# Decimal Floating-Point Arithmetic

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Decimal Floating-Point Arithmetic
Dongdong Chen EE800, U of S

Objectives IEEE standard for Decimal Floating-Point (DFP) arithmetic (Lecture 1) DFP numbers formats DFP number encoding DFP arithmetic operations DFP rounding modes DFP exception handling EE800, U of S

Objectives (Con.) Algorithm, architecture and VLSI circuit design for DFP arithmetic (Lecture 2) DFP adder/substracter DFP multiplier DFP divider DFP transcendental function computation EE800, U of S

Background The decimal computer arithmetic went out of style 25 to 30 years ago; no one uses it now." Is that true? EE800, U of S

Introduction Decimal is still essential for specific applications
Numbers in commercial databases are decimal Extensive use decimal in commercial applications Survey of commercial databases report Decimal fixed-point or floating-point number How to process decimal computation Software computation Convert back to decimal representation Problems EE800, U of S

Introduction (Con.) Errors from decimal and binary conversion
Example 1: represent 0.1 in DFP or BFP Decimal representation (BCD code):0.0001 Binary representation: … 0.09… Example 2: telephone billing Cost: 0.70; Tax: 5% BFP arithmetic: …8*(1.05)= … DFP arithmetic: 0.70*(1.05)=0.74 Decimal integer, fixed-point or floating-point? Decimal hardware or software solutions? EE800, U of S

Current Researches DFP arithmetic defined in IEEE 754-2008
IBM computing systems include DFP hardware IBM Power6, z9, z10 Intel include DFP software solution in system Intel DFP software computation library DFP arithmetic IP blocks: Basic DFP arithmetic IPs: DFP adder/substrcter, multiplier, divider, square root etc. Transcendental DFP arithmetic IPs: DFP CORDIC, Logarithm, antilogarithm, reciprocal etc. EE800, U of S

DFP Arithmetic in IEEE 754-2008
Review BFP arithmetic in IEEE How to define new DFP in IEEE EE800, U of S

BFP Floating-point representation
sign, exponent, significand (or mantissa): (–1)sign × significand × 2exponent more bits for significand gives more accuracy more bits for exponent increases range IEEE 754 floating point standard: single precision: 8 bit exponent, 23 bit significand double precision: 11 bit exponent, 52 bit significand EE800, U of S

BFP floating-point Number
Leading “1” bit of significand is implicit Example: if the significand is …0, the actual significand is …0 This is called a normalized number; there is exactly one non-zero digit to the left of the point. Unique representation of a number We get a little more precision: there are 24 bits in the significand, but only 23 of them are stored. EE800, U of S

Exponent Exponent is “biased” to make sorting easier
all 0s is smallest exponent, all 1s is largest The actual exponent is e-127 for single precision, and e-1023 for double precision Bias of 127 for single precision and 1023 for double precision By biasing the exponent and storing it before the significand, we can compare magnitudes as if they were unsigned integers. If e = (13110), the actual exponent is =4 If e = (9310), the actual exponent is =-34 EE800, U of S

BFP Floating-Point Formats
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BFP Floating-Point Formats (Con.)
Positive and negative zero 1 Biased exponent Fraction Positive and negative infinity Biased exponent Fraction Positive underflow Negative underflow Negative Overflow Expressible negative numbers Expressible positive numbers Positive Overflow -2-127 2-127 - (2 – 2-23)×2128 (2 – 2-23)×2128 exponent = 128 and fraction ≠ 0, It is called “not a number” or NaN EE800, U of S

(–1)sign×(1+significand)×2exponent – bias
Example Summary: FP representation (–1)sign×(1+significand)×2exponent – bias Example: decimal: = -3/4 = -3/22 binary: = -1.1 x 2-1 floating point: exponent = 126 = IEEE single precision: EE800, U of S

DFP Number Representation
sign, exponent, significand (or mantissa): (–1)sign × significand × 10exponent more digits for significand gives more accuracy more bits for exponent increases range representation: DFP formats: decimal32: DFP storage format encoded in 32-bit decimal64: DFP computational format encoded in 64-bit decimal128: DFP computational format encoded in 128-bit EE800, U of S

