Calculating n! mod m when m is not prime - c

I have read a lot of good algos to calculate n! mod m but they were usually valid when m was prime . I wanted to know whether some good algo exists when m is not prime .I would be helpful if someone could write the basic function of the algo too.I have been using
long long factMOD(long long n,long long mod)
{
long long res = 1;
while (n > 0)
{
for (long long i=2, m=n%mod; i<=m; i++)
res = (res * i) % mod;
if ((n/=mod)%2 > 0)
res = mod - res;
}
return res;
}
but getting wrong answer when I try to print factMOD(4,3) even. source of this algo is :
http://comeoncodeon.wordpress.com/category/algorithm/

The basic algorithm is valid for any value of m:
product := 1
for i := 2 to n
product := (product * i) mod m
return product
and an easy optimization is that you can bail out early and return 0 whenever product becomes 0. You can also return 0 at the beginning if n > m, since that guarantees that n! is a multiple of m.

This is what I've come up with:
#include <stdio.h>
#include <stdlib.h>
unsigned long long nfactmod(unsigned long long n, unsigned long long m)
{
unsigned long long i, f;
for (i = 1, f = 1; i <= n; i++) {
f *= i;
if (f > m) {
f %= m;
}
}
return f;
}
int main(int argc, char *argv[])
{
unsigned long long n = strtoull(argv[1], NULL, 10);
unsigned long long m = strtoull(argv[2], NULL, 10);
printf("%llu\n", nfactmod(n, m));
return 0;
}
and this:
h2co3-macbook:~ h2co3$ ./mod 1000000 1001001779
744950559
h2co3-macbook:~ h2co3$
runs in a fraction of a second.

Related

How do I reverse the order of the digits of an integer using recursion in C programming?

