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I am restoring the Ascota 170 antique mechanical programmable computer. It is already working.
Now I’m looking for an algorithm to demonstrate its capabilities — like calculating trigonometric or logarithmic tables. Or something like that.
Unfortunately, from mathematical operations, a computer is only capable of adding and subtracting integers (55 registers from -1E12 to 1E12). There is not even a shift-to-digit operation — so that it can be programmatically implemented to multiply only by very small numbers.
But its logical operations are very well developed.
Could you advise me any suitable algorithm?
So what you're doing is really kinda awesome. And as it happens, I can explain quite a bit about how to implement fractional logarithms using only integer addition and subtraction! This post is going to be long, but there's lots of detail included, and a working implementation at the end, and it should be enough for you to do some fun things with your weird mechanical computer.
Implementing Comparisons
You're going to need to be able to compare numbers. While you said you can perform comparisons == 0 and > 0, that's not really quite enough for most of the interesting algorithms you'll want to implement. You need relative comparisons, which can be determined via subtraction:
isLessThan(a, b):
diff = b - a
if diff > 0 then return true
else return false
isGreaterThan(a, b):
diff = a - b
if diff > 0 then return true
else return false
isLessThanOrEqual(a, b):
diff = a - b
if diff > 0 then return false
else return true
isGreaterThanOrEqual(a, b):
diff = b - a
if diff > 0 then return false
else return true
For the rest of this post, I'm just going to write the simpler form of a > b, but if you can't do that directly, you can substitute in one of the operations above.
Implementing Shifts
Now, since you don't have digit-shifting hardware, you'll have to create "routines" to implement it. A left-shift is easy: Add a number to itself, and again, and again, and then add the original number, and then add it one more time; and that's the equivalent of shifting left by 1 digit.
So shift left by one digit, or multiply-by-ten:
shiftLeft(value):
value2 = value + value
value4 = value2 + value2
value5 = value4 + value
return value5 + value5
Shifting by many digits is just repeated invocation of shiftLeft():
shl(value, count):
repeat:
if count <= 0 then goto done
value = shiftLeft(value)
count = count - 1
done:
return value
Shifting right by one digit is a little harder: We need to do this with repeated subtraction and addition, as in the pseudocode below:
shr(value, count):
if count == 0 then return value
index = 11
shifted = 0
repeat1:
if index < 0 then goto done
adder = shl(1, index - count)
subtractor = shl(adder, count)
repeat2:
if value <= subtractor then goto next
value = value - subtractor
shifted = shifted + adder
goto repeat2
next:
index = index - 1
goto repeat1
done:
return count
Conveniently, since it's hard to shift right in the first place, the algorithm lets us directly choose how many digits to shift by.
Multiplication
It looks like your hardware might have multiplication? But if it doesn't, you can implement multiplication using repeated addition and shifting. Binary multiplication is the easiest form to implement that's actually efficient, and that requires us to first implement multiplyByTwo() and divideByTwo(), using the same basic techniques that we used to implement shiftLeft() and shr().
Once you have those implemented, multiplication involves repeatedly slicing off the last bit of one of the numbers, and if that bit is a 1, then adding a growing version of the other number to the running total:
multiply(a, b):
product = 0
repeat:
if b <= 0 then goto done
nextB = divideByTwo(b)
bit = b - multiplyByTwo(nextB)
if bit == 0 then goto skip
product = product + a
skip:
a = a + a
b = nextB
goto repeat
done:
return product
A full implementation of this is included below, if you need it.
Integer Logarithms
We can use our ability to shift right by a digit to calculate the integer part of the base-10 logarithm of a number — this is really just how many times you can shift the number right before you reach a number too small to shift.
integerLogarithm(value):
count = 0
repeat:
if value <= 9 then goto done
value = shiftRight(value)
count = count + 1
goto repeat
done:
return count
So for 0-9, this returns 0; for 10-99, this returns 1; for 100-999 this returns 2, and so on.
Integer Exponents
The opposite of the above algorithm is pretty trivial: To calculate 10 raised to an integer power, we just shift the digits left by the power.
integerExponent(count):
value = shl(1, count)
return value
So for 0, this returns 1; for 1, this return 10; for 2, this returns 100; for 3, this returns 1000; and so on.
Splitting the Integer and Fraction
Now that we can handle integer powers and logarithms, we're almost ready to handle the fractional part. But before we can really talk about how to compute the fractional part of the logarithm, we have to talk about how to divide up the problem so we can compute the fractional part separately from the integer part. Ideally, we only want to deal with computing logarithms for numbers in a fixed range — say, from 1 to 10, rather than from 1 to infinity.
We can use our integer logarithm and exponent routines to slice up the full logarithm problem so that we're always dealing with a value in the range of [1, 10), no matter what the input number was.
First, we calculate the integer logarithm, and then the integer exponent, and then we subtract that from the original number. Whatever is left over is the fractional part that we need to calculate: And then the only remaining exercise is to shift that fractional part so that it's always in a consistent range.
normalize(value):
intLog = integerLogarithm(value) // From 0 to 12 (meaningful digits)
if intLog <= 5 then goto lessThan
value = shr(value, intLog - 5)
goto done
lessThan:
value = shl(value, 5 - intLog)
done:
return value
You can convince yourself with relatively little effort that no matter what the original value was, its highest nonzero digit will be moved to column 7: So "12345" will become "000000123450" (i.e., "0000001.23450"). This allows us to pretend that there's always an invisible decimal point a little more than halfway down the number, so that now we only need to solve the problem of calculating logarithms of values in the range of [1, 10).
(Why "more than halfway"? We will need the upper half of the value to always be zero, and you'll see why in a moment.)
Fractional Logarithms
Knuth explains how to do this in The Art of Computer Programming, section 1.2.2. Our goal will be to calculate log10(x) so that for some values of b1, b2, b3 ... , where n is already 0 (because we split out the integer portion above):
log10(x) = n + b1/2 + b2/4 + b3/8 + b4/16 + ...
