Math

(n as f64).sqrt() as u64 is the classic hack — and it silently gives the wrong answer for large values. Rust 1.84 stabilized isqrt on every integer type: exact, float-free, no precision traps.

The floating-point trap

Converting to f64, calling .sqrt(), and casting back is the go-to pattern. It looks fine. It isn’t.

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let n: u64 = 10_000_000_000_000_000_000;
let bad = (n as f64).sqrt() as u64;
// bad == 3_162_277_660, but floor(sqrt(n)) is 3_162_277_660 — or is it?
// For many large u64 values, the f64 round-trip is off by 1.

f64 only has 53 bits of mantissa, so for u64 values above 2^53 the conversion loses precision before you even take the square root.

The fix: isqrt

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let n: u64 = 10_000_000_000_000_000_000;
let root = n.isqrt();
assert_eq!(root * root <= n, true);
assert_eq!((root + 1).checked_mul(root + 1).map_or(true, |sq| sq > n), true);

It’s defined on every integer type — u8, u16, u32, u64, u128, usize, and their signed counterparts — and always returns the exact floor of the square root. No casts, no rounding, no surprises.

Signed integers too

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let x: i32 = 42;
assert_eq!(x.isqrt(), 6); // 6*6 = 36, 7*7 = 49

// Negative values would panic, so check first:
let maybe_neg: i32 = -4;
assert_eq!(maybe_neg.checked_isqrt(), None);

Use checked_isqrt on signed types when the input might be negative — it returns Option<T> instead of panicking.

When you’d reach for it

Perfect-square checks, tight loops over divisors, hash table sizing, geometry on integer grids — anywhere you were reaching for f64::sqrt purely to round down, reach for isqrt instead. It’s faster, exact, and one character shorter.

Need to split items into fixed-size pages or chunks? The classic (n + size - 1) / size trick silently overflows. div_ceil does it correctly.

The classic footgun

Paging, chunking, allocating — any time you divide and need to round up, this pattern shows up:

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fn pages_needed(total: u64, per_page: u64) -> u64 {
    (total + per_page - 1) / per_page // ⚠️ overflows when total is large
}

It works until total + per_page - 1 wraps around. With u64::MAX items and a page size of 10, you get a wrong answer instead of a panic or correct result.

The fix: div_ceil

Stabilized in Rust 1.73, div_ceil handles the rounding without intermediate overflow:

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fn pages_needed(total: u64, per_page: u64) -> u64 {
    total.div_ceil(per_page)
}

One method call, no overflow risk, intent crystal clear.

Real-world examples

Allocating pixel rows for a tiled renderer:

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let image_height: u32 = 1080;
let tile_size: u32 = 64;
let tile_rows = image_height.div_ceil(tile_size);
assert_eq!(tile_rows, 17); // 16 full tiles + 1 partial

Splitting work across threads:

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let items: usize = 1000;
let threads: usize = 6;
let chunk_size = items.div_ceil(threads);
assert_eq!(chunk_size, 167); // each thread handles at most 167 items

It works on all unsigned integers

div_ceil is available on u8, u16, u32, u64, u128, and usize. Signed integers also have it (since Rust 1.73), but watch out — it rounds toward positive infinity, which for negative dividends means rounding away from zero.

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let signed: i32 = -7;
assert_eq!(signed.div_ceil(2), -3); // rounds toward +∞, not toward 0

Next time you reach for (a + b - 1) / b, stop — div_ceil already exists and it won’t betray you at the boundaries.

Ever wondered what the next representable floating point number after 1.0 is? Since Rust 1.86, f64::next_up and f64::next_down let you step through the number line one float at a time.

The problem

Floating point numbers aren’t evenly spaced — the gap between representable values grows as the magnitude increases. Before next_up / next_down, figuring out the next neighbor required bit-level manipulation of the IEEE 754 representation:

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fn main() {
    // The hard way (before 1.86): manually decode bits
    let x: f64 = 1.0;
    let bits = x.to_bits();
    let next_bits = bits + 1;
    let next = f64::from_bits(next_bits);

    assert!(next > x);
    assert_eq!(next, 1.0000000000000002);
}

Error-prone, unreadable, and doesn’t handle edge cases like negative numbers, zero, or special values.

