## General Speed of Built-in Functions

Generally, only `exp2`, `log2`, `inversesqrt`, `sin`, `cos` and `rcp` can be assumed to be implemented in hardware, every other function is made up of those parts. These are called "special functions" and they are slower than arithmetic. They are generally assumed to take 4 cycles

### SFU

On Nvidia, special functions are run on a separate lane, so they do not cost anything if mixed in with arithmetic, but they still run at 1/4 rate (1/8 on old fermi cards) so using multiple special functions in a row will still cost the same as on more classical cards.

### Implementation of Built-in Functions

``` a / b == a * rcp(b) ```

```1./a == rcp(a) ```

```sqrt(a) == a * inversesqrt(a) ```

```pow(a, b) == exp2(log2(a) * b) ```

```exp(a) == exp2(a * constant) // constant == log2(M_E) ```

```normalize(a) == a * inversesqrt(dot(a,a)) ```

```mix(a, b, c) == (b-a) * c + a ```

### Vectors and Matrices

Vectors are a collection of multiple scalars, the cost every operation on them is multiplied by the number of components of the vector.

So `vec3 * vec3` is 3x more expensive than `float * float`

vec3 * float is as expensive as vec3 * vec3

Matrix multiplications are not the same as simple vector / scalar multiplications, they are way more expensive

• `vec2 * mat2` is 4 cycles
• `vec3 * mat3` is 9 cycles
• `vec4 * mat4` is 16 cycles
• `mat2 * mat2` is 8 cycles
• `mat3 * mat3` is 27 cycles
• `mat4 * mat4` is 64 cycles!

## Identities

``` exp(a+b) == exp(a) * exp(b) ```

```pow(pow(a,b),c) == pow(a, b*c) ```

```a / pow(b, c) == a * pow(b,-c) ```

```log(a) + log(b) == log(a*b) ```

```log(a/b) == log(a) - log(b) ```

```log(pow(a,b)) == b * log(a) ```

```log(sqrt(a)) = log(a) * 0.5 ```

```cross(a, cross(b, c)) = b * dot(a,c) - c * dot(a,b) ```