Knuth says that we can obtain b1, b2, b3 ... like this:
To obtain b1, b2, ..., we now set x0 = x / 10^n and, for k >= 1,
b[k] = 0, x[k] = x[k-1] ^ 2, if x[k-1] ^ 2 < 10;
b[k] = 1, x[k] = x[k-1] ^ 2 / 10, if x[k-1] ^ 2 >= 10.
That is to say, each step uses pseudocode loop something like this:
fractionalLogarithm(x):
for i = 1 to numberOfBinaryDigitsOfPrecision:
nextX = x * x
if nextX < 10 then:
b[i] = 0
else:
b[i] = 1
nextX = nextX / 10
In order for this to work using the fixed-point numbers we have above, we have to implement x * x using a shift to move the decimal point back into place, which will lose some digits. This will cause error to propagate, as Knuth says, but it will give enough accuracy that it's good enough for demonstration purposes.
So given a fractional value generated by normalize(value), we can compute its fractional binary logarithm like this:
fractionalLogarithm(value):
for i = 1 to 20:
value = shr(value * value, 6)
if value < 1000000 then:
b[i] = 0
else:
b[i] = 1
value = shr(value, 1)
But a binary fractional logarithm — individual bits! — isn't especially useful, especially since we computed an decimal version of the integer part of the logarithm in the earlier step. So we'll modify this one more time, to calculate a decimal fractional logarithm, to five places, instead of calculating an array of bits; for that, we'll need a table of 20 values that represent the conversions of each of those bits to decimal, and we'll store them as fixed-point as well:
table[1] = 1/(2^1) = 1/2 = 500000
table[2] = 1/(2^2) = 1/4 = 250000
table[3] = 1/(2^3) = 1/8 = 125000
table[4] = 1/(2^4) = 1/16 = 062500
table[5] = 1/(2^5) = 1/32 = 031250
table[6] = 1/(2^6) = 1/64 = 015625
...
table[17] = 1/(2^17) = 1/131072 = 000008
table[18] = 1/(2^18) = 1/262144 = 000004
table[19] = 1/(2^19) = 1/514288 = 000002
table[20] = 1/(2^20) = 1/1048576 = 000001
So now with that table, we can produce the whole fractional logarithm, using pure integer math:
fractionalLogarithm(value):
log = 0
for i = 1 to 20:
value = shr(value * value, 6)
if value >= 1000000 then:
log = log + table[i]
value = shr(value, 1)
return log
Putting It All Together
Finally, for a complete logarithm of any integer your machine can represent, this is the whole thing, which will compute the logarithm with six digits of precision, in the form "0000XX.XXXXXX":
log(value):
intPart = integerLogarithm(value)
value = normalize(value)
fracPart = fractionalLogarithm(value)
result = shl(intPart, 6) + fracPart
return result
Demonstration
To show that the math works — and that it works pretty well! — below is a JavaScript implementation of the above algorithm. It uses pure integer math: Only addition, subtraction, and relative comparison. Functions are used to organize the code, but they behave like subroutines: They're not recursive, and don't nest very deeply.
You can try it out live (click the 'Run' button and type 12345 in the input field). Compare the result to the standard Math.log() function, and you'll see how close the pure-integer version gets:
function shiftLeft(value) {
var value2 = value + value;
var value4 = value2 + value2;
var value5 = value4 + value;
return value5 + value5;
}
function shl(value, count) {
while (count > 0) {
value = shiftLeft(value);
count = count - 1;
}
return value;
}
function shr(value, count) {
if (count == 0) return value;
var index = 11;
var shifted = 0;
while (index >= 0) {
var adder = shl(1, index - count);
var subtractor = shl(adder, count);
while (value > subtractor) {
value = value - subtractor;
shifted = shifted + adder;
}
index = index - 1;
}
return shifted;
}
//-----------------------------------
function multiplyByTwo(value) {
return value + value;
}
function multiplyByPowerOfTwo(value, count) {
while (count > 0) {
value = value + value;
count = count - 1;
}
return value;
}
function divideByPowerOfTwo(value, count) {
if (count == 0) return value;
var index = 39; // lg(floor(pow(10, 12)))
var shifted = 0;
while (index >= 0) {
var adder = multiplyByPowerOfTwo(1, index - count);
var subtractor = multiplyByPowerOfTwo(adder, count);
while (value >= subtractor) {
value = value - subtractor;
shifted = shifted + adder;
}
index = index - 1;
}
return shifted;
}
function divideByTwo(value) {
return divideByPowerOfTwo(value, 1);
}
function multiply(a, b) {
var product = 0;
while (b > 0) {
nextB = divideByTwo(b);
bit = b - multiplyByTwo(nextB);
if (bit != 0) {
product += a;
}
a = a + a;
b = nextB;
}
return product;
}
//-----------------------------------
var logTable = {
"1": 500000,
"2": 250000,
"3": 125000,
"4": 62500,
"5": 31250,
"6": 15625,
"7": 7813,
"8": 3906,
"9": 1953,
"10": 977,
"11": 488,
"12": 244,
"13": 122,
"14": 61,
"15": 31,
"16": 15,
"17": 8,
"18": 4,
"19": 2,
"20": 1,
};
//-----------------------------------
function integerLogarithm(value) {
var count = 0;
while (value > 9) {
value = shr(value, 1);
count = count + 1;
}
return count;
}
function normalize(value) {
var intLog = integerLogarithm(value);
if (intLog > 5)
value = shr(value, intLog - 5);
else
value = shl(value, 5 - intLog);
return value;
}
function fractionalLogarithm(value) {
var log = 0;
for (i = 1; i < 20; i++) {
var squaredValue = multiply(value, value);
value = shr(squaredValue, 5);
if (value >= 1000000) {
log = log + logTable[i];
value = shr(value, 1);
}
}
return log;
}
function log(value) {
var intPart = integerLogarithm(value);
value = normalize(value);
var fracPart = fractionalLogarithm(value);
var result = shl(intPart, 6) + fracPart;
return result;
}
//-----------------------------------
// Just a little jQuery event handling to wrap a UI around the above functions.
