1. 1 COMS 161 Introduction to Computing Title: Numeric Processing Date: November 08, 2004 Lecture Number: 30

2. 2 Announcements

3. 3 Review Real numbers –Representation –Limitations

4. 4 Outline Real numbers –Representation –Limitations

5. 5 IEEE Standard 754 Provides two floating point types –Single 24-bits of significand precision –Double 53-bits of significand precision

6. 6 Single Precision IEEE standard 754 –Floating point number representation –32-bit s eeeeeeee fffffff ffffffffffffffff –s: (1) sign bit 0 means positive, 1 means negative sexponentsignificand 31 302322 0

7. 7 Single Precision s eeeeeeee fffffff ffffffffffffffff –e: (8) exponent bits [-126 … 127] A bias of 127 is added to the exponent –f: (24) fractional part [23 bits + 1 implied bit] Normalize the fractional part 1 will always be on the left side of the binary point

8. 8 Special Single Cases Two zeros –Signed zero –e = 0, f = 0 (exponent and fractional bits are all 0) –(-1) s x 0.0 0000 0000 0000 0000 –0x0000 0000 (+0) 1000 0000 0000 0000 0000 0000 0000 0000 –0x8000 0000 (-0)

9. 9 Special Single Cases Positive infinity –+INF –s = 0, e = 255, f = 0 (all fractional bits are all 0) 0111 1111 1000 0000 0000 0000 0000 0000 0x7f80 0000 Negative infinity –-INF –s = 1, e = 255, f = 0 (all fractional bits are all 0) 1111 1111 1000 0000 0000 0000 0000 0000 0xff80 0000

10. 10 Special Single Cases Not-A-Number (NaN) –s = 0 | 1, e = 255, f != 0 (at least one fractional bit is NOT 0) –There are many representations for NaN –Here is one example 0111 1111 1100 0000 0000 0000 0000 0000 0x7fc0 0000

11. 11 Special Single Cases Maximum single number –0111 1111 0111 1111 1111 1111 1111 1111 –0x7f7f ffff –3.40282347 x 10 38 Minimum positive single number –0000 0000 1000 0000 0000 0000 0000 0000 –0x00800000 –1.17549435 x 10 -38 To represent larger numbers

12. 12 Double Precision IEEE standard 754 –Floating point number representation –64-bit s eeeeeeeeeeeffffffffffffffffffffffffffffffffffffffffffffffffff –s: (1) sign bit 0 means positive, 1 means negative sexponentsignificand 63 625251 32 significand 310

13. 13 Single Precision s eeeeeeeeeeeffffffffffffffffffffffffffffffffffffffffffffffffff –e: (11) exponent bits [-1022 … 1023] A bias of 1023 is added to the exponent –f: (53) fractional part [52 bits + 1 implied bit] Normalize the fractional part 1 will always be on the left side of the binary point

14. 14 Real (Decimal) Number Storage Double precision floating point numbers –s: (1) sign bit –e: (11) exponent bits [-1022 … 1023] –f: (53) fractional part [52 bits + 1 implied bit] seeeeeeeeee f f f ff f f f Byte 0 1 2 3 f f f f Byte 4 5 6 7

15. 15 Special Double Cases Two zeros –Signed zero –e = 0, f = 0 (exponent and fractional bits are all 0) –(-1) s x 0.0 64 bits 0000 0000 0000 0000 0000 0000 0000 … 0000 –0x0000 0000 0000 0000 (+0) 1000 0000 0000 0000 0000 0000 0000 … 0000 –0x8000 0000 0000 0000 (-0)

16. 16 Special Double Cases Positive infinity –+INF –s = 0, e = 2047, f = 0 (all fractional bits are all 0) 0111 1111 1111 0000 0000 0000 0000 … 0000 0x7ff0 0000 0000 0000 Negative infinity –-INF –s = 1, e = 2047, f = 0 (all fractional bits are all 0) 1111 1111 1111 0000 0000 0000 0000 … 0000 0xfff0 0000 0000 0000

17. 17 Special Double Cases Not-A-Number (NaN) –s = 0 | 1, e = 2047, f != 0 (at least one fractional bit is NOT 0) –There are many representations for NaN –Here is one example 0111 1111 1111 1000 0000 0000 0000 … 0000 0x7ff8 0000 0000 0000

18. 18 Special Double Cases Maximum double number –0111 1111 1110 1111 1111 1111 1111 … 1111 –0x7fef ffff ffff ffff –1.7976931348623157 x 10 308 Minimum positive single number –0000 0000 0001 0000 0000 0000 0000 … 0000 –0x0010 0000 0000 0000 –2.2250738585072014 x 10 -308 –Don’t forget about the implied 1 bit!!

19. 19 Decimal to Float Conversion Show –24.12510 in IEEE single precision format –First, save sign (negative so 1) and convert to binary… –24.12510 = 11000.001 2 x 2 0 –Normalize… – = 1.1000001 2 x 2 4 –Strip 1 off the mantissa and extend to form significand – =.10000010000000000000000 –Bias the exponent… –Exp + Bias = 4 + 127 = 131 = 10000011 2

20. 20 Real (Decimal) Number Storage 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 09 08 07 06 05 04 03 02 01 00 1 1 0 0 0 0 0 1 1 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 Hex value : 0xC1C10000 Link me baby