Made some speed improvements

Author: fredbr

math/prime_count.cpp

@@ -43,28 +43,33 @@ vector<int> sieve(int n) {
 
 // at least sqrt(n) for pi(n)
 // bigger values may be faster
-int const sqrt_limit = isqrt(1e12) + 1;
+// for values up to 1e12 the fastest is around 1e8
+// for values up to 1e11 the fastest is around 3e7
+int const sqrt_limit = 30000000;
 
 auto primes = sieve(sqrt_limit);
 
-using ii = pair<ll, int>;
+using ii = pair<ll, ll>;
 
-map<ii, ll> phi_cache;
+// at least pi(sqrt(sqrt(n)))
+unordered_map<ll, ll> phi_cache[200];
 
 ll phi(ll x, int a) {
-    if (phi_cache.count({x, a})) return phi_cache[{x, a}];
-
     if (a == 1) {
         return (x+1)/2;
     }
 
+    if (phi_cache[a].count(x)) {
+        return phi_cache[a][x];
+    }
+
     ll res = phi(x, a-1) - phi(x / primes[a-1], a-1);
-    phi_cache[{x, a}] = res;
+    phi_cache[a][x] = res;
 
     return res;
 }
 
-map<ll, ll> pi_cache;
+unordered_map<ll, ll> pi_cache;
 
 ll pi(ll x) {
     if (pi_cache.count(x)) return pi_cache[x];
@@ -81,7 +86,7 @@ ll pi(ll x) {
 
     for (ll i = a+1; i <= b; i++) {
         ll w = x / primes[i-1];
-        ll b_i = pi(sqrt(w));
+        ll b_i = pi(isqrt(w));
         res -= pi(w);
 
         if (i <= c) {
@@ -94,6 +99,7 @@ ll pi(ll x) {
     pi_cache[x] = res;
     return res;
 }
+
 int main() {
     ll n;
     cin >> n;