T mod_exp(T a, T b, T n){
T res = 1;
while(b){
if(b&1)
res = mod_mul(res, a, n);
a = mod_mul(a, a, n);
b >>= 1;
}
return res;
}
T mod_pow(T base, T exp, T m) {
assert(base >= 0 && exp >= 0);
base %= m;
T result = 1;
while (exp) {
if (exp & 1)
result = mod_mul(result, base, m);
base = mod_mul(base, base, m);
exp >>= 1;
}
return result;
}
/*!
* Gets \a n0 digits of \f$ \pi \f$ starting with digit \a n. The first digit after the decimal point is
* digit 0. Because the digits are returned in a double, the number of digits in the result cannot be higher
* than the precision of a double, regardless of n0.
*
* This method only returns accurate results when \f$ n \ge 4 \cdot n0 \f$.
*
* \param n Offset of digit to retrieve
* \param n0 Number of digits to retrieve; also the precision of the result
*
* \return A decimal number whose integer part is 0 and whose decimal part corresponds to the nth digit of
* pi. Another way to put it is, this is the decimal part of \f$ \pi \cdot 10^n \f$ , accurate to
* \a n0 decimal places.
*/
double get_pi_digits(long n, long n0)
{
long k;
long M, N, m, s;
double b, c, x, t;
int sign;
double log_n;
log_n = log(n);
M = 2 * (long) (3 * n / log_n / log_n / log_n);
N = (long) ((n + n0 + 1) * (log(10) / (log(2 * M_E * M)))) + 1;
N += N % 2;
b = 0;
for(k = 0; k < (M + 1) * N; k += 2) {
x = 0;
m = 2 * k + 1;
s = expt_mod(10, n, m);
s = mod_mul(4, s, m);
x += (double) s / (double) m;
m = 2 * k + 3;
s = expt_mod(10, n, m);
s = mod_mul(4, s, m);
x -= (double) s / (double) m;
b += x;
if(b <= -0.5) {
b += 1;
} else if(b >= 1) {
b -= 2;
}
}
c = 0;
sign = -1;
for(k = 0; k < N; k++) {
m = 2 * M * N + 2 * k + 1;
s = mod_sum_binom(k, N, m);
s = mod_mul(s, expt_mod(5, N, m), m);
s = mod_mul(s, expt_mod(10, n - N, m), m);
s = mod_mul(4, s, m);
b += sign * (double) s / (double) m;
b = b - floor(b);
sign = -sign;
}
return modf(b, &t);
}
/* Multiply two transition matrices; result may alias either/both inputs. */
static void mrg_multiply(const mrg_transition_matrix* m, const mrg_transition_matrix* n, mrg_transition_matrix* result) {
uint_least32_t rs = mod_mac(mod_mac(mod_mac(mod_mac(mod_mul(m->s, n->d), m->t, n->c), m->u, n->b), m->v, n->a), m->w, n->s);
uint_least32_t rt = mod_mac(mod_mac(mod_mac(mod_mac(mod_mul_y(mod_mul(m->s, n->s)), m->t, n->w), m->u, n->v), m->v, n->u), m->w, n->t);
uint_least32_t ru = mod_mac(mod_mac(mod_mac(mod_mul_y(mod_mac(mod_mul(m->s, n->a), m->t, n->s)), m->u, n->w), m->v, n->v), m->w, n->u);
uint_least32_t rv = mod_mac(mod_mac(mod_mul_y(mod_mac(mod_mac(mod_mul(m->s, n->b), m->t, n->a), m->u, n->s)), m->v, n->w), m->w, n->v);
uint_least32_t rw = mod_mac(mod_mul_y(mod_mac(mod_mac(mod_mac(mod_mul(m->s, n->c), m->t, n->b), m->u, n->a), m->v, n->s)), m->w, n->w);
result->s = rs;
result->t = rt;
result->u = ru;
result->v = rv;
result->w = rw;
mrg_update_cache(result);
}
T chinese_remainder_theorem(const std::vector<T>& a, const std::vector<T>& m) {
auto solve2 = [](T a0, T m0, T a1, T m1) {
T t = mod_inverse(m0 % m1, m1);
assert(t != m1); // Otherwise no solution exists.
