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— よã™ã½ (@yosupot) 2025å¹´6月15æ—¥
src/lib.rs を見ã¾ã™ã€‚
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四則演算以外ã®é–¢æ•°ã¨ã—㦠.sqrt(), .round(), .powi(2), .max(0.0) ãŒä½¿ã‚ã‚Œã¦ã„ã¾ã™ãŒã€ã“ã‚Œã¯å•é¡Œãªã„ã§ã—ょã†ã€‚
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$\gdef\libmpowf#1#2{\operatorname{\texttt{powf}}(#1, #2)}$
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cat <<\EOF | base64 -d > powf_10_0x1c3b16fe640f44p-51 f0VMRgIBAQAAAAAAAAAAAAMAPgABAAAAmGABAAAAAABAAAAAAAAAAAAAAAAAAAAAAAAAAEAAOAADAAAA AAAAAAEAAAAGAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAABAAAAAAAACY4AAAAAAAAAAQAAAAAAAA AQAAAAUAAAAAAAAAAAAAAADwAAAAAAAAAPAAAAAAAAB/hAAAAAAAAH+EAAAAAAAAABAAAAAAAABR5XRk BgAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAQAAAAAAAAAI7L8T9VUFgh 8BMOFgAAAADA2AAA9nMAAOACAACtAAAADgAAABoDAD+RRYRoPYmm2orhgzJO2QdONpiOE6E+T8eYjn6W 4kf8ClWN7V9kIE4vapRbk364KEcho5iL34dNhUw1tLhyYMtJmuoT+ikmY1xeEhUoqIMXu6BrLPh2JgwW eqk2Xjts8gU0dU4hKr7p3hr16jMeow2bNO72D838lCQstkvt1nKRwrAX36nDUVV1rCQlaPVrgfX1zBCD ya+7JSDRPfHSal2sb+Z+apcxzj0wcAsAAMoCAAAOAAAAGgMAF5sJJjNxpgmACx0yTzUav5YmIZ0oDxV9 ljtwrLQdE2EEbj5w5/3iFQrHdD88CR/Bf1TTwTfvjpCPB+cRZygIgEUeeZjYCF2cCNpGuUThjfu1jSiN FV4I3icm+6VC/vV4jmLOC0Fc2uAMJOfvpmGDyCtWqDLcZ6xevc+jNihykih1NZ51oVWkxZszVNWQn0v/ HG/kpf8kVjhT/B7pl9vRPpRwJ8zOj6nkODEYgjbGdPWkGwRkrJXcQyMZaUShjPxdeG/h3XR8Io1WEpZX 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ODt9YOdKGLxrFpW0IeI76tdjtcyrkOE2h2HuO8vwQ8rB83HJkqr35XLqvhqJd6lkCHg7zF/kr2mID4N3 K/YAAAAAAFVQWCEAAAAAAFVQWCEOFg4KaZFkvFt6wnW4CAAAIAIAAMDYAABJAgBK9AAAAA== EOF chmod u+x powf_10_0x1c3b16fe640f44p-51 ./powf_10_0x1c3b16fe640f44p-51 GLIBC_TUNABLES=glibc.cpu.hwcaps=-FMA ./powf_10_0x1c3b16fe640f44p-51
コードãŠã‚ã‚Š $\eod$
GLIBC_TUNABLES=glibc.cpu.hwcaps=-FMA ã«ã¤ã„ã¦ã¯ä¸‹è¨˜ã®è¨˜äº‹ã«æ›¸ãã¾ã—ãŸã€‚「AtCoder 環境ã§å¤‰ãªã“ã¨ã‚’ã—ãŸã‚‰è¨ˆç®—çµæžœãŒå¤‰ã‚ã£ãŸã‚ˆã€ã¨ã„ã†ã“ã¨ã§ã¯ãªãã¦ã€ã€ŒFMA 命令ãŒã‚µãƒãƒ¼ãƒˆã•ã‚Œã¦ã„ãªã„環境ã§ã¯å®Ÿéš›ã«ãã†ãªã‚‹ã€ã¨ã„ã†æ„Ÿã˜ã§ã™ã€‚
å…ƒã¨ãªã‚‹ã‚³ãƒ¼ãƒ‰è‡ªä½“ã¯ä¸‹è¨˜ã§ã™ã€‚
fn main() { let x: f64 = "10.0".parse().unwrap(); let y: f64 = "3.528852450779512".parse().unwrap(); println!("{}", x.powf(y)); }
10.0_f64.powf(3.528852450779512) ã¨æ›¸ãã¨ã‚³ãƒ³ãƒ‘イル時ã«è¨ˆç®—ã•ã‚ŒãŸå€¤ãŒåŸ‹ã‚è¾¼ã¾ã‚Œã¦ã—ã¾ã£ãŸã®ã§ã€ãれを防ããŸã‚ã®ã‚¹ãƒ†ãƒƒãƒ—ã‚’åŠ ãˆã¾ã—ãŸã€‚パースã®éƒ¨åˆ†ã§å·®ç•°ãŒèµ·ãã¦ã„ã‚‹ã‚ã‘ã§ã¯ãªã„ã“ã¨ã¯ã€ãƒ‡ãƒãƒƒã‚¬ã‚’使ã„ãªãŒã‚‰ __ieee754_pow_fma ã‚„ __ieee754_pow_sse2 を直接呼んã ã‚Šã™ã‚‹ã“ã¨ãªã©ã§ç¢ºã‹ã‚られã¾ã™ã€‚æ–‡å—列ã‹ã‚‰ã®å¤‰æ›ã¯ correctly rounded を仮定ã§ãã‚‹ã¨æ€ã£ã¦ã„ã¾ã™ãŒã€å«Œãªäººã¯ f64::from_bits(_) ã¨ã‹ã‚’使ã£ã¦ç¢ºã‹ã‚ã‚‹ã¨ã„ã„ã‹ã‚‚ã—ã‚Œã¾ã›ã‚“。
下記ã«ã¤ã„ã¦ãµã‚“ã‚ã‚Šã—ã¦ã„ã‚‹ã®ã§ã€æ’ƒå¢œã¨å‘¼ã¹ã‚‹ã®ã‹ã¯ã‚ã‹ã‚Šã¾ã›ã‚“。
gen() ã®éƒ¨åˆ†ã¾ã§ã¯ã€Œé…布ツールもジャッジプãƒã‚°ãƒ©ãƒ ã¨å®Œå…¨ã«åŒä¸€ã®æŒ™å‹•ã‚’ã™ã‚‹ã‚ˆã†ã«å®Ÿè£…ã•ã‚Œã¦ã„ã‚‹ã€ãŒä¿è¨¼ã•ã‚Œã¦ã„ã‚‹ã‚ã‘ã§ã¯ãªã„？rand_chacha::ChaCha20Rng::seed_from_u64(seed) ã§è©²å½“ã®ã‚±ãƒ¼ã‚¹ã‚’生æˆã™ã‚‹ã‚ˆã†ãª seed ã®å˜åœ¨ã¯è¬Žæ°—ã«ãªã£ãŸéƒ¨åˆ†ã‚’挙ã’ã¦ã„ãã¾ã™ã€‚
コードä¸ã§ x * 1e-5 ã¨ã„ã†æ›¸ãæ–¹ãŒã—ã°ã—ã°å‡ºã¦ãã¾ã™ãŒã€ã“れ㯠$x \otimes \roundcirc{10^{-5}}$ ã®æ„味ã§ã‚ã£ã¦ã€$x \oslash 10^5 = \roundcirc{x\cdot 10^{-5}}$ ã¨ã¯ç•°ãªã‚Šã¾ã™ã€‚本æ¥ã¯é¿ã‘られる丸ã‚ãŒä½™åˆ†ã«ä¸€å›žã‚ã‚‹ã¨ã„ã†ã“ã¨ã§ã™ã€‚
実際ã€ãŸã¨ãˆã° $$ 3 \oslash {10^5} = (3 + 22432\cdot 2^{-68})\cdot 10^{-5} $$ ã§ã™ãŒã€ $$ 3 \otimes \roundcirc{10^{-5}} = (3 + 122432\cdot 2^{-68})\cdot 10^{-5} $$ ã§ã™ã€‚æ°—ã«ã™ã‚‹ã»ã©ã®èª¤å·®ã§ã¯ãªã„ã¨ã„ã†ç«‹å ´ã‚‚ã‚ã‚Šã¤ã¤ã€ç›¸å¯¾èª¤å·®ã§è¨€ãˆã° $5.46$ å€ç¨‹åº¦ã«å¤§ãããªã£ã¦ã„ã‚‹ã®ã§ã€ãªã‚“ã ã‹ãªã€œã¨ã„ã†æ°—æŒã¡ãŒã‚ã‚Šã¾ã™ã€‚ 精度よりも速度をé‡è¦–ã—ã¦ã„ã¾ã™ã¨è¨€ã‚れれã°ï¼ˆå°‘ãªãã¨ã‚‚ã“ã®æ–‡è„ˆã§ã¯ï¼‰ç´å¾—ã¯ã§ãã¾ã™ã€‚
次ã§ã™ã€‚
æ“作å¯å¦ã®åˆ¤å®š
$v\lt 1-10^{-6}$ ã®å ´åˆ [...] æ“作 2 ã¯å®Ÿè¡Œã§ããªã„。
$1-10^{-6}$ ã¨æ›¸ã„ã¦ã„る部分ã¯å®Ÿéš›ã«ã¯ $1\ominus \roundcirc{10^{-6}}$ ã‚„ $\roundcirc{1-10^{-6}}$ ã®æ„味ãªã®ã‹ï¼Ÿã¨ã„ã†ã¨ã“ã‚ãŒæ°—ã«ãªã‚Šã¾ã™*1。
ã¨ã„ã†ã®ã‚‚ã€ã‚³ãƒ¼ãƒ‰ä¸ã§ v < 1.0 - 1e-6 ã¨åˆ¤å®šã—ã¦ã„ã¾ã™ãŒã€ã“ã®å³è¾ºã®å®Ÿéš›ã®å€¤ã¯
$$
1 \ominus \roundcirc{10^{-6}} = (999999-4047\cdot 2^{-47})\cdot 10^{-6} \lt 1-10^{-6}
$$
ã§ã™ã€‚よã£ã¦ã€$v = 1 \ominus \roundcirc{10^{-6}}$ ã ã£ãŸå ´åˆã€$v\lt 1-10^{-6}$ ã§ã‚ã‚ŠãªãŒã‚‰ã‚‚æ“作 2 ãŒå®Ÿè¡Œã§ãã‚‹ã“ã¨ã«ãªã‚Šã¾ã™ã€‚
ã“れも「ãã†ã„ã†ã¨ã“ã‚を厳密ã«ã‚„ã£ã¦æ°—æŒã¡ã‚ˆããªã‚‹ã®ã¯ãˆã³ã¡ã‚ƒã‚“ã ã‘ã§ã™ã‚ˆç¬‘ã€ã¨è¨€ã‚れるã¨ã€é™ºã—ã„æ°—æŒã¡ã«ãªã‚Šã¤ã¤ç´å¾—ã¯ä¸€å¿œã§ãã¾ã™ã€‚
三ã¤ç›®ã§ã™ã€‚
å€ç²¾åº¦æµ®å‹•å°æ•°ï¼ˆdouble）
浮動å°æ•°ç‚¹ (floating-point) æ•° (number) ãªã®ã§ã€æµ®å‹•å°æ•°ã¨ã„ã†å‘¼ã³æ–¹ã«ã¯å…±æ„Ÿã§ããªã„ã¨ã“ã‚ãŒã‚ã‚Šã¾ã™ã€‚「浮動å°æ•°ç‚¹æ•°ã€ãªã®ã‹ã€Œæµ®å‹•å°æ•°ç‚¹æ•°ã€ãªã®ã‹ã©ã£ã¡ã§ã™ã‹ï¼Ÿ
浮動点数*2ã¨ã‹æµ®ç‚¹æ•°ã¨ã‹ã®æ–¹ãŒç´å¾—ã§ãã¾ã™ã€‚実際ã€ä¸æ–‡ã§ã¯æµ®ç‚¹æ•°ï¼ˆæµ®é»žæ•¸ï¼‰ã¨å‘¼ã¶ã¿ãŸã„ã§ã™ (cf. 浮点数è¿ç®— - 维基百科)。
floating-point datum ã‚„ floating-point number ã¨ã„ã£ãŸç”¨èªžã«ã¤ã„ã¦ã‚‚ IEEE 754 ã§å®šç¾©ã•ã‚Œã¦ã„ã¾ã™ã€‚éŽåŽ»ã«æ›¸ã„ãŸè¨˜äº‹ã§ã‚‚å°‘ã—触れã¾ã—ãŸã€‚
å››ã¤ç›®ã§ã™ã€‚
$\mathrm{rand\_double}(L,U)$: $L$ 以上 $U$ 以下ã®å®Ÿæ•°å€¤ã‚’一様ランダムã«ç”Ÿæˆã™ã‚‹ã€‚
[...] $x_k = -\ln \mathrm{rand\_double}(0, 1)$ ã«ã‚ˆã‚Šç”Ÿæˆã™ã‚‹ã€‚
$0$ ãŒç”Ÿæˆã•ã‚ŒãŸã¨ã $-{\ln 0}$ ã«ãªã£ã¦ã“ã‚ã‚Œãã†ã ãªã¨æ€ã„ã¾ã—ãŸã€‚
ãªãŠã€rng.gen::<f64>() ã®å†…部ã§ä½¿ã‚ã‚Œã¦ã„ã‚‹ rand::distributions::Standard を見るã¨ã€$[0\lldot 1]$ ã§ã¯ãªã $[0\lldot 1)$ ã«è¦‹ãˆã¾ã—ãŸã€‚
境界付近ã®å€¤ã‚’全探索ã—ã¦ã€FMA èµ·å› ã§å¤‰ã‚るケースãŒãŸã¾ãŸã¾å‡ºã¦ããŸã¨ããŒä¸€ç•ªãƒ†ãƒ³ã‚·ãƒ§ãƒ³ãŒé«˜ãã¦ã€ã‚ã¨ã¯ (-’’-;) ã¨ã„ã†æ„Ÿã˜ã§ã—ãŸã€‚
libc ã®å®Ÿè£…次第ã§ã€ã‚‚ã£ã¨ã„ã‚ã„ã‚見ã¤ã‹ã‚‹ã‹ã‚‚ã—ã‚Œã¾ã›ã‚“？ $T$ ã®æ–¹ã¯è¦‹ã¤ã‹ã‚Šã¾ã›ã‚“ã§ã—ãŸãŒã€$D$ ã®æ–¹ã§ M2 Mac ã¨å·®ç•°ãŒã‚るケースã¨ã—ã¦ã¯ä¸‹è¨˜ãŒã‚ã‚Šã¾ã—ãŸã€‚
$$
\begin{aligned}
10^{\texttt{1}.\texttt{7057235B3EE78}_{(16)}\times 2^1} &= \texttt{1}.\texttt{{\small 794000000000}{\footnotesize 07F9E18EE17C}{\scriptsize AEF86997B0EA}{\tiny\ldots}}_{(16)}\times 2^9 \\
10^{\texttt{1}.\texttt{FE70F8B86E1B1}_{(16)}\times 2^1} &= \texttt{1}.\texttt{{\small 2FDBFFFFFFFF}{\footnotesize F8055416F30B}{\scriptsize CB808AF4D765}{\tiny\ldots}}_{(16)}\times 2^{13} \\
\end{aligned}
$$
.round() ã®çµæžœã€å‰è€…㯠$(754, 755)$ ã§å¾Œè€…㯠$(9723, 9724)$ ã§ã—ãŸã€‚
探索ã«ä½¿ã£ãŸã‚³ãƒ¼ãƒ‰
py=' from binascii import hexlify from hashlib import sha256 from math import inf, log10, log2, nextafter from struct import pack # let mut D = f64::round(f64::powf(10.0, rng.gen_range(1.0..4.0))) as i32; # let mut T = f64::round(4000.0 * f64::powf(2.0, rng.gen_range(0.0..4.0))) as usize; nextdown = lambda x: nextafter(x, -inf) nextup = lambda x: nextafter(x, inf) xs = [] n = 10**4 # n = 4000 * 2**4 for i in range(n): y = i + 0.5 x = log10(y) # x = log2(y) xs.extend([nextdown(x), x, nextup(x)]) xs = [*filter(lambda x: 1.0 <= x < 4.0, xs)] # xs = [*filter(lambda x: 0.0 <= x < 4.0, xs)] # print(len(xs)) ys = [] for x in xs: y = round(10**x) # y = round(4000 * 2**x) ys.append((x, y)) # b = pack(f"<{len(ys)}d", *ys) # digest = sha256(b).digest() # print(hexlify(digest).decode().upper()) print(*ys, sep="\n") ' python3 -c "$py" > out-fma.txt GLIBC_TUNABLES=glibc.cpu.hwcaps=-FMA python3 -c "$py" > out-no-fma.txt diff out-fma.txt out-no-fma.txt
log2 ã‚„ log10 ã®å‡ºåŠ›ãŒé•ã†ã¨é¢å€’ãªæ„Ÿã˜ã«ãªã‚Šã¾ã™ãŒã€ãã†ã„ã†å ´åˆã¯åˆ¥é€”よã—ãªã«ã—ã¾ã™ã€‚$\eod$
ã‚ã¾ã‚Š M2 Mac ã®ã“ã¨ã‚’ã‚ã‹ã£ã¦ã„ãªã„ã®ã§ã™ãŒã€libsystem_m.dylib`pow ã¨ã‹ /usr/lib/libSystem.B.dylib (compatibility version 1.0.0, current version 1319.0.0) ã¨ã‹æ›¸ã„ã¦ã‚ã‚Šã¾ã—ãŸã€‚
$\log_e(x)$ ã‚„ $e^x$ ã«ã¤ã„ã¦ã¯ TMD ãŒè§£æ±ºæ¸ˆã¿ã ã£ãŸã¨è¨˜æ†¶ã—ã¦ã„ã¾ã™ãŒã€äºŒå¤‰æ•°ã§ã‚ã‚‹ã¨ã“ã‚ã® $x^y$ ã¯ã¨ã¦ã‚‚大変ã§ã€ã¾ã ã¾ã 未解決ã ã£ãŸã¨è¨˜æ†¶ã—ã¦ã„ã¾ã™ã€‚記憶ã§ã—ゃã¹ã‚Šã™ãŽã€‚解決ã•ã‚Œã‚‹æ—¥ã¯æ¥ã‚‹ã®ã§ã—ょã†ã‹ã€‚
ãŠã‚ã‚Šã§ã™ã€‚
人「浮動å°æ•°ç‚¹æ•°ã®æ¼”ç®—ãªã‚“ã ã‹ã‚‰ç’°å¢ƒã«ã‚ˆã£ã¦çµæžœãŒå¤‰ã‚ã‚Šã†ã‚‹ã®ã¯å½“然ã ã‚ã€
ãã†ã‹...？
$\gdef\libmsin#1{\operatorname{\texttt{s\hspace{-0.03em}i\hspace{-0.03em}n}}(#1)}$ $\gdef\libmcos#1{\operatorname{\texttt{c\hspace{-0.03em}o\hspace{-0.03em}s}}(#1)}$
次ã®ã‚ˆã†ãªå‰æãŒã‚ã‚Šã¾ã™ã€‚
特ã«ã€å‡¦ç†ç³»ãŒ IEEE 754 æº–æ‹ ã§ã‚ã‚‹å‰æã§ã€å››å‰‡æ¼”ç®—ãƒ»å¹³æ–¹æ ¹ãƒ»èžåˆç©å’Œã«ã¤ã„ã¦ã¯ c.r. ãŒä¿è¨¼ã•ã‚Œã¦ã„ã‚‹ã“ã¨ã‚’å‰æã«ã§ãã¾ã™ã€‚ã“ã“ã§ã€èžåˆç©å’Œ (fused multiply-add; FMA) ã¯æµ®å‹•å°æ•°ç‚¹æ•° $x$, $y$, $z$ ã«å¯¾ã—㦠$\roundcirc{xy+z}$ ã‚’è¿”ã™æ¼”ç®—ã§ã™ã€‚$\roundcirc{\roundcirc{xy}+z}$ ã¨ã¯ä¸€èˆ¬ã«ä¸€è‡´ã—ã¾ã›ã‚“。ã¾ãŸã€$\roundcirc{x}$ ã¯å®Ÿæ•° $x$ を浮動å°æ•°ç‚¹åž‹ã«ä¸¸ã‚ãŸå€¤ã‚’æ„味ã—ã¾ã™ã€‚
ã¨ã“ã‚ã§ã€table-maker’s dilemma ã¨ã„ã†åå‰ã§çŸ¥ã‚‰ã‚Œã¦ã„ã‚‹å•é¡ŒãŒã‚ã‚Šã€ä¸‰è§’関数を始ã‚ã¨ã™ã‚‹æ•°å¦é–¢æ•°ãŸã¡ã® c.r. ãªå€¤ã‚’å¾—ã‚‹ã®ã¯è‡ªæ˜Žã§ã¯ã‚ã‚Šã¾ã›ã‚“。
ãã®ãŸã‚ã€sin ã®ã‚ˆã†ãªé–¢æ•°ã¯ c.r. ã§ã‚ã‚‹ã“ã¨ã‚’期待ã§ãã¾ã›ã‚“。実際多ãã®å‡¦ç†ç³»ã§ c.r. ã§ãªã„値ãŒè¿”ã£ã¦ãるケースãŒå˜åœ¨ã—ã¾ã™ã—ã€ãŸã¨ãˆã° double ã«ãŠã‘ã‚‹ sin ã® TMD ã¯ï¼ˆãˆã³ã¡ã‚ƒã‚“ã®çŸ¥ã‚‹é™ã‚Šï¼‰æœªè§£æ±ºã§ã™*1。
一方ã§ã€ä¸Šè¨˜ã§æŒ™ã’ãŸä¸€éƒ¨ã®æ¼”ç®—ãŸã¡ã¯ c.r. ã§ã‚ã‚‹ã“ã¨ã«æ³¨æ„ã—ã¾ã™ã€‚ãŸã¨ãˆã° cos ã‚’èžåˆç©å’Œã‚’用ã„㦠$\libmcos x = \roundcirc{(-x)(x/2) + 1}$ ã¨ã—ã¦å®šç¾©ã—ã¦ã„るライブラリãŒã‚ã£ãŸã¨ã—ã¾ã—ょã†*2。使ã£ã¦ã„る演算ã«å¯æ¬æ€§ãŒã‚ã‚‹ã®ã§ã€å½“然 cos 自体ã«ã‚‚（åŒã˜ãƒ©ã‚¤ãƒ–ラリを使ã£ã¦ã„る環境åŒå£«ãªã‚‰ï¼‰å¯æ¬æ€§ãŒã‚ã‚Šã¾ã™ã€‚実際ã®ãƒ©ã‚¤ãƒ–ラリã«ãŠã„ã¦ã‚‚（ã“ã“ã¾ã§å˜ç´”ã§ã¯ãªã„ã‚‚ã®ã®ï¼‰å››å‰‡æ¼”ç®—ãªã©ã‚’用ã„ã¦è¿‘似をã—ã¦ã„ã‚‹ãŸã‚ã€åŒæ§˜ã®ä¸»å¼µã¯ã§ãã¾ã™ã€‚
ãªã®ã§ã€å…±é€šã®ãƒ©ã‚¤ãƒ–ラリを使ã£ã¦ã„る環境åŒå£«ã§å·®ç•°ãŒã‚ã£ãŸå ´åˆã€ã‚ã¾ã‚Šè‡ªç„¶ã«ã€Œå½“然ã ã‚ã€ã¨ã¯è¨€ãˆãªã„ã§ã—ょã†ã¨ã„ã†æ°—æŒã¡ãŒã‚ã‚Šã¾ã™ã€‚
コンパイル時ã®è¨ˆç®—ã¨å®Ÿè¡Œæ™‚ã®è¨ˆç®—ã§ã¯ç²¾åº¦ãŒç•°ãªã‚‹å ´åˆãŒã‚ã‚Šã¾ã™ã€‚ãã®ãŸã‚ã€å®Ÿè¡Œæ™‚ã®ç²¾åº¦ã«ã¤ã„ã¦èª¿ã¹ã‚ˆã†ã¨ã—ã¦ã„ã‚‹ã¨ãã¯ã€ã‚³ãƒ³ãƒ‘イル時ã«è¨ˆç®—ã—ãŸå€¤ãŒä½¿ã‚ã‚Œã¦ã„ãªã„ã“ã¨ã‚’（é©å®œ disassemble ã—ã¤ã¤ï¼‰ç¢ºã‹ã‚ã¦ãŠãå¿…è¦ãŒã‚ã‚Šã¾ã™ã€‚
ãŸã¨ãˆã°ã€ç«‹æ–¹æ ¹ã‚’計算ã™ã‚‹ä»¥ä¸‹ã®ã‚³ãƒ¼ãƒ‰ã‚’考ãˆã¾ã™ã€‚
#include <math.h> #include <stdio.h> #include <stdlib.h> int main(void) { printf("%.60g\n", cbrt(3375.0)); printf("%.60g\n", cbrt(strtod("3375.0", NULL))); }
å‰è€…ã¯ã‚³ãƒ³ãƒ‘イル時ã«è¨ˆç®—ã•ã‚Œã€å¾Œè€…ã¯ï¼ˆæ–‡å—列ã‹ã‚‰å¤‰æ›ã™ã‚‹éƒ¨åˆ†ãŒå®Ÿè¡Œæ™‚ã«è¡Œã‚ã‚Œã¦ï¼‰å®Ÿè¡Œæ™‚ã«è¨ˆç®—ã•ã‚Œã¦ã€ä¸‹è¨˜ã®å‡ºåŠ›ã«ãªã‚‹å ´åˆãŒã‚ã‚Šã¾ã™ (wandbox)。
15 14.9999999999999982236431605997495353221893310546875
ã•ã¦ã€æ¬¡ã®å…¥åŠ›ã‚’考ãˆã¾ã™ã€‚
$$ \begin{aligned} x &= \texttt{1}.\texttt{\small 00000000004AD}_{(16)}\times 2^{0} \\ &= {\small 1.000000000000265}{\footnotesize 787392095262475}{\scriptsize 777417421340942}{\tiny 3828125}. \end{aligned} $$
AtCoder 環境ã§ã¯æ¬¡ã®ã‚ˆã†ã«ãªã‚Šã¾ã—ãŸã€‚ $$ \begin{aligned} \libmsin x &= \texttt{1}.\texttt{\small AED548F0911FB}_{(16)}\times 2^{-1} \\ &= {\small 0.841470984808040}{\footnotesize 056712741261435}{\scriptsize 439810156822204}{\tiny 58984375}. \end{aligned} $$ 手元環境㯠M2 Mac 上㮠colima ã§å‹•ã‹ã—ã¦ã„ã‚‹ Ubuntu ã§ã™*3。
FROM ubuntu:25.04@sha256:dc4565c7636f006c26d54c988faae576465e825ea349fef6fd3af6bf5100e8b6
% colima start --cpu 2 --memory 2 --disk 30
ã“ã®ç’°å¢ƒã§ã¯æ¬¡ã®ã‚ˆã†ã«ãªã‚Šã¾ã—ãŸã€‚ $$ \begin{aligned} \libmsin x &= \texttt{1}.\texttt{\small AED548F0911FC}_{(16)}\times 2^{-1} \\ &= {\small 0.841470984808040}{\footnotesize 167735043723951}{\scriptsize 093852519989013}{\tiny 671875}. \end{aligned} $$
真ã®å€¤ã¯ $$ \sin(x) = \texttt{1}.\texttt{{\small AED548F0911FB}{\footnotesize 7FEDF452F4881}{\scriptsize \ldots}}_{(16)}\times 2^{-1} $$ ãªã®ã§ã€AtCoder å´ã§ã¯ c.r. ãªå€¤ãŒå¾—られã¦ã„ã‚‹ã“ã¨ã«ãªã‚Šã¾ã™ã€‚
note: $1.0$ ã‹ã‚‰ $2^{-52}$ ãšã¤å¢—ã‚„ã—ã¦ã„ã£ã¦æœ€åˆã«è¦‹ã¤ã‹ã£ãŸã‚±ãƒ¼ã‚¹ãŒã“ã‚Œã ã£ãŸã¨ã„ã†ã ã‘ã§ã€ä¸€è‡´ã—ãªã„ケースã¯ãŸãã•ã‚“ã‚ã‚Šã¾ã™ã€‚
note: M2 Mac å´ãŒ c.r. ã«ãªã‚‹ã‚±ãƒ¼ã‚¹ã¨ã—ã¦ã¯ $x = \texttt{1}.\texttt{00000000088BC}_{(16)}\times 2^{0}$ ãŒã‚ã‚Šã¾ã—ãŸã€‚
ã†ã€œã€œã€œã‚“。ã“ã®å·®ç•°ã¯ã©ã®ã‚ˆã†ã«ã—ã¦ç”Ÿã˜ã‚‹ã‚‚ã®ãªã®ã§ã—ょã†ï¼Ÿ
実際ã«èª¿ã¹ãŸæ‰‹é †ã«æ²¿ã£ã¦æ›¸ã„ã¦ã„ãã¾ã™ã€‚目次ã§ãƒã‚¿ãƒãƒ¬ã«ãªã‚‹ã®ã‚’é¿ã‘ã‚‹ãŸã‚番å·ã§ç« ç«‹ã¦ã—ã¦ã„ã¾ã™ã€‚
安直ãªäºˆæƒ³ã¨ã—ã¦ã€libc ã®ãƒãƒ¼ã‚¸ãƒ§ãƒ³ãŒé•ã†ã®ã§ã¯ãªã„ã‹ï¼Ÿã¨ã„ã†ã®ãŒæœ€åˆã«æ€ã„当ãŸã‚‹éƒ¨åˆ†ã§ã—ãŸã€‚
実際ã«ã¯ sin ã®å®Ÿä½“㯠libm ã®ä¸ã«ã‚ã‚‹ã¯ãšã§ã™ãŒã€ã‚³ãƒ¼ãƒ‰ã®ãƒãƒ¼ã‚¸ãƒ§ãƒ³ã¨ã—ã¦ã¯ libc ã¨å…±é€šã ã¨æ€ã‚れるã®ã§ã€libc ã®ãƒãƒ¼ã‚¸ãƒ§ãƒ³ã§èª¿ã¹ã‚‹ã“ã¨ã«ã—ã¾ã™ã€‚
AtCoder ã®ã‚³ãƒ¼ãƒ‰ãƒ†ã‚¹ãƒˆã§ä¸‹è¨˜ã‚’実行ã—ã¾ã—ãŸã€‚
cp /lib/x86_64-linux-gnu/libc.so.6 . chmod u+x libc.so.6 ./libc.so.6
元㮠libc.so.6 ã®å®Ÿè¡Œæ¨©é™ãŒãªã‹ã£ãŸã“ã¨ã¨ã€ãã‚Œã«å¯¾ã—ã¦ã® chmod ãŒã§ããªã‹ã£ãŸã®ã§ã€ã‚³ãƒ”ーã—ã¦ããŸã‚‚ã®ã‚’実行ã—ã¾ã—ãŸã€‚ã‚ã‚‹ã„ã¯ä¸‹è¨˜ã‚’実行ã™ã‚‹ã®ã§ã‚‚よã‹ã£ãŸã‹ã‚‚ã—ã‚Œã¾ã›ã‚“。
/usr/lib64/ld-linux-x86-64.so.2 /lib/x86_64-linux-gnu/libc.so.6
得られãŸçµæžœã¯æ¬¡ã®é€šã‚Šã§ã™ã€‚
GNU C Library (Ubuntu GLIBC 2.36-0ubuntu4) stable release version 2.36. Copyright (C) 2022 Free Software Foundation, Inc. This is free software; see the source for copying conditions. There is NO warranty; not even for MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. Compiled by GNU CC version 12.2.0. libc ABIs: UNIQUE IFUNC ABSOLUTE Minimum supported kernel: 3.2.0 For bug reporting instructions, please see: <https://bugs.launchpad.net/ubuntu/+source/glibc/+bugs>.
手元ã§ã¯æ¬¡ã®é€šã‚Šã§ã—ãŸã€‚
% /lib/x86_64-linux-gnu/libc.so.6
GNU C Library (Ubuntu GLIBC 2.41-6ubuntu1) stable release version 2.41. Copyright (C) 2025 Free Software Foundation, Inc. This is free software; see the source for copying conditions. There is NO warranty; not even for MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. Compiled by GNU CC version 14.2.0. libc ABIs: UNIQUE IFUNC ABSOLUTE Minimum supported kernel: 3.2.0 For bug reporting instructions, please see: <https://bugs.launchpad.net/ubuntu/+source/glibc/+bugs>.
ãŸã—ã‹ã«ãƒãƒ¼ã‚¸ãƒ§ãƒ³ã¯é•ã£ã¦ã„ã¾ã™ãŒã€ã“ã®ãƒãƒ¼ã‚¸ãƒ§ãƒ³ã®é•ã„ã§æŒ™å‹•ã«å·®ãŒå‡ºã‚‹ã®ã‹ã¯ã‚ã‹ã‚Šã¾ã›ã‚“。ãã“ã§ã€åˆ¥ã®ã‚¸ãƒ£ãƒƒã‚¸ã§ã‚‚調ã¹ã¦ã¿ã‚‹ã“ã¨ã«ã—ã¾ã™ã€‚
Library Checker ã§ã¯ã‚³ãƒ¼ãƒ‰ãƒ†ã‚¹ãƒˆãŒãªã„（ã¨æ€ã†ï¼‰ã®ã§ã€A + B を用ã„ã¦èª¿ã¹ã¾ã™*4。標準エラー出力ãŒè¦‹ã‚Œã‚‹ã®ã§åŠ©ã‹ã‚Šã¾ã™ã€‚
#include <stdio.h> #include <stdlib.h> int main() { int a, b; scanf("%d %d", &a, &b); printf("%d\n", a + b); system("/lib/x86_64-linux-gnu/libc.so.6 >&2"); }
得られãŸçµæžœã¯æ¬¡ã®é€šã‚Šã§ã™ã€‚
GNU C Library (Debian GLIBC 2.36-9+deb12u10) stable release version 2.36. Copyright (C) 2022 Free Software Foundation, Inc. This is free software; see the source for copying conditions. There is NO warranty; not even for MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. Compiled by GNU CC version 12.2.0. libc ABIs: UNIQUE IFUNC ABSOLUTE Minimum supported kernel: 3.2.0 For bug reporting instructions, please see: <http://www.debian.org/Bugs/>.
ã¾ãŸã€åŒã‚¸ãƒ£ãƒƒã‚¸ã‚µãƒ¼ãƒã® Dockerfile.GCC を手元ã§å‹•ã‹ã—ã¦ã¿ã¾ã™ã€‚
% /lib/x86_64-linux-gnu/libc.so.6
åŒã˜å‡ºåŠ›ãŒå¾—られã¾ã—ãŸã€‚
GNU C Library (Debian GLIBC 2.36-9+deb12u10) stable release version 2.36. Copyright (C) 2022 Free Software Foundation, Inc. This is free software; see the source for copying conditions. There is NO warranty; not even for MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. Compiled by GNU CC version 12.2.0. libc ABIs: UNIQUE IFUNC ABSOLUTE Minimum supported kernel: 3.2.0 For bug reporting instructions, please see: <http://www.debian.org/Bugs/>.
