We may need to calculate factorials from time to time. But we will easily get into a very large number as increases. We all know Stirling’s formula for calculating factorials:

str[n_]:=n^n E^(-n) Sqrt[2 Pi n]

or

Error in Stirling’s formula

But how close is the Stirling’s formula? I did this in mathematica:

For[n=1, n<100, n=n+1, Print[n,"    ", N[n!/str[n]]]]

and the result is:

  1 1.08444
  2 1.04221
  3 1.02806
  4 1.02101
  5 1.01678
  6 1.01397
  7 1.01197
  8 1.01047
  9 1.0093
  10 1.00837
  11 1.0076
  12 1.00697
  13 1.00643
  14 1.00597
  15 1.00557
  16 1.00522
  17 1.00491
  18 1.00464
  19 1.0044
  20 1.00418
  21 1.00398
  22 1.00379
  23 1.00363
  24 1.00348
  25 1.00334
  26 1.00321
  27 1.00309
  28 1.00298
  29 1.00288
  30 1.00278
  31 1.00269
  32 1.00261
  33 1.00253
  34 1.00245
  35 1.00238
  36 1.00232
  37 1.00225
  38 1.0022
  39 1.00214
  40 1.00209
  41 1.00203
  42 1.00199
  43 1.00194
  44 1.0019
  45 1.00185
  46 1.00181
  47 1.00177
  48 1.00174
  49 1.0017
  50 1.00167
  51 1.00164
  52 1.0016
  53 1.00157
  54 1.00154
  55 1.00152
  56 1.00149
  57 1.00146
  58 1.00144
  59 1.00141
  60 1.00139
  61 1.00137
  62 1.00134
  63 1.00132
  64 1.0013
  65 1.00128
  66 1.00126
  67 1.00124
  68 1.00123
  69 1.00121
  70 1.00119
  71 1.00117
  72 1.00116
  73 1.00114
  74 1.00113
  75 1.00111
  76 1.0011
  77 1.00108
  78 1.00107
  79 1.00106
  80 1.00104
  81 1.00103
  82 1.00102
  83 1.001
  84 1.00099
  85 1.00098
  86 1.00097
  87 1.00096
  88 1.00095
  89 1.00094
  90 1.00093
  91 1.00092
  92 1.00091
  93 1.0009
  94 1.00089
  95 1.00088
  96 1.00087
  97 1.00086
  98 1.00085
  99 1.00084

So you can see,

When n!/str[n]-1
n >= 9
n >= 17
n >= 28
n >= 42
n >= 84

C-implementation

But actually, we can do more precise. According to MathWorld, we can have the inequality form for Stirling’s approximation:

also, we have the Gosper’s inequality which better approximate:

To compare, we can write a program as the one below:

/* Factorial approximations
 * This program runs for n=1 to 1550, calculating approximations of n!
 * Approximations used here are:
 *   1. Stirling's formula
 *   2. Lower bound of Stirling's inequality
 *   3. Upper bound of Stirling's inequality
 *   4. Gosper's approximation
 * This program will show that, long double can only hold up to 1546!
 */
#include <stdio.h>
#include <math.h>

# define M_El		2.7182818284590452353602874713526625L  /* e */
# define M_PIl		3.1415926535897932384626433832795029L  /* pi */

long double factorial(int n) {
	// Real factorial
	long double val=1;
	for(;n>1;n--) {
		val *= n;
	};
	return val;
};

long double stirling(int n) {
	// Stirling's approximation
	return powl(n,n)*expl(-n)*sqrtl(2*M_PIl*n);
};

long double stirling_l(int n) {
	// Stirling's lower bound
	return powl(n,n)*expl(-n+1/(long double)(12*n+1))*sqrtl(2*M_PIl*n);
};

long double stirling_u(int n) {
	// Stirling's lower bound
	return powl(n,n)*expl(-n+1/(long double)(12*n))*sqrtl(2*M_PIl*n);
};

long double gosper(int n) {
	// Gosper's approximation
	return powl(n,n)*expl(-n)*sqrtl(M_PIl*(2*n+1/(long double)3));
};

int main(int argc, char**argv)
{
	// Print a table showing the derivation between different factorial approximation
	int n;
	printf("%5s %15s %15s %15s %15s %15s\n",
			"n","n!","Stirling","Stirling LB","Stirling UB","Gosper");
	for(n=1;n<1550;n++) {
		printf("%5d %15Lg %15Lg %15Lg %15Lg %15Lg\n",n,factorial(n),
				stirling(n)/factorial(n)-1,
				stirling_l(n)/factorial(n)-1,
				stirling_u(n)/factorial(n)-1,
				gosper(n)/factorial(n)-1);
	};
	return 0;
};

This program generates different approximations and compare it to the correct value of factorial. We can also shown by this program that, the long double precision in C can only hold up to factorial of 1546.

