DECIMALE VS BINARY
11010101
1*10000000 | (128) | 2^7 | 128 *1 | 0 *2 +1
1*1000000 | (64) | 2^6 | + 64 *1 | 1 *2 +1
0*100000 | (32) | 2^5 | | 11 *2
1*10000 | (16) | 2^4 | + 16 *1 | 110 *2 +1
0*1000 | (8) | 2^3 | | 1101 *2
1*100 | (4) | 2^2 | + 4 *1 | 11010 *2 +1
0*10 | (2) | 2^1 | | 110101 *2
1*1 | (1) | 2^0 | + 1 *1 | 1101010 *2 +1
| ______ |
| 213 |
213
2*100 |(1100100)| 10^2 | 1100100 *2 | 0 *10 +2
1*10 | (1010)| 10^1 | + 1010 *1 | 2 *10 +1
3*1 | (1)| 10^0 | + 1 *3 | 21 *10 +3
| | | ___________ | 213
| | | 11010101 |
|
|
1100100 11001000 1 11010010 |
* 2 + 1010 * 3 + 11 |
_________ _________ ___ _________ |
11001000 11010010 11 11010101 |
213 -> 11010101
213 -128 = 85 | 213 /2 = 106 r 1
85 - 64 = 2 | 106 /2 = 53 r 0
| 53 /2 = 26 r 1
21 - 16 = 5 | 26 /2 = 13 r 0
| 13 /2 = 6 r 1
5 - 4 = 1 | 6 /2 = 3 r 0
| 3 /2 = 1 r 1
1 - 1 = 0 | 1 /2 = 0 r 1
11010101 -> 213
| 0*2 + 1 = 1
| 1*2 + 1 = 3
| 3*2 = 6
| 6*2 + 1 = 13
| 13*2 = 26
| 26*2 + 1 = 53
| 53*2 = 106
| 106*2 + 1 = 213
NEGATIVS INTEGERS
5 0101 0 0000 0 0000 0000 0000 0000 0000 0000 0000 0000
1010 + 1 1 0001 1 0000 0000 0000 0000 0000 0000 0000 0001
-5 1011 2 0010 2 0000 0000 0000 0000 0000 0000 0000 0010
3 0011 3 0000 0000 0000 0000 0000 0000 0000 0011
4 0100 4 0000 0000 0000 0000 0000 0000 0000 0100
-> 5 0101 5 0000 0000 0000 0000 0000 0000 0000 0101
6 0110 6 0000 0000 0000 0000 0000 0000 0000 0110
7 0111 7 0000 0000 0000 0000 0000 0000 0000 0111
...
0111 1111 1111 1111 1111 1111 1111 1111 MAXINT
1000 0000 0000 0000 0000 0000 0000 0000 MININT
...
8 1000 -8 1111 1111 1111 1111 1111 1111 1111 1000
9 1001 -7 1111 1111 1111 1111 1111 1111 1111 1001
10 1010 -6 1111 1111 1111 1111 1111 1111 1111 1010
-> 11 1011 -5 1111 1111 1111 1111 1111 1111 1111 1011
12 1100 -4 1111 1111 1111 1111 1111 1111 1111 1100
13 1101 -3 1111 1111 1111 1111 1111 1111 1111 1101
14 1110 -2 1111 1111 1111 1111 1111 1111 1111 1110
15 1111 -1 1111 1111 1111 1111 1111 1111 1111 1111
FLOATS
https://stackoverflow.com/questions/7644699/how-are-floating-point-numbers-stored-in-memory
1.....................23
8......1
seeeeeeeemmmmmmmmmmmmmmmmmmmmmmm meaning
31 0 bit #
signe
exponent
mantis
1.175494351 * 10^-38 < ... > 3.40282347 * 10^+38
2x10^-1 = 0.2x10^0 = 0.02x10^1 = 0.2
-
x 00000000 xxxxxxxx xxxxxxxx xxxxxxx special meanings (see bellow)
-
x 11111111 xxxxxxxx xxxxxxxx xxxxxxx //
-
x 00000001 xxxxxxxx xxxxxxxx xxxxxxx smallest exponent (-126)
-
x 01111111 xxxxxxxx xxxxxxxx xxxxxxx 0 is middle exponent (254 / 2 = 127)
-
x 11111110 xxxxxxxx xxxxxxxx xxxxxxx biggest exponent (+127)
-
x 00000000 00000000 00000000 0000000 0 (x can still be + or -)
-
x 11111111 00000000 00000000 0000000 +/- infinity
-
x 11111111 xxxxxxxx xxxxxxxx xxxxxx1 NaN (at least one non-zero mantissa digit)
-
x 00000000 xxxxxxxx xxxxxxxx xxxxxx1 denormalized numbers (same for mantissa)
CONVERSIONS
5.75
5 :
5 / 2 = 2 -> 1
2 / 2 = 1 -> 0
1 / 2 = 0 -> 1
75 :
.75 *2^2 = 2,88 ~= 3
(*2^x parce que c le moyen dont on fait "bouger" la virgule
en base 2 (*10^x) en base 10)
3 / 2 = 1 -> 1
1 / 2 = 0 -> 1
101.11 = 2^2 + 2^0 + 2^-1 + 2^-2
= 4 + 1 + 0.5 + 0.25
0.875
.875 * 2^3 = 7 -> 0.111 * 2^3 = 111.0
7 / 2 = 3 -> 1
3 / 2 = 1 -> 1
1 / 1 = 0 -> 1
0.111
0.1875
-1 1875
0.1875 = 1.875 * 10^-1 = 01 11101010011
0.1875 = 0011
0*2^-1 + 082^-2 + 1*2^-3 + 1*2^-4
0 + 0 + .125 + .0625 = .1875