åŠ ãˆã¦ã€å„環境㧠sin(0x1.00000000004ADp+0) ã®å‡ºåŠ›ã‚‚確èªã—ã¾ã—ãŸã€‚
手元ã®ã‚‚ã®ã¯æœ€åˆã«è¦‹ã¦ã„ãŸã‚‚ã®ã‚’「手元 â‘ ã€ã€Dockerfile.GCC を使ã£ãŸã‚‚ã®ã‚’「手元 â‘¡ã€ã¨ã—ã¦è¨˜è¼‰ã—ã¾ã™ã€‚
| 環境 | libc ã®ãƒãƒ¼ã‚¸ãƒ§ãƒ³ | sin(0x1.00000000004ADp+0) |
|---|---|---|
| AtCoder | Ubuntu GLIBC 2.36-0ubuntu4 | 0x1.AED548F0911FBp-1 |
| Library Checker | Debian GLIBC 2.36-9+deb12u10 | 0x1.AED548F0911FBp-1 |
| 手元 ①| Ubuntu GLIBC 2.41-6ubuntu1 | 0x1.AED548F0911FCp-1 |
| 手元 ② | Debian GLIBC 2.36-9+deb12u10 | 0x1.AED548F0911FCp-1 |
Library Checker ã¨æ‰‹å…ƒ â‘¡ 㯠libc ã®ãƒãƒ¼ã‚¸ãƒ§ãƒ³ã¯åŒã˜ã§ã™ãŒã€sin ã®è¨ˆç®—çµæžœã¯ç•°ãªã£ã¦ã„ã‚‹ãŸã‚ã€libc ã®ãƒãƒ¼ã‚¸ãƒ§ãƒ³èµ·å› ã§ã¯ãªã„ã¨è€ƒãˆã‚‹ã®ãŒå¦¥å½“ãã†ã§ã™ã€‚ã¨ã„ã†ã“ã¨ã§ä»®èª¬ 1 ã¯æ£„å´ã§ã™ã€‚
コード㯠sin_0x100000000004adp-52.c ã¨ã—ã¦ä¸‹è¨˜ã‚’想定ã—ã¦ã„ã¾ã™ã€‚
#include <math.h> #include <stdio.h> #include <stdlib.h> int main(void) { double x = strtod("1.0000000000002658", NULL); printf("%.13a\n", sin(x)); }
ã“れを手元ã§ã‚³ãƒ³ãƒ‘イルã—ãŸã¨ã㨠AtCoder 環境ã§å®Ÿè¡Œã—ãŸã¨ãã¨ã§ã€åˆ¥ã®ãƒã‚¤ãƒŠãƒªãŒã§ãã¦ã„ã‚‹ã®ã§ã¯ãªã„ã‹ï¼Ÿã¨ã„ã†ã®ã‚’ç–‘ã„ã¾ã—ãŸï¼ˆå„種都åˆã§åˆ¥ã®ãƒã‚¤ãƒŠãƒªã«ãªã‚‹ã“ã¨ã¯ã‚ã‚Šå¾—ã‚‹ã¨æ€ã†ãŒã€ä»Šå›žã®è¨ˆç®—ã«å½±éŸ¿ã™ã‚‹éƒ¨åˆ†ã«å·®ç•°ãŒã‚ã‚‹ã®ã§ã¯ï¼Ÿã¨ã„ã†ã“ã¨ï¼‰ã€‚ãã“ã§ã€æ‰‹å…ƒã§ã‚³ãƒ³ãƒ‘イルã—ãŸã‚‚ã®ã‚’ AtCoder 環境ã§å®Ÿè¡Œã—ã¦ã¿ã¾ã™ã€‚
% gcc -o sin_0x100000000004adp-52{,.c} -O2 -lm % printf 'cat <<\\EOC | base64 -d | gzip -d > a.out\n%s\nEOC\nchmod u+x a.out\n./a.out\n' "$(cat sin_0x100000000004adp-52 | gzip -9 | base64)"
下記ãŒå¾—られã¾ã—ãŸã€‚
0x1.aed548f0911fbp-1
ã“ã‚Œã¯å…ƒã€… AtCoder 環境ã§ã‚³ãƒ³ãƒ‘イルã—ãŸã¨ãã¨åŒã˜çµæžœã§ã™ã‹ã‚‰ã€ä»®èª¬ 2 も棄å´ã§ãã¾ã™ã€‚
libc ã®ãƒãƒ¼ã‚¸ãƒ§ãƒ³ã¯åŒã˜ã§ã‚‚ã€ãƒ“ルド時ã®è¨å®šï¼ˆï¼Ÿï¼‰ï¼ˆ./configure ã§ãªã‚“ã‹ã™ã‚‹ã‚„ã¤ï¼‰ã«ã‚ˆã£ã¦å·®ç•°ãŒã‚ã‚‹ã®ã§ã¯ï¼Ÿã¨æ€ã„ã¾ã—ãŸã€‚Library Checker ã¨æ‰‹å…ƒ â‘¡ ã§ä¸‹è¨˜ã‚’実行ã—ã¾ã™ã€‚
% sha256sum /lib/x86_64-linux-gnu/libc.so.6
ã©ã¡ã‚‰ã‚‚åŒã˜å€¤ã«ãªã£ãŸã®ã§ã€åŒã˜ãƒ•ã‚¡ã‚¤ãƒ«ã§ã‚ã‚‹ã¨æ€ã‚ã‚Œã¾ã™*5。
1d25fd63234b59e4c581564c7a6d8f5c6cf36eee757e3d26f4b0808dd36a4896 /lib/x86_64-linux-gnu/libc.so.6
よã£ã¦ã€ä»®èª¬ 3 も棄å´ã§ã™ã€‚libc ã«å·®ç•°ãŒã‚ã‚‹ã‚ã‘ã§ã¯ãªã„ã¨ã„ã†ã“ã¨ã§ã™ã。ã“ã“を最åˆã«è¦‹ã¦ã„ã‚Œã°ä»®èª¬ 1 を調ã¹ã‚‹å¿…è¦ã¯ãªã‹ã£ãŸã‹ã‚‚ã§ã™ãŒã€ã¾ããã®ã¸ã‚“ã¯ä»•æ–¹ãªã„ã§ã—ょã†ã€‚
途方ã«æš®ã‚Œã¤ã¤ã€gdb ã§ã‚¹ãƒ†ãƒƒãƒ—実行ã—ã¦ã¿ã¾ã™ã€‚
(gdb) file ./sin_0x100000000004adp-52 (gdb) b sin@plt (gdb) r (gdb) si 2
__sin_sse2 (x=1.0000000000002658) at ../sysdeps/ieee754/dbl-64/s_sin.c:201
ã©ã†ã‚„らã€__sin_sse2 ã¨ã„ã†ã®ãŒå‘¼ã°ã‚Œã¦ã„ã‚‹ã¿ãŸã„ã§ã™ã€‚AtCoder 環境ã§ã¯é•ã†ã‚‚ã®ãŒå‘¼ã°ã‚Œã¦ã„ã‚‹ã®ã‹ãªï¼Ÿ 仮説 3 ã‹ã‚‰ libc ã®å®Ÿä½“ã¯å…±é€šã§ã‚ã‚ã†ã®ã§ã€libc ã®ä¸ã« __sin_sse2 ã‚„ __sin_??? ãªã©ãŒã‚ã‚Šã€å®Ÿè¡Œæ™‚ã«ä½•ã‚‰ã‹ã®åŸºæº–ã«ã‚ˆã£ã¦é©åˆ‡ã¨æ€ã‚れるもã®ã‚’呼んã§ã„ã‚‹ã®ã‹ãªï¼Ÿã¨ã„ã†æ°—æŒã¡ã«ãªã£ã¦ãã¾ã™ã€‚
ãã†ã„ãˆã° colima 㧠AVX2 ã¨ã‹ã‚’有効ã«ã™ã‚‹ã“ã¨ãŒã§ããŸã‚ˆã†ãª...？ã¨ã„ã†ã®ã‚’æ€ã„出ã—ãŸã®ã§ã€æœ‰åŠ¹ã«ã—ã¦ã¿ã¾ã™ã€‚
% colima stop % colima start --cpu 2 --memory 2 --disk 30 --cpu-type max,+avx,+avx2
ã‹ã‚‰ã®ã€å®Ÿè¡Œã§ã™ã€‚
% ./sin_0x100000000004adp-52
#> 0x1.aed548f0911fbp-1
ãŠã£ã¨......？？ ã“れ㯠AtCoder 環境ã§ã®å‡ºåŠ›ã¨åŒã˜ã‚‚ã®ã§ã™ã。ã•ã£ãã¨åŒæ§˜ã« gdb ã§ã‚‚確èªã—ã¦ã¿ã‚‹ã¨ã€ä¸‹è¨˜ã®ã‚‚ã®ãŒå‘¼ã°ã‚Œã¦ã„るよã†ã§ã™ã€‚
__sin_fma (x=1.0000000000002658) at ../sysdeps/ieee754/dbl-64/s_sin.c:201
ã©ã†ã‚„らã€__sin_fma 㨠__sin_sse2 ã¨ã„ã†ã®ãŒã‚ã‚Šã€ãƒãƒ¼ãƒ‰ã‚¦ã‚§ã‚¢ã®æ©Ÿèƒ½ã«å¿œã˜ã¦åˆ¥ã®ã‚‚ã®ãŒå‘¼ã°ã‚Œã¦ã„ãã†ã§ã™ã€‚FMA を使ã†å ´åˆã¨ãã†ã§ãªã„å ´åˆã§è¨ˆç®—çµæžœã¯ä¸€èˆ¬ã«ä¸€è‡´ã—ãªã„ã®ã§ã€è¨ˆç®—çµæžœãŒç•°ãªã‚‹ã®ã¯ç´å¾—ã§ãã¾ã™ã€‚
ã¨ã„ã†ã“ã¨ã§ã€ã©ã†ã„ã†é–¢æ•°ãŒç”¨æ„ã•ã‚Œã¦ã„ã¦ã€ã©ã†ã„ã†æ¡ä»¶ã§èª°ãŒå‘¼ã³åˆ†ã‘ã¦ã„ã‚‹ã®ã‹ï¼Ÿã¨ã„ã†ã¨ã“ã‚ãŒæ°—ã«ãªã£ã¦ãã¾ã™ã€‚
glibc ã®ã‚³ãƒ¼ãƒ‰ã‚’見ã¦ã„ãã¾ã™ã€‚ã‚¿ã‚°ã¯æœ€æ–°ã® glibc-2.41.9000 を見ã¾ã—ãŸãŒã€ä»–ã‚‚ãŸã¶ã‚“ã‚ã¾ã‚Šå¤‰ã‚らãªã„ã§ã—ょã†*6。手元 â‘ ã®ç’°å¢ƒã¨åˆã‚ã›ãŸãƒãƒ¼ã‚¸ãƒ§ãƒ³ã§ã™ã。
ã¾ãšã€__sin_??? ã¯ä¸‹è¨˜ã®ãƒ•ã‚¡ã‚¤ãƒ«ã§å®šç¾©ã•ã‚Œã¦ã„ã¾ã™ã€‚
__sin_sse2 in sysdeps/x86_64/fpu/multiarch/s_sin.c__sin_fma in sysdeps/x86_64/fpu/multiarch/s_sin-fma.c__sin_fma4 in sysdeps/x86_64/fpu/multiarch/s_sin-fma4.c__sin_avx in sysdeps/x86_64/fpu/multiarch/s_sin-avx.cãã‚Œãžã‚Œã€ä¸‹è¨˜ã®ã‚ˆã†ãªè¨˜è¿°ãŒã‚ã‚Šã¾ã™ã€‚
#define __sin __sin_fma #define SECTION __attribute__ ((section (".text.fma"))) #include <sysdeps/ieee754/dbl-64/s_sin.c>
é©å®œç«¯æŠ˜ã‚Šã¾ã™ãŒã€sysdeps/ieee754/dbl-64/s_sin.c ã§ã¯æ¬¡ã®ã‚ˆã†ã«ãªã£ã¦ã„ã¾ã™ã€‚
double SECTION __sin (double x) { /* ... */ } libm_alias_double (__sin, sin)
マクãƒã‚’展開ã™ã‚‹ã¨æ¬¡ã®ã‚ˆã†ãªæ„Ÿã˜ã§ã™ã。
double __attribute__ ((section (".text.fma"))) __sin_fma (double x) { /* ... */ } libm_alias_double (__sin_fma, sin)
sysdeps/x86_64/fpu/multiarch/Makefile 内ã§ä¸‹è¨˜ã®ã‚ˆã†ãªè¨˜è¿°ãŒã‚ã‚Šã€ãã‚Œãžã‚Œã«åˆã‚ã›ãŸæœ€é©åŒ–ã‚’ã—ã¦ã„ã‚‹ã®ã ã‚ã†ã¨æ€ã£ã¦ã„ã¾ã™ã€‚
CFLAGS-s_sin-fma.c = -mfma -mavx2 CFLAGS-s_sin-fma4.c = -mfma4 CFLAGS-s_sin-avx.c = -msse2avx -DSSE2AVX
ã¾ãŸã€sysdeps/x86_64/fpu/multiarch/s_sin.c ã«ä¸‹è¨˜ã®è¨˜è¿°ãŒã‚ã‚Šã¾ã™ã€‚
extern double __redirect_sin (double); # define SYMBOL_NAME sin # include "ifunc-avx-fma4.h" libc_ifunc_redirected (__redirect_sin, __sin, IFUNC_SELECTOR ()); libm_alias_double (__sin, sin)
libm_alias_double (__sin, sin) ã«é–¢ã—ã¦ã¯ã€sysdeps/generic/libm-alias-double.h ãªã©ã‚’見ã¤ã¤ã€çµå±€ weak_alias(__sin, sin) ãªã©ãŒå‘¼ã°ã‚Œã¦ã„ãã†ã§ã™ã€‚
libc_ifunc_redirected (...) ã«é–¢ã—ã¦ã¯ã€include/libc-symbols.h ã«èª¬æ˜Žã®ã‚³ãƒ¡ãƒ³ãƒˆãŒæ›¸ã‹ã‚Œã¦ã„ã¾ã™ã€‚“The following macros are used for indirect function symbols in libc.so.†ã‹ã‚‰å§‹ã¾ã‚‹ã€ã‚µãƒ³ãƒ—ルã¤ã㧠80 行程度ã®ã‚‚ã®ã§ã™ã€‚
int foo (int __bar) ã¨ã„ã†é–¢æ•°ã«å¯¾ã—ã¦ã€libc.so ãŒä½¿ã‚れる全部ã®ç’°å¢ƒã§ä½¿ãˆã‚‹ã¨ã¯é™ã‚‰ãªã„（実行時ã«ã¯åˆ¤å®šã§ãる）ãƒãƒ¼ãƒ‰ã‚¦ã‚§ã‚¢æ©Ÿèƒ½ã‚’用ã„ãŸå®Ÿè£… int __foo_special (int __bar) ã¨ã€ãƒ•ã‚©ãƒ¼ãƒ«ãƒãƒƒã‚¯ã®å®Ÿè£… int __foo_default (int __bar) ã‚’æä¾›ã™ã‚‹å ´åˆã®ã“ã¨ãŒæ›¸ã‹ã‚Œã¦ã„ã¾ã™ã€‚ã¾ã•ã«ã€œï¼Ÿã¨ã„ã†æ„Ÿã˜ãŒã—ã¾ã™ã€‚
sysdeps/generic/ifunc-init.h ã«ã€REDIRECT_NAME, OPTIMIZE, IFUNC_SELECTOR ã® #define ãŒã‚ã‚Šã¾ã™ã€‚ã¾ãŸã€IFUNC_SELECTOR (sin_ifunc_selector？) ã®å®šç¾©ã«ã¤ã„ã¦ã¯ sysdeps/x86_64/fpu/multiarch/ifunc-avx-fma4.h ã«è¨˜è¿°ãŒã‚ã‚Šã¾ã™ã€‚
CPU_FEATURE_USABLE_P (cpu_features, FMA) ã«ã¤ã„ã¦ã¯ã€sysdeps/x86/include/cpu-features.h ã«å®šç¾©ãŒã‚ã‚Šã€è«¸ã€…ã‚ã£ã¦
((ptr->features[index_cpu_FMA].active.reg_FMA & bit_cpu_FMA) != 0)
ã‹ã‚‰ã®
((ptr->features[0].active.ecx & (1u << 12)) != 0)
ã«å±•é–‹ã•ã‚Œã¦ã„ãã†ã§ã™ã€‚
__get_cpu_features () ã«ã¤ã„ã¦ã¯ã€ã–ã£ãり飛ã°ã—ã¤ã¤ã€é–¢é€£ã™ã‚‹ãƒ•ã‚¡ã‚¤ãƒ«ã¯ä¸‹è¨˜ãŸã¡ã§ã™ã€‚
sysdeps/x86/include/cpu-features.h
#define __get_cpu_features() _dl_x86_get_cpu_features()
sysdeps/x86/dl-get-cpu-features.c
__ifunc (__x86_cpu_features, __x86_cpu_features, NULL, void, _dl_x86_init_cpu_features); const struct cpu_features * _dl_x86_get_cpu_features (void) { return &GLRO(dl_x86_cpu_features); }
sysdeps/generic/ldsodefs.h
# define GLRO(name) _##name
sysdeps/x86/cpu-features.c
static inline void init_cpu_features (struct cpu_features *cpu_features)
実行ファイルを実行ã™ã‚‹ã¨ãã€å®Ÿéš›ã«ã¯ program interpreter ã¨å‘¼ã°ã‚Œã‚‹ãƒ—ãƒã‚°ãƒ©ãƒ ãŒå®Ÿè¡Œã•ã‚Œã¦ã„ãŸã‚Šã—ã¾ã™ã€‚./foo.sh を実行ã™ã‚‹ã¨ãã« shebang ã«æ›¸ã„ãŸãƒ—ãƒã‚°ãƒ©ãƒ ãŒå®Ÿè¡Œã•ã‚Œã‚‹ã®ã¨ä¼¼ã¦ã„ã‚‹æ°—ãŒã—ã¾ã™ï¼Ÿ*7
ãªã‚“ã®ãƒ—ãƒã‚°ãƒ©ãƒ ãŒå®Ÿè¡Œã•ã‚Œã‚‹ã‹ã¯ã€readelf -l ã®å‡ºåŠ›ã® Program Headers ã® INTERP ã®éƒ¨åˆ†ã§ç¢ºèªã™ã‚‹ã“ã¨ãŒã§ãã¾ã™ã€‚
% readelf -l sin_0x100000000004adp-52 #: #> Program Headers: #> Type Offset VirtAddr PhysAddr #> FileSiz MemSiz Flags Align #: #> INTERP 0x00000000000003a4 0x00000000000003a4 0x00000000000003a4 #> 0x000000000000001c 0x000000000000001c R 0x1 #> [Requesting program interpreter: /lib64/ld-linux-x86-64.so.2] #:
ã“ã‚Œã¯ã‚·ãƒ³ãƒœãƒªãƒƒã‚¯ãƒªãƒ³ã‚¯ã«ãªã£ã¦ãŠã‚Šã€/usr/lib/x86_64-linux-gnu/ld-linux-x86-64.so.2 を指ã—ã¦ã„ã¾ã™ã€‚ld.so ã‚‚åŒæ§˜ã®ãƒ•ã‚¡ã‚¤ãƒ«ã‚’指ã—ã¦ã„るよã†ãªã®ã§ã€åå‰ã®çŸã„ã“ã¡ã‚‰ã‚’使ã£ã¦ã„ãã“ã¨ã«ã—ã¾ã™ã€‚実際ã€ä¸‹è¨˜ã§ã‚‚åŒã˜å‡ºåŠ›ãŒå¾—られるã“ã¨ãŒç¢ºã‹ã‚られã¾ã™ã。
% ld.so ./sin_0x100000000004adp-52
#> 0x1.aed548f0911fbp-1
ã¨ã„ã†ã“ã¨ã§ã€ld.so ã«ã‚ãŸã‚‹éƒ¨åˆ†ã®ã‚³ãƒ¼ãƒ‰ã‚’追ã£ã¦ã„ãã¾ã—ょã†ï¼ˆgdb を使ã„ã¤ã¤èª¿ã¹ã¦ã„ã¾ã™ï¼‰ã€‚ã“れもã€ã‚³ãƒ¼ãƒ‰ã¨ã—ã¦ã¯ glibc ã¨ä¸€ç·’ã«æä¾›ã•ã‚Œã¦ã„ã¾ã™ã€‚
sysdeps/x86_64/dl-machine.h ã® RTLD_START ã®éƒ¨åˆ†ï¼ˆé–¢æ•°ã¨ã—ã¦ã¯ _start ã«ã‚ãŸã‚‹ï¼‰ã«ä¸‹è¨˜ã®ã‚³ãƒ¡ãƒ³ãƒˆãŒã‚ã‚Šã¾ã™ã€‚
Initial entry point code for the dynamic linker. The C function ‘_dl_start’ is the real entry point; its return value is the user program's entry point.
_dl_start 㯠elf/rtld.c ã«ã‚ã‚Šã¾ã™ã€‚ファイル冒é 㫠“Run time dynamic linker.†ã¨ã‚ã‚‹ã®ã§ã€åå‰ã¯ãã†ã„ã†æ„味ã ã¨æ€ã„ã¾ã™ã€‚下記ã®ã‚ˆã†ãªå‘¼ã³å‡ºã—ãŒã‚ã‚Šã€CPU ã®å„機能ãŒä½¿ãˆã‚‹ã‹ãŒèª¿ã¹ã‚‰ã‚Œã¾ã™ã€‚
_dl_start (elf/rtld.c:581)_dl_start_final (elf/rtld.c:496)_dl_sysdep_start (sysdeps/unix/sysv/linux/dl-sysdep.c:119)dl_platform_init (sysdeps/x86_64/dl-machine.h:209)_dl_x86_init_cpu_features (sysdeps/x86/dl-get-cpu-features.c:41)init_cpu_features (sysdeps/x86/cpu-features.c:754)note: dl_start 内部㧠dl_start_final を呼ã³ã€ãã®å†…部㧠dl_sysdep_start を呼ã³ã€... ã¨ã„ã†æ„Ÿã˜ã§ã™ã€‚
ã“ã†ã—ã¦å‰åŠãƒ‘ートã§è¦‹ãŸ _dl_x86_cpu_features ã‚’åˆæœŸåŒ–ã—ã¾ã™ã€‚dl_sysdep_start 内部ã‹ã‚‰ã€ä¸‹è¨˜ã®å‘¼ã³å‡ºã—ãŒç¶šãã¾ã™ã€‚
_dl_start (elf/rtld.c:581)_dl_start_final (elf/rtld.c:496)_dl_sysdep_start (sysdeps/unix/sysv/linux/dl-sysdep.c:141)dl_main (elf/rtld.c:2267)_dl_relocate_object (elf/dl-reloc.c:346)_dl_relocate_object_no_relro (elf/dl-reloc.c:296)elf_dynamic_do_Rela (elf/do-rel.h:138)elf_machine_rela (sysdeps/x86_64/dl-machine.h:309)
value = ((ElfW(Addr) (*) (void)) value) ();ãŠãらãã€libm.so.6 ã‹ã‚‰å–å¾—ã§ãã‚‹ sin ã¯ã€
libc_ifunc_redirected (__redirect_sin, __sin, sin_ifunc_selector ()); libm_alias_double (__sin, sin)
ã®ãŠã‹ã’ã§
extern __typeof (__redirect_sin) __sin; __typeof (__redirect_sin) *__sin_ifunc (void) __asm__ ("__sin"); __attribute__ ((__optimize__ ("-fno-stack-protector"))) __typeof (__redirect_sin) *__sin_ifunc (void) { INIT_ARCH (); __typeof (__redirect_sin) *res = sin_ifunc_selector (); return res; } __asm__ (".type " "__sin" ", %gnu_indirect_function"); extern __typeof (__sin) sin __attribute__ ((weak, alias ("__sin"))) __attribute_copy__ (__sin);
ã®ã‚ˆã†ã«ãªã£ã¦ãŠã‚Šã€sin（ã¨ã„ã†åå‰ã§å…¬é–‹ã•ã‚Œã¦ã„ã‚‹ã‚„ã¤ï¼‰ã‚’呼んã§è¿”ã£ã¦ããŸã‚¢ãƒ‰ãƒ¬ã‚¹ã‚’ [email protected] ã«å…¥ã‚Œã‚‹ã¨ã„ã†æ„Ÿã˜ã«ãªã£ã¦ã„るよã†ã«è¦‹ãˆã¾ã™ (cf. 6.12.5 Controlling Names Used in Assembler Code)。
実際㫠[email protected] ã« __sin_fma ãŒå…¥ã‚‹ã®ã¯ä¸‹è¨˜ã§ã™ã€‚RELRO ã®è¨å®šã«ã‚ˆã‚‹ã‹ã‚‚。今回㯠full RELRO ã«ãªã£ã¦ã„ã¾ã™ã€‚
_dl_start (elf/rtld.c:581)_dl_start_final (elf/rtld.c:496)_dl_sysdep_start (sysdeps/unix/sysv/linux/dl-sysdep.c:141)dl_main (elf/rtld.c:2267)_dl_relocate_object (elf/dl-reloc.c:346)_dl_relocate_object_no_relro (elf/dl-reloc.c:296)elf_dynamic_do_Rela (elf/do-rel.h:138)elf_machine_rela (sysdeps/x86_64/dl-machine.h:418)
*(Elf64_Addr *) reloc_addr = (Elf64_Addr) value + reloc->r_addend;ã“ã†ã—ã¦ã€æ¡ä»¶ã«å¿œã˜ã¦ __sin_sse2 ã‚„ __sin_fma ãªã©ãŒ [email protected] ã«ãŠå±Šã‘ã•ã‚Œã‚‹ã‚ã‘ã§ã™ã。
ld.so(8) ã‚„ 38.6 Hardware Capability Tunables を眺ã‚ã¦ã„ã‚‹ã¨ã€glibc.cpu.hwcaps ã¨ã„ã†ã®ã®å˜åœ¨ã«æ°—ã¥ãã¾ã™ã€‚環境変数 GLIBC_TUNABLES ã«ã‚ˆã£ã¦æŒ‡å®šã§ãるよã†ã§ã™ã€‚
% ./sin_0x100000000004adp-52 #> 0x1.aed548f0911fbp-1 % GLIBC_TUNABLES=glibc.cpu.hwcaps=-FMA,-AVX ./sin_0x100000000004adp-52 #> 0x1.aed548f0911fcp-1
ã‚らã‚らã€çµæžœãŒå¤‰ã‚ã‚‹ã“ã¨ãŒç¢ºã‹ã‚られã¾ã—ãŸã。オプション --list-diagnostics ã«ã‚ˆã£ã¦æƒ…å ±ã‚’ç¢ºèªã§ãã‚‹ã®ã§ã€å·®åˆ†ã‚‚見ã¦ã¿ã¾ã—ょã†ã€‚
% diff <(ld.so --list-diagnostics) <(GLIBC_TUNABLES=glibc.cpu.hwcaps=-FMA,-AVX ld.so --list-diagnostics) #: #> 93c94 #> < x86.cpu_features.features[0x0].active[0x2]=0x7ed83203 #> --- #> > x86.cpu_features.features[0x0].active[0x2]=0x6ed82203
ãŸã—ã‹ã«ã€è©²å½“ã™ã‚‹ãƒ“ット (cf. sysdeps/x86/include/cpu-features.h) ã«å½±éŸ¿ãŒã‚ã‚‹ã“ã¨ãŒã‚ã‹ã‚Šã¾ã—ãŸã€‚
GCC ã§ã‚‚ Clang ã§ã‚‚ã€FMA ãŒä½¿ãˆã‚‹ç’°å¢ƒå‘ã‘ã«ã¯ï¼ˆ-ffast-math ãªã—ã§ã‚‚）x * y + z ã®å½¢ã®å¼ã«å¯¾ã—㦠FMA を使ã†ã‚ˆã†ãªæœ€é©åŒ–ã‚’ã™ã‚‹ã‚ˆã†ã§ã™ã€‚
今回触れ㟠__sin_sse2 㨠__sin_fma ã§å€¤ãŒå¤‰ã‚ã‚‹ç†ç”±ã¨ã—ã¦ã€æœ¬è³ªçš„ãªéƒ¨åˆ†ã¯ã“ã®æœ€é©åŒ–ã®ã›ã„ã§ã™ã。
ãã‚“ãªã“ã¨ã—ã¦ã„ã„ã®ã‹...？
ã„や〜満足満足。調査ã‹ã‚‰è¨˜äº‹ã¾ã§ã€åœŸæ—¥ä¸€å›žåˆ†ã§æ¸ˆã‚“ã§ã†ã‚Œã—ã„ã§ã™ã€‚ä»–ã®é€²æ—ã¯ã‚ã‚Šã¾ã›ã‚“。 GOT, PLT, RELRO ãªã©ã®èª¬æ˜Žã‚’çœãã¾ã—ãŸãŒã€èˆˆå‘³ãŒã‚る人ã¯é©å®œèª¿ã¹ã¦ãã ã•ã‚Œã°ã¨æ€ã„ã¾ã™ã€‚CTF ã® pwn ã¨ã‹ã‚’ã™ã‚‹ã¨è©³ã—ã•ãŒå¢—ã™ã‹ã‚‚？ã¨ã„ã†æ°—ã‚‚ã—ã¾ã™ã€‚
C ã§å‘¼ã³å‡ºã™å ´åˆã«é™ã‚‰ãš Rust ã‚„ Python ãªã©ã®è¨€èªžã‹ã‚‰ã§ã‚‚（ã‚ã‚‹ã„ã¯ã‚‚ã¡ã‚ã‚“ã€GCC ã§ã‚³ãƒ³ãƒ‘イルã™ã‚‹å ´åˆã«é™ã‚‰ãš Clang を使ã†å ´åˆã§ã‚‚）libm ã«ä¾å˜ã—ã¦ã„ã‚‹ã¨å½±éŸ¿ã‚’å—ã‘る部分ãªã®ã§ã€æ°—ã‚’ã¤ã‘ãªãゃã ãªã¨æ€ã„ã¾ã—ãŸã€‚
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*1:定義域をå分å°ã•ã絞ã£ãŸå ´åˆã«ã¤ã„ã¦ã¯è§£æ±ºã•ã‚Œã¦ã„ã‚‹ã¨èªè˜ã—ã¦ã„ã¾ã™ã€‚æ•°å万ビット？ã¨ã‹ã®ã‚ã¡ã‚ƒã§ã‹ã„上界ã¯ç¤ºã•ã‚Œã¦ã„ãŸã‚ˆã†ãªæ°—ã‚‚ã—ã¾ã™ã€‚
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apple, banana, cherry, durian ã¨ã‹ã«ã—ã¦ã»ã—ã‹ã£ãŸ...
