In computing, single precision is a computer numbering format that occupies one storage location in computer memory at a given address. Computing is usually defined like the activity of using and developing Computer technology Computer hardware and software. The term computer numbering formats refers to the schemes implemented in Digital computer and Calculator hardware and software to represent numbers A common A single-precision number, sometimes simply a single, may be defined to be an integer, fixed point, or floating point. The integers (from the Latin integer, literally "untouched" hence "whole" the word entire comes from the same origin but via French In Computing, floating point describes a system for numerical representation in which a string of digits (or Bits represents a Real number.
Modern computers with 32-bit words (single precision) provide 64-bit double precision. A bit is a binary digit, taking a value of either 0 or 1 Binary digits are a basic unit of Information storage and communication In Computing, " word " is a term for the natural unit of data used by a particular computer design In Computing, double precision is a Computer numbering format that occupies two adjacent storage locations in computer memory Single precision floating point is an IEEE 754 standard for encoding floating point numbers that uses 4 bytes. The IEEE Standard for Binary Floating-Point Arithmetic ( IEEE 754) is the most widely-used standard for floating-point computation and is followed by many Standardization (or standardisation) is the process of developing and agreeing upon technical standards. In Computing, floating point describes a system for numerical representation in which a string of digits (or Bits represents a Real number. A byte (pronounced "bite" baɪt is the basic unit of measurement of information storage in Computer science.
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Single precision memory format
Sign bit: 1 Exponent width: 8 Significand precision: 23 (24 implicit)
The format is written with an implicit most-significant bit with value 1 unless the written exponent is all zeros. In Computer science, the sign bit is a Bit (usually the Most significant bit) in a Computer numbering format that indicates the sign The significand (also Coefficient or Mantissa) is the part of a floating-point number that contains its significant digits The precision of a value describes the number of digits that are used to express that value 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, log10(224)).
Exponent encodings
Emin (0x01) = -126 Emax (0xfe) = 127 Exponent bias (0x7f) = 127
The 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 infinity and NaNs
All bit patterns are valid encodings. In IEEE 754 Floating point numbers the Exponent is Biased in the engineering sense of the word – the value stored is offset from the actual value by the In Computer science, denormal numbers or denormalized numbers (now often called subnormal numbers) fill the gap around zero in Floating Infinity (symbolically represented with ∞) comes from the Latin infinitas or "unboundedness In Computing, NaN ( N ot a N umber is a value or symbol that is usually produced as the result of an operation on invalid input operands
Single precision examples in hexadecimal
3f80 0000 = 1
c000 0000 = -2
7f7f ffff ~ 3. In Mathematics and Computer science, hexadecimal (also base -, hexa, or hex) is a Numeral system with a 4028234 x 1038 (Max Single)
3eaa aaab ~ 1/3
By default, 1/3 rounds up instead of down like double precision, because of the even number of bits in the significand. In Computing, double precision is a Computer numbering format that occupies two adjacent storage locations in computer memory So the bits beyond the rounding point are 1010. . . which is more than 1/2 of a unit in the last place. In Computer science, Unit in the Last Place or' Unit of Least Precision (ULP is the gap between two very close Floating-point numbers
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 mantissa by 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
Thus
41c8 0000 = 25
See also
- half precision – single precision – double precision – quadruple precision
- Floating point
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