* * * * *
                                        
        Converting IEEE-754 floating point to Color BASIC floating point
                                        
I'm still playing around with floating point on the 6809 [1]—specifically,
support for floating point for the Color Computer. The format for floating
point for Color BASIC (Beginners' All-purpose Symbolic Instruction Code)
(written by Microsoft) predates the IEEE (Institute of Electrical and
Electronics Engineers)-754 Floating Point Standard [2] by a few years and
thus, isn't quite compatible. It's close, though. It's defined as an 8-bit
exponent, biased by 129, a single sign bit (after the exponent) and 31 bits
for the mantissa (the leading one assumed). It also does not support ±∞ nor
NaN (Not a Number). This differs from the IEEE-754 single precision that uses
a single sign bit, an 8-bit exponent biased by 127 and 23 bits for the
mantissa (which also assumes a leafing one) and support for infinities and
NaN. The IEEE-754 double precision uses a single sign bit, an 11-bit exponent
biased by 1023 and 52 bit for the mantissa (leading one assumed) plus support
for infinities and NaN.

So the Color BASIC is about halfway between single precision and double
precision. This lead me to use IEEE-754 double precision for the Color
Computer backend (generating an error for inifinities and NaN) then massaging
the resulting double into the proper format [3]. I double checked this by
finding some floating point constants in the Color BASIC ROM (Read Only
Memory) as shown in the book Color BASIC Unravelled II, (available on the
Computer Computer Archives [4]), like this table:

-----[ Listing ]-----
4634				* MODIFIED TAYLOR SERIES SIN COEFFICIENTS
4635	BFC7 05			LBFC7	FCB	6-1			SIX COEFFICIENTS
4636	BFC8 84 E6 1A 2D 1B	LBFC8	FCB	$84,$E6,$1A,$2D,$1B	* -((2*PI)**11)/11!
4637	BFCD 85 28 07 FB F8	LBFCD	FCB	$86,$28,$07,$FB,$F8	*  ((2*PI)**9)/9!
4638	BFD2 87 99 68 89 01	LBFD2	FCB	$87,$99,$68,$89,$01	* -((2*PI)**7)/7!
4639	BFD7 87 23 35 DF E1	LBFD7	FCB	$87,$23,$35,$DF,$E1	*  ((2*PI)**5)/5!
4640	BFDC 86 A5 5D E7 28	LBFDC	FCB	$86,$A5,$5D,$E7,$28	* -((2*PI)**3)/3!
4641	BFE1 83 49 0F DA A2	LBFE1	FCB	$83,$49,$0F,$DA,$A2	*    2*PI
-----[ END OF LINE ]-----

Then using the byte values to populate a variable and printing it inside
BASIC (this is the expression -2π^3/3!):

-----[ BASIC ]-----
X=0         ' CREATE A VARIABLE
Y=VARPTR(X) ' GET ITS ADDRESS
POKE Y,&H86 ' AND SET ITS VALUE
POKE Y+1,&HA5 ' THE HARD WAY
POKE Y+2,&H5D
POKE Y+3,&HE7
POKE Y+4,&H28
PRINT X ' LET'S SEE WHAT IT IS
-41.3417023
-----[ END OF LINE ]-----

Then using that to create a floating point value:

-----[ Assembly ]-----
	org	$1000
	.float	-41.3417023
	end
-----[ END OF LINE ]-----

Checking the resulting bytes that were generated:

-----[ Listing ]-----
                         | FILE ff.a
                       1 |         org     $1000
1000: 86A55DE735       2 |         .float  -41.3417023
                       3 |         end
-----[ END OF LINE ]-----

And adjusting the floating point constant until I got bytes that matched:

-----[ Listing ]-----
                         | FILE ff.a
                       1 |         org     $1000
1000: 86A55DE728       2 |         .float  -41.341702110
                       3 |         end
-----[ END OF LINE ]-----

I figure it's “close enough.” The parsing code in the Color BASIC ROM is old
and predates the IEEE-754 floating point standard, so a few different digits
at the end I think is okay.