$("#InputValue").on("keydown keyup keypress focus blur", function(e) {
var inputValue = Number(this.value.replace(/[^0-9]+/g, ''));
var outputValue = log(inputValue);
$("#OutputValue").text(outputValue / 1000000);
var trueResult = Math.floor((Math.log(inputValue) / Math.log(10)) * 1000000 + 0.5) / 1000000
$("#TrueResult").text(trueResult);
});
<script src="https://cdnjs.cloudflare.com/ajax/libs/jquery/3.3.1/jquery.min.js"></script>
Input integer: <input type="text" id="InputValue" /><br /><br />
Result using integer algorithm: <span id="OutputValue"></span><br /><br />
True logarithm: <span id="TrueResult"></span><br />
As I mentioned in your Original question on SE/RC for pow,sqrt,n-root,log,exp see:
Power by squaring for negative exponents
and all the sub-links in there.
Once you got working *,/,<<,>> (which the other answer covers well) and may fixed point instead of floating you can also start computing goniometrics. For that the best is use Chebyshev series but as I lack the math behind them I can use only already precomputed ones ... Taylor is a common knowledge so computing that should be easy here what I code for my arithmetics template to cover math for arbitrary math data types (bignums):
// Taylor goniometric https://en.wikipedia.org/wiki/Taylor_series
friend T sin (const T &x) // = sin(x)
{
int i; T z,dz,x2,a,b;
x2=x/(pi+pi); x2-=::integer(x2); x2*=pi+pi;
for (z=x2,a=x2,b=1,x2*=x2,i=2;;)
{
a*=x2; b*=i; i++; b*=i; i++; dz=a/b; z-=dz;
a*=x2; b*=i; i++; b*=i; i++; dz=a/b; z+=dz;
if (::abs(dz)<zero) break;
}
return z;
}
friend T cos (const T &x) // = cos(x)
{
int i; T z,dz,x2,a,b;
x2=x/(pi+pi); x2-=::integer(x2); x2*=pi+pi;
for (z=1,a=1,b=1,x2*=x2,i=1;;)
{
a*=x2; b*=i; i++; b*=i; i++; dz=a/b; z-=dz;
a*=x2; b*=i; i++; b*=i; i++; dz=a/b; z+=dz;
if (::abs(dz)<zero) break;
}
return z;
}
friend T tan (const T &x) // = tan(x)
{
int i; T z0,z1,dz,x1,x2,a,b;
x1=x/pi; x1-=::integer(x1); x1*=pi; x2=x1*x1;
for (z0=1,z1=1,a=1,b=1,i=2;;)
{
a*=x2; b*=i; i++; dz=a/b; z0-=dz; // z0=cos(x)
b*=i; i++; dz=a/b; z1-=dz; // z1=sin(x)/x
a*=x2; b*=i; i++; dz=a/b; z0+=dz;
b*=i; i++; dz=a/b; z1+=dz;
if (::abs(dz)<zero) break;
}
return (x1*z1)/z0;
}
friend T ctg (const T &x) // = cotan(x)
{
int i; T z0,z1,dz,x1,x2,a,b;
x1=x/pi; x1-=::integer(x1); x1*=pi; x2=x1*x1;
for (z0=1,z1=1,a=1,b=1,i=2;;)
{
a*=x2; b*=i; i++; dz=a/b; z0-=dz; // z0=cos(x)
b*=i; i++; dz=a/b; z1-=dz; // z1=sin(x)/x
a*=x2; b*=i; i++; dz=a/b; z0+=dz;
b*=i; i++; dz=a/b; z1+=dz;
if (::abs(dz)<zero) break;
}
return z0/(x1*z1);
}
friend T asin (const T &x) // = asin(x)
{
if (x<=-1.0) return -0.5*pi;
if (x>=+1.0) return +0.5*pi;
return ::atan(x/::sqrt(1.0-(x*x)));
}
friend T acos (const T &x){ T z; z=0.5*pi-::asin(x); return z; } // = acos(x)
friend T atan (const T &x) // = atan(x)
{
bool _shift=false;
bool _invert=false;
bool _negative=false;
T z,dz,x1,x2,a,b; x1=x;
if (x1<0.0) { _negative=true; x1=-x1; }
if (x1>1.0) { _invert=true; x1=1.0/x1; }
if (x1>0.7) { _shift=true; b=::sqrt(3.0)/3.0; x1=(x1-b)/(1.0+(x1*b)); }
for (x2=x1*x1,z=x1,a=x1,b=1;;) // if x1>0.8 convergence is slow
{
a*=x2; b+=2; dz=a/b; z-=dz;
a*=x2; b+=2; dz=a/b; z+=dz;
if (::abs(dz)<zero) break;
}
if (_shift) z+=pi/6.0;
if (_invert) z=0.5*pi-z;
if (_negative) z=-z;
return z;
}
friend T actg (const T &x){ T z; z=::atan(1.0/x); return z; } // = acotan(x)
friend T atan2 (const T &y,const T &x){ return atanxy(x,y); } // = atan(y/x)
friend T atanxy (const T &x,const T &y) // = atan(y/x)
{
int sx,sy; T a;
T _zero=1.0e-30;
sx=0; if (x<-_zero) sx=-1; if (x>+_zero) sx=+1;
sy=0; if (y<-_zero) sy=-1; if (y>+_zero) sy=+1;
if ((sy==0)&&(sx==0)) return 0.0;
if ((sx==0)&&(sy> 0)) return 0.5*x.pi;
if ((sx==0)&&(sy< 0)) return 1.5*x.pi;
if ((sy==0)&&(sx> 0)) return 0.0;
if ((sy==0)&&(sx< 0)) return x.pi;
a=y/x; if (a<0) a=-a;
a=::atan(a);
if ((sx>0)&&(sy>0)) a=a;
if ((sx<0)&&(sy>0)) a=x.pi-a;
if ((sx<0)&&(sy<0)) a=x.pi+a;
if ((sx>0)&&(sy<0)) a=x.pi+x.pi-a;
return a;
}
As I mentioned you need to use floating or fixed point for this as the results are not integers !!!
