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Ingy döt Net 2026-02-01 16:33:20 -08:00
parent 5150844a7d
commit 4bb20c9b71
7735 changed files with 38060 additions and 199180 deletions
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60
Task/Legendre-prime-counting-function/Chapel/legendre-prime-counting-function-1.chapel
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// compile with --fast for maximum speed...
use Time;
proc countPrimes(lmt: uint(64)): int(64) {
if lmt < 9 { // when there are no odd primes less than square root...
if lmt < 3 { if lmt < 2 { return 0; } else { return 1; } }
return (lmt - (lmt >> 1)): int(64);
}
// Chapel doesn't have closures, so emulate them with a class...
class LegendrePi {
var n: uint(64);
var dom: domain(1);
var oprms: [dom] uint(32);
proc init(n: uint(64)) {
// first, an array of odd primes to the square root of n is generated...
this.n = n;
const sqrtn = sqrt(n: real(64)): int(64);
const rtlmt = (sqrtn - 3) / 2; this.dom = {0 .. rtlmt};
this.oprms = 0;
for i in 0 .. rtlmt do this.oprms[i] = (i + i + 3): uint(32);
var i = 0;
for i in (0 ..) { // cull the array
var ci = (i + i) * (i + 3) + 3; if ci > rtlmt { break; }
const bp = i + i + 3;
while (ci <= rtlmt) { this.oprms[ci] = 0; ci += bp; }
}
var psz = 0;
for ti in 0 .. rtlmt { // compress the odd primes array...
const tv = this.oprms[ti];
if tv != 0 { this.oprms[psz] = tv; psz += 1; }
}
this.dom = { 0 ..< psz };
}
proc phi(x: uint(64), a: int): int(64) {
if a <= 0 { return (x - (x >> 1)): int(64); } // take care of prime of 2
const na = a - 1; const p = this.oprms[na]: uint(64);
if x <= p { return 1: int(64); }
return phi(x, na) - phi(x / p, na);
}
proc this(): int(64) {
return phi(n, this.oprms.size) + this.oprms.size: int(64);
}
}
return (new LegendrePi(lmt))();
}
proc main() {
var timer: Timer;
timer.start();
for i in 0 .. 9 {
writeln("π(10**", i, ") = ", countPrimesx(10: uint(64) ** i));
}
timer.stop();
writeln("This took ", timer.elapsed(TimeUnits.milliseconds), " milliseconds.");
}

78
Task/Legendre-prime-counting-function/Chapel/legendre-prime-counting-function-2.chapel
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// tiny Phi Look Up for `a` of small degree...
const tinyPhiPrimes = [ 2, 3, 5, 7, 11, 13 ]; // degree six
const cC = tinyPhiPrimes.size - 1;
proc product(a: [] int): int {
var acc = 1; for v in a { acc *= v; }; return acc >> 1; }
const tinyPhiOddCirc = product(tinyPhiPrimes);
proc tot(a: [] int): int {
var acc = 1; for v in a { acc *= v - 1; }; return acc; }
const tinyPhiOddTot = tot(tinyPhiPrimes);
proc makeTinyLUT(ps: [] int, sz: int): [] uint(32) {
var arr: [0 .. sz - 1] uint(32) = 1;
for p in ps {
if p <= 2 { continue; }
arr[p >> 1] = 0;
for c in ((p * p) >> 1) ..< sz by p { arr[c] = 0; }
}
var acc = 0: uint(32);
for i in 0 ..< sz { acc += arr[i]; arr[i] = acc; }
return arr;
}
const tinyPhiLUT = makeTinyLUT(tinyPhiPrimes, tinyPhiOddCirc);
inline proc tinyPhi(x: uint(64)): int(64) {
const ndx = (x - 1) >> 1; const numtot = ndx / tinyPhiOddCirc: uint(64);
return (numtot * tinyPhiOddTot +
tinyPhiLUT[(ndx - numtot * tinyPhiOddCirc): int]): int(64);
}
proc countPrimes(lmt: uint(64)): int(64) {
if lmt < 169 { // below 169 whose sqrt is 13 is where TinyPhi doesn't work...
if lmt < 3 { if lmt < 2 { return 0; } else { return 1; } }
// adjust for the missing "degree" base primes
if lmt <= 13 {
return ((lmt - 1): int(64) >> 1) + (if (lmt < 9) then 1 else 0); }
return 5 + tinyPhiLUT[(lmt - 1): int >> 1]: int(64);
}
// Chapel doesn't have closures, so emulate them with a class...
class LegendrePi {
var n: uint(64);
var dom: domain(1);
var oprms: [dom] uint(32);
proc init(n: uint(64)) {
// first, an array of odd primes to the square root of n is generated...