Problem statement :
Given a 32-bit signed integer, reverse digits of an integer.
Note: Assume we are dealing with an environment that could only store
integers within the 32-bit signed integer range: [ −2^31, 2^31 − 1]. For
the purpose of this problem, assume that your function returns 0 when
the reversed integer overflows.
I'm trying to implement the recursive function reverseRec(), It's working for smaller values but it's a mess for the edge cases.
int reverseRec(int x)
{
if(abs(x)<=9)
{
return x;
}
else
{
return reverseRec(x/10) + ((x%10)*(pow(10, (floor(log10(abs(x)))))));
}
}
I've implemented non recursive function which is working just fine :
int reverse(int x)
{
long long val = 0;
do{
val = val*10 + (x%10);
x /= 10;
}while(x);
return (val < INT_MIN || val > INT_MAX) ? 0 : val;
}
Here I use variable val of long long type to check the result with MAX and MIN of signed int type but the description of the problem specifically mentioned that we need to deal within the range of 32-bit integer, although somehow it got accepted but I'm just curious If there is a way to implement a recursive function using only int datatype ?
One more thing even if I consider using long long I'm failing to implement it in the recursive function reverseRec().
If there is a way to implement a recursive function using only int datatype ?
(and) returns 0 when the reversed integer overflows
Yes.
For such +/- problems, I like to fold the int values to one side and negate as needed. The folding to one side (- or +) simplifies overflow detection as only a single side needs testing
I prefer folding to the negative side as there are more negatives, than positives. (With 32-bit int, really didn't make any difference for this problem.)
As code forms the reversed value, test if the following r * 10 + least_digit may overflow before doing it.
An int only recursive solution to reverse an int. Overflow returns 0.
#include <limits.h>
#include <stdio.h>
static int reverse_recurse(int i, int r) {
if (i) {
int least_digit = i % 10;
if (r <= INT_MIN / 10 && (r < INT_MIN / 10 || least_digit < INT_MIN % 10)) {
return 1; /// Overflow indication
}
r = reverse_recurse(i / 10, r * 10 + least_digit);
}
return r;
}
// Reverse an int, overflow returns 0
int reverse_int(int i) {
// Proceed with negative values, they have more range than + side
int r = reverse_recurse(i > 0 ? -i : i, 0);
if (r > 0) {
return 0;
}
if (i > 0) {
if (r < -INT_MAX) {
return 0;
}
r = -r;
}
return r;
}
Test
int main(void) {
int t[] = {0, 1, 42, 1234567890, 1234567892, INT_MAX, INT_MIN};
for (unsigned i = 0; i < sizeof t / sizeof t[0]; i++) {
printf("%11d %11d\n", t[i], reverse_int(t[i]));
if (t[i] != INT_MIN) {
printf("%11d %11d\n", -t[i], reverse_int(-t[i]));
}
}
}
Output
0 0
0 0
1 1
-1 -1
42 24
-42 -24
1234567890 987654321
-1234567890 -987654321
1234567892 0
-1234567892 0
2147483647 0
-2147483647 0
-2147483648 0
You could add a second parameter:
int reverseRec(int x, int reversed)
{
if(x == 0)
{
return reversed;
}
else
{
return reverseRec(x/10, reversed * 10 + x%10);
}
}
And call the function passing the 0 for the second parameter. If you want negative numbers you can check the sign before and pass the absolute value to this function.
In trying to learn C programming I programed this question and get some correct results and some incorrect. I don't see the reason for the difference.
#include <stdio.h>
#include <string.h>
#include <stdlib.h>
#include <math.h> // requires adding link to math -lm as in: gcc b.c -lm -o q11
int ReverseInt(int startValue, int decimalPlace)
{
if(decimalPlace == 0) // if done returns value
{
return startValue;
}
int temp = startValue % 10; // gets units digit
int newStart = (startValue -temp)/10; // computes new starting value after removing one digit
int newDecimal = decimalPlace -1;
int value = temp*pow(10,decimalPlace);
return value + ReverseInt(newStart,newDecimal); // calls itself recursively until done
}
int main()
{
int x, decimalP, startValue;
printf("Input number to be reversed \n Please note number must be less than 214748364 :");
scanf("%d", &x);
if (x > 214748364)
{
printf("Input number to be reversed \n Please note number must be less than 214748364 :");
scanf("%d", &x);
}
decimalP = round(log10(x)); // computes the number of powers of 10 - 0 being units etc.
startValue = ReverseInt(x, decimalP); // calls function with number to be reversed and powers of 10
printf("\n reverse of %d is %d \n", x, startValue);
}
Output is: reverse of 1234 is 4321 but then reverse of 4321 is 12340
It's late and nothing better does not come into my mind. No float calculations. Of course, integer has to be big enough to accommodate the result. Otherwise it is an UB.
int rev(int x, int partial, int *max)
{
int result;
if(x / partial < 10 && (int)(x / partial) > -10)
{
*max = partial;
return abs(x % 10) * partial;
}
result = rev(x, partial * 10, max) + abs(((x / (int)(*max / partial)) % 10) * partial);
return result;
}
int reverse(int x)
{
int max;
return rev(x, 1, &max) * ((x < 0) ? -1 : 1);
}
int main(void){
printf("%d", reverse(-456789));
}
https://godbolt.org/z/M1eezf
unsigned rev(unsigned x, unsigned partial, unsigned *max)
{
unsigned result;
if(x / partial < 10)
{
*max = partial;
return (x % 10) * partial;
}
result = rev(x, partial * 10, max) + (x / (*max / partial) % 10) * partial;
return result;
}
unsigned reverse(unsigned x)
{
unsigned max;
return rev(x, 1, &max);
}
int main(void){
printf("%u", reverse(123456));
}
when using long long to store the result all possible integers can be reversed
long long rev(int x, long long partial, long long *max)
{
long long result;
if(x / partial < 10 && (int)(x / partial) > -10)
{
*max = partial;
return abs(x % 10) * partial;
}
result = rev(x, partial * 10, max) + abs(((x / (int)(*max / partial)) % 10) * partial);
return result;
}
long long reverse(int x)
{
long long max;
return rev(x, 1, &max) * ((x < 0) ? -1 : 1);
}
int main(void){
printf("%d reversed %lld\n", INT_MIN, reverse(INT_MIN));
printf("%d reversed %lld\n", INT_MAX, reverse(INT_MAX));
}
https://godbolt.org/z/KMfbxz
I am assuming by reversing an integer you mean turning 129 to 921 or 120 to 21.
You need an initial method to initialize your recursive function.
Your recursive function must figure out how many decimal places your integer uses. This can be found by using log base 10 with the value and then converting the result to a integer.
log10 (103) approx. 2.04 => 2
Modulus the initial value by 10 to get the ones place and store it in a variable called temp
Subtract the ones place from the initial value and store that in a variable called newStart.
divide this value by 10
Subtract one from the decimal place and store in another variable called newDecimal.
Return the ones place times 10 to the power of the decimal place and add it to the function where the initial value is newStart and the decimalPlace is newDecimal.
#include <stdio.h>
#include <math.h>
int ReverseInt(int startValue, int decimalPlace);
int main()
{
int i = -54;
int positive = i < 0? i*-1 : i;
double d = log10(positive);
int output = ReverseInt(positive,(int)d);
int correctedOutput = i < 0? output*-1 : output;
printf("%d \n",correctedOutput);
return 0;
}
int ReverseInt(int startValue, int decimalPlace)
{
if(decimalPlace == 0)
{
return startValue;
}
int temp = startValue % 10;
int newStart = (startValue -temp)/10;
int newDecimal = decimalPlace -1;
int value = temp*pow(10,decimalPlace);
return value + ReverseInt(newStart,newDecimal);
}