コンテストä¸ã«ã‚„ã£ãŸè€ƒå¯Ÿã«æ²¿ã£ã¦è©±ã‚’進ã‚ã¾ã™ã€‚
一番å³ã® A ã¨ä¸€ç•ªå³ã® B ã«æ³¨ç›®ã—ã¾ã™ã€‚ãれらã®é–“ã« C ãŒå«ã¾ã‚Œã‚‹ã‚±ãƒ¼ã‚¹ã¯æ¬¡ã®ã‚ˆã†ã«ãªã‚Šã¾ã™ã€‚
AAAABBBABBA ++ A ++ BCCBB ++ B ++ CDCCCCDCDD
ã™ãªã‚ã¡ã€æ¬¡ã‚’連çµã—ãŸã‚‚ã®ã«ãªã‚Šã¾ã™ã€‚
よã£ã¦ã€ã“ã®é€šã‚Šæ•°ã¯æ¬¡ã®å¼ã§è¡¨ã›ã¾ã™ã€‚ $$ \begin{aligned} &\phantom{{}={}} \sum_{i=0}^{B-1} \sum_{j=1}^C \binom{A-1+i}{i} \binom{B-i-1+j}{j} \binom{C-j+D}{D} \\ &= \sum_{i=0}^{B-1} \binom{A-1+i}{i} \sum_{j=1}^C \binom{B-i-1+j}{j} \binom{C-j+D}{D} \end{aligned} $$
ã¾ãŸã€ãれらã®é–“ã« C ãŒå«ã¾ã‚Œãªã„ケースã¯æ¬¡ã®ã‚ˆã†ã«ãªã‚Šã¾ã™ã€‚
AABABBAABAABABBA ++ CDDCDCDCCCCDCC
ã™ãªã‚ã¡ã€æ¬¡ã‚’連çµã—ãŸã‚‚ã®ã«ãªã‚Šã¾ã™ã€‚
特ã«ã€A 㨠C ã®åˆ¶ç´„ã‹ã‚‰ã€ä¸€ç•ªå³ã® B よりも一番å³ã® A ã®æ–¹ãŒå³ã«æ¥ã‚‹ã‚±ãƒ¼ã‚¹ã‚‚ã“ã¡ã‚‰ã«å«ã¾ã‚Œã¾ã™ã€‚ ã“ã®é€šã‚Šæ•°ã¯æ¬¡ã®ã‚ˆã†ã«ãªã‚Šã¾ã™ã€‚
$$ \binom{A+B}{B} \binom{C+D}{D} $$
後者ã®ã‚±ãƒ¼ã‚¹ã¯ã„ã„ã¨ã—ã¦ã€å‰è€…ã®ã‚±ãƒ¼ã‚¹ã®å†…å´ã® $\sum$ を高速ã«è¨ˆç®—ã™ã‚‹å¿…è¦ãŒã‚ã‚Šã¾ã™ã€‚ $$ \sum_{i=0}^{B-1} \binom{A-1+i}{i} \underbrace{\sum_{j=1}^C \binom{B-i-1+j}{j} \binom{C-j+D}{D}}_{b_i} $$ ã™ãªã‚ã¡ã€ã“ã® $b_i$ ã®éƒ¨åˆ†ã§ã™ã€‚ã¨ã‚Šã‚ãˆãš Wolfram|Alpha ã«æŠ•ã’ã¾ã™ã€‚ $$ b_i = \frac{C+D}C\binom{C+D-1}{D} \cdot({}_2 F_1(-C, B-i; -C-D; 1)-1) $$ äºŒé …ä¿‚æ•°ã¨éšŽä¹—ã®éƒ¨åˆ†ã‚’æ•´ç†ã™ã‚‹ã¨ä¸‹è¨˜ã«ãªã‚Šã¾ã™ã€‚ $$ b_i = \frac{(C+D)!}{D!\,C!} \cdot({}_2 F_1(-C, B-i; -C-D; 1)-1) $$
ã“ã“ã§ã€${}_2 F_1(a, b; c; z)$ 㯠超幾何関数 (hypergeometric function) ã¨å‘¼ã°ã‚Œã€ä¸‹è¨˜ã§ä¸Žãˆã‚‰ã‚Œã‚‹ã‚‰ã—ã„ã§ã™*1。
$$
{}_2 F_1(a, b; c; z) = \sum_{n=0}^{\infty} \frac{(a)_n\,(b)_n}{(c)_n}\cdot\frac{z^n}{n!}
$$
$(a)_n$ 㯠Pochhammer ã®ä¸Šæ˜‡éšŽä¹—冪ã§ã€$(a)_n = a\cdot(a+1)\cdot\cdots\cdot(a+n-1)$ ã§ã™ã€‚Wolfram|Alpha ã§ã¯ Hypergeometric2F1 ã¨ã„ã†åå‰ã§å®šç¾©ã•ã‚Œã¦ã„ã¾ã™ã€‚
å›°ã£ã¦ã„るよã†ã™
$c\lt a\lt 0\le b$ ã¨ã—ã¦ã€ $$ \begin{aligned} {}_2 F_1(a, b; c; 1) &= \sum_{n=0}^{\infty} \frac{(a)_n\,(b)_n}{(c)_n}\cdot\frac{1}{n!} \\ &= \sum_{n=0}^{|a|} \frac{(a)_n\,(b)_n}{(c)_n}\cdot\frac{1}{n!} \\ &= \sum_{n=0}^{|a|} \frac{(a)_n}{(c)_n}\cdot\frac{(b)_n}{n!} \\ &= \sum_{n=0}^{|a|} \frac{a\cdot(a+1)\cdot\cdots\cdot(a+n-1)}{c\cdot(c+1)\cdot\cdots\cdot(c+n-1)}\cdot\frac{(b)_n}{n!} \\ &= \sum_{n=0}^{|a|} \frac{|a|\cdot(|a|-1)\cdot\cdots\cdot(|a|-(n-1) )}{|c|\cdot(|c|+1)\cdot\cdots\cdot(|c|-(n-1) )}\cdot\frac{(b)_n}{n!} \\ &= \sum_{n=0}^{|a|} \frac{|a|!}{(|a|-n)!} \frac{(b+n-1)!}{(b-1)!} \frac{(|c|-n)!}{|c|!} \frac1{n!} \\ &= \sum_{n=0}^{|a|} \frac{|a|!}{(|a|-n)!\,n!} \frac{(b+n-1)!}{(b-1)!\,n!} \frac{(|c|-n)!\,n!}{|c|!} \\ &= \sum_{n=0}^{|a|} \binom{|a|}{n} \binom{b+n-1}{n} \binom{|c|}{n}^{-1} \end{aligned} $$ ã“ã“ã‹ã‚‰ã©ã†ã™ã‚Œã°...？ リンク先ã®å¼ (74) ã«ã‚ˆã‚Œã° $$ {}_2 F_1(a, b; c; 1) = \frac{\Gamma(c)\,\Gamma(c-a-b)}{\Gamma(c-a)\,\Gamma(c-b)} $$ らã—ã„ã§ã™ã€‚ã“ã“ã§ã¯ $c$ ãŒè² ãªã®ã§ãƒãƒãƒ£ãƒ¡ãƒãƒ£ã«ç™ºæ•£ã—ãã†ã§ã™ã€‚å›°ã£ãŸã€‚
$a\lt 0$ ã‚„ $\Gamma(n) = (n-1)\,\Gamma(n-1)$ ãªã©ã‹ã‚‰ $$ \begin{aligned} {}_2 F_1(a, b; c; 1) &= \frac{\Gamma(c)\,\Gamma(c-a-b)}{\Gamma(c-a)\,\Gamma(c-b)} \\ &= \frac{\Gamma(c)}{\Gamma(c-a)}\frac{\Gamma(c-a-b)}{\Gamma(c-b)} \\ &= \frac{\Gamma(c)}{(c-a-1)\,\Gamma(c-a-1)}\frac{(c-a-b-1)\,\Gamma(c-a-b-1)}{\Gamma(c-b)} \\ &= \cdots \\ &= \frac{(c-a-b-1)\cdot\cdots\cdot(c-b)}{(c-a-1)\cdot\cdots\cdot c} \end{aligned} $$ ã¨ã—ã¦ã‚ˆã„ã®ã§ã‚ã‚Œã°ã€è¨ˆç®—ã¯ã§ããã†ã§ã™ã€‚今回計算ã—ãŸã„ã®ã¯ ${}_2 F_1(-C, B-i; -C-D; 1)$ ã ã£ãŸã®ã§ã€ $$ \begin{aligned} &\phantom{{}={}} {}_2 F_1(-C, B-i; -C-D; 1) \\ &= \frac{-C-D-(-C)-(B-i)-1}{-C-D-(-C)-1}\cdot\cdots\cdot\frac{-C-D-(B-i)}{-C-D} \\ &= \frac{-D-B+i-1}{-D-1}\cdot\cdots\cdot\frac{-C-D-B+i}{-C-D} \\ &= \frac{(B-i)+(D+1)}{D+1}\cdot\cdots\cdot\frac{(B-i)+(D+C)}{D+C} \\ &= \frac{( (B-i)+(D+C) )!}{( (B-i)+D)!}\cdot\frac{D!}{(D+C)!} \end{aligned} $$ ã¨ãªã‚Šãã†ã§ã™ã€‚
ãŸã¨ãˆã°ã€$(B, C, D, i) = (10, 5, 2, 3)$ ã¨ã—ã¦ã¿ã‚‹ã¨ã€${}_2 F_1(-5, 7; -7; 1) = \frac{286}3$ ã¨ãªã‚Š*2〠$$ \begin{aligned} b_i &= \frac{7!}{2!\,5!}\cdot({}_2 F_1(-5, 7; -7; 1)-1) \\ &= 21\cdot \left(\frac{286}3-1\right) \\ &= 7\cdot(286-3) \\ &= 1981 \end{aligned} $$ ã¨ãªã‚Šã€å…ƒã€…ã® $\sum$ ã§è¨ˆç®—ã—ãŸå€¤ã¨ä¸€è‡´ã—ã¾ã™ã€‚$\eod$
コンテスト本番ã§ã¯ã€è¶…幾何関数ã¯ã©ã†ã—よã†ã‚‚ãªã„ã¨æ€ã£ã¦ã„ãŸã®ã§åˆ¥ã®æ–¹é‡ã‚’考ãˆã¾ã—ãŸã€‚ $B$, $C$, $D$ ã‚’ $1$ ãšã¤å¤‰ãˆãªãŒã‚‰å€¤ã‚’眺ã‚ã‚‹ã“ã¨ã§ã€ä¸‹è¨˜ã«æ°—ã¥ãã¾ã™ã€‚
Observation 1: $b_B = 0$ ã‹ã¤ã€å„ $0\le i\lt B$ ã«å¯¾ã—ã¦ä¸‹è¨˜ãŒæˆã‚Šç«‹ã¤ã€‚ $$ b_i = b_{i+1} + \binom{(B-1)-i+C+D}{C-1} $$
$(B-1)-i$ ã®éƒ¨åˆ†ã‚„漸化å¼ã®å‘ããªã©ã‹ã‚‰ã€$b'_i$ を下記ã§å®šç¾©ã—ã¾ã™ã€‚ $$ b'_i = \sum_{j=1}^C \binom{i+j}{i} \binom{C-j+D}{D} $$ ãªãŠã€$b'_i = b_{(B-1)-i}$ ã§ã™ã€‚
Conjecture 2: $b_{-1} = 0$ ã‹ã¤ã€å„ $0\le i\le B-1$ ã«å¯¾ã—ã¦ä¸‹è¨˜ãŒæˆã‚Šç«‹ã¤ã€‚ $$ b'_i = b'_{i-1} + \binom{i+C+D}{C-1} $$
More observations
$i=0$ ã®ã¨ãã€Hockey-stick identity ã‹ã‚‰ $$ \begin{aligned} b'_0 &= \sum_{j=1}^C \binom jj\binom{C-j+D}D \\ &= \sum_{j=1}^C \binom{C-j+D}D \\ &= \sum_{j'=0}^{C-1} \binom{D+j'}D \\ &= \binom{D+C}{D+1} \end{aligned} $$ ãŒæˆã‚Šç«‹ã¤ã€‚一方〠$$ \begin{aligned} b'_0 &= b'_{-1} + \binom{C+D}{C-1} \\ &= 0 + \binom{D+C}{D+1} \end{aligned} $$ よりã€$i = 0$ ã®ã¨ãã¯æˆã‚Šç«‹ã¤ã€‚
$$ \begin{aligned} b'_i - b'_{i-1} &= \sum_{j=1}^C \binom{i+j}j\binom{C-j+D}D - \sum_{j=1}^C \binom{i-1+j}j\binom{C-j+D}D \\ &= \sum_{j=1}^C \left( \binom{i+j}j - \binom{i-1+j}j\right) \binom{C-j+D}D \\ % &= \sum_{j=1}^C \left( \frac{(i+j)!}{i!\, j!} - \frac{(i+j-1)!}{(i-1)!\, j!}\right) \binom{C-j+D}D \\ % &= \sum_{j=1}^C \left( \frac{(i+j)!}{i!\, j!} - \frac{i}{i+j}\frac{(i+j)!}{i!\, j!}\right) \binom{C-j+D}D \\ % &= \sum_{j=1}^C \frac{j}{i+j}\frac{(i+j)!}{i!\, j!} \binom{C-j+D}D \\ % &= \sum_{j=1}^C \frac{(i+j-1)!}{i!\, (j-1)!} \binom{C-j+D}D \\ &= \sum_{j=1}^C \binom{i-1+j}{j-1} \binom{C-j+D}D \\ &= \sum_{j=0}^{C-1} \binom{i+j}{j} \binom{C-1-j+D}D \\ &= \sum_{j=0}^{C-1} \binom{i+j}{i} \binom{C-1-j+D}D \\ % &= \sum_{j=0}^{C-1} \binom{i+j}{i} \binom{C-1-j+D}D \\ % &= \binom{i+C+D}{i+D+1} \\ % &= \binom{i+C+D}{C-1}. \quad\qed \end{aligned} $$
よã£ã¦ã€ä¸‹è¨˜ãŒæˆã‚Šç«‹ã¦ã°ã‚ˆã„。 $$ \sum_{j=0}^{C-1} \binom{i+j}{i} \binom{C-1-j+D}D = \binom{i+D+C}{i+D+1}. $$
Lemma 3: 下記ãŒæˆã‚Šç«‹ã¤ã€‚
$$ \sum_{i=0}^{k-1} \binom{n+i}{n} \binom{m+k-1-i}{m} = \binom{n+m+k}{n+m+1}. $$
note: Pascal ã®ä¸‰è§’形㮠$n$ 列目㨠$m$ 列目ã‹ã‚‰ $k$ 個æŒã£ã¦ããŸã‚‚ã®ã®ç•³ã¿è¾¼ã¿ã«ç›¸å½“ã™ã‚‹ã€‚
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Proof
$m$, $k$ ã«é–¢ã™ã‚‹å¸°ç´æ³•ã§ç¤ºã™ã€‚
$$ P(m, k) \iff \ForallL{n}{\sum_{i=0}^{k-1} \binom{n+i}{n} \binom{m+k-1-i}{m} = \binom{n+m+k}{n+m+1}} $$ ã¨ã™ã‚‹ã€‚
To-be-proved 1: $P(m, 1)$
$$ \begin{aligned} \sum_{i=0}^{1-1} \binom{n+i}{n} \binom{m+1-1-i}{m} &= \binom{n}{n} \binom{m}{m} \\ &= 1, \\ \binom{n+m+1}{n+m+1} &= 1. \end{aligned} $$
To-be-proved 2: $P(m, k+1) \wedge P(m+1, k) \implies P(m+1, k+1)$
$$ \begin{aligned} &\phantom{{}={}} \sum_{i=0}^{(k+1)-1} \binom{n+i}{n} \binom{(m+1)+(k+1)-1-i}{m+1} \\ &= \sum_{i=0}^{(k+1)-1} \binom{n+i}{n} \left(\binom{(m+1)+(k+1)-1-i-1}{(m+1)-1} \right. \\ & \qquad\qquad\qquad\qquad\quad {} + \left. \binom{(m+1)+(k+1)-1-i-1}{m+1}\right) \\ &= \sum_{i=0}^{(k+1)-1} \binom{n+i}{n} \binom{(m+1)+(k+1)-1-i-1}{(m+1)-1} \\ &\qquad\qquad {} + \sum_{i=0}^{k-1} \binom{n+i}{n} \binom{(m+1)+(k+1)-1-i-1}{m+1} \\ &\qquad\qquad {} + \binom{n+k}{n} \binom{(m+1)+(k+1)-1-k-1}{m+1} \\ &= \sum_{i=0}^{(k+1)-1} \binom{n+i}{n} \binom{m+(k+1)-1-i}{m} + \sum_{i=0}^{k-1} \binom{n+i}{n} \binom{(m+1)+k-1-i}{m+1} + 0 \\ &= \binom{n+m+(k+1)}{n+m+1} + \binom{n+(m+1)+k}{n+(m+1)+1} \\ &= \binom{n+(m+1)+(k+1)}{n+(m+1)+1}. \quad\qed \end{aligned} $$
ãªã‚“ã‹ãµã¤ã†ã«ãŠå‡ºã—ã•ã‚ŒãŸã€‚最åˆã‹ã‚‰æ•™ãˆã¦ã‚ˆã€œ www.wolframalpha.com
変数を $k$ ã‹ã‚‰ $d$ ã«å¤‰ãˆãŸã‚‰å‡ºã—ã¦ãã‚Œãªããªã‚Šã€ã“れマジ？ã¨ãªã‚Šã¾ã—ãŸã€‚ www.wolframalpha.com
比較ã®ãŸã‚ã«è¦‹ã¦ãŠãã¨ã€Vandermonde identity ã¯ä¸‹è¨˜ã®ã‚ˆã†ãªç•³ã¿è¾¼ã¿ã§ã™ã€‚ã†ã¾ã„ã“ã¨å¸°ç€ã§ããŸã‚Šã™ã‚‹ã®ã‹ãªï¼Ÿ
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A B C D .
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Corollary 4: ä»»æ„ã® $n\ge 1$ ã«å¯¾ã—ã¦ä¸‹è¨˜ãŒæˆã‚Šç«‹ã¤ã€‚
$$ \sum_{i=0}^{k-1} \binom{n+i}{n} \binom{m+k-1-i}{m} = \sum_{i=0}^{k-1} \binom{n-1+i}{n-1} \binom{m+k-i}{m+1} $$
Proof: Lemma 3 より明らã‹ã€‚
ã“れを示ã—㦠$n = 0$ ã«å¸°ç€ã•ã›ã‚‹ã“ã¨ã§ Hockey-stick identity ã‹ã‚‰ Lemma 3 を示ã™æ–¹é‡ã‚’考ãˆã¦ã„ã¾ã—ãŸãŒã€ã“ã¡ã‚‰ãŒ Corollary ã«ãªã‚Šã¾ã—ãŸã€‚
ã“ã“ã¾ã§ã®è€ƒå¯Ÿã§ã€$b_i$ ã‚’ incremental ã«æ±‚ã‚ã‚‹ã“ã¨ã§ $i$ ã”ã¨ã«å®šæ•°æ™‚é–“ã§æ±‚ã‚られるã“ã¨ãŒã‚ã‹ã‚Šã¾ã—ãŸã€‚コンテストä¸ã¯ãã‚Œã§å分ã§ã—ãŸãŒã€ã‚‚ã†å°‘ã—進ã‚ã¦ã¿ã¾ã™ã€‚
下記ãŒæˆã‚Šç«‹ã¡ã¾ã™ã€‚ $$ \begin{aligned} b_i &= \sum_{j=1}^C \binom{B-i-1+j}{j} \binom{C-j+D}{D} \\ &= \sum_{j=1}^C \binom{B-i+j-1}{B-i-1} \binom{C-j+D}{D} \\ &= \sum_{j=0}^{C-1} \binom{B-i+j}{B-i-1} \binom{C-1-j+D}{D} \\ &= \left( \sum_{j=0}^{C} \binom{B-i-1+j}{B-i-1} \binom{C-j+D}{D} - \binom{B-i-1}{B-i-1} \binom{C+D}{D} \right) \\ &= \binom{(B-i-1)+D+(C+1)}{(B-i-1)+D+1} - \binom{C+D}{D} \\ &= \binom{B+C+D-i}{B+D-i} - \binom{C+D}{D} \\ &= \binom{B+C+D-i}{C} - \binom{C+D}{D} \\ \end{aligned} $$
よã£ã¦ã€å†’é ã«æŒ™ã’ãŸå‰è€…ã®ã‚±ãƒ¼ã‚¹ã®é€šã‚Šæ•°ã¯ $$ \begin{aligned} &\phantom{{}={}} \sum_{i=0}^{B-1} \binom{A-1+i}{i} \sum_{j=1}^C \binom{B-i-1+j}{j} \binom{C-j+D}{D} \\ &= \sum_{i=0}^{B-1} \binom{A-1+i}{i} \left( \binom{B+C+D-i}{C} - \binom{C+D}{D} \right) \\ &= \sum_{i=0}^{B-1} \binom{A-1+i}{A-1} \binom{B+C+D-i}{C} - \binom{C+D}{D} \sum_{i=0}^{B-1} \binom{A-1+i}{A-1} \\ &= \sum_{i=0}^{B-1} \binom{A-1+i}{A-1} \binom{B+C+D-i}{C} - \binom{A-1+B}{A} \binom{C+D}{D} \end{aligned} $$ ã¨ãªã‚Šã¾ã™ã€‚後者ã®ã‚±ãƒ¼ã‚¹ã¨åˆã‚ã›ã¦ã€å…¨ä½“ã®ç”ãˆã¯ $$ \begin{aligned} &\phantom{{}={}} \sum_{i=0}^{B-1} \binom{A-1+i}{A-1} \binom{B+C+D-i}{C} \\ &\qquad\quad {} - \binom{A-1+B}{A} \binom{C+D}{D} + \binom{A+B}{A} \binom{C+D}{D} \\ &= \sum_{i=0}^{B-1} \binom{A-1+i}{A-1} \binom{B+C+D-i}{C} + \binom{A+B-1}{A-1} \binom{C+D}{D} \\ &= \sum_{i=0}^{B} \binom{A-1+i}{A-1} \binom{B+C+D-i}{C} \end{aligned} $$ ã¨ã§ãã¾ã™ã。
ã¾ãŸã€å¯¾ç§°æ€§ã‹ã‚‰ã€å…¥åŠ›ãŒ $(A, B, C, D)$ ã®ã¨ãã®ç”ãˆã¨ $(D, C, B, A)$ ã®ã¨ãã®ç”ãˆãŒä¸€è‡´ã™ã‚‹ã“ã¨ã‚‚ã‚ã‹ã‚Šã¾ã™ã€‚実際〠$$ \begin{aligned} &\phantom{{}={}} \sum_{i=0}^{B} \binom{A-1+i}{A-1} \binom{B+C+D-i}{C} \\ &= \sum_{i=0}^{C} \binom{D-1+i}{D-1} \binom{C+B+A-i}{B} \\ &= \sum_{i=0}^{C} \binom{D-1+C-i}{D-1} \binom{B+A+i}{B} \\ &= \sum_{i=0}^{C} \binom{A+B+i}{B} \binom{D-1+C-i}{D-1} \\ \end{aligned} $$ ã¨ãªã‚Šã€å…¬å¼è§£èª¬ ã¨åŒã˜å¼ãŒå¾—られã¾ã—ãŸã€‚
ã¨ã„ã†ã“ã¨ã§ã€ã“ã®ç•³ã¿è¾¼ã¿ã‚‚高速化ã§ããªã„ã‚‚ã®ã‹ã？ã¨ã„ã†æ°—æŒã¡ã«ãªã‚Šã¾ã™ã€‚下記ãŒæˆã‚Šç«‹ã¤ã‚‰ã—ã„ã§ã™ã€‚ $$ \begin{aligned} &\phantom{{}={}} \sum_{i=0}^{k-1} \binom{n+r+i}{n} \binom{m+k-1-i}{m} \\ &= \binom{k+m-1}{m}\binom{n+r}{r} {}_3 F_2 (1, 1-k, n-r+1; -k-m+1, r+1; 1). \end{aligned} $$ ãªã‚“ã‚„ã‹ã‚“ã‚„ã§ç”ãˆã¯ä¸‹è¨˜ã«ãªã‚Šã¾ã™ã€‚ $$ \begin{aligned} \binom{A+B}{A} \binom{C+D-1}{C} {}_3 F_2(1, -C, A+B+1; A+1, -C-D+1; 1) \end{aligned} $$ ${}_3 F_2$ を高速ã«æœ‰ç†æ•°è¡¨ç¤ºã§è¨ˆç®—ã™ã‚‹æ–¹æ³•ï¼ˆã®å˜åœ¨ï¼‰ãŒã‚ã‹ã‚‰ãšã€ãŠã‚ã‚Šã§ã™...
定義通り㮠$$ {}_p F_q(a_1, \dots, a_p; b_1, \dots, b_q; z) = \sum_{n=0}^{\infty} \frac{(a_1)_n\cdot\cdots\cdot(a_p)_n}{(b_1)_n\cdot\cdots\cdot(b_q)_n}\frac{z^n}{n!} $$ ã® $n$ é …ç›®ã®å„ factor ã‚’æŒã£ã¦ãŠã‘ã°ã€æ¬¡ã®é …㯠$O(p+q)$ 時間程度ã§è¨ˆç®—ã§ãã¾ã™ã€‚$n\ge C$ ã®ã¨ã $(-C)_n = 0$ ãªã®ã§å…ˆé $C$ é …ã ã‘見れã°ã‚ˆã„ã“ã¨ã‹ã‚‰ $O(C)$ 時間ã§æ±‚ã‚られã¯ã—ã¾ã™ãŒã€ãªã‚“ã ã‹ãªã€œã¨ã„ã†æ„Ÿã˜ã§ã™ã。
ã‚‚ã—ã‹ã—ã¦äºŒé …係数関連ã®ã©ã†ã«ã‚‚ãªã‚‰ãªã„ç©å’Œã‚’超幾何関数ã¨ã—ã¦è¿½ã„ã‚„ã£ã¦ã„ã‚‹ã ã‘ã§ã™ã‹ï¼Ÿ
$$ \begin{aligned} {}_3 F_2 (1, 1-k, n+1; -k-m+1, 1; 1) &= {}_2 F_1 (1-k, n+1; -k-m+1; 1) \end{aligned} $$ ãŒæˆã‚Šç«‹ã¤ã‚‰ã—ã„ã®ã§ã€$r=0$ を考ãˆã¦ $$ \begin{aligned} {}_2 F_1 (1-k, n+1; -k-m+1; 1) = \binom{n+m+k}{n+m+1} \binom{k+m-1}{m}^{-1} \end{aligned} $$ ã§ã‚ã‚Šã€$(k-1, n+1, m-a) = (a, b, c)$ ã¨ã—㦠$$ \begin{aligned} {}_2 F_1 (-a, b; -c; 1) = \binom{b+c}{b+c-a} \binom{c}{c-a}^{-1} \end{aligned} $$ ã‚’å¾—ã¾ã™ã€‚
ãŸã¶ã‚“失敗方é‡ã§ã¯ã‚ã‚Šãã†ãªæ°—ã‚‚ã—ã¤ã¤ã€ãŠã‚‚ã—ã‚ã„ç‰å¼ã‚’知るã“ã¨ãŒã§ããŸã®ã§ã‚ˆã‹ã£ãŸãªã€œã¨ã„ã†æ„Ÿã˜ã§ã™ã€‚ 超幾何関数ã¨ã‚‚知りåˆã„ã«ãªã‚Œã¾ã—ãŸã€‚
ã“ã®æ‰‹ã®å•é¡Œã§ã€ã€Œã¨ã‚Šã‚ãˆãšæ„šç›´ã‚’書ã„ã¦ã€å·®åˆ†ã‚’ç¨ã‚“ã§ã„ãŸã‚‰äºŒé …ä¿‚æ•°ãŒå‡ºã¦ãã‚‹ã ã‚ã†ã‹ã‚‰ã€ã‚¨ã‚¹ãƒ‘ーã—ã¦åˆã‚ã›ã‚Œã°ãªã‚“ã¨ã‹ãªã‚‹ã€ã¿ãŸã„ãªã‚„ã‚Šæ–¹ã°ã‹ã‚Šã‚„ã£ã¦ã„ã‚‹ã®ã¯ã‚„ã£ã±ã‚Šã‚ˆããªã„ã¨æ€ã„ã¾ã™ã€‚ã§ã‚‚é を使ã‚ãšã«æ¸ˆã‚€ã‹ã‚‰æ¥½ãªã‚“ã§ã™ã‚ˆã。é を使ã„ãŸããªã„ãªã‚‰ ABC ã‚’ã‚„ã‚ã‚Œã°ã„ã„ã®ã«... ã¨ã‚‚æ€ã„ã¾ã™ã—ã€ABC ã§é を使ã£ã¦ã„る時点ã§è² ã‘ã¨ã„ã†æ°—ã‚‚ã—ã¾ã™ã€‚
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ãŠã‚ã‚Šã§ã™ã€‚
*1:$,$ 㨠$;$ ã¯èª¤æ¤ã§ã¯ãªãã€ä¸€èˆ¬ã«ã¯ ${}_p F_q(a_1, \dots, a_p; b_1, \dots, b_q; z)$ ã¨ã„ã†å½¢ã‚’ã—ã¦ã„ã‚‹ã¿ãŸã„ã§ã™ã€‚
*2:実際ã€Wolfram|Alpha ã«è¨ˆç®—ã•ã›ã¦ã‚‚ãã®å€¤ã«ãªã‚Šã¾ã—ãŸã€‚ã¾ãŸã€æ•´æ•°ã‹ã‚‰ã¡ã‚‡ã£ã¨ã ã‘ãšã‚‰ã—㦠$\Gamma$ を計算ã™ã‚‹ã¨ã€ãŸã—ã‹ã«ãã‚Œã£ã½ã„値ã«è¿‘ã¥ããã†ã§ã¯ã‚ã‚Šã¾ã—ãŸã€‚
ã‚„ã‚Šã¾ã—ょã†ã€‚
å‰å›žã€double-double ã‚„ triple-double ã®è©±ã‚’ã—ã¾ã—ãŸãŒã€ã€Œtriple-double ã¯å¿…è¦ãªã®ã‹ï¼Ÿã€ã¨ã‹ã€Œquadruple-double ã¯ä¸è¦ãªã®ã‹ï¼Ÿã€ã¨ã‹ã«ã¤ã„ã¦è§¦ã‚ŒãŸæ–¹ãŒã„ã„ã‹ãªã¨æ€ã£ãŸã®ã§ã€å…ˆã«ãã†ã„ã†è©±ã‚’ã—ã¾ã™ã€‚é‡è¦ãªãŠè©±ã§ã™ã€‚
disclaimer: é‡è¦ãªãŠè©±ã§ã¯ã‚ã‚‹ã‚‚ã®ã®ã€è¨ˆç®—ã™ã‚‹ã«ã‚ãŸã£ã¦ã®ã‚¢ãƒ«ã‚´ãƒªã‚ºãƒ ãªã©ã®è©±ã§ã¯ãªã„ã§ã™ã€‚
多ãã®è¨˜æ³•ã«ã¤ã„ã¦ã¯å‰å›žå®šç¾©ã—ãŸã‚‚ã®ã‚’使ã„ã¾ã™ã€‚特ã«ã€double ã§è¡¨ã›ã‚‹æµ®å‹•å°æ•°ç‚¹æ•°å…¨ä½“ã‹ã‚‰ãªã‚‹é›†åˆã‚’ $F$ ã¨ã—ã¾ã™ã€‚
今回以é™ã€å®Ÿæ•°ã®è¡¨è¨˜ã¨ã—ã¦ä¸»ã«æŒ‡æ•°è¡¨è¨˜ã¤ãã® 16 進法や 2 進法を使ã£ã¦ã„ãã¾ã™ã€‚ãŸã¨ãˆã° $$ \pi = {\small 3.14159265358979}{\footnotesize 323846264338327}{\scriptsize 950288419716939}{\tiny 937510582097494{\ldots}} $$ ã§ã¯ãªã $$ \begin{aligned} \pi &= \texttt{1}.\texttt{{\small 921FB54442D18}{\footnotesize 469898CC51701}{\scriptsize B839A252049C1}{\tiny 114CF98E80417{\ldots}}}_{(16)}\times 2^{1} \\ &= 1. \underbrace{\footnotesize 1001}_{\texttt{9}} {\footnotesize\,} \underbrace{\footnotesize 0010}_{\texttt{2}} {\footnotesize\,} \underbrace{\footnotesize 0001}_{\texttt{1}} {\footnotesize\,} \underbrace{\footnotesize 1111}_{\texttt{F}} {\footnotesize\,} \underbrace{\footnotesize 1011}_{\texttt{B}} {\footnotesize\,} \underbrace{\footnotesize 0101}_{\texttt{5}} {\footnotesize\,} \underbrace{\footnotesize 0100}_{\texttt{4}} {\footnotesize\,} \underbrace{\footnotesize 0100}_{\texttt{4}} {\footnotesize\,} \underbrace{\footnotesize 0100}_{\texttt{4}} {\footnotesize\,} \underbrace{\footnotesize 0010}_{\texttt{2}} {\footnotesize\,} \underbrace{\footnotesize 1101}_{\texttt{D}} {\footnotesize\,} \underbrace{\footnotesize 0001}_{\texttt{1}} {\footnotesize\,} \underbrace{\footnotesize 1000}_{\texttt{8}} {\footnotesize\,} \\ & \qquad \underbrace{\footnotesize 0100}_{\texttt{4}} {\footnotesize\,} \underbrace{\footnotesize 0110}_{\texttt{6}} {\footnotesize\,} \underbrace{\footnotesize 1001}_{\texttt{9}} {\footnotesize\,} \underbrace{\footnotesize 1000}_{\texttt{8}} {\footnotesize\,} \underbrace{\footnotesize 1001}_{\texttt{9}} {\footnotesize\,} \underbrace{\footnotesize 1000}_{\texttt{8}} {\footnotesize\,} \underbrace{\footnotesize 1100}_{\texttt{C}} {\footnotesize\,} \underbrace{\footnotesize 1100}_{\texttt{C}} {\footnotesize\,} \underbrace{\footnotesize 0101}_{\texttt{5}} {\footnotesize\,} \underbrace{\footnotesize 0001}_{\texttt{1}} {\footnotesize\,} \underbrace{\footnotesize 0111}_{\texttt{7}} {\footnotesize\,} \underbrace{\footnotesize 0000}_{\texttt{0}} {\footnotesize\,} \underbrace{\footnotesize 0001}_{\texttt{1}} {\footnotesize\,} \\ & \qquad \underbrace{\footnotesize 1011}_{\texttt{B}} {\footnotesize\,} \underbrace{\footnotesize 1000}_{\texttt{8}} {\footnotesize\,} \underbrace{\footnotesize 0011}_{\texttt{3}} {\footnotesize\,} \underbrace{\footnotesize 1001}_{\texttt{9}} {\footnotesize\,} \underbrace{\footnotesize 1010}_{\texttt{A}} {\footnotesize\,} \underbrace{\footnotesize 0010}_{\texttt{2}} {\footnotesize\,} \underbrace{\footnotesize 0101}_{\texttt{5}} {\footnotesize\,} \underbrace{\footnotesize 0010}_{\texttt{2}} {\footnotesize\,} \underbrace{\footnotesize 0000}_{\texttt{0}} {\footnotesize\,} \underbrace{\footnotesize 0100}_{\texttt{4}} {\footnotesize\,} \underbrace{\footnotesize 1001}_{\texttt{9}} {\footnotesize\,} \underbrace{\footnotesize 1100}_{\texttt{C}} {\footnotesize\,} \underbrace{\footnotesize 0001}_{\texttt{1}} {\footnotesize\,} \\ & \qquad \underbrace{\footnotesize 0001}_{\texttt{1}} {\footnotesize\,} \underbrace{\footnotesize 0001}_{\texttt{1}} {\footnotesize\,} \underbrace{\footnotesize 0100}_{\texttt{4}} {\footnotesize\,} \underbrace{\footnotesize 1100}_{\texttt{C}} {\footnotesize\,} \underbrace{\footnotesize 1111}_{\texttt{F}} {\footnotesize\,} \underbrace{\footnotesize 1001}_{\texttt{9}} {\footnotesize\,} \underbrace{\footnotesize 1000}_{\texttt{8}} {\footnotesize\,} \underbrace{\footnotesize 1110}_{\texttt{E}} {\footnotesize\,} \underbrace{\footnotesize 1000}_{\texttt{8}} {\footnotesize\,} \underbrace{\footnotesize 0000}_{\texttt{0}} {\footnotesize\,} \underbrace{\footnotesize 0100}_{\texttt{4}} {\footnotesize\,} \underbrace{\footnotesize 0001}_{\texttt{1}} {\footnotesize\,} \underbrace{\footnotesize 0111}_{\texttt{7}} {\footnotesize\,} {\footnotesize \,\dots\,}_{(2)} \times 2^1 \end{aligned} $$ ã®ã‚ˆã†ã«æ›¸ãã¾ã™ã€‚指数ã®åº•ã¯ $2$ ã§ã™ã€‚2 進法ã§æ›¸ãå ´åˆã¯ 4 æ¡ã”ã¨ã«åŒºåˆ‡ã£ã¦ 16 進法ã§ã®å€¤ã‚’æ·»ãˆã‚‹ã‚ˆã†ã«ã—ã¾ã™ã€‚
