &operator=(const long double &val)
    {
      return fromLDBL(val);
    }

    BDFP operator+(const BDFP &val)
    {
      return BDFP(*this).add(val);
    }

    BDFP operator+(int32_t val) const
    {
      return BDFP(*this).add(val);
    }

    BDFP operator+(uint32_t val) const
    {
      return BDFP(*this).add(val);
    }

    BDFP operator+(const int64_t &val) const
    {
      return BDFP(*this).add(val);
    }

    BDFP operator+(const uint64_t &val) const
    {
      return BDFP(*this).add(val);
    }

    BDFP operator+(const double &val) const
    {
      return BDFP(*this).add(val);
    }

    BDFP operator+(const long double &val) const
    {
      return BDFP(*this).add(val);
    }

    BDFP &operator+=(const BDFP &val)
    {
      return add(val);
    }

    BDFP &operator+=(int32_t val)
    {
      return add(BDFP(val));
    }

    BDFP &operator+=(uint32_t val)
    {
      return add(BDFP(val));
    }

    BDFP &operator+=(const int64_t &val)
    {
      return add(BDFP(val));
    }

    BDFP &operator+=(const uint64_t &val)
    {
      return add(BDFP(val));
    }

    BDFP &operator+=(const double &val)
    {
      return add(BDFP(val));
    }

    BDFP &operator+=(const long double &val)
    {
      return add(BDFP(val));
    }

    BDFP operator-(const BDFP &val) const
    {
      return BDFP(*this).sub(val);
    }

    BDFP operator-(int32_t val) const
    {
      return BDFP(*this).sub(val);
    }

    BDFP operator-(uint32_t val) const
    {
      return BDFP(*this).sub(val);
    }

    BDFP operator-(const int64_t &val) const
    {
      return BDFP(*this).sub(val);
    }

    BDFP operator-(const uint64_t &val) const
    {
      return BDFP(*this).sub(val);
    }

    BDFP operator-(const double &val) const
    {
      return BDFP(*this).sub(val);
    }

    BDFP operator-(const long double &val) const
    {
      return BDFP(*this).sub(val);
    }

    BDFP &operator-=(const BDFP &val)
    {
      return sub(val);
    }

    BDFP &operator-=(int32_t val)
    {
      return sub(BDFP(val));
    }

    BDFP &operator-=(uint32_t val)
    {
      return sub(BDFP(val));
    }

    BDFP &operator-=(const int64_t &val)
    {
      return sub(BDFP(val));
    }

    BDFP &operator-=(const uint64_t &val)
    {
      return sub(BDFP(val));
    }

    BDFP &operator-=(const double &val)
    {
      return sub(BDFP(val));
    }

    BDFP &operator-=(const long double &val)
    {
      return sub(BDFP(val));
    }

    BDFP operator*(const BDFP &val) const
    {
      return BDFP(*this).mul(val);
    }

    BDFP operator*(int32_t val) const
    {
      return BDFP(*this).mul(val);
    }

    BDFP operator*(uint32_t val) const
    {
      return BDFP(*this).mul(val);
    }

    BDFP operator*(const int64_t &val) const
    {
      return BDFP(*this).mul(val);
    }

    BDFP operator*(const uint64_t &val) const
    {
      return BDFP(*this).mul(val);
    }

    BDFP operator*(const double &val) const
    {
      return BDFP(*this).mul(val);
    }

    BDFP operator*(const long double &val) const
    {
      return BDFP(*this).mul(val);
    }

    BDFP &operator*=(const BDFP &val)
    {
      return mul(val);
    }

    BDFP &operator*=(int32_t val)
    {
      return mul(BDFP(val));
    }

    BDFP &operator*=(uint32_t val)
    {
      return mul(BDFP(val));
    }

    BDFP &operator*=(const int64_t &val)
    {
      return mul(BDFP(val));
    }

    BDFP &operator*=(const uint64_t &val)
    {
      return mul(BDFP(val));
    }

    BDFP &operator*=(const double &val)
    {
      return mul(BDFP(val));
    }

    BDFP &operator*=(const long double &val)
    {
      return mul(BDFP(val));
    }

    BDFP operator/(const BDFP &val) const
    {
      return BDFP(val).div(val);
    }

    BDFP operator/(int32_t val) const
    {
      return BDFP(*this).div(val);
    }

    BDFP operator/(uint32_t val) const
    {
      return BDFP(*this).div(val);
    }

    BDFP operator/(const int64_t &val) const
    {
      return BDFP(*this).div(val);
    }

    BDFP operator/(const uint64_t &val) const
    {
      return BDFP(*this).div(val);
    }

    BDFP operator/(const double &val) const
    {
      return BDFP(*this).div(val);
    }

    BDFP operator/(const long double &val) const
    {
      return BDFP(*this).div(val);
    }

    BDFP &operator/=(const BDFP &val)
    {
      return div(val);
    }

    BDFP &operator/=(int32_t val)
    {
      return div(BDFP(val));
    }

    BDFP &operator/=(uint32_t val)
    {
      return div(BDFP(val));
    }

    BDFP &operator/=(const int64_t &val)
    {
      return div(BDFP(val));
    }

    BDFP &operator/=(const uint64_t &val)
    {
      return div(BDFP(val));
    }

    BDFP &operator/=(const double &val)
    {
      return div(BDFP(val));
    }

    BDFP &operator/=(const long double &val)
    {
      return div(BDFP(val));
    }

    BDFP operator%(const BDFP &val) const
    {
      return BDFP(*this).mod(val);
    }

    BDFP operator%(int32_t val) const
    {
      return BDFP(*this).mod(val);
    }

    BDFP operator%(uint32_t val) const
    {
      return BDFP(*this).mod(val);
    }

    BDFP operator%(const int64_t &val) const
    {
      return BDFP(*this).mod(val);
    }

    BDFP operator%(const uint64_t &val) const
    {
      return BDFP(*this).mod(val);
    }

    BDFP operator%(const double &val) const
    {
      return BDFP(*this).mod(val);
    }

    BDFP operator%(const long double &val) const
    {
      return BDFP(*this).mod(val);
    }

    BDFP &operator%=(const BDFP &val)
    {
      return mod(val);
    }

    BDFP &operator%=(int32_t val)
    {
      return mod(BDFP(val));
    }

    BDFP &operator%=(uint32_t val)
    {
      return mod(BDFP(val));
    }

    BDFP &operator%=(const int64_t &val)
    {
      return mod(BDFP(val));
    }

    BDFP &operator%=(const uint64_t &val)
    {
      return mod(BDFP(val));
    }

    BDFP &operator%=(const double &val)
    {
      return mod(BDFP(val));
    }

    BDFP &operator%=(const long double &val)
    {
      return mod(BDFP(val));
    }

    BDFP operator<<(const BDFP &val)
    {
      BDFP n(*this);
      return n.mul(val);
    }

    BDFP &operator<<=(const BDFP &val)
    {
      BDFP n(val);
      return mul(n);
    }

    BDFP operator>>(const BDFP &val)
    {
      BDFP n(*this);
      return n.div(n);
    }

    BDFP &operator>>=(const BDFP &val)
    {
      BDFP n(val);
      return div(n);
    }

    BDFP &operator++(void)
    {
      return add(ONE);
    }

    BDFP operator++(int)
    {
      BDFP save(*this);
      add(ONE);
      return save;
    }

    BDFP &operator--(void)
    {
      return sub(ONE);
    }

    BDFP operator--(int)
    {
      BDFP save(*this);
      sub(ONE);
      return save;
    }