But as I mentioned before CORDIC is better suited for computing on integers (if you search there where some QAs here on SE/SO with C++ code for this).
IIRC it exploit some (arc)tan angle summation identity that leads to a nicely computable on integers delta angle something like sqrt(1+x*x) which is easily computable on integers. With binary search or approximation/iteration you can compute the tan of any angle and using goniometric identities you can compute any cotan sin and cos ... But I might be wrong as I do not use CORDIC and read about it a long time ago
Anyway once you got some function its inverse can be usually computed with binary search.
What I am trying to do is make it so that the game I am creating will randomly change characters every 5 seconds.
I got this working via a timer, the only problem is I don't want them repeating, I'm currently working on dummy code so it's just changing the screen colour, but how can I make it so that it doesn't repeat the number it just called?
if (timer <= 0)
{
num = rand.Next(2);
timer = 5.0f;
}
That is the current code and then in the draw I've literally just done "if num equals a certain number then change background colour".
I tried adding a prev_num checker but I can't get it to work properly (here it is)
if (timer <= 0)
{
prev_number = num;
num = rand.Next(2);
if (prev_number == num)
{
num = rand.Next(2);
}
else
{
timer = 5.0f;
}
}
Consider that if you're picking (for example) a random number from 1-5 then there are five possible outcomes, so you would use rand.Next(5) to select the zero-based "ordinal" or index of the outcome, then convert it into the range you actually want (in this case, by adding one).
If you want a random number from 0-4, excluding the number you just picked, then there are only four possible outcomes, not five - if the previous number was 3, then the possible outcomes are 0, 1, 2 or 4. You can then simplify your algorithm by choosing one of those four outcomes (rand.Next(4)) and mapping that ordinal to your desired range. A simple mapping would be to say if the new random number is below the previous number, return it as-is, otherwise (if equal or greater) add one.
int new_num = rand.Next(4);
if(new_num >= prev_num)
{
new_num++;
}
Your new number is now guaranteed to be in the same range as the previous number, but not equal to it.
Maybe just put it into a loop instead of a single check?
Also, I think because your timer was inside the else then it was not always
updated correctly.
if (timer <= 0)
{
tempNum = rand.Next(2);
do
{
tempNum = rand.Next(2);
}
while (tempNum == num)
num = tempNum;
timer = 5.0f;
}
Create an array of sequential numbers and then shuffle them (like a deck of cards) when your application begins.
int[] numbers = new int[100];
for(int i = 0; i < numbers.Length; i++)
numbers[i] = i;
Shuffle(numbers);
Using a function to shuffle the list:
public static void Shuffle<T>(IList<T> list)
{
Random rng = new Random();
int n = list.Count;
while (n > 1) {
n--;
int k = rng.Next(n + 1);
T value = list[k];
list[k] = list[n];
list[n] = value;
}
}
You can then access them sequentially out of the list. They will be random as the list was shuffled, but you won't have any repetitions since each number only exists once in the list.
if (timer <= 0)
{
num = numbers[index];
index++;
timer = 5.0f;
}
What is the best method to find the number of digits of a positive integer?
I have found this 3 basic methods:
conversion to string
String s = new Integer(t).toString();
int len = s.length();
for loop
for(long long int temp = number; temp >= 1;)
{
temp/=10;
decimalPlaces++;
}
logaritmic calculation
digits = floor( log10( number ) ) + 1;
where you can calculate log10(x) = ln(x) / ln(10) in most languages.
First I thought the string method is the dirtiest one but the more I think about it the more I think it's the fastest way. Or is it?
There's always this method:
n = 1;
if ( i >= 100000000 ) { n += 8; i /= 100000000; }
if ( i >= 10000 ) { n += 4; i /= 10000; }
if ( i >= 100 ) { n += 2; i /= 100; }
if ( i >= 10 ) { n += 1; }
Well the correct answer would be to measure it - but you should be able to make a guess about the number of CPU steps involved in converting strings and going through them looking for an end marker
Then think how many FPU operations/s your processor can do and how easy it is to calculate a single log.
edit: wasting some more time on a monday morning :-)
String s = new Integer(t).toString();
int len = s.length();
One of the problems with high level languages is guessing how much work the system is doing behind the scenes of an apparently simple statement. Mandatory Joel link
This statement involves allocating memory for a string, and possibly a couple of temporary copies of a string. It must parse the integer and copy the digits of it into a string, possibly having to reallocate and move the existing memory if the number is large. It might have to check a bunch of locale settings to decide if your country uses "," or ".", it might have to do a bunch of unicode conversions.
Then finding the length has to scan the entire string, again considering unicode and any local specific settings such as - are you in a right->left language?.
Alternatively:
digits = floor( log10( number ) ) + 1;
Just because this would be harder for you to do on paper doesn't mean it's hard for a computer! In fact a good rule in high performance computing seems to have been - if something is hard for a human (fluid dynamics, 3d rendering) it's easy for a computer, and if it's easy for a human (face recognition, detecting a voice in a noisy room) it's hard for a computer!
You can generally assume that the builtin maths functions log/sin/cos etc - have been an important part of computer design for 50years. So even if they don't map directly into a hardware function in the FPU you can bet that the alternative implementation is pretty efficient.
I don't know, and the answer may well be different depending on how your individual language is implemented.
So, stress test it! Implement all three solutions. Run them on 1 through 1,000,000 (or some other huge set of numbers that's representative of the numbers the solution will be running against) and time how long each of them takes.
