## Why Is This Factorial Algorithm Not Accurate

Question

Sorry I feel stupid asking this and am prepared to lose half of my points asking this but why does this algorithm not work? It works up to a point. After the number 13 the factorials are a little off. For instance the numbers do not entirely match in the hundreds thousands place and onward.

``````#include <stdio.h>

float factorial(unsigned int i) {

if (i <= 1) {
return 1;
}
return i * factorial(i - 1);
}

int  main() {
int i = 13;
printf("Factorial of %d is %f\n", i, factorial(i));
return 0;
}
``````

Here's the output:

``````Factorial of 13 is 6227020800.000000
``````

Here is an example of inaccurate output:

``````Factorial of 14 is 87178289152.000000
``````

The output for the number 14 should actually be this (from mathisfun.com)

14        87,178,291,200

I changed the return type to float to obtain more accurate output but I obtained this code for the most part from here: https://www.tutorialspoint.com/cprogramming/c_recursion.htm

EDIT: If I change to the return type to double the output is accurate up to 21.I am using the %Lf string formatter for the output in the printf function.

Show source

## Answers to Why Is This Factorial Algorithm Not Accurate ( 5 )

1. Simple. `float` cannot accurately store integers above 16777216 without loss of precision.

`int` is better than float. But try `long long` so you can properly store 19 digits.

2. Why Is This Factorial Algorithm Not Accurate

There's nothing wrong in your `algorithm` as such. It is just that the data types you use have a limit for the highest number they can store. This will be a problem no matter which algorithm you choose. You can change the data types from `float` to something like `long double` to hold something bigger. But eventually it will still start failing once the factorial value exceeds the capacity of that data type. In my opinion, you should put an a condition in your factorial function to return without calculating anything if the passed in argument is greater than a value that your chosen datatype can support.

3. `float` can represent a wider range of numbers than `int`, but it cannot represent all the values within that range - as you approach the edge of the range (i.e., as the magnitudes of the values increase), the gap between representable values gets wider.

For example, if you cannot represent values between 0.123 and 0.124, then you also cannot represent values between 123.0 and 124.0, or 1230.0 and 1240.0, or 12300.0 and 12400.0, etc. (of course, IEEE-754 single-precision `float` gives you a bit more precision than that).

Having said that, `float` should be able to represent all integer values up to 224 exactly, so I'm going to bet the issue is in the `printf` call - `float` parameters are "promoted" to `double`, so there's a representation change involved, and that may account for the lost precision.

Try changing the return type of `factorial` to `double` and see if that doesn't help.

<gratuitous rant>

Every time I see a recursive factorial function I want to scream. Recursion in this particular case offers no improvement in either code clarity or performance over an iterative solution:

``````double fac( int x )
{
double result = 1.0;
while ( x )
{
result *= x--;
}
return result;
}
``````

and can in fact result in worse performance due to the overhead of so many function calls.

Yes, the definition of a factorial is recursive, but the implementation of a factorial function doesn't have to be. Same for Fibonacci sequences. There's even a closed form solution for Fibonacci numbers

``Fn = ((1 + √5)n - (1 - √5)n) / (2n * √5)``

that doesn't require any looping in the first place.

Recursion's great for algorithms that partition their data into relatively few, equal-sized subsets (Quicksort, tree traversals, etc.). For something like this, where the partitioning is N-1 subsets of 1 element? Not so much.

</gratuitous rant>

4. OP is encountering the precision limits of `float`. For typical `float`, whole number values above `16777216.0f` are not all exactly representable. Some factorial results above this point are exactly representable.

Let us try this with different types.
At `11!`, the `float` results exceeds `16777216.0f` and is exactly correct.
At `14!`, the `float` result is imprecise because of limited precision.
At `23!`, the `double` result is imprecise because of limited precision.

At `21!`, the answer exceeds my `uintmax_t` range.
At `35!`, the answer exceeds my `float` range.
At `171!`, the answer exceeds my `double` range.