t = mod_mul(mod_sub(a1, a0 % m1, m1), t, m1);
return std::make_pair(a0 + m0 * t, m0 * m1);
};
std::pair<T, T> reduced{a[0], m[0]};
for (size_t i = 1; i < a.size(); ++i) {
reduced = solve2(reduced.first, reduced.second, a[i], m[i]);
assert(reduced.first >= 0 && reduced.first < reduced.second);
}
return reduced.first;
}
static void mrg_apply_transition(const mrg_transition_matrix* mat, const mrg_state* st, mrg_state* r) {
#ifdef __MTA__
uint_fast64_t s = mat->s;
uint_fast64_t t = mat->t;
uint_fast64_t u = mat->u;
uint_fast64_t v = mat->v;
uint_fast64_t w = mat->w;
uint_fast64_t z1 = st->z1;
uint_fast64_t z2 = st->z2;
uint_fast64_t z3 = st->z3;
uint_fast64_t z4 = st->z4;
uint_fast64_t z5 = st->z5;
uint_fast64_t temp = s * z1 + t * z2 + u * z3 + v * z4;
r->z5 = mod_down(mod_down_fast(temp) + w * z5);
uint_fast64_t a = mod_down(107374182 * s + t);
uint_fast64_t sy = mod_down(104480 * s);
r->z4 = mod_down(mod_down_fast(a * z1 + u * z2 + v * z3) + w * z4 + sy * z5);
uint_fast64_t b = mod_down(107374182 * a + u);
uint_fast64_t ay = mod_down(104480 * a);
r->z3 = mod_down(mod_down_fast(b * z1 + v * z2 + w * z3) + sy * z4 + ay * z5);
uint_fast64_t c = mod_down(107374182 * b + v);
uint_fast64_t by = mod_down(104480 * b);
r->z2 = mod_down(mod_down_fast(c * z1 + w * z2 + sy * z3) + ay * z4 + by * z5);
uint_fast64_t d = mod_down(107374182 * c + w);
uint_fast64_t cy = mod_down(104480 * c);
r->z1 = mod_down(mod_down_fast(d * z1 + sy * z2 + ay * z3) + by * z4 + cy * z5);
/* A^n = [d s*y a*y b*y c*y] */
/* [c w s*y a*y b*y] */
/* [b v w s*y a*y] */
/* [a u v w s*y] */
/* [s t u v w ] */
#else
uint_fast32_t o1 = mod_mac_y(mod_mul(mat->d, st->z1), mod_mac4(0, mat->s, st->z2, mat->a, st->z3, mat->b, st->z4, mat->c, st->z5));
uint_fast32_t o2 = mod_mac_y(mod_mac2(0, mat->c, st->z1, mat->w, st->z2), mod_mac3(0, mat->s, st->z3, mat->a, st->z4, mat->b, st->z5));
uint_fast32_t o3 = mod_mac_y(mod_mac3(0, mat->b, st->z1, mat->v, st->z2, mat->w, st->z3), mod_mac2(0, mat->s, st->z4, mat->a, st->z5));
uint_fast32_t o4 = mod_mac_y(mod_mac4(0, mat->a, st->z1, mat->u, st->z2, mat->v, st->z3, mat->w, st->z4), mod_mul(mat->s, st->z5));
uint_fast32_t o5 = mod_mac2(mod_mac3(0, mat->s, st->z1, mat->t, st->z2, mat->u, st->z3), mat->v, st->z4, mat->w, st->z5);
r->z1 = o1;
r->z2 = o2;
r->z3 = o3;
r->z4 = o4;
r->z5 = o5;
#endif
}
LL func(LL x,LL n){ return(mod_mul(x,x,n)+1)%n; }
template<class value_type> value_type mod_mul(value_type x, value_type k, ll m) { if(k == 0) { return 0; } if(k % 2 == 0) { return mod_mul((x+x) % m, k/2, m); } else { return (x + mod_mul(x, k-1, m)) % m; } }
static long mod_sum_binom(long k, long n, long m)
{
long j;
long A, B, C, C_acc;
long a, b, a_star, b_star;
size_t num_factors_m;
struct factor *prime_factors_m;
struct factor *cur_fact;
struct factor *r = NULL;
struct factor *cur_r = NULL;
if(k > n / 2) {
return int_modulus(expt_mod(2, n, m) - mod_sum_binom(n - k - 1, n, m), m);
}
/* Step 1 */
prime_factors_m = calc_prime_factors(m, k, &num_factors_m);
/* Step 2 */
A = 1; B = 1; C = 1;
for(j = 0; j < num_factors_m; j++) {
if(!cur_r) {
r = malloc(sizeof(struct factor));
cur_r = r;
} else {
cur_r->next = malloc(sizeof(struct factor));
cur_r = cur_r->next;
}
cur_r->value = 1;
cur_r->next = NULL;
}
for(j = 1; j <= k; j++) {
/* Step 3a */
a = n - j + 1;
b = j;
/* Steps 3b and 3c */
cur_fact = prime_factors_m;
cur_r = r;
a_star = a;
b_star = b;
while(cur_fact) {
while(a_star % cur_fact->value == 0) {
a_star /= cur_fact->value;
cur_r->value *= cur_fact->value;
}
while(b_star % cur_fact->value == 0) {
b_star /= cur_fact->value;
cur_r->value /= cur_fact->value;
}
cur_fact = cur_fact->next;
cur_r = cur_r->next;
}
/* Step 3d */
A = mod_mul(A, a_star, m);
B = mod_mul(B, b_star, m);
C = mod_mul(C, b_star, m);
C_acc = A;
for(cur_r = r; cur_r; cur_r = cur_r ->next) {
C_acc = mod_mul(C_acc, cur_r->value, m);
}
C = int_modulus(C + C_acc, m);
}
free_factors(prime_factors_m);
free_factors(r);
/* Step 4 */
return mod_mul(C, mod_inv(B, m), m);
}
int main(){
int x = mod_mul(2, 5, 7);
int y = mod_exp(2, 5, 7);
printf("%d, %d\n", x, y);
return 0;
}