    BDFP operator-(void) const
    {
      BDFP save(*this);
      save._data.sign ^= 1;
      return save;
    }

    BDFP operator+(void) const
    {
      BDFP save(*this);
      save._data.sign = 0;
      return save;
    }

    bool operator==(const BDFP &val) const
    {
      return cmp(val) == 0;
    }

    bool operator==(int32_t val) const
    {
      return cmp(BDFP(val)) == 0;
    }

    bool operator==(uint32_t val) const
    {
      return cmp(BDFP(val)) == 0;
    }

    bool operator==(const int64_t &val) const
    {
      return cmp(BDFP(val)) == 0;
    }

    bool operator==(const uint64_t &val) const
    {
      return cmp(BDFP(val)) == 0;
    }

    bool operator==(const double &val) const
    {
      return cmp(BDFP(val)) == 0;
    }

    bool operator==(const long double &val) const
    {
      return cmp(BDFP(val)) == 0;
    }

    bool operator!=(const BDFP &val) const
    {
      return cmp(val) != 0;
    }

    bool operator!=(int32_t val) const
    {
      return cmp(BDFP(val)) != 0;
    }

    bool operator!=(uint32_t val) const
    {
      return cmp(BDFP(val)) != 0;
    }

    bool operator!=(const int64_t &val) const
    {
      return cmp(BDFP(val)) != 0;
    }

    bool operator!=(const uint64_t &val) const
    {
      return cmp(BDFP(val)) != 0;
    }

    bool operator!=(const double &val) const
    {
      return cmp(BDFP(val)) != 0;
    }

    bool operator!=(const long double &val) const
    {
      return cmp(BDFP(val)) != 0;
    }

    bool operator<(const BDFP &val) const
    {
      return cmp(val) < 0;
    }

    bool operator<(int32_t val) const
    {
      return cmp(BDFP(val)) < 0;
    }

    bool operator<(uint32_t val) const
    {
      return cmp(BDFP(val)) < 0;
    }

    bool operator<(const int64_t &val) const
    {
      return cmp(BDFP(val)) < 0;
    }

    bool operator<(const uint64_t &val) const
    {
      return cmp(BDFP(val)) < 0;
    }

    bool operator<(const double &val) const
    {
      return cmp(BDFP(val)) < 0;
    }

    bool operator<(const long double &val) const
    {
      return cmp(BDFP(val)) < 0;
    }

    bool operator<=(const BDFP &val) const
    {
      return cmp(val) <= 0;
    }

    bool operator<=(int32_t val) const
    {
      return cmp(BDFP(val)) <= 0;
    }

    bool operator<=(uint32_t val) const
    {
      return cmp(BDFP(val)) <= 0;
    }

    bool operator<=(const int64_t &val) const
    {
      return cmp(BDFP(val)) <= 0;
    }

    bool operator<=(const uint64_t &val) const
    {
      return cmp(BDFP(val)) <= 0;
    }

    bool operator<=(const double &val) const
    {
      return cmp(BDFP(val)) <= 0;
    }

    bool operator<=(const long double &val) const
    {
      return cmp(BDFP(val)) <= 0;
    }

    bool operator>(const BDFP &val) const
    {
      return cmp(val) > 0;
    }

    bool operator>(int32_t val) const
    {
      return cmp(BDFP(val)) > 0;
    }

    bool operator>(uint32_t val) const
    {
      return cmp(BDFP(val)) > 0;
    }

    bool operator>(const int64_t &val) const
    {
      return cmp(BDFP(val)) > 0;
    }

    bool operator>(const uint64_t &val) const
    {
      return cmp(BDFP(val)) > 0;
    }

    bool operator>(const double &val) const
    {
      return cmp(BDFP(val)) > 0;
    }

    bool operator>(const long double &val) const
    {
      return cmp(BDFP(val)) > 0;
    }

    bool operator>=(const BDFP &val) const
    {
      return cmp(val) >= 0;
    }

    bool operator>=(int32_t val) const
    {
      return cmp(BDFP(val)) >= 0;
    }

    bool operator>=(uint32_t val) const
    {
      return cmp(BDFP(val)) >= 0;
    }

    bool operator>=(const int64_t &val) const
    {
      return cmp(BDFP(val)) >= 0;
    }

    bool operator>=(const uint64_t &val) const
    {
      return cmp(BDFP(val)) >= 0;
    }

    bool operator>=(const double &val) const
    {
      return cmp(BDFP(val)) >= 0;
    }

    bool operator>=(const long double &val) const
    {
      return cmp(BDFP(val)) >= 0;
    }

    bool operator!(void) const
    {
      return isZero();
    }

    explicit operator bool(void) const
    {
      return !isZero();
    }

    operator storage_t(void) const
    {
      storage_t rv;
      to(rv);
      return rv;
    }

    operator long double(void) const
    {
      long double rv;
      to(rv);
      return rv;
    }

    operator std::string(void) const
    {
      return toString();
    }
};

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bdfp.h - Billion Decimal Floating Point Calculation Library

This program is free software: you can redistribute it and/or

modify it under the terms of the GNU General Public License as

published by the Free Software Foundation, either version 3 of

the License, or at your option) any later version.

This program is distributed in the hope that it will be useful,

but WITHOUT ANY WARRANTY; without even the implied warranty of

MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.

See the GNU General Public License for more details.

You should have received a copy of the GNU General Public License

along with this program. If not, see <https://www.gnu.org/licenses/>.

#pragma once

#include <stdint.h>

#include <cstdlib>

#include <cctype>

#include <cstring>

#include <string>

#include <cmath>

* [specification]

* 128-bit (fraction: 120-bit, exponent: 7-bit, sign: 1-bit)

* e-(64-3)*9 = e-549

* e+(63+3)*9 = e+594

* 256-bit (fraction: 240-bit, exponent: 15-bit, sign: 1-bit)

* e-(16384-7)*9 = e-147393

* e+(16383+7)*9 = e+147510

* 512-bit (fraction: 480-bit, exponent: 31-bit, sign: 1-bit)

* e-(1073741824-15)*9 = e-1073741767

* e+(1073741823+15)*9 = e+1073741958

#ifndef BDFP_SIZE

#define BDFP_SIZE 4 /* 4 * 4 = 16-bytes (128-bit) */

//#define BDFP_SIZE 8 /* 8 * 4 = 32-bytes (256-bit) */

//#define BDFP_SIZE 16 /* 16 * 4 = 64-bytes (512-bit) */

#endif

#define BDFP_SIZE_MIN 4

#define BDFP_SIZE_MAX 16

#if (BDFP_SIZE < BDFP_SIZE_MIN) || (BDFP_SIZE > BDFP_SIZE_MAX)

#error "invalid decimal size."