言語ã«ã‚ˆã£ã¦ã¯ã€æµ®å‹•å°æ•°ç‚¹åž‹ã® 16 進法リテラルãŒã‚µãƒãƒ¼ãƒˆã•ã‚Œã¦ã„ã¦ã€0x1.921FB54442D18p+1 ã®ã‚ˆã†ã«æ›¸ãã“ã¨ã‚‚ã§ãã¾ã™ã€‚
C ã§ã¯ãƒªãƒ†ãƒ©ãƒ«ã¨ã—ã¦å¯èƒ½ã§ã€Python ã§ã‚‚ float.fromhex() を使ã†ã“ã¨ã§å¯èƒ½ã§ã™ã€‚Rust ã§ã¯æ¨™æº–ã§ã¯ã§ããšã€hexfloat2 ãªã©ã®ã‚¯ãƒ¬ãƒ¼ãƒˆã‚’使ã†å¿…è¦ãŒã‚ã‚Šã¾ã™*1。ã¾ãŸã€å‡ºåŠ›ã«é–¢ã—ã¦ã‚‚ã€C ã§ã¯ printf("%a", x) ã¨ã—ãŸã‚Šã€Python ã§ã¯ x.hex() ã¨ã™ã‚‹ã“ã¨ã§å¯èƒ½ã§ã™ã€‚Rust ã¯ãŠå¯Ÿã—ã§ã™ã€‚
$\sin(x)$ を計算ã—ãŸããªã£ãŸã“ã¨ã«ã—ã¾ã—ょã†ã€‚
ã¨ã„ã†ã“ã¨ã§ã€æ¬¡ã®æ‰‹ç¶šã $\textsc{Fr54-Sin}$ を考ãˆã¾ã™*2。
$\roundcirc{\sin(x)} = \roundcirc{\textsc{Fr54-Sin}(x)}$ ã¨ã—ã¦æ±‚ã‚よã†ã¨ã„ã†ã‚ã‘ã§ã™ã€‚
ãŸã¨ãˆã° $x = \texttt{1}.\texttt{D037CB27EE6DF}_{(16)}\times 2^{-3}\in F$ を考ãˆã¾ã™ã€‚ 実際ã®è¨ˆç®—çµæžœã¯ä¸‹è¨˜ã®é€šã‚Šã§ã™ã€‚ $$ \begin{aligned} &\phantom{{}={}} \textsc{Fr54-Sin}(x) \\ &= \texttt{1}.\texttt{CC40C3805229A8}_{(16)}\times 2^{-3} \\ &= 1. \underbrace{\footnotesize 1100}_{\texttt{C}} {\footnotesize\,} \underbrace{\footnotesize 1100}_{\texttt{C}} {\footnotesize\,} \underbrace{\footnotesize 0100}_{\texttt{4}} {\footnotesize\,} \underbrace{\footnotesize 0000}_{\texttt{0}} {\footnotesize\,} \underbrace{\footnotesize 1100}_{\texttt{C}} {\footnotesize\,} \underbrace{\footnotesize 0011}_{\texttt{3}} {\footnotesize\,} \underbrace{\footnotesize 1000}_{\texttt{8}} {\footnotesize\,} \underbrace{\footnotesize 0000}_{\texttt{0}} {\footnotesize\,} \underbrace{\footnotesize 0101}_{\texttt{5}} {\footnotesize\,} \underbrace{\footnotesize 0010}_{\texttt{2}} {\footnotesize\,} \underbrace{\footnotesize 0010}_{\texttt{2}} {\footnotesize\,} \underbrace{\footnotesize 1001}_{\texttt{9}} {\footnotesize\,} \underbrace{\footnotesize 1010}_{\texttt{A}} {\footnotesize\,} \underbrace{\footnotesize 1\hphantom{000}}_{\texttt{8}} {}_{(2)} \times 2^{-3} \end{aligned} $$ 最後ã®éƒ¨åˆ†ã«ã¤ã„ã¦ã€$\underbrace{1010}_{\texttt{A}}\,100\ldots01\ldots$ 㨠$\underbrace{1010}_{\texttt{A}}\,011\ldots10\ldots$ ã®ã©ã¡ã‚‰ã‚’丸ã‚ãŸçµæžœãªã®ã‹ã‚’判別ã§ãã¾ã›ã‚“。å‰è€…ã§ã‚ã‚Œã°ã€$\roundcirc{\sin(x)} = \texttt{1}.\texttt{CC40C3805229B}_{(16)}\times 2^{-3}$ ã§ã™ã—ã€å¾Œè€…ã§ã‚れ㰠$\roundcirc{\sin(x)} = \texttt{1}.\texttt{CC40C3805229A}_{(16)}\times 2^{-3}$ ãªã®ã§ã€$\textsc{Fr54-Sin}(x)$ を用ã„ã¦æ±‚ã‚ã‚‹ã“ã¨ã¯ã§ãã¾ã›ã‚“。
note: $x\in\Q\smallsetminus\{0\}$ より $\sin(x)\notin\Q$ ãªã®ã§ã€$\sin(x) = \texttt{1}.\texttt{CC40C3805229A8}_{(16)}\times 2^{-3}$ ã¨ãªã‚‹ã“ã¨ã¯ã‚ã‚Šã¾ã›ã‚“。
ã¨ã„ã†ã“ã¨ã§ã€ã‚‚ã†å°‘ã—高ã„精度ã§è¨ˆç®—ã—ã¦ã¿ã¾ã—ょã†ã€‚åŒæ§˜ã« $\textsc{Fr55-Sin}(x)$ を定義ã—ã¦è¨ˆç®—ã—ã¾ã™ã€‚ $$ \begin{aligned} &\phantom{{}={}} \textsc{Fr55-Sin}(x) \\ &= \texttt{1}.\texttt{CC40C3805229A8}_{(16)}\times 2^{-3} \\ &= 1. \underbrace{\footnotesize 1100}_{\texttt{C}} {\footnotesize\,} \underbrace{\footnotesize 1100}_{\texttt{C}} {\footnotesize\,} \underbrace{\footnotesize 0100}_{\texttt{4}} {\footnotesize\,} \underbrace{\footnotesize 0000}_{\texttt{0}} {\footnotesize\,} \underbrace{\footnotesize 1100}_{\texttt{C}} {\footnotesize\,} \underbrace{\footnotesize 0011}_{\texttt{3}} {\footnotesize\,} \underbrace{\footnotesize 1000}_{\texttt{8}} {\footnotesize\,} \underbrace{\footnotesize 0000}_{\texttt{0}} {\footnotesize\,} \underbrace{\footnotesize 0101}_{\texttt{5}} {\footnotesize\,} \underbrace{\footnotesize 0010}_{\texttt{2}} {\footnotesize\,} \underbrace{\footnotesize 0010}_{\texttt{2}} {\footnotesize\,} \underbrace{\footnotesize 1001}_{\texttt{9}} {\footnotesize\,} \underbrace{\footnotesize 1010}_{\texttt{A}} {\footnotesize\,} \underbrace{\footnotesize 10\hphantom{00}}_{\texttt{8}} {}_{(2)} \times 2^{-3}. \end{aligned} $$ ã¾ã åŒæ§˜ã®ç†ç”±ã§ã ã‚ã§ã™ã。ã•ã‚‰ã«é«˜ã„精度ã§ã‚„ã‚Šã¾ã—ょã†ã€‚
〜ã—ã°ã‚‰ãç¹°ã‚Šè¿”ã—〜
$$ \begin{aligned} &\phantom{{}={}} \textsc{Fr105-Sin}(x) \\ &= \texttt{1}.\texttt{CC40C3805229A8000000000000}_{(16)}\times 2^{-3} \\ &= 1. \underbrace{\footnotesize 1100}_{\texttt{C}} {\footnotesize\,} \underbrace{\footnotesize 1100}_{\texttt{C}} {\footnotesize\,} \underbrace{\footnotesize 0100}_{\texttt{4}} {\footnotesize\,} \underbrace{\footnotesize 0000}_{\texttt{0}} {\footnotesize\,} \underbrace{\footnotesize 1100}_{\texttt{C}} {\footnotesize\,} \underbrace{\footnotesize 0011}_{\texttt{3}} {\footnotesize\,} \underbrace{\footnotesize 1000}_{\texttt{8}} {\footnotesize\,} \underbrace{\footnotesize 0000}_{\texttt{0}} {\footnotesize\,} \underbrace{\footnotesize 0101}_{\texttt{5}} {\footnotesize\,} \underbrace{\footnotesize 0010}_{\texttt{2}} {\footnotesize\,} \underbrace{\footnotesize 0010}_{\texttt{2}} {\footnotesize\,} \underbrace{\footnotesize 1001}_{\texttt{9}} {\footnotesize\,} \underbrace{\footnotesize 1010}_{\texttt{A}} {\footnotesize\,} \underbrace{\footnotesize 1000}_{\texttt{8}} \\ & \qquad \underbrace{\footnotesize 0000}_{\texttt{0}} {\footnotesize\,} \underbrace{\footnotesize 0000}_{\texttt{0}} {\footnotesize\,} \underbrace{\footnotesize 0000}_{\texttt{0}} {\footnotesize\,} \underbrace{\footnotesize 0000}_{\texttt{0}} {\footnotesize\,} \underbrace{\footnotesize 0000}_{\texttt{0}} {\footnotesize\,} \underbrace{\footnotesize 0000}_{\texttt{0}} {\footnotesize\,} \underbrace{\footnotesize 0000}_{\texttt{0}} {\footnotesize\,} \underbrace{\footnotesize 0000}_{\texttt{0}} {\footnotesize\,} \underbrace{\footnotesize 0000}_{\texttt{0}} {\footnotesize\,} \underbrace{\footnotesize 0000}_{\texttt{0}} {\footnotesize\,} \underbrace{\footnotesize 0000}_{\texttt{0}} {\footnotesize\,} \underbrace{\footnotesize 0000}_{\texttt{0}} {\footnotesize\,} {}_{(2)} \times 2^{-3}. \end{aligned} $$ ã¾ã€ã¾ã ã ã‚ã§ã™...。
$$ \begin{aligned} &\phantom{{}={}} \textsc{Fr106-Sin}(x) \\ &= \texttt{1}.\texttt{CC40C3805229A7FFFFFFFFFFFF8}_{(16)}\times 2^{-3} \\ &= 1. \underbrace{\footnotesize 1100}_{\texttt{C}} {\footnotesize\,} \underbrace{\footnotesize 1100}_{\texttt{C}} {\footnotesize\,} \underbrace{\footnotesize 0100}_{\texttt{4}} {\footnotesize\,} \underbrace{\footnotesize 0000}_{\texttt{0}} {\footnotesize\,} \underbrace{\footnotesize 1100}_{\texttt{C}} {\footnotesize\,} \underbrace{\footnotesize 0011}_{\texttt{3}} {\footnotesize\,} \underbrace{\footnotesize 1000}_{\texttt{8}} {\footnotesize\,} \underbrace{\footnotesize 0000}_{\texttt{0}} {\footnotesize\,} \underbrace{\footnotesize 0101}_{\texttt{5}} {\footnotesize\,} \underbrace{\footnotesize 0010}_{\texttt{2}} {\footnotesize\,} \underbrace{\footnotesize 0010}_{\texttt{2}} {\footnotesize\,} \underbrace{\footnotesize 1001}_{\texttt{9}} {\footnotesize\,} \underbrace{\footnotesize 1010}_{\texttt{A}} {\footnotesize\,} \underbrace{\footnotesize 0111}_{\texttt{7}} {\footnotesize\,} \\ & \qquad \underbrace{\footnotesize 1111}_{\texttt{F}} {\footnotesize\,} \underbrace{\footnotesize 1111}_{\texttt{F}} {\footnotesize\,} \underbrace{\footnotesize 1111}_{\texttt{F}} {\footnotesize\,} \underbrace{\footnotesize 1111}_{\texttt{F}} {\footnotesize\,} \underbrace{\footnotesize 1111}_{\texttt{F}} {\footnotesize\,} \underbrace{\footnotesize 1111}_{\texttt{F}} {\footnotesize\,} \underbrace{\footnotesize 1111}_{\texttt{F}} {\footnotesize\,} \underbrace{\footnotesize 1111}_{\texttt{F}} {\footnotesize\,} \underbrace{\footnotesize 1111}_{\texttt{F}} {\footnotesize\,} \underbrace{\footnotesize 1111}_{\texttt{F}} {\footnotesize\,} \underbrace{\footnotesize 1111}_{\texttt{F}} {\footnotesize\,} \underbrace{\footnotesize 1111}_{\texttt{F}} {\footnotesize\,} \underbrace{\footnotesize 1\hphantom{000}}_{\texttt{8}} {\footnotesize\,} {}_{(2)} \times 2^{-3}. \end{aligned} $$ ã‚ã€ã‚ˆã†ã‚„ã区別ãŒã¤ãã¾ã—ãŸã€‚ ã“ã‚Œã«ã‚ˆã‚Š $$ \sin(x) \lt \texttt{1}.\texttt{CC40C3805229A8}_{(16)}\times 2^{-3} $$ ãŒè¨€ãˆã‚‹ã®ã§ã€ $$ \roundcirc{\sin(x)} = \texttt{1}.\texttt{CC40C3805229A}_{(16)}\times 2^{-3} $$ ã§ã‚ã‚‹ã¨ã‚ã‹ã£ãŸã“ã¨ã«ãªã‚Šã¾ã™ã€‚ã‚ã§ãŸã—ã‚ã§ãŸã—*3。実際ã®ã¨ã“ã‚〠$$ \sin(x) = \texttt{1}.\texttt{{\small CC40C3805229A}{\footnotesize 7FFFFFFFFFFFF}{\scriptsize 83E76BBF0FCBE}{\tiny 19AFB5AC38007{\ldots}}} {\,}_{(16)} \times 2^{-3} $$ ã§ã™ã€‚
ã•ã¦ã€ä¸Šè¨˜ã®ä¾‹ã§ã¯ 106-bit ã§ã®è¨ˆç®—ã§å分ã§ã—ãŸãŒã€ã“ã‚ŒãŒã€Œ106-bit ã§å分ã§ã™ã‚ˆã€ã¨ã„ã†ã“ã¨ã‚’予ã‚知ã£ã¦ãŠãã“ã¨ã¯å¯èƒ½ã§ã—ょã†ã‹ï¼Ÿ å¹³æ–¹æ ¹ãªã©ä¸€éƒ¨ã®ä¾‹ã§ã¯å¯èƒ½ã§ã™ãŒã€ä¸‰è§’関数を始ã‚ã¨ã™ã‚‹å¤šãã®é–¢æ•°ã§ã¯ã“ã‚Œã¯é›£ã—ã„ã¨ã„ã†ã“ã¨ã«ãªã£ã¦ã„ã¾ã™ã€‚
William M. Kahan*4ã¯æ¬¡ã®ã‚ˆã†ã«è¨˜ã—ã¦ã„ã¾ã™[6]。
Why can’t YW be rounded within half an ulp like SQRT ? Because nobody knows how much computation it would cost to resolve what I long ago christened “The Table-Maker’s Dilemmaâ€.
拙訳：ãªãœ $Y^W$ ã¯ã€å¹³æ–¹æ ¹ã®ã‚ˆã†ã« 0.5 ULP 以内（ã®èª¤å·®ã§åŽã¾ã‚‹ã‚ˆã†ï¼‰ã«ä¸¸ã‚られãªã„ã®ã‹ï¼Ÿ ãã‚Œã¯ã€ä»¥å‰ç§ãŒ 数表作æˆè€…ã®ã‚¸ãƒ¬ãƒ³ãƒž (Table-Maker’s Dilemma) ã¨åã¥ã‘ãŸäº‹è±¡ã‚’解決ã™ã‚‹ãŸã‚ã«ã©ã‚Œã ã‘ã®è¨ˆç®—ãŒå¿…è¦ã‹ã‚’誰も知らãªã„ã‹ã‚‰ã 。
note: table-maker’s dilemma ã¯ã€ã—ã°ã—ã° TMD ã¨ç•¥ã•ã‚Œã¾ã™ã€‚
ã“ã®å¾Œã€$\tfrac18\cosh(\pi\sqrt{163}) - (2^{53}-1)$ ã‚’ 53 ビットã«æ£ç¢ºã«ä¸¸ã‚ãŸå€¤ã‚’計算ã—よã†ã¨ã—ã¦ã€ç²¾åº¦ã‚’上ã’ã¤ã¤ ${\small 7401389035307055}.{\footnotesize 49999978}$ ã‚„ ${\small 7401389035307055}.{\footnotesize 5000000000015}$ を得る話ãŒç¶šãã¾ã™ã€‚å…ˆã»ã©ã® $\textsc{Fr106-Sin}(x)$ ã®ä¾‹ã¨åŒã˜ã‚ˆã†ãªæ„Ÿã˜ã§ã™ã€‚
No general way exists to predict how many extra digits will have to be carried to compute a transcendental expression and round it correctly to some preassigned number of digits.
拙訳：超越å¼ï¼ˆè¶…越関数をå«ã‚€å¼ï¼Ÿï¼‰ã‚’計算ã—ã¦äºˆã‚決ã‚られãŸæ¡æ•°ã«æ£ç¢ºã«ä¸¸ã‚ã‚‹ã«ã‚ãŸã‚Šã€ã©ã‚Œã ã‘ã®æ¡æ•°ã‚’余分ã«æŒã£ã¦ãŠãå¿…è¦ãŒã‚ã‚‹ã‹ã‚’予測ã™ã‚‹ä¸€èˆ¬çš„ãªæ–¹æ³•ã¯å˜åœ¨ã—ãªã„。
Table ã¨è¨€ã£ã¦ã„ã‚‹ã®ã¯ã€ãŸã¶ã‚“教科書ã®å·»æœ«ã«è¼‰ã£ã¦ã„る三角関数表ã®ã‚ˆã†ãªã‚‚ã®ã®ã“ã¨ã ã¨æ€ã„ã¾ã™ã€‚下記ã®ã‚ˆã†ãªã‚¤ãƒ¡ãƒ¼ã‚¸ã§ã™ã。
| $x$ | $\roundcirc{\sin(x)}$ |
|---|---|
| $\texttt{1}.\texttt{D037CB2700000}_{(16)}\times 2^{-3}$ | $\texttt{1}.\texttt{CC40C37F69D51}_{(16)}\times 2^{-3}$ |
| $\vdots$ | $\vdots$ |
| $\texttt{1}.\texttt{D037CB27EE6DF}_{(16)}\times 2^{-3}$ | $\texttt{1}.\texttt{CC40C3805229A}_{(16)}\times 2^{-3}$ |
| $\vdots$ | $\vdots$ |
| $\texttt{1}.\texttt{D037CB2800000}_{(16)}\times 2^{-3}$ | $\texttt{1}.\texttt{CC40C3806348B}_{(16)}\times 2^{-3}$ |
ã“ã‚Œã®å„è¡Œã«ã¤ã„ã¦ã€æ£ç¢ºã«ä¸¸ã‚ãŸå€¤ã‚’計算ã™ã‚‹ã«ã‚ãŸã£ã¦ã©ã‚Œã ã‘ã®ç²¾åº¦ãŒå¿…è¦ãªã®ã‹ã‚’予測ã§ããšã€å¾ã€…ã«æ¡æ•°ã‚’上ã’ãªãŒã‚‰ç¹°ã‚Šè¿”ã™*5ã«ã—ã¦ã‚‚ã©ã®ç¨‹åº¦å¾…ã¦ã°ã„ã„ã‹ã‚ã‹ã‚‰ãªã„ããˆã¨ã„ã†æ„Ÿã˜ã§ã—ょã†ã€‚
ãŸã¨ãˆã°ã€ãŸã‹ãŒ 53-bit ã§ã® correctly-rounded ãªå€¤ã‚’å¾—ã‚‹ãŸã‚ã®é€”ä¸è¨ˆç®—ã« $10^{3000}$ bits ã®ç²¾åº¦ãŒå¿…è¦ãªã„ã¨ã©ã†ã—ã¦è¨€ã„切れるã§ã—ょã†ã‹ï¼Ÿ 実際ã€å…ˆã® $\sin(x)$ ã®ã‚ˆã†ãªä¾‹ãŒå˜åœ¨ã™ã‚‹ã“ã¨ã‚‚予想ã—ã¦ã„ãªã‹ã£ãŸäººãŒå¤šã„ã®ã§ã¯ãªã„ã‹ã¨æ€ã„ã¾ã™ã€‚ 下記ã®ã‚ˆã†ãªä¾‹ã‚‚ã‚ã‚Šã¾ã™ã€‚ $$ \begin{aligned} &\phantom{{}={}} \tan(\texttt{1}.\texttt{50486B2F87014}_{(16)}\times 2^{-5}) \\ &= 1. \underbrace{\footnotesize 0101}_{\texttt{5}} {\footnotesize\,} \underbrace{\footnotesize 0000}_{\texttt{0}} {\footnotesize\,} \underbrace{\footnotesize 0111}_{\texttt{7}} {\footnotesize\,} \underbrace{\footnotesize 1000}_{\texttt{8}} {\footnotesize\,} \underbrace{\footnotesize 1100}_{\texttt{C}} {\footnotesize\,} \underbrace{\footnotesize 1110}_{\texttt{E}} {\footnotesize\,} \underbrace{\footnotesize 1011}_{\texttt{B}} {\footnotesize\,} \underbrace{\footnotesize 1111}_{\texttt{F}} {\footnotesize\,} \underbrace{\footnotesize 1111}_{\texttt{F}} {\footnotesize\,} \underbrace{\footnotesize 1001}_{\texttt{9}} {\footnotesize\,} \underbrace{\footnotesize 1100}_{\texttt{C}} {\footnotesize\,} \underbrace{\footnotesize 0111}_{\texttt{7}} {\footnotesize\,} \underbrace{\footnotesize 0010}_{\texttt{2}} {\footnotesize\,} \underbrace{\footnotesize 1000}_{\texttt{8}} {\footnotesize\,} \\ & \qquad \underbrace{\footnotesize 0000}_{\texttt{0}} {\footnotesize\,} \underbrace{\footnotesize 0000}_{\texttt{0}} {\footnotesize\,} \underbrace{\footnotesize 0000}_{\texttt{0}} {\footnotesize\,} \underbrace{\footnotesize 0000}_{\texttt{0}} {\footnotesize\,} \underbrace{\footnotesize 0000}_{\texttt{0}} {\footnotesize\,} \underbrace{\footnotesize 0000}_{\texttt{0}} {\footnotesize\,} \underbrace{\footnotesize 0000}_{\texttt{0}} {\footnotesize\,} \underbrace{\footnotesize 0000}_{\texttt{0}} {\footnotesize\,} \underbrace{\footnotesize 0000}_{\texttt{0}} {\footnotesize\,} \underbrace{\footnotesize 0000}_{\texttt{0}} {\footnotesize\,} \underbrace{\footnotesize 0000}_{\texttt{0}} {\footnotesize\,} \underbrace{\footnotesize 0000}_{\texttt{0}} {\footnotesize\,} \underbrace{\footnotesize 0000}_{\texttt{0}} {\footnotesize\,} \underbrace{\footnotesize 0010}_{\texttt{2}} {\footnotesize\,} {\footnotesize \, \ldots} {\,}_{(2)} \times 2^{-5}. \end{aligned} $$
double-double ã ã‘ã 㨠106-bit å¼±ã«ãªã£ã¦ã—ã¾ã†ã®ã§ã€å…ˆã®ã‚ˆã†ãªä¾‹ã§ã¯è² ã‘ã¦ã—ã¾ã„ã¾ã™ã€‚triple-double ã§ã¯ã©ã†ã§ã—ょã†ã‹ï¼Ÿ 足りã¾ã™ã‹ï¼Ÿ ã‚ã‹ã‚Šã¾ã›ã‚“... ã„ã‹ãŒã§ã—ãŸã‹ï¼Ÿ
「100 å°ç¨‹åº¦ã®ãƒ¯ãƒ¼ã‚¯ã‚¹ãƒ†ãƒ¼ã‚·ãƒ§ãƒ³ã‚’使ã£ã¦ 4 å¹´ã‹ã‘ã¦æœ€æ‚ªã‚±ãƒ¼ã‚¹ã®æŽ¢ç´¢ã‚’ã—ã¾ã—ãŸã€ã¨ã„ã†è«–æ–‡[2]㌠2000 å¹´ã«å‡ºã¦ã„ã¾ã™ãŒã€ä¸‹è¨˜ã®ã‚ˆã†ã«æ›¸ã‹ã‚Œã¦ã„ã¾ã™ã€‚
The results [...] give the worst cases for functions $\sin$, $\arcsin$, $\cos$, $\arccos$, $\tan$ and $\arctan$. For those functions, we have worst cases in some bounded domain only, because trigonometric functions are more difficult to handle than the other functions.
拙訳・è¦ç´„：三角関数ã¯ä»–ã®é–¢æ•°ãŸã¡ã‚ˆã‚Šæ‰±ã„ãŒé›£ã—ãã€ã„ãã¤ã‹ã®åŒºé–“ã«ãŠã‘る最悪ケースã—ã‹å¾—られã¦ãŠã‚Šã¾ã›ã‚“。
ãã‚“ãªã‚。
