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Decimal型も浮動小数点型も演算誤差があります...再現実演でとちった

会議室で定期的に登場する問題です。なぜそうなるかを理解していない人が一定数存在します。理系でなくてもコンピュータ系の学校では教えているんですがね。会社のOJTで欠落するのかな?
浮動小数点演算の有効桁に依存する誤差と変数の型に依存する表現誤差かあるので、難しく考えてしまうのでしょうか?
有効桁数の話は初等数学で登場するので省略します。ご承知のように数字変数の型には整数型・固定小数点型・浮動小数点型がありました。
これらには二進数で表現できない数 (2 ^ n)は保持できないわけで、代入した時点で精度が落ちます。
桁落ちしにくい保持の方法には二進化十進(BCD)・パック10進数・ゾーン10進数・3増し符号・固定小数点数などがあります。汎用機(COBOL)ではPack/Zoneが使われてますね。昔80系アセンブラが全盛だったころ( <-いつだ)BCDで悪戦苦闘した記憶があります。MSの世界では、COleCurrencyが固定小数点と称していましたが、.netではDecimal型になってます。固定小数点ではなく固定長の有効桁数と小数点部桁数を持つ数値型と称し
てます。固定小数点型は System.Data.SqlTypes.SqlMoney型がありますが、標準では何故か無いんでよね。

Decimal型は固定小数点だから誤差は生じないと誤解している人が少なからずいます。Decimal型といえども二進表現で保持するので誤差が生じることがあります。Decimal型は固定小数点ではありません。もちろん純粋の固定小数点でも除算による誤差は生じます。実演して説明しようとして墓穴を掘っちゃいました。

  電卓で 1 / 3  の後 * 3 をしたら 0.99999999 になりますね。( 1になる電卓もあるみたいですね。)
ではプログラムで
    Dim da As Decimal = 1
    Dim db As Decimal = (da / 3)
    Dim dc As Decimal = db * 3

     Dim fa As Double = 1
     Dim fb As Double = (fa / 3)
     Dim fc As Double = fb * 3

     MessageBox.Show(dc.ToString + " : " + fc.ToString)

結果は共に "0.999・・ : 0.999・・" を期待していたのですが、意に反して "0.9999999999999999999999999999 : 1"でした。 ( doubleの結果こそ 0.999・・であって欲しかったのに。最終桁の足しこみ誤差が繰り上がって結果として1.0になったのでしょうか。)
「このように異なった結果になります。Decimalでも期待と違った結果になりますね。」・・・誤魔化しちゃった。.wwwwww)

気を取り直して

     Dim da As Decimal = 1
     Dim db As Decimal = 0.01D
     Dim dc As Decimal = 0
      For i As Integer = 0 To 99
           dc += db
      Next

      Dim fa As Single = 1
      Dim fb As Single = 0.01
      Dim fc As Single = 0

       For  i As Integer = 0 To 99
            fc += fb
       Next

      Dim ta As Double = 1
      Dim tb As Double = 0.01
      Dim tc As Double = 0

       For i As Integer = 0 To 99
           tc += tb
       Next

これも、すべて 0.999・・・を期待してたのですが、結果は "1.00 : 0.9999993 : 1" でした。
「DecimalとDoubleは1になりますが、singleは1になりませんね。」・・・再び誤魔化しちゃった。(なぜだろう..未解決)
このように小数点演算は誤差を伴います。型の特性をよく理解して使用してください。Decimal型と言えども同じです。(強引に結論)

 起こって欲しい時に起こらないマーフィー則なのかな。<---事前準備がなってないだけだ。

投稿日時 : 2007年5月12日 1:52

Feedback

# re: Decimal型も浮動小数点型も演算誤差があります...再現実演でとちった 2007/05/12 2:38 囚人

それは100回ループしているからでは…。

# re: Decimal型も浮動小数点型も演算誤差があります...再現実演でとちった 2007/05/12 13:04 通りすがり

あれ? Decimal は10進数ですよね?
それと、固定小数とも一般的な浮動小数とも異なり、小数点の位置が可変な固定小数みないなものですよね。

10進数なので、Double 型等のように2進数では表現できない数値の場合の丸め誤差は生じないですが、割り切れない値を有効桁で表す場合に生じる誤差は仕方がないのでは?と思います。

ところで 0.01D って double じゃないの?って思って VB のヘルプを見ると、VB では Decimal が D なんですね(C# では M です)。
そしてそこでさらに @ の存在を知ったのですが、以下を試しても意義がわかりませんでした...

