15-213 Floating Point Arithmetic August 31, 2009 Topics class03.ppt IEEE Floating Point Standard Rounding Floating Point Operations Mathematical properties Floating Point Puzzles For each of the following C expressions, either: Argue that it is true for all argument values Explain why not true x == (int)(float) x int x = ; x == (int)(double) x float f = ; f == (float)(double) f double d = ; d == (float) d f == -(-f); Assume neither d nor f is NaN 2/3 == 2/3.0 d < 0.0 ((d*2) < 0.0) d > f -f > -d

d * d >= 0.0 (d+f)-d == f 2 15-213: Intro to Computer Systems Fall 2009 IEEE Floating Point IEEE Standard 754 Established in 1985 as uniform standard for floating point arithmetic Before that, many idiosyncratic formats Supported by all major CPUs Driven by Numerical Concerns Nice standards for rounding, overflow, underflow Hard to make go fast Numerical analysts predominated over hardware types in defining standard 3 15-213: Intro to Computer Systems Fall 2009 Fractional Binary Numbers 2i 2i1 bi bi1 b2 b1 4 2

1 b0 . b1 b2 b3 1/2 1/4 1/8 bj 2j Representation Bits to right of binary point represent fractional powers of 2 Represents rational number: i bk 2 k k j 4 15-213: Intro to Computer Systems Fall 2009 Frac. Binary Number Examples Value 5-3/4 2-7/8 63/64 Representation 101.112 10.1112 0.1111112 Observations

Divide by 2 by shifting right Multiply by 2 by shifting left Numbers of form 0.1111112 just below 1.0 1/2 + 1/4 + 1/8 + + 1/2i + 1.0 Use notation 1.0 5 15-213: Intro to Computer Systems Fall 2009 Representable Numbers Limitation Can only exactly represent numbers of the form x/2k Other numbers have repeating bit representations Value 1/3 1/5 1/10 6 Representation 0.0101010101[01]2 0.001100110011[0011]2 0.0001100110011[0011]2 15-213: Intro to Computer Systems Fall 2009 Floating Point Representation Numerical Form 1s M 2E Sign bit s determines whether number is negative or positive Significand M normally a fractional value in range [1.0,2.0). Exponent E weights value by power of two Encoding s

7 exp frac MSB is sign bit exp field encodes E frac field encodes M 15-213: Intro to Computer Systems Fall 2009 Floating Point Precisions Encoding s exp frac MSB is sign bit exp field encodes E frac field encodes M Sizes Single precision: 8 exp bits, 23 frac bits 32 bits total Double precision: 11 exp bits, 52 frac bits 64 bits total Extended precision: 15 exp bits, 63 frac bits Only found in Intel-compatible machines Stored in 80 bits 1 bit wasted 8 15-213: Intro to Computer Systems

Fall 2009 Normalized Numeric Values Condition exp 0000 and exp 1111 Exponent coded as biased value E = Exp Bias Exp : unsigned value denoted by exp Bias : Bias value Single precision: 127 (Exp: 1254, E: -126127) Double precision: 1023 (Exp: 12046, E: -10221023) in general: Bias = 2e-1 - 1, where e is number of exponent bits Significand coded with implied leading 1 M = 1.xxxx2 xxxx: bits of frac Minimum when 0000 (M = 1.0) Maximum when 1111 (M = 2.0 ) Get extra leading bit for free 9 15-213: Intro to Computer Systems Fall 2009 Normalized Encoding Example Value Float F = 15213.0; 15213 13 10 = 11101101101101 2 = 1.1101101101101 2 X 2 Significand M = frac = 1.11011011011012 110110110110100000000002 Exponent E = Bias = Exp =

13 127 140 = 100011002 Floating Point Representation: Hex: Binary: 0000 140: 10 15213: 4 6 6 D B 4 0 0 0100 0110 0110 1101 1011 0100 0000 100 0110 0 1110 1101 1011 01 15-213: Intro to Computer Systems Fall 2009 Denormalized Values Condition exp = 0000 Value Exponent value E = Bias + 1 Significand value M = 0.xxxx2 xxxx: bits of frac Cases

exp = 0000, frac = 0000 Represents value 0 Note that have distinct values +0 and 0 exp = 0000, frac 0000 Numbers very close to 0.0 Lose precision as get smaller Gradual underflow 11 15-213: Intro to Computer Systems Fall 2009 Special Values Condition exp = 1111 Cases exp = 1111, frac = 0000 Represents value(infinity) Operation that overflows Both positive and negative E.g., 1.0/0.0 = 1.0/0.0 = +, 1.0/0.0 = exp = 1111, frac 0000 Not-a-Number (NaN) Represents case when no numeric value can be determined E.g., sqrt(1), 12 15-213: Intro to Computer Systems Fall 2009 Summary of Floating Point Real Number Encodings NaN

