IEEE Standard for Floating-Point Arithmetic: 754
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1 IEEE Standard for Floating-Point Arithmetic: 754 G.E. Antoniou G.E. Antoniou () IEEE Standard for Floating-Point Arithmetic: / 34
2 Floating Point Standard: IEEE /2008 Established in 1985 (2008) as homogeneous standard for binary floating point arithmetic In scientific calculations the range of the numbers can be very large or very small... These numbers can be expressed with Floating Point Notation Floating Point Numbers (FPN) are processed in the Floating Point Unit (FPU). G.E. Antoniou () IEEE Standard for Floating-Point Arithmetic: / 34
3 It floats... The decimal point in FPN is variable or is automatically adjusted or it floats A Floating Point Number (FPN) is defined as, FPN = Fraction Base Exponent. G.E. Antoniou () IEEE Standard for Floating-Point Arithmetic: / 34
4 Example: Decimal Numbers = = = = = and = General decimal Floating Point Number formula: FPN 10 = F 10 Exponent. G.E. Antoniou () IEEE Standard for Floating-Point Arithmetic: / 34
5 Example: Binary Numbers Therefore the binary Floating Point Number formula is: FPN 2 = F 2 Exponent. G.E. Antoniou () IEEE Standard for Floating-Point Arithmetic: / 34
6 Signed (positive and negative) Numbers The leftmost bit of a signed binary number determines the sign of the number: If the leftmost bit is 0 :... [ The number is: Positive ]... If the leftmost bit is 1:... [ The number is: Negative ]... Therefore our FPN formula becomes: where, S = Sign F = Fraction. FPN = ( 1) S F 2 Exponent G.E. Antoniou () IEEE Standard for Floating-Point Arithmetic: / 34
7 Problem of Uniqueness A binary number can be represented as: There is no unique representation... For a unique representation (IEEE 754 standard): Use only one digit to the left of the binary point = Normalization. G.E. Antoniou () IEEE Standard for Floating-Point Arithmetic: / 34
8 FPN Normalization FPM Normalization ensures a unique floating point representation of each number Normalized number: A number in scientific notation that has no leading zeros. G.E. Antoniou () IEEE Standard for Floating-Point Arithmetic: / 34
9 Examples Not Normalized numbers Normalized numbers (Decimal) (Binary) G.E. Antoniou () IEEE Standard for Floating-Point Arithmetic: / 34
10 Packing, Hidden 1 Principle To pack (include) more bits... the IEEE 754 standard makes the leading 1 bit of the normalized binary numbers implicit. Hidden 1 principle... Using the normalization, the leading bit is always nonzero or 1 Since the leading bit is always 1, why carry it? (There is no need to store it)... To have one extra representation bit we can do the following: Shift left by one bit Leading 1 is discarded To get back the initial number... put back the 1. G.E. Antoniou () IEEE Standard for Floating-Point Arithmetic: / 34
11 Updated Formula To represent the hidden 1 and the Fraction use the formula: F = 1 + significant F = 1.significant Using the above Hidden 1 principle our FPN formula now becomes: FPN = ( 1) S (1.significant) 2 Exponent FPN = ( 1) S (1 + significant) 2 Exponent. G.E. Antoniou () IEEE Standard for Floating-Point Arithmetic: / 34
12 Precision Formats: 32-bit IEEE 754 Floating Point Standard Single (32-bit) Precision Format (Occupies one four byte word) S = 1 E = 8 F = 23 G.E. Antoniou () IEEE Standard for Floating-Point Arithmetic: / 34
13 Precision Formats: 64-bit IEEE 754 Floating Point Standard Double (64-bit) Precision Format S = 1 E = 11 F = 52 G.E. Antoniou () IEEE Standard for Floating-Point Arithmetic: / 34
14 Biased Notation Biasing the exponent improves accuracy with very small numbers We represent... the most negative exponent as: the most positive exponent as: G.E. Antoniou () IEEE Standard for Floating-Point Arithmetic: / 34
15 Ordered Binary numbers (Exponent). Biased 127 Exponent Real Exponent (Biased 127 Exponent) (Reserved) (1-127) (2-127) ( ) ( ) ( ) ( ) ( ) (Reserved) For a 32 bit single-precision number, an exponent in the range: ,..., is biased by adding 127 (bias) to get a value in the range 1,..., 254 (0 and 255 are reserved). G.E. Antoniou () IEEE Standard for Floating-Point Arithmetic: / 34
16 IEEE 754 Standard; (Single Precision) 1 The IEEE 754 standard specifies the exponent in the excess 127 format or bias for Single Precision 2 In this format the number 127 is added to the value of the actual Unbiased Exponent so that Biased Exponent: E = Unbiased Exponent Solving for the Unbiased Exponent yields, Unbiased Exponent = E 127 Therefore our final FPN 32 formula takes the form: FPN 32 = ( 1) S (1 + significand) 2 E 127. G.E. Antoniou () IEEE Standard for Floating-Point Arithmetic: / 34
