mith College Computer Science Fixed & Floating Point Formats CSC231 Fall 2017 Week #15 Dominique Thiébaut

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1 mith College Computer Science Fixed & Floating Point CSC231 Fall 2017 Week #15 Formats Dominique Thiébaut

2 Trick Question for ( double d = 0; d!= 0.3; d += 0.1 ) System.out.println( d );

3 We stopped here last time

4 Normalization (in decimal) y = (normal = standard form) y = x 10 2 y = x 10 11

5 Normalization (in binary) y = ( d) y = x 2 3 y = x 10 11

6 Normalization (in binary) y = decimal y = x 2 3

7 Normalization (in binary) y = decimal y = x 2 3 y = x binary

8 x sign mantissa exponent

9 But, remember, all* numbers have a leading 1, so, we can pack the bits even more efficiently! *really?

10 implied bit! x sign mantissa exponent

11 IEEE Format S Exponent (8) Mantissa (23) 24 bits stored in 23 bits!

12 y = y = y = bbbbbbbb

13 y = x bbbbbbbb Why not ?

14 How is the exponent coded?

15 bbbbbbbb bias of 127

16 bbbbbbbb

17 y = x * 2^3 Ah! 3 represented by 130 = =

18 Verification in IEEE FP

19 Exercises How is 1.0 coded as a 32-bit floating point number? What about 0.5? 1.5? -1.5? what floating-point value is stored in the 32-bit number below?

20 1.0 = = = = = = - (stored exp = 131 > real exp = 4) = x 2^4 = x 16 = -31

21 what about 0.1?

22 0.1 decimal, in 32-bit precision, IEEE Format:

23 0.1 decimal, in 32-bit precision, IEEE Format: Value in double-precision:

24 NEVER NEVER COMPARE FLOATS OR DOUBLEs FOR EQUALITY! N-E-V-E-R!

25 for ( double d = 0; d!= 0.3; d += 0.1 ) System.out.println( d );

26 bbbbbbbb

27 Zero Why is it special? 0.0 =

28 bbbbbbbb {if mantissa is 0: number = 0.0

29 Very Small Numbers Smallest numbers have stored exponent of 0. In this case, the implied 1 is omitted, and the exponent is -126 (not -127!)

30 bbbbbbbb {if mantissa is 0: number = 0.0 if mantissa is!0: no hidden 1

31 Very Small Numbers Example: (2-126 ) x ( ) binary + (2-126 ) x ( ) =

32 bbbbbbbb?

33 Very large exponents stored exponent = if the mantissa is = 0 ==> +/-

34 Very large exponents stored exponent = if the mantissa is = 0 ==> +/-

35 Very large exponents stored exponent = if the mantissa is = 0 ==> +/- if the mantissa is!= 0 ==> NaN

36 Very large exponents stored exponent = if the mantissa is = 0 ==> +/- if the mantissa is!= 0 ==> NaN = Not-a-Number

Very large exponents stored exponent = 1111 1111 if

37 Very large exponents stored exponent = if the mantissa is = 0 ==> +/- if the mantissa is!= 0 ==> NaN

38 = = = NaN

39 Operations that create NaNs ( The divisions 0/0 and ± /± The multiplications 0 ± and ± 0 The additions + ( ), ( ) + and equivalent subtractions The square root of a negative number. The logarithm of a negative number The inverse sine or cosine of a number that is less than 1 or greater than +1

40 // import java.util.*; import static java.lang.double.nan; import static java.lang.double.positive_infinity; import static java.lang.double.negative_infinity; Generating NaNs public class GenerateNaN { public static void main(string args[]) { double[] allnans = { 0D / 0D, POSITIVE_INFINITY / POSITIVE_INFINITY, POSITIVE_INFINITY / NEGATIVE_INFINITY, NEGATIVE_INFINITY / POSITIVE_INFINITY, NEGATIVE_INFINITY / NEGATIVE_INFINITY, 0 * POSITIVE_INFINITY, 0 * NEGATIVE_INFINITY, Math.pow(1, POSITIVE_INFINITY), POSITIVE_INFINITY + NEGATIVE_INFINITY, NEGATIVE_INFINITY + POSITIVE_INFINITY, POSITIVE_INFINITY - POSITIVE_INFINITY, NEGATIVE_INFINITY - NEGATIVE_INFINITY, Math.sqrt(-1), Math.log(-1), Math.asin(-2), Math.acos(+2), }; System.out.println(Arrays.toString(allNaNs)); System.out.println(NaN == NaN); System.out.println(Double.isNaN(NaN)); } } [NaN, NaN, NaN, NaN, NaN, NaN, NaN, NaN, NaN, NaN, NaN, NaN, NaN, NaN, NaN, NaN] false true

41 NaN is sticky!

42 Range of Floating-Point Numbers

43 Range of Floating-Point Numbers Remember that!

44 Resolution of a Floating-Point Format Check out table here:

46 What does it have to do with Art?

47 Rosetta Landing on Comet 10-year trajectory

48 We stopped here last time

49 Why not using 2 s Complement for the Exponent? = = = =

http://www.h-schmidt.net/floatconverter/ieee754.html Exercises Does this converter support NaN, and? Are there several different representations of +?

