일반화하다

Question 1

괜찮나요?

dash$ printf "%.99f\n" 3.472727272727276
3.472727272727276215391611913219094276428222656250000000000000000000000000000000000000000000000000000

또는

dash$ printf "%.99f\n" 3.472727272727276 | sed -e 's/0*$//'
3.47272727272727621539161191321909427642822265625

이것은 Dash의 출력으로, 간단한 C 프로그램인 Zsh ^(*) 또는 Perl( )을 사용하여 perl -e 'printf "%.99f\n", 3.472727272727276'얻은 결과와 일치합니다 . 반면에 Bash는 더 정밀한 기능을 제공하는 것 같습니다. 부동 소수점 숫자에 대해 printf가 수행하는 작업을 살펴보지 않았습니다.

_{(*x86_64 Debian에서 기본 옵션으로 컴파일되었으므로 부동 소수점 연산에 SSE를 사용합니다.)}

폭은 99물론 모자에서 뽑아냅니다. double에는 53개의 유효 숫자가 있고 17개의 밑수 10 숫자는 명확한 표현을 얻기에 충분하지만, 정확한 값을 얻으려면 약 1의 정수 부분이 있는 숫자의 경우 다시 53개의 밑수 10 숫자가 필요합니다. 규모를 확대하거나 축소하는 경우 더 많은 숫자, 99개 이상의 숫자 또는 지수 표기법( %g아마도)이 필요할 수 있습니다.

이건 제 상상을 초월하는 일이고, 버그나 코너케이스가 있을 수 있습니다. 나는 부동 소수점 수학을 깊이 파고들지는 않을 것입니다. 여기에는 나보다 똑똑한 사람들이 부동 소수점에 관해 쓴 기사가 있어야 합니다.

Answer

괜찮나요?

dash$ printf "%.99f\n" 3.472727272727276
3.472727272727276215391611913219094276428222656250000000000000000000000000000000000000000000000000000

또는

dash$ printf "%.99f\n" 3.472727272727276 | sed -e 's/0*$//'
3.47272727272727621539161191321909427642822265625

이것은 Dash의 출력으로, 간단한 C 프로그램인 Zsh ^(*) 또는 Perl( )을 사용하여 perl -e 'printf "%.99f\n", 3.472727272727276'얻은 결과와 일치합니다 . 반면에 Bash는 더 정밀한 기능을 제공하는 것 같습니다. 부동 소수점 숫자에 대해 printf가 수행하는 작업을 살펴보지 않았습니다.

_{(*x86_64 Debian에서 기본 옵션으로 컴파일되었으므로 부동 소수점 연산에 SSE를 사용합니다.)}

폭은 99물론 모자에서 뽑아냅니다. double에는 53개의 유효 숫자가 있고 17개의 밑수 10 숫자는 명확한 표현을 얻기에 충분하지만, 정확한 값을 얻으려면 약 1의 정수 부분이 있는 숫자의 경우 다시 53개의 밑수 10 숫자가 필요합니다. 규모를 확대하거나 축소하는 경우 더 많은 숫자, 99개 이상의 숫자 또는 지수 표기법( %g아마도)이 필요할 수 있습니다.

이건 제 상상을 초월하는 일이고, 버그나 코너케이스가 있을 수 있습니다. 나는 부동 소수점 수학을 깊이 파고들지는 않을 것입니다. 여기에는 나보다 똑똑한 사람들이 부동 소수점에 관해 쓴 기사가 있어야 합니다.

Question 2

일반화하다

짧은 답변정밀한숫자는 다음과 같습니다(대시를 사용하여 나타냄).더블 플로트):

$ dash -c 'printf "%a\n" 3.472727272727276'
0x1.bc8253c8253dp+1

bash에서 우리는 얻는다확장된 이중 플로트(80비트), 더 많은 숫자:

$ bash -c 'printf "%a\n" 3.4727272727272760001'
0xd.e4129e4129e7c1fp-2

awk 및 zsh에는 형식이 없으며 %aksh에서는 변환이 실패할 수 있습니다.

