Newton-Raphson Reciprocal Square Root | AMD x86 specification

22007E/0 — November 1999

AMD Athlon™ Processor x86 Code Optimization

Newton-Raphson Reciprocal Square Root

The general Newton-Raphson reciprocal square root recurrence is:

Zi+1 = 1/2 • Zi • (3 – b • Zi2)

To reduce the number of iterations, the initial approximation read from a table. The 3DNow! reciprocal square root approximation is accurate to at least 15 bits. Accordingly, to obtain a single-precision 24-bit reciprocal square root of an input operand b, one Newton-Raphson iteration is required, using the following sequence of 3DNow! instructions:

X0 = PFRSQRT(b)

X1 = PFMUL(X0,X0)

X2 = PFRSQIT1(b,X1)

X3 = PFRCPIT2(X2,X0)

X4 = PFMUL(b,X3)

The 24-bit final reciprocal square root value is X3. In the AMD Athlon processor 3DNow! implementation, the estimate contains the correct round-to-nearest value for approximately 87% of all arguments. The remaining arguments differ from the correct round-to-nearest value by one unit-in-the-last-place. The square root (X4) is formed in the last step by multiplying by the input operand b.

Use MMX™ PMADDWD Instruction to Perform Two 32-Bit Multiplies in Parallel

The MMX PMADDWD instruction can be used to perform two signed 16x16→32 bit multiplies in parallel, with much higher performance than can be achieved using the IMUL instruction. The PMADDWD instruction is designed to perform four 16x16→32 bit signed multiplies and accumulate the results pairwise. By making one of the results in a pair a zero, there are now just two multiplies. The following example shows how to multiply 16-bit signed numbers a,b,c,d into signed 32-bit products a×c and b×d:

Use MMX™ PMADDWD Instruction to Perform Two 32-Bit Multiplies in Parallel

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AMD x86 manual Newton-Raphson Reciprocal Square Root

