QFormat number representation. Lecture 5 Fixed Point vs Floating Point. How to store Q30 number to 16bit memory? Qformat notation.


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1 Lecture 5 Fixed Point vs Floating Point Objectives: Understand fixed point representations Understand scaling, overflow and rounding in fixed point Understand Qformat Understand TM32C67xx floating point representations Understand relationship between the two in C6x architecture Reference: "What Every Computer cientist hould Know About FloatingPoint Arithmetic" by David GoldbergACM Computing urveys 23, 5 (March 1991). Lecture 5  Fixed point vs Floating point 51 QFormat number representation Nbit fixed point, 2 s complement number is given by: x = b N1 N 1 N 2 1 N 12 + bn b1 2 + b 2 imaginary binary point Difficult to work with due to possible overflow & scaling problems Often normalise number to some fractional representation (e.g. between ± 1) x = b N1 1 N 2 N 1 N 12 + bn b1 2 + b 2 imaginary binary point Lecture 5  Fixed point vs Floating point 52 Qformat notation How to store Q3 number to 16bit memory? Qformat representation: if N=16, 15 bit fractional representation Q15 format Rule: Q m + Q m Q m Q m x Q n Q m+n Assume 16bit data format, Q15 x Q15 Q3 X Q15 Q15 Q toring Q3 number to 16bit memory requires rounding or truncation: Q3 Rounding: if r =, round down, r = 1, round up rounding by addition a '1' here 15 r + 1 MPY A3,A4,A6 ; A3 x A4 A6 NOP ; Delay slot K 4h,A6 ; rounding add HR A6,15,A6 ; truncate bottom 15 bits TH A6,*A7 ; A6 mem[a7] Lecture 5  Fixed point vs Floating point 53 Lecture 5  Fixed point vs Floating point 54
2 Avoid overflow with afe add routine in C to avoid overflow  saturation add instruction Always clip to max (or min) possible et bit 9 of the CR register to indicate saturation has occurred Lecture 5  Fixed point vs Floating point 55 Lecture 5  Fixed point vs Floating point 56 ingle Precsion Floating Point number Easy (and lazy) way of dealing with scaling problem 32bit single precision floating point: single precision x = 8bit exp s exp bit frac 1. frac MB is signbit (same as fixed point) 8bit exponent in bias127 integer format (i.e., add 127 to it) 23bit to represent only the fractional part of the mantissa. The MB of the mantissa is ALWAY 1, therefore it is not stored < x < Double Precision Floating Point number 64bit double precision floating point: double precision bit exp 52bit frac Odd register (e.g. A5) Even register (e.g. A4) x s = < exp frac x < MB is signbit (same as fixed point) 11bit exponent in bias123 integer format (i.e., add 123 to it) 52bit to represent only the fractional part of the mantissa. The MB of the mantissa is ALWAY 1, therefore it is not stored 38 Lecture 5  Fixed point vs Floating point 57 Lecture 5  Fixed point vs Floating point 58
3 Convert 5.75 to P FP Examples 5.75 to binary: x 2 2 exponent in bias127 is = 129 = 1 b The fractional part is after we drop the hidden 1 bit. Answer: = 4B8 (hex) Convert.1 to DP FP.1 to binary: (11 repeats) x 24 exponent in bias123 is = 119 = b The fractional part is after we drop the hidden 1 bit and rounding Answer: = 3FB A (hex). Problems of Qformat Wrong Qformat representation will give totally wrong results Even correct use of Qformat notation may reduce precision For this example, Q12 result is totally wrong, and Q8 result is imprecise: Q Q * Q Q Q Lecture 5  Fixed point vs Floating point 59 Lecture 5  Fixed point vs Floating point 51 Data types used by C6x DPs pecial P numbers IEEE floating point standard has a set of special numbers: pecial ign (s) Exponent (e) Fraction (f) Hex Value Decimal + x.  1 x x3f x4 2. +Inf 255 x7f8 + Inf xff8  NaN x 255 Nonzero x7fff FFFF not a number LFPN 254 All 1 s x7f7f FFFF e+38 FPN 1 All s x e38 Lecture 5  Fixed point vs Floating point 511 Lecture 5  Fixed point vs Floating point 512
4 pecial DP numbers TM32C67x Internal ystem Architecture Double precision floating point special numbers: pecial Exponent (e) Fraction (f) Hex Value Decimal + x.  x x3ff x4 2. +Inf 247 x7ff + Inf 247 xfff  NaN 247 Nonzero x7fff FFFF FFFF FFFF not a number LFPN 246 All 1 s x7fef FFFF FFFF FFFF e+38 FPN 1 All s x e38 External Memory P E R I P H E R A L Regs (AA15/31) Internal Memory Internal Buses.D1.M1.L1.1 CPU.D2.M2.L2.2 Regs (BB15/31) Lecture 5  Fixed point vs Floating point 513 Lecture 5  Fixed point vs Floating point 514 Four functional units for each datapath Mapping of instructions to functional units..l.d.m K 2 AND B CLR EXT C K KH AB LDB LDDW. Unit NOT OR ET HL HR HL UB UB2 XOR ZERO ABP ABDP CMPGTP CMPEQP CMPLTP CMPGTDP CMPEQDP CMPLTDP RCPP RCPDP RQRP RQRDP PDP.D Unit (B/H/W) TB (B/H/W) (B/H/W) UB UBAB (B/H/W) ZERO AB AND CMPEQ CMPGT CMPLT LMBD NORM MPY MPYH MPYLH MPYHL.L Unit NOT P OR DP UBP AT UBDP UB INTP UB INTDP UBC PINT XOR DPINT ZERO PRTUNC DPTRUNC DPP.M Unit MPY MPYH MPYP MPYDP MPYI MPYID No Unit Used NOP IDLE Lecture 5  Fixed point vs Floating point 515 Lecture 5  Fixed point vs Floating point 516
5 Detailed internal datapaths Data path A Data path B Lecture 5  Fixed point vs Floating point 517
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