10진수로 변환해야 하는 경우 자릿수는 최대 16450자리가 될 수 있습니다.정밀한전환하다. 명령은 다음과 같아야 합니다.

$ printf '%.16450g\n' 3.472727272727276
3.47272727272727600006906045759791368254809640347957611083984375

형식이 모든 후행 0을 제거한다는 사실을 활용합니다 g(그러나 대부분의 경우 과학 지수를 사용합니다).

1e-4931(0에 가까워짐)까지 내려가는 것은유효한 확장 부동 소수점 수f이 번호는 16445 자리를 생성합니다.¹

이 숫자는 이진 소수점 이하 자릿수에 가깝습니다.

$ printf '%a\n' 1e-4931
0xb.e5b66ecbce0b7b1p-16384

11516자리(마지막 6자는 지수), g형식은 다음과 같습니다.

$ printf '%.16445g\n' 1e-4931 | wc -c
11522

자세한 설명

@ilkkachu의 답변이 부정확합니다//불완전합니다

예, 더 많은 숫자(경우에 따라)를 요청하여 더 많은 숫자(십진수 또는 정수가 아닌 숫자)를 얻을 수 있습니다.

 $ printf '%.10f\n' 0.5; printf '%.25f\n' 0.5
 0.500000
 0.5000000000000000000000000

그런데 왜 25, 99, 250, 2500에서 멈추나요? 한계는 어디에 있습니까? 한계는 어디에 있어야 합니까?

더블

Linux의 GNU awk에 printf를 구현하면 0.5=<x<1 또는 [0.5,1) 범위의 숫자에 대해 최대 53자리의 숫자를 얻을 수 있습니다. 따라서 가수의 첫 번째 숫자는 1이고 지수는 "0"(영)}입니다.

$ awk 'BEGIN{ printf "%.56f\n", 0.6 }'
0.59999999999999997779553950749686919152736663818359375000
  123456789_123456789-123456789¯123456789|123456789_123456789-

이는 awk가 기본적으로 사용하는 C의 일반 double에 53비트 가수가 있기 때문입니다(예, 십진수는 십진수 이진수만큼 많습니다).

확장 플로트

그러나 Bash(및 일부 다른 구현)에서 이중 부동 소수점의 크기는 수학 보조 프로세서 내부의 크기만큼 크고 80비트 부동 소수점의 가수는 64비트입니다.

$ printf "%.70f\n", 0.6
0.6000000000000000000216840434497100886801490560173988342285156250000000
  123456789-123456789-123456789-123456789-123456789-123456789-123456789-

하지만 조심하세요. dash는 C 이중 부동 소수점을 사용하고, ksh는 소수 부분을 18자리로 자릅니다.

$ dash -c 'printf "%.70f\n" 0.6
0.5999999999999999777955395074968691915273666381835937500000000000000000
  123456789-123456789-123456789-123456789-123456789-123456789-123456789-

$ ksh -c 'printf "%.70f\n" 0.12345678901234567890'
0.1234567890123456790000000000000000000000000000000000000000000000000000
  123456789-123456789-123456789-123456789-123456789-123456789-123456789-

긴 소수

인쇄할 숫자가 범위를 벗어난 경우 [0.5,1). 실제로 숫자가 이 범위 아래로 떨어지면 printf생성된 자릿수(정확히 말하면)를 늘려야 하며 작은 지수 -12만으로도 소수점 이하 99자리 이상을 얻을 수 있습니다.