Below are the output of the C program up to factorial of 100. The number is the relative difference between approximation and actual value, i.e. approx/actual:

    n        Stirling     Stirling LB     Stirling UB          Gosper
    1       -0.077863     -0.00412984      0.00227445     -0.00397819
    2      -0.0404978      -0.0013398     0.000326024     -0.00131847
    3      -0.0272984    -0.000650687     9.98571e-05    -0.000644108
    4       -0.020576    -0.000382436     4.26621e-05    -0.000379603
    5      -0.0165069    -0.000251217     2.19757e-05    -0.000249749
    6      -0.0137803    -0.000177483     1.27601e-05    -0.000176626
    7      -0.0118262    -0.000131995     8.05199e-06    -0.000131453
    8      -0.0103573    -0.000101982     5.40142e-06    -0.000101616
    9     -0.00921276     -8.1147e-05     3.79708e-06    -8.08896e-05
   10     -0.00829596    -6.60984e-05      2.7699e-06    -6.59103e-05
   11     -0.00754507     -5.4877e-05     2.08209e-06    -5.47353e-05
   12     -0.00691879    -4.62873e-05     1.60434e-06     -4.6178e-05
   13      -0.0063885    -3.95667e-05     1.26222e-06    -3.94805e-05
   14      -0.0059337    -3.42098e-05     1.01084e-06    -3.41407e-05
   15     -0.00553933    -2.98712e-05     8.22004e-07     -2.9815e-05
   16     -0.00519412    -2.63084e-05     6.77414e-07    -2.62621e-05
   17      -0.0048894    -2.33469e-05     5.64836e-07    -2.33082e-05
   18     -0.00461846    -2.08586e-05      4.7588e-07     -2.0826e-05
   19     -0.00437596    -1.87478e-05     4.04663e-07    -1.87201e-05
   20     -0.00415765     -1.6942e-05     3.46975e-07    -1.69181e-05
   21     -0.00396009    -1.53849e-05      2.9975e-07    -1.53643e-05
   22     -0.00378045    -1.40331e-05     2.60719e-07    -1.40152e-05
   23     -0.00361641    -1.28518e-05     2.28181e-07    -1.28362e-05
   24       -0.003466    -1.18137e-05     2.00839e-07    -1.17999e-05
   25     -0.00332761    -1.08964e-05     1.77697e-07    -1.08842e-05
   26     -0.00319984     -1.0082e-05     1.57977e-07    -1.00711e-05
   27     -0.00308152    -9.35556e-06      1.4107e-07    -9.34585e-06
   28     -0.00297164     -8.7049e-06     1.26493e-07    -8.69619e-06
   29     -0.00286932    -8.11982e-06     1.13856e-07    -8.11197e-06
   30     -0.00277382     -7.5918e-06     1.02848e-07    -7.58471e-06
   31     -0.00268447    -7.11366e-06     9.32145e-08    -7.10723e-06
   32     -0.00260069     -6.6793e-06     8.47474e-08    -6.67346e-06
   33     -0.00252199    -6.28354e-06     7.72755e-08    -6.27822e-06
   34     -0.00244791    -5.92194e-06     7.06567e-08    -5.91707e-06
   35     -0.00237806    -5.59068e-06     6.47727e-08    -5.58621e-06
   36     -0.00231208    -5.28645e-06     5.95243e-08    -5.28235e-06
   37     -0.00224966     -5.0064e-06     5.48279e-08    -5.00262e-06
   38     -0.00219053    -4.74802e-06     5.06129e-08    -4.74453e-06
   39     -0.00213442    -4.50915e-06      4.6819e-08    -4.50591e-06
   40     -0.00208112    -4.28785e-06      4.3395e-08    -4.28485e-06
   41     -0.00203042    -4.08245e-06     4.02969e-08    -4.07967e-06
   42     -0.00198212    -3.89147e-06     3.74869e-08    -3.88888e-06
   43     -0.00193607    -3.71358e-06     3.49321e-08    -3.71117e-06
   44     -0.00189211    -3.54762e-06     3.26043e-08    -3.54536e-06
   45     -0.00185011    -3.39253e-06     3.04789e-08    -3.39043e-06
   46     -0.00180993     -3.2474e-06     2.85342e-08    -3.24543e-06
   47     -0.00177145    -3.11138e-06     2.67515e-08    -3.10954e-06
   48     -0.00173458    -2.98374e-06     2.51142e-08      -2.982e-06