Pit your solutions against one another and let them fight it out. Like intellectual gladiators. Three algorithms enter! One algorithm leaves!
Test conditions
Decimal numeral system
Positive integers
Up to 10 digits
Language: ActionScript 3
Results
digits: [1,10],
no. of runs: 1,000,000
random sample: 8777509,40442298,477894,329950,513,91751410,313,3159,131309,2
result: 7,8,6,6,3,8,3,4,6,1
CONVERSION TO STRING: 724ms
LOGARITMIC CALCULATION: 349ms
DIV 10 ITERATION: 229ms
MANUAL CONDITIONING: 136ms
Note: Author refrains from making any conclusions for numbers with more than 10 digits.
Script
package {
import flash.display.MovieClip;
import flash.utils.getTimer;
/**
* #author Daniel
*/
public class Digits extends MovieClip {
private const NUMBERS : uint = 1000000;
private const DIGITS : uint = 10;
private var numbers : Array;
private var digits : Array;
public function Digits() {
// ************* NUMBERS *************
numbers = [];
for (var i : int = 0; i < NUMBERS; i++) {
var number : Number = Math.floor(Math.pow(10, Math.random()*DIGITS));
numbers.push(number);
}
trace('Max digits: ' + DIGITS + ', count of numbers: ' + NUMBERS);
trace('sample: ' + numbers.slice(0, 10));
// ************* CONVERSION TO STRING *************
digits = [];
var time : Number = getTimer();
for (var i : int = 0; i < numbers.length; i++) {
digits.push(String(numbers[i]).length);
}
trace('\nCONVERSION TO STRING - time: ' + (getTimer() - time));
trace('sample: ' + digits.slice(0, 10));
// ************* LOGARITMIC CALCULATION *************
digits = [];
time = getTimer();
for (var i : int = 0; i < numbers.length; i++) {
digits.push(Math.floor( Math.log( numbers[i] ) / Math.log(10) ) + 1);
}
trace('\nLOGARITMIC CALCULATION - time: ' + (getTimer() - time));
trace('sample: ' + digits.slice(0, 10));
// ************* DIV 10 ITERATION *************
digits = [];
time = getTimer();
var digit : uint = 0;
for (var i : int = 0; i < numbers.length; i++) {
digit = 0;
for(var temp : Number = numbers[i]; temp >= 1;)
{
temp/=10;
digit++;
}
digits.push(digit);
}
trace('\nDIV 10 ITERATION - time: ' + (getTimer() - time));
trace('sample: ' + digits.slice(0, 10));
// ************* MANUAL CONDITIONING *************
digits = [];
time = getTimer();
var digit : uint;
for (var i : int = 0; i < numbers.length; i++) {
var number : Number = numbers[i];
if (number < 10) digit = 1;
else if (number < 100) digit = 2;
else if (number < 1000) digit = 3;
else if (number < 10000) digit = 4;
else if (number < 100000) digit = 5;
else if (number < 1000000) digit = 6;
else if (number < 10000000) digit = 7;
else if (number < 100000000) digit = 8;
else if (number < 1000000000) digit = 9;
else if (number < 10000000000) digit = 10;
digits.push(digit);
}
trace('\nMANUAL CONDITIONING: ' + (getTimer() - time));
trace('sample: ' + digits.slice(0, 10));
}
}
}
This algorithm might be good also, assuming that:
Number is integer and binary encoded (<< operation is cheap)
We don't known number boundaries
var num = 123456789L;
var len = 0;
var tmp = 1L;
while(tmp < num)
{
len++;
tmp = (tmp << 3) + (tmp << 1);
}
This algorithm, should have speed comparable to for-loop (2) provided, but a bit faster due to (2 bit-shifts, add and subtract, instead of division).
As for Log10 algorithm, it will give you only approximate answer (that is close to real, but still), since analytic formula for computing Log function have infinite loop and can't be calculated precisely Wiki.
Use the simplest solution in whatever programming language you're using. I can't think of a case where counting digits in an integer would be the bottleneck in any (useful) program.
C, C++:
char buffer[32];
int length = sprintf(buffer, "%ld", (long)123456789);
Haskell:
len = (length . show) 123456789
JavaScript:
length = String(123456789).length;
PHP:
$length = strlen(123456789);
Visual Basic (untested):
length = Len(str(123456789)) - 1
conversion to string: This will have to iterate through each digit, find the character that maps to the current digit, add a character to a collection of characters. Then get the length of the resulting String object. Will run in O(n) for n=#digits.
for-loop: will perform 2 mathematical operation: dividing the number by 10 and incrementing a counter. Will run in O(n) for n=#digits.
logarithmic: Will call log10 and floor, and add 1. Looks like O(1) but I'm not really sure how fast the log10 or floor functions are. My knowledge of this sort of things has atrophied with lack of use so there could be hidden complexity in these functions.
So I guess it comes down to: is looking up digit mappings faster than multiple mathematical operations or whatever is happening in log10? The answer will probably vary. There could be platforms where the character mapping is faster, and others where doing the calculations is faster. Also to keep in mind is that the first method will creats a new String object that only exists for the purpose of getting the length. This will probably use more memory than the other two methods, but it may or may not matter.
You can obviously eliminate the method 1 from the competition, because the atoi/toString algorithm it uses would be similar to method 2.
Method 3's speed depends on whether the code is being compiled for a system whose instruction set includes log base 10.
For very large integers, the log method is much faster. For instance, with a 2491327 digit number (the 11920928th Fibonacci number, if you care), Python takes several minutes to execute the divide-by-10 algorithm, and milliseconds to execute 1+floor(log(n,10)).
import math
def numdigits(n):
return ( int(math.floor(math.log10(n))) + 1 )
Regarding the three methods you propose for "determining the number of digits necessary to represent a given number in a given base", I don't like any of them, actually; I prefer the method I give below instead.
