Last updated: 2026-06-29

Amdahl's Law

Amdahl's law predicts how much faster a fixed task can run when part of it is executed in parallel across several processors. It answers a question every engineer faces when scaling software onto more cores: if only some of the work can be parallelized, what is the realistic speedup, and where does adding hardware stop helping? The calculator takes the parallelizable fraction of the work and a processor count and returns the resulting speedup, the theoretical ceiling, and the parallel efficiency.

The serial bottleneck

Most programs contain a mix of work. Some parts — independent iterations of a loop, rendering separate image tiles, processing distinct records — can run simultaneously. Other parts — reading a configuration file, building a data structure that everything else depends on, a final reduction step — must run in sequence. The serial parts cannot be accelerated by adding processors, so they set a floor on the total run time.

Amdahl's law makes this precise. Let $p$ be the fraction of the original run time that is parallelizable and $1 - p$ the serial fraction. With $N$ processors the parallel part finishes in $p/N$ of the original time, while the serial part still takes $1 - p$. The overall speedup is the original time divided by the new time:

$S = \dfrac{1}{(1 - p) + \dfrac{p}{N}}$

The ceiling

As $N$ grows, the term $p/N$ shrinks toward zero, but the serial term $1 - p$ stays put. Taking the limit gives the maximum speedup the program can ever reach:

$S_{\max} = \dfrac{1}{1 - p}$

A program that is 95% parallel can never exceed a 20× speedup, no matter how many processors are thrown at it. A program that is 50% parallel tops out at 2×. This is why reducing the serial fraction — not just buying more cores — is often the higher-leverage optimization.

Worked example

Suppose 95% of a job is parallelizable and you run it on 16 processors. The speedup is:

So 16 processors deliver about a 9.1× speedup — well short of a perfect 16×. The theoretical ceiling here is $1/0.05 = 20\times$, and the parallel efficiency is $9.14/16 \approx 57\%$, meaning nearly half of the added compute capacity is lost to the serial bottleneck.

Why efficiency matters

Parallel efficiency, $E = S / N$, captures how well the extra processors are being used. Efficiency near 100% means almost linear scaling; efficiency that falls quickly as $N$ rises signals that the serial fraction dominates and that more hardware is poor value. Amdahl's law describes strong scaling — a fixed workload on more processors. When the workload itself grows with the hardware, the more optimistic Gustafson's Law Calculator applies instead.

Treat the parallel fraction as an estimate: it is usually measured by profiling and often shifts with problem size and hardware, so the predicted speedup is a guide to expected behavior rather than a guarantee.