# Computer arithmetics: integers, binary floating-point, and decimal floating-point

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1 n!= 0 && -n == n z+1 == z Computer arithmetics: integers, binary floating-point, and decimal floating-point v+w-w!= v x+1 < x Peter Sestoft y!= y p == n && 1/p!= 1/n 1 Computer arithmetics Computer numbers are cleverly designed, but Very different from high-school mathematics There are some surprises Choose representation with care: When to use int, short, long, byte, When to use double or float When to use decimal floating-point 2 1

2 Integers Taxonomy Unsigned, binary representation Signed Signed-magnitude Two s complement (Java and C# int, short, byte, ) Arithmetic modulo 2 n Floating-point numbers IEEE 754 binary32 and binary64 Which you know as float and double in Java and C# IEEE 754 decimal128 and also C# s decimal type and also Java s java.math.bigdecimal 3 Positive integers, binary representation Decimal notation = 8* * *10 0 = 805 Every place is worth 10 times that to the right Binary notation = 1* * * *2 0 = 13 Every place is worth 2 times the right one Positional number systems: Base is 10 or 2 or 16 or Any non-positional number systems?

3 Binary numbers A bit is a binary digit: 0 or 1 Easy to represent in electronics (But some base-10 hardware in the 1960es) Counting with three bits: 000, 001, 010, 011, 100, 101, 110, 111 Computing: = = 101 There are 10 kinds of people: those who understand binary and those who don t 5 Hexadecimal numbers Hexidecimal numbers have base 16 Digits: A B C D E F = 3 * * *16 0 = 805 Each place is worth 16 times that... Useful alternative to binary Because 16 = 2 4 So one hex digit = 4 bits Computing in hex: A + B = 15 AA + 1 = AB AA + 10 = BA A 1010 B 1011 C 1100 D 1101 E 1110 F

4 Negative integers Signed magnitude: A sign bit and a number Problem: Then we have both +0 and -0 Two s complement: Negate all bits and add 1 Only one zero Easy to compute with Requires known size of number, e.g. 8, 16, 32, 64 bits Examples of two s complement (4 bits) -3 is represented by 1101 because 3 = so complement is 1100; add 1 to get -3 = is represented by 1111 because 1 = so complement is 1110; add 1 to get -1 = is represented by 1000 because 8 = so complement is 0111; add 1 to get -8 = Clicker: Two's complement What decimal number does the 8-bit two's complement number represent? A) 137 = B) -118 = -( ) C) 119 = D) -119 = -( ) 8 4

5 Integer arithmetics modulo 2 n Java and C# int is 32-bit two s-complement Max int is = Min int is (2 31 ) = If x = then x+1 = < x If n = then n = n = = = = = = = = = An obviously non-terminating loop? int i = 1; while (i > 0) i++; System.out.println(i); Does terminate! Values of i:

6 Binary fractions Before the point:, 16, 8, 4, 2, 1 After the point: 1/2, 1/4, 1/8, 1/16, 0.5 = = = = = = = But how many digits are needed before the point? how many digits are needed after the point? Answer: Binary floating-point (double, float) The point is placed dynamically 11 Clicker: Binary fractions If the binary fraction represents , what does represent? A) = * 2 B) = / 2 C) = * 10 D) = /

7 Clicker: Binary fractions What is represented as a binary fraction? A) B) C) D) Some nasty fractions Some numbers not representable as finite decimal fractions: 1/7 = Same problem with binary fractions: 1/10 = Quite unfortunate: Cannot represent 0.10 krone or \$0.10 exactly Nor 0.01 krone or \$0.01 exactly Do not use binary floating-point (float, double) for accounting! 14 7

8 An obviously terminating loop? double d = 0.0; while (d!= 1.0) d += 0.1; d never equals 1.0 Does not terminate! Values of d: History of floating-point numbers Until 1985: Many different designs, anarchy Difficult to write portable numerical software Standard IEEE binary fp Implemented by all modern hardware Assumed by modern programming languages Designed primarily by William Kahan for Intel Revised standard IEEE binary floating-point as in IEEE decimal floating-point (new) IEEE = Eye-triple-E = Institute of Electrical and Electronics Engineers (USA) 16 8

9 IEEE floating point representation Signed-magnitude Sign, exponent, significand: s * 2 e * c Representation: Sign s, exponent e, fraction f (= significand minus 1) s eeeeeeee fffffffffffffffffffffff = 1.0 float Java, C# float, binary32 double, binary64 bits e bits f bits range e bias sign. digits ±10-44 to ± ± to ± Intel ext ± to ± Understanding the representation Normalized numbers Choose exponent e so the significand is 1.ffffff Hence we need only store the.ffffff not 1 (!) Exponent is unsigned but a bias is added For 32-bit float the bias is 127 s eeeeeeee fffffffffffffffffffffff = = = = = = =