Re your method #1 (strings): Anything involving converting back-and-forth between strings and numbers is usually very slow.
Re your method #2 (temp/=10): This is fatally flawed because it assumes that x/10 always means "x divided by 10". But in many programming languages (eg: C, C++), if "x" is an integer type, then "x/10" means "integer division", which isn't the same thing as floating-point division, and it introduces round-off errors at every iteration, and they accumulate in a recursive formula such as your solution #2 uses.
Re your method #3 (logs): it's buggy for large numbers (at least in C, and probably other languages as well), because floating-point data types tend not to be as precise as 64-bit integers.
Hence I dislike all 3 of those methods: #1 works but is slow, #2 is broken, and #3 is buggy for large numbers. Instead, I prefer this, which works for numbers from 0 up to about 18.44 quintillion:
unsigned NumberOfDigits (uint64_t Number, unsigned Base)
{
unsigned Digits = 1;
uint64_t Power = 1;
while ( Number / Power >= Base )
{
++Digits;
Power *= Base;
}
return Digits;
}
Keep it simple:
long long int a = 223452355415634664;
int x;
for (x = 1; a >= 10; x++)
{
a = a / 10;
}
printf("%d", x);
You can use a recursive solution instead of a loop, but somehow similar:
#tailrec
def digits (i: Long, carry: Int=1) : Int = if (i < 10) carry else digits (i/10, carry+1)
digits (8345012978643L)
With longs, the picture might change - measure small and long numbers independently against different algorithms, and pick the appropriate one, depending on your typical input. :)
Of course nothing beats a switch:
switch (x) {
case 0: case 1: case 2: case 3: case 4: case 5: case 6: case 7: case 8: case 9: return 1;
case 10: case 11: // ...
case 99: return 2;
case 100: // you get the point :)
default: return 10; // switch only over int
}
except a plain-o-array:
int [] size = {1,1,1,1,1,1,1,1,1,2,2,2,2,2,... };
int x = 234561798;
return size [x];
Some people will tell you to optimize the code-size, but yaknow, premature optimization ...
log(x,n)-mod(log(x,n),1)+1
Where x is a the base and n is the number.
Here is the measurement in Swift 4.
Algorithms code:
extension Int {
var numberOfDigits0: Int {
var currentNumber = self
var n = 1
if (currentNumber >= 100000000) {
n += 8
currentNumber /= 100000000
}
if (currentNumber >= 10000) {
n += 4
currentNumber /= 10000
}
if (currentNumber >= 100) {
n += 2
currentNumber /= 100
}
if (currentNumber >= 10) {
n += 1
}
return n
}
var numberOfDigits1: Int {
return String(self).count
}
var numberOfDigits2: Int {
var n = 1
var currentNumber = self
while currentNumber > 9 {
n += 1
currentNumber /= 10
}
return n
}
}
Measurement code:
var timeInterval0 = Date()
for i in 0...10000 {
i.numberOfDigits0
}
print("timeInterval0: \(Date().timeIntervalSince(timeInterval0))")
var timeInterval1 = Date()
for i in 0...10000 {
i.numberOfDigits1
}
print("timeInterval1: \(Date().timeIntervalSince(timeInterval1))")
var timeInterval2 = Date()
for i in 0...10000 {
i.numberOfDigits2
}
print("timeInterval2: \(Date().timeIntervalSince(timeInterval2))")
Output
timeInterval0: 1.92149806022644
timeInterval1: 0.557608008384705
timeInterval2: 2.83262193202972
On this measurement basis String conversion is the best option for the Swift language.
I was curious after seeing #daniel.sedlacek results so I did some testing using Swift for numbers having more than 10 digits. I ran the following script in the playground.
let base = [Double(100090000000), Double(100050000), Double(100050000), Double(100000200)]
var rar = [Double]()
for i in 1...10 {
for d in base {
let v = d*Double(arc4random_uniform(UInt32(1000000000)))
rar.append(v*Double(arc4random_uniform(UInt32(1000000000))))
rar.append(Double(1)*pow(1,Double(i)))
}
}
print(rar)
var timeInterval = NSDate().timeIntervalSince1970
for d in rar {
floor(log10(d))
}
var newTimeInterval = NSDate().timeIntervalSince1970
print(newTimeInterval-timeInterval)
timeInterval = NSDate().timeIntervalSince1970
for d in rar {
var c = d
while c > 10 {
c = c/10
}
}
newTimeInterval = NSDate().timeIntervalSince1970
print(newTimeInterval-timeInterval)
Results of 80 elements
0.105069875717163 for floor(log10(x))
0.867973804473877 for div 10 iterations
Adding one more approach to many of the already mentioned approaches.
The idea is to use binarySearch on an array containing the range of integers based on the digits of the int data type.
The signature of Java Arrays class binarySearch is :
binarySearch(dataType[] array, dataType key) which returns the index of the search key, if it is contained in the array; otherwise, (-(insertion point) – 1).
The insertion point is defined as the point at which the key would be inserted into the array.
Below is the implementation:
static int [] digits = {9,99,999,9999,99999,999999,9999999,99999999,999999999,Integer.MAX_VALUE};
static int digitsCounter(int N)
{
int digitCount = Arrays.binarySearch(digits , N<0 ? -N:N);
return 1 + (digitCount < 0 ? ~digitCount : digitCount);
}
Please note that the above approach only works for : Integer.MIN_VALUE <= N <= Integer.MAX_VALUE, but can be easily extended for Long data type by adding more values to the digits array.
For example,
I) for N = 555, digitCount = Arrays.binarySearch(digits , 555) returns -3 (-(2)-1) as it's not present in the array but is supposed to be inserted at point 2 between 9 & 99 like [9, 55, 99].
As the index we got is negative we need to take the bitwise compliment of the result.
At last, we need to add 1 to the result to get the actual number of digits in the number N.