Contents

AMD AthlonTM Processor Trademarks Contents Instruction Decoding Optimizations Cache and Memory Optimizations Scheduling Optimizations Floating-Point Optimizations General x86 Optimization Guidelines 127 Appendix B Pipeline and Execution Unit Resources Overview Appendix E Programming the Mtrr and PAT List of Figures Xii List of Tables Xiv Revision History Xvi About this Document Introduction Source Level Optimizations. Describes optimizations that AMD Athlon Processor Family AMD Athlon Processor Microarchitecture Summary AMD Athlon Processor Microarchitecture Summary AMD Athlon Processor x86 Code Optimization Top Optimizations Use the 3DNow! Prefetch and Prefetchw Instructions Memory Size and Alignment IssuesOptimization Star Group I Optimizations Essential Optimizations Use Load-Execute Instructions Group II Optimizations-Secondary OptimizationsSelect DirectPath Over VectorPath Instructions Load-Execute Instruction Usage Use 3DNow! Instructions Take Advantage of Write CombiningAvoid Branches Dependent on Random Data Avoid Placing Code and Data in the Same 64-Byte Cache Line 22007E/0 November Use 32-Bit Data Types for Integer Code Source Level Optimizations Example 1 Avoid Consider the Sign of Integer OperandsExample Preferred Use signed types for Use Array Style Instead of Pointer Style CodeExample Avoid Use unsigned types for Vertex Example 2 Preferred Avoid Unnecessary Store-to-Load Dependencies Completely Unroll Small Loops Avoid Unnecessary Store-to-Load Dependencies Consider Expression Order in Compound Branch Conditions Optimize Switch Statements Switch Statement UsageUse Prototypes for All Functions Generic Loop Hoisting Use Const Type QualifierExample Generalization for Multiple Constant Control Code Declare Local Functions as Static Introduce Explicit Parallelism into Code Dynamic Memory Allocation Consideration Explicitly Extract Common Subexpressions Preferred Language Structure Component ConsiderationsExample Avoid Original ordering Avoid Sort Local Variables According to Base Type SizeNew ordering, with padding Preferred Improved ordering Preferred Accelerating Floating-Point Divides and Square Roots Accelerating Floating-Point Divides and Square Roots Avoid Unnecessary Integer Division AMD Athlon Processor x86 Code Optimization Overview Instruction Decoding Optimizations Load-Execute Instruction Usage Select DirectPath Over VectorPath InstructionsUse Load-Execute Integer Instructions TOP Use Short Instruction Lengths Align Branch Targets in Program Hot Spots Example 2 Avoid Avoid Partial Register Reads and Writes Replace Certain Shld Instructions with Alternative Code Use 8-Bit Sign-Extended Immediates Code Padding Using Neutral Code Fillers Use 8-Bit Sign-Extended Displacements Recommendations for the AMD Athlon Processor Code Padding Using Neutral Code Fillers AMD Athlon Processor x86 Code Optimization NOP6EDI AMD Athlon Processor x86 Code Optimization Cache and Memory Optimizations Memory Size and Alignment IssuesAvoid Memory Size Mismatches Use the 3DNow! Prefetch and Prefetchw Instructions Align Data Where Possible Code Example Multiple Prefetches MOV ECX, -LARGENUM Prefetch Distance = 200 DS/C bytes Determining Prefetch Distance Avoid Placing Code and Data in the Same 64-Byte Cache Line Take Advantage of Write Combining Store-to-Load Forwarding Pitfalls -True Dependencies Store-to-Load Forwarding Restrictions Example 4 Avoid Example 3 Avoid Example 6 Avoid Example 5 PreferredExample 7 Avoid Summary of Store-to-Load Forwarding Pitfalls to Avoid Stack Alignment Considerations Align Tbyte Variables on Quadword Aligned Addresses Sort Variables According to Base Type Size Avoid Branches Dependent on Random Data Branch Optimizations Blended AMD-K6and AMD Athlon Processor Code AMD Athlon Processor Specific Code Example 6 Increment Ring Buffer Offset Always Pair Call and ReturnExample 7 Integer Signum Function Muxing Constructs Replace Branches with Computation in 3DNow! Code Code Sample Code Translated into 3DNow! Code3DNow! code MM5 Pfsub Psrad Avoid Far Control Transfer Instructions Avoid the Loop Instruction Avoid Recursive Functions Complete Loop Unrolling Scheduling OptimizationsSchedule Instructions According to their Latency Unrolling