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and solved before. Four hundred years ago, when scientists were taking the results from plotting the path of the planets about the sun and trying to come up with a general mathematical theory about the motion of the planets, they had to work with trigonometric tables. The problem with using these tables was that the decimal point had not yet been invented. Instead of having values based as fractions of 1, these tables returned the numerical fraction over 6,000. For existence, if you look up the sine of 45 degrees, you would get the value 0.707107. In the 6,000-based table, the sine of 45 degrees would be 4,243. This same principal can be applied to the nding of fractions in the PIC microcontroller. Instead with coming up with a result that is less than 1, the division method that I am going to present here calculates the fraction as a result less than 65,5236 (0x010000). By doing this, 1/3 is not processed as 0.33333 but as 21845 (0x05555 or 0b00101010101010101), 1/5 is 13,107 (0x03333 or 0b00011001100110011), 1/7 is 9,362 (0x02492 or 0b00010010010010010), and so on. It is important to note that these fractions as binary strings are repeating or open series, which complicates the division operations somewhat. Powers of 2 will result in a closed series, but for the most part, the fractional values will not be closed. To ensure that the result is as correct as possible, the fractional bits should be taken as far as possible, which is why I divided by 65,536 (2 ** 16) and not 256 (2 ** 8). Now that I have the fraction, I can develop the arithmetic series. For 8-bit division, this series is
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A / B = (((65,536 / B) & (1 << 15)) != 0) * (A >> 0) + (((65,536 / B) & (1 << 14)) != 0) * (A >> 1) + (((65,536 / B) & (1 << 13)) != 0) * (A >> 2) + ... + (((65,536 / B) & (1 << 0)) != 0) * (A >> 15)
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The reason why I shift A down for each element in the series is because each test of the shifting down bit in the fraction requires that the dividend be shifted down as well. This operation could be written in a high level language as
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int Divide(int A, int B) { int int int int Quotient = 0; Divisor; TempDividend; i; // Carry out Eight Bit Division
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// Get the Fractional Divisor // Get the Dividend to be // used. for (i = 0; i < 8; i++ ) { // Repeat for sixteen cycles if ((Divisor & 0x08000) != 0) // Have to add Quotient // Fraction Quotient = Quotient + TempDividend; TempDividend = TempDividend >> 1; Divisor = (Divisor & 0x07FFF) << 1;
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Divisor = 65536 / B; TempDividend = Dividend << 8;
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MULDIV, CONSTANT MULTIPLICATION AND DIVISION
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} if ((Divisor & 0x080) != 0) Divisor = Divisor + 0x0100; Divisor = Divisor >> 8; return Divisor; } // End Divide // Calculate Rounded Divisor
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As you work through this function, there should be two things that are unexpected in the code. The rst is that I use 16-bit values for an 8-bit result. This was done because I wanted to get a rounded (to the nearest one) result. If the result has a fraction of 0.5 or greater (in which bit 7 of the result is set), then I increment the returned Divisor. The second is that I shift up the dividend and divisor by 8 bits. This is done so that as I shift down the dividend, I do not loose the fractional bits of the result and cannot produce an accurate rounding of the result. When I implemented this function, I had not written it out as straight PIC microcontroller assembler for use in an application. The multiplication operation, because it is closed, can be carried out within straight code. The division operation laid out above does not have this advantage because the result is most likely open. This open result means that the PIC microcontroller s internal functions cannot be used for calculating the fraction of the divisor. To calculate the divisor fraction, I have written the following macro with the divisor calculated by the macro calculator (explained later in this book):
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divide macro Register, Value variable i = 0, TValue TValue = 0x010000 / Value movf Register, w movwf Temporary + 1 ; Divide 8 bit value by a constant ; Get the Constant Divider ; Use Temporary as the Shifted ; Value
clrf Temporary clrf Quotient clrf Quotient + 1 while (i < 8) bcf STATUS, C ; Shift Down the Temporary rrf Temporary + 1, f rrf Temporary, f if ((TValue & 0x08000) != 0) ; If LSB Set, Add the Value movf Temporary + 1, w addwf Quotient + 1, f movf Temporary, w addwf Quotient, f btfsc STATUS, C incf Quotient + 1, f endif