10 A detailed example Consider x = We know that = Normalize to 2 6 * So exponent e = 6, represented by = 133 significand is so fraction f = sign is 1 for negative s eeeeeeee fffffffffffffffffffffff = Clickers: floating points If is represented as the binary fraction , what is the 32-bit floating point representation of ? s!eeeeeeeee! fffffffffffffffffffffff! A!0! = 4! ! B!1! = 131! ! C!0! = 5! ! D!1! = 130! ! E!1! = 2! ! 20 10

11 Special numbers Denormal numbers, resulting from underflow Infinite numbers, resulting from 1/0, Math.log(0), NaNs (not-a-number), resulting from 0/0, Math.sqrt(-1), Exponent e Represented number Normal: ±10-38 to ± Denormal (underflow): ±10-44 to ±10-38 and Infinities, when f= NaNs, when f=1b b s eeeeeeee fffffffffffffffffffffff = = 7.346E = Infinity = -Infinity s = NaN 21 Why denormal numbers? To allow gradual underflow, small numbers To ensure that x y==0 if and only if x==y Example: Smallest non-zero normal number is So choose x= *2-126 and y= *2-126 : s eeeeeeee fffffffffffffffffffffff = x = y = x-y What would happen without denormal? Since x-y is it is less than So result of x-y would be represented as 0.0 But clearly x!=y, so this would be confusing 22 11

12 Why infinities? 1: A simple solution to overflow Math.exp(100000) gives +Infinity 2: To make sensible expressions work Example: Compute f(x) = x/(x 2 +1) But if x is large then x 2 may overflow Better compute: f(x) = 1/(x+1/x) But if x=0 then 1/x looks bad, yet we want f(x)=0 Solution: Let 1/0 be Infinity Let 0+Infinity be Infinity Let 1/Infinity be 0 Then 1/(0+1/0) gives 0 as desired for x=0 23 Why NaNs? A simple and efficient way to report error Languages like C do not have exceptions Exceptions are 10,000 times slower than (1.2+x) Even weird expressions must have a result 0/0 gives NaN Infinity Infinity gives NaN Math.sqrt(-1) gives NaN Math.log(-1) gives NaN Operations must preserve NaNs NaN + 17 gives NaN Math.sqrt(NaN) gives NaN and so on 24 12

13 What about double (binary64)? The same, just with 64= bits instead of 32 s eeeeeeeeeee ffffffffffffffffffffffffffffffffffffffffffffffffffff = = = = = = Infinity = -Infinity s = NaN = 1.39E = -Infinity = = , clearly not equal to IEEE addition 26 13

14 IEEE subtraction 27 IEEE multiplication 28 14

15 IEEE division 29 IEEE equality and ordering Equality (==,!=) A NaN is not equal to anything, not even itself So if y is NaN, then y!= y Ordering: < 2.0 < 0.0 == 0.0 < 2.0 < + All ordering comparisons involving NaNs give false 30 15

16 Java and C# mathematical functions In general, functions behave sensibly Give +Infinity or Infinity on extreme arguments Give NaN on invalid arguments Preserve NaN arguments sqrt(-2.0) = NaN log(0.0) = -Inf log(-1.0) = NaN sin(inf) = NaN asin(2.0) = NaN exp( ) = Inf exp(-inf) = 0.0 pow(0, -1) = Inf sqrt(nan) = NaN log(nan) = NaN sin(nan) = NaN exp(nan) = NaN pow(nan, 0) = 1 in Java 31 Rounding modes High-school: round 0.5 upwards Rounds 0,1,2,3,4 down and rounds 5,6,7,8,9 up Looks fair But it s dangerous: may introduce drift IEEE rounds 0.5 to nearest even number So both 1.5 and 2.5 round to

17 Basic principle of IEEE floating-point Each of the computational operations shall be performed as if it first produced an intermediate result correct to infinite precision and unbounded range, and then rounded that intermediate result to fit in the destination s format (IEEE ) So the machine result of x*y is the rounding of the real result of x*y This is simple and easy to reason about and quite surprising that it can be implemented in finite hardware 33 Clickers: IEEE surprises For which values of x is x = 1.0 with 32-bit floating point arithmetic? A) 1.0x10-6 B) 1.0x10 8 C) 1.0x10-8 D) 1.0x10 6 Hint: 23/log 2 (10) 6.92 significant digits 34 17

18 Loss of precision 1 (ex: double) Let double z=2 53, then z+1==z because only 52 digits in fraction =z =z+1 35 Loss of precision 2 (ex: double) Catastrophic cancellation Let v= and w= Big and nearly equal; correct to 16 decimal places But their difference v w is correct only to 6 places Because fractions were correct only to 6 places v = w = v-w = Garbage digits = v = w = v-w Would be non-zero in full-precision

