Apple II manual Floating Point Package, Floating Point Representation LOW

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TOPIC -- Apple II -- WOZPAK Floating point routines description

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Wozpak ][, November 1979, pages 109-115.

FLOATING POINT PACKAGE

The mantissa-exponent, or 'floating point' numerical representation is widely used by computers to express values with a wide dynamic range. With floating point representation, the number 7.5 x 10^22 requires no more memory to store than the number 75 does. We have allowed for binary floating point arithmetic on the APPLE ][ computer by providing a useful subroutine package in ROM, which performs the common arithmetic functions. Maximum precision is retained by these routines and overflow conditions such as 'divide by zero' are trapped for the user. The 4-byte floating point number representation is compatible with future APPLE products such as floating point BASIC.

A small amount of memory in Page Zero is dedicated to the floating point workspace, including the two floating-point accumulators, FP1 and FP2. After placing operands in these accumulators, the user calls subroutines in the ROM which perform the desired arithmetic operations, leaving results in FP1. Should an overflow condition occur, a jump to location $3F5 is executed, allowing a user routine to take appropriate action.

FLOATING POINT REPRESENTATION

 

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HI

LOW

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Exponent

 

Signed Mantissa

 

1.Mantissa

The floating point mantissa is stored in two's complement representation with the sign at the most significant bit (MSB) position of the high-order mantissa byte. The mantissa provides 24 bits of precision, including sign, and can represent 24-bit integers precisely. Extending precision is simply a matter of adding bytes at the low order end of the mantissa.

Except for magnitudes less than 2^-128 (which lose precision) mantissa are normalized by the floating point routines to retain maximum precision.

That is, the numbers are adjusted so that the upper two high-order mantissa bits are unequal.

HIGH-ORDER MANTISSA BYTE 01.XXXXXX Positive mantissa. 10.XXXXXX Negative mantissa. 00.XXXXXX Unnormalized mantissa. 11.XXXXXX Exponent = -128.

2.Exponent.