Segmentation fault for numbers too big for Ackermann's function

Why is this failing? I have written Ackermann's function in C and used longs to make sure that no number will be too small. Yet, when I go above (including) 4 for m and n, it gives me a segmentation fault: 11. Does anyone know why?
#include <stdio.h>
int ackermann(long m, long n) {
if (m == 0)
return n + 1;
else if (m > 0 && n == 0)
return ackermann(m - 1, 1);
else if (m > 0 && n > 0)
return ackermann(m - 1, ackermann(m, n - 1));
}
int main() {
long result = ackermann(4, 4);
printf("%lu", result);
}
I have written Ackermann's function in C and used longs to make sure
that no number will be too small.
The size of an unsigned long long is 2^6 (64) bits. The size of the result for ackermann(4, 2) is greater than 2^16 (65536) bits. You can compute ackermann(4, 1) and ackermann(5, 0) but not much with larger values of m and n.
Code-wise, you use a signed long when an unsigned long might serve you better and you declare the ackermann() function itself to return a signed int which is inconsistent. (Your ackermann() function also has a fourth exit point that isn't properly defined, compiler-wise.) Here's a rework of your code using unsigned long long which still won't get you very far:
#include <stdio.h>
unsigned long long ackermann(unsigned long long m, unsigned long long n) {
if (m == 0) {
return n + 1;
}
if (m > 0 && n == 0) {
return ackermann(m - 1, 1);
}
return ackermann(m - 1, ackermann(m, n - 1));
}
int main() {
unsigned long long result = ackermann(5, 0);
printf("%llu\n", result);
return 0;
}

Efficient algorithm to calculate the sum of number of base2 digits (number of bits) over an interval of positive integers