ã¨ã„ã†ã“ã¨ã§ã€æ¬¡ã‹ã‚‰ã®è¨˜äº‹ã¯ log ã‚„ exp ã® correct rounding ãªå®Ÿè£…ã®è©±ã«ãªã‚‹ã‹ã‚‚ã—ã‚Œãªã„ã—ã€ä¸‰è§’関数関連ã®è©±ã‚’ã™ã‚‹ã‹ã‚‚ã—ã‚Œãªã„ã—ã€ã‚µãƒ¼ãƒ™ã‚¤ã®æ—…ã«å‡ºãŸã£ãり帰ã£ã¦ã“ãªã„ã‹ã‚‚ã—ã‚Œã¾ã›ã‚“。
Kahan ã‚‚ [6] ã§è§¦ã‚Œã¦ã„ã¾ã™ãŒã€Bessel 関数 $J_n(x)$ ã®ã‚ˆã†ãªåå‰ã®ã¤ã„ãŸéžåˆç‰é–¢æ•°ã‚„ã€$y^w$ ã®ã‚ˆã†ã«è¤‡æ•°ã®å¼•æ•°ãŒã‚る関数ã«ã¤ã„ã¦ã®æœ€æ‚ªã‚±ãƒ¼ã‚¹ã®æŽ¢ç´¢ã¯ã»ã¼æœ›ã¿è–„ã§ã—ょã†ã€‚身近ãªã¨ã“ã‚ã 㨠atan2 ã¨ã‹ã¯ã©ã†ãªã‚‹ã‚“ã§ã—ょã†ã。hypot ã¯è¶…越関数ã§ã¯ãªã„ã®ã§ãã“ã¾ã§æ‚²æƒ¨ãªã“ã¨ã«ã¯ãªã‚‰ãªã„ã‹ã‚‚ã—ã‚Œã¾ã›ã‚“。
ã“ã‚Œã¯ã‚ãã¾ã§æ„Ÿæƒ…å´ã®è©±ãªã®ã§ã™ãŒã€æ¨™æº–ライブラリã®å®Ÿè£…ãŒã€Œã¾ã€œ 1–10 ULP ãらã„ã®èª¤å·®ã«åŽã¾ã£ã¦ã„ã‚‹æ°—ãŒã™ã‚‹ã—ã„ã„ã§ã—ょ（未証明）ã€ã¨ã„ã†æ„Ÿã˜ã ã£ãŸã‚Šè¦æ ¼å´ãŒã€Œåˆ¥ã«ç²¾åº¦ã«ã¤ã„ã¦ã¯è¦å®šã—ãªã„よã€ã¨ã„ã†æ„Ÿã˜ã ã£ãŸã‚Šã™ã‚‹ã®ã¯å›°ã‚Šã¾ã™ã€‚一方ã§ã€è¦æ ¼å´ãŒã€Œï¼ˆãŸã¨ãˆã°ï¼‰3 ULP ã¾ã§ã®èª¤å·®ã¯è¨±å®¹ã™ã‚‹ã€ã¨è¨€ã£ã¦ãã¦ã‚‚「ãã®å€¤ã®æ ¹æ‹ ã¯ã©ã“ã‹ã‚‰...？ã€ã¨ãªã‚Šã¾ã™ã—ã€ãã‚‚ãも「å¯æ¬æ€§ã¯...？ã€ã¨ãªã‚Šã¾ã™ã€‚値自体ã®å¯æ¬æ€§ã¯ãªã„ã‘ã©äº‹å¾Œæ¡ä»¶ã«é–¢ã—ã¦ã¯æ‹…ä¿ã™ã‚‹ã‚ˆ*6ã¿ãŸã„ãªï¼Ÿ å¯æ¬æ€§ã‚’第一目標ã«ã™ã‚‹ãªã‚‰ã€Œã“ã®æ•°å¦é–¢æ•°ã¯ã“ã†ã„ã†ã‚¢ãƒ«ã‚´ãƒªã‚ºãƒ ã§è¨ˆç®—ã™ã‚‹ã€ã¨ã§ã‚‚ã™ã‚Œã°ã‚ˆã„ã§ã™ãŒã€ã‚ˆã‚Šã‚ˆã„精度・速度ã®ã‚¢ãƒ«ã‚´ãƒªã‚ºãƒ ãŒæ案ã•ã‚ŒãŸã¨ãã«ã©ã†ã™ã‚‹ã®ï¼Ÿã¨ã„ã†è©±ã«ã‚‚ãªã£ã¦ãã¾ã™ã€‚
ã¨ã„ã†ã‚ˆã†ãªæ—¨ã®ã“ã¨ãŒ [7] ã®æ–‡çŒ®ã§æ›¸ã‹ã‚Œã¦ã„ã¦ã€æ¬¡ã®ã‚ˆã†ãªä¸»å¼µ*7ã®ç¯€ãŒã‚ã‚Šã¾ã™ã€‚
3.2 $\quad$ The only specification that makes sense is correct rounding
ãªã‹ãªã‹å¼·ã‚ã®æ€æƒ³ãªæ°—ã‚‚ã—ãªã„ã§ã¯ãªã„ã§ã™ãŒã€ã¾ãå¦å®šã‚‚ã§ããªã„ã‹ãªã¨ã„ã†æ„Ÿã˜ã§ã™ã€‚
ã¨ã„ã£ãŸæ„Ÿã˜ã§ã€ä¸‰è§’関数ã«é–¢ã—ã¦ã¯ã¡ã‚ƒã‚“ã¨ã—ãŸæœ€æ‚ªã‚±ãƒ¼ã‚¹ãŒã¾ã ã‚ã‹ã£ã¦ã„ãªã„ã¨æ€ã£ã¦ã„ã‚‹ã®ã§ã™ãŒã€correct rounding を謳ã†ãƒ©ã‚¤ãƒ–ラリãŸã¡ã¯ã©ã†ã‚„ã£ã¦ã„ã‚‹ã®ã§ã—ょã†ã€‚Ziv’s rounding test ã‚’ã—ãªãŒã‚‰æœ€å¤§ 768-bit ãらã„ã®ç²¾åº¦ã¾ã§è©¦ã™ã¿ãŸã„ãªè©±ã¯ã©ã“ã‹ã§è¦‹ãŸã‚ˆã†ãªæ°—ã¯ã—ã¤ã¤ã€ãã‚Œã§å分ã§ã‚る証明ãªã—ã«ã¯ correct ã¨ä¸»å¼µã§ããªã„ã¯ãšã§ã™ã—。
次数 $d\ge 2$ ã®ä»£æ•°çš„æ•° $\alpha$ ã«å¯¾ã—ã¦ã¯ã€ã‚る定数 $C_{\alpha}$ ãŒå˜åœ¨ã—ã¦ä»»æ„ã®æ•´æ•° $p$, $q$ ($q\ge 1$) ã«å¯¾ã—㦠$|\alpha-\tfrac pq| \ge \tfrac{C_{\alpha}}{q^d}$ ãŒæˆã‚Šç«‹ã¤ã“ã¨ãŒ Liouville ã«ã‚ˆã£ã¦ç¤ºã•ã‚Œã¦ã„るらã—ã„ã§ã™ãŒã€è¶…越数ã¯...？ã¨ã„ã†æ„Ÿã˜ã§ã€å‚ã£ã¦ã„ã¾ã™ã€‚ã©ã†ã—ã¾ã—ょã†ï¼Ÿ ã‚„ã¯ã‚Šæžåˆˆã‚Šå…¨æŽ¢ç´¢ã™ã‚‹ã—ã‹ï¼Ÿ
ãŸã¨ãˆã°ä½•ã‚‰ã‹ã®ã‚¢ãƒ«ã‚´ãƒªã‚ºãƒ ã«ã‚ˆã£ã¦é«˜ã€… $2^{-768}$ 程度ã®ç›¸å¯¾èª¤å·®ã—ã‹ãªã„値を得られるã®ã§ã‚れ㰠53-bit ã® correct rounding ã«ã“ã ã‚ã‚‹å¿…è¦ã¯ãªã„ã®ã§ã¯ï¼Ÿã¨ã„ã†ä¸»å¼µã¯çœŸã£å½“ãªæ°—ãŒã—ã¤ã¤ã€ã¾ããã‚Œã¯ãã‚Œã‹ãªã¨ã„ã†æ°—æŒã¡ã§ã‚„ã£ã¦ã„ã¾ã™ã€‚
Vincent Lefèvre ãŒå¤§æ´»èºã™ãŽã€‚ã¾ãŸã€æœ€å¾Œã®ã‚µãƒ¼ãƒ™ã‚¤è«–æ–‡ã«é–¢ã—ã¦ã¯ 2025 年（åˆç‰ˆã¯ 2024 年）ã§ã€ã³ã£ãã‚Šã§ã™ã€‚
ã¨ã„ã†ã“ã¨ã§ãˆã³ã¡ã‚ƒã‚“ã¯ã“ã‚Œã‹ã‚‰ãŠå‹‰å¼·ã§ã™ã€‚フォãƒãƒ®ã‚‚ã”ãã’んよã†ã€‚
ãŠã‚ã‚Šã§ã™ã€‚
*1:標準入りã—ã¦ãれ〜〜〜。
*2:$\sin(x)$ ã® 54-bit 仮数部ã¸ã® faithful-rounded ãªå€¤ã‚’è¿”ã™ã¨ã„ã†æ„味åˆã„ã§ã®å‘½åã§ã™ã€‚
*3:普通ã®äººã«ã¨ã£ã¦ã¯ $\sin(x) \approx \texttt{1}.\texttt{CC40C3805229B}_{(16)}\times 2^{-3}$ ã§å分ã‚ã§ãŸããªã„？ã¨ã„ã†æ°—ã‚‚ã—ã¾ã™ã€‚普通ã®äººã®ã“ã¨ã¯çŸ¥ã‚Šã¾ã›ã‚“ãŒã€‚
*4:IEEE 754 ã®ç–定ã«ã‚‚é–¢ã‚ã£ã¦ãŠã‚‰ã‚Œã‚‹æ–¹ã§ã€ã¨ã¦ã‚‚ãˆã‚‰ã„æ–¹ã§ã™ã。
*5:Ziv’s strategy ã¨ã‹ Ziv’s rounding test ã¨ã‹å‘¼ã°ã‚Œã‚‹ã‚‚ã®ãŒã‚るよã†ã§ã™ã€‚
*6:人々ã«ã¯ãã‚‚ãも浮動å°æ•°ç‚¹åž‹ã£ã¦ãã†ã„ã†ã‚‚ã®ã ã¨æ€ã‚ã‚Œã¦ã„ã¾ã™ã‹ï¼Ÿ ã†ã€œã€œã€œã€œã‚“。
*7:ã„ã‚ゆる数å¦ã®æ„味åˆã„ã§ã® Claim ã§ã¯ãªãã¦ã€ãŠæ°—æŒã¡è¡¨æ˜Žã«è¿‘ã„ã¨æ€ã„ã¾ã™ã€‚
å°ã•ãªèª¤å·®ã‚’ç©ã¿é‡ãã‚‹æ“作ã¨ã€ç©ã¿é‡ãªã£ãŸèª¤å·®ã«å½±éŸ¿ã•ã‚Œã‚‹æ“作を区別ã—ã¦è€ƒãˆã¾ã—ょã†ã€‚
åŒã˜ç¬¦å·ã®æµ®å‹•å°æ•°ç‚¹æ•°ã‚’åŠ ç®—ã‚„ä¹—ç®—ã‚’è¨ˆ $k$ 回繰り返ã—ã¦ã‚‚ã€$(1+2^{-53})^k-1 \approx 2^{-53}\cdot k$ 程度ã®ç›¸å¯¾èª¤å·®ã—ã‹ç”Ÿã¾ã‚Œã¾ã›ã‚“。 競プãƒã«ãŠã‘る演算回数ã¯ã›ã„ãœã„ $2^{30}$ 回程度ã§ã—ょã†ã‹ã‚‰ã€ã ã„ãŸã„ $2^{-23}\approx 1.19\times 10^{-7}$ 程度ã®ç›¸å¯¾èª¤å·®ã«åŽã¾ã‚Šã¾ã™ã€‚ $2^{20}$ 回ãªã‚‰ $2^{-33}\approx 1.16\times 10^{-10}$ 程度ã§ã™ã€‚
ã“れらã®æ“作ãŒå•é¡Œã«ãªã‚‹ã“ã¨ã¯åŸºæœ¬çš„ã«ãªã„ã§ã—ょã†ã€‚ å•é¡Œã«ãªã‚‹æ“作ã¯ä¸»ã«ä¸‰ã¤ã§ã™ã€‚æ•´æ•°ã¸ã®ä¸¸ã‚ã€æ¯”較ã€å¼•ãç®—ã§ã™ã€‚
具体例を交ãˆãªãŒã‚‰è¦‹ã¦ã„ãã¾ã™ã€‚
IEEE 754 ã®è¦æ ¼ã«æº–æ‹ ã—ãŸå‡¦ç†ç³»ã‚’考ãˆã¾ã™ã€‚
C ã«ãŠã‘ã‚‹ double ã®ãƒ‡ãƒ•ã‚©ãƒ«ãƒˆã®æŒ™å‹•ã«ã¤ã„ã¦è€ƒãˆã€overflow ã‚„ underflow ã«ã¤ã„ã¦ã¯èµ·ããªã„ã“ã¨ã¨ã—ã¾ã™ã€‚
ã™ãªã‚ã¡ã€è€ƒãˆã‚‹æµ®å‹•å°æ•°ç‚¹æ•° $x$ ã¯æœ‰é™ã§ã‚ã‚Šã€ä¸‹è¨˜ã‚’満ãŸã—ã¾ã™ã€‚
デフォルトã®ä¸¸ã‚方㯠tiesToEven ã¨å‘¼ã°ã‚Œã‚‹ã‚‚ã®ã§ã€ã€Œdouble ã§è¡¨ã›ã‚‹å€¤ã®ã†ã¡ã€ä¸Žãˆã‚‰ã‚ŒãŸå®Ÿæ•°ã«æœ€ã‚‚è¿‘ã„ã‚‚ã®ã‚’è¿”ã™ã€ã¨ã„ã†ã‚‚ã®ã§ã™ã€‚表ã›ã‚‹ 2 æ•°ã®ã¡ã‚‡ã†ã©ä¸é–“ã®å ´åˆã® tiebreak ã¯åŽ³å¯†ã«å®šã‚られã¦ã„ã¾ã™ãŒã€è¾¼ã¿å…¥ã‚‰ãªã„誤差評価ã«ãŠã„ã¦ã¯å•é¡Œã«ãªã£ã¦ã“ãªã„ã®ã§ã€ä»Šå›žã¯å‰²æ„›ã—ã¾ã™ã€‚
実数 $x$ を丸ã‚ãŸã‚‚ã®ã‚’ $\roundcirc x$ ã¨æ›¸ãã¾ã™ã€‚$|x|\gt 0$ ã®ã¨ã下記ãŒæˆã‚Šç«‹ã¡ã¾ã™ã€‚
浮動å°æ•°ç‚¹æ•°ã«ãŠã‘る四則演算を $\oplus$, $\ominus$, $\otimes$, $\oslash$ ã¨æ›¸ãã€$x \oplus y = \roundcirc{x+y}$ ãªã©ãŒæˆã‚Šç«‹ã¡ã¾ã™ï¼ˆä»–ã®å„演算ã«ã¤ã„ã¦ã‚‚åŒæ§˜ï¼‰ã€‚ ã™ãªã‚ã¡ã€å››å‰‡æ¼”ç®—ã«ãŠã„ã¦ã¯ã€Œç„¡é™ç²¾åº¦ã§è¨ˆç®—ã—ãŸã‚‚ã®ã‚’æ£ç¢ºã«ä¸¸ã‚ãŸã‚‚ã®ãŒå¾—られるã€ã¨ã„ã†ã“ã¨ã§ã™ã€‚
ãŸã¨ãˆã° $\tfrac43 = 1.333{\ldots}$ ã§ã€ã“ã‚Œã¨éš£ã‚Šåˆã† double ã®å€¤ã¯
$$
\begin{aligned}
6004799503160661\times 2^{-52} &= {\small 1.333333333333333}{\footnotesize 259318465024989}{\scriptsize 563971757888793}{\tiny 9453125}, \\
6004799503160662\times 2^{-52} &= {\small 1.333333333333333}{\footnotesize 481363069950020}{\scriptsize 872056484222412}{\tiny 109375}
\end{aligned}
$$
ã§ã™ã€‚è¿‘ã„ã®ã¯å‰è€…ã§ã™ã‹ã‚‰ã€$4\oslash 3 = 6004799503160661\times 2^{-52}$ ã¨ãªã‚Šã¾ã™ã€‚丸ã‚モードを変ãˆãªã„é™ã‚Šã€å‹æ‰‹ã«å¾Œè€…ã«ãªã£ã¦ã—ã¾ã†ã“ã¨ã¯ã‚ã‚Šã¾ã›ã‚“。
特ã«ã€åŽ³å¯†ã«
$$
\begin{aligned}
4\oslash 3 &= \frac{4-2^{-52}}3 \\
&= \frac43\cdot(1-2^{-54})
\end{aligned}
$$
ãŒæˆã‚Šç«‹ã¡ã¾ã™ã€‚相対誤差㌠$2^{-53}$ 以下ã«ãªã£ã¦ã„ã‚‹ã“ã¨ãŒã‚ã‹ã‚Šã¾ã™ã。
ã•ã¦ã€å•é¡Œã«ãªã‚‹äºŒã¤ã®æ“作ã«ã¤ã„ã¦è¦‹ã¦ã„ãã¾ã—ょã†ã€‚
床関数 $\floor x$ を考ãˆã¾ã™ã€‚次ã®ã‚ˆã†ãªå•é¡Œã‚’見ã¦ã¿ã¾ã—ょã†ã€‚
何らã‹ã®å¼ã§è¡¨ã•ã‚Œã‚‹å®Ÿæ•° $x^{\ast}$ ãŒã‚ã‚‹ã¨ã—ã¾ã™ã€‚ã“ã‚Œã«å¯¾ã—㦠$\floor{x^{\ast}}$ を出力ã—ã¦ãã ã•ã„。
ãªãŠã€double ã§è¡¨ã›ã‚‹ä»»æ„ã®å€¤ $x$ ã«å¯¾ã—㦠$\roundcirc{\floor x} = \floor x$ ãŒæˆã‚Šç«‹ã¡ã¾ã™ã€‚ã™ãªã‚ã¡ã€åºŠé–¢æ•°ã«ã‚ˆã£ã¦èª¤å·®ãŒç”Ÿã˜ã‚‹ã“ã¨ã¯ã‚ã‚Šã¾ã›ã‚“*1。
ç°¡å˜ãªä¾‹ã¨ã—ã¦ã€$x^{\ast} = 0.1 \times 10$ を考ãˆã¦ã¿ã¾ã™ã€‚ ã“ã“ã§ã€$\roundcirc{0.1} = \tfrac1{10}\cdot(1+2^{-54})$ ã‹ã¤ $\roundcirc{10} = 10$ ã§ã™ã€‚ $x^{\ast}$ ã®è¿‘似値ã«ç›¸å½“ã™ã‚‹æµ®å‹•å°æ•°ç‚¹æ•°ã‚’ $x$ ã¨ã—ã¦æ›¸ãã¾ã™ã€‚ $$ \begin{aligned} x &= \roundcirc{0.1} \otimes \roundcirc{10} \\ &= \left(\tfrac1{10}\cdot (1+2^{-54})\right) \otimes 10 \\ &= \roundcirc{1+2^{-54}} \\ &= 1 \end{aligned} $$ ã§ã™ã€‚一方ã€$0.1\times 10 = 0.1+0.1+\dots+0.1$ ã¨ã—ã¦è¨ˆç®—ã—ãŸå ´åˆã‚’考ãˆã¦ã¿ã¾ã™ã€‚ $$ \begin{aligned} x &= \underbrace{\roundcirc{0.1} \oplus \roundcirc{0.1} \oplus \dots \oplus \roundcirc{0.1}}_{10} \\ &= \left(\tfrac1{10}\cdot (\hphantom{0}1+\hphantom{0}1\cdot 2^{-54})\right) \oplus \underbrace{\roundcirc{0.1} \oplus \roundcirc{0.1} \oplus \dots \oplus \roundcirc{0.1}}_9 \\ &= \left(\tfrac1{10}\cdot (\hphantom{0}2+\hphantom{0}2\cdot 2^{-54})\right) \oplus \underbrace{\roundcirc{0.1} \oplus \roundcirc{0.1} \oplus \dots \oplus \roundcirc{0.1}}_8 \\ &= \left(\tfrac1{10}\cdot (\hphantom{0}3+\hphantom{0}8\cdot 2^{-54})\right) \oplus \underbrace{\roundcirc{0.1} \oplus \roundcirc{0.1} \oplus \dots \oplus \roundcirc{0.1}}_7 \\ &= \left(\tfrac1{10}\cdot (\hphantom{0}4+\hphantom{0}4\cdot 2^{-54})\right) \oplus \underbrace{\roundcirc{0.1} \oplus \roundcirc{0.1} \oplus \dots \oplus \roundcirc{0.1}}_6 \\ &= \left(\tfrac1{10}\cdot (\hphantom{0}5+\hphantom{0}0\cdot 2^{-54})\right) \oplus \underbrace{\roundcirc{0.1} \oplus \roundcirc{0.1} \oplus \dots \oplus \roundcirc{0.1}}_5 \\ &= \left(\tfrac1{10}\cdot (\hphantom{0}6-\hphantom{0}4\cdot 2^{-54})\right) \oplus \underbrace{\roundcirc{0.1} \oplus \roundcirc{0.1} \oplus \dots \oplus \roundcirc{0.1}}_4 \\ &= \left(\tfrac1{10}\cdot (\hphantom{0}7-\hphantom{0}8\cdot 2^{-54})\right) \oplus \roundcirc{0.1} \oplus \roundcirc{0.1} \oplus \roundcirc{0.1} \\ &= \left(\tfrac1{10}\cdot (\hphantom{0}8-12\cdot 2^{-54})\right) \oplus \roundcirc{0.1} \oplus \roundcirc{0.1} \\ &= \left(\tfrac1{10}\cdot (\hphantom{0}9-16\cdot 2^{-54})\right) \oplus \roundcirc{0.1} \\ &= \left(\tfrac{1}{10}\cdot(10-20\cdot 2^{-54})\right) \\ &= 1 - 2^{-53}. \end{aligned} $$ ã™ãªã‚ã¡ã€$x^{\ast}$ ã¨ã®ç›¸å¯¾èª¤å·®ã¯ $-2^{-53}\approx -1.11\times 10^{-16}$ ã§ã™ã€‚（無駄ãªèª¤å·®ãŒå‡ºã¦ã„ã‚‹ã¨ã¯ã„ãˆï¼‰å分ã«è¿‘ã„値ãŒå¾—られã¦ã„ã‚‹ã¨è¨€ã£ã¦æ¦‚ãå·®ã—支ãˆãªã„ã§ã—ょã†ã€‚
ã—ã‹ã—ã€$\floor{x^{\ast}}$ を計算ã™ã‚‹ã¨ãªã‚‹ã¨è©±ã¯åˆ¥ã§ã™ã€‚ æ•´æ•°ã«ã¤ã„ã¦è€ƒãˆãŸã„ã¨ãã¯ç›¸å¯¾èª¤å·®ãŒèˆˆå‘³ã®å¯¾è±¡ã§ã‚ã‚‹ã“ã¨ã¯ã‚ã¾ã‚Šãªãã€çµ¶å¯¾èª¤å·®ãŒæ°—ã«ãªã‚‹ã“ã¨ãŒå¤šã„ã§ã™ã€‚ã™ãªã‚ã¡ã€$1$ 以上ã®çµ¶å¯¾èª¤å·®ã¯è¨±å®¹ã—ãªã„ã¨ã„ã†ã“ã¨ã§ã™ã€‚ $\floor{1-2^{-53}} = 0$ ã§ã™ã‹ã‚‰ã€ã“ã‚Œã¯è¨±å®¹ã§ãã¾ã›ã‚“。 相対誤差ãŒã©ã‚Œã ã‘å°ã•ã抑ãˆã‚‰ã‚Œã¦ã„ãŸã¨ã—ã¦ã‚‚ã€$x$ ㌠$\floor{x^{\ast}}$ より少ã—ã§ã‚‚å°ã•ã„ã¨ç ´ç¶»ã—ã¦ã—ã¾ã„ã¾ã™ã€‚
$\floor{x^{\ast}}-x$ ã®ä¸Šç•Œã‚’ $\delta$ ã¨ã—ã¾ã™ã€‚$x\oplus\delta\lt \floor{x^{\ast}}+1$ ãŒå¸¸ã«æˆã‚Šç«‹ã¤ã®ã§ã‚ã‚Œã°ã€$\floor{x\oplus\delta}$ を計算ã™ã‚Œã°ã‚ˆã„ã§ã™*2。
åŒç¬¦å·ã®è¨ˆç®— $k$ 回程度ã®å ´åˆã§ã‚ã‚Œã°ã€$\delta\approx kx^{\ast}\cdot 2^{-53}$ 程度ã«ã¨ã£ã¦ã¿ã¦ã€å•é¡Œãªã„ã‹ã‚’調ã¹ã‚‹ã®ãŒã‚ˆã„ã§ã—ょã†ã‹ã€‚
一応ã€ã€Œã‚ã–ã‚ã–掛ã‘ç®—ã§æ¸ˆã‚€ã¨ã“ã‚を足ã—ç®—ã«ã—ãŸã‹ã‚‰ãŠã‹ã—ããªã£ãŸã€ã¨ã„ã†ä¸»å¼µã«å¯¾ã™ã‚‹å例も挙ã’ã¦ãŠãã¾ã™*3。 $1\oslash 49 = \tfrac1{49}\cdot(1-23\cdot 2^{-58})$ を使ã„ã¾ã™ã€‚ $$ \begin{aligned} (1\oslash 49)\otimes 49 &= 1-2^{-53}, \\ \underbrace{(1\oslash 49)\oplus\dots\oplus(1\oslash 49)}_{49} &= 1+6\cdot 2^{-53} \end{aligned} $$ ã§ã™ã€‚ã“ã‚Œã¯ã€è¶³ã—ç®—ã‚’ã—ãŸæ–¹ã®ã¿ãŒï¼ˆãŸã¾ãŸã¾ï¼‰ã†ã¾ãã„ã£ã¦ã„ã¾ã™ã。
ã„ãšã‚Œã«ã›ã‚ˆã€ã€ŒãŸã¾ãŸã¾ã†ã¾ãã„ãã®ã‚’祈るã€ã®ã§ã¯ãªãã¦ã€èª¤å·®è©•ä¾¡ã‚’ã—ã¦è£œæ£é …を入れるãªã‚Šã—ã¦ã€Œã†ã¾ãã„ãã“ã¨ã‚’示ã—ã¦ã‚„ã£ã¦ã„ã‚‹ã€ã«ã™ã‚‹ã®ãŒé‡è¦ã§ã™ã€‚ 実際ã®å…·ä½“例ã¯å¾ŒåŠã®ç« ã§ç´¹ä»‹ã—ã¾ã™ã€‚
本質的ã«ã¯æ•´æ•°ã¸ã®ä¸¸ã‚ã¨ã»ã¼åŒæ§˜ã ã¨æ€ã„ã¾ã™ã€‚ãŸã¨ãˆã°æœ¬æ¥ $x^{\ast} = 1$ ã‹ã¤ $y^{\ast} = 1$ ã¨ãªã‚‹ã¹ãã¨ã“ã‚㧠$x = 1+2^{-52}$ ã‹ã¤ $y = 1$ ã¨ãªã£ã¦ã—ã¾ã£ãŸã ã‘㧠$x\le y$ ã¯æˆã‚Šç«‹ãŸãªã„ã§ã™ã。
ã„ã‚ゆる eps を足ã—引ãã—ã¦èª¿æ•´ã™ã‚‹ã®ãŒå°æ‰‹å…ˆãƒ†ã‚¯ãƒ‹ãƒƒã‚¯ã¨ã—ã¦åºƒãŒã£ã¦ã„ã¾ã™ãŒã€é©åˆ‡ãªå€¤ã®æ±ºã‚æ–¹ã¯å‰é …ã® $\delta$ ã¨åŒæ§˜ã§ã™ã€‚
実数 $x^{\ast}$, $y^{\ast}$ ã®ã¤ã‚‚ã‚Šã®å€¤ã¨ã—ã¦æµ®å‹•å°æ•°ç‚¹æ•° $x$, $y$ ã‚’å¾—ãŸã¨ã—ã¾ã™ã€‚$x^{\ast} \le y^{\ast}$ ã®åˆ¤å®šã®ãŸã‚ã« $x\le y$ ã¨ã™ã‚‹ã®ã¯å¦¥å½“ã‹ãªï¼Ÿã¨ã„ã†å•é¡Œã§ã™ã€‚ãªã«ã‚‚考ãˆãªã„ã¨ä¸‹è¨˜ã®ã‚ˆã†ã«ãªã‚Šå¾—ã¾ã™ã€‚
| $x\ll y$ | $x\le y$ | $x\gt y$ | $x\gg y$ | |
|---|---|---|---|---|
| $x^{\ast} \le y^{\ast}$ | true | true | false (?) | N/A |
| $x^{\ast} \gt y^{\ast}$ | N/A | true (?) | false | false |
N/A ã¨æ›¸ã„ã¦ã„ã‚‹ã®ã¯ã€ãã®æ¬„ã®æ¡ä»¶ã‚’満ãŸã™ã‚ˆã†ãªå…¥åŠ›ãŒå˜åœ¨ã—ãªã„ã¨ã„ã†æ„図ã§ã™ã€‚ $x$ 㨠$y$ ãŒå分ã«é›¢ã‚Œã¦ã„ã‚‹ã¨ãã¯æœŸå¾…通り判定ã§ãã¾ã™ãŒã€ãã†ã§ãªã„ã¨ãã¯èª¤ã£ã¡ã‚ƒã†ã‹ã‚‚？ã¨ã„ã†ã“ã¨ã§ã™ã。
ãã“ã§ã€ã‚ã‚‹ $\delta$ ãŒå˜åœ¨ã—ã¦ä¸‹è¨˜ã®ã‚ˆã†ã«ãªã£ãŸã‚‰ã†ã‚Œã—ã„ã§ã™ã€‚
| $x\le y+\delta$ | $x\gt y+\delta$ | |
|---|---|---|
| $x^{\ast} \le y^{\ast}$ | true | N/A |
| $x^{\ast} \gt y^{\ast}$ | N/A | false |
$x$ ㌠$y$ より少ã—ã ã‘大ãã‹ã£ãŸã¨ã㯠$x^{\ast}\le y^{\ast}$ ã¨ã—ã¦åˆ¤å®šã—ãŸã„ã¨ã„ã†ã“ã¨ã§ã™ã€‚ å³ä¸Šã‚„左下㌠N/A ã«ãªã‚‰ãšã€ä¸‹è¨˜ã®ã‚ˆã†ã«ãªã£ã¦ã—ã¾ã†å ´åˆã¯ã†ã‚Œã—ããªã„ã§ã™ã€‚
| $x\le y+\delta$ | $x\gt y+\delta$ | |
|---|---|---|
| $x^{\ast} \le y^{\ast}$ | true | false (?) |
| $x^{\ast} \gt y^{\ast}$ | N/A | false |
$x\le y+\delta$ ã®åˆ¤å®šã¯ $x\ominus y\le \delta$ ã¨ã—ã¦è¡Œãˆã°ã€$\tfrac y2\le x\le 2y$ ãªã‚‰ã° $x\ominus y=x-y$ ãŒæˆã‚Šç«‹ã¤ã®ã§ã‚ˆã„ã§ã™ã。 ã“れ㯠Sterbenz lemma ã¨å‘¼ã°ã‚Œã‚‹æœ‰åãªè£œé¡Œã‹ã‚‰å¾“ã„ã¾ã™ã€‚ $x\gt 2y$ ã‚„ $x\lt \tfrac y2$ ã®ã¨ã㯠$x\gg y$ ã‚„ $x\ll y$ ã®æ‰±ã„ã«ãªã£ã¦ã„ã‚‹ã¨ã†ã‚Œã—ã„ã§ã™ã€‚
評価ã®æ–¹æ³•ã¨ã—ã¦ã¯ã€æ•´æ•°ã¸ã®ä¸¸ã‚ã®éš›ã® $\delta$ ã¨åŒæ§˜ã«è¡Œãˆã°ã‚ˆã„ã§ã™ã。
ã‚‚ã¡ã‚ã‚“é©åˆ‡ãª $\delta$ ã¯çŠ¶æ³ã«ã‚ˆã‚Šã¾ã™ã€‚「テンプレート用ã«ãŠã™ã™ã‚ã® eps ã¯ã„ãらã§ã™ã‹ï¼Ÿã€ã®ã‚ˆã†ãªè³ªå•*4ã¯ãƒŠãƒ³ã‚»ãƒ³ã‚¹ã§ã‚ã‚‹ã“ã¨ãŒã‚ã‹ã‚‹ã¨æ€ã„ã¾ã™ã€‚
ã¯ã„ã€æœ¬é¡Œã®ä¸ã®æœ¬é¡Œã§ã™ã€‚æ¡è½ã¡ (catastrophic cancellation (of significant digits)) ã¨å‘¼ã°ã‚Œã‚‹ã‚‚ã®ã«ã¤ã„ã¦è©±ã—ã¾ã™ã€‚ æ•°ãŒæ‰“ã¡æ¶ˆã—åˆã†ã“ã¨ã‚’指ã—㦠cancellation ã¨å‘¼ã‚“ã§ã„ã¾ã™*5。 主è¦ãªæ¡åŒå£«ãŒæ‰“ã¡æ¶ˆã—åˆã£ãŸã¨ãã«èª¤å·®ã®éƒ¨åˆ†ãŒç„¡è¦–ã§ããªããªã‚‹ã®ãŒå•é¡Œã¨ãªã‚Šã¾ã™ã€‚ ã‚‚ã¡ã‚ã‚“ã€æµ®å‹•å°æ•°ç‚¹æ•°å´ã¯ã€Œè‡ªåˆ†ã¯èª¤å·®ã‚’ã“ã®ãらã„å«ã‚“ã§ã„ã¾ã™ã€ã¨ã„ã†ã®ã‚’自覚ã—ã¦ã„ã‚‹ã‚ã‘ã§ã¯ãªã„ã®ã§ã€ã€Œã“ã®é …ã¯ã“ã®ãらã„誤差をå«ã‚“ã§ã„ã‚‹ããˆã€ã¨ã„ã†ã®ã¯äººé–“å´ãŒè§£æžã™ã‚‹ã—ã‹ãªã„ã§ã™ã。
実数 $x^{\ast}$, $y^{\ast}$ ã®è¿‘似値ã¨ã—ã¦è¨ˆç®—ã—ãŸæµ®å‹•å°æ•°ç‚¹æ•°ã‚’ $x$, $y$ ã¨ã—ã¾ã™ã€‚$x^{\ast} = x\cdot(1+\varepsilon_x)$ ã‹ã¤ $y^{\ast} = y\cdot(1+\varepsilon_y)$ ã¨ã—ã¾ã—ょã†ã€‚$|\varepsilon_x|$, $|\varepsilon_y|$ ã®ä¸Šç•Œã¯çŠ¶æ³ã«ã‚ˆã‚‹ã‚‚ã®ã§ã™ã€‚
ã“ã®ã¨ãã€$x^{\ast}-y^{\ast} = (x-y) + (x\varepsilon_x - y\varepsilon_y)$ ã¨ãªã‚Šã¾ã™ã€‚$x-y$ ãŒã¨ã¦ã‚‚å°ã•ã„ã¨ã€$x\varepsilon_x - y\varepsilon_y$ ãŒç„¡è¦–ã§ããªããªã£ã¦ãã¾ã™ã€‚
Exercise 1: $x = 2^{52}+2$ ã‹ã¤ $y = 2^{52}+1$ ã«å¯¾ã—㦠$\sqrt x-\sqrt y$ を計算ã›ã‚ˆã€‚
note: $\sqrt{1+x} = 1 + \tfrac12x - \tfrac18x^2 + \tfrac1{16}x^3 + O(x^4)$ ãŒæˆã‚Šç«‹ã¤ã€‚
Bad solution
$$ \begin{aligned} \sqrt{2^{52}+2} &= 2^{26}\cdot\sqrt{1+2^{-25}} \\ &= 2^{26}\cdot(1 + 2^{-26} - 2^{-28} + 2^{-29} + \cdots), \\ \sqrt{2^{52}+1} &= 2^{26}\cdot\sqrt{1+2^{-26}} \\ &= 2^{26}\cdot(1 + 2^{-27} - 2^{-29} + 2^{-30} + \cdots) \\ \end{aligned} $$ ãªã®ã§ $$ \begin{aligned} \roundcirc{\sqrt{2^{52}+2}} &= 2^{26} + 2^{-26}, \\ \roundcirc{\sqrt{2^{52}+1}} &= 2^{26} \end{aligned} $$ ãŒæˆã‚Šç«‹ã¤ã€‚よã£ã¦ $$ \begin{aligned} \roundcirc{\sqrt{2^{52}+2}} \ominus \roundcirc{\sqrt{2^{52}+1}} &= (2^{26}+2^{-26})\ominus 2^{-26} \\ &= 2^{-26} \end{aligned} $$ ã§ã‚る。よã£ã¦ $\sqrt x-\sqrt y \approx 2^{-26}$ ã§ã‚ã‚‹ (?)。$\eod$
Better solution
$$ \begin{aligned} &\phantom{{}={}} \sqrt{2^{52}+2} - \sqrt{2^{52}+1} \\ &= \frac{(\sqrt{2^{52}+2} - \sqrt{2^{52}+1})\cdot(\sqrt{2^{52}+2} + \sqrt{2^{52}+1})}{\sqrt{2^{52}+2} + \sqrt{2^{52}+1}} \\ &= \frac{(2^{52}+2) - (2^{52}+1)}{\sqrt{2^{52}+2} + \sqrt{2^{52}+1}} \\ &= \frac1{\sqrt{2^{52}+2} + \sqrt{2^{52}+1}} \end{aligned} $$ ã§ã‚り〠$$ \begin{aligned} 1\oslash (\roundcirc{\sqrt x}\oplus \roundcirc{\sqrt y}) &= 1\oslash ( (2^{26}+2^{-26})\oplus 2^{26} ) \\ &= 1\oslash 2^{27} \\ &= 2^{-27} \end{aligned} $$ ã§ã‚る。よã£ã¦ $\sqrt x-\sqrt y \approx 2^{-27}$ ã§ã‚る。ã¾ãŸã€ç›¸å¯¾èª¤å·®ã¯é«˜ã€… $4\cdot 2^{-53}$ 程度ã§ã‚る。$\eod$
Exercise 2: $x =2^{-53}$ ã«å¯¾ã—㦠$\tfrac1x\cdot(e^x-1)$ を計算ã›ã‚ˆã€‚
Bad solution
$$ \exp(2^{-53}) = {\small 1.000000000000000}{\footnotesize 111022302462515}{\scriptsize 660205338988848}{\tiny 236989105051344{\dots}} $$ ã§ã‚り〠$$ \roundcirc{\exp(2^{-53})} = 1 + 2^{-52} $$ ã§ã‚ã‚‹*6。実際ã«è¨ˆç®—ã—〠$$ \begin{aligned} &\phantom{{}={}} (\roundcirc{\exp(2^{-53})}\ominus 1)\oslash 2^{-53} \\ &= 2^{-52}\oslash 2^{-53} \\ &= 2 \end{aligned} $$ を得る。よã£ã¦ $\tfrac1x\cdot(e^x-1) \approx 2$ ã§ã‚ã‚‹ (?)。$\eod$
Better solution（むãšã‹ã—ã„）
$e^x-1$ を計算ã™ã‚‹éš›ã®æ¡è½ã¡ã‚’é¿ã‘ãŸã„。
$y = \roundcirc{\exp(x)}$ ã¨ã™ã‚‹ã€‚ã“ã“ã§ã€ã‚る実数 $x'$ ã«å¯¾ã—㦠$y = \exp(x')$ ãŒæˆã‚Šç«‹ã¤ã€‚ ã“ã‚Œã«ã‚ˆã‚Šã€$\roundcirc{\exp(x)}\ominus 1 = \exp(x')-1$ ãŒæˆã‚Šç«‹ã¤ãŸã‚ã€$\tfrac1x\cdot(e^x-1)$ ã®ä»£ã‚ã‚Šã« $\tfrac1{x'}\cdot(e^{x'}-1)$ を計算ã™ã‚‹ã“ã¨ã«ã™ã‚‹ã€‚
数値計算ã«ã‚ˆã‚Š $\roundcirc{\log(y)} = 2^{-52} - 2^{-105}$ を得る。よã£ã¦ã€ $$ \begin{aligned} (y\ominus 1)\oslash \roundcirc{\log(y)} &= 2^{-52} \oslash (2^{-52} - 2^{-105}) \\ &= 1+2^{-52} \end{aligned} $$ ã¨ãªã‚‹ã€‚
