Understanding Binary Conversion and Computer Performance

Binary Conversion

Single-Precision Floating-Point

Format: 1 sign bit, 8 exponent bits, 23 fraction bits

Bias: +127 (decimal to binary), -127 (binary to decimal)

Double-Precision Floating-Point

Format: 1 sign bit, 11 exponent bits, 52 fraction bits

Bias: +1023 (decimal to binary), -1023 (binary to decimal)

Example: -17.6875 (Single-Precision)

1. Sign Bit: + = 0, - = 1 (In this case, 1)
2. Integer Part to Binary: Repeatedly divide the integer part by 2 and record the remainders.
3. Fractional Part to Binary: Repeatedly multiply the fractional part by 2 and record the integer parts.
4. Combine and Normalize: Combine the binary integer and fractional parts. Normalize to the form 1.xxx * 2^exponent.
5. Calculate Exponent: Add the bias to the exponent.

Binary Representation: 1 10001011 00011011000000000000000

Explanation:

• Sign bit: 1 (negative)
• Exponent: 4 + 127 (bias) = 131 = 10000011
• Fraction: 000110110...

Example: 0 01111100 01110000... (Single-Precision)

1. Sign Bit: 0 (positive)
2. Exponent: 01111100 = 124. Subtract the bias (127): 124 - 127 = -3
3. Fraction: 01110000... This represents the fractional part of the normalized binary number (1.fraction).
4. Calculate Value: 1.0111 * 2^-3 = 1.4375 * 2^-3 = 0.1796875

Modern Trends in Computer Performance

Annual performance improvement is slowing down to around 20%. Transistor count is still increasing, but at a slower rate. Multi-thread parallelism and hardware accelerators are becoming increasingly important for boosting performance.

Historical Contributions to Performance

• Better processes (faster devices): ~20%
• Better circuits/pipelines: ~15%
• Better organization/architecture: ~15%

Key Performance Equations