Double-Precision Floating-Point Numbers Douglas Wilhelm Harder Department of Electrical and Computer Engineering University of Waterloo Copyright © 2007 by Douglas Wilhelm Harder. All rights reserved. ECE 204 Numerical Methods for Computer Engineers

Double-Precision Floating-Point Numbers This topic introduces binary numbers –requirements –a poor means of storage –a good means of storage

Double-Precision Floating-Point Numbers We will now use this same floating-point format, but we will apply it to binary numbers

Double-Precision Floating-Point Numbers In our example, we used six decimal digits The double-precision floating-point format uses 64 bits (or eight bytes) Like our format, they are broken up into –a leading sign bit, –an exponent (with a bias), and –a mantissa

Double-Precision Floating-Point Numbers Like our six-digit version, the bits are stored in the order: SEEEEEEEEEEEMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMM The bias is , or 1023 This allows us to represent numbers in the range to , though the floating- point standard IEEE 754 reserves the use of the lowest (all zeros) and highest (all ones) exponents

Double-Precision Floating-Point Numbers Recall that the leading bit in a floating- point representation must be non-zero, thus, the bit must be 1 We therefore do not store the leading digit, thus, the mantissa actually represents 1.MMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMM

Double-Precision Floating-Point Numbers Rather than printing out a lot of 1s and 0s, instead, we will use hexadecimal numbers: a b c d e f151111

Double-Precision Floating-Point Numbers To convert a binary number into hexadecimal, simply group the bits into groups of four (starting a a radix point if it exists) and replace each group with the corresponding hexadecimal value To convert from hexadecimal to binary, replace each hexadecimal digit with its four-bit equivalent (including leading zeros)

Double-Precision Floating-Point Numbers Some of the more common numbers are: >> format hex >> 1 ans = 3ff >> 2 ans = >> -1 ans = bff >> -2 ans = c Recall that 3ff 16 = which is our bias

Double-Precision Floating-Point Numbers Some operations are quite straight- forward: –multiplication by 2 adds 1 to the exponent and leaves the mantissa unchanged –division by 2 subtracts 1 from the exponent and leaves the mantissa unchanged

Double-Precision Floating-Point Numbers Rounding rules are simplified Given a binary number which has more than 53 bits of precision, then to round it to a 53 bit number –if the 54 th bit is 0, then truncate (round down) –if all bits after the 53 rd bit are 1000··· then round up if the 53 rd bit is 1, otherwise truncate, and –otherwise, round up (add 1 to the 53 rd bit)

Double-Precision Floating-Point Numbers Remember, we deal with 53 bits because we store 52 bits together with the implicit leading 1

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