Okay this is just something me and my coworker are playing with.
We know PHP has it's own PI function but this came forth out of a theory and curiosity.
So we were wondering if and how PHP was able to calculate pi.
Formule of pi = π= 4/1 - 4/3 + 4/5 - 4/7 + 4/9...
Here is what we did:
$theValue = 100;// the max
for ($i=1; $i<$theValue; $i++){
if ($i % 2 == 1){
$iWaardes[] = 4 / $i; // divide 4 by all uneven numbers and store them in an array
}
}
// Use the array's $keys as incrementing numbers to calculate the $values.
for ($a=0, $b=1, $c=2; $a<$theValue; $a+=3, $b+=3, $c+=3 ){
echo ($iWaardes[$a] - $iWaardes[$b] + $iWaardes[$c]).'<br>';
}
So now we have a loop that calculated the first series of 4/1 - 4/3 + 4/5 but it stops after that and starts over with the following 3 sequences.
How can we make it run the entire $theValue and calculate the whole series?
Please keep in mind that this is nothing serious and just a fun experiment for us.
You're overthinking this. Just use the modulo to decide if you want to add or subtract and do it in place.
$theValue = 100;// the max
$pi = 0;
for ($i=1; $i<$theValue; $i++){
if ($i % 2 == 1){
$pi += 4.0 / ($i * 2 - 1);
} else {
$pi -= 4.0 / ($i * 2 - 1);
}
}
Just use one loop. Have a $bottom variable that you add 2 on each iteration, divide by it, and add it/subtract it depending on the modulo:
$theValue = 10000; // the max
$bottom = 1;
$pi = 0;
for ($i = 1; $i < $theValue; $i++) {
if ($i % 2 == 1) {
$pi += 4 / $bottom;
} else {
$pi -= 4 / $bottom;
}
$bottom += 2;
}
var_dump($pi); // 3.14169266359
Demo
What's wrong with your code (other than not dividing by the appropriate number) is the second loop. You're for some reason printing out the stored numbers 3 by 3. This, until $a, that increases by 3, is lower than $theValue which is much higher. So, for example, if $theValue is 10, you only need 2 loops before you start getting out of bound errors.
pi() Returns an approximation of pi. The returned float has a precision based on the precision directive in php.ini, which defaults to 14. Also, you can use the M_PI constant which yields identical results to pi()
source
Using PHP we can also calculate Pi, albeit very slowly.
$pi = 4; $top = 4; $bot = 3; $minus = TRUE;
$accuracy = 1000000;
for($i = 0; $i < $accuracy; $i++)
{
$pi += ( $minus ? -($top/$bot) : ($top/$bot) );
$minus = ( $minus ? FALSE : TRUE);
$bot += 2;
}
print "Pi ~=: " . $pi;
This method of calculating Pi is slow, but it is easy to read code.
You can read more about this method here:
http://en.wikipedia.org/wiki/Leibniz_formula_for_%CF%80
If you increase the $accuracy variable, Pi will be calculated more and more accurately. Depending on how fast your web server is, you can calculate the first 6 digits of Pi fairly quickly.
The time it takes to calculate each succeeding number goes up exponentially however. To calculate 20 digits of Pi using this method method could take years.
Related
By stumbling on this so thread i decided to write similar test in PHP.
My test code is this:
// Slow version
$t1 = microtime(true);
for ($n = 0, $i = 0; $i < 20000000; $i++) {
$n += 2 * ($i * $i);
}
$t2 = microtime(true);
echo "n={$n}\n";
// Optimized version
$t3 = microtime(true);
for ($n = 0, $i = 0; $i < 20000000; $i++) {
$n += $i * $i;
}
$n *= 2;
$t4 = microtime(true);
echo "n={$n}\n";
$speedup = round(100 * (($t2 - $t1) - ($t4 - $t3)) / ($t2 - $t1), 0);
echo "speedup: {$speedup}%\n";
Results
in PHP 2 * ($i * $i) version runs quite similar like 2 * $i * $i,
so PHP interpreter isn't optimizing bytecode as JVM in Java
Even when I optimized code manually - I've got ~ 8% speedup, when
Java's version gets ~ 16% speedup. So PHP version gets about 1/2 speedup factor of that in Java's code.
