浮点数的二进制表示学习笔记
基础知识：
十进制转十六进制；
十六进制转二进制；
IEEE制定的浮点数表示规则；
了解：
目前C/C++编译器标准都遵照IEEE制定的浮点数表示法来进行float,double运算。这种结构是一种科学计数法，用符号、指数和尾数来表示，底数定为2——即把一个浮点数表示为尾数乘以2的指数次方再添上符号。下面是具体的规格：
符号位 阶码 尾数 长度
float 1 8 23 32
double 1 11 52 64
以下通过几个例子讲解浮点数如何转换为二进制数
例一：
已知：double类型38414.4。
求：其对应的二进制表示。
分析：double类型共计64位，折合8字节。由最高到最低位分别是第63、62、61、……、0位：
最高位63位是符号位，1表示该数为负，0表示该数为正；
62-52位，一共11位是指数位；
51-0位，一共52位是尾数位。
步骤：按照IEEE浮点数表示法，下面先把38414.4转换为十六进制数。
把整数部和小数部分开处理:整数部直接化十六进制：960E。小数的处理:
0.4=0.5*0+0.25*1+0.125*1+0.0625*0+……
实际上这永远算不完！这就是著名的浮点数精度问题。所以直到加上前面的整数部分算够53位就行了。隐藏位技术：最高位的1不写入内存（最终保留下来的还是52位）。
如果你够耐心，手工算到53位那么因该是：38414.4(10)=1001011000001110.0110011001100110011001100110011001100(2)
科学记数法为：1.001011000001110 0110011001100110011001100110011001100，右移了15位，所以指数为15。或者可以如下理解：
1.001011000001110 0110011001100110011001100110011001100×2^15
于是来看阶码，按IEEE标准一共11位，可以表示范围是-1024 ~ 1023。因为指数可以为负，为了便于计算，规定都先加上1023(2^10-1)，在这里，阶码：15+1023=1038。二进制表示为：100 00001110；
符号位：因为38414.4为正对应 为0；
合在一起（注：尾数二进制最高位的1不要）：
01000000 11100010 11000001 110 01100 11001100 11001100 11001100 11001100
例二：
已知：整数3490593(16进制表示为0x354321)。
求：其对应的浮点数3490593.0的二进制表示。
解法如下：
先求出整数3490593的二进制表示：
H: 3 5 4 3 2 1 （十六进制表示）
B: 0011 0101 0100 0011 0010 0001 （二进制表示）
│←──────21─────→│
即：
1.1010101000011001000012×221
可见，从左算起第一个1后有21位，我们将这21为作为浮点数的小数表示，单精度浮点数float由符号位1位，指数域位k=8位，小数域位(尾数)n=23位构成，因此对上面得到的21位小数位我们还需要补上2个0，得到浮点数的小数域表示为：
1 0101 0100 0011 0010 0001 00
float类型的偏置量Bias=2k-1-1=28-1-1=127，但还要补上刚才因为右移作为小数部分的21位，因此偏置量为127+21=148，就是IEEE浮点数表示标准：
V = (-1)s×M×2E
E = e-Bias
中的e，此前计算Bias=127，刚好验证了E=148-127=21。
将148转为二进制表示为10010100，加上符号位0，最后得到二进制浮点数表示1001010010101010000110010000100，其16进制表示为：
H: 4 A 5 5 0 C 8 4
B: 0100 1010 0101 0101 0000 1100 1000 0100
|←──── 21 ─────→ |
1|←─8 ─→||←───── 23 ─────→ |
这就是浮点数3490593.0(0x4A550C84)的二进制表示。
例三：
0.5的二进制形式是0.1
它用浮点数的形式写出来是如下格式
0 01111110 00000000000000000000000
符号位 阶码 小数位
正数符号位为0，负数符号位为1
阶码是以2为底的指数
小数位表示小数点后面的数字
下面我们来分析一下0.5是如何写成0 01111110 00000000000000000000000
首先0.5是正数所以符号位为0
再来看阶码部分,0.5的二进制数是0.1,而0.1是1.0*2^(-1),所以我们总结出来:
要把二进制数变成(1.f)*2^(exponent)的形式,其中exponent是指数
而由于阶码有正负之分所以阶码=127+exponent;
即阶码=127+(-1)=126 即 01111110
余下的小数位为二进制小数点后面的数字,即00000000000000000000000
由以上分析得0.5的浮点数存储形式为0 01111110 00000000000000000000000
注：如果只有小数部分,那么需要右移小数点. 比如右移3位才能放到第一个1的后面, 阶码就是127-3=124.
