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CHAPTER 7 THE FLOATING-POINT PROCESSOR In this chapter, we'll refer to the various Central Processing Units (CPUs) as the "86". Thus "86" refers to either the 8088, 8086, 80186, 80286, etc. We'll refer to the various coprocessors as the "87". Thus "87" refers to either the 8087 or the 287. The 8087 and 287 Coprocessors All IBM-PC's, and most clones, contain a socket for a floating point coprocessor. If you shell out between $80 and $300, and plug the appropriate chip into that socket, then a host of floating point instructions is added to the assembly language instruction set. The original IBM-PC, and the XT, accept the original floating point chip, the 8087. The AT accepts a later update, the 287. From a programming standpoint, the two chips are nearly identical: the 287 adds the instructions FSETPM and FSTSW AX, and ignores the instructions FENI and FDISI. There is, however, a rather nasty design flaw in the 8087, that was corrected in the 287. To understand the flaw, you must understand how the 86 and 87 work as coprocessors. Whenever the 86 sees a floating point instruction, it communicates the instruction, and any associated memory operands, to the 87. Then the 86 goes on to its next instruction, operating in parallel with the 87. That's OK, so long as the following instructions don't do one of the following: 1. Execute another floating point instruction; or 2. Try to read the results of the still-executing floating point instruction. If they do, then you must provide an instruction called WAIT (or synonymously FWAIT), which halts the 86 until the 87 is finished. For almost all floating point instructions, it should not be necessary to provide an explicit FWAIT; the 86 ought to know that it should wait. For the 8087, it IS necessary to give an explicit FWAIT before each floating point instruction: that is the flaw. Because of the flaw, all assemblers supporting the 8087 will silently insert an FWAIT code (hex 9B) before all 87 instructions, except those few (the FN instructions other than FNOP) not requiring the FWAIT. A86 provides the switch +F (the F must be capitalized), to signal that the 287 is the target processor. A86 also provides the directive ".287", compatible with Microsoft's assembler, that you can insert into your programs to accomplish the same thing as +F. However, the actions taken by A86 and Microsoft when seeing .287 are completely disjoint! To wit: * A86 ceases outputting FWAIT directives that are unnecessary for the 287. For reasons beyond my comprehension, Microsoft continues to put them out. Can someone enlighten me as to why Microsoft is putting out those codes? 7-2 * A86 ignores the instructions FENI, FDISI, FNENI, and FNDISI after it sees a .287 directive. Microsoft continues to assemble these instructions. * Microsoft recognizes the new 287 instructions, if and only if it sees the .287 directive. A86 recognizes them even if .287 is not given. In general, I don't attempt to police your instruction usage-- if you use an instruction available on a limited number of processors, I trust that you are programming for one of those processors. In summary, if your program will be running only on machines with a 287, you can give ".287" directive. Your programs will be significantly shorter than if they were assembled by Microsoft. If you want your programs to run on all machines containing a floating point chip, you should refrain from specifying .287. WARNING: The most common mistake 87 programmers make is to try to read the results of an 87 operation in 86 memory, before the results are ready. At least on my AT, the system often crashes when you do this! If your program runs correctly when single stepped, but crashes when set loose, then chances are you need an extra explicit FWAIT somewhere. Emulating the 8087 by Software There is a software package provided with many compilers (Borland's Turbo C and most Microsoft compilers, for example) that emulates the 8087 instruction set. The emulator is very cleverly implemented so that the programmer need not know whether a floating