Rationale for optimization
I will not go into many details, but ratio of multiplications in optimized and un-optimized code is ->
1 summation : 3/4
2 summations: 4/6
3 summations: 5/8
4 summations: 6/10
...
And in general:
where n is number of summations in a loop. To be formula useful to us - we need to calculate limit of it when N approaches infinity (to replicate situation that we do A LOT of summations in a loop). So :
So we get conclusion that in optimized code there must be 50% less multiplications.
Questions
Why PHP interpreter isn't applying code optimization ?
Why PHP speedup factor is just half of that in Java ?
It's time to analyze PHP opcodes which are generated by PHP interpreter. For that you need to install VLD extension and use it from command line to generate opcodes of php script at hand.
Opcode analysis
Seems that $i++ is not the same as ++$i in terms of opcodes and memory usage. Statement $i++; generates opcodes:
POST_INC ~4 !1
FREE ~4
Increases counter by 1 and saves previous value into memory slot #4. Then, because this value is never used - frees it from memory. Question - why do we need to store value if it is never used ?
Seems that indeed there is a loop penalty, so we can gain additional performance by performing loop unrolling.
Optimized test code
Changing POST_INC into ASSIGN_ADD (which don't saves additional info in memory) and performing loop unrolling, gives use such test code :
while (true) {
// Slow version
$t1 = microtime(true);
for ($n = 0, $i = 0; $i < 2000; $i+=10) {
// loop unrolling
$n += 2 * (($i+0) * ($i+0));
$n += 2 * (($i+1) * ($i+1));
$n += 2 * (($i+2) * ($i+2));
$n += 2 * (($i+3) * ($i+3));
$n += 2 * (($i+4) * ($i+4));
$n += 2 * (($i+5) * ($i+5));
$n += 2 * (($i+6) * ($i+6));
$n += 2 * (($i+7) * ($i+7));
$n += 2 * (($i+8) * ($i+8));
$n += 2 * (($i+9) * ($i+9));
}
$t2 = microtime(true);
echo "{$n}\n";
// Optimized version
$t3 = microtime(true);
for ($n = 0, $i = 0; $i < 2000; $i+=10) {
// loop unrolling
$n += ($i+0) * ($i+0);
$n += ($i+1) * ($i+1);
$n += ($i+2) * ($i+2);
$n += ($i+3) * ($i+3);
$n += ($i+4) * ($i+4);
$n += ($i+5) * ($i+5);
$n += ($i+6) * ($i+6);
$n += ($i+7) * ($i+7);
$n += ($i+8) * ($i+8);
$n += ($i+9) * ($i+9);
}
$n *= 2;
$t4 = microtime(true);
echo "{$n}\n";
$speedup = round(100 * (($t2 - $t1) - ($t4 - $t3)) / ($t2 - $t1), 0);
$table[$speedup]++;
echo "****************\n";
foreach ($table as $s => $c) {
if ($s >= 0 && $s <= 20)
echo "$s,$c\n";
}
}
Results
Script aggregates number of times CPU hit into one or other speedup value.
When CPU hits vs Speedup is drawn as a graph, we get such picture:
So it is most likely that script will get 10% speedup. This means that our optimizations resulted in +2% speedup (compared to original scripts 8%).
Expectations
I'm pretty sure that all these things i've done - could be done automatically by a PHP JIT'er. I don't think that it's hard to change automatically a pair of POST_INC/FREE opcodes into one PRE_INC opcode when generating binary executable. Also it's not a miracle that PHP JIT'er could apply loop unrolling. And this is just a start of optimizations !
Hopefully there will be a JIT'er in PHP 8.0
My rand(0,1) php function returns me the 0 and 1 randomly when I call it.
Can I define something in php, so that it makes 30% numbers will be 0 and 70% numbers will be 1 for the random calls? Does php have any built in function for this?
Sure.
$rand = (float)rand()/(float)getrandmax();
if ($rand < 0.3)
$result = 0;
else
$result = 1;
You can deal with arbitrary results and weights, too.
$weights = array(0 => 0.3, 1 => 0.2, 2 => 0.5);
$rand = (float)rand()/(float)getrandmax();
foreach ($weights as $value => $weight) {
if ($rand < $weight) {
$result = $value;
break;
}
$rand -= $weight;
}
You can do something like this:
$rand = (rand(0,9) > 6 ? 1 : 0)
rand(0,9) will produce a random number between 0 and 9, and whenever that randomly generated number is greater than 6 (which should be nearly 70% time), it will give you 1 otherwise 0...