#endif

class BDFP

{

private:

static constexpr int32_t POW10[] = {

10,

100,

1000,

10000,

100000,

1000000,

10000000,

100000000,

1000000000

};

static constexpr int32_t NEWTON_LOOP = 1000; /* max loop count (stopper) */

static constexpr int DIGIT_SIZE = 9;

static constexpr int32_t DIGIT_BASE = POW10[DIGIT_SIZE];

static constexpr uint32_t EXPONENT_MSB = 1UL << ((BDFP_SIZE << 1) - 1);

static constexpr uint32_t EXPONENT_MAX = ((EXPONENT_MSB >> 1) - 1);

static constexpr uint32_t EXPONENT_MIN = (~(EXPONENT_MSB - 1) | (EXPONENT_MSB >> 1));

static const std::string FORMAT1;

static const std::string FORMAT2;

static const std::string ZEROSTR;

typedef union

{

struct

{

#if defined(_BIT_FIELDS_HTOL)

uint32_t exponent: 2;

uint32_t fraction: 30;

#else

uint32_t fraction: 30;

uint32_t exponent: 2;

#endif

} bit;

int32_t bin;

} fraction_t;

struct

{

fraction_t fractions[BDFP_SIZE];

int32_t exponent;

int8_t sign;

} _data;

int lowest(void) const

{

int idx;

for (idx = 0; idx < BDFP_SIZE; ++idx)

{

if (_data.fractions[idx].bin)

break;

}

return idx;

}

int highest(void) const

{

int idx;

for (idx = BDFP_SIZE - 1; idx >= 0; --idx)

{

if (_data.fractions[idx].bin)

break;

}

return idx;

}

int cmp0(const BDFP &val) const

{

if (isZero())

return val.isZero() ? 0 : -1;

if (val.isZero())

return 1;

int32_t off = val._data.exponent - _data.exponent;

if (off)

return off < 0 ? -1 : 1;

for (int i = BDFP_SIZE - 1; i >= 0; --i)

{

if ((off = _data.fractions[i].bin - val._data.fractions[i].bin))

return off < 0 ? -1 : 1;

}

return 0;

}

BDFP &add0(const BDFP &val)

{

if (isZero())

from(val);

else if (!val.isZero())

{

int32_t off = _data.exponent - val._data.exponent;

int32_t cy = 0;

int i;

if (off <= 0)

{

for (i = 0; i < BDFP_SIZE; ++i)

{

cy += _data.fractions[i].bin;

if (i - off < BDFP_SIZE)

cy += val._data.fractions[i - off].bin;

if (cy >= DIGIT_BASE)

{

_data.fractions[i].bin = cy - DIGIT_BASE;

cy = 1;

}

else

{

_data.fractions[i].bin = cy;

cy = 0;

}

else

{

for (i = 0; i < BDFP_SIZE; ++i)

{

cy += val._data.fractions[i].bin;

if (i + off < BDFP_SIZE)

cy += _data.fractions[i + off].bin;

if (cy >= DIGIT_BASE)

{

_data.fractions[i].bin = cy - DIGIT_BASE;

cy = 1;

}

else

{

_data.fractions[i].bin = cy;

cy = 0;

}

_data.exponent = val._data.exponent;

}

if (cy)

{

for (i = 1; i < BDFP_SIZE; ++i)

_data.fractions[i - 1].bin = _data.fractions[i].bin;

_data.fractions[BDFP_SIZE - 1].bin = 1;

--_data.exponent;

}

return *this;

}

BDFP &sub0(const BDFP &val)

{

if (isZero())

from(val);

else if (!val.isZero())

{

int i = cmp0(val);

if (i == 0)

zero();

else

{

int32_t off = _data.exponent - val._data.exponent;

int32_t cy = 0;

if ((off < 0) || ((off == 0) && (i >= 0)))

{

for (i = 0; i < BDFP_SIZE; ++i)

{

cy += _data.fractions[i].bin;

if (i - off < BDFP_SIZE)

cy -= val._data.fractions[i - off].bin;

if (cy < 0)

{

_data.fractions[i].bin = cy + DIGIT_BASE;

cy = -1;

}

else

{

_data.fractions[i].bin = cy;

cy = 0;

}

else

{

for (i = 0; i < BDFP_SIZE; ++i)

{

cy += val._data.fractions[i].bin;

if (i + off < BDFP_SIZE)

cy -= _data.fractions[i + off].bin;

if (cy < 0)

{

_data.fractions[i].bin = cy + DIGIT_BASE;

cy = -1;

}

else

{

_data.fractions[i].bin = cy;

cy = 0;

}

_data.exponent = val._data.exponent;

}

off = highest();

if (off < 0)

zero();

else

{

off = BDFP_SIZE - 1 - off;

for (i = BDFP_SIZE - 1; i >= 0; --i)

_data.fractions[i].bin = (i - off >= 0 ? _data.fractions[i - off].bin : 0);

_data.exponent += off;

}

return *this;

}

BDFP &mul0(const BDFP &val)

{

if (isZero() || val.isZero())

zero();

else

{

fraction_t result[BDFP_SIZE * 2] = {};

uint32_t part1, part2;

uint64_t cy;

int i, j = 0;

for (i = 0; i < BDFP_SIZE; ++i)

{

if ((part1 = _data.fractions[i].bin))

{

cy = 0;

for (j = i; j < BDFP_SIZE * 2; ++j)

{

if ((j - i) < BDFP_SIZE)

{

cy += result[j].bin;

if ((part2 = val._data.fractions[j - i].bin))

cy += (uint64_t)part1 * part2;

}

else if (cy == 0)

break;

if (cy >= DIGIT_BASE)

{

std::lldiv_t x = std::lldiv(cy, DIGIT_BASE);

result[j].bin = x.rem;

cy = x.quot;

}

else

{

result[j].bin = cy;

cy = 0;

}

_data.sign ^= val._data.sign;

_data.exponent = (_data.exponent + val._data.exponent) - (BDFP_SIZE - 1) - (j == BDFP_SIZE * 2);

for (i = BDFP_SIZE; i > 0; )

_data.fractions[--i].bin = result[--j].bin;

}

return *this;

}

int mul2(fraction_t *dst, fraction_t *src) const

{

int32_t cy = 0;

for (int i = 0; i < BDFP_SIZE + 1; ++i)

{

if ((cy += (src[i].bin << 1)) >= DIGIT_BASE)

{

dst[i].bin = cy - DIGIT_BASE;

cy = 1;

}

else

{

dst[i].bin = cy;

cy = 0;

}

return cy;

}

int mul10(fraction_t *dst, fraction_t *src) const

{

int cy = 0;

for (int i = 0; i < BDFP_SIZE + 1; ++i)

{

int64_t d = src[i].bin << 1;

d += (d << 2) + cy;

for (cy = 0; d >= DIGIT_BASE; d -= DIGIT_BASE, ++cy)

continue;

dst[i].bin = d;

}

return cy;

}

bool sub(fraction_t *a, fraction_t *b) const

{

int cy, zc, i;

for (cy = zc = i = 0; i < BDFP_SIZE + 1; ++i)

{

if ((a[i].bin -= b[i].bin + cy) < 0)

{

a[i].bin += DIGIT_BASE;

cy = 1;

}

else

{

zc += a[i].bin == 0;

cy = 0;

}

return zc == BDFP_SIZE + 1;

}

int32_t cmp(fraction_t *a, fraction_t *b) const

{

int32_t rv;

for (int i = BDFP_SIZE + 1 - 1; i >= 0; --i)

{

if ((rv = a[i].bin - b[i].bin))

break;

}

return rv;

}

int msd(int32_t val) const

{

int i;

for (i = DIGIT_SIZE - 1; i >= 0; --i)

{

if (POW10[i] <= val)

break;

}

return i + 1;

}

BDFP &div0(const BDFP &val, BDFP *rem = nullptr)

{

int i = msd(_data.fractions[BDFP_SIZE - 1].bin);

int j = msd(val._data.fractions[BDFP_SIZE - 1].bin);

int k, r = 0;

if (rem)

rem->zero();

if ((i == 0) || (j == 0))

zero();

else

{

/* sign and exponent */

_data.sign ^= val._data.sign;

int32_t e = (_data.exponent - val._data.exponent) + (BDFP_SIZE - 1) + (i < j);

if (rem)

{

if (e >= BDFP_SIZE)

{

rem->from(*this);

zero();

return *this;