Let's say I've been given two integers a, b where a is a positive integer and is smaller than b. I have to find an efficient algorithm that's going to give me the sum of number of base2 digits (number of bits) over the interval [a, b]. For example, in the interval [0, 4] the sum of digits is equal to 9 because 0 = 1 digit, 1 = 1 digit, 2 = 2 digits, 3 = 2 digits and 4 = 3 digits.
My program is capable of calculating this number by using a loop but I'm looking for something more efficient for large numbers. Here are the snippets of my code just to give you an idea:
int numberOfBits(int i) {
if(i == 0) {
return 1;
}
else {
return (int) log2(i) + 1;
}
}
The function above is for calculating the number of digits of one number in the interval.
The code below shows you how I use it in my main function.
for(i = a; i <= b; i++) {
l = l + numberOfBits(i);
}
printf("Digits: %d\n", l);
Ideally I should be able to get the number of digits by using the two values of my interval and using some special algorithm to do that.
Try this code, i think it gives you what you are needing to calculate the binaries:
int bit(int x)
{
if(!x) return 1;
else
{
int i;
for(i = 0; x; i++, x >>= 1);
return i;
}
}
The main thing to understand here is that the number of digits used to represent a number in binary increases by one with each power of two:
+--------------+---------------+
| number range | binary digits |
+==============+===============+
| 0 - 1 | 1 |
+--------------+---------------+
| 2 - 3 | 2 |
+--------------+---------------+
| 4 - 7 | 3 |
+--------------+---------------+
| 8 - 15 | 4 |
+--------------+---------------+
| 16 - 31 | 5 |
+--------------+---------------+
| 32 - 63 | 6 |
+--------------+---------------+
| ... | ... |
A trivial improvement over your brute force algorithm would then be to figure out how many times this number of digits has increased between the two numbers passed in (given by the base two logarithm) and add up the digits by multiplying the count of numbers that can be represented by the given number of digits (given by the power of two) with the number of digits.
A naive implementation of this algorithm is:
int digits_sum_seq(int a, int b)
{
int sum = 0;
int i = 0;
int log2b = b <= 0 ? 1 : floor(log2(b));
int log2a = a <= 0 ? 1 : floor(log2(a)) + 1;
sum += (pow(2, log2a) - a) * (log2a);
for (i = log2b; i > log2a; i--)
sum += pow(2, i - 1) * i;
sum += (b - pow(2, log2b) + 1) * (log2b + 1);
return sum;
}
It can then be improved by the more efficient versions of the log and pow functions seen in the other answers.
First, we can improve the speed of log2, but that only gives us a fixed factor speed-up and doesn't change the scaling.
Faster log2 adapted from: https://graphics.stanford.edu/~seander/bithacks.html#IntegerLogLookup
The lookup table method takes only about 7 operations to find the log
of a 32-bit value. If extended for 64-bit quantities, it would take
roughly 9 operations. Another operation can be trimmed off by using
four tables, with the possible additions incorporated into each. Using
int table elements may be faster, depending on your architecture.
Second, we must re-think the algorithm. If you know that numbers between N and M have the same number of digits, would you add them up one by one or would you rather do (M-N+1)*numDigits?
But if we have a range where multiple numbers appear what do we do? Let's just find the intervals of same digits, and add sums of those intervals. Implemented below. I think that my findEndLimit could be further optimized with a lookup table.
Code
#include <stdio.h>
#include <limits.h>
#include <time.h>
unsigned int fastLog2(unsigned int v)
{
static const char LogTable256[256] =
{
#define LT(n) n, n, n, n, n, n, n, n, n, n, n, n, n, n, n, n
-1, 0, 1, 1, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3,
LT(4), LT(5), LT(5), LT(6), LT(6), LT(6), LT(6),
LT(7), LT(7), LT(7), LT(7), LT(7), LT(7), LT(7), LT(7)
};
register unsigned int t, tt; // temporaries
if (tt = v >> 16)
{
return (t = tt >> 8) ? 24 + LogTable256[t] : 16 + LogTable256[tt];
}
else
{
return (t = v >> 8) ? 8 + LogTable256[t] : LogTable256[v];
}
}
unsigned int numberOfBits(unsigned int i)
{
if (i == 0) {
return 1;
}
else {
return fastLog2(i) + 1;
}
}
unsigned int findEndLimit(unsigned int sx, unsigned int ex)
{
unsigned int sy = numberOfBits(sx);
unsigned int ey = numberOfBits(ex);
unsigned int mx;
unsigned int my;
if (sy == ey) // this also means sx == ex
return ex;
// assumes sy < ey
mx = (ex - sx) / 2 + sx; // will eq. sx for sx + 1 == ex
my = numberOfBits(mx);
while (ex - sx != 1) {
mx = (ex - sx) / 2 + sx; // will eq. sx for sx + 1 == ex
my = numberOfBits(mx);
if (my == ey) {
ex = mx;
ey = numberOfBits(ex);
}
else {
sx = mx;
sy = numberOfBits(sx);
}
}
return sx+1;
}
int main(void)
{
unsigned int a, b, m;
unsigned long l;
clock_t start, end;
l = 0;
a = 0;
b = UINT_MAX;
start = clock();
unsigned int i;
for (i = a; i < b; ++i) {
l += numberOfBits(i);