The clean way

next_up returns the smallest f64 greater than self. next_down returns the largest f64 less than self:

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fn main() {
    let x: f64 = 1.0;

    let up = x.next_up();
    let down = x.next_down();

    assert!(up > x);
    assert!(down < x);
    assert_eq!(up, 1.0000000000000002);
    assert_eq!(down, 0.9999999999999998);

    // There's no float between x and its neighbors
    assert_eq!(up.next_down(), x);
    assert_eq!(down.next_up(), x);
}

They handle all the edge cases you’d rather not think about — negative numbers, subnormals, and infinity:

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fn main() {
    // Works across zero
    assert_eq!(0.0_f64.next_up(), 5e-324);   // smallest positive f64
    assert_eq!(0.0_f64.next_down(), -5e-324); // largest negative f64

    // Infinity is the boundary
    assert_eq!(f64::MAX.next_up(), f64::INFINITY);
    assert_eq!(f64::INFINITY.next_up(), f64::INFINITY);

    // NaN stays NaN
    assert!(f64::NAN.next_up().is_nan());
}

Practical use: precision-aware comparisons

The gap between adjacent floats is called an ULP (unit in the last place). next_up lets you build tolerance-aware comparisons without guessing at epsilon values:

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fn almost_equal(a: f64, b: f64, max_ulps: u32) -> bool {
    if a == b { return true; }

    let mut current = a;
    for _ in 0..max_ulps {
        current = if a < b { current.next_up() } else { current.next_down() };
        if current == b { return true; }
    }
    false
}

fn main() {
    let a = 0.1 + 0.2;
    let b = 0.3;

    // They're not equal...
    assert_ne!(a, b);

    // ...but they're within 1 ULP of each other
    assert!(almost_equal(a, b, 1));
}

Also available on f32 with the same API. These methods are const fn, so you can use them in const contexts too.

Tired of defining your own golden ratio or Euler-Mascheroni constant? As of Rust 1.94, std ships them out of the box — no more copy-pasting magic numbers.

Before: Roll Your Own

If you needed the golden ratio or the Euler-Mascheroni constant before Rust 1.94, you had to define them yourself:

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const PHI: f64 = 1.618033988749895;
const EULER_GAMMA: f64 = 0.5772156649015329;

This works, but it’s error-prone. One wrong digit and your calculations drift. And every project that needs these ends up with its own slightly-different copy.

After: Just Use std

Rust 1.94 added GOLDEN_RATIO and EULER_GAMMA to the standard consts modules for both f32 and f64:

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use std::f64::consts::{GOLDEN_RATIO, EULER_GAMMA};

fn main() {
    // Golden ratio: (1 + √5) / 2
    let phi = GOLDEN_RATIO;
    assert!((phi * phi - phi - 1.0).abs() < 1e-10);

    // Euler-Mascheroni constant
    let gamma = EULER_GAMMA;
    assert!((gamma - 0.5772156649015329).abs() < 1e-10);

    println!("φ = {phi}");
    println!("γ = {gamma}");
}

These sit right alongside the constants you already know — PI, TAU, E, SQRT_2, and friends.

Where You’d Actually Use Them

Golden ratio shows up in algorithm design (Fibonacci heaps, golden-section search), generative art, and UI layout proportions:

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use std::f64::consts::GOLDEN_RATIO;

fn golden_section_dimensions(width: f64) -> (f64, f64) {
    let height = width / GOLDEN_RATIO;
    (width, height)
}

fn main() {
    let (w, h) = golden_section_dimensions(800.0);
    assert!((w / h - GOLDEN_RATIO).abs() < 1e-10);
    println!("Width: {w}, Height: {h:.2}");
}

Euler-Mascheroni constant appears in number theory, harmonic series approximations, and probability:

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use std::f64::consts::EULER_GAMMA;

/// Approximate the N-th harmonic number using the
/// asymptotic expansion: H_n ≈ ln(n) + γ + 1/(2n)
fn harmonic_approx(n: u64) -> f64 {
    let nf = n as f64;
    nf.ln() + EULER_GAMMA + 1.0 / (2.0 * nf)
}

fn main() {
    // Exact H_10 = 1 + 1/2 + 1/3 + ... + 1/10
    let exact: f64 = (1..=10).map(|i| 1.0 / i as f64).sum();
    let approx = harmonic_approx(10);

    println!("Exact H_10:  {exact:.6}");
    println!("Approx H_10: {approx:.6}");
    assert!((exact - approx).abs() < 0.01);
}

The Full Lineup

With these additions, std::f64::consts now includes: PI, TAU, E, SQRT_2, SQRT_3, LN_2, LN_10, LOG2_E, LOG2_10, LOG10_2, LOG10_E, FRAC_1_PI, FRAC_1_SQRT_2, FRAC_1_SQRT_2PI, FRAC_2_PI, FRAC_2_SQRT_PI, FRAC_PI_2, FRAC_PI_3, FRAC_PI_4, FRAC_PI_6, FRAC_PI_8, GOLDEN_RATIO, and EULER_GAMMA. That’s a pretty complete toolkit for numerical work — all with full f64 precision, available at compile time.