}

rem->_data.sign = _data.sign;

rem->_data.exponent = (_data.exponent < val._data.exponent) || (i > j)

? val._data.exponent : _data.exponent;

r = ((_data.exponent == val._data.exponent) || (i > j) ? (i - j) : 0);

}

_data.exponent = e;

/* copy data to working variables */

fraction_t dividend[BDFP_SIZE + 1];

fraction_t divisor[4][BDFP_SIZE + 1];

for (k = 0; k < BDFP_SIZE; ++k)

{

divisor[0][k] = val._data.fractions[k];

dividend[k] = _data.fractions[k];

_data.fractions[k].bin = 0;

}

dividend[BDFP_SIZE].bin = 0;

divisor[0][BDFP_SIZE].bin = 0;

/* Align the most significant digit */

for (k = j; i > k; ++k)

mul10(divisor[0], divisor[0]);

for (k = i; k < j; ++k)

mul10(dividend, dividend);

/* multiply the divisor by 2, 4, or 8 */

for (k = 1; k < (int)(sizeof(divisor) / sizeof(divisor[0])); ++k)

mul2(divisor[k], divisor[k - 1]);

/* digit position */

int nod = i - j;

if (nod < 0)

nod += DIGIT_SIZE;

/* optimize of floating point position */

bool declining = (nod == 0);

/* division */

for (i = BDFP_SIZE - 1;; --nod)

{

if (nod < 0)

{

nod = DIGIT_SIZE - 1;

if (--i < 0)

break;

}

if (cmp(dividend, divisor[0]) >= 0)

{

int32_t n = POW10[nod];

for (j = sizeof(divisor) / sizeof(divisor[0]) - 1; j >= 0; --j)

{

if (cmp(dividend, divisor[j]) >= 0)

{

_data.fractions[i].bin += n << j;

if (sub(dividend, divisor[j]))

{

i = -1;

break;

}

if (i < 0)

break;

}

else if (declining)

{

/* optimize of floating point position */

++_data.exponent;

nod = DIGIT_SIZE;

}

declining = false;

if (rem && ((nod == 0) || (nod == DIGIT_SIZE)) && (i == _data.exponent))

{

for (i = 0; (i < BDFP_SIZE) && (dividend[BDFP_SIZE - 1 - i].bin == 0); ++i)

continue;

if (i >= BDFP_SIZE)

rem->zero();

else

{

rem->_data.exponent += i;

for (j = 0; j < BDFP_SIZE; ++j)

rem->_data.fractions[j].bin = (j - i >= 0 ? dividend[j - i].bin : 0);

if (r)

rem->div0(BDFP(POW10[r]));

}

break;

}

mul10(dividend, dividend);

}

return *this;

}

void dump(fraction_t *val, const char *msg = nullptr) const

{

printf(msg ? "%s [" : "[", msg);

for (int i = 0; i < BDFP_SIZE + 1; ++i)

printf(i ? ", %09d" : " %09d", val[BDFP_SIZE - i].bin);

printf(" ]\n");

}

public:

static const BDFP TEN;

static const BDFP FIVE;

static const BDFP TWO;

static const BDFP ONE;

static const BDFP ZERO;

static const BDFP E;

static const BDFP LOG2;

static const BDFP LOG10;

typedef struct

{

fraction_t fractions[BDFP_SIZE];

} storage_t;

bool isZero(void) const

{

return _data.fractions[BDFP_SIZE - 1].bin == 0;

}

BDFP &zero(void)

{

_data = { {}, BDFP_SIZE };

return *this;

}

bool sign(void) const

{

return _data.sign;

}

BDFP &neg(void)

{

_data.sign = isZero() ? 0 : _data.sign ^ 1;

return *this;

}

BDFP &abs(void)

{

_data.sign = 0;

return *this;

}

bool isOdd(void) const

{

return (_data.exponent >= 0) && (_data.exponent < BDFP_SIZE) && (_data.fractions[_data.exponent].bin & 1);

}

int cmp(const BDFP &val) const

{

int sign = val._data.sign - _data.sign;

if (sign)

return sign;

sign = cmp0(val);

return sign == 0 ? 0 : ((sign < 0) ^ _data.sign ? -1 : 1);

}

BDFP &add(const BDFP &val)

{

if (isZero())

from(val);

else if (sign() == val.sign())

add0(val);

else

{

_data.sign = sign() ? cmp0(val) >= 0 : cmp0(val) < 0;

sub0(val);

}

return *this;

}

BDFP &sub(const BDFP &val)

{

if (isZero())

from(val).neg();

else if (sign() != val.sign())

add0(val);

else

{

_data.sign = sign() ? cmp0(val) >= 0 : cmp0(val) < 0;

sub0(val);

}

return *this;

}

BDFP &mul(const BDFP &val)

{

return mul0(val);

}

BDFP &div(const BDFP &val, BDFP *rem = nullptr)

{

return div0(val, rem);

}

BDFP &mod(const BDFP &val)

{

BDFP rem;

div0(val, &rem);

return from(rem);

}

BDFP &modf(BDFP *iptr = nullptr)

{

int msd = -1;

if (iptr)

iptr->from(*this);

for (int i = 0; i < BDFP_SIZE; ++i)

{

if (i >= _data.exponent)

_data.fractions[i].bin = 0;

else

{

if (_data.fractions[i].bin)

msd = i;

if (iptr)

iptr->_data.fractions[i].bin = 0;

}

if (iptr)

{

if (iptr->highest() < 0)

iptr->zero();

}

if (msd < 0)

zero();

else if (BDFP_SIZE - 1 > msd)

{

for (int i = 0; i < BDFP_SIZE; ++i)

_data.fractions[(BDFP_SIZE - 1) - i].bin = (msd - i >= 0 ? _data.fractions[msd - i].bin : 0);

_data.exponent += (BDFP_SIZE - 1) - msd;

}

return *this;

}

BDFP &floor(void)

{

BDFP i;

if (!modf(&i).isZero() && sign())

i.add0(ONE);

return from(i);

}

BDFP &ceil(void)

{

BDFP i;

if (!modf(&i).isZero() && !sign())

i.add0(ONE);

return from(i);

}

BDFP &trunc(void)

{

for (int i = 0; i < BDFP_SIZE; ++i)

{

if (i < _data.exponent)

_data.fractions[i].bin = 0;

}

return *this;

}

typedef enum

{

ROUND_UP,

ROUND_DOWN,

ROUND_CEILING,

ROUND_FLOOR,

ROUND_HALF_UP,

ROUND_HALF_EVEN,

ROUND_HALF_DOWN,

} ROUND_MODE;

BDFP &round(const BDFP &val = 0, ROUND_MODE mode = ROUND_HALF_EVEN)

{

BDFP d, r, q;

d.from(TEN).pow(BDFP(val).trunc().neg());

switch (mode)

{

case ROUND_UP:

add0(BDFP(d).sub0(BDFP(d).div(TEN)));

BDFP(*this).div(d, &r);

sub0(r);

break;

case ROUND_DOWN:

BDFP(*this).div(d, &r);

sub0(r);

break;

case ROUND_CEILING:

if (!sign())

add0(BDFP(d).sub0(BDFP(d).div(TEN)));

BDFP(*this).div(d, &r);

sub0(r);

break;

case ROUND_FLOOR:

if (sign())

add0(BDFP(d).sub0(BDFP(d).div(TEN)));

BDFP(*this).div(d, &r);

sub0(r);

break;

case ROUND_HALF_UP:

add0(BDFP(d).div(TWO));

