x86 Assembly/Floating Point
The ALU 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.
The original x86 family members had a separate math coprocessor that handled floating point arithmetic. The original coprocessor was the 8087, and all FPUs since have been dubbed “x87” chips. Later variants integrated the FPU 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, 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 Register Stack the “x87 Stack”, and the FPU operations will frequently be called the “x87 instruction set”.
The presence of an integrated x87 FPU can be checked using the
; after you have verified ; that the cpuid instruction is indeed available: mov eax, 1 ; argument request feature report cpuid xor rax, rax ; wipe clean accumulator register bt edx, rax ; CF ≔ edx[rax] retrieve bit 0 setc al ; al ≔ CF
FPU Register Stack
The FPU 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 registers.
The x87 stack can only be accessed by FPU 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 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 stack registers are all 80 bits wide.
Most calling conventions return floating point values in the
The following program (using NASM syntax) calculates the square root of 123.45.
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: ; 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 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
- load a constant on top of the stack
fistp: store integer
fisttp: store a truncated integer
fabs: absolute value
fchs: change sign
fxtract: split exponent and significant
fisubr: reverse subtraction
fsqrt: square root
fidiv: division (see also
fdivbug on Wikipedia)
frndint: round to integer
FPU Internal and Other Instructions
finit: initialize FPU
fdecstp: increment or decrement top
ffree: tag a register as free
fcompp: compare floating-point values
ficomp: compare with an integer
fxam: examine a register
fclex: clear exceptions
fwaitdoes the same as
Added in specific processors
Added with 80287
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
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)
ffree st(i)and pops the stack