ã•ã¦ã€èª¤å·®è©•ä¾¡ã‚’ãŒã‚“ã°ã‚‹ã€‚ $$ \begin{aligned} \exp(x) &= \sum_{i=0}^{\infty} \frac{x^i}{i!}, \\ \frac{\exp(x)-1}x &= \sum_{i=0}^{\infty} \frac{x^i}{(i+1)!} \end{aligned} $$ ã§ã‚り〠$$ \begin{aligned} &\phantom{{}={}} \frac{\exp(x+h)-1}{x+h} - \frac{\exp(x)-1}x \\ &= \sum_{i=0}^{\infty} \frac{(x+h)^i}{(i+1)!} - \sum_{i=0}^{\infty} \frac{x^i}{(i+1)!} \\ &= \sum_{i=0}^\infty \frac1{(i+1)!}\sum_{j=1}^i \binom ij h^j x^{i-j} \\ % &= \frac1{2!} \left(\binom 11 h^1 x^0 \right) \\ % & \qquad\quad {} + \frac1{3!} \left(\binom 21 h^1 x^1 + \binom 22 h^2 x^0 \right) \\ % & \qquad\quad {} + \frac1{4!} \left(\binom 31 h^1 x^2 + \binom 32 h^2 x^1 + \binom 33 h^3 x^0 \right) \\ % & \qquad\quad {} + \cdots \\ % &= \left(\frac1{2!} \binom11 h^1 + \frac1{3!} \binom 22 h^2 + \frac1{4!} \binom 33 h^3 + \cdots \right) x^0 \\ % & \qquad\quad {} + \left(\frac1{3!} \binom21 h^1 + \frac1{4!} \binom 32 h^2 + \cdots \right) x^1 \\ % & \qquad\quad {} + \cdots \\ &= \sum_{i=0}^{\infty} \sum_{j=0}^{\infty} \frac1{(i+j+2)!} \binom{i+j+1}{j+1} h^{j+1} x^i \\ &= \sum_{j=0}^{\infty} \frac1{(j+2)!} h^{j+1} + \sum_{j=0}^{\infty} \frac{j+2}{(j+3)!} h^{j+1} x + O(x^2) \\ &= \left(\frac{\exp(h)-(1+h)}h\right) + \left(\frac{\exp(h)\cdot(h-1)+1}{h^2}-\frac12\right)x + O(x^2) \\ % &= \left(\frac{\tfrac12h^2 + O(h^3)}h\right) + \left(\frac{(1+h+\tfrac12h^2+O(h^3) )\cdot(h-1)+1}{h^2}-\frac12\right)x + O(x^2) \\ % &= \left(\frac12h + O(h^2)\right) + \left(\frac{(h+h^2+\tfrac12h^3 - 1-h-\tfrac12h^2-\tfrac16h^3 + O(h^4) )+1}{h^2}-\frac12\right)x + O(x^2) \\ % &= \left(\frac12h + O(h^2)\right) + \left(\frac{(\tfrac12h^2+\tfrac13h^3 + O(h^4) )}{h^2}-\frac12\right)x + O(x^2) \\ &= \left(\tfrac12h + O(h^2)\right) + \left(\tfrac13h + O(h^2)\right)x + O(x^2) \\ &= \tfrac h2 + O(h(x+h) ) \end{aligned} $$ ã«ãªã‚‹ã¨æ€ã†ã€‚ã¾ãŸã€$\tfrac1{x'}(y-1)$ 㨠$(y\ominus 1)\oslash \roundcirc{\log(y)}$ ã®ç›¸å¯¾èª¤å·®ã¯ $2\cdot 2^{-53}$ 程度ã§ã‚る。真ã®å€¤ã¯ $1$ 程度ã§ã‚ã‚‹ã‹ã‚‰ã€çµ¶å¯¾èª¤å·®ã‚‚ $2\cdot 2^{-53}$ 程度ã§ã‚る。 $h$ ã¯é«˜ã€… $2^{-53}$ 程度ãªã®ã§ã€åˆã‚ã›ã¦ $2.5\cdot 2^{-53}$ 程度ã®çµ¶å¯¾èª¤å·®ã§ã‚ã‚‹ã“ã¨ãŒã‚ã‹ã‚‹ã€‚$\eod$
Sterbenz lemma ã¯ã€$x$ 㨠$y$ ã®æ¯”㌠$\tfrac12$ 以上 $2$ 以下ã®ã¨ã㯠$x\ominus y = x-y$ ã«ãªã‚‹ï¼ˆå·®ã«ã‚ˆã‚‹èª¤å·®ãŒç”Ÿã˜ãªã„）ã¨ã„ã†ã“ã¨ã‚’主張ã—ã¦ã„ã¾ã™ã€‚ã“ã‚Œã¯ã‚ãã¾ã§ã€Œå…¥åŠ›ã®æµ®å‹•å°æ•°ç‚¹æ•°ã‹ã‚‰ã®èª¤å·®ãŒãªã„ã€ã¨ã„ã†ã ã‘ã§ã€å…¥åŠ›ã®æµ®å‹•å°æ•°ç‚¹æ•°ãŒèª¤å·®ã‚’å«ã‚€å ´åˆã¯ã€Œå‡ºåŠ›ã‚‚ãã®ã¾ã¾ãã®èª¤å·®ã‚’å«ã‚€ã€ã¨ã„ã†ã“ã¨ã«ãªã‚Šã¾ã™ã€‚ 逆ã«ã€å…¥åŠ›ã®èª¤å·®ãŒãªã„å ´åˆã¯èª¤å·®ãªã引ãç®—ã§ãã‚‹ã®ã§ã€ã€Œè¿‘ã„値ã®å¼•ãç®—ã¯æ€–ã„ã‹ã‚‰...ã€ãªã©ã¨å¤‰ãªæ€¯ãˆæ–¹ã‚’ã™ã‚‹å¿…è¦ã¯ã‚ã‚Šã¾ã›ã‚“。
ã“ã®ã‚ˆã†ãªã€å…¥åŠ›ã«èª¤å·®ãŒãªã„ã“ã¨ãŒã‚ã‹ã£ã¦ã„る状æ³ã§ã® cancellation ã¯ã€benign cancellation ã¨å‘¼ã°ã‚Œã‚‹ã“ã¨ãŒã—ã°ã—ã°ã‚ã‚Šã¾ã™ã€‚ 日本語ã§ã®å®šè¨³ãŒã‚ã‚‹ã‹ã¯ã‚ã‹ã‚Šã¾ã›ã‚“。
解æžã®ã‚„ã‚Šæ–¹ã¨ã‹ã‚‚å«ã‚ã¦è¿½ã£ã¦ã„ãã¾ã—ょã†ã€‚コンテストä¸ã§ã‚‚ã§ãるよã†ã«ã€æ‰‹è»½ãªæ„Ÿã˜ã®æ‰‹æ³•ã‚’心ãŒã‘ã¾ã™ã€‚
reference: AGC 047 A - Integer Product
整数部分㌠4 æ¡ä»¥ä¸‹ã€å°æ•°éƒ¨åˆ†ãŒ 9 æ¡ä»¥ä¸‹ã§ã‚るよã†ãªå®Ÿæ•° $A_i$ ãŒä¸Žãˆã‚‰ã‚Œã¾ã™ã€‚解法ã®è©³ç´°ã«ã¯è§¦ã‚Œã¾ã›ã‚“ãŒã€$10^9\cdot A_i$ を計算ã—ãŸããªã‚Šã¾ã™ã€‚ ã—ã‹ã—ã€$\floor{10^9\otimes \roundcirc{A_i}}$ ã¨ã—ã¦è¨ˆç®—ã™ã‚‹ã¨å¤±æ•—ã™ã‚‹ã‚ˆã†ãªä¾‹ã¯ã„ãらã§ã‚‚ã‚ã‚Šã¾ã™ã€‚ $$ \begin{aligned} \roundcirc{1024.000000006} &= {\small 1024.000000005999936}{\footnotesize 547596007585525}{\scriptsize 5126953125} \\ &= 4503599627396884\cdot 2^{-42}, \\ \roundcirc{1024.000000006} \otimes 10^9 &= {\small 1024000000005.9998779296875} \\ &= 8388608000049151 \cdot 2^{-13}. \end{aligned} $$
ã‚ã‚‹ $\delta$ ã‚’å°Žå…¥ã—㦠$\floor{(10^9\otimes \roundcirc{A_i})\oplus\delta}$ を考ãˆã‚‹æ–¹æ³•ã‚’考ãˆã¾ã—ょã†ã€‚
以下ã€$x = \roundcirc{A_i}$ ã¨ã—ã¾ã™ã€‚ $$ \begin{aligned} |x - A_i| &\le x\cdot 2^{-53}, \\ |10^9\otimes x - 10^9\cdot x| &\le 10^9\cdot x\cdot 2^{-53} \end{aligned} $$ ãŒæˆã‚Šç«‹ã¤ã®ã§ã€ä¸‰è§’ä¸ç‰å¼ã‚ˆã‚Š $$ \begin{aligned} |10^9\otimes x - 10^9\cdot A_i| &= |10^9\otimes x - 10^9\cdot x| + |10^9\cdot x - 10^9\cdot A_i| \\ &\le 10^9\cdot x\cdot 2^{-53} + 10^9\cdot x\cdot 2^{-53} \\ &\lt 2\cdot 10^9\cdot 10^5\cdot 2^{-53} \\ &\lt 0.022204\\ &\lt \roundcirc{0.02221} \end{aligned} $$ ãŒæˆã‚Šç«‹ã¡ã¾ã™ã€‚よã£ã¦ã€ $$ 10^9\cdot A_i - 0.022204 \lt 10^9\otimes x \lt 10^9\cdot A_i + 0.022204 $$ ã§ã‚り〠$$ \begin{aligned} 10^9\cdot A_i \lt 10^9\otimes x + \roundcirc{0.02221} &\lt 10^9\cdot A_i + 0.022204 + \roundcirc{0.02221} \\ &\lt 10^9\cdot A_i + 1 \end{aligned} $$ ã§ã™ã€‚ã¾ãŸã€$10^9\cdot A_i\lt 2^{52}$ ã‹ã‚‰ã€$(10^9\otimes x) \oplus \roundcirc{0.02221} \lt 10^9\cdot A_i + 1$ ãŒæˆã‚Šç«‹ã¡ã¾ã™ã€‚ よã£ã¦ã€$\floor{(10^9\otimes x) \oplus \roundcirc{0.02221}} = 10^9\cdot A_i$ ãŒå¾“ã„ã¾ã™ã€‚ã‚ã–ã‚ã–ç´°ã‹ã„æ¡ã¾ã§æŒ‡å®šã›ãšã€$\delta = \roundcirc{0.03}$ ã¨ã—ã¦å•é¡Œãªã„ã§ã—ょã†ã€‚
note: ãŸã¨ãˆã° $2^{52} + \roundcirc{0.6} \lt 2^{52}+1$ ã§ã™ãŒã€$2^{52}\oplus\roundcirc{0.6} = 2^{52}+1$ ãªã®ã§ã€ãã®ã‚ˆã†ãªéƒ¨åˆ†ãŒå•é¡Œãªã„ã‹ã‚‚æ°—ã‚’ã¤ã‘ã‚‹å¿…è¦ãŒã‚ã‚Šã¾ã™ã€‚
note: 差㌠$0.5$ 未満ã§æŠ‘ãˆã‚‰ã‚Œã¦ã„ã‚‹ã“ã¨ã‹ã‚‰ã€.round() を用ã„る解法もæ£å½“ã§ã‚ã‚‹ã“ã¨ãŒã‚ã‹ã‚Šã¾ã™ã。
æ出コード：æ出 #65357163
次ã®å•é¡Œã§ã™ã€‚
reference: ABC 169 C - Multiplication 3
ã“れ㯠double ã ã‘ã§è§£ã‘ã‚‹ã§ã—ょã†ã‹ï¼Ÿ
ãŸã¨ãˆã° $(A, B) = (999689151469700, 9.01)$ ãªã©ã®ã‚±ãƒ¼ã‚¹ã§ $\floor{AB}$ 自体を double ã§è¡¨ã™ã“ã¨ãŒã§ããªã„ã®ã§ã€ã€Œç”ãˆã‚’ double ã§è¿”ã™ã€ã¨ã„ã†å‰æã§ã¯åŽ³ã—ãã†ã§ã™ã。
$100B$ ã‚’ $901$ 以上 $999$ 以下ã®å¥‡æ•°ã¨ã—ã¦å›ºå®šã—ã€
$$
A = \left(2\Floor{\frac12\Ceil{\frac{2^{53}}{100B}}}+1\right)\cdot 100
$$
ã¨ã™ã‚‹ã“ã¨ã§ã€æ‰€æœ›ã® $(A, B)$ を構æˆã™ã‚‹ã“ã¨ãŒã§ãã¾ã™ã€‚
ã“ã®ã‚ˆã†ã«ã€ã€Œãã‚‚ãã‚‚ç”ãˆã‚’ double ã§è¡¨ã›ã‚‹ã®ã‹ï¼Ÿã€ã¨è€ƒãˆã‚‹ã®ã¯é‡è¦ã§ã™ã€‚$2^{53}+1$ 以上ã®æ•´æ•°ã¯ double ã§è¡¨ã›ã‚‹ã¨ã¯é™ã‚‰ãªã„ã®ã§ã€ã€Œç”ãˆãŒãã†ã„ã†å€¤ã«ãªã‚‹å…¥åŠ›ã‚’構æˆã§ãã‚‹ã‹ï¼Ÿã€ã¨ã„ã†è¦–点ã§è€ƒãˆã¦ã¿ã‚‹ã®ãŒã‚ˆã„ã§ã™ã。
å…ˆã®ä¾‹ã¨åŒæ§˜ã«ã—㦠xx.yy を補æ£ã—ã¤ã¤ 100 å€ã™ã‚‹ãªã©ã—㦠xxyy ã«ã—ã¦ã‹ã‚‰æ•´æ•°åž‹ã¨ã—ã¦æ‰±ã†ã®ãŒè³¢æ˜Žã§ã—ょã†*7。
仮数部㌠64-bit ã‚„ 113-bit ã®æµ®å‹•å°æ•°ç‚¹åž‹ã‚’使ãˆã‚‹è¨€èªžã§ã‚ã‚Œã°ã€ãれを使ãˆã°å¯èƒ½ãªæ°—ã¯ã—ã¾ã™ã€‚ã‚‚ã¡ã‚ã‚“ç´ æœ´ã« floor(a * b) ã¨ã™ã‚‹ã ã‘ã§ã¯ã ã‚ã§ã™ãŒã€å‰è¿°ã®ã‚ˆã†ã«è£œæ£é …を入れるãªã©ã§å¯¾å¿œã™ã‚‹ã“ã¨ã¯ã§ãã¾ã™ã。「誤差ã¨ã‹ã‚ˆãã‚ã‹ã‚“ãªã„ã‘ã© long double を使ã£ã¦å¼·å¼•ã«é€šã—ãŸã€ã§ã¯ãªãã¦ã€Œlong double ã ã¨ä»®æ•°éƒ¨ã®ãƒ“ット長ãŒè¶³ã‚Šã‚‹ã®ã§ long double を使ã„ã¾ã—ãŸã€ã¨è¨€ãˆã‚‹ã‚ˆã†ã«ãªã£ã¦ã»ã—ã„ã‚‚ã®ã§ã™*8。
ã¨ã„ã†ã“ã¨ã§ã€ã“ã“ã§ã¯ã€64-bit 仮数部ã®åž‹ã‚’使ã„ã¾ã—ょã†ã€‚ 求ã‚ãŸã„ã®ã¯ $\floor{AB}$ ã§ã€è¨ˆç®—å¯èƒ½ãªã®ã¯ $A\otimes\roundcirc B$ ã§ã™ã€‚
実数 $|\varepsilon_B|, |\varepsilon_{\otimes}|\le 2^{-64}$ ãŒå˜åœ¨ã—㦠$$ A\otimes\roundcirc B = \left(A\cdot (B\cdot (1+\varepsilon_B) )\right)\cdot (1+\varepsilon_{\otimes}) $$ ãŒæˆã‚Šç«‹ã¤ã®ã§ã€ $$ \begin{aligned} |A\otimes\roundcirc B - AB| &= AB\cdot ( (1+2^{-64})^2-1) \\ &\lt 10^{15}\cdot 9.99\cdot 1.085\times 10^{-19} \\ &\lt 1.084\times 10^{-3} \end{aligned} $$ ãŒæˆã‚Šç«‹ã¡ã¾ã™ã€‚
今回ã¯ã€$AB$ ãŒæ•´æ•°ã¨ã¯é™ã‚‰ãªã„ãŸã‚ã€å…ˆã®ã‚‚ã®ã‚ˆã‚Šã‚‚上界ã¯ãŽã‚ŠãŽã‚Šã§ã™ã€‚ $$A\otimes \roundcirc B - \floor{AB} \in (-0.001084\lldot 0.991084)$$ ã§ã‚り〠$$(A\otimes \roundcirc B)\oplus \roundcirc{0.002} - \floor{AB} \in (0.00092\lldot 0.99309)$$ ãªã®ã§ã€$\floor{(A\otimes \roundcirc B)\oplus \roundcirc{0.002}}$ を計算ã™ã‚Œã°ã‚ˆã„ã§ã™ã。
æ出コード：æ出 #65358049
大ã¾ã‹ãªè€ƒãˆæ–¹ã¯å‰é …ã¨åŒã˜ã§ã—ょã†ã‹ã‚‰ã€ã“ã“ã§ç‰¹ã«è©±ã™ã“ã¨ã¯ãªã„よã†ãªæ°—ãŒã—ã¦ã„ã¾ã™ã€‚æ°—ãŒå¤‰ã‚ã£ãŸã‚‰è¿½è¨˜ã—ã¾ã™ã€‚
二ã¤ã®å®Ÿæ•°ã‚’ç•°ãªã‚‹å¼ã§è¨ˆç®—ã—ã€ã€Œæœ¬æ¥ãれらã¯å·®ãŒ $0$ ãªã®ã§å•é¡Œãªã„ã¯ãšã ã€ã¨ã„ã†ã®ã‚’å‰æã¨ã—ã¦ã„るよã†ãªä¾‹ã§ã¯ã€æ¡è½ã¡ã§æ’ƒå¢œã§ãã‚‹ã“ã¨ãŒå¤šã„ã§ã™ã€‚
減算をã—ãªã„å½¢*9ã‚’æ„è˜ã—ã¦å¼å¤‰å½¢ã‚’è¡Œã†ã®ãŒé‡è¦ã§ã—ょã†ã€‚ 分数ã§ã‚ã‚Œã°ã€åˆ†å $n$ ã¨åˆ†æ¯ $d$ ã‚’æ•´æ•°ã§è¨ˆç®—ã—ã€$\roundcirc n\oslash \roundcirc d$ ã¨ã—ã¦è¨ˆç®—ã™ã‚‹ã®ãŒã‚ˆã•ãã†ã§ã™ã€‚
実数 $|\varepsilon_n|, |\varepsilon_d|, |\varepsilon_{\oslash}|\le 2^{-53}$ ãŒå˜åœ¨ã—㦠$$ \roundcirc n\oslash \roundcirc d = \frac{n\cdot(1+\varepsilon_n)}{d\cdot(1+\varepsilon_d)}\cdot(1+\varepsilon_{\oslash}) $$ ã¨æ›¸ã‘ã‚‹ã®ã§ã€ $$ \begin{aligned} |(\roundcirc n\oslash \roundcirc d) - (n\div d)| &= \frac nd\cdot \left| \frac{(1+\varepsilon_n)(1+\varepsilon_{\oslash})}{1+\varepsilon_d} -1 \right| \\ &\le \frac nd\cdot \left| \frac{(1+2^{-53})^2}{1-2^{-53}} -1 \right| \\ % &= \frac nd\cdot \left| 1 - (1+2^{-53})^2 \cdot \sum_{i=0}^{\infty} {(2^{-53})^i} \right| \\ % &= \frac nd\cdot \left| 1 - (1+2^{-53})^2 \cdot \left(1+2^{-53}+\sum_{i=2}^{\infty} {(2^{-53})^i}\right) \right| \\ % &= \frac nd\cdot \left| 1 - (1+2^{-53})^3 - (1+2^{-53})^2\cdot \sum_{i=2}^{\infty} {(2^{-53})^i} \right| \\ % &= \frac nd\cdot \left| 1 - (1+2^{-53})^3 - (1+2^{-53})^2\cdot \frac{(2^{-53})^2}{1-2^{-53}} \right| \\ % &= \frac nd\cdot \left| 1 - (1+2^{-53})^3 - (2^{-53})^2\cdot \frac{(1+2^{-53})^2}{1-2^{-53}} \right| \\ &= \frac nd\cdot \left( (1+2^{-53})^3 - 1 + (2^{-53})^2\cdot \frac{(1+2^{-53})^2}{1-2^{-53}} \right) \\ &\lt \frac nd\cdot (3+10^{-15})\cdot 2^{-53} \end{aligned} $$ ã¨ãªã‚Šã€$3\cdot 2^{-53}$ 程度ã®ç›¸å¯¾èª¤å·®ã§æŠ‘ãˆã‚‰ã‚Œã¾ã™ã€‚
分åや分æ¯ã«æ ¹å·ãŒå«ã¾ã‚Œã‚‹å ´åˆã¯ã€åˆ†åã¨åˆ†æ¯ã®é©åˆ‡ãªæ–¹ã®æœ‰ç†åŒ–ã‚’è¡Œã†ã“ã¨ã§æ¸›ç®—ã‚’é¿ã‘ã‚‹ã®ãŒã‚ˆã•ãã†ã§ã™ã€‚ãれ以外ã®ç„¡ç†æ•°ãŒã‚ã‚‹å ´åˆã¯ã€ãŸã¶ã‚“å›°ã‚Šã¾ã™ã€‚ 三角関数ã§ã‚ã‚Œã°å’Œç©å…¬å¼ãªã©ãŒä½¿ãˆã‚‹å ´åˆã‚‚ã‚ã‚‹ã‹ã‚‚。
具体的ã«æ’ƒå¢œã™ã‚‹ã‚±ãƒ¼ã‚¹ã®æ§‹ç¯‰ã‚„å¼å¤‰å½¢ã®æ–¹æ³•ã®ä¾‹ã«ã¤ã„ã¦ã¯ã€åˆ¥ã®è¨˜äº‹ã§éŽåŽ»ã«æ›¸ã„ãŸã®ã§ãã¡ã‚‰ã‚’å‚考ã«ã—ã¦ãã ã•ã„。
例ã¨ã—ã¦æœŸå¾…値㮠DP ã¨ã‹ã‚’考ãˆã¦ã¿ã¾ã—ょã†ã€‚
reference: ABC 402 E - Payment Required
ã¾ãšæœ€åˆã«ã€æ•´æ•° $p$ ã«å¯¾ã—ã¦ã® $(100 - p) \oslash 100$ 㨠$1 \ominus (p\oslash 100)$ ã®èª¤å·®ã®é•ã„ã«ã¤ã„ã¦è€ƒãˆã¦ã¿ãŸã‚Šã—ã¾ã™ã‹ï¼Ÿ å‰è€…ã¯èª¤å·®ã®å‡ºã‚‹æ“作㌠1 回ãªã®ã§ $2^{-53}$ 程度ã®ç›¸å¯¾èª¤å·®ãªã®ã«å¯¾ã—ã€å¾Œè€…㯠catastrophic cancellation ã®å½±éŸ¿ã§ã‚‚ã£ã¨å¤§ããªèª¤å·®ãŒå‡ºãã†ã§ã™ã。実際*10〠$$ \begin{aligned} % 1\oslash 100 &= \frac{1+3\cdot 2^{-57}}{100}, \\ 99\oslash 100 &= \frac{99-2^{-50}}{100} \end{aligned} $$ ãªã®ã§ã€Sterbenz lemma を使ã„ã¤ã¤ $$ \begin{aligned} 1\ominus (99\oslash 100) &= 1 - \frac{99-2^{-50}}{100} \\ &= \frac{1+2^{-50}}{100} \\ &= 0.01 \cdot (1+8\cdot 2^{-53}) \end{aligned} $$ ã¨ãªã‚Šã€æœ¬æ¥ã®ä¸Šç•Œã® $8$ å€ç¨‹åº¦ã®èª¤å·®ãŒç„¡é§„ã«å‡ºã¦ã„ã¾ã™ã€‚実際ã«ã“ã®èª¤å·®ãŒå•é¡Œã«ãªã‚‹ã“ã¨ã¯å°‘ãªã„ã§ã—ょã†ãŒã€ã‚ã–ã‚ã–誤差ã®å‡ºã‚‹æ–¹ã®å½¢ã‚’好む必è¦ã‚‚ãªã„ã§ã—ょã†ã。
ã•ã¦ã€DP ã§è¨ˆç®—ã•ã‚Œã‚‹å€¤ï¼ˆã“ã“ã§ã¯æœŸå¾…値）ã®èª¤å·®ã¯ã©ã†ãªã‚‹ã§ã—ょã†ï¼Ÿ 雑ã«è¦‹ç©ã‚‚ã£ã¦ã‚‚ã€é·ç§»ã¯ $2^8\cdot 5000$ 回ã§æŠ‘ãˆã‚‰ã‚Œã¾ã™ã€é·ç§»ã¯ $$ \mathrm{dp}[i] = (p_0\otimes \mathrm{dp}[i_0]) \oplus (p_1\otimes \mathrm{dp}[i_1]) $$ ã®ã‚ˆã†ãªå¼ã«ãªã‚Šã€$p_0$ ã‚„ $p_1$ ã«ãã‚Œãžã‚Œè¨ˆç®— $1$ 回分ã®ç›¸å¯¾èª¤å·®ãŒã‚ã‚‹ã®ã§ã€é·ç§»ã‚ãŸã‚Šè¨ˆç®— $5$ 回分ã®ç›¸å¯¾èª¤å·®ã§æŠ‘ãˆã‚‰ã‚Œã¾ã™*11。
ã¨ã„ã†ã“ã¨ã§ã€è¨ˆ $2^8\cdot 5000\cdot 5 = 6.4\times 10^6$ 回分ã«ãªã‚Šã€$2^{-53}\cdot 6.4\times 10^6 \approx 7.1\times 10^{-10}$ 程度ã®ç›¸å¯¾èª¤å·®ã§æŠ‘ãˆã‚‰ã‚Œã‚‹ã“ã¨ãŒã‚ã‹ã‚Šã¾ã™ã€‚許容誤差㯠$10^{-6}$ ãªã®ã§å•é¡Œãªã•ãã†ã§ã™ã。
よã $x_U - x_L \ge \roundcirc{10^{-9}}$ ãªã©ã‚’ループ継続æ¡ä»¶ã«ã—ã¦å¤±æ•—ã™ã‚‹äººãŒã„ã¾ã™ã。 ãŸã¨ãˆã° $(x_L, x_U) = (10^8, x_L+2^{-26})$ を考ãˆã¦ã¿ã¾ã™ã€‚ $x_L \lt x\lt x_U$ を満ãŸã™ã‚ˆã†ãª $x$ ã¯å˜åœ¨ã—ã¾ã›ã‚“ãŒã€$2^{-26}\gt 10^{-9}$ ãŒæˆã‚Šç«‹ã¡ã¾ã™ã€‚ ã“ã®ã¨ã $(x_L\oplus x_U)\oslash 2 = x_L$ ãªã®ã§ã€æ›´æ–°å¼ã®æ¡ä»¶çš„ã« $x_U \gets x_L$ ã§æ›´æ–°ã•ã‚Œãªã„å ´åˆã¯ç„¡é™ãƒ«ãƒ¼ãƒ—ã¨ãªã‚Šã¾ã™ã€‚
ãã“ã§ã€ã‚ˆã「浮動å°æ•°ç‚¹æ•°ã§äºŒåˆ†æŽ¢ç´¢ã‚’ã™ã‚‹ã¨ãã¯ã€å·®ã§åˆ¤å®šã™ã‚‹ã‚“ã˜ã‚ƒãªãã¦å›žæ•°ã§ãƒ«ãƒ¼ãƒ—ã™ã‚‹ã¨ã„ã„ã‚“ã よã€ã¨ã„ã†ã“ã¨ãŒè¨€ã‚ã‚Œã¦ã„ã¾ã™ã€‚ æžœãŸã—ã¦å›žæ•°ã§ã®ãƒ«ãƒ¼ãƒ—ã§å•é¡Œãªã„ã“ã¨ã¯æ˜Žã‚‰ã‹ã§ã—ょã†ã‹ï¼Ÿ ãã®å›žæ•°ã¯ã©ã®ç¨‹åº¦ï¼Ÿ 得られる精度ã¯ã©ã®ãらã„？ $[x_L\lldot x_U]$ ã®ç¯„囲ã§ãªã„値ãŒã†ã£ã‹ã‚Šè¿”ã£ã¦ã—ã¾ã†ã“ã¨ã¯ãªã„？ ãªã©ã€ã€Œãã‚Œã£ã½ã„ã‹ã‚‰å¤§ä¸ˆå¤«ãã†ã€ã§ã‚„ã£ã¦ã„る人ã¯ã‚„ã£ã±ã‚Šå¤šã„ã®ã§ã¯ãªã„ã§ã—ょã†ã‹ã€‚
$|x\oplus y| \lt \infty$ ãªã‚‰ã° $(x\oplus y)\oslash 2 = \roundcirc{\tfrac{x+y}2}$ ãŒæˆã‚Šç«‹ã¡ã¾ã™ã€‚ よã£ã¦ã€$x\le y$ ãªã‚‰ã° $x \le (x\oplus y)\oslash 2\le y$ ã¨ãªã‚Šã¾ã™ã。
ãŸã¨ãˆã° $(x_L\oplus x_U)\oslash 2\notin \{x_L, x_U\}$ をループ継続æ¡ä»¶ã«ã—ã¦ã‚‚よã•ãã†ãªæ°—ãŒã—ã¦ã„ã¾ã™ãŒã€ã©ã†ã§ã—ょã†ã€‚ ã©ã®ãらã„ã®å›žæ•°ã§åŽæŸã—ã¦ãれるã§ã—ょã†ã‹ã€‚
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ãŠã‚ã‚Šã§ã™ã€‚
*1:ã‚‚ã¡ã‚んライブラリã®å®Ÿè£…ã«ã¯ã‚ˆã‚‹ã®ã§ã™ãŒã€å®¹æ˜“ã«å®Ÿç¾ã§ãã‚‹ãŸã‚ã€ã¾ã¨ã‚‚ãªå®Ÿè£…ã§ã‚ã‚‹ã“ã¨ã‚’仮定ã™ã‚‹ã®ã¯å分ç¾å®Ÿçš„ã§ã—ょã†ã€‚
*2:$\floor{x+\delta}$ ã§ã¯ãªã $\floor{x\oplus\delta}$ を考ãˆã‚‹å¿…è¦ãŒã‚ã‚‹ã“ã¨ã«æ³¨æ„。
*3:$x=1$ ã¨ã™ã‚Œã°ã„ã„ã®ã«ã‚ã–ã‚ã–四則演算をã—ã¦ã„ã‚‹ã‹ã‚‰ãŠã‹ã—ããªã£ãŸ ↠ãã‚Œã¯ãã†ã€‚
*4:ã“ã†ã„ã†è³ªå•ã‚’ã™ã‚‹äººãŒæ˜”ã—ã°ã—ã°ã„ãŸã‚ˆã†ãªæ°—ã‚‚ã™ã‚‹ã—ã€æž¶ç©ºã®äººã ã£ãŸã‚ˆã†ãªæ°—ã‚‚ã—ã¾ã™ã€‚è€äººã®è¨˜æ†¶ã¯ã‚¢ãƒ†ã«ãªã‚Šã¾ã›ã‚“。
*5:打ã¡æ¶ˆã—ã®æ“作を主語ã¨ã™ã‚‹æ–‡ã§ã¯ã€ãŸã¨ãˆã° “In this situation, the cancellation is catastrophic.†ã¨ã‹è¨€ã‚ã‚ŒãŸã‚Šã—ã¾ã™ã€‚ä»–ã«ã‚‚ “the terms are cancelled catastrophically.†ã¨æ›¸ã‹ã‚Œã¦ã„ãŸã¨ã—ã¦ã‚‚æ¡è½ã¡ã®ã“ã¨ã‚’指ã™ã§ã—ょã†ã€‚catastrophic ã®ä»£ã‚ã‚Šã« severe ã¨è¨€ã‚れるã“ã¨ã‚‚ã‚ã‚Šã¾ã™ã€‚
*6:correct rounding ãªå®Ÿè£…ã§ãªã„ã¨ã“ã“㧠$1$ ã‚’è¿”ã™ã‹ã‚‚ã—ã‚Œã¾ã›ã‚“。correct rounding ã§ãªã„実装をå‰æã«ã—ã¦ã‚‚仕方ãŒãªã„ã§ã™ã—ã€correct rounding ãªå®Ÿè£…ã‚’å‰æã«ã—ã¦ã„ã¦ã‚‚æ¡è½ã¡ã§ã“ã‚ã‚Œã¾ã™ã‚ˆã€œã¨ã„ã†ã®ã‚’話ã—ãŸã„ã®ã§ã€å¤§å‹¢ã«ã¯å½±éŸ¿ãŒãªã„ã¨æ€ã„ã¾ã™ã€‚
*7:浮動å°æ•°ç‚¹æ•°ã‚’é¿ã‘ã‚‹ã®ãŒè³¢æ˜Žï¼Ÿ ã¾ããã†ã„ã†ç«‹å ´ã‚‚ã‚ã‚‹ã§ã—ょã†ã。
*8:ç”ãˆã‚’表ã™ã®ã«å¿…è¦ãªãƒ“ット長ãŒã‚ã‚Œã°å分ã¨ã„ã†æ„味ã§ã¯ãªãã¦ã€è¨ˆç®—途ä¸ã®ç²¾åº¦ãªã©ã‚‚å½“ç„¶åŠ å‘³ã—ãŸä¸Šã§ã®ä¸»å¼µã§ã™ã€‚
*9:ã‚‚ã¡ã‚ã‚“åŠ ç®—ã ã‘ã®å½¢ã§ã‚ã£ã¦ã‚‚異符å·ã«ãªã‚Šã†ã‚‹ãªã‚‰ç„¡æ„味ã§ã™ã€‚
*10:ã“ã†ã„ã†ã®ã£ã¦èªè€…ã¸ã®èª²é¡Œã¨ã—ã¦æ®‹ã—ãŸæ–¹ãŒã„ã„ã§ã™ã‹ï¼Ÿ
*11:$\oplus$ ã®éƒ¨åˆ†ã‚’ã¡ã‚ƒã‚“ã¨è€ƒãˆã‚‹ã¨ $3$ 回分ã§æŠ‘ãˆã‚‰ã‚Œã‚‹æ°—ãŒã—ã¾ã™ãŒã€ã“ã“ã§ã¯ä¸è¦ã ã¨æ€ã†ã®ã§å¤§é›‘把ã«ã„ãã¾ã™ã€‚
å€ç²¾åº¦æµ®å‹•å°æ•°ç‚¹åž‹ã§ã‚ã‚‹ã¨ã“ã‚ã® double ã§ã™ãŒã€ç¾ä»£ã«ãŠã„ã¦ã¯ã‚‚ã¯ã‚„ double ãŒã€Œãµã¤ã†ã€ã«ãªã‚Šã€å˜ç²¾åº¦ã¨å‘¼ã°ã‚Œã¦ã„ã‚‹ float ã¯å®Ÿè³ªçš„ã«ã€ŒåŠç²¾åº¦ã€ã®ã‚ˆã†ãªæ„Ÿè¦šã«ãªã£ã¦ã„ã‚‹æ°—ãŒã—ã¾ã™ã€‚
ãªã«ã‹ã—らを計算ã—ãŸã„ã¨ãã«ã€è¨ˆç®—çµæžœã¯ double ã ã¨ã—ã¦ã‚‚計算途ä¸ã¯ double よりも高ã„精度ã§è¡Œã†å¿…è¦ãŒã‚ã‚‹å ´åˆã¯ã—ã°ã—ã°ã‚ã‚Šã¾ã™*1。
ãã†ã—ãŸå ´åˆã« 64-bit ã‚„ 113-bit ã®ä»®æ•°éƒ¨ã‚’æŒã¤ long double ã‚„ __float128 ãªã©ã®åž‹ã‚’扱ãˆã‚‹ç’°å¢ƒãªã‚‰ãれを使ã†ã®ã‚‚手ã§ã™ãŒã€ãã†ã§ãªã„環境もã—ã°ã—ã°ã‚ã‚Šã¾ã™ã—ã€å°†æ¥ __float128 ãŒã€Œãµã¤ã†ã€ã«ãªã£ãŸã¨ãã«ã€Œ__float256 ãªã—ã«ã¯ãªã«ã‚‚打ã¤æ‰‹ãŒãªã„...ã€ã¨ãªã£ã¦ã—ã¾ã„ã¾ã™ã€‚
内部実装ã¨ã—ã¦å¤šå€é•·æ•´æ•°ã‚’使ã†ã‚ˆã†ãªä»»æ„精度ã®åž‹ã‚’使ã†æ–¹æ³•ã‚‚ã‚ã‚Šã¾ã™ãŒã€ã“れも標準ã§ç”¨æ„ã•ã‚Œã¦ã„ãªã„言語もã—ã°ã—ã°ã‚ã‚Šã¾ã™*2ã—ã€é€Ÿåº¦é¢ã‚‚æ°—ã«ãªã£ã¦ãã¾ã™ã€‚
ã¨ã‚‚ã‹ãã€ä»Šå›žã‹ã‚‰ã®æ•°å›žã¯ã€Œãµã¤ã†ã€ã®æµ®å‹•å°æ•°ç‚¹æ•°ã ã‘を使ã£ã¦ã€ã‚ˆã‚Šé«˜ã„精度ã§è¨ˆç®—ã™ã‚‹æ–¹æ³•ã«ã¤ã„ã¦è§¦ã‚Œã‚ˆã†ã¨æ€ã„ã¾ã™ã€‚
一旦ã€å††å‘¨çŽ‡ $\pi$ ã«ã¤ã„ã¦è€ƒãˆã¦ã¿ã¾ã—ょã†ã€‚
$$
\pi = {\small 3.141592653589793}{\footnotesize 238462643383279}{\scriptsize 502884197169399}{\tiny 375105820974944592307{\dots}}.
$$
ã“れを普通㫠double ã§è¡¨ãã†ã¨ã™ã‚‹ã¨ä»¥ä¸‹ã®ã‚ˆã†ã«ãªã‚Šã¾ã™ã€‚
$$
\begin{aligned}
\roundcirc{\pi} &= {\small 3.141592653589793}{\footnotesize 115997963468544}{\scriptsize 185161590576171}{\tiny 875} \\
&= 7074237752028440\times 2^{-51}.
\end{aligned}
$$
$\hat\pi_1 = \roundcirc{\pi}$ ã¨ã—ã¦ã€ä¸‹è¨˜ãŒæˆã‚Šç«‹ã¡ã¾ã™ã€‚
$$
\pi - \hat\pi_1 = {\small 1.224646799147353}{\footnotesize 177226065932275}{\scriptsize 001058209749445}{\tiny 923078164062862{\dots}}\times 10^{-16}
$$
よã£ã¦ã€$\hat\pi_2 = \roundcirc{\pi - \hat\pi_1}$ ã¨ã™ã‚‹ã¨ä¸‹è¨˜ã®ã‚ˆã†ã«ãªã‚Šã¾ã™ã€‚
$$
\begin{aligned}
\hat\pi_2 &= ({\small 1.224646799147353}{\footnotesize 207173764029458}{\scriptsize 396604625692124}{\tiny 677580063796256} \\
&\qquad\quad + {\tiny 12680683843791484832763671875}\times 10^{-60})\times 10^{-16} \\
&= 4967757600021511\times 2^{-105}.