Dim v1 As Decimal = 10.5D '普通(C# の 10.5m と同じ)
Dim v2 As Decimal = 10D 'New Decimal(CLong(10)) に変換される
Dim v3 As Decimal = 10@ 'v2 と同じ
Dim v4 As Decimal = 10.5@ 'v1 と同じ(@ の意義って何?)
Dim v5 As Decimal = 10.5 'v1 と同じ
Dim v6 As Decimal = 10 'v2 と同じ

D と @ の違いの説明のところで倍精度浮動小数点型 (Decimal) と整数型 (Decimal) の2つの表現が出てくるんですが、これらって何者ですか...?
もしかして、内部表現のスケールファクタが非ゼロかゼロかを区別しているとか??

# re: Decimal型も浮動小数点型も演算誤差があります...再現実演でとちった 2007/05/12 15:18 siokoshou

はじめまして。
固定小数点は表示を工夫することでintなどを流用できます。C#の例ですがvalueがintだとして、
value.ToString( "##0'.'0" );
として文字列にして表示するわけです。カスタム数値書式指定ってやつです。

# re: Decimal型も浮動小数点型も演算誤差があります...再現実演でとちった 2007/05/12 16:15 Ognac

>それは100回ループしているからでは…。
ええ。誤差が出ることを実証してみせたかったので、ループ加算が確実かなと考えたわけてして。

>あれ? Decimal は10進数ですよね?
>それと、固定小数とも一般的な浮動小数とも異なり、小数点の位置が可変な固定小数みないなものですよね。

うわぁ、私が間違ってる...orz
 Helpによると http://msdn2.microsoft.com/ja-jp/library/system.decimal(VS.80).aspx
 「Decimal 構造体 で10 進数を表します。 」とあるのでこれが 10進浮動小数点数の実装形態なんでしょうね。

Decimalでも誤差が出るのは
「 Dim db As Decimal = (1D / 3) この値は 0.3333‥となり、循環小数は二進で表現できないので誤差を生じます。」
 と話したほうがスッキリしましたね。(反省)


>VB では Decimal が D なんですね(C# では M です)。
そうなんですよ。VBとC#の両方をコーディングしていると混乱します。

>D と @ の違いの説明のところで
これは、Strict onの状態でも
Dim z@ = 33.5@
Dim K@ = 33.5D
Dim y As Decimal = 3.4@
Dim p As Decimal = 33.5D
という記述を許しているために、 '@'と'D' を同等に処理しているのではないかと想像するのですが。

siokoshou さん。こちらこそ宜しくお願いいたします。

>カスタム数値書式指定ってやつです。
そうですね。出力段階で加工するしかないでしょうね。

誤差がでることの実証も難しいもので、勉強になりました。

# re: Decimal型も浮動小数点型も演算誤差があります...再現実演でとちった 2007/05/14 13:04 通りすがり

> Dim z@
あ、そういう意味でしたか~
MSDN を読み返すと、確かに D の方は「あるリテラルに型文字 D を付けると…」で、@ の方は「ある識別子に識別子の型文字 @ を付けると…」のようにちゃんと区別されていました。
# つまり私の読み間違い...
ただ、「倍精度浮動小数点型 (Decimal)」と「整数型 (Decimal)」の違いは依然謎なんですが、原文(英語の MSDN)を見てみてもこんなこと書かれてないみたい。あれ??

# re: Decimal型も浮動小数点型も演算誤差があります...再現実演でとちった 2007/05/15 0:26 Ognac

>倍精度浮動小数点型 (Decimal)」と「整数型 (Decimal)」の違いは依然謎
二種類のDecimalの存在は? ですね。説明上の違いの気が
しますが謎ですね。

# re: Decimal型も浮動小数点型も演算誤差があります...再現実演でとちった 2007/05/29 18:25 おせっかい

「浮動小数点 0.1」とかでぐぐるといいかも。

倍精度浮動小数点型 (Decimal)は10進浮動小数点(Decimal)の間違い?

# re: Decimal型も浮動小数点型も演算誤差があります...再現実演でとちった 2007/05/30 1:28 Ognac

>「浮動小数点 0.1」とかでぐぐるといいかも。
ピッタシの資料がヒットしました。有難うございました。

>倍精度浮動小数点型 (Decimal)は10進浮動小数点(Decimal)の間違い?
ミスプリ(ミスType? ミスプリでは変だな)でしょうね。

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