1. Computer Science 210 Computer Organization Floating Point Representation

2. Real Numbers Format 1:. Examples: 0.25, 3.1415 … Format 2 (normalized form):. × Example: 2.5 × 10 -1 In mathematics, infinite range and infinite precision (“uncountably infinite”)

3. math.pi >>> import math >>> math.pi 3.141592653589793 >>> print(math.pi) 3.14159265359 >>> print("%.50f" % math.pi) 3.14159265358979311599796346854418516159057617187500 Looks like about 48 places of precision (in base 10 )

4. IEEE Standard Single precision: 32 bits Double precision: 64 bits 2.5 × 10 -1 Reserve some bits for the significand (the digits to the left of ×) and some for the exponent (the stuff to the right of ×) Double precision uses 53 bits for the significand, 11 bits for the exponent, and one sign bit Approximate double precision range is 10 -308 to 10 308

5. IEEE Single Precision Format 32 bits Roughly (-1) S x F x 2 E F is related to the significand E is related to the exponent Rough range Small fractions 2 x 10 -38 Large fractions 2 x 10 38 S Exponent Significand 1 8 23

6. Fractions in Binary In general, 2 -N = 1/2 N 0.1 2 = 1 × 2 -1 = 1 × ½ = 0.5 10 0.01 2 = 1 × 2 -2 = 1 × ¼ = 0.25 10 0.11 2 = 1 × ½ + 1 × ¼ = 0.75 10

7. Decimal to Binary Conversion (Whole Numbers) While N > 0 do Set N to N/2 (whole part) Record the remainder (1 or 0) Set A to remainders in reverse order

8. Decimal to Binary - Example Example: Convert 324 10 to binary N Rem N Rem 324 162050 81021 40110 20001 100 324 10 = 101000100 2

9. Decimal to Binary - Fractions While N > 0 (or enough bits) do Set N to N*2 (whole part) Record the whole number part (1 or 0) Set N to fraction part Set bits to sequence of whole number parts (in order obtained)

10. Decimal fraction to binary - Example Example: Convert.65625 10 to binary N Whole Part.65625 1.312501 0.62500 1.2501 0.500 1.01.65625 10 =.10101 2

11. Decimal fraction to binary - Example Example: Convert.45 10 to binary N Whole Part.45 0.90 1.81 1.61 1.21 0.40 0.80 1.61.45 10 =.011100110011… 2

12. Round-Off Errors >>> 0.1 0.1 >>> print("%.48f" % 0.1) 0.100000000000000005551115123125782702118158340454 >>> print("%.48f" % 0.25) 0.250000000000000000000000000000000000000000000000 >>> print("%.48f" % 0.3) 0.299999999999999988897769753748434595763683319092 Caused by conversion of decimal fractions to binary

13. Scientific Notation - Decimal Number Normalized Scientific 0.0000000011.0 x 10 -9 5,326,043,000 5.326043 x 10 9

14. Floating Point IEEE Single Precision Standard (32 bits) Roughly (-1) S x F x 2 E –F is related to significand –E is related to exponent Rough range –Small fractions 2 x 10 -38 –Large fractions 2 x 10 38 S Exponent Significand 1 8 23

15. Floating Point – Exponent Field This comes before significand for sorting purposes With 8 bit exponent range would be –128 to 127 Note: -1 would be 11111111 and with simple sorting would appear largest. For this reason, we take the exponent, add 127 and represent this as unsigned. This is called bias 127. Then exponent field 11111111 (255) would represent 255 - 127 = 128. Also 00000000 (0) would represent 0 - 127 = -127. Range of exponents is -127 to 128

16. Floating Point – Significand Normalized form: 1.1011… x 2 E Hidden bit trick: Since the bit to left of binary point is always 1, why store it? We don’t. Number = (-1) S x (1+Significand) x 2 E-127

17. Floating Point Example: Convert 312.875 to IEEE Step 1. Convert to binary: 100111000.111 Step 2. Normalize: 1.00111000111 x 2 8 Step 3. Compute biased exponent in binary: 8 + 127 = 135  10000111 Step 4. Write the floating point representation: 0 10000111 00111000111000000000000 or 439C7000 in hexadecimal

18. Floating Point Example: Convert IEEE 11000000101000… to decimal Step 1. Sign bit is 1; so number is negative Step 2. Exponent field is 10000001 or 129; so actual exponent is 2 Step 3. Significand is 010000…; so 1 + Significand is 1.0100… Step 4. Number = (-1) S x (1+Significand) x 2 E-127 = (-1) 1 x (1.010) x 2 2 = -101 = -5