Difference between revisions of "Denormalising a number"
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| − | [[Category:6502]] | + | [[Category:6502]][[Category:BASIC]] |
| + | 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 :REM Returned integer | |
| − | + | exp=int+4:real=exp+1 :REM Pointer to real | |
| − | int=&70:exp=int+4 | ||
| − | |||
: | : | ||
\ Denormalise - Denormalise a number (convert real to integer) | \ Denormalise - Denormalise a number (convert real to integer) | ||
| Line 38: | Line 39: | ||
==Explanation== | ==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. | + | 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: | + | 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 | 100 is 1*10^2 | ||
| Line 46: | Line 50: | ||
0.5 is 5*10^-1. | 0.5 is 5*10^-1. | ||
| − | Exactly the same can be done using base 2, expressing numbers as m*2^e, | + | Exactly the same can be done using base 2, expressing numbers as m*2^e, for |
| − | + | example: | |
4 is 1*2^2 | 4 is 1*2^2 | ||
| Line 54: | Line 58: | ||
-0.5 is -1*2^-1 | -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: | + | 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 | 4 is 0.5*2^3 | ||
| Line 61: | Line 67: | ||
-0.5 is -0.5*2^0 | -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: | + | 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 | 4 is exponent &83, mantissa &00, &00, &00, &00 | ||
| Line 70: | Line 79: | ||
Note that the mantissa is stored the opposite way round to an integer. | 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, | + | 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, &80, &00, &00, &00 is 128 (&80) | ||
&00, &FE, &FF, &FF, &FF is -2 (&FFFFFFFE) | &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. | + | 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. | + | 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. | ||
[[User:Jgharston|Jgharston]] 00:03, 30 April 2008 (BST) | [[User:Jgharston|Jgharston]] 00:03, 30 April 2008 (BST) | ||
Latest revision as of 06:09, 30 June 2018
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 :REM Returned integer exp=int+4: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)