BDFP(*this).div(d, &r);

sub0(r);

break;

case ROUND_HALF_DOWN:

add0(BDFP(d).div(TWO).sub0(BDFP(d).div(TEN)));

BDFP(*this).div(d, &r);

sub0(r);

break;

case ROUND_HALF_EVEN:

default:

q.from(*this).div(d, &r);

if (r.isZero())

break;

sub0(r);

switch (r.div(BDFP(d).div(TEN)).trunc().cmp0(FIVE))

{

case 0:

if (q.isOdd())

add0(d);

break;

case 1:

add0(d);

break;

}

break;

}

return *this;

}

BDFP &root(const BDFP &val)

{

bool s = sign();

BDFP a, b(ONE), c(val), d, e;

if (val.sign() || (!val.isOdd() && s))

zero();

else

{

a.from(abs());

e.from(c.abs().trunc()).sub(ONE);

for (int i = 0; i < NEWTON_LOOP; ++i)

{

from(BDFP(b).sub(BDFP(b).powi(c).sub(a).div(BDFP(b).powi(e).mul(c))));

if (b.sub(*this).isZero() || (b.cmp0(d) == 0))

break;

d.from(b.from(*this));

}

if (s)

neg();

/* delete the error */

if (_data.fractions[0].bin)

_data.fractions[0].bin -= _data.fractions[0].bin % 10;

}

return *this;

}

BDFP &sqrt(void)

{

BDFP a(*this), b, c, d;

if (a.sign())

zero();

else

{

for (int i = 0; i < NEWTON_LOOP; i++)

{

mul(b.from(c.from(*this))).add(a).div(c.add(c));

if (b.sub(*this).isZero() || (b.cmp0(d) == 0))

break;

d.from(b);

}

return *this;

}

BDFP &exp(void)

{

static const BDFP PREC("1e-" + std::to_string((BDFP_SIZE - 1) * DIGIT_SIZE));

BDFP a(*this), b, c, d;

from(ONE);

if (!a.isZero())

{

bool sign = a.sign();

for (b.from(a.abs()).modf(&a); !a.isZero(); a.sub0(ONE))

mul(E);

if (!b.isZero())

{

d.from(c.from(ONE));

for (a.from(ONE); c.cmp0(PREC) > 0; a.add0(ONE))

d.add0(c.mul(b).div(a));

mul(d);

}

if (sign)

from(BDFP(ONE).div(*this));

}

return *this;

}

BDFP &exp2(void)

{

return mul(LOG2).exp();

}

BDFP &exp10(void)

{

return mul(LOG10).exp();

}

BDFP &exp(const BDFP &val)

{

return mul(log(val)).exp();

}

BDFP &expm1(void)

{

return exp().sub(ONE);

}

BDFP &ln(void)

{

static const BDFP PREC(1e16);

if (cmp(ZERO) <= 0)

zero();

else

pow(BDFP(ONE).div(PREC)).sub(ONE).mul(PREC);

return *this;

}

BDFP &log2(void)

{

return ln().div(LOG2);

}

BDFP &log10(void)

{

return ln().div(LOG10);

}

BDFP &log(const BDFP &val)

{

return ln().div(BDFP(val).ln());

}

BDFP &log1p(void)

{

return add(ONE).ln();

}

BDFP &pow(const BDFP &val)

{

BDFP d(val), n;

d.fraction(n);

return root(d).powi(n);

}

BDFP &powi(const BDFP &val)

{

BDFP f(1);

if (!val.isZero())

{

if (isZero())

f.zero();

else

{

uint32_t n[BDFP_SIZE];

BDFP(val).abs().to(n, BDFP_SIZE);

int i, j;

for (j = BDFP_SIZE; j > 0; --j)

{

if (n[j - 1])

break;

}

for (j <<= 5, i = 0; i < j; ++i)

{

if (n[i >> 5] & (1 << (i & 31)))

{

if (val._data.sign)

f.div0(*this);

else

f.mul0(*this);

}

mul0(*this);

}

return from(f);

}

BDFP &fraction(BDFP &val)

{

BDFP i, n(modf(&i));

zero();

for (int j = 0; j < BDFP_SIZE; ++j)

{

int32_t d = n._data.fractions[j].bin;

if (d == 0)

continue;

if (n._data.exponent > j)

{

from((n._data.exponent - j) * DIGIT_SIZE + 1);

for (j = 0; j < DIGIT_SIZE; ++j)

{

if (d % POW10[j])

break;

sub0(ONE);

}

break;

}

from(BDFP(TEN).powi(*this));

if (i.from(BDFP(*this).gcd(val.from(i.mul(*this)).add0(n.mul(*this)))).cmp0(ONE))

{

val.div(i);

div(i);

}

return *this;

}

BDFP &gcd(const BDFP &val)

{

BDFP a(*this), b(val), r;

if (a.trunc().isZero() || b.trunc().isZero())

zero();

else

{

int i = cmp0(val);

if (i == 0)

from(ONE);

else

{

if (i > 0)

from(a);

else

{

from(b);

b.from(a);

}

while (!b.isZero())

{

r.from(mod(b));

from(b);

b.from(r);

}

return *this;

}

void to(storage_t &val) const

{

uint32_t exponent = (_data.sign ? EXPONENT_MSB : 0) | (_data.exponent & (EXPONENT_MSB - 1));

for (int i = 0; i < BDFP_SIZE; ++i)

{

val.fractions[i] = _data.fractions[i];

val.fractions[i].bit.exponent = exponent >> (i << 1);

}

void to(long double &val) const

{

val = std::stod(toExponential());

}

bool to(uint32_t *buf, size_t len) const

{

if (buf)

{

size_t i;

int32_t j;

uint64_t cy;

for (i = 0; i < len; ++i)

buf[i] = 0;

for (j = BDFP_SIZE - 1; j >= _data.exponent; --j)

{

cy = (j >= 0 ? _data.fractions[j].bin : 0);

for (i = 0; i < len; ++i)

{

if ((cy += (uint64_t)buf[i] * DIGIT_BASE) == 0)

break;

buf[i] = cy;

cy >>= 32;

}

if (cy)

return false; /* overflow */

}

if (_data.sign)

{

cy = 1;

for (i = 0; i < len; ++i)

{

cy += ~buf[i];

if (cy > (uint64_t)UINT32_MAX)

{

buf[i] = 0;

cy = 1;

}

else

{

buf[i] = cy;

cy = 0;

}

return true;

}

return false;

}

BDFP &from(const BDFP &val)

{

_data = val._data;

return *this;

}

BDFP &from(const storage_t &val)

{

zero();

_data.exponent = 0;

for (int i = 0; i < BDFP_SIZE; ++i)

{

fraction_t part = val.fractions[i];

_data.fractions[i].bin = part.bit.fraction;

_data.exponent |= part.bit.exponent << (i << 1);

}

_data.sign = _data.exponent & EXPONENT_MSB ? 1 : 0;

if (_data.exponent & (EXPONENT_MSB >> 1))

_data.exponent |= ~(EXPONENT_MSB - 1);

return *this;

}

BDFP &fromI64(const int64_t &val)

{

zero();

_data.sign = val < 0;

if (!_data.sign && (val <= DIGIT_BASE))

{

_data.fractions[BDFP_SIZE - 1].bin = val;

_data.exponent = BDFP_SIZE - 1;

}

else

{

std::lldiv_t x = std::lldiv(val, DIGIT_BASE);

if (_data.sign)

{

x.quot = std::abs(x.quot);

x.rem = std::abs(x.rem);

}

if (x.quot >= DIGIT_BASE)

