The details in this post are acquired from IEEE 764 floating point standards. Usually a real number in binary will be represented in the following format- ImIm-1…I2I1I0.F1F2…FnFn-1 where Im and Fn will be either 0 or 1 of integer and fraction parts respectively.
A finite number can also be represented by four integer components : a sign (s), a base (b), a significand (m), and an exponent (e). Then the numerical value of the number is evaluated as (-1)s x m x be.In the binary32 format the floating point numbers are stored in the single precision format. In the single precision format there are 23 bits for the significand 8 bits for the exponent and 1 bit for the sign.
For example, the rational number 9÷2 can be converted to single precision float format as following,
In other words, the above result can be written as (-1)0 x 1.001(2) x 22 which yields the integer components as s = 0, b = 2, significand (m) = 1.001, mantissa = 001 and e = 2. The corresponding single precision floating number can be represented in binary as shown below,
Where the exponent field is supposed to be 2, yet encoded as 129 (127+2) called biased exponent.
One of the goals of the IEEE floating point standards was that you could treat the bits of a floating point number as a (signed) integer of the same size, and if you compared them that way, the values will sort into the same order as the floating point numbers they represented.
If you used a twos-complement representation for the exponent, a small positive number (i.e., with a negative exponent) would look like a very large integer because the second MSB would be set. By using a bias representation instead, you don't run into that -- a smaller exponent in the floating point number always looks like a smaller integer.
A bias of (2n-1 – 1), where n is # of bits used in exponent, is added to the exponent (e) to get biased exponent (E). So, the biased exponent (E) of single precision number can be obtained as
Double Precision Format:
Precision:
The smallest change that can be represented in floating point representation is called as precision. The fractional part of a single precision normalized number has exactly 23 bits of resolution, (24 bits with the implied bit). This corresponds to log(10) (223) = 6.924 = 7 (the characteristic of logarithm) decimal digits of accuracy. Similarly, in case of double precision numbers the precision is log(10) (252) = 15.654 = 16 decimal digits.
Accuracy:
Accuracy in floating point representation is governed by number of significand bits, whereas range is limited by exponent. Not all real numbers can exactly be represented in floating point format. For any numberwhich is not floating point number, there are two options for floating point approximation, say, the closest floating point number less than x as x- and the closest floating point number greater than x as x+. A rounding operation is performed on number of significant bits in the mantissa field based on the selected mode. The round down mode causes x set to x_, the round up mode causes x set to x+, the round towards zero mode causes x is either x- or x+ whichever is between zero and. The round to nearest mode sets x to x- or x+ whichever is nearest to x. Usually round to nearest is most used mode. The closeness of floating point representation to the actual value is called as accuracy.
Special Bit Patterns:
The standard defines few special floating point bit patterns. Zero can’t have most significant 1 bit, hence can’t be normalized. The hidden bit representation requires a special technique for storing zero. We will have two different bit patterns +0 and -0 for the same numerical value zero. For single precision floating point representation, these patterns are given below,
0 00000000 00000000000000000000000 = +0
1 00000000 00000000000000000000000 = -0
Similarly, the standard represents two different bit patters for +INF and -INF. The same are given below,
0 11111111 00000000000000000000000 = +INF
1 11111111 00000000000000000000000 = -INF
All of these special numbers, as well as other special numbers (below) are subnormal numbers, represented through the use of a special bit pattern in the exponent field. This slightly reduces the exponent range, but this is quite acceptable since the range is so large.
An attempt to compute expressions like 0 x INF, 0 ÷ INF, etc. make no mathematical sense. The standard calls the result of such expressions as Not a Number (NaN). Any subsequent expression with NaN yields NaN. The representation of NaN has non-zero significand and all 1s in the exponent field. These are shown below for single precision format (x is don’t care bits),
x 11111111 1m0000000000000000000000
Where m can be 0 or 1. This gives us two different representations of NaN.
0 11111111 110000000000000000000000 _____________ Signaling NaN (SNaN)
0 11111111 100000000000000000000000 _____________Quiet NaN (QNaN)
Usually QNaN and SNaN are used for error handling. QNaN do not raise any exceptions as they propagate through most operations. Whereas SNaN are which when consumed by most operations will raise an invalid exception.
Overflow and Underflow:
Overflow is said to occur when the true result of an arithmetic operation is finite but larger in magnitude than the largest floating point number which can be stored using the given precision. Underflow is said to occur when the true result of an arithmetic operation is smaller in magnitude (infinitesimal) than the smallest normalized floating point number which can be stored. Overflow can’t be ignored in calculations whereas underflow can effectively be replaced by zero.
