## Wednesday, February 28, 2024

### Converting IEEE-754 floating point to Color BASIC floating point

I'm still playing around with floating point on the 6809—specifically,
support for floating point for the Color Computer.
The format for floating point for Color BASIC
(written by Microsoft)
predates the IEEE-754 Floating Point Standard 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.
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.
I double checked this by finding some floating point constants in the Color BASIC ROM as shown in the book *Color BASIC Unravelled II*,
(available on the Computer Computer Archives),
like this table:

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

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

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

Then using that to create a floating point value:

org $1000 .float -41.3417023 end

Checking the resulting bytes that were generated:

| FILE ff.a 1 | org $1000 1000: 86A55DE735 2 | .float -41.3417023 3 | end

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

| FILE ff.a 1 | org $1000 1000: 86A55DE728 2 | .float -41.341702110 3 | end

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:

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

The results were:

-41.3417023 -41.3417023 6.23418531

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 ~~Walter K. Zydhek,~~ 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 and make it available.