Chapter 2 Float Point Arithmetic. Real Numbers in Decimal Notation. Real Numbers in Decimal Notation

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1 Chapter 2 Float Point Arithmetic Topics IEEE Floating Point Standard Fractional Binary Numbers Rounding Floating Point Operations Mathematical properties Real Numbers in Decimal Notation Representation d i d i 1 d 2 d 1 d 0. 1 d 1 1/10 d 2 d 3 d j 1/100 1/ j 10 i 10 i x x x x Real Numbers in Decimal Notation Representation x x x x 10-2 Digit to the left of the symbol. are weighted by positive powers of ten, while digits to the right are weighted by negative powers of ten, giving fractional values. 4

2 Fractional Binary Numbers b i b i 1 b 2 b 1 b 0. b 1 b 2 b 3 b j 1/2 1/4 1/8 Representation Bits to right of binary point represent fractional numbers, weighted by negative power of 2. 2 j 2 i 2 i Fractional Binary Number Examples Value Representation 5.3/ / / Observation Divide by 2 by shifting right Numbers of form just below 1.0 Use notation 1.0 ε 6 Limitation Limitation Can only exactly represent numbers of the form x/2 k Other numbers have repeating bit representations Value Representation 1/ [01] 2 1/ [0011] 2 1/ [0011] 2 7

3 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 8 Floating Point Representation Numerical Form 1 s M 2 E 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 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 9 Normalized Encoding Example Value Float F = ; = = X 2 13 Significand M = frac = Exponent E = 13 Bias = 127 Exp = 140 = Floating Point Representation (Class 02): Hex: D B Binary: : :

4 Normalized Numeric Values Condition exp and exp Exponent coded as biased value E = Exp Bias Exp : unsigned value denoted by exp Bias : Bias value» Single precision: 127 (Exp: 1 254, E: )» Double precision: 1023 (Exp: , E: » in general: Bias = 2 m-1-1, where m is the number of exponent bits Significand coded with implied leading 1 m = 1.xxx x 2 xxx x: bits of frac Minimum when (M = 1.0) Maximum when (M = 2.0 ε) Get extra leading bit for free 11 Denormalized Values Condition exp = Value Exponent value E = Bias + 1 Significand value m = 0.xxx x 2 xxx x: bits of frac Cases exp = 000 0, frac = Represents value 0 Note that have distinct values +0 and 0 exp = 000 0, frac Numbers very close to 0.0 Lose precision as get smaller Gradual underflow 12 Floating Point Representation 1 s x M x 2 E s exp frac 14

5 Normalized Numbers 1 s x M x 2 E s exp frac x 1.1 x 2 1 = Denormalized Numbers 1 s x M x 2 E s exp frac x 0.1 x = x Condition exp = Special Values Cases exp = 111 1, frac = 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 = 111 1, frac Not-a-Number (NaN) Represents case when no numeric value can be determined E.g., sqrt( 1),, 0/0, / 17

6 Summary of Floating Point Real Number Encodings -Normalized -Denorm +Denorm +Normalized + NaN 0 +0 NaN 18 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 s exp frac 0 20 Values related to the exponent Exp exp E 2 E /64 (denorms) / / / / / / n/a (inf, Nan) 21

7 Dynamic Range (8 bits encoding) S exp frac E Value Denormalized numbers Normalized numbers /8*1/64 = 1/ /8*1/64 = 2/ /8*1/64 = 6/ /8*1/64 = 7/ /8*1/64 = 8/ /8*1/64 = 9/ /8*1/2 = 14/ /8*1/2 = 15/ /8*1 = /8*1 = 9/ /8*1 = 10/ /8*128 = /8*128 = n/a inf closest to zero largest denorm smallest norm closest to 1 below closest to 1 above largest norm 22 Interesting Numbers Description exp frac Numeric Value Zero Smallest Pos. Denorm {23,52} X 2 {126,1022} Single 1.4 X 10 45, Double 4.9 X Largest Denormalized (1.0 ε) X 2 {126,1022} Single 1.18 X 10 38, Double 2.2 X Smallest Pos. Normalized X 2 {126,1022} Just larger than largest denormalized One Largest Normalized (2.0 ε) X 2 {127,1023} Single 3.4 X 10 38, Double 1.8 X 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 24