DFP Number format 1-bit Sign (S) is defined as same as BFP format
w+5-bit combination (G) to two subfield: 5-bit (G0…G4) to encode: 2 MSBs of exponent; 1 MSD of significand; Not-a-Number (NaN); Inf; W-bit(G5…Gw+4) as a suffix 2 MSBs derived from G0…G4, which consists of w+2-bit nonnegative biased exponent. EE800, U of S

DFP Exponent Exponent is “biased” to make sorting easier
Binary format (not decimal) The actual exponent is e-101 for decimal32, e-398 for decimal64, e-6167 for decimal128 Range of exponent is (emin−q+1) ≤ e ≤ (emax−q+1); EE800, U of S

DFP Number format (Con.)
J×10-bit Trailing Significand (T) Field: Densely packed decimal (DPD) encoding 3-digit decimal number encoded to 10-bit binary number DPD converted to binary coded decimal (BCD) Binary integer decimal (BID) encoding decimal number encoded by binary integer Non-normalized decimal significand (-1)0 × × (-1)0 × × 101 DFP number’s Cohort EE800, U of S

Parameters in DFP Format
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Example Summary: DFP representation
(–1)sign×(significand)×10exponent-bias Convert -8.35×10-2 to decimal64 Sign bit: “1” negative, “0” positive (sign 1) Exponent: =396 (8-bit “ ”) Significand: 835(50-bit DPD coding “0… D”) Encoding of 5-bit MSBs (G0…G4) of Combinational field “01000” Decimal-64 : “ …..00… ” “A D” (binary/hex) EE800, U of S

DFP special values Not-a-Number: G0…G4 “11111”;
Infinite Number: G0…G4 “11110”, sign of Inf according to the sign bit; Overflow: If DFP numbers with absolute values are larger than the largest DFP number (|vmax|=(10q - 1)×10emax-q+1) then overflow occurs. Underflow: If DFP number are less than the smallest DFP number (|vmin|=10emin-q+1) then underflow occurs. If the absolute value of DFP number is less than 10emin and larger than 10emax-q+1, it produces subnormal. Normal number: The remaining exponent values and significands represent normal numbers. EE800, U of S

DFP Arithmetic Operations
Basic DFP arithmetic operations Two decimal-specific DFP operations SameQuantum(DFP1,DFP2) Quantize(DFP1,DFP2) DFP comparison operations do not distinguish between redundant of the same number DFP conversion operations DFP to BFP conversion (correctly rounded); DFP to integer conversion Recommended DFP operations EE800, U of S

DFP Arithmetic Operations
Basic DFP arithmetic operations Two decimal-specific DFP operations SameQuantum(DFP1,DFP2) Quantize(DFP1,DFP2) DFP comparison operations do not distinguish between redundant of the same number DFP conversion operations DFP to BFP conversion (correctly rounded); DFP to integer conversion Recommended DFP operations EE800, U of S

DFP Number’s Cohort Non-normalized decimal significand
Standard defines the preferred (required) exponent (quantum) Exact operation results: the cohort member is selected based on the preferred exponent (quantum) for a DFP result of that operation Inexact operation results: the cohort member of least possible exponent is used to get the maximum number of significant digits EE800, U of S

DFP Rounding Modes Five types of active rounding modes
roundTiesToEven roundTiesToAway roundTiesToPositive roundTiesToNegative roundTowardZero Correct rounding and Faithful rounding IEEE require to satisfy the correct rounded results for all DFP arithmetic operations DFP operations should satisfy all rounding modes EE800, U of S

DFP Exception Handling
Invalid operation: Operand is NaN; 0×Inf; quare-root of negative operand; default result is NaN Division by zero: if the dividend is a finite non-zero number and the divisor is zero. The default result is a +inf or −inf. Overflow operation: if the magnitude of a result exceeds the largest finite number representable in the format of the operation. Underflow operation: if the magnitude of a result is below 10emin. Inexact: the correctly rounded result of an operation differs from the infinite precision result. EE800, U of S

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DFP Add/Sub Data flow EE800, U of S

DFP Addition Step 1: equalize the exponents
add the mantissas only when exponents are the same. the number with smaller exponent should be shifting its point to the left, and the number with larger exponent should be shifting its point to right. Rewriting the operand with the smaller exponent could result in a loss of the least significant digits keep guard digit, round digit, and stick digit for the operand with smaller exponent EE800, U of S