{

std::lldiv_t y = std::lldiv(x.quot, DIGIT_BASE);

_data.fractions[BDFP_SIZE - 3].bin = x.rem;

_data.fractions[BDFP_SIZE - 2].bin = y.rem;

_data.fractions[BDFP_SIZE - 1].bin = y.quot;

_data.exponent = BDFP_SIZE - 3;

}

else if (x.quot)

{

_data.fractions[BDFP_SIZE - 2].bin = x.rem;

_data.fractions[BDFP_SIZE - 1].bin = x.quot;

_data.exponent = BDFP_SIZE - 2;

}

else

{

_data.fractions[BDFP_SIZE - 1].bin = x.rem;

_data.exponent = BDFP_SIZE - 1;

}

return *this;

}

BDFP &fromU64(const uint64_t &val)

{

if (val <= INT64_MAX)

fromI64(val);

else

{

fromI64(INT64_MIN);

_data.sign = 0;

add(BDFP(val + INT64_MIN));

}

return *this;

}

BDFP &fromDBL(const double &val, int len = __DBL_DIG__)

{

return fromLDBL(val, len);

}

BDFP &fromLDBL(const long double &val, int len = __LDBL_DIG__)

{

char buf[64];

snprintf(buf, sizeof(buf), "%*Lg", len, val);

return fromStr(std::string(buf));

}

BDFP &fromStr(const std::string &str)

{

std::string num, dec;

std::string::size_type len, pos, off = 0;

int32_t exp = 0;

int i;

char c;

zero();

c = str[off++];

while (c == ' ');

if ((c == '-') || (c == '+'))

{

if (c == '-')

_data.sign = 1;

c = str[off++];

}

while ((c == ' ') || (c == '0'))

c = str[off++];

while (std::isdigit(c))

{

num += c;

c = str[off++];

}

if (c == '.')

{

len = 0;

c = str[off++];

while (std::isdigit(c))

{

dec += c;

if (c != '0')

len = dec.length();

c = str[off++];

}

dec.resize(len);

}

if ((c == 'E') || (c == 'e'))

exp = std::strtol(&str[off], nullptr, 10);

if (num.length() + dec.length() == 0)

zero();

else

{

if (exp < 0)

{

std::string::size_type len = num.length();

if ((exp += len) <= 0)

{

dec.insert(0, num);

num.clear();

}

else

{

dec.insert(0, num.substr(exp));

num.resize(exp);

exp = 0;

}

else if (exp > 0)

{

std::string::size_type len = dec.length();

if ((exp -= len) >= 0)

{

num += dec;

dec.clear();

}

else

{

num += dec.substr(0, len + exp);

dec = dec.substr(len + exp);

exp = 0;

}

_data.exponent = i = BDFP_SIZE;

if (num.length())

{

if ((len = (num.length() + exp) % DIGIT_SIZE) == 0)

len = DIGIT_SIZE;

if (num.length() < len)

{

exp -= (off = len - num.length());

num += ZEROSTR.substr(0, off);

}

else if (num.length() > len)

{

if ((off = (num.length() - len) % DIGIT_SIZE))

{

exp -= (off = DIGIT_SIZE - off);

num += ZEROSTR.substr(0, off);

}

for (off = 0; i > 0; off += pos)

{

pos = num.length() - off;

if (pos == 0)

break;

if (pos > len)

pos = len;

len = DIGIT_SIZE;

_data.fractions[--i].bin = std::stoul(num.substr(off, pos));

}

_data.exponent = i - (num.length() - off + exp) / DIGIT_SIZE;

}

if (dec.length() && (i > 0))

{

exp = -exp;

if ((off = exp % DIGIT_SIZE))

{

exp -= off;

dec.insert(0, ZEROSTR.substr(0, off));

}

for (off = 0; i > 0; off += DIGIT_SIZE)

{

if ((len = dec.length() - off) == 0)

break;

if (len < DIGIT_SIZE)

dec += ZEROSTR.substr(0, DIGIT_SIZE - len);

int32_t part = std::stoul(dec.substr(off, DIGIT_SIZE));

if ((i == BDFP_SIZE) && (part == 0))

++_data.exponent;

else

_data.fractions[--i].bin = part;

}

_data.exponent += exp / DIGIT_SIZE;

}

return *this;

}

std::string toString(void) const

{

std::string str;

if (!isZero())

{

const char *fmt = FORMAT1.c_str();

const char *fmt2 = FORMAT2.c_str();

int32_t sign, dot, cnt, idx;

sign = _data.sign;

dot = 0;

cnt = -1;

if (_data.exponent >= BDFP_SIZE)

{

dot = 1;

cnt = _data.exponent - BDFP_SIZE;

fmt = fmt2;

}

for (idx = BDFP_SIZE - 1; idx >= 0; --idx)

{

int32_t part = _data.fractions[idx].bin;

if (part)

{

if (sign)

str += '-';

if (dot > 0)

{

if (str.length() == (std::string::size_type)sign)

str += '0';

str += '.';

dot = -1;

}

sign = 0;

for (; cnt > 0; --cnt)

str += ZEROSTR;

char buf[32];

snprintf(buf, sizeof(buf), fmt, part);

str += buf;

fmt = fmt2;

}

else if (cnt >= 0)

++cnt;

else if (fmt == fmt2)

str += ZEROSTR;

if (idx == _data.exponent)

{

dot = 1;

cnt = 0;

fmt = fmt2;

}

if (dot < 0)

{

std::string::size_type len = str.length();

char c;

c = str[--len];

while (c == '0');

str.resize(c == '.' ? len : len + 1);

}

if (fmt == fmt2)

{

for (idx = 0; idx > _data.exponent; --idx)

str += ZEROSTR;

}

return str.length() ? str : "0";

}

std::string toExponential(void) const

{

std::string str;

if (!isZero())

{

const char *fmt = FORMAT1.c_str();

const char *fmt2 = FORMAT2.c_str();

int32_t sign, dot, cnt, idx;

sign = _data.sign;

dot = 0;

cnt = -1;

for (idx = BDFP_SIZE - 1; idx >= 0; --idx)

{

int32_t part = _data.fractions[idx].bin;

if (part)

{

if (sign)

str += '-';

sign = 0;

for (; cnt > 0; --cnt)

str += ZEROSTR;

char buf[32];

snprintf(buf, sizeof(buf), fmt, part);

fmt = fmt2;

if (dot == 0)

{

str += buf[0];

str += '.';

str += buf + 1;

dot += std::strlen(buf);

}

else

str += buf;

cnt = 0;

}

else if (cnt >= 0)

++cnt;

}

std::string::size_type len = str.length();

char c;

c = str[--len];

while (c == '0');

str.resize(c == '.' ? len : len + 1);

if ((dot = ((BDFP_SIZE - 1) - _data.exponent) * DIGIT_SIZE + dot - 1))

{

str += 'e';

str += std::to_string(dot);

}

return str.length() ? str : "0";

}

std::string dump(void) const

{

char buf[64];

std::string str;

str += "fractions: [";

for (int32_t i = 0; i < BDFP_SIZE; ++i)

{

str += i ? ", " : " ";

snprintf(buf, sizeof(buf), FORMAT2.c_str(), _data.fractions[(BDFP_SIZE - 1) - i].bin);

str += buf;

}

snprintf(buf, sizeof(buf), " ], exponent: %d, sign: %d", _data.exponent, _data.sign);

str += buf;

return str;

}

/* constructor's */

BDFP(void)

{

zero();

}

BDFP(const BDFP &val)

{

from(val);

}

BDFP(const storage_t &val)

{

from(val);

