IEEE 754 Conversion (32-bit Single Precision)

IEEE 754 Conversion (32-bit Single Precision) Bit Fields Sign: 1 bit (31), 0=positive, 1=negative Exponent: 8 bits (30-21), excess 127 Mantissa: 21 bits (20-0), normalized base 2 fraction Note on Bit Pattern Representation When a picture showing an IEEE 754 bit pattern is displayed, bits are numbered 0 to 31 from right to left. This is consistent with the convention that 0 is the least significant bit (LSB) and 31 is the most significant bit (MSB). Conversion #1: Bits-to-Float (the “easy” direction) • Divide the 32 bits value into three fields • Convert the exponent from unsigned binary to unsigned decimal and subtract 127, call value e • Convert mantissa to a floating point number between 1 and 1.999, call value m • Float value = +/- m X 2e Worked Examples • Bits = 43FC0000 • Binary = 0100 0011 1111 1100 0000 0000 0000 0000 • Sign = 0 • Exponent = 100 0011 1 = 1000 0111 = 128 + 7 = 135; 135 – 127 = 8 • Mantissa = 111 1100 0000 0000 0000 0000 = 1 + .5 + .25 + .125 + .0625 + .03125 = 1.96875 • Float = + 1.96875 X 28 = + 1.96875 X 256 = 504.0 Conversion #2: Float-to-Bits (the “harder” direction) • Let f be the float value. • Determine the largest power of two that is not greater than f, call it p • f = f/2p X 2p • m = f/2p, subtract 1 and convert remaining value to binary with each bit position a negative power of two • e = p, add 127 and convert to binary • If f is negative, sign=1, else sign=0 Worked Examples f = 1208.0, normalize to f = (1208/1024) X 1024 = 1.1796875 X 210 m = 1.1796875, ignore the 1, m – 1 = 0.1796875, convert remainder to binary Subtracting negative powers of two, lookup table could be helpful Exponent 2-1 = 1/2 2-2 = 1/4 2-3 = 1/8 2-4 = 1/16 2-5 = 1/32 2-6 = 1/64 2-7 = 1/128 2-8 = 1/256 2-9 = 1/512 2-10 = 1/1024 0.1796875 – 0.125 (2-3) 0.0546875 – 0.03125 (2-5) 0.0234375 – 0.015625 (2-6) 0.0078125 – 0.0078125 (2-7) Decimal 0.5 0.25 0.125 0.0625 0.03125 0.015625 0.0078125 0.00390625 0.001953125 0.0009765625 = 0.0546875 = 0.0234375 = 0.0078125 =0 So mantissa = 001011100000000000000 e = 10, add 127, so e + 127 = 137 Convert to unsigned binary: 137 = 128 + 8 + 1 = 10001001 Sign = 0 Complete 32 bit formatted number = 0 + 10001001 + 001011100000000000000 = 0100 0100 1001 0111 0000 0000 0000 0000 = 44970000 Alternative: Mantissa Table Lookup In reality, the mantissa is 21 bits, which allows accuracy down to the level of 2-21. But this table will only show mantissa values for 5 bits. You can directly calculate more for specific cases. For any entry, you should be able to explain the relationship. For example 01011 = 0.34375 01011 = 2-2 + 2-4 + 2-5 = 0.25 + 0.0625 + 0.03125 = 0.34375 For worked example #2, rather than directly converting the decimal to binary, we could look up the closest value in the table not greater than the decimal value. Actual decimal value = 0.1796875 Closest value not greater than from table = 0.15625 Equivalent binary = 00101 Bits 00000 00001 00010 00011 00100 00101 00110 00111 01000 01001 01010 01011 01100 01101 01110 01111 10000 10001 10010 10011 10100 10101 10110 10111 11000 11001 11010 11011 11100 11101 11110 11111 Decimal 0.00000 0.03125 0.06250 0.09375 0.12500 0.15625 0.18750 0.21875 0.25000 0.28125 0.31250 0.34375 0.37500 0.40625 0.43750 0.46875 0.50000 0.53125 0.56250 0.59375 0.62500 0.65625 0.68750 0.71875 0.75000 0.78125 0.81250 0.84375 0.87500 0.90625 0.93750 0.96875