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Contents Apple II Original ROM Information Data Qjjj0005A0 FDA9A920 EDFDA98D 4CEDFDA9 8D4CEDFD Page Page Page JI@ Page 001AE0 E8EEE7E8 Efefeeef Eeefeeee Efeeeeee 001AA0 F3FFE8E1 E8E8EFEB FFFFE0FF Ffefeeef001AC0 FFFFE1E1 EFEEE7E8 EEE7F3FB FBEEE1EF 001DB0 9CDD9CDE DD9EC3DD Cfcacdcb 00479AAD 001FB0 Ffffffff FFFF2071 E14CBFEF 2003EEA9 Page DDBAF9F0 0DBDBAF9 FA60FFFF02E61FBD JJJ 60FFFFFF .JJAEAEA8AD Page Page Return Page System Monitor Copyright LOC0LOC1 BaslMask INY NO, INC Index X-COORD LDY PCH PRMN1 LDA FMT1 DFB Format Above EOR STA A3H FAE2 A2 FB Tabv STA Vtabs to ROW in A-REG STA Bash BCC Clreol Clear to END of Line NXTA4 INC A4L Incr 2-BYTE A4 STA BASL,Y Replace Flashing Screen Fdad A5 3C LT2 LDA A2L,X Copy A2 2 Bytes to Ioprt LDA A2L SET RAM IN/OUT Vectors BCC RD3 Loop Until Done STA A3H,X DFB LIST-1 FFF3 CC Save BRTBL,XLSR Prepare Carry for BC, BNC Stat LDA R0L JSR STAT2 Push LOW PC Byte VIA R12 LDA IN,Y CMPBNE Nomove JSRMloop DCRBNZ Mloop RTNINR BNCBNZ BM1Example Load IndirectStore Indirect Load DOUBLE-BYTE IndirectStore POP Indirect Store DOUBLE-BYTE IndirectPOP Indirect POP DOUBLE-BYTE Indirect ACCSubtract Compare BF A0 SET $A0BFLOOP1 SUB BNC LOOP1Return to 6502 Mode DecrementLOOP2 BNZ LOOP2Branch if no Carry Branch if Carry SETBranch if Plus LOOP3 SUBBranch if Minus ONE Branch if not Minus ONEBreak Return from Sweet 16 SubroutineSTA IND JMP LDA #ADRHSTA IND+1 LDA OPTBL,X Page Page Sweet 16 Introduction Page Source DSP Print Dbload Sweet REM Ctrl DInput Enter String a $ , a $ NVAPoke 776 , a REM Poke Destination Print Print GotoCall -605 Return REM XAM8 in Monitor SweetMINI-ASSEMBLER BaumMneml MnemrSBC PCL Form ADDR-PC-2 JSR Cout Position FORM7 STX A1H Save Index ORG CLCSTA BNENorm LDA EXP1 ZERO? Shift Dividend Left Floating Point Package Floating Point Representation LOWStored High LOW EXP MantissaFloating Point Representation Examples Number Exponent MantissaFloating Point Subroutine Descriptions Page Page Page Page STY Init MANT1 LDY STYCalling Sequence HIGH-ORDERPage Page Page Page Page Page Fsub SeeeeeeeMmmmmmmm ORG SET Base page Adresses STA X2,X Common LOG of MANT/EXP1 Result in MANT/EXP1 Ovflw BRK Overflow 1EBB STA X2,X Load EXP/MANT2 with Z 1EBD DEX 1EBE BPL LF3 YES,RETURN with MANT1 Normalized Right Shift MANT1 or Swap with MANT2 on CarryCMP AddendNormx BCC Norm If EXEN, Normalize PRODUCT, Else Complement LDA STA Errata for Rankins Floating Point RoutinesLDX Load X for High Byte of Exponent LOG FaddFsub FmulPage Mantissa MOD 9/76 Load X for Later M2MHL Contin 1EAD LFA LDA Z,X L2E DCM Normx RTLOG1Mdend Object Code Dump Page Tapdel HsbdryToutfl LstcomOutvec Urcvec=UBRKVC =UIRQVCTAX ADCORA Stocom STABPL HASHUS+2 Restore Except A,F BEQ STXCPX JMPExwrap RTS Chksad PHPOutxah PHA TXANibasc and #$0F Nibble in a to Ascii in a BCS INST1 RTI Ldbyte JSR Inbyte Verify MEM E1PARM JMP STD2 MORED2 LDY JSR Brtt INC Ercnt UP to FF MEM3EX CLC SBC P3L BNE Exitgk Ksconf LDA Txtmov LDA BNE INRT2 Outone Dftxfr LDX Newdev LDALDA Toutfl 8BFF 0D Msfirst LOADT2 BufadhDdrdig STA Mode BNE Lcerr Last Char not Kgettr LDA H8DFF BCC PACKT3 Dumpt DUMPT1 TXADMPT1A LDA DUMPT4 LDYOutbth LDX Register Name Patch Outcht STXHS Tape Boundary Rockwell International UdrahUddra NowlnOldlen LengthMovad TMASK1 =MOVADCurpos ChecksumTabuff TABUF2Write Edge Detect Control not Used Because KB CRB EQSDB TO,EQS CFD2A0A0A0A0SEI Disable Interrupt RS6 JSR Crlow Clear Display E1AC Have Valid Command Nxtadd TYA Space CommFor LOAD1A LDALOADK5 LDX JMP LOADK5 Next E46A EE INC Bkflg SET FLG Outcks LDA OUTCK1 PHADumpta LDX DUMPT1 LDAE5EA TOGTA1 LDA TOGTA2 LDA BCC JD3 MSG Whereo LDY ANY Other CPYSTA Outflg Output PHA Outall PHAROL Priflg Restore FLG Wrax JSR Numa EAA3 CA DEX Memerr JSR STA STIY+2 Dehalf LDA EC5E 4A LSR Only ROW Onekey LDA Debkey LDXED3B GET a Char from Tape Subroutine Edba LSR Cpiy Make Room for MSB Outtty PHA OUTDD1 PHA OUTD1A CPXOUTD2A TXA LDA First Chip Select Prndot LDA LDA #PRTIME/256 Start T2 for 1.7 Msec BIT Imask See if DOT is SET NEW Print ROW Newcol LDACkbuff LDA Bkcksm LDAOUTTA1 PHA OUTTA2 LDXF2E1 3E7E7F3E7F7FCOL0 Zero Leftmost ColumnDB BGYU76HN Rtmode Getfmt TAXMNNDX1 LSR MNNDX2 LSRPRADR4 