$ printf '%.99f\n%.110f\n' 1e-12 1e-12
0.00000000000100000000000000000000260071239141640525192192084057152268
  1727856252109631896018981933594
0.00000000000100000000000000000000260071239141640525192192084057152268
  172785625210963189601898193359375000000000

정상

가수를 첫 번째로 조정한 경우바이너리비트 1, IEEE-754 사양에 따라 지수는 이진수로 1022(또는 십진수로 약 308)만큼 클 수 있으며, 이를 위해서는 정확히 1022 + 53 = 1075개의 십진수를 인쇄해야 합니다(대시를 사용하여 C double을 나타냄). 수레):

$ dash -c 'printf "%.1080f" 3e-308'
0.00000000000000000000000000000000000000000000000000000000000000000000000
0000000000000000000000000000000000000000000000000000000000000000000000000
0000000000000000000000000000000000000000000000000000000000000000000000000
0000000000000000000000000000000000000000000000000000000000000000000000000
0000000000000000030000000000000002220455218529919301629802590703175094461
8147776231970806960868915174651286204183765077257126691146784826970001182
3377333217966211682361285986313366061022263946269900386622216365532374597
7178280811925489426908651787672688372272916552149728525251396407136469620
5940554355008967838740787733222562854087120536535403422962140580707820507
1667528534455025639283299437942142025335784795886380202690541431449127657
8638018769522816220159645085115105872872349792947642929427853637000376104
4766449778622710769965927963092807311828491559718885849370616714002362927
2391454493281372654969535173285036362816688696655266375211946090175421780
9466790102733774958189714097426504991570081907927842237039897170724535065
396560619562470184629177083479589782655239105224609375000000

하지만 너비가 고정된 다음 형식을 사용하는 것이 훨씬 쉽습니다 %a.

$ dash -c 'printf "%a" 3e-308'
0x1.59283684dba77p-1022

정확히 같은 숫자이지만 위에 필요한 1080비트보다 훨씬 적습니다.

비정상

그러나 IEEE754도 허용합니다.비정상십진수 범위를 더 작은 숫자(음수가 아닌 0에 가까움)로 확장합니다. 더블 플로트에서는 ^2-1074에 도달하는 것이 가능합니다 . 즉 1022(최대 지수) + 52(가수 이진수)입니다.

$ dash -c 'printf "%a\n" 0x1p-1074' 
0x0.0000000000001p-1022

숫자는 십진수 1075자리 이상이어야 합니다(시도해 보세요).dash -c 'printf "%.1080f\n" 0x1p-1074'

수학

그렇게 많은 숫자가 필요한 이유는 단지 수학적일 뿐입니다.

0x1p-60(소수 구분 기호 오른쪽으로 60자리 1자리)의 정확한 결과는 1/2^60 또는 2 ^-60 입니다 .

$ echo 'scale=65;1/2^60' | bc
.00000000000000000086736173798840354720596224069595336914062500000

수학은 이러한 결과를 강요합니다. 다시 말하지만, 소수점 이하 자릿수는 이진수 소수 자릿수만큼 많아야 합니다.

정밀한?

하지만 숫자가 생성해야 하는 부동소수점을 계산하는 간단한 방법을 제공하려면 다음을 시도해 보세요.

$ number=3.472727272727276; 
$ echo "x=$number"'; f=floor(x*2^53); l=f/2^53; u=(f+1)/2^53;l;x-l;u;u-x' | bc

3.47272727272727599334       # one ulp below
 .00000000000000000666
3.47272727272727610436       # one ulp above
 .00000000000000010436

그리고 오류가 더 작은 것을 선택합니다(이 경우 3.47272727272727599334). 이는 정확한 IEEE754 표현과 동일하지 않습니다. 하지만 이 숫자가 정확한 결과를 제공할 것이라고 확신할 수 있습니다.

$ dash -c 'printf "%.100g\n" "3.47272727272727599334"'
3.472727272727275771302402063156478106975555419921875

ulp는 마지막 숫자의 단위를 의미합니다.

IEEE-754

위의 대부분은 부동 소수점 표현이 IEEE-754 규칙 "가장 가까운 숫자(0.5에 가까운 숫자)로 반올림하고 짝수를 기준으로 합니다(가장 가까운 숫자까지의 거리가 위와 아래가 같은 경우, 그런 다음 짝수로 반올림합니다(간단히 말하면 마지막 숫자 0).

