A collection of tricks with bitwise operations. If you know any other that are not included in this list, share them in comments!

## Integers

**Set n ^{th} bit**

```
x | (1<<n)
```

**Unset n ^{th} bit**

```
x & ~(1<<n)
```

**Toggle n ^{th} bit**

```
x ^ (1<<n)
```

**Round up to the next power of two**

```
unsigned int v; //only works if v is 32 bit
v--;
v |= v >> 1;
v |= v >> 2;
v |= v >> 4;
v |= v >> 8;
v |= v >> 16;
v++;
```

**Get the maximum integer**

```
int maxInt = ~(1 << 31);
int maxInt = (1 << 31) - 1;
int maxInt = (1 << -1) - 1;
```

**Get the minimum integer**

```
int minInt = 1 << 31;
int minInt = 1 << -1;
```

**Get the maximum long**

```
long maxLong = ((long)1 << 127) - 1;
```

**Multiply by 2**

```
n << 1; // n*2
```

**Divide by 2**

```
n >> 1; // n/2
```

**Multiply by the m ^{th} power of 2**

```
n << m;
```

**Divide by the m ^{th} power of 2**

```
n >> m;
```

**Check Equality**

_{This is 35% faster in Javascript}

```
(a^b) == 0; // a == b
!(a^b) // use in an if
```

**Check if a number is odd**

```
(n & 1) == 1;
```

**Exchange (swap) two values**

```
a ^= b;
b ^= a;
a ^= b;
```

**Get the absolute value**

```
//version 1
x < 0 ? -x : x; //version 2 (x ^ (x >> 31)) - (x >> 31);
```

**Get the max of two values**

```
b & ((a-b) >> 31) | a & (~(a-b) >> 31);
```

**Get the min of two values**

```
a & ((a-b) >> 31) | b & (~(a-b) >> 31);
```

**Check whether both numbers have the same sign**

```
(x ^ y) >= 0;
```

**Flip the sign**

```
i = ~i + 1; // or
i = (i ^ -1) + 1; // i = -i
```

**Calculate 2 ^{n}**

```
2 << (n-1);
```

**Whether a number is power of 2**

```
n > 0 && (n & (n - 1)) == 0;
```

**Modulo 2 ^{n} against m**

```
m & (n - 1);
```

**Get the average**

```
(x + y) >> 1;
((x ^ y) >> 1) + (x & y);
```

**Get the m ^{th} bit of n (from low to high)**

```
(n >> (m-1)) & 1;
```

**Set the m ^{th} bit of n to 0 (from low to high)**

```
n & ~(1 << (m-1));
```

**Check if n ^{th} bit is set**

```
if (x & (1<<n)) {
n-th bit is set
} else {
n-th bit is not set
}
```

**Isolate (extract) the right-most 1 bit**

```
x & (-x)
```

**Isolate (extract) the right-most 0 bit**

```
~x & (x+1)
```

**Set the right-most 0 bit to 1**

```
x | (x+1)
```

**n + 1**

```
-~n
```

**n – 1**

```
~-n
```

**Get the negative value of a number**

```
~n + 1;
(n ^ -1) + 1;
```

`if (x == a) x = b; if (x == b) x = a;`

```
x = a ^ b ^ x;
```

## Floats

These are techniques inspired by the fast inverse square root method. Most of these are original.

**Turn a float into a bit-array (unsigned uint32_t)**

```
#include <stdint.h>
typedef union {float flt; uint32_t bits} lens_t;
uint32_t f2i(float x) {
return ((lens_t) {.flt = x}).bits;
}
```

_{Caveat: Type pruning via unions is undefined in C++; use std::memcpy instead.}

**Turn a bit-array back into a float**

```
float i2f(uint32_t x) {
return ((lens_t) {.bits = x}).flt;
}
```

**Approximate the bit-array of a positive float using frexp**

`frexp`

gives the 2^{n} decomposition of a number, so that `man, exp = frexp(x)`

means that man * 2^{exp} = x and 0.5 <= man < 1.

```
man, exp = frexp(x);
return (uint32_t)((2 * man + exp + 125) * 0x800000);
```

_{Caveat: This will have at most 2-16 relative error, since man + 125 clobbers the last 8 bits, saving the first 16 bits of your mantissa.}

**Fast Inverse Square Root**

`return i2f(0x5f3759df - f2i(x) / 2);`

_{Caveat: We’re using the i2f and the f2i functions from above instead.}

See this Wikipedia article for reference.

