Floating-point Opcodes

The FPU offers a lot of operations not available to classic x86 CPU, like SIN, COS, TAN, EXP, SQRT, LN and so on. SIMPLY FPU by Raymond Filiatreault has a compact overview of all FPU commands. Usage and communication with the FPU is a bit uncommon and takes a bit to get used to. It's recommended to read the creation of the snippet we want to modify first, this is how it looks like originally :

cwd             	; "clear" DX for perfect alignment
mov 	al,0x13
X: 		int 0x10	; set video mode AND draw pixel
mov 	ax,cx		; get column in AH
add		ax,di		; offset by framecounter	          <-- REPLACE THIS WITH FPU CODE
xor 	al,ah		; the famous XOR pattern
and 	al,32+8		; a more interesting variation of it
mov 	ah,0x0C		; set subfunction "set pixel" for int 0x10
loop 	X			; loop 65536 times
inc 	di			; increment framecounter
in 		al,0x60		; check keyboard ...
dec 	al			; ... for ESC
jnz 	X			; rinse and repeat
ret					; quit program

and this is how it looks if we replace the instruction with FPU code :

cwd             	; "clear" DX for perfect alignment
mov 	al,0x13
X: 		int 0x10	; set video mode AND draw pixel
mov 	ax,cx		; get column in AH

fninit				; init FPU first
mov		[si],ax		; write first addend to a memory location
fild	word [si]	; F(pu) I(nteger) L(oad)D a WORD from memory location to the FPU stack
mov		[si],di		; write second addend to a memory location
fiadd	word [si]	; Directly add the word in the memory location to the top FPU stack
fist	word [si]	; F(pu) I(nteger) ST(ore) the result into a memory location
mov		ax,[si]		; Get the word from the memory location into AX

xor 	al,ah		; the famous XOR pattern
and 	al,32+8		; a more interesting variation of it
mov 	ah,0x0C		; set subfunction "set pixel" for int 0x10
loop 	X			; loop 65536 times
inc 	di			; increment framecounter
in 		al,0x60		; check keyboard ...
dec 	al			; ... for ESC
jnz 	X			; rinse and repeat
ret					; quit program

The usual interaction with the FPU is as follows

• F(N)INIT : Initialization of the FPU
• store register content in memory location(s)
• transfer from memory location onto FPU stack
• actual calculations on the FPU (more on this soon)
• transfer from FPU stack into memory location(s)
• get register from memory location

That would be a lot for a single integer addition, but once more complex floating point operations are involved, it starts to pay off. For more advanced FPU operation, let's start from scratch with an unoptimized program which plots the distance of each pixel to the screen center as color, in 49 bytes.

push 	0a000h			
pop 	es				; get start of video memory in ES
mov 	al,0x13			; switch to video mode 13h
int 	0x10			; 320 * 200 in 256 colors
fninit					; -	
						; it's useful to comment what's on the
						; stack after each FPU operation
						; to not get lost ;) start is : empty (-)
X:
xor 	dx,dx			; reset the high word before division
mov 	bx,320			; 320 columns
mov 	ax,di			; get screen pointer in AX
div 	bx				; construct X,Y from screen pointer into AX,DX
sub 	ax,100			; subtract the origin
sub 	dx,160			; = (160,100) ... center of 320x200 screen	
mov 	[si],ax			; move X into a memory location
fild 	word [si]		; X
fmul 	st0				; X²
mov 	[si],dx			; move Y into a memory location
fild 	word [si]		; Y X²
fmul 	st0				; Y² X²
fadd 	st0,st1			; Y²+X²
fsqrt					; R
fistp 	word [si]		; -
mov 	ax,[si]			; get the result from memory
stosb					; write to screen (DI) and increment DI
jmp short X				; next pixel

A few words on this :

• The FPU registers (st0, st1, ...) are organized as a stack. When you load something to the FPU, everything else will be moved one location further away from the top (implicitly!) Some FPU instructions work only on the top, other allow the explicit parametrization with arbitrary FPU registers.
• Depending on what you do, sometimes F(N)INIT can be omitted. Real hardware will refuse to work more often than emulators, but it's always worth the try.
• Accessing memory (size) efficiently can be a real pain. The safest way is to reference absolute memory locations (f.e [1234]) but that's two bytes more per instruction than referencing memory with [BX], [SI], [BX+SI], [BP+DI], [BP+SI], [DI] or [BX+DI]. When working with FPU and this classic approach of FPU communication, you have to design your codeflow to have one or some of these locations available.
• Accessing the memory is always with regard to the segment register DS unless you perform segment overrides. When accessing memory with [BP+??] be aware that this access memory in regard to the segment register SS (see here, at 4.6.2.2 The Register Indirect Addressing Modes
• There are a few conventions which help you identify FPU commands. "i" stands for integer (WORD or DWORD), "p" means "pop stack afterwards", so FST means just "store" while FISTP means "store as integer, then pop the stack"

Now let's unleash the state of the art sizecoding arsenal onto this, to bring it down to 37 bytes (40 bytes with aspect correction)

Distant to center optimized

push 	0a000h - 70		; modified to center to 160,100
aas						; aspect ratio constant part
pop 	es				; get start of video memory in ES
mov 	al,0x13			; switch to video mode 13h
int 	0x10			; 320 * 200 in 256 colors
X:
mov 	ax,0xCCCD		; perform the famous...
mul		di				; ... Rrrola trick =)
sub 	dh,[si]			; align vertically
pusha 					; push all registers on stack
fild 	word [bx-8]		; X
fmul 	st0				; X²
fild 	word [bx-9]		; Y X²
fmul	dword [bx+si]	; aspect ratio correction
fmul 	st0				; Y² X²
fadd 	st0,st1			; Y²+X²
fsqrt					; R
fistp 	dword [bx-5]	; -
popa					; pop all registers from stack
stosb					; write to screen (DI) and increment DI
jmp short X				; next pixel

The resulting images are almost identical ...

to be continued