this.n = n;
const sqrtn = sqrt(n: real(64)): int(64);
const rtlmt = (sqrtn - 3) / 2; this.dom = {0 .. rtlmt};
this.oprms = 0;
for i in 0 .. rtlmt do this.oprms[i] = (i + i + 3): uint(32);
var i = 0;
for i in (0 ..) { // cull the array
var ci = (i + i) * (i + 3) + 3; if ci > rtlmt { break; }
const bp = i + i + 3;
while (ci <= rtlmt) { this.oprms[ci] = 0; ci += bp; }
}
var psz = 0;
for ti in 0 .. rtlmt { // compress the odd primes array...
const tv = this.oprms[ti];
if tv != 0 { this.oprms[psz] = tv; psz += 1; }
}
this.dom = { 0 ..< psz };
}
proc lvl(pilmt: int, m: uint(64)): int(64) {
var acc = 0: int(64);
for pi in cC ..< pilmt {
const p = this.oprms[pi]: uint(64); const nm = m * p;
if this.n <= nm * p { return acc + (pilmt - pi); }
if pi > cC { acc -= this.lvl(pi, nm); }
const q = this.n / nm; acc += tinyPhi(q);
}
return acc;
}
proc this(): int(64) {
return tinyPhi(this.n) - this.lvl(this.oprms.size, 1)
+ this.oprms.size: int(64);
}
}
return (new LegendrePi(lmt))();
}

66
Task/Legendre-prime-counting-function/Chapel/legendre-prime-counting-function-3.chapel
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const masks = for i in 0 .. 7 do (1 << i): uint(8); // faster bit twiddling
proc countPrimes(lmt: uint(64)): int(64) {
if lmt < 3 { if lmt < 2 { return 0; } else { return 1; } } // odds only!
inline proc half(x: int): int { return (x - 1) >> 1; } // convenience function
inline proc divide(nm: uint(64), d: uint(64)): int {
return (nm: real(64) / d: real(64)): int; } // floating point div faster
const sqrtn = sqrt(lmt: real(64)): uint(64);
const mxndx = (sqrtn - 1): int / 2;
const dom = {0 .. mxndx}; const csz = (mxndx + 8) / 8;
var smalls = for i in dom do i: uint(32);
var roughs = for i in dom do (i + i + 1): uint(32);
var larges = for i in dom do ((lmt / (i + i + 1)) - 1) >> 1;
var cullbuf: [0 ..< csz] uint(8);
// partial sieve loop, adjusting larges/smalls, compressing larges/roughs...
var nobps = 0; var rilmt = mxndx;
for bp in 3: uint(64) .. by 2 {
const i = (bp >> 1): int; const sqri = (i + i) * (i + 1);
if sqri > mxndx { break; } // up to quad root of counting range
if (cullbuf[i >> 3] & masks[i & 7]) != 0 { continue; } // loop not prime
cullbuf[i >> 3] |= masks[i & 7]; // cull bp itself as not a rough
for ci in sqri .. mxndx by bp { // do partial sieving pass for `bp`...
cullbuf[ci >> 3] |= masks[ci & 7]; } // cull all multiples of `bp`
// now adjust `larges` for latest partial sieve pass...
var ori = 0; // compress input rough index to output one
for iri in 0 .. rilmt {
const r = roughs[iri]: uint(64); const rci = (r >> 1): int;
if (cullbuf[rci >> 3] & masks[rci & 7]) != 0 {
continue; } // skip culled roughs in last partial sieving pass
const d = bp: uint(64) * r;
larges[ori] = larges[iri] -
(if d <= sqrtn then
larges[smalls[(d >> 1): int] - nobps]
else smalls[half(divide(lmt, d))]: uint(64)) + nobps;
roughs[ori] = r: uint(32); ori += 1;
}
var si = mxndx; // and adjust `smalls` for latest partial sieve pass...
for bpm in bp .. (sqrtn / bp - 1) | 1 by -2 {
const c = smalls[(bpm >> 1): int] - nobps: uint(32);
const e = ((bpm * bp) >> 1): int;
while si >= e { smalls[si] -= c; si -= 1; }
}
nobps += 1; rilmt = ori - 1;
}
var ans = larges[0]; // answer from larges, adjusting for over subtraction...
for i in 1 .. rilmt { ans -= larges[i]; } // combine!
ans += (rilmt + 1 + 2 * (nobps - 1)) * rilmt / 2; // adjust!
// add final adjustment for pairs of current roughs to cube root of range...
for ri in (1 ..) { // break when reaches cube root of counting range...
const p = roughs[ri]: uint(64); const q = lmt / p;
const ei = smalls[half(divide(q, p))]: int - nobps;
if ei <= ri { break; } // break here when no more pairs!
for ori in ri + 1 .. ei { // for all pairs never the same prime!
ans += smalls[half(divide(q, roughs[ori]))]: int(64); }
// adjust for over subtractions above...
ans -= (ei - ri): uint(64) * (nobps: uint(64) + ri: uint(64) - 1);
}
return ans: int(64) + 1; // add one for only even prime of two!
}