Step 3: Normalize the result if necessary EE800, U of S

DFP addition Step 4: Round the number if needed
x100 = x100 Step 5: Repeat step 3 if the result is no longer normalized The final result is The correct answer is EE800, U of S

Guard bits To help minimize rounding problems, IEEE specifies that intermediate steps of operations must store guard digits - additional internal digits that increase the precision of the operations. Previous example: add one extra digit. IEEE requires one guard digit, one rounded digit and one sticky digit to make rounding more accurate. EE800, U of S

DFP add/sub EE800, U of S

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Example: Addition EE800, U of S

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DFU: IBM POWER6 and Z10 EE800, U of S

High performance Implementation
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High performance Implementation
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High performance Implementation
[12] A. Vázquez and E. Antelo“A High-performance Significand BCD Adder with IEEE Decimal Rounding” ARITH19, Portland. June EE800, U of S

Evaluation Results and Comparison
[Proposed]: A. Vázquez and E. Antelo“A High-performance Significand BCD Adder with IEEE Decimal Rounding” ARITH19, Portland. June EE800, U of S

DFP Multiplication EE800, U of S

Scheme of decimal multiplier
x : × y : = xy0: x xy1: x −x xy2 : x xy3: 10x −2x EE800, U of S

Partial product generation
Generate XYi Yi {1,2,3…7,8,9} XYi is carry save format EE800, U of S

Partial product generation
Solid Circles: BCD Sum (digit) Hollow Circles: Carry (bit) n-digit radix-10 CSA m-digit radix-10 counter EE800, U of S

Carry Save Adder Tree CSA Tree to Generate Multiplication Result
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Flowchart of DFP Multiplier

Architecture of DFP Multiplier

Exception Detection & Handling
Invalid operation sNaN (pass significand of sNaN) 0 x ∞ (produce qNaN with significand 0) Overflow (and Inexact) IEIP – SLA > Emax Increase SLA until all LZs removed Underflow (and possibly Inexact) IEIP – SLA < Emin Decrease SLA until 0, then shift right Inexact

Implementation Highlights
Leverage operands' LZCs SC, SLA, and IESIP Handle NaNs with minimal overhead No dataflow modification Coerce multiplicand or multiplier to 1 Support gradual underflow Simply extend number of iterations Simple, control-based rounding scheme

Synthesis Results 64-bit (16 digit) operands, DPD encoded
LSI Logic's gflxp 0.11um CMOS, 55ps FO4 Synopsys Design Compiler Results Fixed-point 119,653 um FO4s Floating-point 237,607 um FO4s Critical path Fixed-point 4:2 compressor (accumulator) Floating-point 128-bit barrel shifer

Applicability to Parallel Designs
IE and IP shift generation Rounding scheme NaN handling Exception detection and handling On-the-fly sticky bit generation... NO

Sequential vs. Parallel
Less area Potentially better cycle time Parallel Less latency Higher throughput

DFP Division EE800, U of S

DFP Division Data Flow Unpacking Decimal Floating-Point Number
Check for zeros and infinity Subtract exponents Divide Mantissa Normalize and detect overflow and underflow Perform rounding Replace sign Packing EE800, U of S

Unpacking and Sign Logic
Step1: Unpacking Floating-Point Number Check for zeros and infinity (if F=0, Stop) Step2: Sign Process EE800, U of S

Exponent Subtraction Step3: Exponent Subtract EE800, U of S

Mantissa Division Algorithms Choose here? 1. Restoring division
2. Non-restoring division 3. High-Radix division 4. Convergence division Step4: Mantissa Division EE800, U of S

Normalization Step5 : Left shift over one bit is needed to make Mantissa result Normalized, also need to detect overflow and underflow For example: “0934… ” Left shift one bit  “934… Should tell exponent and Ea=Eb-1 EE800, U of S

Rounding and Packing Step6 : Truncate, Round-up, Round-to-nearest. Sometimes, the Rounding Policy above is not fair, according to IEEE Rounding standard: “Round to nearest even” is more better. Step7: Packing the Sign bit and Exponent bits and Significand bits together, detect the NaN, Infinity, EE800, U of S

High performance Implementation
[1] L.-K. Wang and M. J. Schulte, “Decimal Floating-Point Division Using Newton-Raphson Iteration,” Proceedings of the IEEE International Conference on Application-Specific Systems, Architectures and Processors, pp , Sep EE800, U of S

