松浦 知也 Matsuura Tomoya
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122 lines
4.0 KiB
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122 lines
4.0 KiB
Markdown
---
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date: 2025-07-24 17:54
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---
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#memo #mathematics
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例えば、[[双方向プログラミング]]的なもので、数値をスライダーで調整できるようになっていた場合、実際のところは5/7とかわかりやすい有理数だったりとか、整数に近い値を表したいのにめちゃくちゃ長い小数点の数値が残ったりする。
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これを、適当なグリッドにスナップするUIの一つとして、許容できる誤差範囲と整数の複雑さを指定して分数に変換するのが良いのではないか。
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ChatGPTに聞いた。[[連分数展開]]というのを使うと良いらしい。[[Rust]]のコードを書いてもらった。
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```rust
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/// 任意の実数 x を近似する収束分数・半収束分数を求める
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/// x: 近似したい実数
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/// eps: 絶対誤差許容値
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/// max_den: 分母の上限
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/// max_num: 分子の上限
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fn rational_approx(
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x: f64,
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eps: f64,
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max_den: u64,
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max_num: u64,
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) -> Vec<(u64, u64)> {
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// 連分数展開の a_k リストを構築
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let mut a: Vec<u64> = Vec::new();
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let mut r = x;
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for _ in 0..64 {
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let ak = r.floor() as u64;
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a.push(ak);
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let frac = r - (ak as f64);
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if frac.abs() < f64::EPSILON {
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break;
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}
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r = 1.0 / frac;
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}
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let mut candidates: Vec<(u64, u64)> = Vec::with_capacity(a.len());
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let (mut p_nm2, mut q_nm2) = (0u128, 1u128);
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let (mut p_nm1, mut q_nm1) = (1u128, 0u128);
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for &ak in &a {
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// 収束分数
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let p = ak as u128 * p_nm1 + p_nm2;
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let q = ak as u128 * q_nm1 + q_nm2;
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if q > max_den as u128 { break; }
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let num = p as u64;
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let den = q as u64;
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if num <= max_num {
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let approx = (num as f64) / (den as f64);
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if (x - approx).abs() <= eps {
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candidates.push((num, den));
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}
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}
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// 半収束分数
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for d in 1..=(ak / 2) {
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let ap = ak - d;
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let p2 = ap as u128 * p_nm1 + p_nm2;
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let q2 = ap as u128 * q_nm1 + q_nm2;
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if q2 <= max_den as u128 && p2 <= max_num as u128 {
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let num2 = p2 as u64;
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let den2 = q2 as u64;
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let approx2 = (num2 as f64) / (den2 as f64);
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if (x - approx2).abs() <= eps {
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candidates.push((num2, den2));
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}
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}
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}
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p_nm2 = p_nm1;
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q_nm2 = q_nm1;
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p_nm1 = p;
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q_nm1 = q;
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}
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candidates.sort_by(|&(p1, q1), &(p2, q2)| {
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let e1 = (x - (p1 as f64)/(q1 as f64)).abs();
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let e2 = (x - (p2 as f64)/(q2 as f64)).abs();
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e1.partial_cmp(&e2).unwrap()
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});
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candidates.dedup();
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candidates
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}
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/// x を percent% 精度で近似するラッパー
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/// x: 近似したい実数
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/// percent: 相対誤差許容値（%）
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/// max_num: 分母、分子の上限
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fn rational_approx_pct(
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x: f64,
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percent: f64,
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max_num: u64,
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) -> Vec<(u64, u64)> {
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let eps = (percent / 100.0) * x.abs();
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rational_approx(x, eps, max_num, max_num)
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}
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fn main() {
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let x = 2.7182818284;
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let percent = 1.0; // 1%
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let max_num = 500;
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let results = rational_approx_pct(x, percent, max_num);
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println!("近似候補（percent={}%）:", percent);
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for (p, q) in results {
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let approx = p as f64 / q as f64;
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let err = (x - approx).abs();
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println!("{}/{} = {:.8}, 誤差 {} ({:.4}%差)",
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p, q, approx, err, err / x.abs() * 100.0);
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}
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}
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```
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```
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近似候補（percent=1%）:
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193/71 = 2.71830986, 誤差 0.00002803075492963103 (0.0010%差)
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106/39 = 2.71794872, 誤差 0.00033311045128181505 (0.0123%差)
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87/32 = 2.71875000, 誤差 0.00046817160000012237 (0.0172%差)
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68/25 = 2.72000000, 誤差 0.0017181716000003178 (0.0632%差)
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49/18 = 2.72222222, 誤差 0.003940393822222443 (0.1450%差)
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19/7 = 2.71428571, 誤差 0.003996114114285465 (0.1470%差)
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```
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悪くなさそう（ここから5個ぐらいまでを提示するとして、） |