   49     -0.00169921    -2.86379e-06     2.36079e-08    -2.86216e-06
   50     -0.00166526    -2.75093e-06     2.22197e-08     -2.7494e-06
   51     -0.00163263    -2.64462e-06     2.09382e-08    -2.64317e-06
   52     -0.00160126    -2.54435e-06     1.97534e-08    -2.54298e-06
   53     -0.00157107    -2.44967e-06     1.86563e-08    -2.44838e-06
   54       -0.001542    -2.36019e-06      1.7639e-08    -2.35897e-06
   55     -0.00151399    -2.27551e-06     1.66943e-08    -2.27436e-06
   56     -0.00148697    -2.19532e-06     1.58159e-08    -2.19423e-06
   57     -0.00146091    -2.11929e-06      1.4998e-08    -2.11825e-06
   58     -0.00143574    -2.04714e-06     1.42356e-08    -2.04616e-06
   59     -0.00141142    -1.97862e-06      1.3524e-08    -1.97768e-06
   60     -0.00138791    -1.91348e-06     1.28591e-08    -1.91259e-06
   61     -0.00136518     -1.8515e-06      1.2237e-08    -1.85065e-06
   62     -0.00134317    -1.79249e-06     1.16544e-08    -1.79168e-06
   63     -0.00132187    -1.73625e-06     1.11082e-08    -1.73548e-06
   64     -0.00130123    -1.68262e-06     1.05956e-08    -1.68189e-06
   65     -0.00128122    -1.63144e-06     1.01141e-08    -1.63074e-06
   66     -0.00126182    -1.58255e-06     9.66134e-09    -1.58188e-06
   67       -0.001243    -1.53583e-06     9.23518e-09    -1.53519e-06
   68     -0.00122473    -1.49115e-06     8.83373e-09    -1.49054e-06
   69     -0.00120699    -1.44839e-06     8.45521e-09    -1.44781e-06
   70     -0.00118976    -1.40745e-06     8.09801e-09    -1.40689e-06
   71     -0.00117301    -1.36822e-06     7.76065e-09    -1.36768e-06
   72     -0.00115673     -1.3306e-06     7.44177e-09    -1.33009e-06
   73     -0.00114089    -1.29452e-06     7.14012e-09    -1.29402e-06
   74     -0.00112549    -1.25988e-06     6.85456e-09     -1.2594e-06
   75     -0.00111049    -1.22661e-06     6.58403e-09    -1.22616e-06
   76     -0.00109588    -1.19465e-06     6.32754e-09    -1.19421e-06
   77     -0.00108166    -1.16392e-06     6.08421e-09     -1.1635e-06
   78      -0.0010678    -1.13436e-06      5.8532e-09    -1.13395e-06
   79     -0.00105429    -1.10591e-06     5.63373e-09    -1.10552e-06
   80     -0.00104112    -1.07851e-06     5.42511e-09    -1.07814e-06
   81     -0.00102827    -1.05213e-06     5.22665e-09    -1.05177e-06
   82     -0.00101574     -1.0267e-06     5.03776e-09    -1.02635e-06
   83     -0.00100351    -1.00218e-06     4.85786e-09    -1.00184e-06
   84    -0.000991567    -9.78528e-07     4.68643e-09    -9.78203e-07
   85    -0.000979907    -9.55704e-07     4.52297e-09    -9.55391e-07
   86    -0.000968519     -9.3367e-07     4.36702e-09    -9.33367e-07
   87    -0.000957392    -9.12389e-07     4.21816e-09    -9.12096e-07
   88    -0.000946517    -8.91827e-07     4.07599e-09    -8.91545e-07
   89    -0.000935887    -8.71952e-07     3.94014e-09     -8.7168e-07
   90    -0.000925494    -8.52735e-07     3.81026e-09    -8.52471e-07
   91    -0.000915328    -8.34146e-07     3.68603e-09    -8.33891e-07
   92    -0.000905383    -8.16158e-07     3.56713e-09    -8.15911e-07
   93    -0.000895653    -7.98746e-07      3.4533e-09    -7.98507e-07
   94    -0.000886129    -7.81886e-07     3.34426e-09    -7.81654e-07
   95    -0.000876805    -7.65553e-07     3.23976e-09    -7.65329e-07
   96    -0.000867676    -7.49727e-07     3.13957e-09     -7.4951e-07
   97    -0.000858735    -7.34387e-07     3.04347e-09    -7.34176e-07
   98    -0.000849976    -7.19512e-07     2.95125e-09    -7.19308e-07
   99    -0.000841394    -7.05086e-07     2.86272e-09    -7.04887e-07
  100    -0.000832983    -6.91088e-07      2.7777e-09    -6.90896e-07