Loops Partial Loop Unrolling With Partial Loop Unrolling Without Loop Unrolling Example 1 rolled loop Deriving LoopControl For Partially Unrolled Loops Overview Use Function Inlining Always Inline Functions if Called from One Site Avoid Address Generation Interlocks Minimize Pointer Arithmetic in Loops Use Movzx and Movsx MOV ECX, Maxsize Example 3 Preferred Push Memory Data Carefully Push Memory Data Carefully Replace Divides with Multiplies Integer OptimizationsMultiplication by Reciprocal Division Utility Unsigned Division by Multiplication of Constant Simpler Code for Signed Division by Multiplication of ConstantRestricted Dividend Integer 2 or Signed Division BySigned Division By 2n Remainder of Signed Integer 2n or -2n Use Alternative Code When Multiplying by a Constant ADD REG1, REG1 REG1, REG2 SHL Use MMX Instructions for Integer-Only Work Latency of Repeated String Instructions Repeated String Instruction UsageGuidelines for Repeated String Instructions Destination with Using MovqEnsure DF=0 UP Align Source Efficient 64-Bit Integer Arithmetic Use XOR Instruction to Clear Integer Registers Example 5 Right shift Example 4 Left shiftExample 6 Multiplication EBX, ESP+12 Dividendlo Example 7 Division Example 8 Remainder SHR EDX Step Efficient Implementation of Population Count Function Bit field. Thus the following computation is performed MOV EDX, EDX SHR Derivation of Multiplier Used for Integer Division by Utility sdiv.exe was compiled using the following code MOV ECX EDX Imul ADD SHR SAR Ensure All FPU Data is Aligned Floating-Point OptimizationsUse Multiplies Rather than Divides Use Ffreep Macro to Pop One Register from the FPU Stack Floating-Point Compare Instructions Use the Fxch Instruction Rather than FST/FLD Pairs Avoid Using Extended-Precision Data Example 1 Fast Minimize Floating-Point-to-Integer Conversions Example 2 Potentially faster Example 4 Fastest Example 3 Potentially faster Floating-Point Subexpression Elimination 104 Take Advantage of the Fsincos Instruction 106 Use 3DNow! Instructions 3DNow! and MMX OptimizationsUse Femms Instruction Optimized 14-Bit Precision Divide Use 3DNow! Instructions for Fast DivisionOptimized Full 24-Bit Precision Divide Newton-Raphson Reciprocal Pipelined Pair of 24-Bit Precision Divides Optimized 24-Bit Precision Square Root Optimized 15-Bit Precision Square Root Newton-Raphson Reciprocal Square Root Blended Code 3DNow! and MMX Intra-Operand SwappingAMD Athlon Specific Code Use MMX Pxor to Negate 3DNow! Data Fast Conversion of Signed Words to Floating-Point Positive Use MMX Pcmp Instead of 3DNow! PfcmpBoth Numbers Positive One Negative, One AMD-K6and AMD Athlon Processor Blended Code Use MMX Instructions for Block Copies and Block Fills 116 Processor Specific Use MMX Pxor to Clear All Bits in an MMX Register Use MMX Pcmpeqd to Set All Bits in an MMX Register Optimized Matrix Multiplication MOV EBX, RES Optimized Matrix Multiplication 121 Data Use 3DNow! Pavgusb for MPEG-2 Motion Compensation MM0=QWORD1 Stream of Packed Unsigned Bytes Complex Number Arithmetic Short Forms General x86 Optimization Guidelines Register Operands DependenciesStack Allocation Introduction AMD Athlon Processor Microarchitecture Superscalar Processor AMD Athlon Processor Microarchitecture AMD Athlon Processor Microarchitecture 131 Branch Prediction Predecode Early Decoding Data Cache Instruction Control Unit Integer Execution Unit Integer Scheduler Floating-Point Scheduler 12 to Floating-Point Execution Unit Load/Store Unit Load-Store Unit LSU Write Combining L2 Cache ControllerAMD Athlon System Bus 140 AMD Athlon Processor Microarchitecture Fetch and Decode Pipeline Stages Pipeline and Execution Unit Resources Overview C T L R O M Cycle 3 VectorPath Cycle 1-FETCHCycle 2-SCAN Cycle 3 DirectPath Integer Pipeline Stages Integer Pipeline Stages Cycle 10 -DCACC Cycle 7-SCHEDCycle 8-EXEC Cycle 9-ADDGEN Floating-Point Pipeline Stages Floating-Point Pipeline Stages Cycle 10 -SCHED Cycle 7-STKRENCycle 8-REGREN Cycle 9-SCHEDW Results Execution Unit ResourcesTerminology Operands Integer Pipeline Operation Types Integer Pipeline OperationsInteger Decode Types Floating-Point Pipeline Operation Types Floating-Point Pipeline OperationsFloating-Point Decode