17 Final IEEE-754 standard formula for FPN 32 FPN 32 = ( 1) S (1 + significand) 2 E 127 FPN 32 = ( 1) S (1.significand) 2 E 127. G.E. Antoniou () IEEE Standard for Floating-Point Arithmetic: / 34
18 Example Exponents 1 Unbiased Exponent 2 is stored as ( ) = 129 = (Biased Exponent) 2 Unbiased Exponent -2 is stored as (127 + (-2) = 125 = (Biased Exponent). G.E. Antoniou () IEEE Standard for Floating-Point Arithmetic: / 34
19 IEEE 754 Standard; FPN 64, (Double Precision) 1 The IEEE 754 standard specifies the exponent in the excess 1023 format or bias for Double Precision 2 In this format the number 1023 is added to the value of the actual exponent so that, FPN 64 = ( 1) S (1 + significand) 2 E 1023 FPN 64 = ( 1) S (1.significand) 2 E 1023 G.E. Antoniou () IEEE Standard for Floating-Point Arithmetic: / 34
20 Largest and Smallest Values 1 Single Precision, FPN 32 Largest normalized value: = ± Smallest normalized value: = ± Double Precision, FPN 64 Largest normalized value: = ± Smallest normalized value: G.E. Antoniou () IEEE Standard for Floating-Point Arithmetic: / 34
21 Examples... Illustrative examples follow... Decimal to FPN 32 FPN 32 to Decimal G.E. Antoniou () IEEE Standard for Floating-Point Arithmetic: / 34
22 Decimal to FPN 32 Given: Bit 8Bits {}}{{}}{ Find: Why? 23Bits {}}{ }{{} 21Bits G.E. Antoniou () IEEE Standard for Floating-Point Arithmetic: / 34
23 Example-1: 5 10 Given the Decimal Number: Find the FPN 32 representation in single precision notation (127 bias) = = = Unbiased Exponent = 2 Biased Exponent: E = = The unsigned binary equivalent of is Since, 5 10, is negative: S = 1 Therefore: 1Bit {}}{ 1 8Bits {}}{ Bits {}}{ }{{} 21Bits G.E. Antoniou () IEEE Standard for Floating-Point Arithmetic: / 34
24 Example-2: Given the Decimal Number: Find the FPN 32 representation in single precision notation (127 bias) in single precision notation (127 bias) = = = Biased Exponent: E = = The unsigned binary equivalent of is S = 1 Therefore: }{{} 21 Bits G.E. Antoniou () IEEE Standard for Floating-Point Arithmetic: / 34
25 Example-3: Given the Decimal Number: Find the FPN 32 representation in single precision notation (127 bias). First let us find the binary equivalent of = = = 0.00 [stop] Therefore, top bottom... the answer is: (0.11) 2 Read more: G.E. Antoniou () IEEE Standard for Floating-Point Arithmetic: / 34
26 Example-4: = = Biased Exponent: E = = The unsigned binary equivalent of is: S = 0 Therefore: }{{} Bits 22 G.E. Antoniou () IEEE Standard for Floating-Point Arithmetic: / 34
27 Example-5: 1 10 Given the Decimal Number: Find the FPN 32 representation in single precision notation (127 bias) = Biased Exponent: E = = The unsigned binary equivalent of is: S = 0 Therefore: }{{} Bits 22 G.E. Antoniou () IEEE Standard for Floating-Point Arithmetic: / 34
28 FPN 32 to Decimal Given: }{{} 21 Bits Find: Why? G.E. Antoniou () IEEE Standard for Floating-Point Arithmetic: / 34
29 Example-1: FPN 32 Given the Floating Point Number: }{{} 21 Bits Find the Decimal representation. The sign (S) is: 1 The biased Exponent (E) is: = Fraction: FPN Formula: FPN = ( 1) S (1.significand) 2 E 127 Result = ( 1) 1 (1.11) = ( 1) Result = (1.11) 2 3 Result = Result= G.E. Antoniou () IEEE Standard for Floating-Point Arithmetic: / 34
30 2 minutes Given: 1Bit {}}{ 1 Decimal number? 8Bits {}}{ Bits {}}{ }{{} 21Bits G.E. Antoniou () IEEE Standard for Floating-Point Arithmetic: / 34
31 Example-2: FPN 32 Given the Floating Point Number: }{{} 21 Bits Find the Decimal representation. The sign (S) is: 1 The biased Exponent (E) is: = Fraction: FPN Formula: FPN = ( 1) S (1.significand) 2 E 127 Result = ( 1) 1 (1.01) = ( 1) Result = (1.01) 2 2 Result = Result = 5 10 G.E. Antoniou () IEEE Standard for Floating-Point Arithmetic: / 34
32 ARM Cortex RISC CPU A57 FP operations FP operations are performed within a special unit called; Floating Point Unit (FPU). Communication via the CPU and the FPU is done by special FP registers. The CPU and FPU can work in parallel on different data. (instruction-level parallelism) Today FPU and CPU can be in the same chip G.E. Antoniou () IEEE Standard for Floating-Point Arithmetic: / 34
33 ARM Cortex RISC CPU A57 G.E. Antoniou () IEEE Standard for Floating-Point Arithmetic: / 34
34 Floating-Point in Java, C and C++ Java, C and C++ have two kinds of floating-point numbers (IEEE 754): Float (32-bit; 4-bytes) Double (64-bit; 8-bytes) (end) G.E. Antoniou () IEEE Standard for Floating-Point Arithmetic: / 34
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