ecture 25 Floating Point Friedland and Weaver Computer Science 61C Spring 2017 March 17th, 2017

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1 ecture 25 Computer Science 61C Spring 2017 March 17th, 2017 Floating Point 1

2 New-School Machine Structures (It s a bit more complicated!) Software Hardware Parallel Requests Assigned to computer e.g., Search Katz Parallel Threads Assigned to core e.g., Lookup, Ads Harness Parallelism & Achieve High Performance Warehouse Scale Computer Computer Smart Phone Parallel Instructions >1 one time e.g., 5 pipelined instructions Parallel Data Core Memory Input/Output (Cache) Core Core >1 data one time e.g., Add of 4 pairs of words Instruction Unit(s) Functional Unit(s) Hardware descriptions A 0 +B 0 A 1 +B 1 A 2 +B 2 A 3 +B 3 All one time Programming Languages Cache Memory Logic Gates 2

3 New-School Machine Structures (It s a bit more complicated!) Software Hardware Parallel Requests Assigned to computer e.g., Search Katz Parallel Threads Assigned to core e.g., Lookup, Ads Harness Parallelism & Achieve High Performance Warehouse Scale Computer How do we know? Computer Smart Phone Parallel Instructions >1 one time e.g., 5 pipelined instructions Parallel Data Core Memory Input/Output (Cache) Core Core >1 data one time e.g., Add of 4 pairs of words Instruction Unit(s) Functional Unit(s) Hardware descriptions A 0 +B 0 A 1 +B 1 A 2 +B 2 A 3 +B 3 All one time Programming Languages Cache Memory Logic Gates 2

4 Review of Numbers What can we represent in N bits? 2 N things, and no more! They could be 3

5 Review of Numbers What can we represent in N bits? 2 N things, and no more! They could be Unsigned integers: 3

6 Review of Numbers What can we represent in N bits? 2 N things, and no more! They could be Unsigned integers: 0 to 2 N - 1 (for N=32, 2 N 1 = 4,294,967,295) Signed Integers (Two s Complement) 3

7 Review of Numbers What can we represent in N bits? 2 N things, and no more! They could be Unsigned integers: 0 to 2 N - 1 (for N=32, 2 N 1 = 4,294,967,295) Signed Integers (Two s Complement) -2 (N-1) to 2 (N-1) - 1 (for N=32, 2 (N-1) = 2,147,483,648) 3

8 What about other numbers? 1. Very large numbers? (seconds/millennium) 31,556,926,000 ten ( x ) 2. Very small numbers? (Bohr radius) ten ( x ) 3. Numbers with both integer & fractional parts? 1.5 4

9 What about other numbers? 1. Very large numbers? (seconds/millennium) 31,556,926,000 ten ( x ) 2. Very small numbers? (Bohr radius) ten ( x ) 3. Numbers with both integer & fractional parts? 1.5 First consider #3. our solution will also help with #1 and #2. 4

10 Representation of Fractions Binary Point like decimal point signifies boundary between integer and fractional parts: Example 6-bit representation: xx.yyyy

11 Representation of Fractions Binary Point like decimal point signifies boundary between integer and fractional parts: Example 6-bit representation: xx.yyyy two = 1x x x2-3 = ten 5

12 Representation of Fractions Binary Point like decimal point signifies boundary between integer and fractional parts: Example 6-bit representation: xx.yyyy two = 1x x x2-3 = ten If we assume fixed binary point, range of 6-bit representations with this format: 0 to (almost 4) 5

13 Fractional Powers of 2 i 2 -i / / / / /

14 Representation of Fractions with Fixed Pt. What about addition and multiplication? Addition is straightforward: ten ten ten 7

15 Representation of Fractions with Fixed Pt. What about addition and multiplication? Addition is straightforward: ten ten ten ten ten Multiplication a bit more complex:

16 Representation of Fractions with Fixed Pt. What about addition and multiplication? Addition is straightforward: ten ten ten ten ten Multiplication a bit more complex: Where s the answer, 0.11? (need to remember where point is) 7

17 Representation of Fractions: Floating Pt What we really want is to float the binary point. Why? 8

18 Representation of Fractions: Floating Pt What we really want is to float the binary point. Why? Floating binary point most effective use of our limited bits (and thus more accuracy in our number representation): 8