In Swift 5.x, you get the number of digit in integer as below :
Convert to string and then count number of character in string
let nums = [1, 7892, 78, 92, 90]
for i in nums {
let ch = String(describing: i)
print(ch.count)
}
Calculating the number of digits in integer using loop
var digitCount = 0
for i in nums {
var tmp = i
while tmp >= 1 {
tmp /= 10
digitCount += 1
}
print(digitCount)
}
let numDigits num =
let num = abs(num)
let rec numDigitsInner num =
match num with
| num when num < 10 -> 1
| _ -> 1 + numDigitsInner (num / 10)
numDigitsInner num
F# Version, without casting to a string.
We're given a string and a permutation of the string.
For example, an input string sandeep and a permutation psdenae.
Find the position of the given permutation in the sorted list of the permutations of the original string.
The total number of permutation of a given string of length n would be n! (if all characters are different), thus it would not be possible to explore all the combinations.
This question is actually like the mathematics P & C question
Find the rank of the word "stack" when arranged in dictionary order.
Given the input string as NILSU
Take a word which we have to find the rank. Take "SUNIL" for example.
Now arrange the letter of "SUNIL" in alphabetical order.
It will be. "I L N S U".
Now take the first letter. Its "I". Now check, is the letter "I" the
first letter of "SUNIL"? No. The number of words that can be formed
starting with I will be 4!, so we know that there will be 4! words
before "SUNIL".
I = 4! = 24
Now go for the second letter. Its "L". Now check once again if this
letter we want in first position? No. So the number of words can be
formed starting with "L" will be 4!.
L = 4! = 24
Now go for "N". Is this we want? No. Write down the number of words
can be formed starting with "N", once again 4!
N = 4! = 24
Now go for "S". Is this what we want? Yes. Now remove the letter from
the alphabetically ordered word. It will now be "I L N U"
Write S and check the word once again in the list. Is we want SI? No.
So the number of words can be formed starting with SI will be 3!
[S]:I-> 3! = 6
Go for L. is we want SL? No. So it will be 3!.
[S]:L-> 3! = 6
Go for N. is we want SN? No.
[S]:N-> 3! = 6
Go for SU. Is this we want? Yes. Cut the letter U from the list and
then it will be "I L N". Now try I. is we want SUI? No. So the number
of words can be formed which starts from SUI will be 2!
[SU]:I-> 2! = 2 Now go for L. Do we want "SUL". No. so the number of
words starting with SUL will be 2!.
[SU]:L-> 2! = 2
Now go for N. Is we want SUN? Yes, now remove that letter. and this
will be "I L". Do we want "SUNI"? Yes. Remove that letter. The only
letter left is "L".
Now go for L. Do we want SUNIL? Yes. SUNIL were the first options, so
we have 1!. [SUN][I][L] = 1! = 1
Now add the whole numbers we get. The sum will be.
24 + 24 + 24 + 6 + 6 + 6 + 2 + 2 + 1 = 95.
So the word SUNIL will be at 95th position if we count the words that can be created using the letters of SUNIL arranged in dictionary order.
Thus through this method you could solve this problem quite easily.
Building off #Algorithmist 's answer, and his comment to his answer, and using the principle discussed in this post for when there are repeated letters, I made the following algorithm in JavaScript that works for all letter-based words even with repeated letter instances.
function anagramPosition(string) {
var index = 1;
var remainingLetters = string.length - 1;
var frequencies = {};
var splitString = string.split("");
var sortedStringLetters = string.split("").sort();
sortedStringLetters.forEach(function(val, i) {
if (!frequencies[val]) {
frequencies[val] = 1;
} else {
frequencies[val]++;
}
})
function factorial(coefficient) {
var temp = coefficient;
var permutations = coefficient;
while (temp-- > 2) {
permutations *= temp;
}
return permutations;
}
function getSubPermutations(object, currentLetter) {
object[currentLetter]--;
var denominator = 1;
for (var key in object) {
var subPermutations = factorial(object[key]);
subPermutations !== 0 ? denominator *= subPermutations : null;
}
object[currentLetter]++;
return denominator;
}
var splitStringIndex = 0;
while (sortedStringLetters.length) {
for (var i = 0; i < sortedStringLetters.length; i++) {
if (sortedStringLetters[i] !== splitString[splitStringIndex]) {
if (sortedStringLetters[i] !== sortedStringLetters[i+1]) {
var permutations = factorial(remainingLetters);
index += permutations / getSubPermutations(frequencies, sortedStringLetters[i]);
} else {
continue;
}
} else {
splitStringIndex++;
frequencies[sortedStringLetters[i]]--;
sortedStringLetters.splice(i, 1);
remainingLetters--;
break;
}
}
}
return index;
}
anagramPosition("ARCTIC") // => 42
I didn't comment the code but I did try to make the variable names as explanatory as possible. If you run it through a debugger process using your dev tools console and throw in a few console.logs you should be able to see how it uses the formula in the above-linked S.O. post.
I tried to implement this in js. It works for string that have no repeated letters but I get a wrong count otherwise. Here is my code:
function x(str) {
var sOrdinata = str.split('').sort()
console.log('sOrdinata = '+ sOrdinata)
var str = str.split('')
console.log('str = '+str)
console.log('\n')
var pos = 1;
for(var j in str){
//console.log(j)
for(var i in sOrdinata){
if(sOrdinata[i]==str[j]){
console.log('found, position: '+ i)
sOrdinata.splice(i,1)
console.log('Nuovo sOrdinata = '+sOrdinata)
console.log('\n')
break;
}
else{
//calculate number of permutations
console.log('valore di j: '+j)
//console.log('lunghezza stringa da permutare: '+str.slice(~~j+1).length);
if(str.slice(j).length >1 ){sub = str.slice(~~j+1)}else {sub = str.slice(j)}
console.log('substring to be used for permutation: '+ sub)
prep = nrepC(sub.join(''))
console.log('prep = '+prep)
num = factorial(sub.length)
console.log('num = '+num)
den = denom(prep)
console.log('den = '+ den)
pos += num/den
console.log(num/den)
console.log('\n')
}
}
}
console.log(pos)
return pos
}
/* ------------ functions used by main --------------- */
function nrepC(str){
var obj={}
var repeats=[]
var res= [];
for(x = 0, length = str.length; x < length; x++) {
var l = str.charAt(x)
obj[l] = (isNaN(obj[l]) ? 1 : obj[l] + 1);
}
//console.log(obj)
for (var i in obj){
if(obj[i]>1) res.push(obj[i])
}
if(res.length==0){res.push(1); return res}
else return res
}
function num(vect){
var res = 1
}
function denom(vect){
var res = 1
for(var i in vect){
res*= factorial(vect[i])
}
return res
}
function factorial (n){
if (n==0 || n==1){
return 1;
}
return factorial(n-1)*n;
}
A bit too late but just as reference... You can use this C# code directly.
