Denormalising a number

BBC BASIC stores reals (non-integers) in five-byte floating point format. The following code will convert a real to its 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.

Jgharston 00:03, 30 April 2008 (BST)