Types Load/Store Unit Stages Load/Store Pipeline OperationsStage 1 Cycle Stage 2 Cycle Stage 3 Cycle Code Sample Analysis ADD EDI, EBX SHL Imul EAX, ECX INC ESI MOVINC EBX ADD ESI, EDX DEC EDX MOV EDI, ECX SUB Sample 2 Integer Register and Memory Load OperationsSAR Implementation Write Combining Appendix C Write-Combining Definitions and Abbreviations What is Write Combining?Programming Details Write-Combining Operations INIT, Halt Write Combining Completion Events Sending Write-Buffer Data to the System AMD Athlon System Bus Commands Generation Rules 160 Write-Combining Operations Performance Counter Usage Performance-Monitoring Counters PerfEvtSel30 Registers PerfEvtSel30 MSRs MSR Addresses C0010000h-C0010003h Performance Counter Usage 163 Performance-Monitoring Counters 74h 65h73h Snoop hits Instruction cache misses Event Source Event DescriptionWaited to use the L2 Instruction cache fetches PerfCtr30 MSRs MSR Addresses C0010004h-C0010007h Starting and Stopping the Performance-Monitoring Counters Event and Time-Stamp Monitoring Software Monitoring Counter Overflow 170 Monitoring Counter Overflow Memory Type Range Register Mtrr Mechanism Programming the Mtrr 172 Memory Type Range Register Mtrr Mechanism FFFFFFFFh Fixed Ranges100000h Kbytes each Fixed Ranges C0000h 80000h Register Format Memory TypesMemory Type Encodings Mtrr Capability Memory Type Range Register Mtrr Mechanism 175 Standard Mtrr Types and Properties Attribute Table MSR 277h Attribute Table PAT PATi PAT Entry Reset ValuePATi 3-Bit Encodings MTRRs and PAT PAT Memory Type Mtrr Memory Type Effective Memory Type Based on PAT and MTRRs Input Memory Type Final Output Memory Types Attribute Table PAT 181 C7FFF C6FFF C5FFF C4FFF C3FFF C2FFF C1FFF C0FFF 7FFFF 6FFFF 5FFFF 4FFFF 3FFFF 2FFFF 1FFFF 0FFFF9FFFF 9BFFF 97FFF 93FFF 8FFFF 8BFFF 87FFF 83FFF Bffff Bbfff B7FFF B3FFF Affff Abfff A7FFF A3FFF Attribute Table PAT 183 MTRRphysMaskn Register Format MTRR-Related Model-Specific Register MSR Map 186 Instruction Dispatch Execution Resources Appendix F AAM Integer InstructionsAAA AAD ADC mreg16/32, reg16/32 ModR/M Decode ByteADC mreg8, reg8 ADC mem8, reg8 Bswap EDX BoundBswap EAX Bswap ECX CLI CBW/CWDECLC CLD CMOVG/CMOVNLE reg16/32, mem16/32 0Fh CMOVE/CMOVZ reg16/32, reg16/32 0FhCMOVE/CMOVZ reg16/32, mem16/32 0Fh CMOVG/CMOVNLE reg16/32, reg16/32 0Fh DAS CpuidCWD/CDQ DAA EAX, DX EnterAL, DX AX, DX Invlpg Invd Leave Lahf LSL reg16/32, mem16/32 0Fh 03h LOOPE/LOOPZ disp8 E1hLOOPNE/LOOPNZ disp8 E0h LSL reg16/32, mreg16/32 0Fh 03h NOP Xchg EAX, EAX POP ES OUT DX, ALOUT DX, AX OUT DX, EAX POP ESI POP EBXPOP ESP POP EBP Rdpmc RdmsrRdtsc Sahf SBB reg8, mem8 SBB mreg16/32, reg16/32SBB mem16/32, reg16/32 SBB reg8, mreg8 Setns mem8 Sets mreg8Sets mem8 Setns mreg8 STD STCSTI Sysret SyscallSysenter Sysexit Xchg EAX, EBX Xchg EAX, EAXXchg EAX, ECX Xchg EAX, EDX Emms MMX Instructions Pcmpeqb mmreg1, mmreg2 Pandn mmreg1, mmreg2DFh Pandn mmreg, mem64 FADD/FMUL FPU MMX Extensions Floating-Point Instructions Fcos FcomppFdecstp Finit FincstpFLD1 FLDLN2 FLDL2EFLDL2T FLDLG2 Fucomp Fstsw AXFtst Fucom Femms DNow! Instructions DNow! Extensions DirectPath Instructions DirectPath versus VectorPath Instructions BT mreg16/32, reg16/32 BT mreg16/32, imm8 BT mem16/32, imm8 DirectPath Integer InstructionsCBW/CWDE CLC CMC INC mreg8 INC mem8 INC mreg16/32 INC mem16/32 JO short disp8 DEC mreg8 DEC mem8 DEC mreg16/32 DEC mem16/32 222 DirectPath Instructions DirectPath Instructions 223 224 DirectPath Instructions Wait Xchg EAX, EAX 226 DirectPath Instructions DirectPath MMX Instructions DirectPath MMX Extensions FLD1 FLDL2E FLDL2T FLDLG2 FLDLN2 Fldpi Fldz DirectPath Floating-Point InstructionsFcompp Fdecstp Fist mem16int Ftst Fucom Fucomp Fucompp Fwait Fxch CLD CLI Clts VectorPath InstructionsVectorPath Integer Instructions AAA AAD AAM AAS AL, DX AX, DX EAX, DX Invd Invlpg Instruction Mnemonic DIV EAX, mem16/32 Rdmsr Rdpmc Rdtsc POP mreg 16/32 POP mem 16/32Push mreg16/32 Push mem16/32 PUSHA/PUSHAD PUSHF/PUSHFD Wbinvd Wrmsr VectorPath MMX InstructionsVectorPath MMX Extensions Syscall Sysenter Sysexit Sysret Fxam Fxtract FYL2X FYL2XP1 VectorPath Floating-Point InstructionsFptan Fpatan Frndint Fscale Fsin Fsincos 236 Index 238 Index Index 239 240 Index