50 Exercises Does this converter support NaN, and? Are there several different representations of +? What is the largest float representable with the 32-bit format? What is the smallest normalized float (i.e. a float which has an implied leading 1. bit)?

51 How do we add 2 FP numbers?

52 1.111 x x x x 2 8 after expansion x x x 2 8 locate largest number shift mantissa of smaller compute sum x 2 8 = x 2 8 round & truncate normalize = x 2 9

53 fp1 = s1 m1 e1 fp2 = s2 m2 e2 fp1 + fp2 =? denormalize both numbers (restore hidden 1) assume fp1 has largest exponent e1: make e2 equal to e1 and shift decimal point in m2 > m2 compute sum m1 + m2 truncate & round result renormalize result (after checking for special cases)

54 What are the "dangers" of adding 2 FP numbers?

Algorithm for adding a Series of FP numbers? https://www.

55 Algorithm for adding a Series of FP numbers?

56 How do we Multiply 2 FP Numbers?

57 1.111 x 2 5 x x x 2 5 x x x = x 2 1 after expansion multiply mantissas normalize round & truncate = x 2 1 normalize = 14 add exponents =

58 fp1 = s1 m1 e1 fp2 = s2 m2 e2 fp1 x fp2 =? Test for multiplication by special numbers (0, NaN, ) denormalize both numbers (restore hidden 1) compute product of m1 x m2 compute sum e1 + e2 truncate & round m1 x m2 adjust e1+e2 and normalize.

59 How do we compare two FP numbers?

60 Check sign bits first, the compare as unsigned integers! No unpacking necessary!

61 Programming FP Operations in Assembly

62 Pentium EAX EBX ECX EDX ALU

63 EAX EBX ECX EDX ALU Cannot do FP computation

66 SP0 SP1 SP2 SP3 SP4 SP5 FLOATING POINT UNIT SP6 SP7

67 Operation: ( )/9.0 SP0 SP1 SP2 SP3 SP4 SP5 FLOATING POINT UNIT SP6 SP7

68 SP0 7 Operation: (7+10)/9 fpush 7 SP1 SP2 SP3 SP4 SP5 SP6 SP7 FLOATING POINT UNIT

69 SP0 SP Operation: (7+10)/9 fpush 7 fpush 10 SP2 SP3 SP4 SP5 SP6 SP7 FLOATING POINT UNIT

70 Operation: (7+10)/9 SP0 SP1 10 fpush 7 fpush 10 fadd SP2 7 SP3 SP4 SP5 FLOATING POINT UNIT SP6 SP7

71 SP0 SP1 17 Operation: (7+10)/9 fpush 7 fpush 10 fadd SP2 SP3 SP4 SP5 SP6 SP7 FLOATING POINT UNIT

72 SP0 SP1 SP Operation: (7+10)/9 fpush 7 fpush 10 fadd fpush 9 SP3 SP4 SP5 SP6 SP7 FLOATING POINT UNIT

73 Operation: (7+10)/9 SP0 fpush 7 SP1 SP fpush 10 fadd fpush 9 fdiv SP3 SP4 SP5 FLOATING POINT UNIT SP6 SP7

74 SP0 SP1 SP2 SP Operation: (7+10)/9 fpush 7 fpush 10 fadd fpush 9 fdiv SP4 SP5 SP6 SP7 FLOATING POINT UNIT

75 The Pentium computes FP expressions using RPN!

76 The Pentium computes FP expressions using RPN! Reverse Polish Notation

77 Nasm Example: z = x+y SECTION.data x dd 1.5 y dd 2.5 z dd 0 ; compute z = x + y SECTION.text fld fld fadd fstp dword [x] dword [y] dword [z]

78 Printing floats in C #include "stdio.h" int main() { float z = e10; printf( "z = %e\n\n", z ); return 0; }

79 Printing floats in C #include "stdio.h" int main() { float z = e10; printf( "z = %e\n\n", z ); return 0; } gcc -o printfloat printfloat.c./printfloat z = e+10

80 More code examples here: CSC231_An_Introduction_to_Fixed-_and_Floating- Point_Numbers#Assembly_Language_Programs

81 THE END!

mith College Computer Science Fixed-Point & Floating-Point Number Formats CSC231 Dominique Thiébaut

mith College Computer Science Fixed-Point & Floating-Point Number Formats CSC231 Dominique Thiébaut dthiebaut@smith.edu Reference http://cs.smith.edu/dftwiki/index.php/ CSC231_An_Introduction_to_Fixed-_and_Floating-