-43.625
-101011.101
fixed :
1 101011 101
-
101011
2^5 + 2^3 + 2^1 + 1
32 + 8 + 2 + 1 = 43
101
2^-1 + 2^-3
.5 + .125 = .625
-53.5 --> -110101.1
[ fixed : ]
1 110101 1
-
110101
2^5 + 2^4 + 2^2 + 1
32 + 16 + 4 + 1 = 53
1
2^-1
.5 = .5
[ float : ]
-1.101011 * 2^5
11.01011 -> 1
110.1011 -> 2
1101.011 -> 3
11010.11 -> 4
110101.1 -> 5
1 101 101011
-
2^2 + 1
4 + 1 = 5
101011
->1101011
.85
.85 * 2 = 1.7 [1] [.7]
.7 * 2 = 1.4 [1] [.4]
.4 * 2 = 0.8 [0] [.8]
.8 * 2 = 1.6 [1] [.6]
.6 * 2 = 1.2 [1] [.2]
.2 * 2 = 0.4 [0] [.4]
.4 * 2 = 0.8 [0] [.8]
.8 * 2 = 1.6 [1] [.6]
.6 * 2 = 1.2 [1] [.2]
.2 * 2 = 0.4 [0] [.4]
.4 * 2 = 0.8 [0] [.8]
.8 * 2 = 1.6 [1] [.6]
.6 * 2 = 1.2 [1] [.2]
.2 * 2 = 0.4 [0] [.4]
.4 * 2 = 0.8 [0] [.8]
.8 * 2 = 1.6 [1] [.6]
.6 * 2 = 1.2 [1] [.2]
.2 * 2 = 0.4 [0] [.4]
.4 * 2 = 0.8 [0] [.8]
.8 * 2 = 1.6 [1] [.6]
.6 * 2 = 1.2 [1] [.2]
.2 * 2 = 0.4 [0] [.4]
.4 * 2 = 0.8 [0] [.8]
...
-> 11011001100110011001100...
1 1 0 1 1 0 0 1 1 0 0 1 1
2^-1 + 2^-2 + 0 + 2^-4 + 2^-5 + 0 + 0 + 2^-8 + 2^-9 + 0 + 0 + 2^-12 + 2^-13
.5 + .25 + .0625 + .03125 + .00390625 + .001953125 + .000244140625 + .0001220703125
= .8499755859375
.5
.75
.8125
.84375
.84765625
.849609375
.849853515625
.8499755859375
.453125
.453125 *2 = 0.90625 [o] [.90625]
.90625 *2 = 1.8125 [1] [.8125]
.8125 *2 = 1.625 [1] [.625]
.625 *2 = 1.25 [1] [.25]
.25 *2 = 0.5 [0] [.5]
.5 *2 = 1 [1] []
-> .011101
REPRESENTATION FLOATS vs INTEGERS
integer : 1 (1)
floater : 0 01111111 00000000000000000000000 (1)
integer : 1010 (10)
floater : 0 10000010 01000000000000000000000 (10)
integer : 1100100 (100)
floater : 0 10000101 10010000000000000000000 (100)
integer : 1111101000 (1000)
floater : 0 10001000 11110100000000000000000 (1000)
integer : 00000000000000000000000000000010 (2)
floater : 0 10000000 00000000000000000000000 (2)
100010.1 (34.5)
floater : 0 10000100 00010 100000000000000000 (34.5)
100010.100011001100110011 (34.55)
floater : 0 10000100 00010 100011001100110011 (34.55)
1.0001100110011001101 (1.1)
floater : 0 01111111 00011001100110011001101 (1.1)
100010.01 (34.25)
floater : 0 10000100 00010 010000000000000000 (34.25)
10101.0011100001010001111 (21.22)
floater : 0 10000011 0101 0011100001010001111 (21.22)
110.11001100110011001101 (6.8)
floater : 0 10000001 10 110011001100110011010 (6.8)
FLOATS -> FIXED
par multiplications binaires :
100010.01 (34.25)
00100010 01000000 (8768)
00000001 00000000 (256)
100010.01000000 34.25
* 1 00000000.00000000 * 256.00
----------------------- --------
100010 01000000 8768.00
34.25 * (1 << _frac)
par decalage binaire :
0 10000100 00010 010000000000000000 (34.25) float
10000100 (132) decaler la virgule de 132 - 127 = 5
100010.01 (34.25) fixe
this->_value = *((int *)&floater); // access float adress content as int
int sign = this->_value & (1 << 31); // extract sign
int exponent = ((unsigned int)(this->_value << 1) >> 24) - 127; // extract exponent
int integer = (this->_value << 8) | (1 << 31); // add left 1
integer = (unsigned int)integer >> (31 - this->_frac - exponent);// align to right
if (sign != 0)
integer = (~integer + 1); // reverse negatif
integer = (integer << (30 - this->_frac - exponent)) | sign; // add sign
integer >>= (30 - this->_frac - exponent); // align right
std::cout << "integer : " << printBitsInt(integer) << " (" << integer << ")\n";