}
if (i == b) {
l += numberOfBits(i);
}
end = clock();
printf("Naive\n");
printf("Digits: %ld; Time: %fs\n",l, ((double)(end-start))/CLOCKS_PER_SEC);
l=0;
start = clock();
do {
m = findEndLimit(a, b);
l += (b-m + 1) * (unsigned long)numberOfBits(b);
b = m-1;
} while (b > a);
l += (b-a+1) * (unsigned long)numberOfBits(b);
end = clock();
printf("Binary search\n");
printf("Digits: %ld; Time: %fs\n",l, ((double)(end-start))/CLOCKS_PER_SEC);
}
Output
From 0 to UINT_MAX
$ ./main
Naive
Digits: 133143986178; Time: 25.722492s
Binary search
Digits: 133143986178; Time: 0.000025s
My findEndLimit can take long time in some edge cases:
From UINT_MAX/16+1 to UINT_MAX/8
$ ./main
Naive
Digits: 7784628224; Time: 1.651067s
Binary search
Digits: 7784628224; Time: 4.921520s
Conceptually, you would need to split the task to two subproblems -
1) find the sum of digits from 0..M, and from 0..N, then subtract.
2) find the floor(log2(x)), because eg for the number 77 the numbers 64,65,...77 all have 6 digits, the next 32 have 5 digits, the next 16 have 4 digits and so on, which makes a geometric progression.
Thus:
int digits(int a) {
if (a == 0) return 1; // should digits(0) be 0 or 1 ?
int b=(int)floor(log2(a)); // use any all-integer calculation hack
int sum = 1 + (b+1) * (a- (1<<b) +1); // added 1, due to digits(0)==1
while (--b)
sum += (b + 1) << b; // shortcut for (b + 1) * (1 << b);
return sum;
}
int digits_range(int a, int b) {
if (a <= 0 || b <= 0) return -1; // formulas work for strictly positive numbers
return digits(b)-digits(a-1);
}
As efficiency depends on the tools available, one approach would be doing it "analog":
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
unsigned long long pow2sum_min(unsigned long long n, long long unsigned m)
{
if (m >= n)
{
return 1;
}
--n;
return (2ULL << n) + pow2sum_min(n, m);
}
#define LN(x) (log2(x)/log2(M_E))
int main(int argc, char** argv)
{
if (2 >= argc)
{
fprintf(stderr, "%s a b\n", argv[0]);
exit(EXIT_FAILURE);
}
long a = atol(argv[1]), b = atol(argv[2]);
if (0L >= a || 0L >= b || b < a)
{
puts("Na ...!");
exit(EXIT_FAILURE);
}
/* Expand intevall to cover full dimensions: */
unsigned long long a_c = pow(2, floor(log2(a)));
unsigned long long b_c = pow(2, floor(log2(b+1)) + 1);
double log2_a_c = log2(a_c);
double log2_b_c = log2(b_c);
unsigned long p2s = pow2sum_min(log2_b_c, log2_a_c) - 1;
/* Integral log2(x) between a_c and b_c: */
double A = ((b_c * (LN(b_c) - 1))
- (a_c * (LN(a_c) - 1)))/LN(2)
+ (b+1 - a);
/* "Integer"-integral - integral of log2(x)'s inverse function (2**x) between log(a_c) and log(b_c): */
double D = p2s - (b_c - a_c)/LN(2);
/* Corrective from a_c/b_c to a/b : */
double C = (log2_b_c - 1)*(b_c - (b+1)) + log2_a_c*(a - a_c);
printf("Total used digits: %lld\n", (long long) ((A - D - C) +.5));
}
:-)
The main thing here is the number and kind of iterations done.
Number is
log(floor(b_c)) - log(floor(a_c))
times
doing one
n - 1 /* Integer decrement */
2**n + s /* One bit-shift and one integer addition */
for each iteration.
Here's an entirely look-up based approach. You don't even need the log2 :)
Algorithm
First we precompute interval limits where the number of bits would change and create a lookup table. In other words we create an array limits[2^n], where limits[i] gives us the biggest integer that can be represented with (i+1) bits. Our array is then {1, 3, 7, ..., 2^n-1}.
Then, when we want to determine the sum of bits for our range, we must first match our range limits a and b with the smallest index for which a <= limits[i] and b <= limits[j] holds, which will then tell us that we need (i+1) bits to represent a, and (j+1) bits to represent b.
If the indexes are the same, then the result is simply (b-a+1)*(i+1), otherwise we must separately get the number of bits from our value to the edge of same number of bits interval, and add up total number of bits for each interval between as well. In any case, simple arithmetic.
Code
#include <stdio.h>
#include <limits.h>
#include <time.h>
unsigned long bitsnumsum(unsigned int a, unsigned int b)
{
// generate lookup table
// limits[i] is the max. number we can represent with (i+1) bits
static const unsigned int limits[32] =
{
#define LTN(n) n*2u-1, n*4u-1, n*8u-1, n*16u-1, n*32u-1, n*64u-1, n*128u-1, n*256u-1
LTN(1),
LTN(256),
LTN(256*256),
LTN(256*256*256)
};
// make it work for any order of arguments
if (b < a) {
unsigned int c = a;
a = b;
b = c;
}
// find interval of a
unsigned int i = 0;
while (a > limits[i]) {
++i;
}
// find interval of b
unsigned int j = i;
while (b > limits[j]) {
++j;
}
// add it all up
unsigned long sum = 0;
if (i == j) {
// a and b in the same range
// conveniently, this also deals with j == 0
// so no danger to do [j-1] below
return (i+1) * (unsigned long)(b - a + 1);
}
else {
// add sum of digits in range [a, limits[i]]
sum += (i+1) * (unsigned long)(limits[i] - a + 1);
// add sum of digits in range [limits[j], b]