point chip will be available, or whether emulation will be necessary. This is done by having the linker replace all floating point machine instructions with INT calls to certain interrupts, dedicated to emulation. The interrupt handlers interpret the operands to the instructions, and emulate the 8087. You can tell A86 that the emulator might be used, by providing a +f switch in the invocation line, or in the A86 environment variable (make sure the f is lower case). Since your program will be linked to the emulator, you must be producing an OBJ file, not a COM file, for emulation support to take effect. Whenever a floating point instruction is assembled, A86 will generate an external reference at the opcode for the instruction. Then, if the emulation package is linked with your program, the opcodes will be replaced by the INT calls. If a special non- emulation module is linked, the opcodes will be left alone, and the floating point instructions will be executed directly. 7-3 The Floating Point Stack The 87 has its own register set, of 8 floating point numbers occupying 10 bytes each, plus 14 bytes of status and control information. Many of the 87's instructions cause the numbers to act like a stack, much like a Hewlett-Packard calculator. For this reason, the numbers are called the floating point stack. The standard name for the top element of the floating point stack is either ST or ST(0); the others are named ST(1) through ST(7). Thus, for example, the instruction to add stack element number 3 into the top stack element is usually coded FADD ST,ST(3). I find this notation painfully verbose. Especially bad are the parentheses, which are hard to type, and which add visual clutter to the program. To alleviate this problem while retaining language compatibility, I name my stack elements simply 0 through 7. I recognize ST as a synonym for 0. I allow expression elements to be concatenated; concatenation is the same as addition. Thus, when A86 sees ST(3), it computes 0+3 = 3. So you can code the old way, FADD ST,ST(3), or you can code the concise way, FADD 0,3 or simply FADD 3. Floating Point Initializations In general, you use the 87 by loading numbers from 86 memory to the 87 stack (using FLD instructions), calculating on the 87 stack, and storing the results back to 86 memory (using FST and FSTP instructions). There are seven constant numbers built into the 87 instruction set: zero, one, Pi, and four logarithmic conversion constants. These can be loaded using the FLD0, FLD1, FLDPI, FLDL2T, FLDL2E, FLDLG2, and FLDLN2 instructions. All other constants must be declared in, then loaded from, 86 memory. Integer constant words and doublewords can be loaded via FILD. Non-integer constant doubleword, quadwords, and ten-byte numbers can be loaded via FLD. A86 allows you to declare constants loaded via FLD as floating point numbers, using scientific notation if you like. As an exclusive feature, A86 allows you to use any of the 4 arithmetic functions +, -, *, / in expressions involving floating point numbers. A86 will even do type conversion if one of the two operands is given as an integer; though for clarity I recommend that you always give floating point constants with their decimal point. Built-In Constant Names A86 offers another exclusive feature: the built-in symbols PI ratio of circumference to diameter of a circle L2T log base 2 of 10 L2E log base 2 of the calculus constant e = 2.71828... LG2 log base 10 of 2 LN2 natural log (base e) of 2 You can use these symbols in expressions, to declare useful constants. For example, you can declare the degrees-to-radians conversion constant: 7-4 DEG_TO_RAD DT PI/180. Special Immediate FLD Form Yet another exclusive A86 feature is the instruction form FLD constant. This form is intended primarily to facilitate "fooling around" with the 87 when using D86; but it is also useful for quick-and-dirty programs. For example, the instruction FLD 12.3 generates the following sequence of code bytes (without explicitly using the local labels given): CS FLD T[M1] JMP >M2 M1 DT 12.3 M2: Obviously, this form is not terrifically