As a final check, I wrote the following bit of code to calculate and display 
2π^3/3!, display the pre-calculated result, as well as display the pre-
calculated value of 2π:

-----[ Assembly ]-----
		include	"Coco/basic.i"
		include	"Coco/dp.i"

CB.FSUBx	equ	$B9B9	; FP0 = X   - FP0	; addresses for
CB.FSUB		equ	$B9BC	; FP0 = FP1 - FP0	; these routines 
CB.FADDx	equ	$B9C2	; FP0 = X   + FP0	; from
CB.FADD		equ	$B9C5	; FP0 = FP1 + FP1	; Color BASIC Unravelled II
CB.FMULx	equ	$BACA	; FP0 = X   * FP0
CB.FMUL		equ	$BAD0	; FP0 = FP0 * FP1
CB.FDIVx	equ	$BB8F	; FP0 = X   / FP0
CB.FDIV		equ	$BB91	; FP0 = FP1 / FP0

CB.FP0fx	equ	$BC14	; FP0 = X
CB.xfFP0	equ	$BC35	; X   = FP0
CB.FP1f0	equ	$BC5F	; FP1 = FP0
CB.FP0txt	equ	$BDD9	; result in X, NUL terminated

		org	$4000
start		ldx	#tau		; point to 2*pi
		jsr	CB.FP0fx	; copy to FP0
		ldx	#tau		; 2PI * 2PI
		jsr	CB.FMULx
		ldx	#tau		; 2PI * 2PI * 2PI
		jsr	CB.FMULx
		jsr	CB.FP1f0	; copy fp acc to FP1
		ldx	#fact3		; point to 3!
		jsr	CB.FP0fx	; copy to FP0
		jsr	CB.FDIV		; FP0 = FP1 / FP0
		neg	CB.fp0sgn	; negate result by flippping FP0 sign
		jsr	CB.FP0txt	; generate string
		bsr	display		; display on screen

		ldx	#answer		; point to precalculated result
		jsr	CB.FP0fx	; copy to FP0
		jsr	CB.FP0txt	; generate string
		bsr	display		; display

		ldx	#tau		; now display 2*pi
		jsr	CB.FP0fx	; just to see how close
		jsr	CB.FP0txt	; it is.
		bsr	display
		rts

display.char	jsr	[CHROUT]	; display character
display		lda	,x+		; get character
		bne	.char		; if not NUL byte, display
		lda	#13		; go to next line
		jsr	[CHROUT]
		rts

tau		.float	6.283185307
fact3		.float	3!
answer		.float	-(6.283185307 ** 3 / 3!)

		end	start
-----[ END OF LINE ]-----

The results were:

-----[ data ]-----
-41.3417023
-41.3417023
 6.23418531
-----[ END OF LINE ]-----

The calculation results in -41.3417023 and the direct result stored in answer
also prints out -41.3417023, so that matches and it reinforces my approach to
this nominally right.

But I think Microsoft had issues with either generating some of the floating
point constants for the larger terms, or transcribing the byte values of the
larger terms. Take for instance -2π^11/11!. The correct answer is -
15.0946426, but the bytes in the ROM define the constant -14.3813907, a
difference of .7. And it's not like Color BASIC can't calculate that
correctly—when I typed in the expression by hand, it was able to come up with
-15.0946426.

Or it could be that [DELETED-Walter K. Zydhek,-DELETED] the author of Color
BASIC Unravelled II, is wrong in his interpretation of the expressions used
to generate the values, or his interpretation of what the values are used
for. I'm not sure who is at fault here.

Update on Friday, March 1^st, 2024

I was wrong about the authorship of Color BASIC Unravelled II. It was not
Walter K. Zydhek, but some unknown author of Spectral Associates, a company
that is no longer in business. All Zydhek did was to transcribe a physical
copy of the book (which is no longer available for purchase anywhere) into a
PDF (Portable Document Format) and make it available.


[1] gopher://gopher.conman.org/0Phlog:2024/02/07.1
[2] https://codedocs.org/what-is/ieee-754
[3] https://github.com/spc476/a09/blob/8682748ac0f98d7f83bbdb0b6e772270cf3fc89f/frsdos.c#L292
[4] https://colorcomputerarchive.com/repo/Documents/Books/Unravelled%20Series/
---

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