DEX PRADR2 LDAPLA BPL PCADJ4 AIM Text Editor LDA END+1 CMP with END AF E7 JMP KEP RUB COMMAND-FIND String JMP CHN1 See if WE RAN Past END of Buffer Limit BEQ REP2 SBC Oldlen GOGO1 RTS Startm JSR ROM TableFB46 8A9AAABACADA Rdaddr LDY Tryjmp LDA IMMED1 LDXConvrt LDA OPCMP1 LDABrnchc JMP Compbr LDX FORMD2 LDXBrcomp LDA Backwd LDA Forwrd LDAMATCH1 JMP Curead TXAPATCH4 LDX CURPO2 Dont do Anything if 8D PATC15 JSR Crlf Decode Command Addblk ADDS1ABkcksm BrcompCRA Goerr Namo E8CF NAMO1 RD2 EA5D RD1 AIM 65 Microcomputer Monitor Program Listing E6ACIntroduction Table of ContentsAIM 65 Microcomputer Basic Language Reference Manual Installing Basic in the AIM ROM Installation Procedure Entering BasicMemory SIZE? WIDTH?Exiting Basic Reentering BasicBasic Cursor Printer ControlCHR$ Getting Started with Basic Basic Command SETASC Direct and Indirect Commands Direct Commands Indirect CommandsOperating on Programs and Lines Listing a ProgramDeleting a Line Replacing a LineDeleting a Program List Printing DataNumber Output Format Print ONE Half Equals 1/2Number Format Variable Names Variables Assigning Variables with AN Input StatementInput R Reserved Words PSTG$Print A, A*2 LET Z=7 Print Z, Z-A Program Using Relations RemarksRelational Tests Input BLooping Square Root ProgramAN Improved Square Root Program Print NsqrnAnother Square Root Program Print Nsqrn Next NCOUNT-BACKWARD Program Some Other Looping OperationsPrint What is the Number Print I,J Next Next JMatrix Operations Sort ProgramPrint BAD GUESS, TRY Again Restore Input NPrint Must be Integer Stopping a ProgramStrings $=ROCKWELL R6500 Print A$ Rockwell R6500LEN Function Print LENA$,LENMICROCOMPUTERRIGHT$ Function MID$ FunctionPrint B$ Basic for Rockwell R6500 Print C$ Basic FOR-ROCKWELL-R6500 VAL and Strs FunctionsCHR$ Function Data AIM 65,DOGAdditional String Considerations Name Example PURPOSE/USE DIMInput Input X$ Print Print X$Operators Symbol Sample Statement ESCCntl Print Print X/1.3Relational Operators Symbol Sample Statement PURPOSE/USENegation Not If not Q3 then Operator Argument Result NotCommands Print Intruder AlertAlert Statement SYNTAX/FUNCTION Example ClearStatement SYNTAX/FUNCTION FRE Statement SYNTAX/FUNCTION ListStatement SYNTAX/FUNCTION Load Statement SYNTAX/FUNCTION NEWStatement SYNTAX/FUNCTION Example RUN Statement SYNTAX/FUNCTION Example SaveProgram Statements Statement SYNTAX/FUNCTION Example DEFStatement SYNTAX/FUNCTION For Statement SYNTAX/FUNCTION DIMStatement SYNTAX/FUNCTION END Statement SYNTAX/FUNCTION Gosub Statement SYNTAX/FUNCTION GotoStatement SYNTAX/FUNCTION IF...GOTO Statement SYNTAX/FUNCTION IF...THENIf X0 then Print ERROR, X Negative Goto Example Statement SYNTAX/FUNCTION LETStatement SYNTAX/FUNCTION Next Statement SYNTAX/FUNCTION ON...GOSUBStatement SYNTAX/FUNCTION Restore Symbol SYNTAX/FUNCTION WaitExample Restore Return Statement SYNTAX/FUNCTION ReturnStatement SYNTAX/FUNCTION Input Statement SYNTAX/FUNCTION POSStatement SYNTAX/FUNCTION Print Print Value ISAStatement SYNTAX/FUNCTION Read Statement SYNTAX/FUNCTION SPCStatement SYNTAX/FUNCTION TAB String Functions Statement SYNTAX/FUNCTION ASCStatement SYNTAX/FUNCTION LEFT$ Statement SYNTAX/FUNCTION LENStatement SYNTAX/FUNCTION MID$ Statement SYNTAX/FUNCTION RIGHT$Arithmetic Functions Statement SYNTAX/FUNCTION ABS Statement SYNTAX/FUNCTION ATNStatement SYNTAX/FUNCTION COS Statement SYNTAX/FUNCTION EXPStatement SYNTAX/FUNCTION Example SGN Print SgnxStatement SYNTAX/FUNCTION Example SIN Print SinxInverse Hyperbolic Secant Error Messages?XX Error PRINTX,Y,Z TAB, SPCSpace Hints Storage Allocation Information Speed HintsThis is Probably the Most Important Speed Hint Converting Basic Programs not Written for AIM 65 BasicOLD AIM OLDAssembly Language Subroutines Ascii Character Codes DecimalNUL SOHAddress Content $AEInputnumbern PrintxStoring AIM 65 Basic Programs on Cassette Recording on Cassette Using the Basic Save CommandRUN NUMBER? Cassette Operations Using the AIM 65 Editor OUT=T F=OUT=T F=FNAME T= IN=ATN Implementation Entered by Alter Memory MOF9C 0FACWIDTH? Bytes Free AIM 65 Basic Saving ATN Object Code on Cassette
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II specifications

