X86 Assembly/Floating Point

The ALU arithmetic logic unit is only capable of dealing with integer values. While integers are sufficient for some applications, it is often necessary to use decimals. A highly specialized coprocessor, all part of the FPU – the floating-point unit –, will allow you to manipulate numbers with fractional parts. x87 Coprocessor The original x86 family members had a separate math coprocessor that handled floating point arithmetic. The original coprocessor was the 8087, and all FPU floating-point units since have been dubbed “x87” chips. Later variants integrated the FPU floating-point unit into the microprocessor itself. Having the capability to manage floating point numbers means a few things: The microprocessor must have space to store floating point numbers. The microprocessor must have instructions to manipulate floating point numbers. The FPU floating-point unit, even when it is integrated into an x86 chip, is still called the “x87” section. For instance, literature on the subject will frequently call the FPU floating-point unit Register Stack the “x87 Stack”, and the FPU floating-point unit operations will frequently be called the “x87 instruction set”. The presence of an integrated x87 FPU floating-point unit can be checked using the cpuid instruction. ; after you have verified ; that the cpuid instruction is indeed available: mov eax, 1 ; argument request feature report cpuid xor eax, eax ; wipe clean accumulator register bt edx, eax ; CF ≔ edx[eax] retrieve bit 0 setc al ; al ≔ CF FPU Register Stack The FPU floating-point unit has an array of eight registers that can be accessed as a stack. There is one top index indicating the current top of the stack. Pushing or popping items to or from the stack will only change the top index and store or wipe data respectively. st(0) or simply st refers to the register that is currently at the top of the stack. If eight values were stored on the stack, st(7) refers to last element on the stack (i. e. the bottom). Numbers are pushed onto the stack from memory, and are popped off the stack back to memory. There is no instruction allowing to transfer values directly to or from ALU arithmetic logic unit registers. The x87 stack can only be accessed by FPU floating-point unit instructions ‒ you cannot write mov eax, st(0) ‒ it is necessary to store values to memory if you want to print them, for example. FPU floating-point unit instructions generally will pop the first two items off the stack, act on them, and push the answer back on to the top of the stack. Floating point numbers may generally be either 32 bits long, the float data type in the programming language C, or 64 bits long, double in C. However, in order to reduce round-off errors, the FPU floating-point unit stack registers are all 80 bits wide. Most calling conventions return floating point values in the st(0) register. Examples The following program (using NASM syntax) calculates the square root of 123.45. [org 0x7c00] [bits 16] global _start section .data val: dq 123.45 ; define quadword (double precision) section .bss res: resq 1 ; reserve 1 quadword for result section .text _start: ;initilizes the FPU, avoids inconsistent behavior fninit ; load value into st(0) fld qword [val] ; treat val as an address to a qword ; compute square root of st(0) and store the result in st(0) fsqrt ; store st(0) at res, and pop it off the x87 stack fstp qword [res] ; the FPU stack is now empty again ; end of program Essentially, programs that use the FPU floating-point unit load values onto the stack with fld and its variants, perform operations on these values, then store them into memory with one of the forms of fst, most commonly fstp when you are done with x87, to clean up the x87 stack as required by most calling conventions. Here is a more complex example that evaluates the Law of Cosines: ;; c^2 = a^2 + b^2 - cos(C)*2*a*b ;; C is stored in ang global _start section .data a: dq 4.56 ;length of side a b: dq 7.89 ;length of side b ang: dq 1.5 ;opposite angle to side c (around 85.94 degrees) section .bss c: resq 1 ;the result ‒ length of side c section .text _start: fld qword [a] ;load a into st0 fmul st0, st0 ;st0 = a * a = a^2 fld qword [b] ;load b into st0 (pushing the a^2 result up to st1) fmul st0, st0 ;st0 = b * b = b^2, st1 = a^2 faddp ;add and pop, leaving st0 = old_st0 + old_st1 = a^2 + b^2. (st1 is freed / empty now) fld qword [ang] ;load angle into st0. (st1 = a^2 + b^2 which we'll leave alone until later) fcos ;st0 = cos(ang) fmul qword [a] ;st0 = cos(ang) * a fmul qword [b] ;st0 = cos(ang) * a * b fadd st0, st0 ;st0 = cos(ang) * a * b + cos(ang) * a * b = 2(cos(ang) * a * b) fsubp st1, st0 ;st1 = st1 - st0 = (a^2 + b^2) - (2 * a * b * cos(ang)) ;and pop st0 fsqrt ;take square root of st0 = c fstp qword [c] ;store st0 in c and pop, leaving the x87 stack empty again ‒ and we're done! ; don't forget to make an exit system call for your OS, ; or execution will fall off the end and decode whatever garbage bytes are next. mov eax, 1 ; __NR_exit xor ebx, ebx int 0x80 ; i386 Linux sys_exit(0) ;end program Floating-Point Instruction Set You may notice that some of the instructions below differ from another in name by just one letter: a P appended to the end. This suffix signifies that in addition to performing the normal operation, they also Pop the x87 stack after execution is complete. Original 8087 instructions FDISI, FENI, FLDENVW, FLDPI, FNCLEX, FNDISI, FNENI, FNINIT, FNSAVEW, FNSTENVW, FRSTORW, FSAVEW, FSTENVW Data Transfer Instructions fld: load floating-point value fild: load integer fbld fbstp load a constant on top of the stack fld1: fldld2e: fldl2t: fldlg2: flln2: fldz: “positive” fst, fstp fist, fistp: store integer fxch: exchange fisttp: store a truncated integer Arithmetic Instructions fabs: absolute value fchs: change sign fxtract: split exponent and significant fadd, faddp, fiadd: addition fsub, fsubp, fisub: subtraction fsubr, fsubrp, fisubr: reverse subtraction fmul, fmulp, fimul fsqrt: square root fdiv, fdivp, fidiv: division (see also fdiv bug on Wikipedia) fdivr, fdivrp, fidivr fprem: partial remainder fptan fpatan frndint: round to integer fscale: multiply/divide by integral powers of 2 f2xm1: fyl2x: fyl2xp1: FPU Internal and Other Instructions finit: initialize FPU floating-point unit fldcw flenv frstor fsave, fnsave fstcw, fnstcw fstenv, fnstenv fstsw, fnstsw finccstp and fdecstp: increment or decrement top ffree: tag a register as free ftst: test fcom, fcomp, fcompp: compare floating-point values ficom, ficomp: compare with an integer fxam: examine a register fclex: clear exceptions fnop fwait does the same as wait. Added in specific processors Added with 80287 FSETPM Added with 80387 FCOS, FLDENVD, FNSAVED, FNSTENVD, FPREM1, FRSTORD, FSAVED, FSIN, FSINCOS, FSTENVD, FUCOM, FUCOMP, FUCOMPP Added with Pentium Pro FCMOVB, FCMOVBE, FCMOVE, FCMOVNB, FCMOVNBE, FCMOVNE, FCMOVNU, FCMOVU, FCOMI, FCOMIP, FUCOMI, FUCOMIP, FXRSTOR, FXSAVE Added with SSE FXRSTOR, FXSAVE These are also supported on later Pentium IIs which do not contain SSE support Added with SSE3 FISTTP (x87 to integer conversion with truncation regardless of status word) Undocumented instructions ffreep: performs ffree st(i) and pops the stack Further Reading X86 Disassembly/Floating Point Numbers Floating Point