8 Floating Point Operations Conceptual View First compute exact result Make it fit into desired precision Possibly overflow if exponent too large or underflow if too small Possibly round to fit into frac Rounding Modes (illustrate with $ rounding) $1.40 $1.60 $1.50 $2.50 $1.50 Zero $1.00 $1.00 $1.00 $2.00 $1.00 Round down (- ) $1.00 $1.00 $1.00 $2.00 $2.00 Round up (+ ) $2.00 $2.00 $2.00 $3.00 $1.00 Nearest Even (default) $1.00 $2.00 $2.00 $2.00 $2.00 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. 25 A 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 When exactly halfway between two possible values Round so that least significant digit is even E.g., round to nearest hundredth (Less than half way) (Greater than half way) (Half way round up) (Half way round down) 26 Rounding Binary Numbers Binary Fractional Numbers Even when least significant bit is 0 Half way when bits to right of rounding position = Examples Round to nearest 1/4 (2 bits right of binary point) Value Binary Rounded Action Rounded Value 2 3/ (<1/2 down) 2 2 3/ (>1/2 up) 2 1/4 2 7/ (1/2 up) 3 2 5/ (1/2 down) 2 1/2 27

9 FP Multiplication Operands ( 1) s1 M1 2 E1 ( 1) s2 M2 2 E2 Exact Result ( 1) s M 2 E 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 Biggest chore is multiplying significands 28 FP Multiplication Example 1 8bits encoding, with 4 bits for exponent and 3 bits for frac (14/16) X (16.0) E1 = -1 M1 = E2 = 4 M2 = Result: (14.0) S = S1 ^ S2 = 0 E = E1 + E2 = 3 M = M1 x M2 = x 1.0 = FP Multiplication Example 2 8bits encoding, with 4 bits for exponent and 3 bits for frac (3/4) X (24.0) E1 = -1 M1 = E2 = 4 M2 = S = S1 ^ S2 = 0 E = E1 + E2 = 3 M = M1 x M2 = x = (> 2) Fix: M shifts to right and increment E E = 4, M = Result: (9/8 * 2^4) = 18) 30

10 FP Addition Operands ( 1) s1 M1 2 E1 ( 1) s2 M2 2 E2 ( 1) s1 m1 Assume E1 > E2 + Exact Result ( 1) s M 2 E Sign s, significand M: Result of signed align & add Exponent E: E1 Fixing 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 ( 1) s m E1 E2 ( 1) s2 m2 31 FP Addition Example 1 Steps 1 Check for Zeros 2 Align the M (Significands) 3 Add/Subtract Significands 4 Normalize the results 32 FP Addition Example 1 8bits encoding, with 4 bits for exponent and 3 bits for frac (24.0) (6) E1 = 4 M1 = E2 = 2 M2 = E = E1 = 4 M2 must shift to the right by 2 and become M = M1+ M2 = = Result: (15/8* 2^4 = 30) 33

11 Algebraic Properties of FP Mult Compare to Commutative Ring Closed under multiplication? But may generate infinity or NaN Multiplication Commutative? Multiplication is Associative? Possibility of overflow, inexactness of rounding 1 is multiplicative identity? Multiplication distributes over addition? Possibility of overflow, inexactness of rounding Montonicity a b & c 0 a *c b *c? Except for infinities & NaNs YES YES NO YES NO ALMOST 34 Floating Point in C C Guarantees Two Levels: float and double 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»generally saturates to TMin or TMax int to double Exact conversion, as long as int has 53 bit word size int to float Will round according to rounding mode 35 Double/Float/Int Conversion Assuming int I; float f; double d; I = ; f = (float) I; Hex: d2 Hex: 4e932c06 I= d f=4e932c (rounding) I = (int) f; I =

12 Floating Point Puzzles For each of the following C expressions, either: Argue that is true for all argument values Explain why not true int x = ; float f = ; double d = ; Assume neither d nor f is NAN x == (int)(float) x x == (int)(double) x f == (float)(double) f d == (float) d f == -(-f); 2/3 == 2/3.0 d < 0.0 ((d*2) < 0.0) d > f -f < -d d * d >= 0.0 (d+f)-d == f 37

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