따라서 귀하의 질문에 나온 숫자를 바탕으로:

$ arr=(
3.4727272727272765 
3.4727272727272764 
3.472727272727276  
3.472727272727275994 
3.472727272727275993
); dash -c 'printf "%a\n" "$@"' _sh "${arr[@]}"

0x1.bc8253c8253d1p+1
0x1.bc8253c8253dp+1
0x1.bc8253c8253dp+1
0x1.bc8253c8253dp+1
0x1.bc8253c8253cfp+1

상위 1자리( 로 끝남 3d1)부터 하위 1자리( 로 끝남 3cf)까지입니다. Bash에서는 64비트 가수의 숫자가 다릅니다.

arr=(
3.47272727272727600018
3.47272727272727600017
3.472727272727276
3.47272727272727599997
3.47272727272727599996
     ); bash -c 'printf "%a\n" "$@"' _sh "${arr[@]}"

0xd.e4129e4129e7c2p-2
0xd.e4129e4129e7c1fp-2
0xd.e4129e4129e7c1fp-2
0xd.e4129e4129e7c1fp-2
0xd.e4129e4129e7c1ep-2

노트¹

1e-4931로 내려가는 것(0을 향해)은유효한 확장 부동 소수점 수이 숫자는 f형식에서 약간 앞으로 및 아래로 16445자리를 생성합니다.

자릿수는 이진수 소수점 이하 자릿수에 가깝습니다.

$ printf '%a\n' 1e-4931
0xb.e5b66ecbce0b7b1p-16384

11516자리(마지막 6자는 지수), g형식은 다음과 같습니다.

$ printf '%.16445g\n' 1e-4931 | wc -c
11522

printf '%.16450f\n' 1e-4931 | fold -w 73
0.00000000000000000000000000000000000000000000000000000000000000000000000
0000000000000000000000000000000000000000000000000000000000000000000000000
0000000000000000000000000000000000000000000000000000000000000000000000000
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0000000000000000000000000000000000000000000000000000000000000000000000000
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462477207183837890625000000

Answer

일반화하다

짧은 답변정밀한숫자는 다음과 같습니다(대시를 사용하여 나타냄).더블 플로트):

$ dash -c 'printf "%a\n" 3.472727272727276'
0x1.bc8253c8253dp+1

bash에서 우리는 얻는다확장된 이중 플로트(80비트), 더 많은 숫자:

$ bash -c 'printf "%a\n" 3.4727272727272760001'
0xd.e4129e4129e7c1fp-2

awk 및 zsh에는 형식이 없으며 %aksh에서는 변환이 실패할 수 있습니다.

10진수로 변환해야 하는 경우 자릿수는 최대 16450자리가 될 수 있습니다.정밀한전환하다. 명령은 다음과 같아야 합니다.

$ printf '%.16450g\n' 3.472727272727276
3.47272727272727600006906045759791368254809640347957611083984375

형식이 모든 후행 0을 제거한다는 사실을 활용합니다 g(그러나 대부분의 경우 과학 지수를 사용합니다).

1e-4931(0에 가까워짐)까지 내려가는 것은유효한 확장 부동 소수점 수f이 번호는 16445 자리를 생성합니다.¹

이 숫자는 이진 소수점 이하 자릿수에 가깝습니다.

$ printf '%a\n' 1e-4931
0xb.e5b66ecbce0b7b1p-16384

11516자리(마지막 6자는 지수), g형식은 다음과 같습니다.

$ printf '%.16445g\n' 1e-4931 | wc -c
11522

자세한 설명

@ilkkachu의 답변이 부정확합니다//불완전합니다

예, 더 많은 숫자(경우에 따라)를 요청하여 더 많은 숫자(십진수 또는 정수가 아닌 숫자)를 얻을 수 있습니다.

 $ printf '%.10f\n' 0.5; printf '%.25f\n' 0.5
 0.500000
 0.5000000000000000000000000

그런데 왜 25, 99, 250, 2500에서 멈추나요? 한계는 어디에 있습니까? 한계는 어디에 있어야 합니까?

더블

Linux의 GNU awk에 printf를 구현하면 0.5=<x<1 또는 [0.5,1) 범위의 숫자에 대해 최대 53자리의 숫자를 얻을 수 있습니다. 따라서 가수의 첫 번째 숫자는 1이고 지수는 "0"(영)}입니다.