19 Representation of Fractions: Floating Pt What we really want is to float the binary point. Why? Floating binary point most effective use of our limited bits (and thus more accuracy in our number representation): Example: put ten into binary. Represent with 5-bits choosing where to put the binary point: Store these bits and keep track of the binary point 2 places to the left of the MSB Any other solution would lose accuracy! 8

20 Representation of Fractions: Floating Pt What we really want is to float the binary point. Why? Floating binary point most effective use of our limited bits (and thus more accuracy in our number representation): Example: put ten into binary. Represent with 5-bits choosing where to put the binary point: Store these bits and keep track of the binary point 2 places to the left of the MSB Any other solution would lose accuracy! With floating-point rep., each numeral carries an exponent field recording the whereabouts of its binary point. The binary point can be outside the stored bits, so very large and small numbers can be represented. 8

21 Scientific Notation (in Decimal) mantissa exponent 6.02 ten x decimal point radix (base) Normalized form: no leading 0s (exactly one digit to left of decimal point) Alternatives to representing 1/1,000,000,000 Normalized: 1.0 x 10-9 Not normalized: 0.1 x 10-8,10.0 x

22 Scientific Notation (in Binary) mantissa exponent 1.01 two x 2-1 binary point radix (base) Computer arithmetic that supports it called floating point, because it represents numbers where the binary point is not fixed, as it is for integers Declare such variable in C as float double for double precision (64bit vs 32bit). 10

23 Floating-Point Representation (1/2) Normal format: +1.xxx x two *2 yyy y two Multiple of Word Size (32 bits) S Exponent Significand 1 bit 8 bits 23 bits S represents Sign Exponent represents y s Significand represents x s (also called Mantissa) Represent numbers as small as 2.0 ten x to as large as 2.0 ten x

24 Floating-Point Representation (2/2) What if result too large? (> 2.0x10 38, < -2.0x10 38 ) Overflow! Exponent larger than represented in 8-bit Exponent field What if result too small? (>0 & < 2.0x10-38, <0 & > -2.0x10-38 ) Underflow! Negative exponent larger than represented in 8-bit Exponent field overflow underflow overflow -2x x x x10 38 What would help reduce chances of overflow and/or underflow? 12

25 IEEE 754 Floating-Point Standard (1/3) IEEE Bit Single Precision Float Sign Exponent Mantissa Sign bit: 1 means negative 0 means positive Significand in sign-magnitude format (not 2 s complement) 13

26 IEEE 754 Floating-Point Standard (1/3) IEEE Bit Single Precision Float Sign Exponent Mantissa Sign bit: 1 means negative 0 means positive Significand in sign-magnitude format (not 2 s complement) To pack more bits, leading 1 implicit for normalized numbers 13

27 IEEE 754 Floating-Point Standard (1/3) IEEE Bit Single Precision Float Sign Exponent Mantissa Sign bit: 1 means negative 0 means positive Significand in sign-magnitude format (not 2 s complement) To pack more bits, leading 1 implicit for normalized numbers bits single, bits double always true: 0 < Mantissa < 1 (for normalized numbers) Note: 0 has no leading 1, so reserve exponent value 0 just for number 0 13

28 IEEE 754 Floating Point Standard (2/3) IEEE 754 uses biased exponent representation: Use just magnitude and offset by half the range 14

29 IEEE 754 Floating Point Standard (2/3) IEEE 754 uses biased exponent representation: Use just magnitude and offset by half the range 14

30 IEEE 754 Floating Point Standard (2/3) IEEE 754 uses biased exponent representation: Use just magnitude and offset by half the range Designers wanted FP numbers to be used even if no FP hardware; e.g., sort records with FP numbers using integer compares 14

31 IEEE 754 Floating Point Standard (2/3) IEEE 754 uses biased exponent representation: Use just magnitude and offset by half the range Designers wanted FP numbers to be used even if no FP hardware; e.g., sort records with FP numbers using integer compares Wanted bigger (integer) exponent field to represent bigger numbers 14

32 IEEE 754 Floating Point Standard (2/3) IEEE 754 uses biased exponent representation: Use just magnitude and offset by half the range Designers wanted FP numbers to be used even if no FP hardware; e.g., sort records with FP numbers using integer compares Wanted bigger (integer) exponent field to represent bigger numbers 2 s complement poses a problem (because negative numbers look bigger) 14

