# The mathematical mystery inside the legendary ’90s shooter Quake 3

**URL:** <https://discourse.julialang.org/t/the-mathematical-mystery-inside-the-legendary-90s-shooter-quake-3/135713>\
**Category:** Offtopic\
**Created:** [February 17, 2026, 7:59pm UTC](https://discourse.julialang.org/t/the-mathematical-mystery-inside-the-legendary-90s-shooter-quake-3/135713 "2026-02-17T19:59:58Z")\
**Posts on this page:** 6\
**Page:** 1

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**Author:** ![mthelm85](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/mthelm85/32/224164_2.png) [@mthelm85](https://discourse.julialang.org/u/mthelm85)\
**Post date:** [February 17, 2026, 7:59pm UTC](https://discourse.julialang.org/t/the-mathematical-mystery-inside-the-legendary-90s-shooter-quake-3/135713/1 "2026-02-17T19:59:58Z")

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I enjoyed this and I think some folks around here will too:

> **[The mathematical mystery inside the legendary ’90s shooter Quake 3](https://www.scientificamerican.com/article/the-mathematical-mystery-inside-the-legendary-90s-shooter-quake-3/)**
>
> Deep within the source code of this online multiplayer game lies an enigmatic number that puzzles and inspires experts to this day

I used Gemini to throw together some Julia code so I could see it in action:

```julia
function fast_inv_sqrt(x::Float32)
       x2 = x * 0.5f0
       y = x

       # 1. The "Bit Hack"
       # Reinterpret float bits as a 32-bit integer
       i = reinterpret(Int32, y)

       # The Magic Number: 0x5f3759df
       # This performs the logarithmic approximation
       i = Int32(0x5f3759df) - (i >> 1)

       # Reinterpret back to a float
       y = reinterpret(Float32, i)

       # 2. Newton-Raphson Iteration
       # One iteration is enough for Quake 3's precision requirements
       # Formula: y = y * (1.5 - (x/2 * y * y))
       y = y * (1.5f0 - (x2 * y * y))

       return y
end

# Usage

julia> x = 64.0f0
64.0f0

julia> result = fast_inv_sqrt(x)
0.124788396f0

julia> actual = 1.0f0 / sqrt(x)
0.125f0

```

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**Author:** ![Benny](https://avatars.discourse-cdn.com/v4/letter/b/49beb7/32.png) [@Benny](https://discourse.julialang.org/u/Benny)\
**Post date:** [February 17, 2026, 8:44pm UTC](https://discourse.julialang.org/t/the-mathematical-mystery-inside-the-legendary-90s-shooter-quake-3/135713/2 "2026-02-17T20:44:42Z")

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Relevant reading with regards to strict aliasing, which is shared by Julia and C/C++ standards:  
[c++ - How to implement fast inverse sqrt without undefined behavior? - Stack Overflow](https://stackoverflow.com/questions/24405129/how-to-implement-fast-inverse-sqrt-without-undefined-behavior)  
C++20 gets `std::bit_cast` to copy bits like Julia’s `reinterpret`. This copy could be optimized to register moves.

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**Author:** ![mbauman](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/mbauman/32/31082_2.png) [@mbauman](https://discourse.julialang.org/u/mbauman)\
**Post date:** [February 17, 2026, 9:08pm UTC](https://discourse.julialang.org/t/the-mathematical-mystery-inside-the-legendary-90s-shooter-quake-3/135713/3 "2026-02-17T21:08:39Z")

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No, that’s not relevant to Julia as far as I’m aware. Julia’s `reinterpret` neither guarantees copied bits nor is it susceptible to UB. Its semantics are quite optimizable and distinct from C++'s behaviors.

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**Author:** ![Benny](https://avatars.discourse-cdn.com/v4/letter/b/49beb7/32.png) [@Benny](https://discourse.julialang.org/u/Benny)\
**Post date:** [February 17, 2026, 10:11pm UTC](https://discourse.julialang.org/t/the-mathematical-mystery-inside-the-legendary-90s-shooter-quake-3/135713/4 "2026-02-17T22:11:19Z")

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It’s irrelevant to Julia by design, it’s the original C implementation that violates strict aliasing.

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**Author:** ![mbauman](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/mbauman/32/31082_2.png) [@mbauman](https://discourse.julialang.org/u/mbauman)\
**Post date:** [February 17, 2026, 10:50pm UTC](https://discourse.julialang.org/t/the-mathematical-mystery-inside-the-legendary-90s-shooter-quake-3/135713/5 "2026-02-17T22:50:06Z")

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That article makes the number far more magical than it is (are they seriously reaching for the decimal 1\,597\,463\,007 and \frac{3}{2} × 2^{23} × (127 - 0.0450465)??). The [actual paper by Lomont](https://www.lomont.org/papers/2003/InvSqrt.pdf) is better, but I think it can be even simpler.

When you do maths on a floating point number as an integer, you end up doing _two_ computations at once: you’re messing with both the exponent and the significand. The most significant part of that is of course not the significand.

Looking at just the exponent maths, `0x5f3759df` has a floating point exponent _value_ of 190. Shifting the integer representation of the input float to the right by a bit divides the exponent _value_ by two. But the value in that exponent slot is offset by 127. And 127 + 127÷2 is… 190! So a slightly simplified real maths approximation of what that bit twiddling is doing is simply:

2^{\left((190 - 127) - (\log\_2(x)+127)\div2) \right)} \\ \exp\big(\frac{-\log(x)}{2}\big)

Which… if you look at it closely, you’ll realize is a convoluted way to spell x^{\frac{-1}{2}}… or \frac{1}{\sqrt{x}}!

But of course, that only gets the exponent right (or close). It’s guaranteed to be within a factor of 2, though, and then the newton step iteratively refines it from there.

So that explains the `0x5f...` leading bits. The mantissa part is indeed quite interesting and I wonder if you could do better by bitmasking out the odd exponents.

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**Author:** ![GunnarFarneback](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/gunnarfarneback/32/1827_2.png) [@GunnarFarneback](https://discourse.julialang.org/u/GunnarFarneback)\
**Post date:** [February 17, 2026, 10:56pm UTC](https://discourse.julialang.org/t/the-mathematical-mystery-inside-the-legendary-90s-shooter-quake-3/135713/6 "2026-02-17T22:56:54Z")

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> number that puzzles and inspires experts

Anyone who finds that puzzling today is hardly an expert. As @mbauman was faster to show there’s obviously a number around that size which gets the result about right, then it’s just a refinement search to find the optimum when combined with a Newton-Raphson step. Maybe that was a bit trickier in the 90s but today you can just brute force it.