sum += (j+1) * (unsigned long)(b - limits[j-1]);
// add sum of digits in range [limits[i], limits[j]]
for (++i; i<j; ++i) {
sum += (i+1) * (unsigned long)(limits[i] - limits[i-1]);
}
return sum;
}
}
int main(void)
{
clock_t start, end;
unsigned int a=0, b=UINT_MAX;
start = clock();
printf("Sum of binary digits for numbers in range "
"[%u, %u]: %lu\n", a, b, bitsnumsum(a, b));
end = clock();
printf("Time: %fs\n", ((double)(end-start))/CLOCKS_PER_SEC);
}
Output
$ ./lookup
Sum of binary digits for numbers in range [0, 4294967295]: 133143986178
Time: 0.000282s
Algorithm
The main idea is to find the n2 = log2(x) rounded down. That is the number of digits in x. Let pow2 = 1 << n2. n2 * (pow2 - x + 1) is the number of digits in the values [x...pow2]. Now find the sun of digits in the powers of 2 from 1 to n2-1
Code
I am certain various simplifications can be made.
Untested code. Will review later.
// Let us use unsigned for everything.
unsigned ulog2(unsigned value) {
unsigned result = 0;
if (0xFFFF0000u & value) {
value >>= 16; result += 16;
}
if (0xFF00u & value) {
value >>= 8; result += 8;
}
if (0xF0u & value) {
value >>= 4; result += 4;
}
if (0xCu & value) {
value >>= 2; result += 2;
}
if (0x2 & value) {
value >>= 1; result += 1;
}
return result;
}
unsigned bit_count_helper(unsigned x) {
if (x == 0) {
return 1;
}
unsigned n2 = ulog2(x);
unsigned pow2 = 1u << n;
unsigned sum = n2 * (pow2 - x + 1u); // value from pow2 to x
while (n2 > 0) {
// ... + 5*16 + 4*8 + 3*4 + 2*2 + 1*1
pow2 /= 2;
sum += n2 * pow2;
}
return sum;
}
unsigned bit_count(unsigned a, unsigned b) {
assert(a < b);
return bit_count_helper(b - 1) - bit_count_helper(a);
}
For this problem your solution is the simplest, the one called "naive" where you look for every element in the sequence or in your case interval for check something or execute operations.
Naive Algorithm
Assuming that a and b are positive integers with b greater than a let's call the dimension/size of the interval [a,b], n = (b-a).
Having our number of elements n and using some notations of algorithms (like big-O notation link), the worst case cost is O(n*(numberOfBits_cost)).
From this we can see that we can speed up our algorithm by using a faster algorithm for computing numberOfBits() or we need to find a way to not look at every element of the interval that costs us n operations.
Intuition
Now looking at a possible interval [6,14] you can see that for 6 and 7 we need 3 digits, with 4 need for 8,9,10,11,12,13,14. This results in calling numberOfBits() for every number that use the same number of digits to be represented, while the following multiplication operation would be faster:
(number_in_subinterval)*digitsForThisInterval
((14-8)+1)*4 = 28
((7-6)+1)*3 = 6
So we reduced the looping on 9 elements with 9 operations to only 2.
So writing a function that use this intuition will give us a more efficient in time, not necessarily in memory, algorithm. Using your numberOfBits() function I have created this solution:
int intuitionSol(int a, int b){
int digitsForA = numberOfBits(a);
int digitsForB = numberOfBits(b);
if(digitsForA != digitsForB){
//because a or b can be that isn't the first or last element of the
// interval that a specific number of digit can rappresent there is a need
// to execute some correction operation before on a and b
int tmp = pow(2,digitsForA) - a;
int result = tmp*digitsForA; //will containt the final result that will be returned
int i;
for(i = digitsForA + 1; i < digitsForB; i++){
int interval_elements = pow(2,i) - pow(2,i-1);
result = result + ((interval_elements) * i);
//printf("NumOfElem: %i for %i digits; sum:= %i\n", interval_elements, i, result);
}
int tmp1 = ((b + 1) - pow(2,digitsForB-1));
result = result + tmp1*digitsForB;
return result;
}
else {
int elements = (b - a) + 1;
return elements * digitsForA; // or digitsForB
}
}
Let's look at the cost, this algorithm costs is the cost of doing correction operation on a and b plus the most expensive one that of the for-loop. In my solution however I'm not looping over all elements but only on numberOfBits(b)-numberOfBits(a) that in the worst case, when [0,n], become log(n)-1 thats equivalent to O(log n).
To resume we passed from a linear operations cost O(n) to a logartmic one O(log n) in the worst case. Look on this diagram the diferinces between the two.
Note
When I talk about interval or sub-interval I refer to the interval of elements that use the same number of digits to represent the number in binary.
Following there are some output of my tests with the last one that shows the difference:
Considered interval is [0,4]
YourSol: 9 in time: 0.000015s
IntuitionSol: 9 in time: 0.000007s
Considered interval is [0,0]
YourSol: 1 in time: 0.000005s
IntuitionSol: 1 in time: 0.000005s
Considered interval is [4,7]
YourSol: 12 in time: 0.000016s
IntuitionSol: 12 in time: 0.000005s
Considered interval is [2,123456]
YourSol: 1967697 in time: 0.005010s
IntuitionSol: 1967697 in time: 0.000015s