efficient: you can always save the JMP by placing the constant outside of the instruction stream; and the CS override might not be needed. But the form is very, very convenient! NOTE that the preceding 2 sections imply that you can get careless and code, for example, FLD PI when you intended FLDPI. Though the two are functionally equivalent, the first form takes a whopping 17 bytes; and second, only 2 bytes. Be careful! Floating Point Operand Types The list of floating point instructions contains a variety of operand types. Here is a brief explanation of those types: 0 stands for the top element of the floating point stack. A synonym for 0 is ST or ST(0). i stands for element number i of the floating point stack. i can range from 0 through 7. A synonym for i is ST(i). mem10r is a 10-byte memory quantity (typically declared with a DT directive) containing a full precision floating point number. Intel recommends that you NOT store your numbers in full precision; that you use the following double precision format instead. Full precision numbers are intended for storage of intermediate results (on the stack); they exist to insure maximum accuracy for calculations on double precision numbers, which is the official external format of 87 numbers. 7-5 mem8r is an 8-byte memory quantity (typically declared with a DQ directive) containing a double precision floating point number. This is the best format for floating point numbers on the 87. The 87 takes the same amount of time on double precision calculations as it does on single precision. The only extra time is the memory access of 4 more bytes; negligible in comparison to the calculation time. mem4r is a 4-byte quantity (typically defined with a DD directive) containing a single precision floating point number. mem10d is a 10-byte quantity (also defined via DT) containing a special Binary Coded Decimal format recognized by the FBLD and FBSTP instructions. This format is useful for input and output of floating point numbers. mem4i is a 4-byte quantity representing a signed integer in two's-complement notation. mem2i is a 2-byte quantity representing a signed integer in two's-complement notation. mem14 and mem94 are 14- and 94-byte buffers containing the 87 machine state. Operand Choices in A86 In the "standard" assembly language, the choice of operands for floating point instructions seems inconsistent to me. For example, to subtract stack i from 0, you must provide two operands; to do the equivalent comparison, you must provide only one operand. A86 smooths out these inconsistencies by allowing more choices for operands: FADD i is equivalent to FADD 0,i. FCOM 0,i is equivalent to FCOM i. The same holds for the other main arithmetic instructions. FXCH 0,i and FXCH i,0 are allowed. So if you wish to retain compatibility with other assemblers, you should use their more restrictive instruction list, not the following one. The 87 Instruction Set Following is the 87 instruction set. The "w" in the opcode field is the FWAIT opcode, hex 9B, which is suppressed if .287 is selected. Again, "0", "1", and "i" stand for the associated floating point stack registers, not constant numbers! Constant numbers in the descriptions are given with decimal points: 0.0, 1.0, 2.0, 10.0. 7-6 Opcode Instruction Description w D9 F0 F2XM1 0 := (2.0 ** 0) - 1.0 w D9 E1 FABS 0 := |0| w DE C1 FADD 1 := 1 + 0, pop w D8 C0+i FADD i 0 := i + 0 w DC C0+i FADD i,0 i := i + 0 w D8 C0+i FADD 0,i 0 := i + 0 w D8 /0 FADD mem4r 0 := 0 + mem4r w DC /0 FADD mem8r 0 := 0 + mem8r w DE C0+i FADDP i,0 i := i + 0, pop w DF /4 FBLD mem10d push, 0 := mem10d w DF /6 FBSTP mem10d mem10d := 0, pop w D9 E0 FCHS 0 := -0 9B DB E2 FCLEX clear exceptions w D8 D1 FCOM compare 0 - 1 w D8 D0+i FCOM 0,i compare 0 - i w D8 D0+i FCOM i compare 0 - i w D8 /2 FCOM mem4r compare 0 - mem4r w DC /2 FCOM mem8r compare 0 - mem8r w D8 D9 FCOMP compare 0 - 1, pop w D8 D8+i FCOMP 0,i compare 0 - i, pop w D8 D8+i FCOMP i compare 0 - i, pop w D8 /3 FCOMP mem4r compare 0 - mem4r, pop w DC /3 FCOMP mem8r compare 0 - mem8r, pop w DE D9 FCOMPP compare 0 - 1, pop both w D9 F6 FDECSTP decrement