\end{aligned}
$$
åŒæ§˜ã«ã—ã¦
$$
\begin{aligned}
\hat\pi_3 &= \roundcirc{\pi - (\hat\pi_1 + \hat\pi_2)} \\
&= -({\small 2.994769809718339}{\footnotesize 665887016354211}{\scriptsize 984016705500952}{\tiny 339037967798563} \\
& \qquad \quad + {\tiny 706161978508415962547495152434873233460166375152766704559326171875}\times 10^{-60}) \times 10^{-33} \\
&= 8753721960665020 \times 2^{-161}
\end{aligned}
$$
ã¨ã™ã‚‹ã¨ã€
$$
\pi - (\hat\pi_1 + \hat\pi_2 + \hat\pi_3) \approx 1.112\times 10^{-49}
$$
ã¨ãªã‚Šã€$\hat\pi_1$ ã¨æ¯”ã¹ã¦ 3 å€ç¨‹åº¦ã®æ¡æ•°ã§ä¸€è‡´ã—ã¦ã„ã‚‹ã“ã¨ãŒã‚ã‹ã‚Šã¾ã™ï¼ˆ3 å€ç¨‹åº¦ã®æ¡æ•°ã‚’使ã£ã¦ã„ã‚‹ã®ã§ãã‚Œã¯ãã†ï¼‰ã€‚
ãŸã ã—ã€ã‚‚ã¡ã‚ã‚“ $\hat\pi_1 \oplus \hat\pi_2 \oplus \hat\pi_3$ ã®ã‚ˆã†ã«è¨ˆç®—ã—ã¦ã—ã¾ã†ã¨ $\hat\pi_1$ ã¨åŒç¨‹åº¦ã®ç²¾åº¦ã«è½ã¡ã¦ã—ã¾ã†ãŸã‚ã€å†…部表ç¾ã¨ã—ã¦ã¯ $(\hat\pi_1, \hat\pi_2, \hat\pi_3)$ ã®ã‚ˆã†ã«åˆ¥å€‹ã§æŒã£ã¦ãŠãã‚‚ã®ã¨ã—ã¾ã™ã€‚ ã“ã® $(\hat\pi_1, \hat\pi_2)$ ã‚„ $(\hat\pi_1, \hat\pi_2, \hat\pi_3)$ ã®ã‚ˆã†ãªè¡¨ç¾ã‚’ã€ãã‚Œãžã‚Œ double-double, triple-double ã¨å‘¼ã‚“ã§ã„ã¾ã™ã€‚
上記ã®ä¾‹ã§ã¯ã€ã€Œä½•ã‚‰ã‹ã®æ‰‹æ®µã§åˆ¥é€”与ãˆãŸå®šæ•°ã‚’ 3 ã¤ã® double ã«åˆ†è§£ã—ã¦è¡¨ã™ã€ã¨ã„ã†ã“ã¨ã ã‘ã‚’ã—ã¾ã—ãŸãŒã€ã‚‚ã¡ã‚ã‚“ã“れを計算ã«ä½¿ã†æ–¹æ³•ã‚‚å¿…è¦ã«ãªã£ã¦ãã¾ã™ã€‚ ã™ãªã‚ã¡ã€ãŸã¨ãˆã° double 㨠double-double ã®å’Œã‚„ç©ã‚’ triple-double ã¨ã—ã¦å¾—る（内部的ã«ã¯ double ã®æ¼”ç®—ã®ã¿ã‚’用ã„る）方法ãªã©ã«ã¤ã„ã¦ã‚‚話ã—ã¦ã„ãã¾ã™ã€‚
今回ã¯ã€double 㨠double ã®å’Œã¨ç©ã‚’ double-double ã¨ã—ã¦è¡¨ã™æ–¹æ³•ã«ã¤ã„ã¦è¿°ã¹ã€ãれ以外ã«ã¤ã„ã¦ã¯æ¬¡å›žä»¥é™ã«ã—ãŸã„ã¨æ€ã„ã¾ã™*3。
ã“れを明らã‹ã¨æ„Ÿã˜ã‚‹ã‹ã¯ã‚ã‹ã‚‰ãªã„ã®ã§ã™ãŒã€æµ®å‹•å°æ•°ç‚¹æ•° $a$, $b$ ã«å¯¾ã—㦠$(a\oplus b)-(a+b)$ ã‚„ $(a\otimes b)-(a\times b)$ ã¯ï¼ˆunderflow ãªã©ãŒçµ¡ã‚€è°è«–ã¯åˆ¥é€”ã™ã‚‹ã¨ã—ã¦ã€åŸºæœ¬çš„ã«ï¼‰åŒã˜ãƒ•ã‚©ãƒ¼ãƒžãƒƒãƒˆã®æµ®å‹•å°æ•°ç‚¹æ•°ã¨ã—ã¦è¡¨ã™ã“ã¨ãŒã§ãã¾ã™ã€‚ ã™ãªã‚ã¡ã€double åŒå£«ã®å’Œãƒ»ç©ã®èª¤å·®ã‚’ double ã§æ£ç¢ºã«è¡¨ã›ã‚‹ã¨ã„ã†ã“ã¨ã§ã™ã€‚よã£ã¦ã€double åŒå£«ã®å’Œãƒ»ç©ã‚’誤差ãªã double-double ã¨ã—ã¦è¡¨ã›ã‚‹ã¨ã„ã†ã“ã¨ã«ãªã‚Šã¾ã™ã€‚
ã•ã¦ã€ã€Œå€¤ã¨ã—ã¦ã¯è¡¨ã›ã‚‹ã€ã¨ã„ã†äº‹å®Ÿã ã‘ã§ã¯å®Ÿéš›ã«è¨ˆç®—ã§ãã‚‹ã“ã¨ã«ãªã‚‰ãªã„ã®ã§ã€æ‰‹ç¶šãã«ã¤ã„ã¦èª¬æ˜Žã—ã¦ã„ãã¾ã™ã€‚ã‚‚ã¡ã‚ã‚“ã€è¡¨ã›ã‚‹ã¨ã„ã†äº‹å®Ÿã«ã¤ã„ã¦ã‚‚ã¡ã‚ƒã‚“ã¨è¨¼æ˜Žã—ã¾ã™ã€‚
ä»»æ„ã®å®Ÿæ•° $x \gt 0$ ã«å¯¾ã—ã¦ã€$2^k\le x\lt 2^{k+1}$ ãªã‚‹æ•´æ•° $k$ ãŒä¸€æ„ã«å˜åœ¨ã™ã‚‹ã®ã§ã€ãã® $k$ を用ã„㦠$\hfloor{x} = 2^k$ ã¨å®šç¾©ã—ã¾ã™ã€‚ ã¾ãŸã€ä¾¿å®œä¸Š $\hfloor0 = 0$ ã¨ã—ã¦ãŠãã¾ã™ã€‚
ä»»æ„ã®å®Ÿæ•° $x\ge 0$ ã¨æ•´æ•° $k$ ã«å¯¾ã—ã¦æ˜Žã‚‰ã‹ã« $\hfloor{x\cdot 2^k} = \hfloor x\cdot 2^k$ ãŒæˆã‚Šç«‹ã¡ã¾ã™ã€‚
Property 1: ä»»æ„ã®å®Ÿæ•° $a$, $b$ ã«å¯¾ã—㦠$\hfloor{|a|}\ge \hfloor{|b|}$ ãªã‚‰ã° $\hfloor{|a+b|}\le 2\hfloor{|a|}$ ãŒæˆã‚Šç«‹ã¤ã€‚
Proof
Case 1: $\hfloor{|a|} = \hfloor{|b|} = 0$
ã“ã®ã¨ãã€$a = b = 0$ ã§ã‚ã‚Šã€$\hfloor{|a+b|} = 2\hfloor{|a|} = 0$ ã§ã‚る。
Case 2: $\hfloor{|a|} = \hfloor{|b|} \gt 0$
æ•´æ•° $k$ ã¨å®Ÿæ•° $a', b' \in (-2\lldot -1]\sqcup[1\lldot 2)$ ãŒä¸€æ„ã«å˜åœ¨ã—㦠$$ \begin{aligned} 2^k &= \hfloor{|a|} = \hfloor{|b|}, \\ a &= a'\cdot 2^k, \\ b &= b'\cdot 2^k \end{aligned} $$ ãŒæˆã‚Šç«‹ã¤ã€‚ã“ã“㧠$|a'+b'|\lt 4$ ã§ã‚ã‚Šã€$\hfloor{|a'+b'|}\le 2$ ã¨ãªã‚‹ã€‚ $$ \begin{aligned} \hfloor{|a+b|} &= \hfloor{|(a'+b')\cdot 2^k|} \\ &= \hfloor{|a'+b'|}\cdot 2^k \\ &\le 2\cdot 2^k \\ &= 2\hfloor{|a|}. \end{aligned} $$
Case 3: $\hfloor{|a|} \gt \hfloor{|b|}$
ã“ã®ã¨ãã€$|a| \gt |b|$ ãŒæˆã‚Šç«‹ã¤ã€‚ $$ \begin{aligned} \hfloor{|a+b|} &\le \hfloor{|a| + |b|} \\ &\le \hfloor{|a| + |a|} \\ &= \hfloor{2|a|} \\ &= 2\hfloor{|a|}. \quad\qed \end{aligned} $$
Property 2: ä»»æ„ã®å®Ÿæ•° $a, b\ge 0$ ã«å¯¾ã—ã¦ã€$a\le b$ ãªã‚‰ã° $\hfloor a\le \hfloor b$ ãŒæˆã‚Šç«‹ã¤ã€‚
Proof
ã“ã®ã¨ãã€$a\le b$ より $\hfloor a\lt 2\hfloor b$ ã§ã‚る。ã™ãªã‚ã¡ã€$\hfloor{a}\le \hfloor b$ ãŒæˆã‚Šç«‹ã¤ã€‚$\qed$
æ£è² ã®ã‚¼ãƒã«ã¤ã„ã¦ã€0.0 ã‚’ $0_+$ã€-0.0 ã‚’ $0_-$ ã¨è¡¨è¨˜ã—ã¾ã™ã€‚以下ã§ã¯å˜ã« $0$ ã¨æ›¸ã„ãŸå ´åˆã¯ $0_+$ ã‚’æ„味ã™ã‚‹ã‚‚ã®ã¨ã—ã¾ã™ã€‚
特㫠$0_-$ ã«ã¤ã„ã¦è€ƒãˆãªãã¦ã„ã„ã‹ãªã¨æ€ã£ã¦ã„る主ãªç†ç”±ã¯ä¸‹è¨˜ã®é€šã‚Šã§ã™ã€‚
ã•ã¦ã€double ã¨ã—ã¦è€ƒãˆã¦ã„る値ãŸã¡ã‚’改ã‚ã¦å®šå¼åŒ–ã—ã¦ãŠãã¾ã™ã€‚
$-1022\le k\le 1023$ ã«å¯¾ã—ã¦ã€ä¸‹è¨˜ã‚’定義ã—ã¾ã™ã€‚ãã‚Œãžã‚Œç¬¦å·ã”ã¨ã®æ£è¦åŒ–æ•°ã®é›†åˆã§ã™ã€‚ $$ \begin{aligned} F_k^- &= \{-m\cdot 2^{k-52} \mid m\in [2^{52}\lldot 2^{53})\cap\Z\}, \\ F_k^+ &= \{+m\cdot 2^{k-52} \mid m\in [2^{52}\lldot 2^{53})\cap\Z\}. \end{aligned} $$ ã¾ãŸã€éžæ£è¦åŒ–æ•°ã®é›†åˆã‚’下記ã§å®šç¾©ã—ã¾ã™*4。 $$ \begin{aligned} F_{-1023}^- &= \{-m\cdot 2^{-1022-52} \mid m\in[1\lldot 2^{52})\cap\Z\}, \\ F_{-1023}^+ &= \{+m\cdot 2^{-1022-52} \mid m\in[1\lldot 2^{52})\cap\Z\}. \end{aligned} $$ ã¾ãŸã€è€ƒãˆãŸã„数全体ã‹ã‚‰ãªã‚‹é›†åˆ $F$ を下記ã§å®šç¾©ã—ã¾ã™ã€‚ $$ F = \left(\bigsqcup_{k=-1023}^{1023} {(F_k^-\sqcup F_k^+)}\right) \sqcup \{-\infty, 0, +\infty\}. $$ ãŸã¨ãˆã°ä¸‹è¨˜ãŒæˆã‚Šç«‹ã¡ã¾ã™ã€‚ $$ \begin{aligned} 1 &\in F_0^+, \\ -2^{-1074} &\in F_{-1023}^-, \\ 2^{1024} &\notin F, \\ 1+2^{-53} &\notin F, \\ 0.1 &\notin F. \end{aligned} $$ ã•ã‚‰ã«ã€ä¸‹è¨˜ã®æ€§è³ªã‚‚æˆã‚Šç«‹ã¡ã¾ã™ã€‚
Lemma 3: ä»»æ„ã® $k\in[-1022\lldot 1023]\cap\Z$ ãŠã‚ˆã³ $m\in[1\lldot 2^{53})\cap\N$ ã«å¯¾ã—ã¦ã€$$\{-m\cdot 2^{k-52}, +m\cdot 2^{k-52}\}\subset F$$ãŒæˆã‚Šç«‹ã¤ã€‚
Proof
ä»»æ„ã® $k\in[-1022\lldot 1023]$ ãŠã‚ˆã³ $m\in[1\lldot 2^{53})\cap\N$ ã«å¯¾ã—㦠$m\cdot 2^{k-52} \in F$ を示ã›ã°å分。下記㧠$P(k)$ を定義ã—ã€$k$ ã«é–¢ã™ã‚‹æ•°å¦çš„帰ç´æ³•ã§ç¤ºã™ã€‚ $$ P(k) \iff \Forall{m\in[1\lldot 2^{53})\cap\N}{m\cdot 2^{k-52}\in F}. $$
To-be-proved 1: $P(-1022)$
ä»»æ„ã® $m\in[1\lldot 2^{-53})\cap\N$ ã«å¯¾ã—ã€$m\cdot 2^{k-52} \in F_{-1023}^+\sqcup F_{-1022}^+$ ãŒæˆã‚Šç«‹ã¤ã€‚
To-be-proved 2: $k\lt 1023 \wedge P(k) \implies P(k+1)$
$m\in[2^{52}\lldot 2^{53})\cap\N$ ã®ã¨ã $m\cdot 2^{(k+1)-52}\in F_{k+1}^+$ ãŒæˆã‚Šç«‹ã¤ã€‚
$m\in[1\lldot 2^{52})\cap\N$ ã®ã¨ãã€$2m\in[2\lldot 2^{53})\cap\N$ ã‹ã¤ $m\cdot 2^{(k+1)-52} = (2m)\cdot 2^{k-52}$ ãŒæˆã‚Šç«‹ã¡ã€$P(k)$ より従ã†ã€‚$\qed$
tiesToEven ã«ã‚ˆã‚‹ä¸¸ã‚ã«ã¤ã„ã¦ã‚‚定義ã—ã¾ã—ょã†ã€‚$\circ\colon \R\to F$ を次ã®ã‚ˆã†ã«å®šç¾©ã—ã¾ã™ã€‚ ãŸã ã—ã€é–¾å€¤ã¨ã—㦠$\Theta_{\infty} = (2-2^{-53})\cdot 2^{1023}$ ãŠã‚ˆã³ $\Theta_0 = 2^{-53}\cdot 2^{-1022}$ ã¨ã—ã¦ãŠãã¾ã™ã€‚ $$ \roundcirc{x} = \begin{cases} +\infty, & \text{if}\: x \ge \Theta_{\infty}; \\ \roundcirc{x\cdot\hfloor{x}^{-1}\cdot 2^{52}}\cdot\hfloor{x}\cdot 2^{-52}, & \text{if}\: x\in[2^{53}\lldot\Theta_{\infty}); \\ \floor{x}, & \text{if}\: x\in[2^{52}\lldot 2^{53}) \wedge (x\bmod 1)\lt \tfrac12; \\ \floor{\frac{2x+1}4}\cdot 2, & \text{if}\: x\in[2^{52}\lldot 2^{53}) \wedge (x\bmod 1) = \tfrac12; \\ \ceil{x}, & \text{if}\: x\in[2^{52}\lldot 2^{53}) \wedge (x\bmod 1)\gt \tfrac12; \\ \roundcirc{x\cdot\hfloor{x}^{-1}\cdot 2^{52}}\cdot\hfloor{x}\cdot 2^{-52}, & \text{if}\: x\in[2^{-1022}\lldot 2^{52}); \\ \roundcirc{x+2^{-1022}}-2^{-1022}, & \text{if}\: x\in(\Theta_0\lldot 2^{-1022}); \\ 0, &\text{if}\: x\in[0\lldot \Theta_0]; \\ \roundcirc{-x}, & \text{if}\: x\lt 0. \end{cases} $$ 特ã«ã€$x\in F\iff \roundcirc x=x$ ã§ã™ã€‚ã¾ãŸ $x\bmod 1=\tfrac12$ ã®ã¨ã $\floor{\frac{2x+1}4}\cdot 2 \in \{\floor x, \ceil x\}$ ã§ã™ã€‚
ã¾ãŸã€$(x, y)\in \R\times F$ ã‹ã¤ $y\le x$ ãªã‚‰ã° $y\le \roundcirc x$（$\le$ ã‚’ $\ge$ ã§ç½®ãæ›ãˆãŸå‘½é¡Œã‚‚åŒæ§˜ï¼‰ã§ã™ã€‚
ä»»æ„ã®å®Ÿæ•° $x$ ã«å¯¾ã—ã¦ã€$|\roundcirc x|\lt \infty$ ãªã‚‰ã°ä¸‹è¨˜ãŒæˆã‚Šç«‹ã¡ã¾ã™ã€‚ $$ |\roundcirc x-x| \le \begin{cases} \hfloor{|x|}\cdot 2^{-53}, & \text{if}\: |x|\ge 2^{-1022}; \\ 2^{-1022-53}, & \text{if}\: |x|\lt 2^{-1022}. \end{cases} $$
ã•ã‚‰ã«ã€ä»»æ„ã®å®Ÿæ•° $x$ ã¨æ•´æ•° $k$ ã«å¯¾ã—㦠$x\equiv 0\pmod{2^k}$ ãªã‚‰ã° $\roundcirc x\equiv 0\pmod{2^k}$ ãŒæˆã‚Šç«‹ã¡ã¾ã™ã€‚
é »å‡ºãªã®ã§ã€æ”¹ã‚ã¦è§¦ã‚Œã¦ãŠãã¾ã™ã€‚
Lemma 4 (Sterbenz): $x, y\in F\smallsetminus{\{-\infty, +\infty\}}$ ã«ã¤ã„ã¦ã€$\frac x2\le y\le 2x$ ãªã‚‰ã° $x\ominus y=x-y$ ãŒæˆã‚Šç«‹ã¤ã€‚
Proof
$x=y$ ã®ã¨ãã¯æ˜Žã‚‰ã‹ï¼ˆç‰¹ã« $x=y=0$ ã‚‚å«ã‚€ï¼‰ã€‚一般性を失ã‚ãš $x\gt y\gt 0$ ã¨ã™ã‚‹*5。ã¾ãŸã€$x-y\le \frac x2$ ã«æ³¨æ„ã™ã‚‹ã€‚
Case 1: $y\in F_{-1023}^+$
$x\le 2y$ よりã€$x\in F_{-1023}^+\sqcup F_{-1022}^+$ ã§ã‚る。
Case 1-1: $x\in F_{-1023}^+$
ã‚ã‚‹æ•´æ•° $m_x$, $m_y$ ($0\lt m_y\lt m_x\lt 2^{52}$) ã«å¯¾ã—㦠$x = m_x\cdot 2^{-1074}$ ãŠã‚ˆã³ $y = m_y\cdot 2^{-1074}$ ãŒæˆã‚Šç«‹ã¤ã€‚ $$ x - y = (m_x-m_y)\cdot 2^{-1074} $$ ã§ã‚ã‚Šã€$m_x-m_y\in[1\lldot 2^{52})\cap\N$ ã§ã‚ã‚‹ã‹ã‚‰ã€$x-y\in F_{-1023}^+$ ã§ã‚る。
Case 1-2: $x\in F_{-1022}^+$
ã‚ã‚‹æ•´æ•° $2^{52}\le m_x\lt 2^{53}$, $1\le m_y\lt 2^{52}$ ã«å¯¾ã—㦠$x = m_x\cdot 2^{-1074}$ ãŠã‚ˆã³ $y = m_y\cdot 2^{-1074}$ ãŒæˆã‚Šç«‹ã¤ã€‚ $$ x - y = (m_x-m_y)\cdot 2^{-1074} $$ ã§ã‚る。ã“ã“ã§ã€$m_x-m_y\in[1\lldot 2^{53})\cap\N$ ã§ã‚ã‚‹ã‹ã‚‰ã€$x-y\in F_{-1023}^+\sqcup F_{-1022}^+$ ã§ã‚る。
Case 2: ã‚ã‚‹æ•´æ•° $-1022\le k\le 1023$ ã«å¯¾ã—㦠$y\in F_k^+$
Case 1 åŒæ§˜ã« $x\in F_k^+\sqcup F_{k+1}^+$ ã§ã‚る。
Case 2-1: $x\in F_k^+$
ã‚ã‚‹æ•´æ•° $2^{52}\le m_y\lt m_x\lt 2^{53}$ ã«å¯¾ã—㦠$x = m_x\cdot 2^{k-52}$ ãŠã‚ˆã³ $y = m_y\cdot 2^{k-52}$ ãŒæˆã‚Šç«‹ã¤ã€‚ $$ x - y = (m_x-m_y)\cdot 2^{k-52} $$ ã§ã‚ã‚Šã€ã‹ã¤ $m_x-m_y\in[1\lldot 2^{52})\cap\N$ ã§ã‚ã‚‹ã‹ã‚‰ã€Lemma 3 より $x-y\in F$ ãŒå¾“ã†ã€‚
Case 2-2: $x\in F_{k+1}^+$
ã‚ã‚‹æ•´æ•° $2^{52}\le m_x\lt 2^{53}$ ãŠã‚ˆã³ $2^{52}\le m_y\lt 2^{53}$ ã«å¯¾ã—㦠$x = m_x\cdot 2^{k+1-52}$ ãŠã‚ˆã³ $y = m_y\cdot 2^{k-52}$ ãŒæˆã‚Šç«‹ã¤ã€‚ $$ \begin{aligned} x-y &= m_x\cdot 2^{k+1-52} - m_y\cdot 2^{k-52} \\ &= (2m_x-m_y)\cdot 2^{k-52} \end{aligned} $$ ã¨æ›¸ã‘る。$x-y\le\tfrac x2$ よりã€$2m_x-m_y\le m_x\lt 2^{53}$ ãŒå¾“ã†ã€‚Case 2-1 åŒæ§˜ã«ã—ã¦ã€$x-y\in F$ ãŒå¾“ã†ã€‚$\qed$
Lemma 5: æ•´æ•° $m\in[1\lldot 2^{106})$ ãŠã‚ˆã³ $k\in[-1022\lldot 1023-53]$ ã«å¯¾ã—㦠$x = m\cdot 2^{k-52}$ ã¨è¡¨ã›ã‚‹ã¨ãã€ã‚ã‚‹ $(x_h, x_l)\in F^2$ ãŒå˜åœ¨ã—㦠$x=x_h+x_l$ ãŒæˆã‚Šç«‹ã¤ã€‚
Proof
$m_h = \floor{m/{2^{53}}}$ ãŠã‚ˆã³ $m_l = (m\bmod 2^{53})$ ã¨ã™ã‚‹ã¨ $m = m_h\cdot 2^{53} + m_l$ ãŒæˆã‚Šç«‹ã¤ã€‚ $$ \begin{aligned} x_h &= (m_h\cdot 2^{53}) \cdot 2^{k-52} \\ &= m_h\cdot 2^{(k+53)-52}, \\ x_l &= m_l\cdot 2^{k-52} \end{aligned} $$ ã¨ã™ã‚‹ã¨ã€$(x_h, x_l)\in F^2$ ã‹ã¤ $x = x_h + x_l$ ãŒæˆã‚Šç«‹ã¤ã€‚$\qed$
$x, y\in F\smallsetminus{\{-\infty, +\infty\}}$ ã«å¯¾ã—ã¦ã€$(x\oplus y)-(x+y)$ ã™ãªã‚ã¡ $x\oplus y$ ã®èª¤å·®ã«ã¤ã„ã¦è€ƒãˆã¾ã™ã€‚
Lemma 6: $x, y\in F$ ㌠$x\gt 0$ ã‹ã¤ $\hfloor x\ge \hfloor{|y|}$ ã‹ã¤ $x+y\lt \Theta_{\infty}$ を満ãŸã™ã¨ã™ã‚‹ã€‚ã“ã®ã¨ãã€$(x\oplus y)-x \in F$ ã‹ã¤ $(x\oplus y)-(x+y)\in F$ ãŒæˆã‚Šç«‹ã¤ã€‚
Proof
Case 1: $\hfloor{|y|} \le \hfloor x\cdot 2^{-55}$
ã“ã®ã¨ã $|y|\lt \hfloor x\cdot 2^{-54}$ ã§ã‚ã‚Šã€$x\oplus y=x$ ã§ã‚ã‚‹ã‹ã‚‰ã€$(x\oplus y)-x = 0$ ã‹ã¤ $(x\oplus y)-(x+y)=-y$ ã§ã‚る。 明らã‹ã« $0, -y\in F$ ã§ã‚る。
Case 2: $\hfloor{|y|} = \hfloor x\cdot 2^{-54}$
ã“ã®ã¨ã $\hfloor x\cdot 2^{-54}\le |y|\lt \hfloor x\cdot 2^{-53}$ ã§ã‚る。ã¾ãŸã€$x\in F$ より $\hfloor{x}\le 2^{1023}$ ã§ã‚ã‚‹ã‹ã‚‰ã€$\hfloor{|y|}\le 2^{1023-54}$ ã§ã‚る。ã¾ãŸã€$x\notin F_{-1023}^+$ ã§ã‚る。
Case 2-1: $x = \hfloor x$ ã‹ã¤ $-\hfloor x\cdot 2^{-54}\lt y\lt -\hfloor x\cdot 2^{-53}$
ã“ã®ã¨ã $x\oplus y = x-\hfloor x\cdot 2^{-53}$ ã§ã‚ã‚‹ã‹ã‚‰ã€ $$ \begin{aligned} (x\oplus y)-x &= -\hfloor x\cdot 2^{-53} \\ &= -2\hfloor{|y|}, \\ (x\oplus y)-(x+y) &= -y-2\hfloor{|y|} \\ &= |y| - 2\hfloor{|y|} \end{aligned} $$ ã§ã‚る。$\hfloor{|y|}\le 2^{969}$ より $2\hfloor{|y|}\in F$ ã§ã‚ã‚Šã€$|y|-2\hfloor{|y|}\in F$ ã‚‚ Sterbenz lemma ã‹ã‚‰å¾“ã†ã€‚
Case 2-2: $x = \hfloor x$ ã‹ã¤ $y = -\hfloor x\cdot 2^{-54}$
$x\oplus y = x$ ã§ã‚ã‚‹ã‹ã‚‰ã€Case 1 åŒæ§˜ã«ã—ã¦å¾“ã†ã€‚
Case 2-3: $x = \hfloor x$ ã‹ã¤ $y\gt 0$
$x\oplus y = x$ ã§ã‚ã‚‹ã‹ã‚‰ã€Case 1 åŒæ§˜ã«ã—ã¦å¾“ã†ã€‚
Case 2-4: $x \gt \hfloor x$
$x\oplus y = x$ ã§ã‚ã‚‹ã‹ã‚‰ã€Case 1 åŒæ§˜ã«ã—ã¦å¾“ã†ã€‚
Case 3: $\hfloor{|y|} = \hfloor x\cdot 2^{-53}$
ã“ã®ã¨ã $\hfloor x\cdot 2^{-53}\le |y|\lt \hfloor x\cdot 2^{-52}$ ã§ã‚る。
$x=\hfloor x$ ã®ã¨ãã€$y\lt 0$ ãªã‚‰ã° $x\oplus y\in\{x-\hfloor x\cdot 2^{-52}, x-\hfloor x\cdot 2^{-53}\}$ ã§ã‚ã‚Šã€$y\gt 0$ ãªã‚‰ã° $x\oplus y = x$ ã§ã‚る。
$x\gt \hfloor x$ ã®ã¨ãã€$y\lt 0$ ãªã‚‰ã° $x\oplus y\in\{x-\hfloor x\cdot 2^{-52}, x\}$ ã§ã‚ã‚Šã€$y\gt 0$ ãªã‚‰ã° $x\oplus y\in\{x, x+\hfloor x\cdot 2^{-52}\}$ ã§ã‚る。
Sterbenz lemma ã«æ°—ã‚’ã¤ã‘ã¤ã¤ã€ç·æ‹¬ã—ã¦ã€$y\lt 0$ ãªã‚‰ã° $$ \begin{aligned} (x\oplus y)-(x+y) &\in \{-y-\hfloor x\cdot 2^{-52}, -y-\hfloor x\cdot 2^{-53}, -y\} \\ &= \{-y-2\hfloor{|y|}, -y-\hfloor{|y|}, -y\} \\ &= \{|y|-2\hfloor{|y|}, |y|-\hfloor{|y|}, |y|\} \subset F \end{aligned} $$ ã§ã‚ã‚Šã€$y\gt 0$ ãªã‚‰ã° $$ \begin{aligned} (x\oplus y)-(x+y) &\in \{-y, -y+\hfloor x\cdot 2^{-52}\} \\ &= \{-y, -y+2\hfloor{|y|}\} \\ &= \{-y, -y+2\hfloor{y}\} \subset F \end{aligned} $$ ã§ã‚る。
Case 4: $\hfloor x\cdot 2^{-52} \le \hfloor{|y|} \le \hfloor x$
$|y|\gt 0$ ã«æ³¨æ„ã™ã‚‹ã€‚
Case 4-1: $y\lt 0$
Case 4-1-1: $y\in F_{-1023}^-$ ã‹ã¤ $x\in F_{-1023}^+$
ã‚ã‚‹æ•´æ•° $m_x, m_y\in[1\lldot 2^{52})$ ã«å¯¾ã—㦠$x = m_x\cdot 2^{-1022-52}$ ã‹ã¤ $y = -m_y\cdot 2^{-1022-52}$ ãŒæˆã‚Šç«‹ã¤ã€‚ $$ x + y = (m_x-m_y)\cdot 2^{-1022-52} $$ ã§ã‚ã‚Šã€$|m_x-m_y| \in [0\lldot 2^{53})\cap\N$ より $x+y = x\oplus y$ ãŒå¾“ã†ã€‚ よã£ã¦ã€$(x\oplus y)-x = y$ ã‹ã¤ $(x\oplus y)-(x+y) = 0$ ãŒæˆã‚Šç«‹ã¤ã€‚
Case 4-1-2: $y\in F_{-1023}^-$ ã‹ã¤ã€ã‚ã‚‹æ•´æ•° $k\ge -1022$ ã«å¯¾ã—㦠$x\in F_k^+$
$$ \begin{aligned} \hfloor x &\le \hfloor{|y|}\cdot 2^{52} \\ &\lt 2^{52}\cdot 2^{-1022-52}\cdot 2^{52} \\ &= 2^{-1022+52} \end{aligned} $$ ã§ã‚ã‚‹ã‹ã‚‰ã€$x\lt 2^{-1022+52}$ ã¨ãªã‚‹ã€‚ã¾ãŸã€$x+y\equiv 0\pmod{2^{-1022-52}}$ ãŒæˆã‚Šç«‹ã¤ã€‚
$x+y\lt 2^{-1022}$ ã®ã¨ã $x+y\in F_{-1023}^+$ より $x\oplus y=x+y$ ãŒå¾“ã†ãŸã‚ã€ä»¥é™ã¯ $x+y\ge 2^{-1022}$ ã¨ã™ã‚‹ã€‚ ã“ã®ã¨ã〠$$ \begin{aligned} |(x\oplus y)-(x+y)| &\le (x+y)\cdot 2^{-53} \\ &\lt x\cdot 2^{-53} \\ &\le 2^{-1022+52}\cdot 2^{-53} \\ &= 2^{-1022-1} \end{aligned} $$ より $(x\oplus y)-(x+y)\in F_{-1023}^+$ ãŒæˆã‚Šç«‹ã¤ã€‚ ã¾ãŸã€$y\in F_{-1023}^-$ より $(x\oplus y)-x = ( (x\oplus y)-(x+y) ) + y \in F$ ãŒå¾“ã†ã€‚
Case 4-1-3: ã‚ã‚‹æ•´æ•° $k\ge -1022$ ã«å¯¾ã—㦠$y\in F_k^-$
ã“ã®ã¨ãã€ã‚ã‚‹æ•´æ•° $k'\ge k$ ã«å¯¾ã—㦠$x\in F_{k'}^+$ ã§ã‚る。Case 4-1-2 åŒæ§˜ $x+y\ge 2^{-1022}$ ã®å ´åˆã«ã¤ã„ã¦è€ƒãˆã‚‹ã€‚
ã¾ãŸã€$\frac x2\le |y|\le 2x$ ã®ã¨ã㯠Sterbenz lemma より $x\oplus y=x+y$ ãŒå¾“ã†ã‹ã‚‰ã€$-y \lt \frac x2$ ã¨ã™ã‚‹ã€‚ ã“ã®ã¨ã $\tfrac x2\lt x+y$ ã§ã‚る。よã£ã¦ $\tfrac x2\le x\oplus y$ ã§ã‚ã‚‹ã‹ã‚‰ã€Sterbenz lemma より $(x\oplus y)-x\in F$ ã§ã‚る。
ã•ã¦ã€ $$ \begin{aligned} \hfloor x &\le \hfloor{|y|}\cdot 2^{52} \\ &= 2^{52}\cdot 2^{k-52}\cdot 2^{52} \\ &= 2^{k+52} \end{aligned} $$ より $x\lt 2^{k+53}$ ã§ã‚る。 $x+y\equiv 0 \pmod{2^{k-52}}$ ã‹ã‚‰ã€ã‚ã‚‹æ•´æ•° $0\le m\lt 2^{105}$ ãŒå˜åœ¨ã—㦠$x+y = m\cdot 2^{k-52}$ ãŒæˆã‚Šç«‹ã¤ã€‚
$x\oplus y\equiv 0\pmod{2^{k-52}}$ ã‹ã¤ $|(x\oplus y)-(x+y)| \le (x+y)\cdot 2^{-53}$ ã§ã‚ã‚‹ã‹ã‚‰ã€ã‚ã‚‹æ•´æ•° $0\le m'\lt 2^{52}$ ãŒå˜åœ¨ã—㦠$(x\oplus y)-(x+y) = m'\cdot 2^{k-52}$ ãŒæˆã‚Šç«‹ã¤ãŸã‚ã€$(x\oplus y)-(x+y)\in F$ ãŒå¾“ã†ã€‚
Case 4-2: $y\gt 0$
Case 4-2-1: $y\in F_{-1023}^+$ ã‹ã¤ $x\in F_{-1023}^+$
Case 4-1-1 åŒæ§˜ã«ã—ã¦å¾“ã†ã€‚
Case 4-2-2: ã‚ã‚‹æ•´æ•° $k\ge -1022$ ã«å¯¾ã—㦠$x\in F_k^+$
ã“ã®ã¨ãã€$x+y\ge 2^{-1022}$ ãŒæˆã‚Šç«‹ã¤ã€‚
$|(x\oplus y)-(x+y)|\le (x+y)\cdot 2^{-53}$ より Sterbenz lemma ãŒå¾“ã„ã€$(x\oplus y)-(x+y)\in F$ ã¨ãªã‚‹ã€‚
$x+y\le 2x$ ã®ã¨ã $x\oplus y\le 2x$ ã§ã‚ã‚‹ã‹ã‚‰ã€Sterbenz lemma より $(x\oplus y)-x\in F$ ãŒå¾“ã†ã€‚ 以é™ã¯ãã†ã§ãªã„å ´åˆã‚’考ãˆã‚‹ã€‚ã™ãªã‚ã¡ $y\gt x$ ã¨ã™ã‚‹ã€‚$\hfloor x=\hfloor y$ ã‹ã¤ $\hfloor{x+y} = 2\hfloor x$ ãŒæˆã‚Šç«‹ã¤ã€‚
よã£ã¦ã€ã‚ã‚‹ $m_x, m_y\in [2^{52}\lldot 2^{53})\cap\N$ ã«å¯¾ã—㦠$x = m_x\cdot 2^{k-52}$ ã‹ã¤ $y = m_y\cdot 2^{k-52}$ ãŒæˆã‚Šç«‹ã¤ã€‚ $$ x\oplus y \in \{(m_x+m_y-1)\cdot 2^{k-52}, (m_x+m_y)\cdot 2^{k-52}, (m_x+m_y+1)\cdot 2^{k-52}\} $$ ã§ã‚り〠$$ (x\oplus y)-x \in \{(m_y-1)\cdot 2^{k-52}, m_y\cdot 2^{k-52}, (m_y+1)\cdot 2^{k-52}\} \subset F_k^+ $$ ãŒæˆã‚Šç«‹ã¤ã€‚$\qed$
ã•ã¦ã€å®Ÿéš›ã«ã‚„ã£ã¦ã„ãã¾ã™ã€‚$a+b$ ã‚„ $a\times b$ ãªã©ã«å¯¾ã—ã¦ã€äºŒã¤ã®æµ®å‹•å°æ•°ç‚¹æ•° $h$, $l$ ã§ã‚ã£ã¦ï¼ˆèª¤å·®ãªã）$h+l$ ã¨è¡¨ã›ã‚‹ã‚‚ã®ã‚’求ã‚ã¾ã™ã€‚
今後ã®å¿œç”¨ã‚’見æ®ãˆã‚‹ã¨ $a\oplus b$ ã‚„ $a\otimes b$ ãŒéžæ£è¦åŒ–æ•°ã«ãªã‚‹å ´åˆã¯é™¤å¤–ã—ã¦ã‚‚ã„ã„æ°—ãŒã—ã¾ã™ãŒã€éžæ£è¦åŒ–æ•°ã¡ã‚ƒã‚“を仲間外れã«ã—ã¦å¿Œé¿æ„Ÿã‚’å¼·ã‚ã‚‹ã®ã‚‚å«Œãªã®ã§ã€ã§ãã‚‹é™ã‚Šã‚±ã‚¢ã—ã¦ã„ãã¾ã™ã€‚