13 -Normalized +Denorm -Denorm 0 +0 15-213: Intro to Computer Systems Fall 2009 +Normalized + NaN Tiny Floating Point Example 8-bit Floating Point Representation the sign bit is in the most significant bit. the next four bits are the exponent, with a bias of 7. the last three bits are the frac Same General Form as IEEE Format normalized, denormalized representation of 0, NaN, infinity 7 6 s 14

0 3 2 exp frac 15-213: Intro to Computer Systems Fall 2009 Values Related to the Exponent 15 Exp exp E 2E 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 0000 0001 0010 0011 0100 0101 0110 0111 1000

1001 1010 1011 1100 1101 1110 1111 -6 -6 -5 -4 -3 -2 -1 0 +1 +2 +3 +4 +5 +6 +7 n/a 1/64 1/64 1/32 1/16 1/8 1/4 1/2 1 2 4 8 16 32 64 128 (denorms) (inf, NaN) 15-213: Intro to Computer Systems Fall 2009 Dynamic Range

E Value 0000 000 0000 001 0000 010 -6 -6 -6 0 1/8*1/64 = 1/512 2/8*1/64 = 2/512 closest to zero 0000 0000 0001 0001 110 111 000 001 -6 -6 -6 -6 6/8*1/64 7/8*1/64 8/8*1/64 9/8*1/64 = = = = 6/512 7/512 8/512 9/512 largest denorm

smallest norm 0110 0110 0111 0111 0111 110 111 000 001 010 -1 -1 0 0 0 14/8*1/2 15/8*1/2 8/8*1 9/8*1 10/8*1 = = = = = 14/16 15/16 1 9/8 10/8 7 7 n/a 14/8*128 = 224 15/8*128 = 240 inf s exp 0 0

Denormalized 0 numbers 0 0 0 0 0 0 Normalized 0 numbers 0 0 0 0 0 16 frac 1110 110 1110 111 1111 000 15-213: Intro to Computer Systems Fall 2009 closest to 1 below closest to 1 above largest norm Distribution of Values 6-bit IEEE-like format e = 3 exponent bits f = 2 fraction bits Bias is 3 Notice how the distribution gets denser toward zero.

-15 -10 -5 Denormalized 17 0 5 Normalized 15-213: Intro to Computer Systems Fall 2009 10 Infinity 15 Distribution of Values (close-up view) 6-bit IEEE-like format e = 3 exponent bits f = 2 fraction bits Bias is 3 -1 -0.5 0 Denormalized 18 Normalized

15-213: Intro to Computer Systems Fall 2009 0.5 Infinity 1 Interesting Numbers Description exp Zero 0000 0000 0.0 Smallest Pos. Denorm. 0000 0001 2 {23,52} X 2 {126,1022} 0000 1111 (1.0 ) X 2 {126,1022} Single 1.18 X 1038 Double 2.2 X 10308 Smallest Pos. Normalized 0001 0000 Numeric Value Single 1.4 X 1045 Double 4.9 X 10324 Largest Denormalized

frac 1.0 X 2 {126,1022} Just larger than largest denormalized One 0111 0000 1.0 Largest Normalized 1110 1111 (2.0 ) X 2{127,1023} 19 Single 3.4 X 1038 Double 1.8 X 10308 15-213: Intro to Computer Systems Fall 2009 Special Properties of Encoding FP Zero Same as Integer Zero All bits = 0 Can (Almost) Use Unsigned Integer Comparison Must first compare sign bits Must consider -0 = 0 NaNs problematic Will be greater than any other values What should comparison yield?

Otherwise OK Denorm vs. normalized Normalized vs. infinity 20 15-213: Intro to Computer Systems Fall 2009 Floating Point Operations Conceptual View First compute exact result Make it fit into desired precision Possibly overflow if exponent too large Possibly round to fit into frac Rounding Modes (illustrate with $ rounding) $1.40 $1.60 $1.50 $2.50 $1.50 Zero $1 $1 $1 $2 $1

Round down (-) $1 $1 $1 $2 $2 Round up (+) $2 $2 $2 $3 $1 Nearest Even (default) $1 $2 $2 $2 $2 Note: 1. Round down: rounded result is close to but no greater than true result. 2. Round up: rounded result is close to but no less than true result. 21 15-213: Intro to Computer Systems Fall 2009 Closer Look at Round-To-Even

Default Rounding Mode Hard to get any other kind without dropping into assembly All others are statistically biased Sum of set of positive numbers will consistently be over- or under- estimated Applying to Other Decimal Places / Bit Positions When exactly halfway between two possible values Round so that least significant digit is even E.g., round to nearest hundredth 1.2349999 1.2350001 1.2350000 1.2450000 22 1.23 1.24 1.24 1.24 (Less than half way) (Greater than half way) (Half wayround up) (Half wayround down) 15-213: Intro to Computer Systems Fall 2009 Rounding Binary Numbers Binary Fractional Numbers Even when least significant bit is 0

