1 / 24

Topic 3d Representation of Real Numbers

This introduction to computer systems engineering covers the representation of various types of real numbers, including large and small numbers, rationals, and irrationals. It explores fixed-point and floating-point data formats and the IEEE 754 standard for floating-point representation.

rickeyb
Download Presentation

Topic 3d Representation of Real Numbers

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Topic 3d Representation of Real Numbers Introduction to Computer Systems Engineering (CPEG 323) cpeg323-08F\Topic3d-323

  2. Recap • What can be represented in n bits? • Unsigned 0 to 2n - 1 • 2’s Complement -2n-1 to 2n-1 - 1 • BCD 0 to 10n/4 - 1 • But, what about? • very large numbers 9,349,398,989,787,762,244,859,087,678 • very small number 0.0000000000000000000000045691 • rationals 2/3 • Irrationals 2.71828…., 3.1415926….., cpeg323-08F\Topic3d-323

  3. ………………. ………………. Conceptual Overview: Finite-precision numbers Positive underflow Negative underflow zero Expressible negative numbers Expressible positive numbers Negative overflow Positive overflow • Is there any “underflow” region in integer representation? • Between two adjacent integer numbers, there is no other integer. cpeg323-08F\Topic3d-323

  4. Representing Real Numbers • How to represent fractional parts to the right of the ``decimal'' point? • A number like 0.12 (i.e., (1/2)10) not represented well by integers 0 or 1! • Two ways to represent real numbers better: • Fixed point • Floating point cpeg323-08F\Topic3d-323

  5. Fixed-Point Data Format 4 2 1 1/2 1/4 1/8 …… 0 S 1 0 0 1 1 . 0 0 0 0 1 0 Sign Integer Fractional hypothetical binary point cpeg323-08F\Topic3d-323

  6. Fixed Point • Pros • Add two reals just by adding the integers • Much less logic than floating point • Faster • Often used in digital signal processing • Cons • The range of numbers is narrow number=400 000 000 000 000 000 000 000 000. 000 It is much more economical to represent as 4*1026 cpeg323-08F\Topic3d-323

  7. Recall Scientific Notation decimal point exponent -23 • Issues: • Arithmetic (+, -, *, / ) • Representation, Normal form • Range and Precision(Determined by?) • Rounding and errors • Exceptions (e.g., divide by zero, overflow, underflow) • Properties 6.02 x 10 Significand radix (base) cpeg323-08F\Topic3d-323

  8. Scientific Notation: Normalized • 12.35 x 10^-9 ? • 1.235 x 10^-8 ? • scientific notation: has a single digit to the left of the decimal point • Normalized scientific notation: such a single digit must be non-zero. cpeg323-08F\Topic3d-323

  9. s exp frac Floating Point Representation • Numerical Form: (–1)s 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 • MSB is sign bit: S=0/1 • exp field encodesE • frac field encodesM cpeg323-08F\Topic3d-323

  10. s exp frac 1 bit 8 bits 23 bits IEEE 754 standard • Three formats: single/double/extended precision (32,64,80 bits). • Single precision: Double precision: see page 192 in P&H book cpeg323-08F\Topic3d-323

  11. IEEE 754 Standard Floating Point Representation • the representation: (–1)s (1+ Fraction) 2E-bias where E is the exponent part in the notation • Sign bit s determines whether number is negative or positive • Fraction isnormally a fractional value in range between 0 and 1 • A leading 1 added to the fraction is “implicit” • Exponent using a “biased” notation” • For single precision – the bias is 127 That is, when you convert the notation to the number it represents, the exponential part used should be reading E out from the IEEE 754 notation, and calculated as E-127. cpeg323-08F\Topic3d-323

  12. Advantage of using the biased notation for exponents • Under the single-precision IEEE 754 standard: bias = 127 • If the real exponent is +1, what is the biased exponent ? Answer: 1+127 = 128! (Note 128-127 = +1!) • How about if the real exponent is -1 ? Answer: -1+127 = 126! (Note 126-127 = -1!) How about if the real exponent is -127 ? E -127 = -127, then E = ? cpeg323-08F\Topic3d-323