Not Getting a Calculated Value of the Constant e

Not really sure what the problem is here. I know I have the factorial function correct because I tested it separately. But the function that calculates e is tripping me up. All I have to do is add all the values after each factorial has been calculated. But I am having trouble translating that into C code. The problem for sure is in my second function. Any help or pointers would be appreciated.
#include <stdio.h>
#include <math.h>
#define NOERROR 0
#define DECIMAL_PLACES 16
#define EXPECTED_E 2.7182818284590452L
long calcFactorial(int);
double calcE(int);
long calcFactorial(int n)
{
long sum = 0;
sum = n;
if(n == 0)
{
return 1;
}
else
{
while(n != 1)
{
sum = sum * (n - 1);
n = n - 1;
}
printf("factorial sum: %ld\n", sum);
return sum;
}
}
double calcE(int n)
{
double e = 0;
int counter = 0;
for (counter = 0; counter < DECIMAL_PLACES; counter++)
{
e = e + (1/calcFactorial(n));
n--;
}
printf("Expected e value: %0.16Lf\n", EXPECTED_E);
printf("Calculated e value: %0.16d\n", e);
return e;
}
int main()
{
calcE(10);
}
You have a lot of errors in your code:
using long to store floating point result. Use double
passing n but looping using a bigger value: n becomes negative after a while: infinite loop
e = e + (1/calcFactorial(counter)); adds 0 to e most of the time because calcFactorial returns an integer (long)
EXPECTED_E constant had L suffix, which means long. Not what you want.
Fixed version:
#include <stdio.h>
#include <math.h>
#define NOERROR 0
#define DECIMAL_PLACES 16
#define EXPECTED_E 2.7182818284590452
long calcFactorial(int);
void calcE(int);
long calcFactorial(int n)
{
long sum = 0;
sum = n;
if(n == 0)
{
return 1;
}
else
{
while(n != 1)
{
sum *= (n - 1);
n = n - 1;
}
return sum;
}
}
void calcE(int n)
{
double e = 0;
int counter = 0;
for (counter = 0; counter < n; counter++)
{
e = e + (1.0/calcFactorial(counter));
}
printf("Expected e value: %0.16lf\n", EXPECTED_E);
printf("Calculated e value: %0.16lf\n", e);
}
int main( )
{
calcE(10);
}
This code outputs:
Expected e value: 2.7182818284590451
Calculated e value: 2.7182815255731922
Note: you are limited to a given maximum for n because after that you'll overflow long. Maybe consider using long long or unsigned long long for the factorial part (and even with that you're severely limited).
Jean-François Fabre highlighted your formal errors quite well, but it is not so far fetched to calculate with integers up to the final division--which must be done with floats, of course. The trick can be done with a method called binary splitting and, to my own surprise, it works very well native doubles, only one decimal digit off. It is also very simple to implement (code below written with legibility in mind).
#include <stdio.h>
#include <stdlib.h>
#include <stdint.h>
#define BS_AFU 0
#define BS_AOK 1
static int exp1_bin_split(uint64_t a, uint64_t b, uint64_t *P, uint64_t *Q){
int err = BS_AOK;
uint64_t p1, q1, p2, q2, t1, one;
one = 1UL;
t1 = b - a;
if(t1 == one){
*P = one;
*Q = b;
return err;
}
t1 = (a + b) >> 1;
err = exp1_bin_split(a, t1, &p1, &q1);
if(err != BS_AOK){
return err;
}
err = exp1_bin_split(t1, b, &p2, &q2);
if(err != BS_AOK){
return err;
}
*P = q2 * p1 + p2;
*Q = q1 * q2;
return err;
}
#include <float.h>
static int exp1(double *a){
int err = BS_AOK;
uint64_t p = 0UL, q = 0UL, zero = 0UL;
double dp, dq;
// DBL_DIG + 2 = 17 here on my machine
// had DBL_DIG + 1 first but found out via T&E that
// one more is still inside the precision of a binary64
err = exp1_bin_split(zero, DBL_DIG + 2, &p, &q);
if(err != BS_AOK){
return err;
}
p = p + q;
dp = (double) p;
dq = (double) q;
*a = dp/dq;
return err;
}
int main(void){
double e = 0.0;
int err = BS_AOK;
err = exp1(&e);
if(err != BS_AOK){
fprintf(stderr,"Something went wrong in computing e\n");
exit(EXIT_FAILURE);
}
printf("exp(1) ~ 2.7182818284590452353602874713526624978\nexp1 ~ %.20g\n",e);
exit(EXIT_SUCCESS);
}
It uses the same algorithm as you do but does not compute the individual fractions and sums them up as floats but does it all at once with integers such that we have a large fraction at the end to resemble the approximation of exp(1). That explanation is a bit over-simplified, please read the linked paper for the details.