例四 （20.59375）10 =（10100.10011 ）2
首先分别将整数和分数部分转换成二进制数：
20.59375＝10100.10011
然后移动小数点，使其在第1，2位之间
10100.10011＝1.010010011×2^4 即e＝4
于是得到：
S＝0， E＝4＋127＝131， M＝010010011
最后得到32位浮点数的二进制存储格式为：
0100 1001 1010 0100 1100 0000 0000 0000＝(41A4C000)16
例五：
-12.5转为单精度二进制表示
12.5:
1. 整数部分12，二进制为1100; 小数部分0.5, 二进制是.1，先把他们连起来，从第一个1数起取24位（后面补0）：
1100.10000000000000000000
这部分是有效数字。（把小数点前后两部分连起来再取掉头前的1，就是尾数）
2. 把小数点移到第一个1的后面，需要左移3位（1.10010000000000000000000*2^3）, 加上偏移量127：127+3=130，二进制是10000010，这是阶码。
3. -12.5是负数，所以符号位是1。把符号位，阶码和尾数连起来。注意，尾数的第一位总是1，所以规定不存这一位的1，只取后23位：
1 10000010 10010000000000000000000
把这32位按8位一节整理一下，得：
11000001 01001000 00000000 00000000
就是十六进制的 C1480000.
例六：
2.025675
1. 整数部分2，二进制为10; 小数部分0.025675, 二进制是.0000011010010010101001，先把他们连起来，从第一个1数起取24位（后面补0）：
10.0000011010010010101001
这部分是有效数字。把小数点前后两部分连起来再取掉头前的1，就是尾数: 00000011010010010101001
2. 把小数点移到第一个1的后面，左移了1位, 加上偏移量127：127+1=128，二进制是10000000，这是阶码。
3. 2.025675是正数，所以符号位是0。把符号位，阶码和尾数连起来：
0 10000000 00000011010010010101001
把这32位按8位一节整理一下，得：
01000000 00000001 10100100 10101001
就是十六进制的 4001A4A9.
例七：
(逆向求十进制整数)一个浮点二进制数手工转换成十进制数的例子：
假设浮点二进制数是 1011 1101 0100 0000 0000 0000 0000 0000
按1，8，23位分成三段：
1 01111010 10000000000000000000000
最后一段是尾数。前面加上"1.", 就是 1.10000000000000000000000
下面确定小数点位置。由E = e-Bias，阶码E是01111010，加上00000101才是01111111（127），
所以他减去127的偏移量得e=-5。（或者化成十进制得122，122-127=-5）。
因此尾数1.10（后面的0不写了）是小数点右移5位的结果。要复原它就要左移5位小数点，得0.0000110, 即十进制的0.046875 。
最后是符号：1代表负数，所以最后的结果是 -0.046875 。
注意：其他机器的浮点数表示方法可能与此不同. 不能任意移植。
再看一例(类似例七)：
比如：53004d3e
二进制表示为：
01010011000000000100110100111110
按照1个符号 8个指数 23个小数位划分
0 10100110 00000000100110100111110
正确的结果转出来应该是551051722752.0
该怎么算？
好，我们根据IEEE的浮点数表示规则划分，得到这个浮点数的小数位是：
00000000100110100111110
那么它的二进制表示就应该是：
1.000000001001101001111102 × 239
这是怎么来的呢？ 别急，听我慢慢道来。
标准化公式中的M要求在规格化的情况下，取值范围1<M<(2-ε)
正因为如此，我们才需要对原始的整数二进制表示做偏移，偏移多少呢？偏移2E。
这个“E”怎么算？上面的239怎么得来的呢？浮点数表示中的8位指数为就是告诉这个的。我们知道：
E = e-Bias
那么根据指数位:
101001102=>16610
即e=166，由此算出E=e-Bias=166-127=39，就是说将整数二进制表示转为标准的浮点数二进制表示的时候需要将小数点左移39位，好，我们现在把它还原得到整数的二进制表示：
1 00000000100110100111110 0000000000000000
1│←───── 23─────→│←─── 16───→│
23+16=39，后面接着就是小数点了。
拿出计算器，输入二进制数1000000001001101001111100000000000000000
转为十进制数，不正是：551051722752么！
通过这例六例七，介绍了将整数二进制表示转浮点数二进制表示的逆过程，还是希望大家不但能掌握转化的方法，更要理解转化的基本原理。
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厳選された短編作品を一冊にまとめた総集編。感動、笑い、驚きが詰まった多彩なストーリーを収録。短時間で読める深い物語の世界へ。 短編集 総集編
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新社会人“新卒ちゃん”が職場で奮闘する日々をユーモラスに描く、お仕事コメディ作品。共感と笑いが詰まった新卒ライフをのぞいてみませんか？ 新卒ちゃん
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彼女のスマホに残された“チョコロ”というアプリが暴く、知られざる真実。恋愛、記憶、テクノロジーが交差する切なくも不思議な物語。 彼女のスマホ チョコロ
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気まぐれ猫“にゃあ”と、ごはんをめぐるほのぼのストーリー。笑いあり癒しあり、ペット好きに贈るやさしい物語。 にゃあのえさ
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南国テイストにパンチの効いた展開をミックス！「ココナッツパンチ」は、甘くて爽快、でもちょっとビターな物語。クセになる世界へ、いざトリップ！ ココナッツパンチ
