IEEE754 floatingpoint


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1 IEEE754 floatingpoint
2 Real and floatingpoint numbers Real numbers R form a continuum  Rational numbers are a subset of the reals  Some numbers are irrational, e.g. π Floatingpoint numbers are an approximation of real numbers  If finite in length, they are a subset of the rationals  Consist of a sign, a significantdigits part  the mantissa or significand, and an exponent of the base (people usually use base 10)
3 Floating Point Floatingpoint numbers are represented by:  a sign  a significand or mantissa  an exponent Sign is easy sign part  0 number is positive  1 number is negative exponent part  numerically, factor = 1 sign significand part Significand and exponent have structure
4 Significand Floating point numbers are normalized  Represent as binary (fixedpoint) number  Multiply by positive or negative power of 2, such that there is a single 1 bit to the left of the radix point Example: = = The leftmost bit (to the left of the radix) is always 1, so it doesn t need to be stored  The 1 is hidden or implicit  Store as the significand Example 2: = = (1.)
5 Exponent Exponent is a power of 2 Exponents can be positive or negative Exponents are stored in ExcessN notation  N is typically 2 (m1) 1 for mbit storage Example: in 5 bits Excess(2 (51) 1) = Excess = = Example: in 8 bits Excess(2 (81) 1) = Excess = =
6 IEEE754 a standard for representing floatingpoint (f.p.) numbers in computer systems  Three binary formats, two decimal formats  additional "storage" formats  adopted in 1985, updated in many operational details All formats share some characteristics  Normalized  Implicit MSb  Signmagnitude representation for significand  ExcessN representation for exponent  Special values for exceptional cases
7 Formats binary16  "Halfprecision"  storage only binary32  "Single precision" binary64  "Double precision" binary128  "Quadruple precision" decimal32  storage only decimal64 decimal128 Decimal formats are new to the 2008 revision IBM zsystems implement these formats
8 IEEE754 Binary Formats
9 Examples = 0x3c00 1, in Binary16: = 0x3c00  sign bit: 0  exponent: = = significand: » leftmost 1bit is implicit 2, in Binary32: = 0xc sign bit: 1  exponent: = = significand: , in Binary32: = 0x3ea sign bit: 0  exponent: = = significand:
10 Exceptional Values small Exponent = all 0 s 0 significand: true zero  positive and negative 0 are both legal nonzero significand: values are subnormal or denormalized no implicit one bit  trade off precision for smaller exponents Binary16 examples: = +0, true (positive) zero = » the largest (negative) subnormal = = , the smallest possible number in Binary16
11 Exceptional Values large Exponent = all 1 s 0 significand: positive or negative infinity nonzero significand: NaN (Not a Number) and indication of an error condition  e.g. division by zero Binary16 examples: = negative infinity, = quiet NaN, e.g. 0/0» indeterminate values the sign doesn t matter = signaling NaN» invalid operations e.g. a machine exception
12 Binary32 Format Again sign exponent significand 1 bit 8 bits 23 bits 1 if negative Excess127 notation, range 126 to +127 normalized to 1 value < 2, leftmost 1 bit not represented All 0 s in the exponent and significand fields represent ± 0 Other values with all 0 s in the exponent field (looks like 127) are subnormal or denormalized values  exponent is hidden bit is 0 Values with all 1 s in the exponent field (looks like 128) and significand 0 (all 0 bits) represent ±infinity Other values with all 1 s in the exponent field represent NaNs "Not a Number" values
13 C types and IEEE754 C's float datatype generally uses "single precision  a.k.a. Binary32 about 7 decimal digits of precision dynamic range roughly to C's double datatype generally uses "double precision  a.k.a. Binary64 about 15 decimal digits of precision dynamic range roughly to double frequently used for scientific calculations
14 Show the Bits in Binary32
15 Intel Processors "Endian"ness Intel, AMD processors are "Littleendian"  Core i7, Opteron, etc. Littleendian: Least Significant Byte (LSB) stored in lowest memory address Bigendian: LSB stored in highest memory address  Most Significant Byte (MSB) stored in lowest memory address Multibyte values are affected by the endianness  That's everything except characters
16 A routine to inspect endianness
17 FloatingPoint 1.0 in 32bit Intel Memory: Memory address 0x7fff1b5a4360 0x7fff1b5a4361 0x7fff1b5a4362 0x7fff1b5a4363 contents 0x _0000 0x _0000 0x _0000 0xbf 1011_1111 f.p. value 1.0 is 1 7f _ _0000_0
18 FloatingPoint 2.0 in 32bit Intel Memory: Memory address 0x7fff1b5a4360 0x7fff1b5a4361 0x7fff1b5a4362 0x7fff1b5a4363 contents 0x _0000 0x _0000 0x _0000 0xc0 1100_0000 f.p. value 2.0 is _ _0000_0
19 FloatingPoint 8.5 in 32bit Intel Memory: Memory address 0x7fff1b5a4360 0x7fff1b5a4361 0x7fff1b5a4362 0x7fff1b5a4363 contents 0x _0000 0x _0000 0x _1000 0x _0001 f.p. value 8.5 is _ _1000_0
20 FloatingPoint 8.99 in 32bit Intel Memory: Memory address 0x7fff1b5a4360 0x7fff1b5a4361 0x7fff1b5a4362 0x7fff1b5a4363 contents 0x0a 0000_1010 0xd7 1101_0111 0x0f 0000_1111 0x _0001 f.p. value 8.99 is fd70a _ _1111_1101_0111_0000_1010
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