311 lines
9.2 KiB
Markdown
311 lines
9.2 KiB
Markdown
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---------------------------------------------------------
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DECIMALE VS BINARY
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---------------------------------------------------------
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11010101
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```
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1*10000000 | (128) | 2^7 | 128 *1 | 0 *2 +1
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1*1000000 | (64) | 2^6 | + 64 *1 | 1 *2 +1
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0*100000 | (32) | 2^5 | | 11 *2
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1*10000 | (16) | 2^4 | + 16 *1 | 110 *2 +1
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0*1000 | (8) | 2^3 | | 1101 *2
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1*100 | (4) | 2^2 | + 4 *1 | 11010 *2 +1
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0*10 | (2) | 2^1 | | 110101 *2
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1*1 | (1) | 2^0 | + 1 *1 | 1101010 *2 +1
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| ______ |
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| 213 |
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```
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213
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```
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2*100 |(1100100)| 10^2 | 1100100 *2 | 0 *10 +2
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1*10 | (1010)| 10^1 | + 1010 *1 | 2 *10 +1
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3*1 | (1)| 10^0 | + 1 *3 | 21 *10 +3
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| | | ___________ | 213
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| | | 11010101 |
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1100100 11001000 1 11010010 |
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* 2 + 1010 * 3 + 11 |
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_________ _________ ___ _________ |
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11001000 11010010 11 11010101 |
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```
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213 -> 11010101
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```
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213 -128 = 85 | 213 /2 = 106 r 1
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85 - 64 = 2 | 106 /2 = 53 r 0
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| 53 /2 = 26 r 1
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21 - 16 = 5 | 26 /2 = 13 r 0
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| 13 /2 = 6 r 1
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5 - 4 = 1 | 6 /2 = 3 r 0
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| 3 /2 = 1 r 1
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1 - 1 = 0 | 1 /2 = 0 r 1
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```
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11010101 -> 213
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```
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| 0*2 + 1 = 1
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| 1*2 + 1 = 3
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| 3*2 = 6
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| 6*2 + 1 = 13
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| 13*2 = 26
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| 26*2 + 1 = 53
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| 53*2 = 106
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| 106*2 + 1 = 213
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```
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---------------------------------------------------------
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NEGATIVS INTEGERS
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---------------------------------------------------------
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```
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5 0101 0 0000 0 0000 0000 0000 0000 0000 0000 0000 0000
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1010 + 1 1 0001 1 0000 0000 0000 0000 0000 0000 0000 0001
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-5 1011 2 0010 2 0000 0000 0000 0000 0000 0000 0000 0010
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3 0011 3 0000 0000 0000 0000 0000 0000 0000 0011
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4 0100 4 0000 0000 0000 0000 0000 0000 0000 0100
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-> 5 0101 5 0000 0000 0000 0000 0000 0000 0000 0101
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6 0110 6 0000 0000 0000 0000 0000 0000 0000 0110
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7 0111 7 0000 0000 0000 0000 0000 0000 0000 0111
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...
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0111 1111 1111 1111 1111 1111 1111 1111 MAXINT
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1000 0000 0000 0000 0000 0000 0000 0000 MININT
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...
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8 1000 -8 1111 1111 1111 1111 1111 1111 1111 1000
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9 1001 -7 1111 1111 1111 1111 1111 1111 1111 1001
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10 1010 -6 1111 1111 1111 1111 1111 1111 1111 1010
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-> 11 1011 -5 1111 1111 1111 1111 1111 1111 1111 1011
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12 1100 -4 1111 1111 1111 1111 1111 1111 1111 1100
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13 1101 -3 1111 1111 1111 1111 1111 1111 1111 1101
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14 1110 -2 1111 1111 1111 1111 1111 1111 1111 1110
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15 1111 -1 1111 1111 1111 1111 1111 1111 1111 1111
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```
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---------------------------------------------------------
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FLOATS
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---------------------------------------------------------
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https://stackoverflow.com/questions/7644699/how-are-floating-point-numbers-stored-in-memory
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```
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1.....................23
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8......1
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seeeeeeeemmmmmmmmmmmmmmmmmmmmmmm meaning
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31 0 bit #
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signe
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exponent
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mantis
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```
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1.175494351 * 10^-38 < ... > 3.40282347 * 10^+38
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2x10^-1 = 0.2x10^0 = 0.02x10^1 = 0.2
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- x 00000000 xxxxxxxx xxxxxxxx xxxxxxx special meanings (see bellow)
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- x 11111111 xxxxxxxx xxxxxxxx xxxxxxx //
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- x 00000001 xxxxxxxx xxxxxxxx xxxxxxx smallest exponent (-126)
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- x 01111111 xxxxxxxx xxxxxxxx xxxxxxx 0 is middle exponent (254 / 2 = 127)
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- x 11111110 xxxxxxxx xxxxxxxx xxxxxxx biggest exponent (+127)
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- x 00000000 00000000 00000000 0000000 0 (x can still be + or -)
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- x 11111111 00000000 00000000 0000000 +/- infinity
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- x 11111111 xxxxxxxx xxxxxxxx xxxxxx1 NaN (at least one non-zero mantissa digit)
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- x 00000000 xxxxxxxx xxxxxxxx xxxxxx1 denormalized numbers (same for mantissa)
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---------------------------------------------------------
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CONVERSIONS
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---------------------------------------------------------
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5.75
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```
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5 :
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5 / 2 = 2 -> 1
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2 / 2 = 1 -> 0
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1 / 2 = 0 -> 1
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75 :
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.75 *2^2 = 2,88 ~= 3
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(*2^x parce que c le moyen dont fait "bouger" la virgule
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en base 2 (*10^x) en base 10)
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3 / 2 = 1 -> 1
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1 / 2 = 0 -> 1
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101.11 = 2^2 + 2^0 + 2^-1 + 2^-2
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= 4 + 1 + 0.5 + 0.25
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```
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0.875
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```
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.875 * 2^3 = 7 -> 0.111 * 2^3 = 111.0
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7 / 2 = 3 -> 1
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3 / 2 = 1 -> 1
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1 / 1 = 0 -> 1
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0.111
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```
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0.1875
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```
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-1 1875
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0.1875 = 1.875 * 10^-1 = 01 11101010011
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0.1875 = 0011
