Daniel Lemire is a software performance expert. He ranks among the top 2% of scientists globally (Stanford/Elsevier 2025) and is one of GitHub's top 1000 most followed developers.
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In software, you frequently need to check whether some objects is in a set. For example, you might have a list of forbidden Web addresses. As someone enters a new Web address, you may want to check whether it is part of your black list. Or maybe you have a large list of already used … Continue reading Xor Filters…
We sometimes have to look for a value in a sorted array. The simplest algorithm consists in just going through the values We sometimes have to look for a value in a sorted array. The simplest algorithm consists in just going through the values one by one, until we encounter the value, or exhaust the … Continue reading…
Let us say that I ask you to find the number I am thinking about between -1000 and 1000, by repeatedly guessing a number. With each guess, I tell you whether your guess is correct, smaller or larger than my number. A binary search algorithm tries to find a value in an interval by repeating … Continue reading Fast…
If I multiply two 64-bit integers (having values in [0, 264)), the product requires 128 bits. Intel and AMD processors (x64) can compute the full (128-bit) product of two 64-bit integers using a single instruction (mul). ARM processors, such as those found in your mobile phone, require two instructions to achieve the…
In software programming, the product between two integers is often computed to a fixed number of bits with overflow. Consider 8-bit integers. If you multiply 127 by 127, you get back the number 1 as an 8-bit unsigned integer, with an overflow. The actual full product is 16129. To represent 16129, you typically use 16…
Most commodity processors support single-precision IEEE 754 floating-point numbers. Though they are ubiquitous, they are often misunderstood. One of my readers left a comment suggesting that picking an integer in [0,232) at random and dividing it by 232, was equivalent to picking a number at random in [0,1). I am not…
Not all instructions on modern processors cost the same. Additions and subtractions are cheaper than multiplications which are themselves cheaper than divisions. For this reason, compilers frequently replace division instructions by multiplications. Roughly speaking, it works in this manner. Suppose that you want to…
I know that floating-point arithmetic is a bit crazy on modern computers. For example, floating-point numbers are not associative: 0.1+(0.2+0.3) == 0.599999999999999978 (0.1+0.2)+0.3 == 0.600000000000000089 But, at least, this is fairly consistent in my experience. You should simply not assume fancy properties like…
The ubiquitous IEEE floating-point standard defines two numbers to represent zero, the positive and the negative zeros. You also have the positive and negative infinity. If you compute the inverse of the positive zero, you get the positive infinity. If you compute the inverse of the negative zero, you get the negative…
Suppose you want to pick an integer at random in a set of N elements. Your computer has functions to generate random 32-bit integers, how do you transform such numbers into indexes no larger than N? Suppose you have a hash table with a capacity N. Again, you need to transform your hash values (typically 32-bit ……
In software, we generate random numbers by calling a function called a "random number generator". Such functions have hidden states, so that repeated calls to the function generate new numbers that appear random. If you know this state, you can predict all future outcomes of the random number generators. O'Neill, a…
You sometimes feel the need to make all of your integers positive, without losing any information. That is, you want to map all of your integers from 'signed' integers (e.g., -1, 1, 3, -3) to 'unsigned integers' (e.g., 3,2,6,7). This could be useful if you have a fast function to compress integers that fails to ……
Computers often need random numbers. Most times, random numbers are not actually random... in the sense that they are the output of a mathematical function that is purely deterministic. And it is not even entirely clear what "really random" would mean. It is not clear that we live in a randomized universe... it seems…
I my previous blog post, I documented how one might proceed to compute the number of digits of an integer quickly. E.g., given the integer 999, you want 3 but given the integer 1000, you want 4. It is effectively the integer logarithm in base 10. On computers, you can quickly compute the integer logarithm … Continue…