{"id":76,"date":"2021-06-06T19:17:54","date_gmt":"2021-06-06T17:17:54","guid":{"rendered":"http:\/\/automatico.freevar.com\/?page_id=76"},"modified":"2022-02-16T14:08:37","modified_gmt":"2022-02-16T13:08:37","slug":"aritmetica-ieee-p754","status":"publish","type":"page","link":"https:\/\/automatico.freevar.com\/index.php\/cp-m-80-2\/hitech-c-per-z80\/aritmetica-ieee-p754\/","title":{"rendered":"IEEE P754 Hi-Tech Arithmetic C 3.09"},"content":{"rendered":"\r\n

Representation<\/h2>\r\n

\r\n\r\n<\/span><\/p>\r\n

The format for the representation of floating point numbers in computers, in general, is that dictated by the IEEE 754 standard. discrete subset of the real field.<\/span><\/span><\/p>\r\n

The IEEE 754 standard defines a form of representation of the real number: the real number is first divided into the integer part and the fractional part and transformed into a binary base, then once normalized\u00a01<\/sup><\/a> is put in the form +\/- 1, mantissa x 2 E .<\/sup><\/span><\/p>\r\n

For example, 1000.0 can be written + 1 \u00d7 10\u00a03<\/sup> in base ten and + 0.1953125\u00a0 x 2 10<\/sup> in base two.<\/span><\/p>\r\n

In our computer the numbers in Floating Point are written in this form:<\/span><\/p>\r\n

\r\n\r\n<\/span><\/p>\r\n

\r\n\r\n\r\n\r\n
S<\/span><\/td>\r\nE<\/span><\/td>\r\nE<\/span><\/td>\r\nE<\/span><\/td>\r\nE<\/span><\/td>\r\nE<\/span><\/td>\r\nE<\/span><\/td>\r\nE<\/span><\/td>\r\nM<\/span><\/td>\r\nM<\/span><\/td>\r\nM<\/span><\/td>\r\nM<\/span><\/td>\r\nM<\/span><\/td>\r\nM<\/span><\/td>\r\nM<\/span><\/td>\r\nM<\/span><\/td>\r\nM<\/span><\/td>\r\nM<\/span><\/td>\r\nM<\/span><\/td>\r\nM<\/span><\/td>\r\nM<\/span><\/td>\r\nM<\/span><\/td>\r\nM<\/span><\/td>\r\nM<\/span><\/td>\r\nM<\/span><\/td>\r\nM<\/span><\/td>\r\nM<\/span><\/td>\r\nM<\/span><\/td>\r\nM<\/span><\/td>\r\nM<\/span><\/td>\r\nM<\/span><\/td>\r\nM<\/span><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/figure>\r\n

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A bit for the\u00a0S<\/strong>ign of Mantissa, 7 bits for the\u00a0 E<\/strong>xponent\u00a0and 24 bits for\u00a0M<\/strong>antissa. This is because the designers of the Hi-tech C compiler slightly modified the IEEE P754 standard (P means single precision) to better handle FP numbers with the Z80 processor machine registers. Using two double 8-bit registers they could hold the entire FP number in registers HL and DE.<\/span><\/p>\r\n

Let’s now see an example of adding numbers in FP.\u00a0The number 3489.0 in decimal base is encoded in hexadecimal as\u00a04C DA 10 00<\/strong>\u00a0.\u00a0The compiler places these values \u200b\u200bin the machine registers: in HL we will have\u00a04C DA<\/strong>\u00a0and in DE we will have\u00a010 00<\/strong>\u00a0.\u00a0So in the machine registers HL and DE we have:<\/span><\/p>\r\n

\r\n\r\n<\/span><\/p>\r\n

\r\n\r\n\r\n\r\n\r\n\r\n
H<\/span><\/td>\r\nL<\/span><\/td>\r\nD<\/span><\/td>\r\nE<\/span><\/td>\r\n<\/tr>\r\n
4C<\/strong><\/span><\/td>\r\nDA<\/strong><\/span><\/td>\r\n10<\/strong><\/span><\/td>\r\n00<\/strong><\/span><\/td>\r\n<\/tr>\r\n
0<\/strong>100 1100<\/span><\/td>\r\n1101 1010<\/strong><\/span><\/td>\r\n0001 0000<\/strong><\/span><\/td>\r\n0000 0000<\/strong><\/span><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/figure>\r\n