Obviously, it seems to be the easiest solution to me, but definitely, it wont give you 1 exactly 70% times, but should be quite near to do that, if done correctly.
But, I doubt that any solution based on rand will give you 1 exactly 70% times...
Generate a new random value between 1 and 100. If the value falls below 30, then use 0, and 1 otherwise:
$probability = rand(1, 100);
if ($probability < 30) {
echo 0;
} else {
echo 1;
}
To test this theory, consider the following loop:
$arr = array();
for ($i=0; $i < 10000; $i++) {
$rand = rand(0, 1);
$probability = rand(1, 100);
if ($probability < 30) {
$arr[] = 0;
} else {
$arr[] = 1;
}
}
$c = array_count_values($arr);
echo "0 = " . $c['0'] / 10000 * 100;
echo "1 = " . $c['1'] / 10000 * 100;
Output:
0 = 29.33
1 = 70.67
Create an array with 70% 1 and 30% 0s. Then random sort it. Then start picking numbers from the beginning of the array to the end :)
$num_array = array();
for($i = 0; $i < 3; $i++) $num_array[$i] = 0;
for($i = 0; $i < 7; $i++) $num_array[$i] = 1;
shuffle($num_array);
Pros:
You'll get exactly 30% 0 and 70% 1 for any such array.
Cons: Might take longer computation time than a rand() only solution to create the initial array.
I searched for an answer to my question and this was the topic I found.
But it didn't answered my question, so I had to figure it out myself, and I did :).
I figured out that maybe this will help someone else as well.
It's regarding what you asked, but for more usage.
Basically, I use it as a "power" calculator for a random generated item (let's say a weapon). The item has a "min power" and a "max power" value in the db. And I wanted to have 80% chances to have the "power" value closer to the lower 80% of the max possible power for the item, and 20% for the highest 20% possible max power (that are stored in the db).
So, to do this I did the following:
$min = 1; // this value is normally taken from the db
$max = 30; // this value is normally taken from the db
$total_possibilities = ($max - $min) + 1;
$rand = random_int(1, 100);
if ($rand <= 80) { // 80% chances
$new_max = $max - ($total_possibilities * 0.20); // remove 20% from the max value, so you can get a number only from the lowest 80%
$new_rand = random_int($min, $new_max);
} elseif ($rand <= 100) { // 20% chances
$new_min = $min + ($total_possibilities * 0.80); // add 80% for the min value, so you can get a number only from the highest 20%
$new_rand = random_int($new_min, $max);
}
echo $new_rand; // this will be the final item power
The only problem you can have, is if the initial $min and $max variables are the same (or obviously, if the $max is bigger than the $min). This will throw an error since the random works like this ($min, $max), not the other way around.
This code can be very easily changed to have more percentages for different purposes, instead of 80% and 20% to put 40%, 40% and 20% (or whatever you need). I think the code is pretty much easy to read and understand.
Sorry if this is not helpful, but I hope it is :).
It can't do any harm either way ;).
The code:
<?php
$start = 0;
$stop = 1;
$step = ($stop - $start)/10;
$i = $start + $step;
while ($i < $stop) {
echo($i . "<br/>");
$i += $step;
}
?>
The output:
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1 <-- notice the 1 printed when it shouldn't
Created a fiddle
One more: if you set $start = 1 and $stop = 2 it works fine.
Using: php 5.3.27
Why is the 1 printed?
Because not only float math is flawed, sometimes its representation is flawed too - and that's the case here.
You don't actually get 0.1, 0.2, ... - and that's quite easy to check:
$start = 0;
$stop = 1;
$step = ($stop - $start)/10;
$i = $start + $step;
while ($i < $stop) {
print(number_format($i, 32) . "<br />");
$i += $step;
}
The only difference here, as you see, is that echo replaced with number_format call. But the results are drastically different:
0.10000000000000000555111512312578
0.20000000000000001110223024625157
0.30000000000000004440892098500626
0.40000000000000002220446049250313
0.50000000000000000000000000000000
0.59999999999999997779553950749687
0.69999999999999995559107901499374
0.79999999999999993338661852249061
0.89999999999999991118215802998748
0.99999999999999988897769753748435
See? Only one time it was 0.5 actually - because that number can be stored in a float container. All the others were only approximations.