最も基本的ãªæ“作ã§ã™ãŒã€Lemma 6 相当ã®å‰æãŒãªã„ã¨è¨¼æ˜Žã¯é¢å€’ã§ã™ã€‚
Claim 7: 上記ã®æ‰‹ç¶šãã®å®Ÿè¡Œå¾Œã€äº‹å¾Œæ¡ä»¶ãŒæˆã‚Šç«‹ã¤ã€‚
Proof
$s_h = a\oplus b$ ã¯æ˜Žã‚‰ã‹ãªã®ã§ã€$s_h+s_l = a+b$ を示ã™ã€‚
Case 1: $a\lt 0$
$a\gt 0$ ã§æˆã‚Šç«‹ã¤ãªã‚‰ã° $a\lt 0$ ã§ã‚‚æˆã‚Šç«‹ã¤ã“ã¨ã‚’示ã™ã€‚ $$ \begin{aligned} s_h^{\pm} &= (\pm a)\oplus (\pm b), \\ z^{\pm} &= s_h^{\pm}\ominus (\pm a), \\ s_l^{\pm} &= (\pm b)\ominus z^{\pm} \end{aligned} $$ ã¨ã™ã‚‹ï¼ˆè¤‡å·åŒé †ï¼‰ã€‚ã“ã®ã¨ã〠$$ \begin{aligned} s_h^- &= (-a)\oplus(-b) \\ &= -(a\oplus b) \\ &= -s_h^+, \\ z^- &= s_h^- \ominus (-a) \\ &= -(s_h^+ \ominus a) \\ &= -z^+, \\ s_l^- &= (-b)\ominus z^- \\ &= -(b\ominus z^+) \\ &= -s_l^+ \end{aligned} $$ ã§ã‚ã‚‹ã‹ã‚‰ã€$s_h^- + s_l^- = -(s_h^+ + s_l^+)$ ãŒæˆã‚Šç«‹ã¤ã€‚ よã£ã¦ã€$s_h^+ + s_l^+ = a+b$ ãªã‚‰ã° $s_h^- + s_l^- = -(a + b) = (-a) + (-b)$ ãŒæˆã‚Šç«‹ã¤ã€‚
Case 2: $a = 0$
$$ \begin{aligned} s_h &= 0\oplus b = b, \\ z &= b\ominus 0 = b, \\ s_l &= b\ominus b = 0 \end{aligned} $$ よりã€$s_h + s_l = b = a + b$ ãŒæˆã‚Šç«‹ã¤ã€‚
Case 3: $a\gt 0$
Lemma 6 よりã€$z = s_h\ominus a = s_h - a$ ãŒå¾“ã†ã€‚ã¾ãŸã€ $$ \begin{aligned} s_l &= b\ominus z \\ &= b - (s_h - a) \\ &= (a+b)-s_h \\ &= (a+b)-(a\oplus b) \end{aligned} $$ ã§ã‚る。$\qed$
$\hfloor{|a|}\ge \hfloor{|b|}$ ãŒäº‹å‰ã«ä¿è¨¼ã§ããªã„å ´åˆã¯ $|a|\ge |b|$ ã§åˆ†å²ã™ã‚Œã°ã‚ˆã„ã§ã™ã€‚ ã‚ã–ã‚ã– $\hfloor{\bullet}$ ã®éƒ¨åˆ†ã‚’実装ã—ãŸã‚Šã™ã‚‹å¿…è¦ã¯ã‚ã‚Šã¾ã›ã‚“。
$a$, $b$ ã®å¤§å°ã«ã‚ˆã‚‰ãªã„亜種ãªã©ã‚‚考案ã•ã‚Œã¦ã„ã¾ã™ãŒã€ã“ã“ã§ã¯å‰²æ„›ã—ã¾ã™ã€‚
次以é™ã§ä½¿ã†ã‚µãƒ–ルーãƒãƒ³ã§ã™ã€‚$\textsc{Round}(a, k)$ ã¨ã—ã¦ä½¿ã„ã¾ã™ã€‚ 「$a$ ã®ä»®æ•°éƒ¨ã‚’ $53-k$ bits ã«ä¸¸ã‚ãŸå€¤ã€ã¨ãã®éš›ã®èª¤å·®ã«åˆ†é›¢ã—ã¾ã™ã€‚
Claim 8: 上記ã®æ‰‹ç¶šãã®å®Ÿè¡Œå¾Œã€äº‹å¾Œæ¡ä»¶ãŒæˆã‚Šç«‹ã¤ã€‚
Proof
$a = 0$ ã®ã¨ãã¯æ˜Žã‚‰ã‹ã€‚$a\gt 0$ ã§æˆã‚Šç«‹ã¦ã° $a\lt 0$ ã§æˆã‚Šç«‹ã¤ã“ã¨ã‚‚明らã‹ãªã®ã§ã€ä»¥é™ $a\gt 0$ ã¨ã™ã‚‹ã€‚
Case 1: $a\ge 2^{-1022}$
$a\cdot (2^k+1) \ge 2^{-1022}$ ã«æ³¨æ„ã—ã€$\hfloor{(2^k+1)\cdot a} = 2^{52}$ ã¨ã™ã‚‹ã€‚ ãã†ã§ãªã„å ´åˆã¯ $a$ ã‚’ $\hfloor{(2^k+1)\cdot a}^{-1}\cdot 2^{52}$ å€ã™ã‚‹ã“ã¨ã§å¸°ç€ã§ãる。 特ã«ã€$|a\cdot(2^k+1)| \le 2^{1023-1}\cdot (2^1+1)\lt \Theta_{\infty}$ ã«æ³¨æ„ã™ã‚‹ã€‚
ã¾ãš $2^{51}+\tfrac12 \le 2^k\cdot a$ を示ã™ã€‚$2^{51}\le \tfrac12(2^k+1)\cdot a$ ã§ã‚ã‚‹ã‹ã‚‰ã€$\tfrac12(2^k+1)\cdot a\le 2^k\cdot a-\tfrac12$ を示ã›ã°å分。 $$ \begin{aligned} &\phantom{{}\iff{}} \tfrac12(2^k+1)\cdot a\le 2^k\cdot a-\tfrac12 \\ % &\iff \tfrac12\cdot 2^k\cdot a + \tfrac12\cdot a \le 2^k\cdot a-\tfrac12 \\ &\iff \tfrac12\cdot a \le \tfrac12\cdot 2^k\cdot a-\tfrac12 \\ &\iff a\le 2^k\cdot a-1 \\ &\iff 1\le (2^k-1)\cdot a \\ &\iff 1\le \tfrac{2^k-1}{2^k+1}\cdot (2^k+1)\cdot a \end{aligned} $$ $k\ge 1$ ã®ã¨ã $\tfrac{2^k-1}{2^k+1}\ge \tfrac13$ ã§ã‚ã‚Šã€$(2^k+1)\cdot a\ge 2^{52}$ より従ã†ã€‚
ã•ã¦ã€ $$ \begin{aligned} (2^k+1)\cdot a &= 2^k\cdot a + a \\ &= \floor{2^k\cdot a + a} + ( (2^k\cdot a + a)\bmod 1) \end{aligned} $$ ã§ã‚る。
$\hfloor{2^k\cdot a} \ge 2^{51}$ ã‹ã¤ $2^k\cdot a\in F$ ã§ã‚ã‚‹ã‹ã‚‰ã€$( (2^k\cdot a)\bmod 1)\in\{0, \tfrac12\}$ ã§ã‚る。
ã¾ãŸã€$\hfloor a\ge 2^{51-k}$ ã§ã‚ã‚‹ã‹ã‚‰ã€$a\equiv 0\pmod{2^{-k-1}}$ ãŒæˆã‚Šç«‹ã¤ã€‚特ã«ã€$a\equiv 0\pmod{2^{-53}}$ ã§ã‚る。
Case 1-1: $( (2^k\cdot a)\bmod 1) = 0$
$$ \begin{aligned} (2^k+1)\cdot a &= 2^k\cdot a + \floor{a} + (a\bmod 1). \end{aligned} $$
$r = (a\bmod 1)$ ã¨ã™ã‚‹ã¨ã€ä¸‹è¨˜ãŒæˆã‚Šç«‹ã¤ã€‚
$r\le \tfrac12$ ã®ã¨ã〠$$ \begin{aligned} a - ( (2^k+1)\otimes a) &= (\floor{a} + r) - (2^k\cdot a + \floor{a}) \\ &= -(2^k\cdot a - r) \end{aligned} $$ ã§ã‚る。ã¾ãŸã€$r\ge \tfrac12$ ã®ã¨ã〠$$ \begin{aligned} a - ( (2^k+1)\otimes a) &= (\floor{a} + r) - (2^k\cdot a + \floor{a} + 1) \\ &= -(2^k\cdot a - (r-1) ) \end{aligned} $$ ã§ã‚る。
Case 1-1-1: $2^k\cdot a\gt 2^{52}$
remark: $( (2^k\cdot a)\bmod 1) = 0$ より $2^k\cdot a-1\ge 2^{52}$ ãŒæˆã‚Šç«‹ã¤ã€‚
$r\le \tfrac12$ ã®ã¨ãã€ã‚ã‚‹ $r'\in\{0, 1\}$ ã«å¯¾ã—㦠$a\ominus ( (2^k+1)\otimes a) = -(2^k\cdot a-r')$ ã‹ã¤ $|r-r'|\le \tfrac12$ ãŒæˆã‚Šç«‹ã¤ã€‚ よã£ã¦ã€ $$ \begin{aligned} &\phantom{{}={}} (a\ominus ( (2^k+1)\otimes a) ) + ( (2^k+1)\otimes a) \\ &= -(2^k\cdot a-r') + (2^k\cdot a + \floor a) \\ &= \floor a + r' \end{aligned} $$ ã¨ãªã‚‹ã€‚$\hfloor a=2^{52-k}$ より $a\le 2^{52}-\frac12$ ã§ã‚ã‚‹ã‹ã‚‰ã€$|\floor a + r'|\le 2^{52}$ ã§ã‚ã‚Šã€$\floor a+r'\in F$ ã§ã‚る。 よã£ã¦ $$ (a\ominus ( (2^k+1)\otimes a) ) \oplus ( (2^k+1)\otimes a) = \floor a+r' $$ ã¨ãªã‚‹ã€‚$\floor a+r'\equiv 0\pmod 1$ ã‹ã¤ $1 = \hfloor{a}\cdot 2^{-(52-k)}$ ã§ã‚る。
よã£ã¦ $a_h \equiv 0 \pmod{\hfloor a\cdot 2^{-(52-k)}}$ ã§ã‚る。ã¾ãŸã€$a_l = a-(\floor a+r') = r-r'$ ã§ã‚ã‚‹ã‹ã‚‰ã€$|a_l|\le \hfloor a\cdot2^{-(53-k)}$ ã§ã‚る。
ã¾ãŸã€$r\ge \tfrac12$ ã®ã¨ãã€ã‚ã‚‹ $r'\in\{0, 1\}$ ã«å¯¾ã—㦠$a\ominus ( (2^k+1)\otimes a) = -(2^k\cdot a-r')$ ã‹ã¤ $|(r-1)-r'|\le \tfrac12$ ãŒæˆã‚Šç«‹ã¡ã€$r\le \tfrac12$ ã®å ´åˆã¨åŒæ§˜ã«ã—ã¦å¾“ã†ã€‚特ã«ã€ $$ \begin{aligned} a_l &= a - a_h \\ &= (\floor a+r) - (\floor a+r'+1) \\ &= (r-1)-r' \end{aligned} $$ ã§ã‚る。
Case 1-1-2: $2^k\cdot a\le 2^{52}$
ã‚ã‚‹ $r'\in\{0, \tfrac12, 1\}$ ã«å¯¾ã—㦠$a\ominus ( (2^k+1)\otimes a) = -(2^k\cdot a-r')$ ãŒæˆã‚Šç«‹ã¤ã€‚ $r\le \tfrac12$ ã®ã¨ã $|r-r'|\le \tfrac14$ã€$r\ge \tfrac12$ ã®ã¨ã $|(r-1)-r'|\le \tfrac14$ ãŒæˆã‚Šç«‹ã¤ã€‚
よã£ã¦ã€Case 1-1-1 åŒæ§˜ã«ã—㦠$$ (a\ominus ( (2^k+1)\otimes a) ) \oplus ( (2^k+1)\otimes a) = \floor a+r' $$ ã¨ãªã‚‹ã€‚$\floor a+r'\equiv 0\pmod{\tfrac12}$ ã‹ã¤ $\tfrac12 = \hfloor{a}\cdot 2^{-(52-k)}$ ã§ã‚る。 ã™ãªã‚ã¡ã€$a_h\equiv 0\pmod{\hfloor a\cdot 2^{-(52-k)}}$ ã‹ã¤ $|a_l|\le \hfloor a\cdot 2^{-(53-k)}$ ãŒæˆã‚Šç«‹ã¤ã€‚
Case 1-2: $( (2^k\cdot a)\bmod 1) = \tfrac12$
ã“ã®ã¨ãã€$\hfloor{2^k\cdot a} = 2^{51}$ ã§ã‚る。 $$ \begin{aligned} (2^k+1)\cdot a &= 2^k\cdot a - \tfrac12 + \floor{\tfrac12+a} + ( (\tfrac12+a)\bmod 1). \end{aligned} $$ $r = ( (\tfrac12+a)\bmod 1)$ ã¨ã™ã‚‹ã¨ã€ä¸‹è¨˜ãŒæˆã‚Šç«‹ã¤ã€‚
note: $\tfrac12+a = \floor{\tfrac12+a} + ( (\tfrac12+a)\bmod 1)$ ã§ã‚る。
$r\le \tfrac12$ ã®ã¨ã〠$$ \begin{aligned} &\phantom{{}={}} a - ( (2^k+1)\otimes a) \\ &= (-\tfrac12 + \floor{\tfrac12+a} + r) - (2^k\cdot a - \tfrac12 + \floor{\tfrac12+a}) \\ &= -(2^k\cdot a-r) \end{aligned} $$ ã§ã‚り〠$r\ge \tfrac12$ ã®ã¨ã〠$$ \begin{aligned} &\phantom{{}={}} a - ( (2^k+1)\otimes a) \\ &= (-\tfrac12 + \floor{\tfrac12+a} + r) - (2^k\cdot a + \tfrac12 + \floor{\tfrac12+a}) \\ &= -(2^k\cdot a-(r-1) ) \end{aligned} $$ ã§ã‚る。$\hfloor a=2^{51-k}$ よりã€$a\le 2^{51}-\tfrac14$ ã§ã‚ã‚‹ã“ã¨ã«æ³¨æ„ã—㦠Case 1-1-2 ã¨åŒæ§˜ã«ã—ã¦å¾“ã†ã€‚
Case 2: $a\lt 2^{-1022}$
Case 2-1: $a\times (2^k+1)\le 2^{-1022}$
ã“ã®ã¨ãã€$a_c = a\times c$ ã‹ã¤ $(a_h, a_l) = (a, 0)$ ãŒæˆã‚Šç«‹ã¤ã€‚ $|a_l| \le \hfloor a\cdot 2^{-(53-k)}$ ã¯æ˜Žã‚‰ã‹ã§ã‚ã‚‹ã‹ã‚‰ã€$a \equiv 0 \pmod{\hfloor a\cdot 2^{k-53}}$ を示ã™ã€‚
$\hfloor a=2^{i-1022-52}$ ã¨ã™ã‚‹ã¨ã€ã‚ã‚‹æ•´æ•° $2^i\le m_a\lt 2^{i+1}$ ãŒå˜åœ¨ã—㦠$a = m_a\cdot 2^{-1022-52}$ ãŒæˆã‚Šç«‹ã¡ã€$m_a\cdot 2^{-1022-52} \equiv 0 \pmod{2^{-1022-52}}$ ã¨ãªã‚‹ã€‚ $a\times 2^k \lt 2^{-1022}$ より $(i-1022-52)+k\lt -1022$ ãŒæˆã‚Šç«‹ã¤ãŸã‚〠$$ \begin{aligned} &\phantom{{}\implies{}} m_a\cdot 2^{-1022-52} \equiv 0 \pmod{2^{-1022-52}} \\ &\implies m_a\cdot 2^{-1022-52} \equiv 0 \pmod{2^{i-1022-52+k-1-52}} \\ &\iff m_a\cdot 2^{-1022-52} \equiv 0 \pmod{\hfloor a\cdot 2^{k-53}} \end{aligned} $$ ãŒå¾“ã†ã€‚
Case 2-2: $a\times (2^k+1)\gt 2^{-1022}$
$\hfloor{(2^k+1)\cdot a}^{-1}\cdot 2^{52}$ å€ã™ã‚‹ã“ã¨ã§ã€Case 1 ã«å¸°ç€ã§ãる。特ã«ã€$r'\in\{0, 1\}$ ã«å¯¾ã—㦠$$ \hfloor{(2^k+1)\cdot a}\cdot 2^{-52}\cdot(\floor{\hfloor{(2^k+1)\cdot a}^{-1}\cdot 2^{52}\cdot a}+r')\in F $$ ãªã©ãŒæˆã‚Šç«‹ã¤ã“ã¨ã«æ³¨æ„ã›ã‚ˆã€‚$\qed$
ã•ã¦ã€ã“ã¡ã‚‰ã‚‚基本的ãªæ“作ã§ã™ã€‚FMA ãŒä½¿ãˆã‚‹å‰æã§ã‚ã‚Œã°æ¬¡ã®ã‚ˆã†ã«ã§ãã¾ã™ã€‚
Claim 9: 上記ã®æ‰‹ç¶šãã®å®Ÿè¡Œå¾Œã€äº‹å¾Œæ¡ä»¶ãŒæˆã‚Šç«‹ã¤ã€‚
Proof
$p_h = a\otimes b$ ã¯æ˜Žã‚‰ã‹ã§ã‚ã‚Šã€$\roundcirc{a\times b + (-p_h)} = a\times b - p_h$ を証明ã™ã‚Œã°å分。 $ab = 0$ ã®å ´åˆã¯æ˜Žã‚‰ã‹ã€‚$ab\gt 0$ ã®å ´åˆã§æˆã‚Šç«‹ã¦ã° $ab\lt 0$ ã®å ´åˆã§æˆã‚Šç«‹ã¤ã®ã‚‚明らã‹ãªã®ã§ã€$a, b\gt 0$ ã¨ã™ã‚‹ã€‚
Case 1: $ab\ge 2^{-1022}$
$\hfloor a=\hfloor b=2^{52}$ ã¨ã™ã‚‹ã€‚ãã†ã§ãªã„å ´åˆã¯ã€ãã‚Œãžã‚Œ $\hfloor a^{-1}\cdot 2^{52}$, $\hfloor b^{-1}\cdot 2^{52}$ å€ã™ã‚‹ã“ã¨ã§å¸°ç€ã§ãる。
実数 $|\varepsilon|\le 2^{-53}$ ãŒå˜åœ¨ã—㦠$a\otimes b = ab+\hfloor{ab}\cdot\varepsilon$ ãŒæˆã‚Šç«‹ã¤ã€‚ $2^{104}\le ab\lt 2^{106}$ ã§ã‚ã‚‹ã‹ã‚‰ã€$\hfloor{ab}\le 2^{105}$ ã§ã‚る。 $$ \begin{aligned} |(a\otimes b) - (a\times b)| &\le \hfloor{ab}\cdot\varepsilon \\ &\le 2^{105}\cdot 2^{-53} \\ &= 2^{52} \end{aligned} $$ ã§ã‚ã‚Šã€Property 3 ã‹ã‚‰ $(a\otimes b)-(a\times b)\in F$ ãŒå¾“ã†ã€‚
Case 2: $ab\lt 2^{-1022}$
ã“ã®ã¨ãã€$a\otimes b = a\times b$ ãŒæˆã‚Šç«‹ã¡ã€$(p_h, p_l) = (ab, 0)$ ã¨ãªã‚‹ã€‚$\qed$
事å‰æ¡ä»¶ã® $a\times b\equiv 0\pmod{2^{-1074}}$ ãŒãªã„å ´åˆã®å例ã¯ã€ãŸã¨ãˆã° $(a, b) = (1.5, 2^{-1074})$ ã§ã™ã€‚ $$ \begin{aligned} a\times b &= 1.5\cdot 2^{-1074}, \\ a\otimes b &= 2\cdot 2^{-1074}, \\ (a\otimes b)-(a\times b) &= 0.5\cdot 2^{-1074}\notin F \end{aligned} $$ ã§ã™ã€‚事å‰æ¡ä»¶ã¨ã—㦠$|a\times b|\gt \Theta_0$ を課ã—ãŸã¨ã—ã¦ã‚‚ã†ã¾ãã„ã‹ãªã„ã§ã™ã。
特ã«ã€$|a\times b|\ge 2^{-1022}$ ã§ã‚‚ã ã‚ã§ã™ã。$(a, b) = ( (2-2^{-52})\cdot 2^{-971}, 2-2^{-52})$ ãªã©ã‚’考ãˆã‚‹ã¨ã€ $$ \begin{aligned} a\times b &= (2-2^{-51}+2^{-105})\cdot 2^{-970}, \\ a\otimes b &= (2-2^{-51})\cdot 2^{-970} \end{aligned} $$ ã§ã‚ã‚Šã€$a\times b \ge 2^{-1022}$ ã§ã™ãŒ $(a\otimes b) - (a\times b) = 2^{-1075}\notin F$ ã§ã™ã€‚
FMA ãŒä½¿ãˆãªã„å ´åˆã§ã‚‚å¯èƒ½ã§ã™ã€‚サブルーãƒãƒ³ Round を用ã„ã¦æ¬¡ã®ã‚ˆã†ã«ã§ãã¾ã™ã€‚
Claim 10: 上記ã®æ‰‹ç¶šãã®å®Ÿè¡Œå¾Œã€äº‹å¾Œæ¡ä»¶ãŒæˆã‚Šç«‹ã¤ã€‚
Proof
Claim 9 åŒæ§˜ $a, b\gt 0$ ã¨ã—ã¦ã€$p_h+p_l = a\times b$ を示ã™ã€‚
Case 1: $ab\ge 2^{-1022}$
Claim 9 åŒæ§˜ã€$\hfloor a=\hfloor b=2^{52}$ ã¨ã™ã‚‹ã€‚Round ã®äº‹å‰æ¡ä»¶ãŒæˆã‚Šç«‹ã£ã¦ã„ã‚‹ã“ã¨ã«æ³¨æ„ã™ã‚‹ã€‚
Round ã®äº‹å¾Œæ¡ä»¶ã‚ˆã‚Š $u_h \equiv v_h \equiv 0 \pmod{2^{27}}$ ã‹ã¤ $|u_l|, |v_l|\le 2^{26}$ ãŒæˆã‚Šç«‹ã¤ã€‚ $$ \begin{aligned} a\times b &= (u_h+u_l) \times (v_h+v_l) \\ &= \underbrace{u_h\times v_h}_{[2^{104}\lldot 2^{106})} + \underbrace{u_h\times v_l}_{[-2^{79}\lldot 2^{79}]} + \underbrace{u_l\times v_h}_{[-2^{79}\lldot 2^{79}]} + \underbrace{u_l\times v_l}_{[-2^{52}\lldot 2^{52}]} \end{aligned} $$ ã¨æ›¸ã‘る。ã‚る実数 $|\varepsilon|\le 2^{-53}$ ãŒå˜åœ¨ã—㦠$|(a\otimes b)-(a\times b)| \le \hfloor{ab}\cdot \varepsilon$ ãŒæˆã‚Šç«‹ã¤ã‹ã‚‰ã€ $$ \begin{aligned} a\otimes b &= \underbrace{u_h\times v_h\vphantom{\hfloor x}}_{[2^{104}\lldot 2^{106})} + \underbrace{u_h\times v_l\vphantom{\hfloor x}}_{[-2^{79}\lldot 2^{79}]} + \underbrace{u_l\times v_h\vphantom{\hfloor x}}_{[-2^{79}\lldot 2^{79}]} + \underbrace{u_l\times v_l\vphantom{\hfloor x}}_{[-2^{52}\lldot 2^{52}]} + \underbrace{\hfloor{ab}\cdot \varepsilon}_{[-2^{52}\lldot 2^{52}]} \end{aligned} $$ ãŒæˆã‚Šç«‹ã¤ã€‚
$m_u = u_h/2^{27}$ ãŠã‚ˆã³ $m_v = v_h/2^{27}$ ã¨ã™ã‚‹ã¨ $u_h\times v_h = (m_u\times m_v)\times 2^{54}$ ã¨ãªã‚‹ã€‚ $m_u, m_v\in[2^{25}\lldot 2^{26})\cap\N$ ã§ã‚ã‚‹ã‹ã‚‰ã€$u_h \otimes v_h = u_h\times v_h$ ãŒæˆã‚Šç«‹ã¤ã€‚
$u_h\times v_l = (m_u \times v_l)\times 2^{27}$ ã§ã‚ã‚Šã€$|m_u\times v_l|\le 2^{51}$ ã§ã‚ã‚‹ã‹ã‚‰ã€$u_h\otimes v_l = u_h\times v_l$ ãŒæˆã‚Šç«‹ã¤ã€‚ åŒæ§˜ã«ã—ã¦ã€$u_l\otimes v_h = u_l\times v_h$ ã§ã‚る。 ã¾ãŸã€$|u_l\times v_l| \le 2^{52}$ ã§ã‚ã‚‹ã‹ã‚‰ã€$u_l\otimes v_l = u_l\times v_l$ ã§ã‚る。
よã£ã¦ã€Sterbenz lemma より $$ \begin{aligned} &\phantom{{}={}} (u_h\otimes v_h) \ominus p_h \\ &= (u_hv_h) - (u_hv_h + u_hv_l + u_lv_h + u_lv_l + \hfloor{ab}\cdot\varepsilon) \\ &= -(u_hv_l + u_lv_h + u_lv_l + \hfloor{ab}\cdot\varepsilon) \end{aligned} $$ ãŒæˆã‚Šç«‹ã¤ã€‚
ã“ã“ã§ã€$a\otimes b = p_h \equiv 0 \pmod{2^{52}}$ ã‹ã¤ $u_hv_h\equiv 0\pmod{2^{54}}$ ã§ã‚ã‚‹ã‹ã‚‰ã€$u_hv_h-p_h\equiv 0\pmod{2^{52}}$ ãŒæˆã‚Šç«‹ã¤ã€‚ ã¾ãŸã€$u_hv_l\equiv 0\pmod{2^{27}}$ ã§ã‚ã‚‹ã‹ã‚‰ã€ $$ \begin{aligned} (u_hv_h-p_h)+u_hv_l &= -(u_lv_h + u_lv_l + \hfloor{ab}\cdot\varepsilon) \\ &\equiv 0 \pmod{2^{27}} \end{aligned} $$ ã‹ã¤ $$ \begin{aligned} |(u_hv_h-p_h)+u_hv_l| &\le |u_lv_h| + |u_lv_l| + \hfloor{ab}\cdot|\varepsilon| \\ &\le 2^{79} + 2^{52} + 2^{52} \\ &\lt 2^{80} \end{aligned} $$ ã§ã‚ã‚Šã€ã‚ã‚‹æ•´æ•° $0\le m\lt 2^{53}$ ã«å¯¾ã—㦠$|(u_hv_h-p_h)+u_hv_l| = m\cdot 2^{27}$ ã¨æ›¸ã‘る。 よã£ã¦ã€ $$ \begin{aligned} ( (u_h\otimes v_h)\ominus p_h)\oplus (u_h\otimes v_l) &= u_hv_h - p_h + u_hv_l \\ &= -(u_lv_h + u_lv_l + \hfloor{ab}\cdot \varepsilon) \end{aligned} $$ ãŒå¾“ã†ã€‚
note: $|u_hv_l|$ ãŒä¸Šç•Œã«è¿‘ã„ã¨ã¯é™ã‚‰ãªã„ã®ã§ã€Sterbenz lemma ã¯ä½¿ãˆãªã„ã“ã¨ã«æ³¨æ„ã›ã‚ˆã€‚
åŒæ§˜ã«ã—ã¦ã€$-(u_lv_h + u_lv_l + \hfloor{ab}\cdot\varepsilon)\equiv 0\pmod{2^{27}}$ ã‹ã¤ $u_lv_h\equiv 0\pmod{2^{27}}$ ã§ã‚ã‚‹ã‹ã‚‰ã€ $$ \begin{aligned} ( (u_hv_h-p_h)+u_hv_l)+u_lv_h &= -(u_lv_l + \hfloor{ab}\cdot\varepsilon \\ &\equiv 0 \pmod{2^{27}} \end{aligned} $$ ã‹ã¤ $$ \begin{aligned} |( (u_hv_h-p_h)+u_hv_l)+u_lv_h| &= |u_lv_l| + \hfloor{ab}\cdot|\varepsilon| \\ &\le 2^{52} + 2^{52} \\ &= 2^{53} \end{aligned} $$ ã§ã‚ã‚Šã€$u_lv_l+\hfloor{ab}\cdot\varepsilon\in F$ ãŒå¾“ã†ã€‚ ã™ãªã‚ã¡ $$ \begin{aligned} &\phantom{{}={}} ( ( (u_h\otimes v_h) \ominus p_h)\oplus (u_h\otimes v_l) )\oplus (u_l\otimes v_h) \\ &= -(u_lv_h + u_lv_l + \hfloor{ab}\cdot\varepsilon) \oplus u_lv_h \\ &= -(u_lv_l + \hfloor{ab}\cdot\varepsilon) \end{aligned} $$ ãŒå¾“ã†ã€‚
最後ã«ã€$\hfloor{ab}\cdot\varepsilon\le 2^{52}$ ã‹ã¤ $\hfloor{ab}\cdot\varepsilon\equiv 1$ より $\hfloor{ab}\cdot\varepsilon\in F$ ã§ã‚ã‚‹ã‹ã‚‰ã€ $$ \begin{aligned} &\phantom{{}={}} ( ( ( (u_h\otimes v_h) \ominus p_h)\oplus (u_h\otimes v_l) )\oplus (u_l\otimes v_h) )\oplus (u_l\otimes v_l) \\ &= -(u_lv_l + \hfloor{ab}\cdot\varepsilon) \oplus u_lv_l \\ &= -\hfloor{ab}\cdot\varepsilon \end{aligned} $$ ã¨ãªã‚‹ã€‚
ã—ãŸãŒã£ã¦ã€$p_h = a\times b + \hfloor{ab}\cdot \varepsilon$ ã‹ã¤ $p_l = -\hfloor{ab}\cdot\varepsilon$ ã§ã‚ã‚‹ã‹ã‚‰ã€$p_h+p_l = a\times b$ ãŒæˆã‚Šç«‹ã¤ã€‚
Case 2: $ab\lt 2^{-1022}$
$a\otimes b = a\times b$ ã«æ³¨æ„ã™ã‚‹ã€‚
$r\ge 0$ ã«å¯¾ã—㦠$(r_h, r_l) = \textsc{Round}(r, 27)$ ã¨ã™ã‚‹ã¨ $r_h\equiv 0\pmod{\hfloor r\cdot 2^{-25}}$ ã‹ã¤ $|r_l| \le \hfloor r\cdot 2^{-26}$ ãŒæˆã‚Šç«‹ã¤ã€‚ よã£ã¦ã€ $$ \begin{aligned} |r_l| &\le \hfloor r\cdot 2^{-26} \\ &\le 2^{-26}\cdot r, \\ r_h &= r - r_l \\ &\le r + r\cdot 2^{-26} \\ &= (1+2^{-26})\cdot r \end{aligned} $$ ãŒæˆã‚Šç«‹ã¤ã€‚よã£ã¦ã€ $$ \begin{aligned} u_hv_h &= ab - (u_hv_l + u_lv_h + u_lv_l) \\ &\le ab + |u_hv_l| + |u_lv_h| + |u_lv_l| \\ &\le ab + (1+2^{-26})\cdot a\cdot 2^{-26}\cdot b + 2^{-26}\cdot a\cdot (1+2^{-26})\cdot b + 2^{-52}\cdot ab \\ &= (1 + 2\cdot (1+2^{-26})\cdot 2^{-26} + 2^{-52})\cdot ab \\ &\lt 2ab \\ &\lt 2^{-1021} \end{aligned} $$ ã§ã‚ã‚Šã€$u_hv_h\in F_{-1023}^+\sqcup F_{-1022}^+$ ãŒå¾“ã†ã€‚
åŒæ§˜ã«ã—ã¦ã€
ã¯ã™ã¹ã¦ $F$ ã®å…ƒã§ã‚ã‚‹ã“ã¨ãŒç¤ºã›ã‚‹ã‹ã‚‰ã€$(p_h, p_l) = (ab, 0)$ ãŒå¾“ã†ã€‚$\qed$
$a\le 2^{1023-27}$ ãŒäº‹å‰ã«ä¿è¨¼ã§ããªã„å ´åˆã¯ã€$a\gt 2^{1023-27}$ ã‹ã©ã†ã‹åˆ¤å®šã—㦠true ãªã‚‰ $a\otimes 2^{-53}$ ãªã©ã‚’渡ã—ã€çµæžœã‚’ $2^{53}$ å€ã™ã‚Œã°ã‚ˆã„ã§ã™ï¼ˆ$b$ ã«ã¤ã„ã¦ã‚‚åŒæ§˜ï¼‰ã€‚
今回ã¯ã€double-double ã‚„ triple-double を実装ã™ã‚‹éš›ã®åŸºæœ¬æ“作ã®ã¿ã‚’å°Žå…¥ã—ã¾ã—ãŸã€‚
éžæ£è¦åŒ–æ•°ã«ã¤ã„ã¦ã‚‚触れã¾ã—ãŸãŒã€ã€Œã‚ˆãã‚ã‹ã‚‰ãªã„ã®ã§é¿ã‘ãŸã„ã‚‚ã®ã€ã¨ã„ã†å°è±¡ã¯æ‹ãˆã¦ã‚‚ã€Œå ´åˆåˆ†ã‘ãŒé¢å€’ã«ãªã‚‹ã‚‚ã®ã€ã¨ã„ã†å°è±¡ãŒå¼·ã¾ã£ã¦ã—ã¾ã£ã¦ã„ã‚‹ã‹ã‚‚ã—ã‚Œã¾ã›ã‚“？ ã‚ã‹ã‚‰ãªã„ã®ã§é¿ã‘ãŸã„ã¨ã„ã†ã®ã‚ˆã‚Šã¯ãƒžã‚·ãªæ°—ã¯ã—ã¾ã™ã。
次回ã„ããªã‚Šæ°—ãŒå¤‰ã‚ã£ã¦ double-double ã§ã¯ãªã„ãªã«ã‹ã®è©±ã‚’始ã‚ã‚‹ã‹ã‚‚ã—ã‚Œã¾ã›ã‚“？ ãã®ã¨ãã®æ°—分ã§ãƒ†ãƒ¼ãƒžãŒæ±ºã¾ã‚‹ã®ã§ä»•æ–¹ãªã„ã§ã™ã。
毎度ã®ã“ã¨ã§ã™ãŒã€Sterbenz lemma ãŒå¤§æ´»èºã™ãŽã¾ã™ã€‚「$\roundcirc x=x$ を示ã—ãŸããªã£ãŸã‚‰ã¾ãš Sterbenz lemma を使ãˆãªã„ã‹ç–‘ãˆã€ã¾ã§ã‚ã‚‹æ°—ãŒã—ã¾ã™ã€‚ ã‚‚ã¡ã‚ã‚“ã€mod ã®æ€§è³ªãªã©ã€ãれ以外ã®ã‚‚ã®ã‚’使ã†å±€é¢ã‚‚ã—ã°ã—ã°ã‚ã‚Šã¾ã™ãŒã€Sterbenz lemma ãŒãŠæ‰‹è»½ã©ã“ã‚ã®é »å‡ºãƒ‘ターンã¨ã„ã†æ„Ÿã˜ã§ã™ã。
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