**Fast n ^{th} Root of positive numbers via Infinite Series**

```
float root(float x, int n) {
#DEFINE MAN_MASK 0x7fffff
#DEFINE EXP_MASK 0x7f800000
#DEFINE EXP_BIAS 0x3f800000
uint32_t bits = f2i(x);
uint32_t man = bits & MAN_MASK;
uint32_t exp = (bits & EXP_MASK) - EXP_BIAS;
return i2f((man + man / n) | ((EXP_BIAS + exp / n) & EXP_MASK));
}
```

See this blog post regarding the derivation.

**Fast Arbitrary Power**

`return i2f((1 - exp) * (0x3f800000 - 0x5c416) + f2i(x) * exp)`

_{Caveat: The 0x5c416 bias is given to center the method. If you plug in exp = -0.5, this gives the 0x5f3759df magic constant of the fast inverse root method.}

See these set of slides for a derivation of this method.

**Fast Geometric Mean**

The geometric mean of a set of `n`

numbers is the n^{th} root of their product.

```
#include <stddef.h>
float geometric_mean(float* list, size_t length) {
// Effectively, find the average of map(f2i, list)
uint32_t accumulator = 0;
for (size_t i = 0; i < length; i++) {
accumulator += f2i(list[i]);
}
return i2f(accumulator / n);
}
```

See here for its derivation.

**Fast Natural Logarithm**

```
#DEFINE EPSILON 1.1920928955078125e-07
#DEFINE LOG2 0.6931471805599453
return (f2i(x) - (0x3f800000 - 0x66774)) * EPSILON * LOG2
```

_{Caveat: The bias term of 0x66774 is meant to center the method. We multiply by ln(2) at the end because the rest of the method computes the log2(x) function.}

See here for its derivation.

**Fast Natural Exp**

`return i2f(0x3f800000 + (uint32_t)(x * (0x800000 + 0x38aa22)))`

_{Caveat: The bias term of 0x38aa22 here corresponds to a multiplicative scaling of the base. In particular, it corresponds to z such that 2z = e}

See here for its derivation.

## Strings

**Convert letter to lowercase:**

```
OR by space => (x | ' ')
Result is always lowercase even if letter is already lowercase
eg. ('a' | ' ') => 'a' ; ('A' | ' ') => 'a'
```

**Convert letter to uppercase:**

```
AND by underline => (x & '_')
Result is always uppercase even if letter is already uppercase
eg. ('a' & '_') => 'A' ; ('A' & '_') => 'A'
```

**Invert letter’s case:**

```
XOR by space => (x ^ ' ')
eg. ('a' ^ ' ') => 'A' ; ('A' ^ ' ') => 'a'
```

**Letter’s position in alphabet:**

```
AND by chr(31)/binary('11111')/(hex('1F') => (x & "\x1F")
Result is in 1..26 range, letter case is not important
eg. ('a' & "\x1F") => 1 ; ('B' & "\x1F") => 2
```

**Get letter’s position in alphabet (for Uppercase letters only):**

```
AND by ? => (x & '?') or XOR by @ => (x ^ '@')
eg. ('C' & '?') => 3 ; ('Z' ^ '@') => 26
```

**Get letter’s position in alphabet (for lowercase letters only):**

```
XOR by backtick/chr(96)/binary('1100000')/hex('60') => (x ^ '`')
eg. ('d' ^ '`') => 4 ; ('x' ^ '`') => 25
```

## Miscellaneous

**Fast color conversion from R5G5B5 to R8G8B8 pixel format using shifts**

```
R8 = (R5 << 3) | (R5 >> 2)
G8 = (R5 << 3) | (R5 >> 2)
B8 = (R5 << 3) | (R5 >> 2)
```

Note: using anything other than the English letters will produce garbage results

In the case of the “Flip the sign” trick, you assume that the compiler uses two’s complement to represent a number, when that might not be the case. The standard leaves that decision to the compiler designer, who may choose among three different representation methods: two’s complement, ones’ complement or sign/magnitude.

Best regards,

Fernando

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