  13. Observations • Using biased notation, the exponential part in the single-precision IEEE 754 notation itself never get nagative. cpeg323-08F\Topic3d-323

  14. Normalized Encoding Example • Value: Float F = 15213.010; 1521310 = 1.1101101101101 X 213 • Significand M = 1.1101101101101 frac = 11011011011010000000000 (23 bits! With leading 1 hiding!) • Exponent E = 1310, Bias = 12710 Exp = 14010 = 10001100 Floating Point Representation: Binary:01000110011011011011010000000000 exp frac cpeg323-08F\Topic3d-323

  15. Denormalized Values • Condition • exp = 000…0 • Significand value M =0.xxx…x • xxx…x: bits of frac • Cases • frac = 000…0 • Represents value 0 • Note that have distinct values +0 and –0 • frac000…0 • Numbers very close to 0.0 • Lose precision as get smaller • “Gradual underflow” cpeg323-08F\Topic3d-323

  16. Special Values • Condition • exp = 111…1 • (infinity) • Operation that overflows • Both positive and negative • E.g., 1.0/0.0 = 1.0/0.0 = +, 1.0/0.0 =  • Not-a-Number (NaN) • Represents case when no numeric value can be determined • E.g., sqrt(–1),  cpeg323-08F\Topic3d-323

  17. IEEE 754: Summary Normalized: Denormalized: zero: Infinite: NaN: cpeg323-08F\Topic3d-323

  18. IEEE754: Summary (Cont.) +  -Normalized +Normalized -Denorm +Denorm NaN 0 NaN +0 Negative overflow Positive underflow Positive overflow Negative underflow cpeg323-08F\Topic3d-323

  19. E1–E2 • (–1)s2 M2 + • (–1)s M FP Addition • Operands (–1)s1 M1 2E1 (–1)s2 M2 2E2 • Assume E1 > E2 • Exact Result: (–1)s M 2E • Exponent E: E1 • Sign s, significand M: Result of signed align & add • (–1)s1 M1 cpeg323-08F\Topic3d-323

  20. Decimal Number Conversion • Convert Binary to Decimal (base 2 to base 10) x x x x x. d d d d d d …2 24 23 22 21 20 2-1 2-2 2-3 2-4 2-5 2-6 … … 1101.0112 = 1*23+1*22+0*21+1*20+0*2-1+1*2-2+1*2-3 = 13.37510 cpeg323-08F\Topic3d-323

  21. Decimal Number Conversion (Cont.) • Convert Decimal Integer to binary Integer divide the decimal value by 2 and then write down the remainder from bottom to top (Divide 2 and Get the Remainders) 3710 = ?2 Quotient reminder 37÷2 = 18 … 1 18÷2 = 9 … 0 9÷2 = 4 … 1 4÷2 = 2 … 0 2÷2= 1 … 0 1÷2 = 0 … 1 1001012 Can it always be converted into an accurate binary number? YES! cpeg323-08F\Topic3d-323

  22. Decimal Number Conversion (Cont.) • Convert Decimal Fraction to Binary Fraction multiply the decimal value by 2 and then write down the integer number from top to bottom (Multiply 2 and Get the Integers) . 0.37510 = ?2 .375 * 2 = 0.75 .75 * 2 = 1.5 .5 * 2 = 1.0 Where to put the binary point? Prior to the First number! 0.0112 Can it always be converted into an accurate binary number? NO! cpeg323-08F\Topic3d-323

  23. Decimal Number Conversion (Cont.) • Convert Decimal Number to Binary Binary Put Together: Integer Part . Fraction Part 37.37510 = 100101.011 cpeg323-08F\Topic3d-323

  24. Summary • Computer arithmetic is constrained by limited precision • Bit patterns have no inherent meaning but standards do exist • two’s complement • IEEE 754 floating point • OPcode determines “meaning” of the bit patterns • Performance and accuracy are important so there are many complexities in real machines (i.e., algorithms and implementation). cpeg323-08F\Topic3d-323

More Related