How arithmetic pipeline increases performance

An arithmetic pipeline splits an operation such as floating-point addition into k segments (compare exponents, align mantissas, add mantissas, normalise). Each segment works on a different pair of operands in the same clock, so results come out at the rate of one per clock.

Performance

Let k = 4 segments, tp = clock time of the slowest segment, n = number of operand pairs.

• Time without pipeline: n · tn, where tn = k · tp (the whole operation done in turn).
• Time with pipeline: (k + n - 1) · tp
• Speedup: S = n·k·tp / ((k + n - 1)·tp) = n·k / (k + n - 1)

Example: k = 4, tp = 10 ns, n = 1000 additions (a vector sum).

• Non-pipelined: 1000 × 4 × 10 = 40,000 ns
• Pipelined: (4 + 999) × 10 = 10,030 ns
• Speedup = 40,000 / 10,030 = 3.99, almost k.

The first result appears after k clocks (latency), and then one result per clock (throughput of 1 / tp). The gain is greatest when the same operation is done on a long stream of data, as in vector processing, and is limited by the slowest segment and the register delay between segments.

 pair n --> [compare][align][add][normalise] --> result
 pair n+1 -----^ each stage busy with a different pair