Denormalising

Denormalising a number
BBC BASIC stores reals (non-integers) in five-byte floating point format. The following code will convert a real to it's integer version, a process known as denormalisation.

 int=&70:exp=int+4                    :REM Returned integer real=exp+1                           :REM Pointer to real :   \ Denormalise - Denormalise a number (convert real to integer) \ ============================================================   \ On entry,  (real) => 5-byte floating point number \                  => exponent, mantissa hi, mid, mid, lo    \ On exit,   (int)  =  denormalised integer version of real \           CC     =  conversion valid, no under/overflow \           CS     =  conversion invalid, under/overflow .Denormalise LDY #0                               :\ (real),Y => exp, man LDX #4                               :\ Five bytes to reorder and copy .DenormLp1 LDA (real),Y:STA int,X:INY           :\ Copy and reverse into store DEX:BPL DenormLp1 LDA exp:BEQ DenormOK                 :\ exp=00, real was zero LDA int+3:PHP:ORA #&80:STA int+3     :\ Save sign and put top bit in    .DenormLp2 LDA exp:CMP #&A0:BCS Denormalised    :\ Loop until denormalised ROR int+3:ROR int+2:ROR int+1:ROR int :\ Multiply mantissa by two BCS DenormOverflow                   :\ Drop out if run out of bits INC exp:BNE DenormLp2 .Denormalised PLP:BPL DenormOK                     :\ Positive, return integer LDX #&FC                             :\ Negate for negative number .DenormNegate LDA #0:SBC int-&FC,X:STA int-&FC,X INX:BMI DenormNegate .DenormOK CLC:RTS                              :\ CLC = conversion valid .DenormOverflow PLP:SEC:RTS                          :\ SEC = conversion invalid 

Explanation
BBC BASIC stores real (non-integer) numbers in five bytes in a format known as "five-byte floating point". This splits the number into two components - a one-byte exponent and a four-byte mantissa.

All numbers, other than zero, can be expressed as m*10^e. You may be familiar with this form known as exponential format. For example:

100 is 1*10^2 5000 is 5*10^3 0.5 is 5*10^-1.

Exactly the same can be done using base 2, expressing numbers as m*2^e, for example:

4 is 1*2^2 -8 is -1*2^3 12 is 1.5*2^3 -0.5 is -1*2^-1

In five-byte floating point format, the manitissa is multiplied or divided by 2, and the exponent reduced or increased, until the mantissa m is in the range 0.5 to 1, excluding 1, for example:

4 is 0.5*2^3 -8 is -0.5*2^4 12 is 0.75*2^4 -0.5 is -0.5*2^0

This means that the first bit of the mantissa is always 1. That means it can be used to hold the sign bit. To allow negative exponents, &80 is added to the exponent. BASIC stores the number in five bytes with the exponent first, followed by the mantissa, high byte to low byte. For example:

4 is exponent &83, mantissa &00, &00, &00, &00 -8 is exponent &84, mantissa &80, &00, &00, &00 12 is exponent &84, mantissa &C0, &00, &00, &00 -0.5 is exponent &80, mantissa &00, &00, &00, &00

Note that the mantissa is stored the opposite way round to an integer.

Zero is a special case and is stored as five zero bytes. Some versions of BBC BASIC extend this and use a zero exponent to indicate that the real actually holds an integer value. For example,

&00, &80, &00, &00, &00 is 128 (&80) &00, &FE, &FF, &FF, &FF is -2 (&FFFFFFFE)

To convert a real to an integer the mantissa must be multiplied by two until the exponent is zero (ie &80). For example, converting 0.5*2^3 back to 4*2^0.

You can only convert a real to an integer if the the real actually represents an integer. If the real is a non-integer, the code returns Carry set to indicate the real could not be converted.

See http://mdfs.net/Info/Comp/6502/ProgTips