It will work but...
The only important thing is that usually, you should have unique values as your starting set. Otherwise you don't have n! permutations. You have something else (less than n!). I have a little doubt of any useful usage when item could be duplicate ones.
using System;
using System.Collections.Generic;
namespace WpfPermutations
{
public class PermutationOuelletLexico3<T>
{
// ************************************************************************
private T[] _sortedValues;
private bool[] _valueUsed;
public readonly long MaxIndex; // long to support 20! or less
// ************************************************************************
public PermutationOuelletLexico3(T[] sortedValues)
{
if (sortedValues.Length <= 0)
{
throw new ArgumentException("sortedValues.Lenght should be greater than 0");
}
_sortedValues = sortedValues;
Result = new T[_sortedValues.Length];
_valueUsed = new bool[_sortedValues.Length];
MaxIndex = Factorial.GetFactorial(_sortedValues.Length);
}
// ************************************************************************
public T[] Result { get; private set; }
// ************************************************************************
/// <summary>
/// Return the permutation relative to the index received, according to
/// _sortedValues.
/// Sort Index is 0 based and should be less than MaxIndex. Otherwise you get an exception.
/// </summary>
/// <param name="sortIndex"></param>
/// <returns>The result is written in property: Result</returns>
public void GetValuesForIndex(long sortIndex)
{
int size = _sortedValues.Length;
if (sortIndex < 0)
{
throw new ArgumentException("sortIndex should be greater or equal to 0.");
}
if (sortIndex >= MaxIndex)
{
throw new ArgumentException("sortIndex should be less than factorial(the lenght of items)");
}
for (int n = 0; n < _valueUsed.Length; n++)
{
_valueUsed[n] = false;
}
long factorielLower = MaxIndex;
for (int index = 0; index < size; index++)
{
long factorielBigger = factorielLower;
factorielLower = Factorial.GetFactorial(size - index - 1); // factorielBigger / inverseIndex;
int resultItemIndex = (int)(sortIndex % factorielBigger / factorielLower);
int correctedResultItemIndex = 0;
for(;;)
{
if (! _valueUsed[correctedResultItemIndex])
{
resultItemIndex--;
if (resultItemIndex < 0)
{
break;
}
}
correctedResultItemIndex++;
}
Result[index] = _sortedValues[correctedResultItemIndex];
_valueUsed[correctedResultItemIndex] = true;
}
}
// ************************************************************************
/// <summary>
/// Calc the index, relative to _sortedValues, of the permutation received
/// as argument. Returned index is 0 based.
/// </summary>
/// <param name="values"></param>
/// <returns></returns>
public long GetIndexOfValues(T[] values)
{
int size = _sortedValues.Length;
long valuesIndex = 0;
List<T> valuesLeft = new List<T>(_sortedValues);
for (int index = 0; index < size; index++)
{
long indexFactorial = Factorial.GetFactorial(size - 1 - index);
T value = values[index];
int indexCorrected = valuesLeft.IndexOf(value);
valuesIndex = valuesIndex + (indexCorrected * indexFactorial);
valuesLeft.Remove(value);
}
return valuesIndex;
}
// ************************************************************************
}
}
My approach to the problem is sort the given permutation.
Number of swappings of the characters in the string will give us the position of the pemutation in the sorted list of permutations.
An inefficient solution would be to successively find the previous permutations until you reach a string that cannot be permuted anymore. The number of permutations it takes to reach this state is the position of the original string.
However, if you use combinatorics you can achieve the solution faster. The previous solution will produce a very slow output if string length exceeds 12.
I'm a jscript newbie and I've a problem.
I'm writing a script to validate an IBAN bank account number in Belgium. I need to replace some letters by their position in a searchstring and afterwards I convert this string into a number to take the modulo 97 test.
The first part goes well, but afterwards with the conversion from string to number, 10 is added to my number. I don't know what I'm doing wrong.
function checkIBAN()
{
var iban = crmForm.all.fp_iban.DataValue;
if (iban != null)
{
iban = iban.substring(4) + iban.substring(0, 4);
iban = iban.toUpperCase();
var searchString = '0123456789ABCDEFGHIJKLMNOPQRSTUVWXYZ';
var pos;
var tmp = '';
for (x = 0; x < iban.length; x++) {
pos = searchString.search(RegExp(iban.charAt(x),'i'));
if (pos == -1)
return false;
else
tmp += pos.toString();
}
alert(tmp); // Here my value is 735320036532111490
var nr =parseInt(tmp);
alert(nr); // Now my value seems to be 735320036532111500
alert(nr % 97);
if (nr % 97 != 1)
{
alert('IBAN number is not correct !');
}
}
}
Yes, 735320036532111490 is simply too great a value to store in an int. It'll always be rounded:
alert(735320036532111490 / 10);
// alerts 73532003653211150
Here's a solution that might work for you.
Always specify the radix when using parseInt.
var nr =parseInt(tmp, 10);
For reference information: https://developer.mozilla.org/en/JavaScript/Reference/Global_Objects/parseInt