Exponentiaion program

Question is this
Nancy hates any and every string that contains the number "13". Clouseau wants to gift her a string and is looking over his options, ofcourse he would never pick a string that has "13" as a substring.
Tell Clouseau the total number of such strings s that are made of exactly N characters. The strings may contain any integer from 0-9, repeated any number of times.
Input :
The first line of input file contains a number T indicating number of test cases. The following T lines, each contain an integer N.
Output :
The output file should contain answer to each query in a new line modulo 1000000009.
Constraints :
1 T 100000 ,
0 N 1000000009
I am not getting the logic correct.
# include <stdio.h>
# define MOD 1000000009
unsigned long long mod_pow(unsigned long long num, unsigned long long pow, unsigned long long mod)
{
unsigned long long test;
unsigned long long n = num;
for(test = 1; pow; pow >>= 1) {
if (pow & 1)
test = ((test % mod) * (n % mod)) % mod;
n = ((n % mod) * (n % mod)) % mod;
}
return test;
}
int main(int argc, char* argv[])
{
long t;
unsigned long long total_no, bad_no, n;
scanf ("%ld", &t);
while (t--) {
scanf ("%lld", &n);
if (n != 1) {
total_no = (10 * (mod_pow (10, n-1, MOD))) % MOD;
bad_no = ((n - 1) * (mod_pow(10, n-2, MOD))) % MOD;
printf ("%lld\n", (((total_no - bad_no) % MOD)));
}
else
printf ("10\n");
}
return 0;
}
You can use this function:
long long int bigmod ( long long a, long long p, long long m )
{
if ( p == 0 )return 1; // If power is 0, then a ^ 0 = 1 for any value of a, And 1 Mod m=1 for any value of m, So return 1
if ( p % 2 ) // If power is odd, Split it : a ^ 5 =( a )* (a ^ 4) --> left and right child respectively.
{
return ( ( a % m ) * ( bigmod ( a, p - 1, m ) ) ) % m;
}
else //If power is even then split it equally and return the result...
{
long long c = bigmod ( a, p / 2, m );
return ( (c%m) * (c%m) ) % m;
}
}
and call this function like this:
int main(){
// take input....
//Calling...
printf("result is %lld\n",bigmod(a,p,MOD)); //
return 0;
}

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