A string representation is accurate endlessly until it reaches buffer limitations.

``````#include <stdint.h>
#include <string.h>
#include <stdio.h>

uintmax_t factorial_uintmax(unsigned int i) {
if (i <= 1) {
return 1;
}
return i * factorial_uintmax(i - 1);
}

float factorial_float(unsigned int i) {
if (i <= 1) {
return 1;
}
return i * factorial_float(i - 1);
}

double factorial_double(unsigned int i) {
if (i <= 1) {
return 1;
}
return i * factorial_double(i - 1);
}

char * string_mult(char *y, unsigned base, unsigned x) {
size_t len = strlen(y);
unsigned acc = 0;
size_t i = len;
while (i > 0) {
i--;
acc += (y[i] - '0') * x;
y[i] = acc % base + '0';
acc /= base;
}
while (acc) {
memmove(&y[1], &y[0], ++len);
y[0] = acc % base + '0';
acc /= base;
}
return y;
}

char *factorial_string(char *dest, unsigned int i) {
strcpy(dest, "1");
for (unsigned m = 2; m <= i; m++) {
string_mult(dest, 10, m);
}
return dest;
}

void factorial_test(unsigned int i) {
uintmax_t u = factorial_uintmax(i);
float f = factorial_float(i);
double d = factorial_double(i);
char s[2000];
factorial_string(s, i);
printf("factorial of %3d is uintmax_t: %ju\n", i, u);
printf("                    float:     %.0f %s\n", f, "*" + (1.0 * f == u));
printf("                    double:    %.0f %s\n", d, "*" + (d == u));
printf("                    string:    %s\n", s);
}

int main(void) {
for (unsigned i = 11; i < 172; i++)
factorial_test(i);
return 0;
}
``````

Output

``````factorial of  11 is uintmax_t: 39916800
float:     39916800
double:    39916800
string:    39916800
factorial of  12 is uintmax_t: 479001600
float:     479001600
double:    479001600
string:    479001600
factorial of  13 is uintmax_t: 6227020800
float:     6227020800
double:    6227020800
string:    6227020800
factorial of  14 is uintmax_t: 87178291200
float:     87178289152 *
double:    87178291200
string:    87178291200
factorial of  20 is uintmax_t: 2432902008176640000
float:     2432902023163674624 *
double:    2432902008176640000
string:    2432902008176640000
factorial of  21 is uintmax_t: 14197454024290336768
float:     51090940837169725440 *
double:    51090942171709440000 *
string:    51090942171709440000
factorial of  22 is uintmax_t: 17196083355034583040
float:     1124000724806013026304 *
double:    1124000727777607680000 *
string:    1124000727777607680000
factorial of  23 is uintmax_t: 8128291617894825984
float:     25852017444594485559296 *
double:    25852016738884978212864 *
string:    25852016738884976640000
factorial of  34 is uintmax_t: 4926277576697053184
float:     295232822996533287161359432338880069632 *
double:    295232799039604119555149671006000381952 *
string:    295232799039604140847618609643520000000
factorial of  35 is uintmax_t: 6399018521010896896
float:     inf *
double:    10333147966386144222209170348167175077888 *
string:    10333147966386144929666651337523200000000
factorial of 170 is uintmax_t: 0
float:     inf *
double:    72574156153079940453996357155895914678961840000000... *
string:    72574156153079989673967282111292631147169916812964...
factorial of 171 is uintmax_t: 0
float:     inf *
double:    inf *
string:    12410180702176678234248405241031039926166055775016...
``````
5. Someone posted a similar question a while back. The consensus was if you're writing it for work use a big number library (like GMP) and if it's a programming exercise write up a solution using a character array.

For example:

``````/* fact50.c

calculate a table of factorials from 0! to 50! by keeping a running sum of character digits
*/

#include <stdio.h>
#include <string.h>

int main (void)
{
printf ("\n                            Table of Factorials\n\n");

// length of arrays = 65 character digits

char str[] =
"00000000000000000000000000000000000000000000000000000000000000000";
char sum[] =
"00000000000000000000000000000000000000000000000000000000000000001";

const int len = strlen (str);
int index;

for ( int i = 0; i <= 50; ++i ) {

memcpy (str, sum, len);

for ( int j = 1; j <= i - 1; ++j ) {

index = len - 1;
int carry = 0;

do {
int digit = (sum[index] - '0') + (str[index] - '0') + carry;
carry = 0;
if ( digit > 9 ) {
carry = 1;
digit %= 10;
}
sum[index] = digit + '0';
--index;
}
while ( index >= 0 );

}

printf ("%2i! = ", i);
for ( index = 0; sum[index] == '0'; ++index )
printf ("%c", '.');
for ( ; index < len; ++index )
printf ("%c", sum[index]);
printf ("\n");

}

return 0;
}
``````