}

BDFP(const std::string &val)

{

fromStr(val);

}

BDFP(int32_t val)

{

fromI64(val);

}

BDFP(uint32_t val)

{

fromI64(val);

}

BDFP(const int64_t &val)

{

fromI64(val);

}

BDFP(const uint64_t &val)

{

fromU64(val);

}

BDFP(const double &val)

{

fromLDBL(val);

}

BDFP(const long double &val)

{

fromLDBL(val);

}

/* destructor */

virtual ~BDFP(void)

{

}

/* operator's */

BDFP &operator=(const storage_t &val)

{

return from(val);

}

BDFP &operator=(const std::string &val)

{

return fromStr(val);

}

BDFP &operator=(const BDFP &val)

{

from(val);

return *this;

}

BDFP &operator=(int32_t val)

{

return fromI64(val);

}

BDFP &operator=(uint32_t val)

{

return fromI64(val);

}

BDFP &operator=(const int64_t &val)

{

return fromI64(val);

}

BDFP &operator=(const uint64_t &val)

{

return fromU64(val);

}

BDFP &operator=(const double &val)

{

return fromLDBL(val);

}

BDFP &operator=(const long double &val)

{

return fromLDBL(val);

}

BDFP operator+(const BDFP &val)

{

return BDFP(*this).add(val);

}

BDFP operator+(int32_t val) const

{

return BDFP(*this).add(val);

}

BDFP operator+(uint32_t val) const

{

return BDFP(*this).add(val);

}

BDFP operator+(const int64_t &val) const

{

return BDFP(*this).add(val);

}

BDFP operator+(const uint64_t &val) const

{

return BDFP(*this).add(val);

}

BDFP operator+(const double &val) const

{

return BDFP(*this).add(val);

}

BDFP operator+(const long double &val) const

{

return BDFP(*this).add(val);

}

BDFP &operator+=(const BDFP &val)

{

return add(val);

}

BDFP &operator+=(int32_t val)

{

return add(BDFP(val));

}

BDFP &operator+=(uint32_t val)

{

return add(BDFP(val));

}

BDFP &operator+=(const int64_t &val)

{

return add(BDFP(val));

}

BDFP &operator+=(const uint64_t &val)

{

return add(BDFP(val));

}

BDFP &operator+=(const double &val)

{

return add(BDFP(val));

}

BDFP &operator+=(const long double &val)

{

return add(BDFP(val));

}

BDFP operator-(const BDFP &val) const

{

return BDFP(*this).sub(val);

}

BDFP operator-(int32_t val) const

{

return BDFP(*this).sub(val);

}

BDFP operator-(uint32_t val) const

{

return BDFP(*this).sub(val);

}

BDFP operator-(const int64_t &val) const

{

return BDFP(*this).sub(val);

}

BDFP operator-(const uint64_t &val) const

{

return BDFP(*this).sub(val);

}

BDFP operator-(const double &val) const

{

return BDFP(*this).sub(val);

}

BDFP operator-(const long double &val) const

{

return BDFP(*this).sub(val);

}

BDFP &operator-=(const BDFP &val)

{

return sub(val);

}

BDFP &operator-=(int32_t val)

{

return sub(BDFP(val));

}

BDFP &operator-=(uint32_t val)

{

return sub(BDFP(val));

}

BDFP &operator-=(const int64_t &val)

{

return sub(BDFP(val));

}

BDFP &operator-=(const uint64_t &val)

{

return sub(BDFP(val));

}

BDFP &operator-=(const double &val)

{

return sub(BDFP(val));

}

BDFP &operator-=(const long double &val)

{

return sub(BDFP(val));

}

BDFP operator*(const BDFP &val) const

{

return BDFP(*this).mul(val);

}

BDFP operator*(int32_t val) const

{

return BDFP(*this).mul(val);

}

BDFP operator*(uint32_t val) const

{

return BDFP(*this).mul(val);

}

BDFP operator*(const int64_t &val) const

{

return BDFP(*this).mul(val);

}

BDFP operator*(const uint64_t &val) const

{

return BDFP(*this).mul(val);

}

BDFP operator*(const double &val) const

{

return BDFP(*this).mul(val);

}

BDFP operator*(const long double &val) const

{

return BDFP(*this).mul(val);

}

BDFP &operator*=(const BDFP &val)

{

return mul(val);

}

BDFP &operator*=(int32_t val)

{

return mul(BDFP(val));

}

BDFP &operator*=(uint32_t val)

{

return mul(BDFP(val));

}

BDFP &operator*=(const int64_t &val)

{

return mul(BDFP(val));

}

BDFP &operator*=(const uint64_t &val)

{

return mul(BDFP(val));

}

BDFP &operator*=(const double &val)

{

return mul(BDFP(val));

}

BDFP &operator*=(const long double &val)

{

return mul(BDFP(val));

}

BDFP operator/(const BDFP &val) const

{

return BDFP(val).div(val);

}

BDFP operator/(int32_t val) const

{

return BDFP(*this).div(val);

}

BDFP operator/(uint32_t val) const

{

return BDFP(*this).div(val);

}

BDFP operator/(const int64_t &val) const

{

return BDFP(*this).div(val);

}

BDFP operator/(const uint64_t &val) const

{

return BDFP(*this).div(val);

}

BDFP operator/(const double &val) const

{

return BDFP(*this).div(val);

}

BDFP operator/(const long double &val) const

{

return BDFP(*this).div(val);

}

BDFP &operator/=(const BDFP &val)

{

return div(val);

}

BDFP &operator/=(int32_t val)

{

return div(BDFP(val));

}

BDFP &operator/=(uint32_t val)

{

return div(BDFP(val));

}

BDFP &operator/=(const int64_t &val)

{

return div(BDFP(val));

}

BDFP &operator/=(const uint64_t &val)

{

return div(BDFP(val));

}

BDFP &operator/=(const double &val)

{

return div(BDFP(val));

}

BDFP &operator/=(const long double &val)

{

return div(BDFP(val));

}

BDFP operator%(const BDFP &val) const

{

return BDFP(*this).mod(val);

}

BDFP operator%(int32_t val) const

{

return BDFP(*this).mod(val);

}

BDFP operator%(uint32_t val) const

{

return BDFP(*this).mod(val);

}

BDFP operator%(const int64_t &val) const

{

return BDFP(*this).mod(val);

}

BDFP operator%(const uint64_t &val) const

{

return BDFP(*this).mod(val);

}

BDFP operator%(const double &val) const

{

return BDFP(*this).mod(val);

}

BDFP operator%(const long double &val) const

{

return BDFP(*this).mod(val);

}

BDFP &operator%=(const BDFP &val)

{

return mod(val);

}

BDFP &operator%=(int32_t val)

{

return mod(BDFP(val));

}

BDFP &operator%=(uint32_t val)

{

return mod(BDFP(val));

}

BDFP &operator%=(const int64_t &val)

{

return mod(BDFP(val));

}

BDFP &operator%=(const uint64_t &val)

{

return mod(BDFP(val));

}

BDFP &operator%=(const double &val)

{

return mod(BDFP(val));

}

BDFP &operator%=(const long double &val)

{

return mod(BDFP(val));

}

BDFP operator<<(const BDFP &val)

{

BDFP n(*this);

return n.mul(val);

}

BDFP &operator<<=(const BDFP &val)

{

BDFP n(val);

return mul(n);

}

BDFP operator>>(const BDFP &val)

{

BDFP n(*this);

return n.div(n);

}

BDFP &operator>>=(const BDFP &val)

{

BDFP n(val);

return div(n);

