- Single precision
In

computing ,**single precision**is acomputer numbering format that occupies one storage location in computer memory at a given`address`. A**single-precision number**, sometimes simply a**single**, may be defined to be aninteger ,fixed point , orfloating point .Modern computers with 32-

bit words (single precision ) provide 64-bitdouble precision . "Single precision floating point" is an IEEE 754 standard for encodingfloating point numbers that uses 4byte s.**Single precision memory format**Sign bit : 1Exponent width: 8Significant precision: 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, $log\_\{10\}(2^\{24\})\; approx\; 7.225$).

:

**Exponent encoding**E

_{min}(0x01) = -126 E_{max}(0xfe) = 127Exponent bias (0x7f) = 127The true exponent = written exponent - exponent bias0x00 and 0xff are reserved exponents 0x00 is used to represent zero and

denormal s 0xff is used to representinfinity andNaN sAll bit patterns are valid encoding.

**Single precision examples in**hexadecimal 3f80 0000 = 1

c000 0000 = -2

7f7f ffff ~ 3.4028234 x 10

^{38}(Max Single) 3eaa aaab ~ 1/3By default, 1/3 rounds up instead of down likedouble 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 aunit 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 0000

_{16}= 0100 0001 1100 1000 0000 0000 0000 0000_{2}then we break it down into three parts; sign bit, exponent and mantissa.

Sign bit: 0 Exponent: 1000 0011

_{2}= 83_{16}= 131 Mantissa: 100 1000 0000 0000 0000 0000_{2}= 480000_{16}We then add the implicit 24th bit to the mantissa

Mantissa:

**1**100 1000 0000 0000 0000 0000_{2}= C80000_{16}and decode the exponent value by subtracting 127

Raw exponent: 83

_{16}= 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 * 2

^{4}=**25**Thus

41c8 0000 = 25

**See also***

half precision –single precision –double precision –quadruple precision

*Floating point

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