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A crazy fast bit-counting technique

Let's say that you've got a lot of numbers that represent bitmasks of some kind, and you want to count how many times each bit is on or off across the entire set.  Maybe you're analyzing game positions represented as bitboards for an AI, or trying to find certain types of weaknesses in random-number generators, like in Forge (a successor to crypto-js) or Cryptocat (read the great write-up at Sophos).

So, you write some very straight-forward code to count the bits.  It grabs one bitmask.  If the lowest-order bit is set, it increments the counter for that bit position.  Then, it right-shifts the bitmask and moves to the counter for the next bit.  Repeat that for each bit in the mask, then repeat that for each bitmask:

const int N = 1000000;
unsigned long x[N]; // Assuming sizeof(unsigned long) == 8, or 64 bits.
int counts[64] = {0};

void count_simple(void) {
    for(int i = 0; i < N; i++) {
        int j = 0;
        while(x[i] != 0) {
            counts[j] += x[i] & 1;
            x[i] >>= 1;

You run your program, and it works correctly, but it's too slow.  I'll show you how to speed this up.  The technique, which applies to languages like Python or Javascript as well as to C, is both crazy, and crazy-fast!

Continue reading "A crazy fast bit-counting technique"

How to hit your goals

There's a simple process you can use to hit your big goals, reliably.

Perhaps you've tried and failed in the past. Maybe you can hit smaller goals, but have trouble maintaining focus on bigger goals, or connecting them to your day-to-day life.

I used to struggle with this, too, until I figured out the right process. Then, I hit every single big goal I had for the next several years—and all of them ahead of schedule!

And, the process is pretty simple.

Continue reading "How to hit your goals"