$ awk 'BEGIN{ printf "%.56f\n", 0.6 }'
0.59999999999999997779553950749686919152736663818359375000
  123456789_123456789-123456789¯123456789|123456789_123456789-

이는 awk가 기본적으로 사용하는 C의 일반 double에 53비트 가수가 있기 때문입니다(예, 십진수는 십진수 이진수만큼 많습니다).

확장 플로트

그러나 Bash(및 일부 다른 구현)에서 이중 부동 소수점의 크기는 수학 보조 프로세서 내부의 크기만큼 크고 80비트 부동 소수점의 가수는 64비트입니다.

$ printf "%.70f\n", 0.6
0.6000000000000000000216840434497100886801490560173988342285156250000000
  123456789-123456789-123456789-123456789-123456789-123456789-123456789-

하지만 조심하세요. dash는 C 이중 부동 소수점을 사용하고, ksh는 소수 부분을 18자리로 자릅니다.

$ dash -c 'printf "%.70f\n" 0.6
0.5999999999999999777955395074968691915273666381835937500000000000000000
  123456789-123456789-123456789-123456789-123456789-123456789-123456789-

$ ksh -c 'printf "%.70f\n" 0.12345678901234567890'
0.1234567890123456790000000000000000000000000000000000000000000000000000
  123456789-123456789-123456789-123456789-123456789-123456789-123456789-

긴 소수

인쇄할 숫자가 범위를 벗어난 경우 [0.5,1). 실제로 숫자가 이 범위 아래로 떨어지면 printf생성된 자릿수(정확히 말하면)를 늘려야 하며 작은 지수 -12만으로도 소수점 이하 99자리 이상을 얻을 수 있습니다.

$ printf '%.99f\n%.110f\n' 1e-12 1e-12
0.00000000000100000000000000000000260071239141640525192192084057152268
  1727856252109631896018981933594
0.00000000000100000000000000000000260071239141640525192192084057152268
  172785625210963189601898193359375000000000

정상

가수를 첫 번째로 조정한 경우바이너리비트 1, IEEE-754 사양에 따라 지수는 이진수로 1022(또는 십진수로 약 308)만큼 클 수 있으며, 이를 위해서는 정확히 1022 + 53 = 1075개의 십진수를 인쇄해야 합니다(대시를 사용하여 C double을 나타냄). 수레):

$ dash -c 'printf "%.1080f" 3e-308'
0.00000000000000000000000000000000000000000000000000000000000000000000000
0000000000000000000000000000000000000000000000000000000000000000000000000
0000000000000000000000000000000000000000000000000000000000000000000000000
0000000000000000000000000000000000000000000000000000000000000000000000000
0000000000000000030000000000000002220455218529919301629802590703175094461
8147776231970806960868915174651286204183765077257126691146784826970001182
3377333217966211682361285986313366061022263946269900386622216365532374597
7178280811925489426908651787672688372272916552149728525251396407136469620
5940554355008967838740787733222562854087120536535403422962140580707820507
1667528534455025639283299437942142025335784795886380202690541431449127657
8638018769522816220159645085115105872872349792947642929427853637000376104
4766449778622710769965927963092807311828491559718885849370616714002362927
2391454493281372654969535173285036362816688696655266375211946090175421780
9466790102733774958189714097426504991570081907927842237039897170724535065
396560619562470184629177083479589782655239105224609375000000

하지만 너비가 고정된 다음 형식을 사용하는 것이 훨씬 쉽습니다 %a.

$ dash -c 'printf "%a" 3e-308'
0x1.59283684dba77p-1022

정확히 같은 숫자이지만 위에 필요한 1080비트보다 훨씬 적습니다.

비정상

그러나 IEEE754도 허용합니다.비정상십진수 범위를 더 작은 숫자(음수가 아닌 0에 가까움)로 확장합니다. 더블 플로트에서는 ^2-1074에 도달하는 것이 가능합니다 . 즉 1022(최대 지수) + 52(가수 이진수)입니다.

