Computer Science 210 Computer Organization Floating Point Representation

Why “Floating Point” ? For many applications (e.g., banking) it’s okay to have a fixed position for the decimal point: $2.99, $101.23, etc., where decimal point is always to places from the right. In applications where we need more precision (e.g., scientific computing), we need the decimal point to “float” around in the number: 1234.56, 123.456, 12.3456, etc.

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

math.pi Looks like about 48 places of precision (in base10) >>> 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 base10)

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 52 bits for the significand, 11 bits for the exponent, and one sign bit Approximate double precision range is 10-308 to 10308

IEEE Single Precision Format 32 bits Roughly (-1)S x F x 2E F is related to the significand E is related to the exponent Rough range Small fractions 2 x 10-38 Large fractions 2 x 1038 S Exponent Significand 1 8 23

Fractions in Binary In general, 2-N = 1/2N 0.12 = 1 × 2-1 = 1 × ½ = 0.510 0.012 = 1 × 2-2 = 1 × ¼ = 0.2510 0.112 = 1 × ½ + 1 × ¼ = 0.7510

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

Decimal to Binary - Example Example: Convert 32410 to binary N Rem N Rem 324 162 0 5 0 81 0 2 1 40 1 1 0 20 0 0 1 10 0 32410 = 1010001002

Decimal to Binary - Fractions While fractional part > 0 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)

Decimal fraction to binary - Example Example: Convert .6562510 to binary N Whole Part .65625 1.31250 1 0.6250 0 1.250 1 0.50 0 1.0 1 .6562510 = .101012

Decimal fraction to binary - Example Example: Convert .4510 to binary N Whole Part .45 0.9 0 1.8 1 1.6 1 1.2 1 0.4 0 0.8 0 1.6 1 .4510 = .011100110011…2

Round-Off Errors Caused by conversion of decimal fractions to binary >>> 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

Scientific Notation - Decimal Number Normalized Scientific 0.000000001 1.0 x 10-9 5,326,043,000 5.326043 x 109

Floating Point IEEE 754 Single Precision Standard (32 bits) Roughly (-1)S x F x 2E F is related to significand E is related to exponent Rough range Small fractions 2 x 10-38 Large fractions 2 x 1038 S Exponent Significand 1 8 23

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

Floating Point – Significand Normalized form: 1.1011… x 2E 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 2E-127

Floating Point Example: Convert 312.875 to IEEE Step 1. Convert to binary: 100111000.111 Step 2. Normalize: 1.00111000111 x 28 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

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 2E-127 = (-1)1 x (1.010) x 22 = -101 = -5

For Next Time Section 2.6: Boolean Algebra Sections 3.1, 3.2: Transistors and Logic Gates