How to solve this? Well, one radical approach is using not floats, but integers in similar situations. It's easy to notice that have you done it this way...
$start = 0;
$stop = 10;
$step = (int)(($stop - $start) / 10);
$i = $start + $step;
while ($i < $stop) {
print(number_format($i, 32) . "<br />");
$i += $step;
}
... it would work ok:
Alternatively, you can use number_format to convert the float into some string, then compare this string with preformatted float. Like this:
$start = 0;
$stop = 1;
$step = ($stop - $start) / 10;
$i = $start + $step;
while (number_format($i, 1) !== number_format($stop, 1)) {
print(number_format($i, 32) . "\n");
$i += $step;
}
The problem is that the number in the variable $i is not 1 (when printed). Its actual just less than 1. So in the test ($i < $stop) is true, the number is converted to decimal (causing rounding to 1), and displayed.
Now why is $i not 1 exactly? It is because you got there by saying 10 * 0.1, and 0.1 cannot be represented perfectly in binary. Only numbers which can be expressed as a sum of a finite number of powers of 2 can be perfectly represented.
Why then is $stop exactly 1? Because it isn't in floating point format. In other words, it is exact from the start -- it isn't calculated within the system used floating point 10 * 0.1.
Mathematically, we can write this as follows:
A 64 bit binary float can only hold the first 27 non-zero terms of the sum which approximates 0.1. The other 26 bits of the significand remain zero to indicate zero terms. The reason 0.1 isn't physically representable is that the required sequence of terms is infinite. On the other hand, numbers like 1 require only a small finite number of terms and are representable. We'd like that to be the case for all numbers. This is why decimal floating point is such an important innovation (not widely available yet). It can represent any number that we can write down, and do so perfectly. Of course, the number of available digits remains finite.
Returning to the given problem, since 0.1 is the increment for the loop variable and isn't actually representable, the value 1.0 (although representable) is never precisely reached in the loop.
If your step will always be a factor of 10, you can accomplish this quickly with the following:
<?php
$start = 0;
$stop = 1;
$step = ($stop - $start)/10;
$i = $start + $step;
while (round($i, 1) < $stop) { //Added round() to the while statement
echo($i . "<br/>");
$i += $step;
}
?>
I need to generate x amount of random odd numbers, within a given range.
I know this can be achieved with simple looping, but I'm unsure which approach would be the best, and is there a better mathematical way of solving this.
EDIT: Also I cannot have the same number more than once.
Generate x integer values over half the range, and for each value double it and add 1.
ANSWERING REVISED QUESTION: 1) Generate a list of candidates in range, shuffle them, and then take the first x. Or 2) generate values as per my original recommendation, and reject and retry if the generated value is in the list of already generated values.
The first will work better if x is a substantial fraction of the range, the latter if x is small relative to the range.
ADDENDUM: Should have thought of this approach earlier, it's based on conditional probability. I don't know php (I came at this from the "random" tag), so I'll express it as pseudo-code:
generate(x, upper_limit)
loop with index i from upper_limit downto 1 by 2
p_value = x / floor((i + 1) / 2)
if rand <= p_value
include i in selected set
decrement x
return/exit if x <= 0
end if
end loop
end generate
x is the desired number of values to generate, upper_limit is the largest odd number in the range, and rand generates a uniformly distributed random number between zero and one. Basically, it steps through the candidate set of odd numbers and accepts or rejects each one based how many values you still need and how many candidates still remain.
I've tested this and it really works. It requires less intermediate storage than shuffling and fewer iterations than the original acceptance/rejection.
Generate a list of elements in the range, remove the element you want in your random series. Repeat x times.
Or you can generate an array with the odd numbers in the range, then do a shuffle
Generation is easy:
$range_array = array();
for( $i = 0; $i < $max_value; $i++){
$range_array[] .= $i*2 + 1;
}
Shuffle
shuffle( $range_array );
splice out the x first elements.