19 Case: Solving a quadratic equation The solutions to ax 2 + bx + c = 0 are when d = b 2 4ac > 0. But subtraction -b± d may lose precision when b 2 is much larger than 4ac; in this case the square root is nearly b. Since d >= 0, compute x 1 first if b<0, else compute x 2 first Then compute x 2 from x 1 (or x 1 from x 2 ) 37 Bad and good quadratic solutions double d = b * b - 4 * a * c; if (d > 0) { double y = Math.sqrt(d); double x1 = (-b - y)/(2 * a); double x2 = (-b + y)/(2 * a); } Bad double d = b * b - 4 * a * c; Good if (d > 0) { double y = Math.sqrt(d); double x1 = b > 0? (-b - y)/(2*a) : (-b + y)/(2*a); double x2 = c / (x1 * a); } else... When a=1, b=10 9, c=1 we get Bad algorithm: x1 = e+09 and x2 = Good algorithm: x1 = e+09 and x2 = e

20 Case: Linear regression Points (2.1, 5.2), (2.2, 5.4), (2.4, 5.8) have regression line y = α + β x with α = 1 and β = 2 39 Bad way to compute α and β double SX = 0.0, SY = 0.0, SSX = 0.0, SXY = 0.0; for (int i=0; i<n; i++) { Point p = ps[i]; Large and SX += p.x; nearly SY += p.y; identical SXY += p.x * p.y; SSX += p.x * p.x; } double beta = (SXY - SX*SY/n) / (SSX - SX*SX/n); double alpha = SY/n - SX/n * beta; This recipe was used for computing by hand OK for points near (0,0) But otherwise may lose precision because it subtracts large numbers SSX and SX*SX/n 40 20

21 Better way to compute α and β double SX = 0.0, SY = 0.0; for (int i=0; i<n; i++) { Point p = ps[i]; SX += p.x; SY += p.y; } double EX = SX/n, EY = SY/n; double SDXDY = 0.0, SSDX = 0.0; for (int i=0; i<n; i++) { Point p = ps[i]; double dx = p.x - EX, dy = p.y - EY; SDXDY += dx * dy; SSDX += dx * dx; } double beta = SDXDY/SSDX; double alpha = SY/n - SX/n * beta; Mathematically equivalent, but much more precise than previous one on the computer 41 Example results Consider (2.1, 5.2), (2.2, 5.4), (2.4, 5.8) And same with or added to each coordinate Move Bad Good Correct Wrong 0 10 Very M wrong!! 50 M α β α β α β

22 An accurate computation of sums Let double[] xs = { 1E12, -1, 1E12, -1, } The true array sum is 9,999,999,999,990,000.0 double S = 0.0; for (int i=0; i<xs.length; i++) S += xs[i]; double S = 0.0, C = 0.0; for (int i=0; i<xs.length; i++) { double Y = xs[i] - C, T = S + Y; } C = (T - S) - Y; S = T; 20,000 elements Naïve sum, error = 992 Kahan sum, error = 0 C is the error in the sum S Note that C = (T-S)-Y = ((S+Y)-S)-Y may be non-zero 43 C# decimal, and IEEE decimal128 C# s decimal type is decimal floating-point Has 28 significant digits Has range ±10-28 to ±10 28 Can represent 0.01 exactly Uses 128 bits; computations are a little slower Use decimal IEEE 754 decimal128 is even better for accounting (dollars, euro, kroner)! Has 34 significant (decimal) digits Has range ± to ± Can represent 0.01 exactly Uses 128 bits in a very clever way (Mike Cowlishaw, IBM) Java s java.math.bigdecimal Has unlimited number of significant digits Has range ± to ± Computations are a lot slower 44 22

23 Floating-point tips and tricks Do not compare floating-point using ==,!= Use Math.abs(x y) < 1E-9 or similar Or use Math.abs(Double.doubleToLongBits(x) Double.doubleToLongBits(y))<16 Do not use floating-point for currency (\$, kr) Use C# decimal or java.math.bigdecimal Or use long, and store amount as cents or øre A double stores integers <= *10 15 exactly To compute with very small positive numbers (probabilities) or very large positive numbers (combinations), use their logarithms 45 Proposed exercises Find types and values such that It holds that x+1 < x It holds that z+1 == z It holds that y!= y It holds that v+w-w!= v It holds that x==y && 1.0/x!= 1.0/y And check that Java arithmetics works like that Find the binary representation of +3 and -3 as byte, int and long of +128 and -127 as byte, int and long of and and 62.5 and 625 as float of 0.01 as double and check results using displayint etc from Numbers.java Compute 1/Infinity and Math.exp(-Infinity) and 1/ (1+1/0) and Math.pow(0, -1) in Java 46 23

24 References David Goldberg: What every computer scientist should know about floating-point arithmetics, 1991 R. Mak: Java Number Cruncher: The Java Programmer's Guide to Numerical Computing. Prentice-Hall Java example code and more: William Kahan notes on IEEE 754: General Decimal Arithmetic (Mike Cowlishaw, IBM) C# specification (Ecma International standard 334): How to compare floating-point numbers (in C):

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