Is it possible to write Quake's fast InvSqrt()

Question

This is just to satisfy my own curiosity.

Is there an implementation of this:

float InvSqrt (float x)
{
   float xhalf = 0.5f*x;
   int i = *(int*)&x;
   i = 0x5f3759df - (i>>1);
   x = *(float*)&i;
   x = x*(1.5f - xhalf*x*x);
   return x;
}

in Rust? If it exists, post the code.

I tried it and failed. I don't know how to encode the float number using integer format. Here is my attempt:

fn main() {
    println!("Hello, world!");
    println!("sqrt1: {}, ",sqrt2(100f64));
}

fn sqrt1(x: f64) -> f64 {
    x.sqrt()
}

fn sqrt2(x: f64) -> f64 {
    let mut x = x;
    let xhalf = 0.5*x;
    let mut i = x as i64;
    println!("sqrt1: {}, ", i);

    i = 0x5f375a86 as i64 - (i>>1);

    x = i as f64;
    x = x*(1.5f64 - xhalf*x*x);
    1.0/x
}

Reference:
1. Origin of Quake3's Fast InvSqrt() - Page 1
2. Understanding Quake’s Fast Inverse Square Root
3. FAST INVERSE SQUARE ROOT.pdf
4. source code: q_math.c#L552-L572

Answer 1

I don't know how to encode the float number using integer format.

There is a function for that: f32::to_bits which returns an u32. There is also the function for the other direction: f32::from_bits which takes an u32 as argument. These functions are preferred over mem::transmute as the latter is unsafe and tricky to use.

With that, here is the implementation of InvSqrt:

fn inv_sqrt(x: f32) -> f32 {
    let i = x.to_bits();
    let i = 0x5f3759df - (i >> 1);
    let y = f32::from_bits(i);

    y * (1.5 - 0.5 * x * y * y)
}

(Playground)

---

This function compiles to the following assembly on x86-64:

.LCPI0_0:
        .long   3204448256        ; f32 -0.5
.LCPI0_1:
        .long   1069547520        ; f32  1.5
example::inv_sqrt:
        movd    eax, xmm0
        shr     eax                   ; i << 1
        mov     ecx, 1597463007       ; 0x5f3759df
        sub     ecx, eax              ; 0x5f3759df - ...
        movd    xmm1, ecx
        mulss   xmm0, dword ptr [rip + .LCPI0_0]    ; x *= 0.5
        mulss   xmm0, xmm1                          ; x *= y
        mulss   xmm0, xmm1                          ; x *= y
        addss   xmm0, dword ptr [rip + .LCPI0_1]    ; x += 1.5
        mulss   xmm0, xmm1                          ; x *= y
        ret

I have not found any reference assembly (if you have, please tell me!), but it seems fairly good to me. I am just not sure why the float was moved into eax just to do the shift and integer subtraction. Maybe SSE registers do not support those operations?

clang 9.0 with -O3 compiles the C code to basically the same assembly. So that's a good sign.

---

It is worth pointing out that if you actually want to use this in practice: please don't. As benrg pointed out in the comments, modern x86 CPUs have a specialized instruction for this function which is faster and more accurate than this hack. Unfortunately, 1.0 / x.sqrt() does not seem to optimize to that instruction. So if you really need the speed, using the _mm_rsqrt_ps intrinsics is probably the way to go. This, however, does again require unsafe code. I won't go into much detail in this answer, as a minority of programmers will actually need it.

Answer 2

This one is implemented with less known union in Rust:

union FI {
    f: f32,
    i: i32,
}

fn inv_sqrt(x: f32) -> f32 {
    let mut u = FI { f: x };
    unsafe {
        u.i = 0x5f3759df - (u.i >> 1);
        u.f * (1.5 - 0.5 * x * u.f * u.f)
    }
}

---

Did some micro benchmarks using criterion crate on a x86-64 Linux box. Surprisingly Rust's own sqrt().recip() is the fastest. But of course, any micro benchmark result should be taken with a grain of salt.

inv sqrt with transmute time:   [1.6605 ns 1.6638 ns 1.6679 ns]
inv sqrt with union     time:   [1.6543 ns 1.6583 ns 1.6633 ns]
inv sqrt with to and from bits
                        time:   [1.7659 ns 1.7677 ns 1.7697 ns]
inv sqrt with powf      time:   [7.1037 ns 7.1125 ns 7.1223 ns]
inv sqrt with sqrt then recip
                        time:   [1.5466 ns 1.5488 ns 1.5513 ns]

Answer 3

You may use std::mem::transmute to make needed conversion:

fn inv_sqrt(x: f32) -> f32 {
    let xhalf = 0.5f32 * x;
    let mut i: i32 = unsafe { std::mem::transmute(x) };
    i = 0x5f3759df - (i >> 1);
    let mut res: f32 = unsafe { std::mem::transmute(i) };
    res = res * (1.5f32 - xhalf * res * res);
    res
}

You can look for a live example here: here