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Floating Point |
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General form: canonical binary scientific notation |
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1.fffftwo * 2eeee |
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ffff |
fraction (signficand) |
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eeee |
exponent (expressed in decimal
for simplicity) |
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Note that both have signs |
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b31 |
b30 b23 |
b22 |
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b0 |
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s |
e |
f |
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IEEE 754 standard |
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(1985) |
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s (1 bit): sign |
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e (8 bits): exponent |
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excess 127 (NOT 128!) |
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-126 <= e <= 127 |
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f (23 bits): fraction |
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called significand or
mantissa |
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assume leading 1
("hidden one") |
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categories of floating
point numbers |
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* normalized numbers: standard floating point numbers. |
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Most bitstring patterns
in IEEE 754 |
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* denormalized numbers: fewer bits of precision, and smaller (in magnitude) |
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than normalized numbers. |
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* zero: a positive and negative representation of 0 (sign bit 0 or 1) |
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* infinity: also a positive and negative infinity |
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For example, 1.0/0.0
produces infinity. |
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* NaN: "not a number"; undefined value like sqrt (-4) |
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S Exp Fraction |
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Value |
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- ---------
---------------------------- |
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0 1111 1111 1111 1111
1111 1111 1111 111 |
NaN |
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0 1111 1111 0000 0000
0000 0000 0000 001 |
NaN |
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0 1111 1111 0000 0000
0000 0000 0000 000 |
Infinity |
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0 1111 1110 1111 1111
1111 1111 1111 111 |
1.11 . . . 1 x 2127 |
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0 1111 1110 0000 0000
0000 0000 0000 000 |
2127 |
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0 1000 0000 0000 0000
0000 0000 0000 000 |
21 |
Normalized |
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0 0111 1111 0000 0000
0000 0000 0000 000 |
1 |
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0 0111 1110 0000 0000
0000 0000 0000 000 |
2-1 |
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0 0000 0001 0000 0000
0000 0000 0000 000 |
2-126 |
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0 0000 0000 1111 1111
1111 1111 1111 111 |
0.11 . . . 1 x 2-126 |
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0 0000 0000 1000 0000
0000 0000 0000 000 |
2-127 |
Denormalized |
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0 0000 0000 0000 0000
0000 0000 0000 001 |
2-149 |
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0 0000 0000 0000 0000
0000 0000 0000 000 |
0 |
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