$ dash -c 'printf "%a\n" 0x1p-1074' 
0x0.0000000000001p-1022

숫자는 십진수 1075자리 이상이어야 합니다(시도해 보세요).dash -c 'printf "%.1080f\n" 0x1p-1074'

수학

그렇게 많은 숫자가 필요한 이유는 단지 수학적일 뿐입니다.

0x1p-60(소수 구분 기호 오른쪽으로 60자리 1자리)의 정확한 결과는 1/2^60 또는 2 ^-60 입니다 .

$ echo 'scale=65;1/2^60' | bc
.00000000000000000086736173798840354720596224069595336914062500000

수학은 이러한 결과를 강요합니다. 다시 말하지만, 소수점 이하 자릿수는 이진수 소수 자릿수만큼 많아야 합니다.

정밀한?

하지만 숫자가 생성해야 하는 부동소수점을 계산하는 간단한 방법을 제공하려면 다음을 시도해 보세요.

$ number=3.472727272727276; 
$ echo "x=$number"'; f=floor(x*2^53); l=f/2^53; u=(f+1)/2^53;l;x-l;u;u-x' | bc

3.47272727272727599334       # one ulp below
 .00000000000000000666
3.47272727272727610436       # one ulp above
 .00000000000000010436

그리고 오류가 더 작은 것을 선택합니다(이 경우 3.47272727272727599334). 이는 정확한 IEEE754 표현과 동일하지 않습니다. 하지만 이 숫자가 정확한 결과를 제공할 것이라고 확신할 수 있습니다.

$ dash -c 'printf "%.100g\n" "3.47272727272727599334"'
3.472727272727275771302402063156478106975555419921875

ulp는 마지막 숫자의 단위를 의미합니다.

IEEE-754

위의 대부분은 부동 소수점 표현이 IEEE-754 규칙 "가장 가까운 숫자(0.5에 가까운 숫자)로 반올림하고 짝수를 기준으로 합니다(가장 가까운 숫자까지의 거리가 위와 아래가 같은 경우, 그런 다음 짝수로 반올림합니다(간단히 말하면 마지막 숫자 0).

따라서 귀하의 질문에 나온 숫자를 바탕으로:

$ arr=(
3.4727272727272765 
3.4727272727272764 
3.472727272727276  
3.472727272727275994 
3.472727272727275993
); dash -c 'printf "%a\n" "$@"' _sh "${arr[@]}"

0x1.bc8253c8253d1p+1
0x1.bc8253c8253dp+1
0x1.bc8253c8253dp+1
0x1.bc8253c8253dp+1
0x1.bc8253c8253cfp+1

상위 1자리( 로 끝남 3d1)부터 하위 1자리( 로 끝남 3cf)까지입니다. Bash에서는 64비트 가수의 숫자가 다릅니다.

arr=(
3.47272727272727600018
3.47272727272727600017
3.472727272727276
3.47272727272727599997
3.47272727272727599996
     ); bash -c 'printf "%a\n" "$@"' _sh "${arr[@]}"

0xd.e4129e4129e7c2p-2
0xd.e4129e4129e7c1fp-2
0xd.e4129e4129e7c1fp-2
0xd.e4129e4129e7c1fp-2
0xd.e4129e4129e7c1ep-2

노트¹

1e-4931로 내려가는 것(0을 향해)은유효한 확장 부동 소수점 수이 숫자는 f형식에서 약간 앞으로 및 아래로 16445자리를 생성합니다.

자릿수는 이진수 소수점 이하 자릿수에 가깝습니다.

$ printf '%a\n' 1e-4931
0xb.e5b66ecbce0b7b1p-16384

11516자리(마지막 6자는 지수), g형식은 다음과 같습니다.

$ printf '%.16445g\n' 1e-4931 | wc -c
11522

printf '%.16450f\n' 1e-4931 | fold -w 73
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일반화하다

답변1

답변2

일반화하다

자세한 설명

더블

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긴 소수

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수학

정밀한?

IEEE-754

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