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深刻化する少子化にどう立ち向かうべきか。日本の少子化対策の現状、課題、成功事例や今後の展望をわかりやすく解説。社会の未来を考えるヒントがここに。 少子化対策
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都市の片隅で密かに動く“トネリ団”とは何者なのか？謎とユーモアが交錯する物語が今、幕を開ける。クセ者たちの奇妙な冒険を目撃せよ！ トネリ団
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事故で命を落とした男が異世界でまさかの“種付け”特化能力を授かる!? モンスターも姫も虜にする最強（問題児）英雄、爆誕。規格外の異世界繁殖ライフ開幕！ 種付け転生マン
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欲望に素直な“つよつよ”たちが繰り広げる、濃厚でちょっぴり笑えるエロティック・ストーリー。理性なんて置いていけ。今夜は、感じるままに。 性欲つよつよ
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高画質・高評価のエロ動画を厳選紹介。ジャンル別・無料視聴あり。欲望を満たす一本がきっと見つかる、大人のための動画セレクション。 エロ動画
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ジャンル別に人気のエロ動画を徹底比較！素人・人妻・コスプレ・AV女優別に厳選紹介。無料視聴リンクつきで、今すぐ楽しめる大人のための動画ガイド。 エロ動画
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最新のエロ情報、人気エロ動画を厳選してお届け。大人の好奇心を満たす動画、記事、レビューを一挙紹介。今すぐ満足するコンテンツがここに。 エロ
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DXLIVEの最新クーポン情報をチェック！お得にライブストリーミングを楽しむための割引コードをゲット。今すぐ使えるお得なプロモーション情報をお見逃しなく！ DXLIVEクーポン
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DXLIVEで使える最新のクーポンコードをチェック！視聴料金が割引になるお得なキャンペーンや特典情報をまとめてご紹介。賢く楽しもう！ DXLIVEクーポン
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ストリップチャットでクレジットカードを使った支払い方法を完全ガイド。安全に、そしてお得にコンテンツを楽しむためのヒントと注意点をご紹介。簡単で安心な決済方法。 ストリップチャット クレジットカード
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多彩なジャンルのエロ動画を厳選して紹介。高画質、人気俳優、シチュエーション別にまとめて、あなたの欲望を満たす最適なコンテンツを提供します。 エロ動画
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ビットコインで遊べるオンラインカジノ（オンカジ）が急増中！匿名性・即時入出金・高還元率など、そのメリットやおすすめの始め方、注意点を初心者向けにわかりやすく紹介します。 ビットコインオンラインカジノオンカジ
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PayPay銀行(ペイペイ銀行)で入金や出金できるオンラインカジノ(オンカジ)を紹介します。 PayPay銀行オンラインカジノオンカジ
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MangaRwとHitomiは危険ですウィルスだらけです！使用はやめましょう。 mangaraw
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リムイットが手がける作品は、美麗なビジュアルと独特のストーリー展開が魅力。エ□同人誌好きにはたまらない、深く刺さる表現と世界観を徹底解説します。 リムイット
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エ□同人誌を無料で読みたいあなたへ。人気の無料サイトを厳選紹介しつつ、安全に利用するための注意点やおすすめの検索方法も解説します。 エロ
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話題の同人誌『欠損メイドと欠損お嬢様』が無料で楽しめる！独特な世界観と緻密な描写が魅力の作品を、テーマ性や演出面から深掘りして紹介します。 無料、同人誌、欠損メイドと欠損お嬢様
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「はろとらら」の人気の理由とは？作風の特徴やおすすめ作品、作家のこだわりポイントまでわかりやすく解説。初めての人にもわかる入門ガイド。 はろとらら
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高画力で人気を集めるリムイットのエ□同人誌を厳選紹介。ジャンル別に分かりやすく、好みのシチュエーションがすぐ見つかる！ダウンロード前にチェックすべきポイントも。 リムイット
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無料エロ漫画べっぴんは、美麗な作画と幅広いジャンルで人気の同人エ□漫画サイト。話題の新作や人気シリーズを完全無料で楽しもう！ 無料エロ漫画べっぴん
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素人AVが近年人気を集めている理由は、プロにはないリアルさと親近感にあります。自然体の表情や素朴なやりとりに惹かれる人が急増中。その魅力を深掘りしてご紹介します。 素人AV
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2025年最新！リムイットの中でも特に評価の高いエ□同人誌を厳選して紹介。読み応え抜群のストーリーと完成度の高い作画に酔いしれよう。 リムイット
