- Single precision
computing, single precision is a computer numbering formatthat occupies one storage location in computer memory at a given address. A single-precision number, sometimes simply a single, may be defined to be an integer, fixed point, or floating point.
Modern computers with 32-
bitwords ( single precision) provide 64-bit double precision. "Single precision floating point" is an IEEE 754 standard for encoding floating pointnumbers that uses 4 bytes.
Single precision memory format
Sign bit: 1 Exponentwidth: 8 Significantprecision: 23 (24 implicit)
The format is written with an implicit most-significant bit with value 1 unless the written exponent is all zeros. Thus only 23 bits of the fraction mantissa appear in the memory format but the total precision is 24 bits (better than 7 decimal digits, ).
Emin (0x01) = -126 Emax (0xfe) = 127
Exponent bias(0x7f) = 127The true exponent = written exponent - exponent bias
0x00 and 0xff are reserved exponents 0x00 is used to represent zero and
denormals 0xff is used to represent infinityand NaNs
All bit patterns are valid encoding.
Single precision examples in
3f80 0000 = 1
c000 0000 = -2
7f7f ffff ~ 3.4028234 x 1038 (Max Single) 3eaa aaab ~ 1/3By default, 1/3 rounds up instead of down like
double precision, because of the even number of bits in the significant.So the bits beyond the rounding point are
1010...which is more than 1/2 of a
unit in the last place.
0000 0000 = 0 8000 0000 = -0
7f80 0000 = Infinity ff80 0000 = -Infinity
Converting from single precision to human readable form
We start with the hexadecimal representation of the value, 41c80000, in this example, and convert it to binary
41c8 000016 = 0100 0001 1100 1000 0000 0000 0000 00002
then we break it down into three parts; sign bit, exponent and mantissa.
Sign bit: 0 Exponent: 1000 00112 = 8316 = 131 Mantissa: 100 1000 0000 0000 0000 00002 = 48000016
We then add the implicit 24th bit to the mantissa
Mantissa: 1100 1000 0000 0000 0000 00002 = C8000016
and decode the exponent value by subtracting 127
Raw exponent: 8316 = 131 Decoded exponent: 131 - 127 = 4
Each of the 24 bits of the mantissa, bit 23 to bit 0, represents a value, starting at 1 and halves for each bit, as follows
bit 23 = 1 bit 22 = 0.5 bit 21 = 0.25 bit 20 = 0.125 bit 19 = 0.0625 . .
The mantissa in this example has three bits set, bit 23, bit 22 and bit 19. We can now decode the mantissaby adding the values represented by these bits.
Decoded mantissa: 1 + 0.5 + 0.0625 = 1.5625
Then we need to multiply with the base, 2, to the power of the exponent to get the final result
1.5625 * 24 = 25
41c8 0000 = 25
half precision– single precision– double precision– quadruple precision
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