$result = array_slice( $range_array, 0, $x );
This is a complete solution.
function mt_rands($min_rand, $max_rand, $num_rand){
if(!is_integer($min_rand) or !is_integer($max_rand)){
return false;
}
if($min_rand >= $max_rand){
return false;
}
if(!is_integer($num_rand) or ($num_rand < 1)){
return false;
}
if($num_rand <= ($max_rand - $min_rand)){
return false;
}
$rands = array();
while(count($rands) < $num_rand){
$loops = 0;
do{
++$loops; // loop limiter, use it if you want to
$rand = mt_rand($min_rand, $max_rand);
}while(in_array($rand, $rands, true));
$rands[] = $rand;
}
return $rands;
}
// let's see how it went
var_export($rands = mt_rands(0, 50, 5));
Code is not tested. Just wrote it. Can be improved a bit but it's up to you.
This code generates 5 odd unique numbers in the interval [1, 20]. Change $min, $max and $n = 5 according to your needs.
<?php
function odd_filter($x)
{
if (($x % 2) == 1)
{
return true;
}
return false;
}
// seed with microseconds
function make_seed()
{
list($usec, $sec) = explode(' ', microtime());
return (float) $sec + ((float) $usec * 100000);
}
srand(make_seed());
$min = 1;
$max = 20;
//number of random numbers
$n = 5;
if (($max - $min + 1)/2 < $n)
{
print "iterval [$min, $max] is too short to generate $n odd numbers!\n";
exit(1);
}
$result = array();
for ($i = 0; $i < $n; ++$i)
{
$x = rand($min, $max);
//not exists in the hash and is odd
if(!isset($result{$x}) && odd_filter($x))
{
$result[$x] = 1;
}
else//new iteration needed
{
--$i;
}
}
$result = array_keys($result);
var_dump($result);
Writing a routine to display data on a horizontal axis (using PHP gd2, but that's not the point here).
The axis starts at $min to $max and displays a diamond at $result, such an image will be around 300px wide and 30px high, like this:
(source: testwolke.de)
In the example above, $min=0, $max=3, $result=0.6.
Now, I need to calculate a scale and labels that make sense, in the above example e.g. dotted lines at 0 .25 .50 .75 1 1.25 ... up to 3, with number-labels at 0 1 2 3.
If $min=-200 and $max=600, dotted lines should be at -200 -150 -100 -50 0 50 100 ... up to 600, with number-labels at -200 -100 0 100 ... up to 600.
With $min=.02and $max=5.80, dotted lines at .02 .5 1 1.5 2 2.5 ... 5.5 5.8 and numbers at .02 1 2 3 4 5 5.8.
I tried explicitly telling the function where to put dotted lines and numbers by arrays, but hey, it's the computer who's supposed to work, not me, right?!
So, how to calculate???
An algorithm (example values $min=-186 and $max=+153 as limits):
Take these two limits $min, $max and mark them if you wish
Calculate the difference between $max and $min: $diff = $max - $min
153 - (-186) = 339
Calculate 10th logarithm of the difference $base10 = log($diff,10) = 2,5302
Round down: $power = round($base10) = 2.
This is your tenth power as base unit
To calculate $step calculate this:
$base_unit = 10^$power = 100;
$step = $base_unit / 2; (if you want 2 ticks per one $base_unit).
Calculate if $min is divisible by $step, if not take the nearest (round up) one
(in the case of $step = 50 it is $loop_start = -150)
for ($i=$loop_start; $i<=$max; $i++=$step){ // $i's are your ticks
end
I tested it in Excel and it gives quite nice results, you may want to increase its functionality,
for example (in point 5) by calculating $step first from $diff,
say $step = $diff / 4 and round $step in such way that $base_unit is divisible by $step;
this will avoid such situations that you have between (101;201) four ticks with $step=25 and you have 39 steps $step=25 between 0 and 999.
ACM Algorithm 463 provides three simple functions to produce good axis scales with outputs xminp, xmaxp and dist for the minimum and maximum values on the scale and the distance between tick marks on the scale, given a request for n intervals that include the data points xmin and xmax:
Scale1() gives a linear scale with approximately n intervals and dist being an integer power of 10 times 1, 2 or 5.
Scale2() gives a linear scale with exactly n intervals (the gap between xminp and xmaxp tends to be larger than the gap produced by Scale1()).
Scale3() gives a logarithmic scale.