A finite number can also be represented by four integer components : a sign (s), a base (b), a significand (m), and an exponent (e). Then the numerical value of the number is evaluated as (-1)s x m x be.In the binary32 format the floating point numbers are stored in the single precision format. In the single precision format there are 23 bits for the significand 8 bits for the exponent and 1 bit for the sign.
For example, the rational number 9÷2 can be converted to single precision float format as following,
9(10) ÷ 2(10) = 4.5(10) = 100.1(2)
The result said to be normalized, if it is represented with leading 1 bit, i.e. 1.001(2) x 22.
Omitting this implied 1 on left extreme gives us the mantissa of float number.In other words, the above result can be written as (-1)0 x 1.001(2) x 22 which yields the integer components as s = 0, b = 2, significand (m) = 1.001, mantissa = 001 and e = 2. The corresponding single precision floating number can be represented in binary as shown below,
Where the exponent field is supposed to be 2, yet encoded as 129 (127+2) called biased exponent.
One of the goals of the IEEE floating point standards was that you could treat the bits of a floating point number as a (signed) integer of the same size, and if you compared them that way, the values will sort into the same order as the floating point numbers they represented.
If you used a twos-complement representation for the exponent, a small positive number (i.e., with a negative exponent) would look like a very large integer because the second MSB would be set. By using a bias representation instead, you don't run into that -- a smaller exponent in the floating point number always looks like a smaller integer.
A bias of (2n-1 – 1), where n is # of bits used in exponent, is added to the exponent (e) to get biased exponent (E). So, the biased exponent (E) of single precision number can be obtained as
E = e + 127
The range of exponent in single precision format is -126 to +127. Other values are used for special symbols.Double Precision Format:
Precision:
The smallest change that can be represented in floating point representation is called as precision. The fractional part of a single precision normalized number has exactly 23 bits of resolution, (24 bits with the implied bit). This corresponds to log(10) (223) = 6.924 = 7 (the characteristic of logarithm) decimal digits of accuracy. Similarly, in case of double precision numbers the precision is log(10) (252) = 15.654 = 16 decimal digits.
Accuracy:
Accuracy in floating point representation is governed by number of significand bits, whereas range is limited by exponent. Not all real numbers can exactly be represented in floating point format. For any numberwhich is not floating point number, there are two options for floating point approximation, say, the closest floating point number less than x as x- and the closest floating point number greater than x as x+. A rounding operation is performed on number of significant bits in the mantissa field based on the selected mode. The round down mode causes x set to x_, the round up mode causes x set to x+, the round towards zero mode causes x is either x- or x+ whichever is between zero and. The round to nearest mode sets x to x- or x+ whichever is nearest to x. Usually round to nearest is most used mode. The closeness of floating point representation to the actual value is called as accuracy.
Special Bit Patterns:
The standard defines few special floating point bit patterns. Zero can’t have most significant 1 bit, hence can’t be normalized. The hidden bit representation requires a special technique for storing zero. We will have two different bit patterns +0 and -0 for the same numerical value zero. For single precision floating point representation, these patterns are given below,
0 00000000 00000000000000000000000 = +0
1 00000000 00000000000000000000000 = -0
Similarly, the standard represents two different bit patters for +INF and -INF. The same are given below,
0 11111111 00000000000000000000000 = +INF
1 11111111 00000000000000000000000 = -INF
All of these special numbers, as well as other special numbers (below) are subnormal numbers, represented through the use of a special bit pattern in the exponent field. This slightly reduces the exponent range, but this is quite acceptable since the range is so large.
An attempt to compute expressions like 0 x INF, 0 ÷ INF, etc. make no mathematical sense. The standard calls the result of such expressions as Not a Number (NaN). Any subsequent expression with NaN yields NaN. The representation of NaN has non-zero significand and all 1s in the exponent field. These are shown below for single precision format (x is don’t care bits),
x 11111111 1m0000000000000000000000
Where m can be 0 or 1. This gives us two different representations of NaN.
0 11111111 110000000000000000000000 _____________ Signaling NaN (SNaN)
0 11111111 100000000000000000000000 _____________Quiet NaN (QNaN)
Usually QNaN and SNaN are used for error handling. QNaN do not raise any exceptions as they propagate through most operations. Whereas SNaN are which when consumed by most operations will raise an invalid exception.
Overflow and Underflow:
Overflow is said to occur when the true result of an arithmetic operation is finite but larger in magnitude than the largest floating point number which can be stored using the given precision. Underflow is said to occur when the true result of an arithmetic operation is smaller in magnitude (infinitesimal) than the smallest normalized floating point number which can be stored. Overflow can’t be ignored in calculations whereas underflow can effectively be replaced by zero.