Half way when bits to right of rounding position = 1002 Examples Round to nearest 1/4 (2 bits right of binary point) Value Binary 2 3/32 10.000112 10.002 (<1/2down) 2 2 3/16 10.001102 10.012 (>1/2up) 2 1/4 2 7/8 10.111002 11.002 (1/2up) 3 2 5/8 10.101002 10.102 (1/2down) 2 1/2 23 Rounded Action 15-213: Intro to Computer Systems Fall 2009

Rounded Value FP Multiplication Operands (1)s1 M1 2E1 (1)s2 M2 2E2 * Exact Result (1)s M 2E Sign s: s1 ^ s2 Significand M: M1 * M2 Exponent E: E1 + E2 Fixing If M 2, shift M right, increment E If E out of range, overflow Round M to fit frac precision Implementation 24 Biggest chore is multiplying significands 15-213: Intro to Computer Systems Fall 2009

FP Addition Operands (1)s1 M1 2E1 E1E2 (1)s2 M2 2E2 (1)s1 M1 Assume E1 > E2 Exact Result (1)s M 2E (1)s2 M2 + Sign s, significand M: (1)s M Result of signed align & add Exponent E: E1 Fixing 25 If M 2, shift M right, increment E if M < 1, shift M left k positions, decrement E by k Overflow if E out of range

Round M to fit frac precision 15-213: Intro to Computer Systems Fall 2009 Mathematical Properties of FP Add Compare to those of Abelian Group Closed under addition? YES But may generate infinity or NaN Commutative? YES Associative? NO Overflow and inexactness of rounding 0 is additive identity? YES Every element has additive inverse ALMOST Except for infinities & NaNs Monotonicity a b a+c b+c? ALMOST Except for infinities & NaNs 26

15-213: Intro to Computer Systems Fall 2009 Math. Properties of FP Mult Compare to Commutative Ring Closed under multiplication? YES But may generate infinity or NaN Multiplication Commutative? YES Multiplication is Associative? NO Possibility of overflow, inexactness of rounding 1 is multiplicative identity? YES Multiplication distributes over addition? NO Possibility of overflow, inexactness of rounding Monotonicity a b & c 0 a *c b *c? Except for infinities & NaNs 27 15-213: Intro to Computer Systems Fall 2009 ALMOST

Creating Floating Point Number Steps 7 6 Normalize to have leading 1 Round to fit within fraction Postnormalize to deal with effects of rounding s exp Case Study 28 Convert 8-bit unsigned numbers to tiny floating point format Example Numbers 128 10000000 15 00001101 33 00010001 35 00010011 138

10001010 63 00111111 15-213: Intro to Computer Systems Fall 2009 0 3 2 frac Normalize 7 6 s 0 3 2 exp frac Requirement Set binary point so that numbers of form 1.xxxxx Adjust all to have leading one Decrement exponent as shift left 29 Value Binary Fraction Exponent

128 10000000 1.0000000 7 15 00001101 1.1010000 3 17 00010001 1.0001000 5 19 00010011 1.0011000 5 138 10001010 1.0001010 7 63 00111111 1.1111100 5

15-213: Intro to Computer Systems Fall 2009 Rounding 1.BBGRXXX Guard bit: LSB of result Sticky bit: OR of remaining bits Round bit: 1 bit removed st Round up conditions 30 Round = 1, Sticky = 1 > 0.5 Guard = 1, Round = 1, Sticky = 0 Round to even Value Fraction GRS Incr? Rounded 128 1.0000000 000 N 1.000 15 1.1010000 100 N

1.101 17 1.0001000 010 N 1.000 19 1.0011000 110 Y 1.010 138 1.0001010 111 Y 1.001 63 1.1111100 111 Y 10.000 15-213: Intro to Computer Systems Fall 2009 Postnormalize Issue 31

Rounding may have caused overflow Handle by shifting right once & incrementing exponent Value Rounded Exp Adjusted 128 1.000 7 128 15 1.101 3 15 17 1.000 4 16 19 1.010 4 20 138

1.001 7 134 63 10.000 5 1.000/6 15-213: Intro to Computer Systems Fall 2009 Result 64 Floating Point in C C Guarantees Two Levels float single precision double double precision Conversions Casting between int, float, and double changes numeric values Double or float to int Truncates fractional part Like rounding toward zero Not defined when out of range or NaN Generally sets to TMin int to double Exact conversion, as long as int has 53 bit word size

int to float Will round according to rounding mode 32 15-213: Intro to Computer Systems Fall 2009 Curious Excel Behavior Default Format Currency Format 33 Number Subtract 16 Subtract .3 Subtract .01 16.31 0.31 0.01 -1.2681E-15 $16.31 $0.31 $0.01 ($0.00) Spreadsheets use floating point for all computations Some imprecision for decimal arithmetic Can yield nonintuitive results to an accountant! 15-213: Intro to Computer Systems Fall 2009 Summary IEEE Floating Point Has Clear Mathematical Properties Represents numbers of form M X 2E

Can reason about operations independent of implementation As if computed with perfect precision and then rounded Not the same as real arithmetic Violates associativity/distributivity Makes life difficult for compilers & serious numerical applications programmers 34 15-213: Intro to Computer Systems Fall 2009