}

BDFP &operator++(void)

{

return add(ONE);

}

BDFP operator++(int)

{

BDFP save(*this);

add(ONE);

return save;

}

BDFP &operator--(void)

{

return sub(ONE);

}

BDFP operator--(int)

{

BDFP save(*this);

sub(ONE);

return save;

}

BDFP operator-(void) const

{

BDFP save(*this);

save._data.sign ^= 1;

return save;

}

BDFP operator+(void) const

{

BDFP save(*this);

save._data.sign = 0;

return save;

}

bool operator==(const BDFP &val) const

{

return cmp(val) == 0;

}

bool operator==(int32_t val) const

{

return cmp(BDFP(val)) == 0;

}

bool operator==(uint32_t val) const

{

return cmp(BDFP(val)) == 0;

}

bool operator==(const int64_t &val) const

{

return cmp(BDFP(val)) == 0;

}

bool operator==(const uint64_t &val) const

{

return cmp(BDFP(val)) == 0;

}

bool operator==(const double &val) const

{

return cmp(BDFP(val)) == 0;

}

bool operator==(const long double &val) const

{

return cmp(BDFP(val)) == 0;

}

bool operator!=(const BDFP &val) const

{

return cmp(val) != 0;

}

bool operator!=(int32_t val) const

{

return cmp(BDFP(val)) != 0;

}

bool operator!=(uint32_t val) const

{

return cmp(BDFP(val)) != 0;

}

bool operator!=(const int64_t &val) const

{

return cmp(BDFP(val)) != 0;

}

bool operator!=(const uint64_t &val) const

{

return cmp(BDFP(val)) != 0;

}

bool operator!=(const double &val) const

{

return cmp(BDFP(val)) != 0;

}

bool operator!=(const long double &val) const

{

return cmp(BDFP(val)) != 0;

}

bool operator<(const BDFP &val) const

{

return cmp(val) < 0;

}

bool operator<(int32_t val) const

{

return cmp(BDFP(val)) < 0;

}

bool operator<(uint32_t val) const

{

return cmp(BDFP(val)) < 0;

}

bool operator<(const int64_t &val) const

{

return cmp(BDFP(val)) < 0;

}

bool operator<(const uint64_t &val) const

{

return cmp(BDFP(val)) < 0;

}

bool operator<(const double &val) const

{

return cmp(BDFP(val)) < 0;

}

bool operator<(const long double &val) const

{

return cmp(BDFP(val)) < 0;

}

bool operator<=(const BDFP &val) const

{

return cmp(val) <= 0;

}

bool operator<=(int32_t val) const

{

return cmp(BDFP(val)) <= 0;

}

bool operator<=(uint32_t val) const

{

return cmp(BDFP(val)) <= 0;

}

bool operator<=(const int64_t &val) const

{

return cmp(BDFP(val)) <= 0;

}

bool operator<=(const uint64_t &val) const

{

return cmp(BDFP(val)) <= 0;

}

bool operator<=(const double &val) const

{

return cmp(BDFP(val)) <= 0;

}

bool operator<=(const long double &val) const

{

return cmp(BDFP(val)) <= 0;

}

bool operator>(const BDFP &val) const

{

return cmp(val) > 0;

}

bool operator>(int32_t val) const

{

return cmp(BDFP(val)) > 0;

}

bool operator>(uint32_t val) const

{

return cmp(BDFP(val)) > 0;

}

bool operator>(const int64_t &val) const

{

return cmp(BDFP(val)) > 0;

}

bool operator>(const uint64_t &val) const

{

return cmp(BDFP(val)) > 0;

}

bool operator>(const double &val) const

{

return cmp(BDFP(val)) > 0;

}

bool operator>(const long double &val) const

{

return cmp(BDFP(val)) > 0;

}

bool operator>=(const BDFP &val) const

{

return cmp(val) >= 0;

}

bool operator>=(int32_t val) const

{

return cmp(BDFP(val)) >= 0;

}

bool operator>=(uint32_t val) const

{

return cmp(BDFP(val)) >= 0;

}

bool operator>=(const int64_t &val) const

{

return cmp(BDFP(val)) >= 0;

}

bool operator>=(const uint64_t &val) const

{

return cmp(BDFP(val)) >= 0;

}

bool operator>=(const double &val) const

{

return cmp(BDFP(val)) >= 0;

}

bool operator>=(const long double &val) const

{

return cmp(BDFP(val)) >= 0;

}

bool operator!(void) const

{

return isZero();

}

explicit operator bool(void) const

{

return !isZero();

}

operator storage_t(void) const

{

storage_t rv;

to(rv);

return rv;

}

operator long double(void) const

{

long double rv;

to(rv);

return rv;

}

operator std::string(void) const

{

return toString();

}

};

/*
  bdfp.cpp - Billion Decimal Floating Point Calculation Library

  Copyright (c) 2024 Sasapea's Lab. All right reserved.

  This program is free software: you can redistribute it and/or
  modify it under the terms of the GNU General Public License as
  published by the Free Software Foundation, either version 3 of
  the License, or at your option) any later version.

  This program is distributed in the hope that it will be useful,
  but WITHOUT ANY WARRANTY; without even the implied warranty of
  MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.
  See the GNU General Public License for more details.

  You should have received a copy of the GNU General Public License
  along with this program. If not, see <https://www.gnu.org/licenses/>.
*/
#include "bdfp.h"

const std::string BDFP::FORMAT1 = "%lu";
const std::string BDFP::FORMAT2 = "%0" + std::to_string(BDFP::DIGIT_SIZE) + "lu";
const std::string BDFP::ZEROSTR = std::to_string(BDFP::DIGIT_BASE).substr(1);

const BDFP BDFP::TEN(10);
const BDFP BDFP::FIVE(5);
const BDFP BDFP::TWO(2);
const BDFP BDFP::ONE(1);
const BDFP BDFP::ZERO(0);
const BDFP BDFP::E("2.7182818284590452353602874713526624977572470936999595749669676277240766303535475945713821785251664274274663919320030599218174135966290435729");
const BDFP BDFP::LOG2(BDFP(TWO).ln());
const BDFP BDFP::LOG10(BDFP(TEN).ln());

bdfp.cpp - Billion Decimal Floating Point Calculation Library

This program is free software: you can redistribute it and/or

modify it under the terms of the GNU General Public License as

published by the Free Software Foundation, either version 3 of

the License, or at your option) any later version.

This program is distributed in the hope that it will be useful,

but WITHOUT ANY WARRANTY; without even the implied warranty of

MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.

See the GNU General Public License for more details.

You should have received a copy of the GNU General Public License

along with this program. If not, see <https://www.gnu.org/licenses/>.

#include "bdfp.h"

const std::string BDFP::FORMAT1 = "%lu";

const std::string BDFP::FORMAT2 = "%0" + std::to_string(BDFP::DIGIT_SIZE) + "lu";

const std::string BDFP::ZEROSTR = std::to_string(BDFP::DIGIT_BASE).substr(1);

const BDFP BDFP::TEN(10);

const BDFP BDFP::FIVE(5);

const BDFP BDFP::TWO(2);

const BDFP BDFP::ONE(1);

const BDFP BDFP::ZERO(0);

const BDFP BDFP::E("2.7182818284590452353602874713526624977572470936999595749669676277240766303535475945713821785251664274274663919320030599218174135966290435729");

const BDFP BDFP::LOG2(BDFP(TWO).ln());

const BDFP BDFP::LOG10(BDFP(TEN).ln());

投稿者: researcher

10億単位で演算する10進浮動小数点ライブラリ