High performance Implementation
[2] Tomás Lang and Alberto Nannarelli, “A Radix-10 Digit-Recurrence Division Unit: Algorithm and Architecture,”IEEE Transactions on Computers, pp727–739, IEEE, June 2007. EE800, U of S

High performance Implementation
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Evaluation Results and Comparison
DFP Divider[1] DFP Divider[2] Precision (digit) 16 (decimal64) Cycle time (ns) 0.57 1 # of cycles 150 20 Latency (ns) 85.5 1: Synthesized with a STM 90-nm standard cell library EE800, U of S

DFP Transcendental Arithmetic
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Contents Introduction Decimal Logarithmic Converter
Decimal Antilogarithmic Converter Conclusions Future Work EE800, U of S 66

32-bit DFP Logarithm coefficient is a non-normalized decimal Integer.
Example: To guarantee a 32-bit DFP Calculation, there need to keep 14-digit FXP logarithmic calculation. EE800, U of S

32-bit DFP Antilogarithm
Here: For 32-bit DFP: Example: To guarantee a 32-bit DFP calculation, there need to keep 8-digit FXP antilog calculation. EE800, U of S

Digit-Recurrence Algorithm (Log)
The corresponding recurrences: Here: ej ∈｛ …0 1…7 8 9｝ selected so that converges to 1 EE800, U of S

Digit-Recurrence Algorithm (Antilog)
Any 7-digit fixed-point decimal input N: The corresponding recurrences: Here: selected so that converges to 0 ej ∈｛ …0 1…7 8 9｝ EE800, U of S

Selection By Rounding (cont.)
A scaled remainder is defined as: Log: Antilog: is achieved by Rounding W [j] e1 is achieved by using look-up table, e2…ej can be obtained with selection by rounding EE800, U of S

Architecture: Decimal Log Converter
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Implementation Results
Logic Utilization Used Available* Utilization # of Occupied Slices 2842 13696 21% Maximum Frequency 47.7 MHz # of Clock Cycles 17 clock cycle *: Xilinx Virtex2p XC2VP30 with package ff1157 and speed -7 Critical Path Detail (ns): Reg2 Mux2 Mult 2 Shifter Mux5 CLA Round Total 1.188 1.564 9.347 1.438 1.350 5.519 0.566 20.97 EE800, U of S 73

Architecture: Dec. Antilog Converter
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Implementation Results
Logic Utilization Used Available* Utilization # of Occupied Slices 2315 13696 17% Maximum Frequency 51.5 MHz # of Clock Cycles 11 clock cycle *: Xilinx Virtex2p XC2VP30 with package ff1157 and speed -7 Critical Path Detail (ns): Reg6 Mult Mux4 Shifter CLA Round Total 1.599 7.839 1.539 1.100 6.794 0.545 19.42 EE800, U of S 75

Comparison (with Binary FXP Log and Exponential Converters)
similar dynamic range for the normalized coefficients. Binary reference available having the same digit-recurrence algorithm with Selection by Rounding. The radix-10 is close to radix-8. EE800, U of S

Comparison (cont.) (with Binary FXP Log and Exponential Converters)
Radix-10 Decimal1 Radix-8 Binary [1] Log. Exp. Precision (digit) 7 16 24 53 Area (fa2) 1630 2640 1370 2260 647 1829 627 1777 Cycle time (T3) 17 19 18 8 # of cycles 11 21 Latency (T3) 136 323 128 306 56 144 77 168 1: Synthesized with a TMSC 0.18-um standard cell library 2: the area of 1-bit full adder 3: the delay of 1-bit full adder EE800, U of S

Conclusions Achieved 32-bit DFP accuracy of decimal log and antilog results. Implemented them on FPGA and ASIC. Compare them with binary converters. EE800, U of S 78

Decimal Log and Antilog Converters
Future Work The 64-bit and 128-bit DFP logarithm and antilog converters. The presented architecture can be optimized to achieve a faster speed or occupy a smaller area. EE800, U of S EE990 April. 2009 Decimal Log and Antilog Converters 79/18 79

Summary IEEE 754-2008 defines a DFP standard that defines
number representation in several precisions correct DFP arithmetic operations rounding modes Implementation of DFP Adder, Multiplier, Divider, Logarithmic and Antilogarithmic Converter Implementing and programming DFP are both really hard. EE800, U of S