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0*2^-1 + 082^-2 + 1*2^-3 + 1*2^-4
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0 + 0 + .125 + .0625 = .1875
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```
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-43.625
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```
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-101011.101
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fixed :
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1 101011 101
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-
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101011
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2^5 + 2^3 + 2^1 + 1
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32 + 8 + 2 + 1 = 43
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101
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2^-1 + 2^-3
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.5 + .125 = .625
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```
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-53.5 --> -110101.1
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```
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[ fixed : ]
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1 110101 1
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-
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110101
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2^5 + 2^4 + 2^2 + 1
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32 + 16 + 4 + 1 = 53
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1
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2^-1
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.5 = .5
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[ float : ]
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-1.101011 * 2^5
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11.01011 -> 1
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110.1011 -> 2
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1101.011 -> 3
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11010.11 -> 4
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110101.1 -> 5
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1 101 101011
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-
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2^2 + 1
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4 + 1 = 5
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101011
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->1101011
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```
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.85
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```
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.85 * 2 = 1.7 [1] [.7]
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.7 * 2 = 1.4 [1] [.4]
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.4 * 2 = 0.8 [0] [.8]
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.8 * 2 = 1.6 [1] [.6]
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.6 * 2 = 1.2 [1] [.2]
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.2 * 2 = 0.4 [0] [.4]
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.4 * 2 = 0.8 [0] [.8]
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.8 * 2 = 1.6 [1] [.6]
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.6 * 2 = 1.2 [1] [.2]
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.2 * 2 = 0.4 [0] [.4]
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.4 * 2 = 0.8 [0] [.8]
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.8 * 2 = 1.6 [1] [.6]
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.6 * 2 = 1.2 [1] [.2]
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.2 * 2 = 0.4 [0] [.4]
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.4 * 2 = 0.8 [0] [.8]
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.8 * 2 = 1.6 [1] [.6]
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.6 * 2 = 1.2 [1] [.2]
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.2 * 2 = 0.4 [0] [.4]
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.4 * 2 = 0.8 [0] [.8]
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.8 * 2 = 1.6 [1] [.6]
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.6 * 2 = 1.2 [1] [.2]
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.2 * 2 = 0.4 [0] [.4]
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.4 * 2 = 0.8 [0] [.8]
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...
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-> 11011001100110011001100...
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1 1 0 1 1 0 0 1 1 0 0 1 1
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2^-1 + 2^-2 + 0 + 2^-4 + 2^-5 + 0 + 0 + 2^-8 + 2^-9 + 0 + 0 + 2^-12 + 2^-13
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.5 + .25 + .0625 + .03125 + .00390625 + .001953125 + .000244140625 + .0001220703125
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= .8499755859375
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.5
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.75
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.8125
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.84375
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.84765625
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.849609375
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.849853515625
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.8499755859375
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```
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.453125
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```
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.453125 *2 = 0.90625 [o] [.90625]
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.90625 *2 = 1.8125 [1] [.8125]
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.8125 *2 = 1.625 [1] [.625]
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.625 *2 = 1.25 [1] [.25]
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.25 *2 = 0.5 [0] [.5]
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.5 *2 = 1 [1] []
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-> .011101
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```
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---------------------------------------------------------
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REPRESENTATION FLOATS vs INTEGERS
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---------------------------------------------------------
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```
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integer : 1 (1)
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floater : 0 01111111 00000000000000000000000 (1)
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integer : 1010 (10)
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floater : 0 10000010 01000000000000000000000 (10)
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integer : 1100100 (100)
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floater : 0 10000101 10010000000000000000000 (100)
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integer : 1111101000 (1000)
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floater : 0 10001000 11110100000000000000000 (1000)
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integer : 00000000000000000000000000000010 (2)
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floater : 0 10000000 00000000000000000000000 (2)
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100010.1 (34.5)
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floater : 0 10000100 00010 100000000000000000 (34.5)
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100010.100011001100110011 (34.55)
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floater : 0 10000100 00010 100011001100110011 (34.55)
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1.0001100110011001101 (1.1)
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floater : 0 01111111 00011001100110011001101 (1.1)
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100010.01 (34.25)
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floater : 0 10000100 00010 010000000000000000 (34.25)
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10101.0011100001010001111 (21.22)
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floater : 0 10000011 0101 0011100001010001111 (21.22)
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110.11001100110011001101 (6.8)
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floater : 0 10000001 10 110011001100110011010 (6.8)
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```
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---------------------------------------------------------
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FLOATS -> FIXED
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---------------------------------------------------------
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par multiplications binaires :
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```
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100010.01 (34.25)
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00100010 01000000 (8768)
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00000001 00000000 (256)
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100010.01000000 34.25
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* 1 00000000.00000000 * 256.00
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----------------------- --------
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100010 01000000.0 8768.00
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34.25 * (1 << _frac)
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```
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par decalage binaire :
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```
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0 10000100 00010 010000000000000000 (34.25) float
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10000100 (132) decaler la virgule de 132 - 127 = 5
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100010.01 (34.25) fixe
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value = 34.25;
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signe = (unsigned int)value;
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exponent = ((unsigned int)(value << 1) >> 24) - 127;
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fixedd = (unsigned int)(value << 9) >> (32 - exponent - 8);
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```
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