stack pointer w DB E1 FDISI disable interrupts (.287 ignore) w DE F9 FDIV 1 := 1 / 0, pop w D8 F0+i FDIV i 0 := 0 / i w DC F8+i FDIV i,0 i := i / 0 w D8 F0+i FDIV 0,i 0 := 0 / i w D8 /6 FDIV mem4r 0 := 0 / mem4r w DC /6 FDIV mem8r 0 := 0 / mem8r w DE F8+i FDIVP i,0 i := i / 0, pop w DE F1 FDIVR 1 := 0 / 1, pop w D8 F8+i FDIVR i 0 := i / 0 w DC F0+i FDIVR i,0 i := 0 / i w D8 F8+i FDIVR 0,i 0 := i / 0 w D8 /7 FDIVR mem4r 0 := mem4r / 0 w DC /7 FDIVR mem8r 0 := mem8r / 0 w DE F0+i FDIVRP i,0 i := 0 / i, pop w DB E0 FENI enable interrupts (.287 ignore) w DD C0+i FFREE i empty i w DE /0 FIADD mem2i 0 := 0 + mem4i w DA /0 FIADD mem4i 0 := 0 + mem2i w DE /2 FICOM mem2i compare 0 - mem2i w DA /2 FICOM mem4i compare 0 - mem4i w DE /3 FICOMP mem2i compare 0 - mem2i, pop w DA /3 FICOMP mem4i compare 0 - mem4i, pop 7-7 w DE /6 FIDIV mem2i 0 := 0 / mem2i w DA /6 FIDIV mem4i 0 := 0 / mem4i w DE /7 FIDIVR mem2i 0 := mem2i / 0 w DA /7 FIDIVR mem4i 0 := mem4i / 0 w DF /0 FILD mem2i push, 0 := mem2i w DB /0 FILD mem4i push, 0 := mem4i w DF /5 FILD mem8i push, 0 := mem8i w DE /1 FIMUL mem2i 0 := 0 * mem2i w DA /1 FIMUL mem4i 0 := 0 * mem4i w D9 F7 FINCSTP increment stack pointer 9B DB E3 FINIT initialize 87 w DF /2 FIST mem2i mem2i := 0 w DB /2 FIST mem4i mem4i := 0 w DF /3 FISTP mem2i mem2i := 0, pop w DB /3 FISTP mem4i mem4i := 0, pop w DF /7 FISTP mem8i mem8i := 0, pop w DE /4 FISUB mem2i 0 := 0 - mem2i w DA /4 FISUB mem4i 0 := 0 - mem4i w DE /5 FISUBR mem2i 0 := mem2i - 0 w DA /5 FISUBR mem4i 0 := mem4i - 0 w D9 C0+i FLD i push, 0 := old i w DB /5 FLD mem10r push, 0 := mem10r w D9 /0 FLD mem4r push, 0 := mem4r w DD /0 FLD mem8r push, 0 := mem8r w D9 E8 FLD1 push, 0 := 1.0 w D9 /5 FLDCW mem2i control word := mem2i w D9 /4 FLDENV mem14 environment := mem14 w D9 EA FLDL2E push, 0 := log base 2.0 of e w D9 E9 FLDL2T push, 0 := log base 2.0 of 10.0 w D9 EC FLDLG2 push, 0 := log base 10.0 of 2.0 w D9 ED FLDLN2 push, 0 := log base e of 2.0 w D9 EB FLDPI push, 0 := Pi w D9 EE FLDZ push, 0 := +0.0 w DE C9 FMUL 1 := 1 * 0, pop w D8 C8+i FMUL i 0 := 0 * i w DC C8+i FMUL i,0 i := i * 0 w D8 C8+i FMUL 0,i 0 := 0 * i w D8 /1 FMUL mem4r 0 := 0 * mem4r w DC /1 FMUL mem8r 0 := 0 * mem8r w DE C8+i FMULP i,0 i := i * 0, pop DB E2 FNCLEX nowait clear exceptions DB E1 FNDISI disable interrupts (.287 ignore) DB E0 FNENI enable interrupts (.287 ignore) DB E3 FNINIT nowait initialize 87 w D9 D0 FNOP no operation 7-8 DD /6 FNSAVE mem94 mem94 := 87 state D9 /7 FNSTCW mem2i mem2i := control word D9 /6 FNSTENV mem14 mem14 := environment DF E0 FNSTSW AX AX := status word DD /7 FNSTSW mem2i mem2i := status word w D9 F3 FPATAN 0 := arctan(1/0), pop w D9 F8 FPREM 0 := REPEAT(0 - 1) w D9 F2 FPTAN push, 1/0 := tan(old 0) w D9 FC FRNDINT 0 := round(0) w DD /4 FRSTOR mem94 87 state := mem94 w DD /6 FSAVE mem94 mem94 := 87 state w D9 FD FSCALE 0 := 0 * 2.0 ** 1 9B DB E4 FSETPM set protection mode w D9 FA FSQRT 0 := square root of 0 w DD D0+i FST i i := 0 w D9 /2 FST mem4r mem4r := 0 w DD /2 FST mem8r mem8r := 0 w D9 /7 FSTCW mem2i mem2i := control word w D9 /6 FSTENV mem14 mem14 := environment w DD D8+i FSTP i i := 0, pop w DB /7 FSTP mem10r mem10r := 0, pop w D9 /3 FSTP mem4r mem4r := 0, pop w DD /3 FSTP mem8r mem8r := 0, pop w DF E0 FSTSW AX AX := status word w DD /7 FSTSW mem2i mem2i := status word w DE E9 FSUB 1 := 1 - 0, pop w D8 E0+i FSUB i 0 := 0 - i w DC E8+i FSUB i,0 i := i - 0 w D8 E0+i FSUB 0,i 0 := 0 - i w D8 /4 FSUB mem4r 0 := 0 - mem4r w DC /4 FSUB mem8r 0 := 0 - mem8r w DE E8+i FSUBP i,0 i := i - 0 w DE E1 FSUBR 1 := 0 - 1, pop w D8 E8+i FSUBR i 0 := i - 0 w DC E0+i FSUBR i,0 i := 0 - i w D8 E8+i FSUBR 0,i 0 := i - 0 w D8 /5 FSUBR mem4r 0 := mem4r - 0 w DC /5 FSUBR mem8r 0 := mem8r - 0 w DE E0+i FSUBRP i,0 i := 0 - i, pop w D9 E4 FTST compare 0 - 0.0 9B FWAIT wait for 87 ready w D9 E5 FXAM C3 -- C0 := type of 0 w D9 C9 FXCH exchange 0 and 1 w D9 C8+i FXCH 0,i exchange 0 and i w D9 C8+i FXCH i exchange 0 and i w D9 C8+i FXCH i,0 exchange 0 and i w D9 F4 FXTRACT push, 1 := expo, 0 := sig w D9 F1 FYL2X 0 := 1 * log base 2.0 of 0, pop w D9 F9 FYL2XP1 0 := 1 * log base 2.0 of (0+1.0), pop