Apr 18, 2025 03:34:48 PM
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Apr 19, 2025 09:32:51 PM
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Hitomiやmangaraw,Rawkumaの無料エロ漫画サイトは危険 です！ 無料エロ漫画
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Hitomi.laやMangarawやRawkumaはウィルスだらけで危険です！ Hitomi.la
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AIによるえ□画像生成がもはやアートの域に！？リアルで官能的なビジュアルの秘密や、どんなジャンルが作れるのかを徹底解説！ AIえろ画像
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エ□同人誌をダウンロードするならどのサイトがいい？安全な使い方、ジャンル別の探し方、人気作家の傾向など、初心者にもわかりやすく徹底解説します。 エロ同人誌
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理性が崩壊していく瞬間を描いた同人誌は、読者の本能を刺激する唯一無二のジャンル。人気作やシチュエーションの傾向、楽しみ方をわかりやすく解説。 同人誌
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えっちなまんがを安全・快適に楽しむには？おすすめの電子書籍サイトやダウンロード方法、ジャンル選びのコツを分かりやすく解説します。 えっちなまんが
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えっちまんがの魅力を初心者にも分かりやすく解説。ジャンルの違いや人気作家の特徴、どこで読めるのかまで詳しく紹介します。 えっちまんが
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全部読めるって聞いたけど本当？そんな疑問に答えながら、「エロ漫画 全部読めます」で楽しめるジャンルや活用法、おすすめの検索術を紹介します！ エロまんが
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「今すぐ同人誌が読みたい！」そんな時に使える、即閲覧OKなオンラインサービスを厳選紹介。高画質＆ジャンル豊富な人気サイトも！ 同人誌
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「エ□漫画をフルで読みたいけど、どこがいい？」そんなあなたにおすすめのサイトを徹底比較。画質・対応ジャンル・使いやすさで選ぶならココ！ エロ漫画
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「これはもう芸術だろ…」と唸るような、超絶作画のエロすぎ漫画たち。極上のビジュアルとディープなシチュに酔いしれたいあなたへ贈る最前線レビュー！ えろまんが
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えろ同人誌の世界に初めて触れる方へ、ジャンルの特徴や選び方、安心して読める方法をやさしく解説。自分好みの作品に出会えるヒントも紹介します。 えろ同人誌
Apr 22, 2025 05:24:14 PM
【R18エロマンガ】人気の美少女キャラまとめ。 無料Hマンガの角田堂
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話題のエ□VR動画サイト「しこあーる」の魅力を徹底解説。高画質・高没入感で人気急上昇中の理由や使い方、対応デバイスまで、初心者にもわかりやすく紹介します。 瀬戸環奈 vr
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あなたの性欲を満たす本が必ずあります♪。 エロ同人誌
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無料フル動画｜AV女優ナビ｜あの素人AV作品に出演しているAV女優は誰？ AV女優 ランキング 最新
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どのエ□動画を選べばいい？失敗しないための選び方＆年おすすめ動画リストをわかりやすく解説。初心者にも◎！ エロ動画
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高画力・高評価のエ□漫画・同人誌を厳選紹介！話題の人気作品から隠れた名作まで、年にチェックしておきたいおすすめタイトルをまとめました。 エロ漫画・同人
Apr 23, 2025 10:53:36 PM
高画質で見たい人向け！ジャンル豊富な無料エ□動画サイトを厳選してご紹介。無修●・同人系・素人系など、好みに合う一本がきっと見つかる。 動画
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話題の女優・笹倉彩の出演動画をジャンル別にご紹介。気になる最新作や人気タイトルの見どころをわかりやすくまとめました。 笹倉彩 動画
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最新映画やアニメを高画質でフル視聴する方法を紹介。無料で見られるサイトから、安心安全な有料配信サービスまで詳しく解説します。 高画質フル動画
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同人エ□動画は漫画やゲーム原作の世界観をそのままに、よりリアルに再現。キャラの声や動きにこだわった作品の見どころを徹底解説！ 動画
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エ□カミドーガは、エ□アニメや同人動画を楽しめる人気の動画配信サイトです。使い方やジャンル、特徴について初心者にもわかりやすく紹介します。 エロカミドーガ
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Missav/AV01などの違法サイトSTOP！ 動画