\r\n\r\n<\/span><\/p>\r\n

In H bit 7 is the sign (1 negative, 0 positive) and the remaining 7 bits are for the exponent that distinguish 128 values.\u00a02\u00a00<\/sup>\u00a0is fixed\u00a0at the decimal value 65 (\u00a0100 0001<\/strong>\u00a0binary, 41 hexadecimal): values \u200b\u200bgreater than 65 are positive powers;\u00a0values \u200b\u200bbelow 65 are negative powers.\u00a0In this case we have that the sign is positive (bit7 = 0), the exponent\u00a0100 1100<\/strong>\u00a0is 76 decimal.\u00a0We said that 65 decimal equals the power 2\u00a00<\/sup>\u00a0, with the difference 76 – 65 we have the exponent of two, so 2\u00a011<\/sup>\u00a0.<\/span><\/p>\r\n

Let us now see the mantissa which is contained in the 3 bytes of the L and DE registers.<\/span><\/p>\r\n

1101 1010 0001 0000 0000 0000<\/span><\/p>\r\n

The first bit on the left is the weight of 2\u00a00<\/sup>\u00a0, the second of 2\u00a0-1<\/sup>\u00a0, the third of 2\u00a0–\u00a0<\/sup>2<\/sup>\u00a0and so on, so multiplying the powers of two by the bits corresponding to the position in the mantissa we obtain (omitting the products in the summation null):<\/span><\/p>\r\n

1 + 1\/2 + 1\/8 + 1\/16 + 1\/64 + 1\/2048 = 1.70361328125<\/span><\/p>\r\n

So we have 1.70361328125 * 2\u00a011<\/sup>\u00a0or 3489.0 in decimal base.<\/span><\/p>\r\n

\r\n\r\n<\/span><\/p>\r\n

Sum and difference<\/h2>\r\n

\r\n\r\n<\/span><\/p>\r\n

Now let’s see how two numbers can be added in FP form.<\/span><\/p>\r\n

Let’s consider the two numbers:<\/span><\/p>\r\n\r\n\r\n\r\n\r\n
4C DA 10 00\u00a0<\/span><\/td>\r\n\u00a0<\/td>\r\n0<\/strong>100<\/span><\/td>\r\n1100\u00a0<\/span><\/td>\r\n1101\u00a0<\/span><\/td>\r\n1010\u00a0<\/span><\/td>\r\n0001<\/span><\/td>\r\n0000<\/span><\/td>\r\n0000<\/span><\/td>\r\n0000<\/span><\/td>\r\n<\/tr>\r\n
CD DB 01 01<\/span><\/td>\r\n\u00a0<\/td>\r\n1<\/span><\/strong>100\u00a0<\/span><\/span><\/td>\r\n1101\u00a0<\/span><\/td>\r\n1101\u00a0<\/span><\/td>\r\n1011\u00a0<\/span><\/td>\r\n0000\u00a0<\/span><\/td>\r\n0001\u00a0<\/span><\/td>\r\n0000\u00a0<\/span><\/td>\r\n0001<\/span><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n

The first number starts with 0<\/span> so the sign of the mantissa is positive, the second starts with 1<\/span> so the sign of the mantissa is negative.<\/span><\/p>\r\n

\r\n\r\n<\/span><\/p>\r\n

The second FP number, the exponent (remember that the exponent are the bits in bold) is 77 decimal. <\/span>We know that the offset is 65 decimal so the exponent of 2 is 12 decimal.\u00a0According to the calculation previously exposed, the mantissa gives us 1.71096813678.<\/span><\/p>\r\n

So 1.71096813678 * 2\u00a01\u00a0<\/sup>2<\/sup> results\u00a0 -7008.1255 (approximately) taking into account the sign.<\/span><\/p>\r\n

Let’s now see an algorithm that allows us to add two numbers in FP format.<\/span><\/p>\r\n