The Apple II, launched in April 1977, was one of the first highly successful mass-produced microcomputer products. It marked a significant leap in personal computing, setting standards for future developments in the industry. Created by Steve Wozniak and Steve Jobs, the Apple II differentiated itself with its user-friendly design, appealing aesthetics, and robust capabilities.

One of the standout features of the Apple II was its open architecture, which allowed users to expand and enhance the computer's functionality. This design enabled hundreds of third-party hardware and software developers to contribute to its ecosystem, resulting in an array of peripherals, including printers, modems, and storage devices. The Apple II utilized a MOS Technology 6502 microprocessor running at a clock speed of 1 MHz. Initially equipped with 4 KB of RAM, the machine could be expanded to 48 KB, accommodating more complex applications and programs.

The Apple II was also notable for its colorful graphics. It was one of the first computers to support color display, offering a 6-color palette with a resolution of 280x192 pixels in 16 colors when using its Color Graphics Card. This feature significantly enhanced the visual appeal of games and educational software developed for the platform, making computing more accessible and entertaining for various audiences.

Apple's commitment to user experience was evident in the design of the machine. It featured an integrated keyboard and a plastic case, which was both durable and visually appealing. The self-contained design included drive bays for floppy disk drives, allowing for quicker data access than traditional tape drives. It also supported audio output, enabling sound effects and music, a novelty at the time.

The introduction of the Apple DOS operating system further underscored the machine's capabilities. DOS streamlined file management and made it easier for users to navigate and manage their data. The combination of hardware and software positioned the Apple II as an educational tool and a gaming platform, fostering a vibrant software ecosystem.

The Apple II family continued to evolve, with variations like the Apple II+, IIe, and IIgs being introduced over the years. These iterations brought enhancements in memory, processing power, and graphics capabilities. The legacy of the Apple II endures, not only as a foundational product in personal computing but also as a symbol of innovation that paved the way for future advancements in technology. Its impact is still felt today, as it inspired countless developers and shaped the trajectory of the computer industry.
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