CSCI206 - Computer Organization & Programming Floating Point Limits zyBook: 10.9, 10.10

IEEE 754 Standard (1985) (normalized) Exponent Mantissa S, E, and M are encoded in the binary word

IEEE754 - Reserved Values Not a Number =

IEEE754 - Example Show 3.14 as a single precision float

3.14 - step 1 write in binary 3.14 == 3 + 0.14 0.14*2 = 0.28 0.28*2 = 0.56 0.56*2 = 1.12 0.12*2 = 0.24 ...... 0.0010......

3.14 - step 1 write in binary Need 24 bits for single (52 for double). In this example, 2 bits before point, 22 bits after. 3.14 == 11.0010001111010111000011

3.14 - step 2 normalize binary Normalized form is 1.yyyyy 3.14 == 11.0010001111010111000011 == Note a total of 24 bits.

3.14 - step 3 write mantissa & sign 3.14 == M = 10010001111010111000011 S = 0 (positive) Note that the mantissa keeps only 23 bits, the leading bit is always 1, so it is omitted in representation (only!!).

3.14 - step 4 encode exponent 3.14 == Exponent = 1, B = 127, (8 bits) E (biased exponent) = 128 = 1000 0000

3.14 - step 5 write result S = 0 (positive) E = 1000 0000 M = 10010001111010111000011 0 1000 0000 10010001111010111000011 to hex = 0x4048f5c3

Endianness On a little-endian system (Intel, etc), the IEEE754 value is byte & word swapped 0x 40 48 f5 c3 (big endian) 4840 c3f5 0x c3f5 4840 (little endian) Swap bytes and words! float f = 3.14; unsigned char* p = (unsigned char*)&f; printf("%02x%02x %02x%02x\n", *p, *(p+1), *(p+2), *(p+3)); // result on Intel: c3f5 4840, on MIPS: 4048 f5c3

Review IEEE754 S Exponent Mantissa Special values, else normalized numbers Exponent Mantissa (fraction) Value +/- zero nonzero denormalized number all 1’s +/- infinity NaN (not a number)

Largest Single Precision Float 8 bit exponent (bias = 127), 23 bit fraction All 1’s in the exponent is reserved for NaN and infinity Maximum biased exponent is 1111 1110 = 254 Maximum fraction is 23 1’s

Largest Single Precision Float 1111 1110 = 254 254-127 = 127

Largest Single Precision Float Move the decimal point 23 digits to the right subtract 23 from exponent

Largest Single Precision Float Convert mantissa

Smallest Nonzero Single What we want is: But that has exponent & fraction = 0 That value is reserved for zero! Therefore, the closest we can get is: either or

Smallest Nonzero Single Normalized In this case, using a normalized number is not ideal, if we could use a denormalized number we could get a much smaller value: This is equivalent to: An extra 22 bits of precision! Denormalized

Smallest Nonzero Single The IEEE realized this and when the exponent is zero and the fraction is > 0, the value is treated as a denormalized number. The smallest nonzero normalized: The smallest nonzero denormalized: exp = 0000 0001 exp = 0000 0000 m = 0000….1

Smallest Nonzero Normalized Single biased exponent = 1 fraction = 0 0000 0001 00000000000000000000000

Smallest Nonzero Denormalized Single biased exponent = 0 fraction = 0.00000000000000000000001 0000 0000 00000000000000000000001 *Note, even though the exponent is encoded as -127, it is computed using the smallest “valid” exponent, which is -126.