33 IEEE 754 Floating Point Standard (3/3) Called Biased Notation, where bias is number subtracted to get final number IEEE 754 uses bias of 127 for single precision Subtract 127 from Exponent field to get actual value for exponent (-1) S x (1 + Mantissa) x 2 (Exponent-127) Double precision identical, except with exponent bias of 1023 (half, quad similar) IEEE Bit Single Precision Float Sign Exponent Mantissa 15

34 Father of the Floating point standard IEEE Standard 754 for Binary Floating-Point Arithmetic. Prof. Kahan 16

35 Father of the Floating point standard IEEE Standard 754 for Binary Floating-Point Arithmetic ACM Turing Award Winner! Prof. Kahan 16

36 Wake-up: iclickers!!! Guess this Floating Point number: A: -1x B: +1x C: -1x 2 1 D: +1.5x 2-1 E: -1.5x

37 Representation for ± In FP, divide by 0 should produce ±, not overflow. 18

38 Representation for ± In FP, divide by 0 should produce ±, not overflow. Why? 18

39 Representation for ± In FP, divide by 0 should produce ±, not overflow. Why? OK to do further computations with E.g., X/0 > Y may be a valid comparison 18

40 Representation for ± In FP, divide by 0 should produce ±, not overflow. Why? OK to do further computations with E.g., X/0 > Y may be a valid comparison IEEE 754 represents ± Most positive exponent reserved for Significant all zeroes 18

41 Representation for 0 Represent 0? exponent all zeroes significant all zeroes What about sign? Both cases valid +0: :

42 Special Numbers 20

43 Special Numbers What have we defined so far? Exponent Significand Object nonzero??? anything +/- fl. pt. # /- 255 nonzero??? 20

44 Special Numbers What have we defined so far? Exponent Significand Object nonzero??? anything +/- fl. pt. # /- 255 nonzero??? Professor Kahan had clever ideas: Wanted to use Exp=0, 255 & Sig!=0 20

45 Representation for Not a Number What do I get if I calculate sqrt(-4.0)or 0/0? If not an error, these shouldn t be either Called Not a Number (NaN) Exponent = 255, Significand nonzero Why is this useful? Hope NaNs help with debugging? They contaminate: op(nan, X) = NaN Can use the significand to identify which! 21

46 Representation for Denorms (1/2) Problem: There s a gap among representable FP numbers around 0 Smallest representable pos num: a = 1.0 two * = Second smallest representable pos num: b = two * = ( two ) * = ( ) * = Gaps! b 0 a + Normalization and implicit 1 is to blame! a - 0 = b - a =

47 Representation for Denorms (2/2) Solution: We still haven t used Exponent = 0, Significand nonzero DEnormalized number: no (implied) leading 1, implicit exponent = Smallest representable pos num: a = Second smallest representable pos num: b =

48 Special Numbers Summary Reserve exponents, significands: Exponent Significand Object nonzero Denorm anything +/- fl. pt. # /- 255 nonzero NaN 24

49 ICSILog: Let s assume you want to calculate logarithms FAST IEEE Bit Single Precision Float Sign Exponent Mantissa General Equations: value = Sign 2 Exponent x x Mantissa log (value) = Exponent + log (Mantissa) => 2 2 log (value) = log (value) x 2 See: Technical Report in reading for this lecture! 25

50 ICSILog: Idea value: 0 Sign Exponent Mantissa 22 q x Quantized Mantissa + Lookup Table in CPU Cache = log(value): 0 With q=8 usually ~ 30 CPU cycles 26

51 ICSILog Lookup Table in $ 27

52 A Quick Comparison Method Speed up Error Standard Log AMD ACML Log ~ e-07 Fast-Math Log ~ e-07 FastLog (Taylor order 3) ~ e-03 ICSILog (q=0) ~ 1.3 ~0.00 ICSILog (q=7) ~ e-06 ICSILog (q=8) ~ ICSILog v2 (q=7) ~ E-06 ICSILog v2 (q=8) ~ ICSILog() ~6 times faster than log() for floats Arbitrarily accurate (depends on cache usage) 28

53 Conclusion Floating Point lets us: Represent numbers containing both integer and fractional parts; makes efficient use of available bits. Store approximate values for very large and very small #s. IEEE 754 Floating-Point Standard is most widely accepted attempt to standardize interpretation of such numbers (Every desktop or server computer sold since ~1997 follows these conventions) Knowledge of the inner workings of this can help efficiency even for a pure" software developer. 29

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