The original 1973 paper is online here, which provides more explanation than the code linked to above.
The code is in Fortran but it is just a set of arithmetical calculations so it is very straightforward to interpret and convert into other languages. I haven't written any PHP myself, but it looks a lot like C so you might want to start by running the code through f2c which should give you something close to runnable in PHP.
There are more complicated functions that give prettier scales (e.g. the ones in gnuplot), but Scale1() would likely do the job for you with minimal code.
(This answer builds on my answer to a previous question Graph axis calibration in C++)
(EDIT -- I've found an implementation of Scale1() that I did in Perl):
use strict;
sub scale1 ($$$) {
# from TOMS 463
# returns a suitable scale ($xMinp, $xMaxp, $dist), when called with
# the minimum and maximum x values, and an approximate number of intervals
# to divide into. $dist is the size of each interval that results.
# #vInt is an array of acceptable values for $dist.
# #sqr is an array of geometric means of adjacent values of #vInt, which
# is used as break points to determine which #vInt value to use.
#
my ($xMin, $xMax, $n) = #_;
#vInt = {1, 2, 5, 10};
#sqr = {1.414214, 3.162278, 7.071068 }
if ($xMin > $xMax) {
my ($tmp) = $xMin;
$xMin = $xMax;
$xMax = $tmp;
}
my ($del) = 0.0002; # accounts for computer round-off
my ($fn) = $n;
# find approximate interval size $a
my ($a) = ($xMax - $xMin) / $fn;
my ($al) = log10($a);
my ($nal) = int($al);
if ($a < 1) {
$nal = $nal - 1;
}
# $a is scaled into a variable named $b, between 1 and 10
my ($b) = $a / 10^$nal;
# the closest permissable value for $b is found)
my ($i);
for ($i = 0; $i < $_sqr; $i++) {
if ($b < $sqr[$i]) last;
}
# the interval size is computed
$dist = $vInt[$i] * 10^$nal;
$fm1 = $xMin / $dist;
$m1 = int($fm1);
if ($fm1 < 0) $m1--;
if (abs(($m1 + 1.0) - $fm1) < $del) $m1++;
# the new minimum and maximum limits are found
$xMinp = $dist * $m1;
$fm2 = $xMax / $dist;
$m2 = $fm2 + 1;
if ($fm2 < -1) $m2--;
if (abs ($fm2 + 1 - $m2) < $del) $m2--;
$xMaxp = $dist * $m2;
# adjust limits to account for round-off if necessary
if ($xMinp > $xMin) $xMinp = $xMin;
if ($xMaxp < $xMax) $xMaxp = $xMax;
return ($xMinp, $xMaxp, $dist);
}
sub scale1_Test {
$par = (-3.1, 11.1, 5,
5.2, 10.1, 5,
-12000, -100, 9);
print "xMin\txMax\tn\txMinp\txMaxp,dist\n";
for ($i = 0; $i < $_par/3; $i++) {
($xMinp, $xMaxp, $dist) = scale1($par[3*$i+0],
$par[3*$i+1], $par[3*$i+2]);
print "$par[3*$i+0]\t$par[3*$i+1]\t$par[3*$i+2]\t$xMinp\t$xMaxp,$dist\n";
}
}
I know that this isn't exactly what you are looking for, but hopefully it will get you started in the right direction.
$min = -200;
$max = 600;
$difference = $max - $min;
$labels = 10;
$picture_width = 300;
/* Get units per label */
$difference_between = $difference / ($labels - 1);
$width_between = $picture_width / $labels;
/* Make the label array */
$label_arr = array();
$label_arr[] = array('label' => $min, 'x_pos' => 0);
/* Loop through the number of labels */
for($i = 1, $l = $labels; $i < $l; $i++) {
$label = $min + ($difference_between * $i);
$label_arr[] = array('label' => $label, 'x_pos' => $width_between * $i);
}
A quick example would be something in the lines of $increment = ($max-$min)/$scale where you can tweak scale to be the variable by which the increment scales. Since you devide by it, it should change proportionately as your max and min values change. After that you will have a function like:
$end = false;
while($end==false){
$breakpoint = $last_value + $increment; // that's your current breakpoint
if($breakpoint > $max){
$end = true;
}
}
At least thats the concept... Let me know if you have troubles with it.